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1,108,101,565,415 | arxiv | \section{Introduction}
A type II supernova explosion is one of the most spectacular
events in astrophysics, with huge energy release of about
$10^{53}$ erg or several tens of MeV per nucleon \cite{Bethe}.
When the core of a massive star collapses, it reaches densities
several times larger than the normal nuclear density $\rho_0=0.15$
fm$^{-3}$. The repulsive nucleon-nucleon interaction gives rise to
a bounce-off and formation of a shock wave propagating through the
in-falling stellar material, predominantly Fe.
Hydrodynamical simulations (see e.g. refs.
\cite{Janka,Thielemann}) show that during the collapse and
subsequent explosion the temperatures $T\approx (0.5\div 10)$ MeV
and baryon densities $\rho_B \approx (10^{-5}\div 2) \rho_0$ can
be reached. A schematic view of the post-collapse star core is
presented in Fig.~1.
\begin{figure}
\includegraphics[width=12cm]{fig1.eps}
\caption{\small{Schematic view of the post-collapse stellar core
230 ms after the bounce-off, as predicted by the hydrodynamical
simulations \cite{Janka}. The neutrino heating and convection
processes help to revive the shock. Region between the
protoneutron star (PNS) and the shock front is called the Hot
Bubble. In-falling matter is represented by thick arrows labelled
by $\dot M$.} }
\end{figure}
For the realistic description of supernova physics one should
certainly use experience accumulated in recent years by studying
intermediate-energy nuclear reactions. In particular,
multifragmentation reactions provide valuable information about
hot nuclei in dense environment. According to present
understanding, based on numerous theoretical and experimental
studies of multifragmentation reactions, prior to the break-up a
transient state of nuclear matter is formed, where hot nuclear
fragments exist in equilibrium with free nucleons. This state is
characterized by a certain temperature $T \sim 3-6$ MeV and a
density which is typically 3-5 times smaller than the nuclear
saturation density $\rho_0$. A very good
description of such systems is achieved with the Statistical
Multifragmentation model (SMM), for a review see ref. \cite{SMM}.
The statistical nature of multifragmentation is confirmed by
numerous experimental observations, e. g. "rise and fall" of
intermediate-mass fragment production \cite{Bot95,Schue}, evolution of the
fragment mass and multiplicity distributions with excitation energy
\cite{Dag, Schar}, fragment correlations revealing the critical behavior
\cite{Dag, Schar}, confirmation of anomaly in the caloric curve \cite{Poch},
isoscaling \cite{Bot02}. Recent experiments \cite{GANIL}
directly confirm the basic assumption of the SMM, namely, that the
primary fragments are hot, their internal excitation energy may
reach up to 3 MeV per nucleon. Therefore, properties of these hot
nuclei can be extracted from multifragmentation reactions and used
for the description of matter under stellar conditions. The first
steps in this direction were made in our papers
\cite{Botvina04,Botvina05}. A similar model was also used in ref.
\cite{Japan} where, however, only cold nuclei in long-lived states
were considered.
\section{Statistical description of supernova matter}
\subsection{General remarks}
In the supernova environment, as compared to the multifragmentation
reactions, several new important ingredients should be taken
into consideration. First, the matter at stellar scales must be
electrically neutral, and therefore electrons should be included to
balance positive nuclear charge.
Second, energetic photons present in hot matter may change
nuclear composition via photonuclear reactions.
And third, the matter is irradiated by a strong neutrino wind from the
protoneutron star.
Below we consider macroscopic volumes of matter consisting of
various nuclear species $(A,Z)$, nucleons $(n=(1,0)$ and
$p=(1,1))$, electrons $(e^-)$ and positrons $(e^+)$ under the
condition of electric neutrality. We expect that in this situation
an equilibrium ensemble of various nuclear species will be generated like in
a liquid-gas coexistence region, as observed in the
multifragmentation reactions. Now our system is
characterized by the temperature $T$, baryon density $\rho_B$ and
electron fraction $Y_e$ (i.e. the ratio of the net electron
density to the baryon density). One may expect that
the new nuclear effects come into force in this environment. For example, the
liquid-drop properties in hot nuclei may be different from those
observed in cold nuclei (see discussion e.g. in refs.
\cite{Bot02,Bot06,Bot08}).
\subsection{Equilibrium conditions}
Composition of stellar matter can safely be studied within the Grand
Canonical Ensemble dealing with chemical potentials of the constituents.
Generally, the chemical potential of a species $i$ with baryon number $B_i$, charge $Q_i$
and lepton number $L_i$, which participates in chemical equilibrium,
can be found from the general expression:
\begin{equation}
\mu_i=B_i\mu_B+Q_i\mu_Q+L_i\mu_L
\end{equation}
where $\mu_B$, $\mu_Q$ and $\mu_L$ are three independent chemical
potentials which are determined from the conservation of total baryon
number $B=\sum_iB_i$ electric charge $Q=\sum_iQ_i$ and lepton number
$L=\sum_iL_i$ of the system. This gives
\begin{equation}
\begin{array}{ll}
\mu_{AZ}=A\mu_B+Z\mu_Q~,~\\
\mu_{e^-}=-\mu_{e^+}=-\mu_Q+\mu_L~,~\\
\mu_\nu=-\mu_{\tilde{\nu}}=\mu_L~.
\end{array}
\end{equation}
These relations are also valid for nucleons, $\mu_n=\mu_B$ and
$\mu_p=\mu_B+\mu_Q$.
If $\nu$ and $\overline{\nu}$ escape freely from the system, the
lepton number conservation is irrelevant and $\mu_L=0$. In this
case two remaining chemical potentials are determined from the
conditions of baryon number conservation and electro-neutrality:
\begin{equation}
\rho_B=\frac{B}{V}=\sum_{AZ}A\rho_{AZ}~,~
\rho_Q=\frac{Q}{V}=\sum_{AZ}Z\rho_{AZ}-\rho_e=0~.\nonumber
\end{equation}
Here $\rho_{AZ}$ is the number density of nuclear species $(A,Z)$,
$\rho_e=\rho_{e^-}-\rho_{e^+}$ is the net electron density.
The pressure of the relativistic electron-positron gas can be
written as
\begin{equation}
P_e=\frac{\mu_e^4}{12\pi^2}\left[1+2\left(\frac{\pi
T}{\mu_e}\right)^2+\frac{7}{15}\left(\frac{\pi T}{\mu_e}\right)^4
-\frac{m_e^2}{\mu_e^2}\left(3+\left(\frac{\pi
T}{\mu_e}\right)^2\right) \right]~,
\end{equation}
where the first order correction due to the finite electron mass is
included. The net number density $\rho_e$ and entropy density
$s_e$ can be obtained now from standard thermodynamic relations as
$\rho_e=\partial P_e/\partial \mu_e$ and $s_e=\partial
P_e/\partial T$. Neutrinos are taken into account in the same way,
but as massless particles, and with the spin factor twice smaller
than the electron one. The photon pressure is $P_{\gamma}=(\pi^2/45)T^4$.
\subsection{Nuclear statistical ensemble}
For describing an ensemble of nuclear species in thermodynamical
equilibrium we use the Grand Canonical version of the SMM
\cite{SMM,Bot85}, properly modified for supernova conditions.
After integrating out translational degrees of freedom the density
of nuclear species with mass $A$ and charge $Z$ is calculated as
\begin{equation} \label{naz}
\rho_{Az}=\frac{N_{AZ}}{V}=g_{AZ}\frac{V_f}{V}\frac{A^{3/2}}{\lambda_T^3}
{\rm exp}\left[-\frac{1}{T}\left(F_{AZ}-\mu_{AZ}\right)\right],
\end{equation}
were $g_{AZ}$ is the g.-s. degeneracy factor of species $(A,Z)$,
$\lambda_T=\left(2\pi\hbar^2/m_NT\right)^{1/2}$ is the nucleon
thermal wavelength, $m_N \approx 939$ MeV is the average nucleon
mass. $V$ is the actual volume of the system and $V_f$ is so
called free volume, which accounts for the finite size of nuclear
species. We assume that all nuclei have normal nuclear density
$\rho_0$, so that the proper volume of a nucleus with mass $A$ is
$A/\rho_0$. At low densities the finite-size correction can be
taken into account within the excluded volume approximation $V_f/V
\approx \left(1-\rho_B/\rho_0\right)$.
The internal excitations of nuclear species $(A,Z)$ play an
important role in regulating their abundance. Sometimes they are
included through the population of nuclear levels known for nearly
cold nuclei (see e.g. \cite{Japan}). However, in the supernova
environment not only the excited states but also the binding
energies of nuclei will be strongly affected by the surrounding
matter. By this reason, we find it more justified to use another
approach which can easily be generalized to include in-medium
modifications of nuclear properties. Namely, the internal free energy of species $(A,Z)$
with $A>4$ is parameterized in the spirit of the liquid drop model
\begin{equation}
F_{AZ}(T,\rho_e)=F_{AZ}^B+F_{AZ}^S+F_{AZ}^{\rm sym}+F_{AZ}^C~~,
\end{equation}
where the right hand side contains, respectively, the bulk,
the surface, the symmetry and the Coulomb terms. The first three terms
are written in the standard form \cite{SMM},
\begin{eqnarray}
F_{AZ}^B(T)=\left(-w_0-\frac{T^2}{\varepsilon_0}\right)A,~
F_{AZ}^S(T)=\beta_0\left(\frac{T_c^2-T^2}{T_c^2+T^2}\right)^{5/4}A^{2/3},~
F_{AZ}^{\rm sym}=\gamma \frac{(A-2Z)^2}{A}. \nonumber
\end{eqnarray}
Here $w_0=16$ MeV, $\varepsilon_0=16$ MeV, $\beta_0=18$ MeV,
$T_c=18$ MeV and $\gamma=25$ MeV are the model parameters which
are extracted from nuclear phenomenology and provide a good
description of multifragmentation data
\cite{SMM,Bot95,Dag,Schar,GANIL}. However, some parameters,
especially $\gamma$, can be different in hot neutron-rich nuclei,
and they need more precise determination in nuclear experiments
(see e. g. ref. \cite{LeFev}). In the Coulomb term we include the
modification due to the screening effect of electrons. By using
the Wigner-Seitz approximation it can be expressed as \cite{Lat}
\begin{equation}
F_{AZ}^C(\rho_e)=\frac{3}{5}c(\rho_e)\frac{(eZ)^2}{r_0A^{1/3}}~,~~
c(\rho_e)=\left[1-\frac{3}{2}\left(\frac{\rho_e}{\rho_{0p}}\right)^{1/3}
+\frac{1}{2}\left(\frac{\rho_e}{\rho_{0p}}\right)\right]~,\nonumber
\end{equation}
where $r_0=1.17$ fm and $\rho_{0p}=(Z/A)\rho_0$ is the proton
density inside the nuclei. The screening function $c(\rho_e)$ is 1
at $\rho_e=0$ and 0 at $\rho_0=\rho_{0p}$. We want to stress that
both the reduction of the surface energy due to the finite
temperature and the reduction of the Coulomb energy due to the
finite electron density favor the formation of heavy nuclei.
Nucleons and light clusters $(A \leq 4)$ are considered as
structureless particles characterized only by mass and proper
volume.
\begin{figure}
\vspace{-0.5cm}
\includegraphics[width=12cm]{fig2.eps}
\caption{\small{ Mean charge-to-mass ratios (left top panel), and
mass distributions of hot nuclei (other panels)
calculated with the SMM generalized for supernova conditions. Left
panels present calculations for temperature $T=3$ MeV and fixed lepton
(electrons+neutrinos) fraction $Y_L=$0.2 per nucleon. Right panels
are calculations for temperature $T=1$ MeV and fixed electron
fractions $Y_e=0.4$ (top) and 0.2 (bottom). Lines show the
fragment mass distributions at different baryon densities (in
units of the normal nuclear density $\rho_0$=0.15 fm$^{-3}$),
indicated in the figure. }}
\end{figure}
The pressure associated with nuclear species is calculated as for
the mixture of ideal gases,
\begin{equation} \label{pre}
P_{\rm nuc}=T\sum_{AZ}\rho_{AZ}\equiv
T\sum_{AZ}g_{AZ}\frac{V_f}{V}\frac{A^{3/2}}{\lambda_T^3} {\rm
exp}\left[-\frac{1}{T}\left(F_{AZ}-\mu_{AZ}\right)\right]~.
\end{equation}
As follows from eq. (\ref{naz}), the fate of heavy nuclei depends
sensitively on the relationship between $F_{AZ}$ and $\mu_{AZ}$.
In order to avoid an exponentially divergent contribution to the
baryon density, at least in the thermodynamic limit ($A
\rightarrow \infty$), inequality $F_{AZ}\,\raisebox{-.5ex}{$\stackrel{>}{\scriptstyle\sim}$}\, \mu_{AZ}$ must hold.
The equality sign here corresponds to the situation when a big,
ultimately infinite, nuclear fragment coexists with the gas of
smaller clusters \cite{Bug}. When $F_{AZ}>\mu_{AZ}$ only small
clusters with nearly exponentially falling mass spectrum are present.
However, there exist thermodynamic conditions corresponding to
$F_{AZ}\approx\mu_{AZ}$ when the mass distribution of nuclear
species is broadest. The advantage of our approach is that we
consider all the fragments present in this transition region,
contrary to the previous calculations \cite{Lamb,Lattimer}, which
consider only one ``average'' nucleus characterizing the liquid
phase.
\begin{figure}
\vspace{-1cm}
\includegraphics[width=12cm]{fig3a.eps}
\caption{\small{Mass fractions of different nuclear species as
functions of temperature for $Y_e=0.4$ calculated for different
baryon densities (indicated in the panels). Neutrons, protons,
$\alpha$-particles and heavier nuclei (A$>$4) are shown by dotted,
dash-dotted, dashed and solid lines, respectively.}}
\end{figure}
\section{Numerical results}
\subsection{Nuclear composition}
In numerical calculations we first fix temperature $T$, baryon
density $\rho_B$ and electron fraction $Y_e$. Then we consider a
box containing the baryon number $B=$1000 and proton number
$Z=Y_e\cdot B$. The box volume is fixed by the average baryon
density, $V=B/\rho_B$. We use an iterative procedure to find
chemical potentials $\mu_B$ and $\mu_Q$. Finally, relative yields
of all nuclei with 1$\leq A \leq$1000 and 0$\leq Z \leq A$ are
calculated from eq. (\ref{naz}). Nuclei with larger masses
($A>$1000) can be produced only at relatively high densities,
$\rho_B>0.1\rho_0$, which are relevant for the regions deep inside
the protoneutron star, and which are not considered here.
\begin{figure}
\vspace{-1cm}
\includegraphics[width=13cm]{fig5.eps}
\caption{\small{Mass distributions of nuclear species along two
isentropes with entropy per baryon equal 1.0 (upper panel) and 4.0
(lower panel). The corresponding temperatures and densities are
indicated in the figure.} }
\end{figure}
First we consider the case when lepton fraction is fixed as
expected inside a neutrinosphere. Figure~2 (left panels) shows the
results for lepton fraction $Y_L$=0.2 and typical temperature
$T=3$ MeV. Mass distributions are shown in the lower left panel.
One can see that the islands of heavy nuclei, $200<A<400$, appear
at relatively high baryon density, $\rho_B=0.1\rho_0$,
corresponding to the vicinity of a protoneutron star. These nuclei
are very neutron-rich, $Z/A\approx 0.27$. The $Z/A$ ratios are
decreasing with $A$ less rapidly than in the nuclear
multifragmentation case \cite{Bot01}. This can be explained by the
screening effect of electrons. The width of the charge
distribution at given $A$ is determined by $T$ and $\gamma$:
$\sigma_Z \approx \sqrt{AT/8\gamma}$ \cite{Bot85,Bot01}. At lower
density, $\rho_B=0.01\rho_0$, the mass distribution is rather flat
up to $A\approx 80$ and then decreases rapidly for larger $A$. For
$\rho_B=10^{-3}\rho_0$ only light clusters are present and the
mass distribution drops exponentially.
Let us consider now the situation more appropriate for a hot bubble at
early times of a supernova explosion, when the electron fraction of matter
did not change significantly by the electron capture reactions.
In this case the electron fraction is fixed to the initial value,
and the electron and proton chemical potentials are determined
independently, without using the equilibrium relation
$\mu_e=-\mu_Q$. Corresponding results for $Y_e=0.4$ and $Y_e=0.2$
at $T=1$ MeV and several baryon densities are presented in Fig.~2
(left top and bottom panels).
\begin{figure}
\vspace{-1cm}
\includegraphics[width=12cm]{fig6.eps}
\caption{\small{Pressure isotherms as functions of relative baryon density for $Y_e$=0.4. Solid lines show the total pressure including the electron, photon and nuclear contributions. Dotted lines show only the nuclear contribution. Results are presented for temperatures 6, 4, 2, 1 and 0.6 MeV (from top to bottom), as indicated at the corresponding lines.}}
\end{figure}
One can see that heavy and even superheavy nuclei, $50<A<400$, can
be formed in this case too. They exist in a very broad range of
densities, $0.1\rho_0>\rho_B>10^{-5}\rho_0$. At given density the
mass distribution of heavy nuclei has a Gaussian shape. In the $Y_e=0.4$
case the most probable nuclei, corresponding to the maxima of
distributions, have $Z/A$ ratios 0.400, 0.406, and 0.439, for
densities $0.1\rho_0$, $10^{-3}\rho_0$, and $10^{-5}\rho_0$,
respectively. The Gaussian mass distributions may in some cases
justify earlier calculations \cite{Lamb,Lattimer}, when only one kind of
nuclear species was considered at each density. As seen from
the bottom panel, changing the electron fraction from 0.4 to 0.2
leads to a significant increase of nuclear masses. Also, the
nuclei become more neutron rich: the corresponding $Z/A$ ratios
are 0.280, 0.359, and 0.420. Our calculations show that even
larger effect can be caused by the reduction of the symmetry
energy of hot fragments (see ref. \cite{Bot01}).
Figure 3 displays the mass fractions of different nuclear species
as functions of temperature for several baryon densities and fixed
$Y_e=0.4$. One can see several interesting trends. First, nuclei
with $A>4$ survive at high temperatures only if the baryon density
is large enough, $\rho_B>10^{-2}\rho_0$, At lower densities they
are destroyed by hard photons already at $T>2$ MeV. On the other
hand, the neutron and proton fractions increase gradually and
dominate at $\rho_B\leq 10^{-2}\rho_0$. A significant change in
the trend is observed at $T>3$ MeV which can be related to the
liquid-gas transition in such a matter. It is interesting to note
that $\alpha$-particles may exist abundantly only in a narrow
range of temperatures, $2<T<4$ MeV (see two lower panels).
\subsection{Isentropic trajectories}
let us consider now how the composition of matter changes along
the isentropic trajectories. Fig.~4 displays the mass
distributions of nuclear species along two isentropes, $S/B=1.0$
and 4.0. One can clearly see the different trends in these
two cases. In the first case the widest distribution corresponds
to the highest temperature and density state, $T=3.39$ MeV,
$\rho_B=10^{-1}\rho_0$. The mass distribution extends up to about
$A=230$ in this case. At lower densities the mass distributions
are peaked at $A\approx 70$. However, at $S/B=4.0$ the nuclei are
generally much lighter, and the widest distribution corresponds to
the lowest density state, $\rho_B=10^{-3}\rho_0$, $T=1.03$ MeV. It
is remarkable and somewhat unexpected that relatively heavy nuclei
with $20<A<80$ can survive at such a high specific entropy.
One should bear in mind that the mass distributions which are
presented here correspond to hot primary nuclei. After ejection
these nuclei will undergo de-excitation. At typical temperatures
considered here ($T\,\raisebox{-.5ex}{$\stackrel{<}{\scriptstyle\sim}$}\, 3$ MeV) the internal excitation energies
are relatively low, less than 1.0 MeV/nucleon. As well known from
calculations \cite{SMM} and nuclear experiments
\cite{Dag,Schar,GANIL}, de-excitation of nuclei with $A\leq$ 200
will go mainly by means of the nucleon emission. Then the
resulting distributions of cold nuclei are not very different from
the primary ones, they are shifted to lower masses by several
units. One should expect that shell effects (which, however, may
be modified by surrounding electrons) will play an important role
at the de-excitation stage leading to the fine structure of the
mass distribution. We believe that after the de-excitation of hot
nuclei, corresponding to the time when the ejected matter reaches
very low densities, the r-process may be responsible for the final
redistribution of the element abundances, leading to the
pronounced peaks around $A\approx$80, 130 and 200 \cite{Qui}.
As well known, nuclear
composition is extremely important for the physics of supernova
explosions. For example, the electron capture on nuclei plays an
important role in supernova dynamics \cite{Hix}. But the electron
capture rates are sensitive to the nuclear composition and details
of nuclear structure (see e.g. \cite{LMP}). The neutrino-induced
reactions are very sensitive to the nuclear structure effects and
properties of weak interactions in nuclei (see e.g. \cite{Hor}).
It is also important that the presence of nuclei favors the
explosion via the energy balance in the bubble \cite{Bethe}, since
more energy can be used for the explosion. All these
considerations show importance of the nuclear physics input in
supernova phenomenon.
\subsection{Equation of state}
Finally, we present results concering thermodynamical properties of supernova matter.
Figure~6 shows the isothermic equation of state on the pressure---density plane. One can clearly see that the pressure is dominated by the relativistic electrons at high baryon densities and by thermal photons at low baryon densities. The nuclear contribution is is always small compared to these two contributions.
On the other hand, the nuclear pressure shows the tendency to saturation at higher densities. This is consequence of the liquid-gas phase transition in nuclear subsystem, which in thermodynamic limit will manifest itself by a constant pressure in the coexistence region. This behavior will significantly influence the thermodynamical properties of matter, in particular, its heat capacity \cite{Bug}.
\section{Conclusions}
\begin{itemize}
\item The statistical equilibrium approach, which was successfully used for describing multifragmentation reactions, can be applied also for calculating the equation of state and nuclear composition of supernova matter.
\item Survival of (hot) heavy nuclei may significantly influence the explosion dynamics through both the energy balance and modification of the weak reaction rates.
\item Statistical mechanism may provide "seed" nuclei for further nuclear transformations in r-, rp, and s- processes.
\item Due to the screening effect of electrons, the alpha-decay and spontaneous fission may be suppressed in supernova environments, that opens the pathway to the production of heavy and superheavy elements.
\end{itemize}
I am grateful to A.S. Botvina with whom most of the presented results were obtained.
This work was supported in part by the DFG grant 436RUS 113/711/0-2 (Germany), and grants RFFR-05-02-04013 and NS-8756.2006.2 (Russia).
|
1,108,101,565,416 | arxiv | \section{Introduction}\label{sec:Introduction}
Stochastic partial differential equations (stochastic PDEs) arise in
many fields, such as biology, physics, engineering, and economics, in which
random phenomena play a crucial role. Complex systems always
contain some element of uncertainty. Uncertainty may arise in the system
parameters, initial and boundary conditions, and external forcing processes.
Moreover, in many situations there is incomplete or partial understanding of the
governing physical laws, and many models are therefore best
formulated using stochastic PDEs.
Recently there has been an interest in studying the effect of
stochastic forcing on nonlinear conservation laws
\cite{Bauzet:2012kx,Biswas:2014gd,ChenKarlsen2012,
DebusscheVovelle2010,Kim2003,HR1997,
FengNualart2008,Vallet:2009uq,Vallet:2000ys}, with particular
emphasis on existence and uniqueness questions (well-posedness).
Deterministic conservation laws possess
discontinuous (shock) solutions, and a
weak formulation coupled with an appropriate entropy condition
is required to ensure the well-posedness \cite{Kruzkov1970,Malek1996}.
The question of well-posedness gets somewhat more difficult
by adding a stochastic source term, due to the
interaction between noise and nonlinearity.
In a different direction, we also mention the recent
works \cite{Lions:2013aa,Lions:2014aa} by
Lions, Perthame, and Souganidis on conservation
laws with (rough) stochastic fluxes.
To be more precise, we are interested in stochastic
conservation laws driven by Gaussian noise
of the following form:
\begin{equation}\label{eq:StochasticBalanceLaw}
\left\{
\begin{aligned}
du(t,x) + \nabla \cdot f(u(t,x))dt & =
\int_Z \sigma(x,u(t,x),z)W(dt,dz), &(t,x) \in \Pi_T, \\
u(0,x) &= u^0(x), & x \in \R^d,
\end{aligned}
\right.
\end{equation}
where $\Pi_T = (0,T) \times \R^d$, $T > 0$ is some fixed
final time, $u^0 = u^0(x,\omega)$ is a given $\F_0$\=/measurable
random function, and the unknown $u = u(t,x,\omega)$ is a
random (scalar) function. The flux function
\begin{equation}\label{assumption:LipOnf}
f:\R \rightarrow \R^d \mbox{ is assumed to
be $C^1$ and globally Lipschitz.} \tag{$\mathcal{A}_{f}$}
\end{equation}
Concerning the source term, we let $(Z,\Zcal,\mu)$ be a $\sigma$-finite
separable measure space and $W$ be a space-time Gaussian white noise
martingale random measure with respect to a filtration
$\seq{\F_t}_{0 \leq t \leq T}$ \cite{Walsh1984}. Its covariance
measure is given by $dt \otimes d\mu$, where $dt$ denotes Lebesgue
measure on $[0,T]$, that is, for $A,B \in \Zcal$,
\begin{displaymath}
\E{W(t,A)W(t,B)} = t\mu(A \cap B).
\end{displaymath}
The noise coefficient $\sigma:\R^d \times \R \times Z \rightarrow \R$ is
a measurable function satisfying
\begin{equation}\label{assumption:LipOnSigma}
\mbox{$\exists M \in L^2(Z)$ s.t.}
\begin{cases}
\abs{\sigma(x,u,z)-\sigma(x,v,z)} \leq \abs{u-v}M(z),\\
\abs{\sigma(x,u,z)} \leq M(z)(1 + \abs{u}),
\end{cases}
\tag{$\mathcal{A}_{\sigma}$}
\end{equation}
for all $(x,z) \in \R^d \times Z$. Note that $W$ induces a cylindrical
Wiener process (with identity covariance operator)
on $\RKHS = L^2(Z,\Zcal,\mu)$ which we also denote
by $W$ \cite[\S~7.1.2]{PeszatZabczyk2007}.
For a nonnegative $\phi \in C(\R^d) \cap L^1(\R^d)$, let $L^2(\R^d,\phi)$
denote the $\phi$-weighted $L^2$-space, cf.~Section~\ref{sec:Entropy_Formulation}.
Define $G(u):\RKHS \rightarrow L^2(\R^d,\phi)$ by
\begin{equation}\label{eq:GDef}
G(u)h(x) = \int_Z \sigma(x,u(x),z)h(z)\,d\mu(z).
\end{equation}
The collection of all Hilbert-Schmidt operators from
$\RKHS$ into $L^2(\R^d,\phi)$ is denoted
by $\Lin_2(\RKHS;L^2(\R^d,\phi))$.
Due to \eqref{assumption:LipOnSigma}, $G$ is a Lipschitz map
from $L^2(\R^d,\phi)$ into $\Lin_2(\RKHS;L^2(\R^d,\phi))$, i.e.,
\begin{displaymath}
\norm{G(u)-G(v)}_{\Lin_2(\RKHS;L^2(\R^d,\phi))}
\leq \norm{M}_{L^2(Z)}\norm{u-v}_{L^2(\R^d,\phi)}.
\end{displaymath}
In the above setting, \eqref{eq:StochasticBalanceLaw}
may be written as
\begin{displaymath}
du + \nabla \cdot f(u)dt = G(u)dW(t),
\end{displaymath}
where the right-hand side is interpreted with
respect to the cylindrical Wiener process \cite{DebusscheVovelle2010}.
In what follows, we will in general stick to the $\sigma$ notation.
We refer to \cite{DalangSardanyons2011} for a comparison of the
different stochastic integrals.
The Malliavin calculus used later
is developed with respect to the
isonormal Gaussian process $W:H \rightarrow L^2(\Omega)$ defined by
\begin{equation}\label{eq:IsoGausProcessDef}
W(h) = \int_0^T\int_Z h(s,z)W(ds,dz) = \int_0^T h(s)dW(s),
\end{equation}
where $H$ denotes the space $L^2([0,T] \times Z,\Borel{[0,T]}
\otimes \Zcal,dt \otimes d\mu)$. Concerning the notation and
basic theory of Malliavin calculus we
refer to \cite{NualartMalliavinCalc2006}.
When the noise term in \eqref{eq:StochasticBalanceLaw}
is additive ($\sigma$ is independent of $u$),
Kim \cite{Kim2003} used Kru{\v{z}}kov's entropy condition
and proved the well-posedness of entropy solutions, see
also Vallet and Wittbold \cite{Vallet:2009uq}.
When the noise term is additive, a change of variable
turns \eqref{eq:StochasticBalanceLaw} into a conservation law with random
flux function and well-known ``deterministic" techniques apply.
When the noise is multiplicative (i.e., $\sigma$ depends on $u$), a
simple adaptation of Kru{\v{z}}kov's
techniques fails to capture a specific ``noise-noise"
interaction term correlating two entropy solutions, and as a consequence
they do not lead to the $L^1$-contraction principle.
This issue was resolved by Feng and Nualart \cite{FengNualart2008}, introducing
an additional condition capturing the missing noise-noise interaction.
These authors employ the Kru{\v{z}}kov entropy condition (on It\^{o} form)
\begin{equation}\label{eq:kruz-intro}
\begin{split}
& \partial_t \abs{u-c} + \partial_x \left[ \sign{u-c}(f(u)-f(c))\right]
\\ & \qquad \le \frac12 \signd{u-c}\sigma(u)^2+\sign{u-c}\sigma(u)\, \partial_t W,
\qquad \forall c \in \R,
\end{split}
\end{equation}
which is understood in the distributional
sense (and via an approximation of $\sign{\cdot}$).
Here, for the sake of simplicity, we take $W$ to be an
ordinary Brownian motion ($Z$ is a point) and $d=1$.
The above family of inequalities, indexed over the ``Kru{\v{z}}kov" constants $c$,
is in \cite{FengNualart2008} augmented with an additional condition
related to certain substitution formulas \cite[\S~3.2.4]{NualartMalliavinCalc2006}, allowing
the authors to recover the above mentioned interaction
term and thus provide, for the first time, a general uniqueness result for stochastic
conservation laws.
The additional condition proposed in \cite{FengNualart2008} is rather technical
and difficult to comprehend at first glance. Furthermore, the existence proof
(passing to the limit in a sequence vanishing viscosity approximations)
becomes increasingly difficult, with several added arguments
revolving around fractional Sobolev spaces,
estimates of the moments of increments, modulus
of continuity of It\^{o} processes,
and the Garcia-Rodemich-Rumsey lemma.
Recently, Bauzet, Vallet, and Wittbold \cite{Bauzet:2012kx} provided a
framework that uses the Kru{\v{z}}kov entropy
inequalities \eqref{eq:kruz-intro} but bypasses the Feng-Nualart condition.
Rather than comparing two entropy solutions
directly, their uniqueness result compares
the entropy solution against the vanishing viscosity solution, which is
generated as the weak limit (as captured by the Young measure) of a
sequence of solutions to stochastic parabolic equations
with vanishing viscosity parameter. Although with this approach
the existence proof becomes simple, many technical difficulties
are added to the uniqueness proof. At this point, let us mention that
Debussche and Vovelle \cite{DebusscheVovelle2010}
have provided an alternative well-posedness theory based
on a kinetic formulation. The kinetic formulation
avoids some of the difficulties alluded to above, thanks to the so-called
entropy defect measure.
The purpose of our work is to propose a slight modification of
the Kru{\v{z}}kov entropy condition \eqref{eq:kruz-intro} that will shed some new
light on \cite{FengNualart2008}, and also \cite{Bauzet:2012kx}.
To this end, we recall that the uniqueness proof
for entropy solutions is based on a technique
known as ``doubling of variables''. Suppose
that $v$ is another entropy solution of \eqref{eq:StochasticBalanceLaw} with
initial condition $v^0$. The key idea is to consider $v$ as
a function of a different set of variables, say $v = v(s,y)$, and then
for each fixed $(s,y) \in \Pi_T$, take $c = v(s,y)$ in the
entropy condition for $u$. In the case that $u$ and $v$ are
stochastic fields, $v(s,y)$ is no longer a constant, but rather a random variable.
Hence it seems natural to utilize an entropy condition
in which the Kru\v{z}kov parameters $c$ in \eqref{eq:kruz-intro}
are random variables rather than constants.
Let us do an informal derivation of an entropy condition based on this idea.
As above, we let $W$ be an ordinary Brownian motion and $d = 1$.
For each fixed $\varepsilon>0$, suppose $\ue$ is a sufficiently regular
solution of the stochastic parabolic equation
\begin{equation*}
\partial_t\ue + \partial_xf(\ue)
= \sigma(\ue)\partial_tW + \varepsilon\partial_x^2\ue,
\end{equation*}
where the time derivative is understood in the sense of distributions.
We apply the \textit{anticipating} It\^o
formula (Theorem~\ref{theorem:AntIto}) to $\abs{\ue-V}$,
with $V$ being an arbitrary \textit{Malliavin}
differentiable random variable.
Taking expectations, we obtain
\begin{align*}
&\Eb{\partial_t \abs{\ue-V} + \partial_x(\sign{\ue-V}(f(\ue)-f(V)))} \\
&\hphantom{XXXX}+\Eb{\signd{\ue-V}\sigma(\ue)D_tV}
-\frac{1}{2}\Eb{\signd{\ue-V}\sigma^2(\ue)} \\
&\hphantom{XXXXXX}
= \varepsilon\Eb{\sign{\ue-V}\partial_x^2\ue},
\end{align*}
where $D_tV$ is the Malliavin derivative of $V$ at $t$. As
$$
\varepsilon\Eb{\sign{\ue-V}\partial_x^2\ue}
\le \varepsilon\Eb{\partial_x^2 \abs{\ue-V}},
$$
it follows that
\begin{align*}
&\Eb{\partial_t \abs{\ue-V} + \partial_x(\sign{\ue-V}(f(\ue)-f(V)))} \\
&\hphantom{XXXX}+\Eb{\signd{\ue-V}\sigma(\ue)D_tV}
-\frac{1}{2}\Eb{\signd{\ue-V}\sigma^2(\ue)} \\
&\hphantom{XXXXXX}\leq \varepsilon\Eb{\partial_x^2 \abs{\ue-V}}.
\end{align*}
Suppose $\ue \rightarrow u$ in a suitable sense
as $\varepsilon \downarrow 0$. Then the limit
$u$ ought to satisfy
\begin{equation}\label{eq:kruz-intro2}
\begin{split}
&\Eb{\partial_t \abs{u-V} + \partial_x(\sign{u-V}(f(u)-f(V)))} \\
& \hphantom{XXXX}+\Eb{\signd{u-V}\sigma(u)D_tV}
-\frac{1}{2}\Eb{\signd{u-V}\sigma^2(u)} \leq 0,
\end{split}
\end{equation}
which is the entropy condition that we
propose should replace \eqref{eq:kruz-intro}.
At least informally, it is easy to see why this entropy condition implies the
$L^1$ contraction principle. Let $u = u(t,x)$ and $v = v(s,y)$ be
two solutions satisfying the entropy condition \eqref{eq:kruz-intro2}.
Suppose $u,v$ are both Malliavin differentiable
and spatially regular. The entropy condition for $u$ yields
\begin{multline*}
\Eb{\partial_t \abs{u-v} + \partial_x(\sign{u-v}(f(u)-f(v)))} \\
+\Eb{\signd{u-v}\sigma(u)D_tv} - \frac{1}{2}\Eb{\signd{u-v}\sigma^2(u)} \leq 0.
\end{multline*}
Similarly, the entropy condition of $v$ yields
\begin{multline*}
\Eb{\partial_s \abs{v-u} + \partial_y(\sign{v-u}(f(v)-f(u)))} \\
+\Eb{\signd{v-u}\sigma(v)D_su} - \frac{1}{2}\Eb{\signd{v-u}\sigma^2(v)} \leq 0.
\end{multline*}
Suppose that $t > s$. Then $D_tv(s) = 0$ as $v$ is adapted (to the underlying filtration).
Adding the last two equations we obtain
\begin{multline*}
\Eb{(\partial_t + \partial_s)\abs{u-v}
+ (\partial_x + \partial_y)(\sign{u-v}(f(u)-f(v)))} \\
+\Eb{\signd{u-v}\sigma(v)D_su}
- \frac{1}{2}\Eb{\signd{u-v}(\sigma^2(u) + \sigma^2(v))} \leq 0.
\end{multline*}
Completing the square yields
\begin{multline}\label{eq:OutlineOfDoubeling}
\Eb{(\partial_t + \partial_s)\abs{u-v} + (\partial_x + \partial_y)(\sign{u-v}(f(u)-f(v)))} \\
+\Eb{\signd{u-v}\sigma(v)(D_su-\sigma(u))}
-\underbrace{\frac{1}{2}\Eb{\signd{u-v}(\sigma(u) - \sigma(v))^2}}_{=0} \leq 0.
\end{multline}
Next we write
$$
D_su(t)-\sigma(u(t)) = (D_su(t)-\sigma(u(s))) + (\sigma(u(s))-\sigma(u(t))),
$$
and attempt to send $t \downarrow s$.
The second term tends to zero almost everywhere.
Formally, for fixed $s$, we observe that $D_su(t)$ satisfies the
initial value problem
\begin{equation}\label{eq:transport}
\left\{
\begin{aligned}
dw + \partial_x\bigl(f'(u)w\bigr) \,dt &
= \bigl(\sigma'(u)w\bigr)\,dW(t), & (t > s), \\
w(s) &= \sigma(u(s)).
\end{aligned}
\right.
\end{equation}
And so, concluding that
$$
D_su(t) \rightarrow \sigma(u(s)), \quad \text{as $t\downarrow s$},
$$
amounts to showing that the solution to \eqref{eq:transport} satisfies the initial
condition (in some weak sense).
Given this result, the $L^1$ contraction property follows
from \eqref{eq:OutlineOfDoubeling} in a standard way:
\begin{equation}\label{eq:Contraction}
\frac{d}{dt} \E{\norm{u(t)-v(t)}_{L^1(\R)}} \leq 0.
\end{equation}
The above argument is hampered
by an obstacle; namely, that the Malliavin
differentiability of an entropy solution
seems hard to establish. This can be seen related to
the \textit{discontinuous} coefficient $f'(u)$ in the stochastic continuity
equation \eqref{eq:transport}, making it difficult to establish
the existence of a (properly defined) weak solution.
In the deterministic context, continuity (and related transport) equations
with low-regularity coefficients have been an active area
of research, see for example
\cite{Ambrosio:2014aa,Bouchut:1998rt,Bouchut:2005aa} (and
also \cite{Flandoli:2010yq} for a particular stochastic setting).
Continuity equations arise in many applications, such as
fluid mechanics. They also appear naturally when linearizing
a nonlinear conservation law $u_t+f(u)_x=0$ into $w_t + (f'(u)w)_x=0$,
see \cite{Ambrosio:2014aa,Bouchut:1998rt,Bouchut:2005aa}.
The present work shows that stochastic continuity equations arise naturally
as well, through linearisation (by the Malliavin derivative) of stochastic
conservation laws driven by semilinear noise. However, the study
of such equations is beyond the present paper and is left for future work.
As alluded to above, to make the $L^1$ contraction argument
rigorous we would need to know that at least one of the
two entropy solutions being compared, is Malliavin differentiable.
To avoid this nontrivial issue, we shall employ a more
indirect approach, motivated by \cite{Bauzet:2012kx}, comparing one
entropy solution against the solution of the viscous problem
linked to the other entropy solution, relying on weak compactness in the
space of Young measures for the viscous approximation.
The Malliavin differentiability of the viscous solution is then
established and its Malliavin derivative is shown to satisfy a linear
stochastic parabolic equation, with an initial condition fulfilled
in the weak sense. Given these results, the proof of the $L^1$ contraction
property follows as outlined above.
Finally, we mention that the approach developed herein appears
useful in the study of error estimates for numerical approximations
of stochastic conservation laws, whenever the
approximation is Malliavin differentiable.
It seems to us that this Malliavin differentiability
is indeed often available. Furthermore, the approach
may be extended so as to cover strongly
degenerate parabolic equations with L\'{e}vy noise, cf.~\cite{Biswas:2014gd},
\cite{NunnoProskeOksendal2009}. It also constitutes a starting point
for developing a well-posedness theory for stochastic conservation laws with
random, possibly anticipating initial data.
Note however that this seems to depend on the Malliavin differentiability of the entropy
solution (Lemma~\ref{lemma:EntIneqViscForAdapted} is no longer applicable).
The remaining part of the paper is organized as follows:
We present the solution framework and gather some
preliminary results in Section \ref{sec:Entropy_Formulation}. Well-posedness results
for the viscous approximations are provided in Section \ref{sec:ViscousApprox}.
Furthermore, we establish the Malliavin differentiability
of these approximations and show that the Malliavin derivative
can be cast as the solution of a linear stochastic parabolic equation.
The question of (weak) satisfaction of the initial condition is addressed.
Sections~\ref{sec:Existence} and~\ref{sec:Uniqueness} supply
detailed proofs for the existence and uniqueness of Young measure-valued
entropy solutions. Finally, some basic results are collected in Section~\ref{sec:Appendix}.
\section{Entropy solutions}\label{sec:Entropy_Formulation}
Under the assumption $\sigma(x,0,z) = 0$, the ordinary $L^p$ spaces
($2\leq p < \infty$) constitute a natural choice for \eqref{eq:StochasticBalanceLaw}.
Without this assumption, a certain class of weighted
$L^p$ spaces seem to be better suited. For non-negative $\phi$ we define
\begin{displaymath}
\norm{u}_{p,\phi} := \left(\int_{\R^d} \abs{u(x)}^p \phi(x)\,dx\right)^{1/p}.
\end{displaymath}
The relevant weights, denoted by $\mathfrak{N}$,
consist of non-zero $\phi \in C^1(\R^d) \cap L^1(\R^d)$ for which
there is a constant $C_\phi$ such that $\abs{\nabla\phi(x)} \leq C_\phi \phi(x)$.
The weighted $L^p$-space associated with $\phi$ is denoted by $L^p(\R^d,\phi)$.
To see that $\mathfrak{N}$ is non-empty, consider
$\phi_N(x) = (1 + \abs{x}^2)^{-N}$ for $N \in \N$.
Then we claim that $\phi_N \in \mathfrak{N}$ for all $N \geq d$.
To this end, observe that
\begin{displaymath}
\nabla \phi_N(x) = -2N\frac{x}{1+\abs{x}^2}\phi_N(x),
\end{displaymath}
so $\abs{\nabla \phi_N(x)} \leq 2N \phi_N(x)$.
Furthermore,
\begin{equation*}
\begin{split}
\int_{\R^d} \phi_N(x) \,dx
= \int_0^\infty\int_{\partial B(0,r)}\left(\frac{1}{1+r^2}\right)^N \,dS dr
< \infty.
\end{split}
\end{equation*}
Another family of functions in $\mathfrak{N}$ is $\phi_\lambda(x)
= \exp(-\lambda \sqrt{1 + \abs{x}^2})$, for $\lambda > 0$ \cite{VolpertHudjaev1969}.
The fact that $\phi \in L^1(\R^d)$
yields $L^q(\R^d,\phi) \subset L^p(\R^d,\phi)$ for all $1 \leq p < q<\infty$.
Indeed, $\norm{u}_{p,\phi} \leq \norm{u}_{q,\phi}\norm{\phi}_{L^1(\R^d)}^{1/p-1/q}$.
We shall also make use of the weighted $L^\infty$-norm
\begin{displaymath}
\norm{h}_{\infty,\phi^{-1}} := \sup_{x \in \R^d}
\left\{\frac{\abs{h(x)}}{\phi(x)}\right\}, \qquad h \in C(\R^d).
\end{displaymath}
Note that any compactly supported $h \in C(\R^d)$ is bounded
in this norm, for $\phi \in \mathfrak{N}$. The norm is convenient
due to the inequality
$\norm{u}_{p,h} \leq \norm{u}_{p,\phi}\norm{h}_{\infty,\phi^{-1}}$.
Denote by $\mathscr{E}$ the set of non-negative
convex functions in $C^2(\R)$ with $S(0) = 0$, $S'$
bounded, and $S''$ compactly supported.
Suppose $Q:\R^2 \rightarrow \R^d$ satisfies
\begin{displaymath}
\Jac_1 Q(u,c) = S'(u-c)f'(u), \qquad Q(c,c) = 0, \qquad u,c \in \R,
\end{displaymath}
where $S \in \mathscr{E}$. Then we call $(S(\cdot-c),Q(\cdot,c))$
an \emph{entropy/entropy-flux pair} (indexed over $c\in \R$).
For short, we say that $(S,Q)$ is
in $\mathscr{E}$ if $S$ is in $\mathscr{E}$.
We denote by $\D^{1,2}$ the space
of Malliavin differentiable random variables
in $L^2(\Omega)$ with Malliavin derivative in
$L^2(\Omega;L^2([0,T] \times Z))$ \cite[p.~27]{NualartMalliavinCalc2006}.
For $(S,Q) \in \mathscr{E}$, $\test \in C^\infty_c([0,T)
\times \R^d)$, and $V \in \D^{1,2}$, we define the functional
\begin{equation*}
\begin{split}
& \Entropy[(S,Q),\test,V](u) := \E{\int_{\R^d} S(u^0(x)-V)\test(0,x) \dx}\\
& \qquad + \E{\iint_{\Pi_T} S(u-V)\partial_t\test + Q(u,V)\cdot \nabla \test \dxdt} \\
& \qquad - \E{\iint_{\Pi_T}\int_Z S''(u-V) \sigma(x,u,z)D_{t,z}V \test \,d\mu(z)\dxdt } \\
& \qquad +\frac{1}{2}\E{ \iint_{\Pi_T}\int_Z S''(u-V)\sigma(x,u,z)^2\test\,d\mu(z)\dxdt},
\end{split}
\end{equation*}
where $D_{t,z}V$ is the Malliavin derivative
of $V$ at $(t,z) \in [0,T] \times Z$.
We claim that $\Entropy$ is well-defined whenever $V \in \D^{1,2}$,
$u \in L^2([0,T] \times \Omega;L^2(\R^d,\phi))$,
$u^0 \in L^2(\Omega;L^2(\R^d,\phi))$, and
\begin{displaymath}
\norm{\test(t)}_{\infty,\phi^{-1}},
\norm{\partial_t\test(t)}_{\infty,\phi^{-1}},
\norm{\nabla\test(t)}_{\infty,\phi^{-1}}
\quad \text{are bounded on $[0,T]$};
\end{displaymath}
note that any $\test \in C^\infty_c([0,T) \times \R^d)$ meets these criteria.
To this end, observe that the first three terms are bounded
due to the Lipschitz condition on $S$. Indeed,
\begin{equation}\label{eq:LipEstOnEntropyFlux}
\abs{Q(u,V)} = \abs{\int_V^u S'(z-V)\Jac f(z)\,dz}
\leq \norm{S}_{\mathrm{Lip}}\norm{f}_{\mathrm{Lip}}\abs{u-V},
\end{equation}
and so
\begin{equation}\label{eq:EstOnFluxTermEntIneq}
\begin{split}
&\abs{\E{\iint_{\Pi_T}Q(u,V)\cdot \nabla \test \dxdt}}
\leq \norm{S}_{\mathrm{Lip}}\norm{f}_{\mathrm{Lip}}
\E{\iint_{\Pi_T}(\abs{u}+\abs{V})\abs{\nabla \test} \dxdt} \\
&\hphantom{X}\leq \norm{S}_{\mathrm{Lip}}\norm{f}_{\mathrm{Lip}}
\int_0^T \left(\E{\norm{u(t)}_{1,\phi}}\norm{\nabla \test(t)}_{\infty,\phi^{-1}}
+\E{\abs{V}}\norm{\nabla \test(t)}_{L^1(\R^d)}\right)dt,
\end{split}
\end{equation}
which is finite. The terms involving $\sigma$ are easily
seen to be well-defined since the Hilbert Schmidt norm
of $G(u)$ (cf.~\eqref{eq:GDef}) is bounded.
To simplify the notation, we set $HS = \Lin_2(L^2(Z);L^2(\R^d,\phi))$.
Due to assumption~\eqref{assumption:LipOnSigma},
\begin{align*}
\norm{G(u)}_{HS}^2 &= \int_{\R^d}\int_Z \sigma^2(x,u(x),z)\phi(x)\,d\mu(z)\,dx \\
&\leq 2\norm{M}_{L^2(Z)}^2\int_{\R^d}(1 + \abs{u(x)}^2)\phi(x)\,dx.
\end{align*}
Boundedness of the last term follows as
\begin{multline}\label{eq:EstOnSquareTermEntIneq}
\abs{\E{ \iint_{\Pi_T}\int_Z S''(u-V)\sigma(x,u,z)^2\test\,d\mu(z)\dxdt}} \\
\leq \norm{S''}_\infty \E{\int_0^T
\norm{\test(t)}_{\infty,\phi^{-1}}\norm{G(u(t))}_{\HS}^2 \,dt}.
\end{multline}
By the sub-multiplicativity of the Hilbert Schmidt norm and
H\"older's inequality, it follows that
\begin{equation}\label{eq:EstOnMallTermEntIneq}
\begin{split}
&\abs{\E{\iint_{\Pi_T}\int_Z S''(u-V) \sigma(x,u,z)D_{t,z}V \varphi \,d\mu(z)\dxdt }} \\
& \quad \leq \norm{S''}_\infty \E{\iint_{\Pi_T}
\norm{\test(t)}_{\infty,\phi^{-1}} \abs{\int_Z\sigma(x,u,z)D_{t,z}V \,d\mu(z)}\phi\dxdt} \\
& \quad \leq \norm{S''}_\infty\norm{\phi}_{L^1(\R^d)}^{1/2}
\E{\int_0^T\norm{\test(t)}_{\infty,\phi^{-1}} \norm{G(u(t))D_tV}_{2,\phi}dt} \\
& \quad \leq \norm{S''}_\infty\norm{\phi}_{L^1(\R^d)}^{1/2}
\E{\int_0^T \norm{\test(t)}_{\infty,\phi^{-1}}^2
\norm{G(u(t))}_{HS}^2\,dt}^{1/2}
\\ & \qquad \qquad\qquad
\times \norm{DV}_{L^2(\Omega;L^2([0,T] \times Z))}<\infty.
\end{split}
\end{equation}
Let $\Pred$ denote the predictable $\sigma$-algebra
on $[0,T] \times \Omega$ with respect to $\seq{\F_t}$ \cite[\S~2.2]{ChungWilliams2014}.
In general we are working with equivalence classes of functions with respect
to the measure $dt \otimes dP$. The equivalence class $u$ is said to be \emph{predictable}
if it has a version $\tilde{u}$ that is $\Pred$-measurable. In some of the arguments, to
avoid picking versions, we consider the completion of $\Pred$ with
respect to $dt \otimes dP$, denoted by $\Pred^*$.
We recall that any jointly measurable and adapted process
is $\Pred^*$-measurable, see \cite[Theorem~3.7]{ChungWilliams2014}.
\begin{definition}[Entropy solution]\label{Def:EntropySolution}
An entropy solution $u = u(t,x;\omega)$
of \eqref{eq:StochasticBalanceLaw}, with initial
condition $u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$,
is a function satisfying:
\begin{itemize}
\item[(i)] $u$ is a predictable process in
$L^2([0,T] \times \Omega;L^2(\R^d,\phi))$.
\item[(ii)] For any random variable $V\in \D^{1,2}$,
any entropy/entropy-flux pair $(S,Q)$ in $\mathscr{E}$,
and all nonnegative test functions
$\test \in C^\infty_c([0,T) \times \R^d)$,
\begin{displaymath}
\Entropy[(S,Q),\test,V](u) \geq 0.
\end{displaymath}
\end{itemize}
\end{definition}
Here $L^2([0,T] \times \Omega;L^2(\R^d,\phi))$ is the
Lebesgue-Bochner space, see Section~\ref{sec:LebBoch}.
\begin{remark}
One consequence of the upcoming results is that
the viscous approximations \eqref{eq:ViscousApprox}
converge (strongly) to the entropy solution in the
sense of Definition~\ref{Def:EntropySolution}. By passing to the limit
in the weak formulation of \eqref{eq:ViscousApprox},
it follows that the entropy solution is also a weak solution.
At an \textit{informal} level, this is linked to the Malliavin integration by
parts formula. To see this, let $d = 1$, $W$ be an ordinary Brownian
motion, and suppose that $u$ is a Malliavin differentiable and
spatially regular entropy solution. We outline a nonrigorous argument
showing that $u$ is a (strong) solution
of \eqref{eq:StochasticBalanceLaw}. Let
\begin{displaymath}
(u)_+ = \begin{cases}
u \mbox{ for $u > 0$,} \\
0 \mbox{ else,}
\end{cases}
\mbox{ and } \quad
\signp{u} = \begin{cases}
1 \mbox{ for $u > 0$,} \\
0 \mbox{ else,}
\end{cases}
\end{displaymath}
so that $(u)_+' = \signp{u}$. Suppose for any $A \in \F_T$
there is a Malliavin differentiable random variable $V$ satisfying
\begin{equation*}
\begin{cases}
u-V > 0 & \mbox{for $\omega \in A$,} \\
u-V < 0 & \mbox{else},
\end{cases}
\end{equation*}
so that $\signpd{u-V} = \car{A}$. Let us point out that since random
variables of the form $\car{A}$ are not Malliavin
differentiable \cite[Proposition~1.2.6]{NualartMalliavinCalc2006}, this
argument is in need of an additional approximation step.
As the argument is already informal, we skip this step.
Let $S(\cdot) = (\cdot)_+$ and
\begin{displaymath}
Q(u,c) = \signp{u-c}(f(u)-f(c)), \qquad u,c \in \R.
\end{displaymath}
The entropy inequality yields
\begin{equation*}
\begin{split}
&\Eb{\partial_t (u-V)_+ + \partial_x(\signp{u-V}(f(u)-f(V)))} \\
& \hphantom{XXXX}+\Eb{\signpd{u-V}\sigma(u)D_tV}
-\frac{1}{2}\Eb{\signpd{u-V}\sigma^2(u)} \leq 0.
\end{split}
\end{equation*}
Apriori, the trace $t \mapsto D_tu(t)$ is not well-defined.
However, due to \eqref{eq:transport}, $D_tu(\tau) \rightarrow \sigma(u(t))$
as $\tau \downarrow t$ (essentially), while $D_tu(\tau) = 0$ for $\tau < t$, and
so it is natural to assign the value
\begin{displaymath}
D_tu(t) = \lim_{\delta \downarrow 0}\frac{1}{2\delta}
\int_{t-\delta}^{t + \delta} D_tu(\tau) \,d\tau = \frac{1}{2}\sigma(u(t)),
\end{displaymath}
cf.~\cite[p.~173]{NualartMalliavinCalc2006}. By the chain
rule for Malliavin derivatives,
$$
D_t\signp{u-V} = \signpd{u-V}\left(\frac{1}{2}\sigma(u)-D_tV\right),
$$
and so
\begin{equation*}
\Eb{\partial_t (u-V)_+ + \partial_x(\signp{u-V}(f(u)-f(V)))}
\leq \Eb{D_t\signp{u-V}\sigma(u)}.
\end{equation*}
The integration by parts formula of Malliavin calculus yields
\begin{displaymath}
\Eb{D_t\signp{u-V}\sigma(u)}
= \Eb{\signp{u-V}\sigma(u)\partial_tW}.
\end{displaymath}
As $\signp{u-V} = \car{A}$, $(u-V)_+ = (u-V)\car{A}$, and
$A$ is arbitrary, it follows that
\begin{equation*}
\partial_tu + \partial_xf(u) \leq \sigma(u)\partial_tW.
\end{equation*}
The reverse inequality follows by considering $S(\cdot) = (\cdot)_-$.
\end{remark}
Let us fix some notation. For $n = 1,2,\ldots$, we
will denote by $J^n$ a non-negative, smooth function satisfying
\begin{displaymath}
\mathrm{supp}(J^n)
\subset B(0,1), \, \int_{\R^n} J^n(x) \dx = 1,
\mbox{ and } J^n(x) = J^n(-x),
\end{displaymath}
for all $x \in \R^n$. For any $r > 0$ we let
$J_r^n(x) = \frac{1}{r^n}J^n(\frac{x}{r})$.
For $n = 1$ we let $J_r^+(x) = J_r(x-r)$ and
note that $\mathrm{supp}(J_r^+) \subset (0,2r)$.
As the value of $n$ is understood
from the context, we will write $J = J^n$.
According to Theorem~\ref{theorem:ExistenceOfSolution}
and Theorem~\ref{theorem:UniquenessOfEntSol}, if
$u^0 \in L^p(\Omega;L^p(\R^d,\phi))$, then the entropy solution
belongs to $L^p(\Omega \times [0,T];L^p(\R^d,\phi))$
for any $2 \leq p < \infty$. As a consequence of the entropy
inequality we obtain the following:
\begin{proposition}
Let $2 \leq p < \infty$ and suppose $u^0 \in L^p(\Omega,\F_0,P;L^p(\R^d,\phi))$.
If $u \in L^p([0,T] \times \Omega;L^p(\R^d,\phi))$ is an
entropy solution of \eqref{eq:StochasticBalanceLaw}, then
\begin{displaymath}
\esssup_{0 \leq t \leq T}\seq{\E{\norm{u(t)}_{p,\phi}^p}} < \infty.
\end{displaymath}
\end{proposition}
\begin{proof}
Set
\begin{displaymath}
\varphi_\delta(t,x) = \left(1
- \int_0^t J_\delta(\sigma-\tau)\,d\sigma\right)\phi(x).
\end{displaymath}
Introduce the entropy function
\begin{equation*}
S_R(u) =
\begin{cases}
R^p + pR^{p-1}(u-R) & \mbox{ for $u \geq R$}, \\
\abs{u}^p & \mbox{ for $-R < u < R$}, \\
R^p -pR^{p-1}(u+R) & \mbox{ for $u \leq -R$},
\end{cases}
\end{equation*}
and denote by $Q_R$ the corresponding entropy-flux.
Strictly speaking, $S_R$ is not in $\mathscr{E}$, but this can be
amended by a simple mollification step (which we ignore). Note that
$S_R \rightarrow \abs{\cdot}^p$ pointwise. Furthermore,
\begin{equation}\label{eq:pNormEstOnEnt}
\left\{ \begin{split}
&\abs{S_R'(u)} \leq p\abs{u}^{p-1}, \\
&\abs{S_R''(u)} \leq p(p-1)\abs{u}^{p-2}, \\
&\abs{Q_R(u,c)} \leq \norm{f}_{\mathrm{Lip}}\abs{u-c}^p.
\end{split}
\right.
\end{equation}
We apply the Lebesgue differentiation and
dominated convergence theorems to make appear
$\lim_{\delta \downarrow 0}\Entropy[(S_R,Q_R),\test_\delta,0](u) \geq 0$.
This yields
\begin{equation*}
\begin{split}
\E{\int_{\R^d} S_R(u(\tau))\phi(x)\dx}
& \leq \E{\int_{\R^d} S_R(u^0(x))\phi(x) \dx} \\
&\qquad +\E{\int_0^\tau \int_{\R^d} Q_R(u,0) \cdot \nabla \phi \dxdt} \\
&\qquad +\frac{1}{2}\E{\int_0^\tau\int_{\R^d}
\int_Z S_R''(u)\sigma(x,u,z)^2\phi\,d\mu(z)\dxdt},
\end{split}
\end{equation*}
for almost all $\tau \in [0,T]$. Due to \eqref{eq:pNormEstOnEnt} it is
straightforward to supply estimates, uniform in $R$, of
the type \eqref{eq:EstOnFluxTermEntIneq}
and \eqref{eq:EstOnSquareTermEntIneq}. By the dominated
convergence theorem, we may send $R \rightarrow \infty$. The result follows.
\end{proof}
It is enough to consider smooth random variables in
Definition~\ref{Def:EntropySolution}, i.e., random variables of the form
\begin{displaymath}
V = f(W(h_1),\dots,W(h_n))
\end{displaymath}
where $f \in C^\infty_c(\R^n)$, $W$ is the isonormal Gaussian process
defined by $\eqref{eq:IsoGausProcessDef}$, and $h_1,\dots, h_n$ are in
$H = L^2([0,T] \times Z)$, see \cite[p.~25]{NualartMalliavinCalc2006}.
We denote the space of smooth random variables by $\Sm$.
\begin{lemma}\label{lemma:ContinuityOfEntWRTV}
Suppose \eqref{assumption:LipOnf} and \eqref{assumption:LipOnSigma} are satisfied.
Fix $u \in L^2([0,T] \times \Omega;L^2(\R^d,\phi))$, an
entropy/entropy-flux pair $(S,Q) \in \mathscr{E}$,
and $\test \in C^\infty_c([0,T) \times \R^d)$. Then
\begin{displaymath}
V \mapsto \Entropy[(S,Q),\test,V](u)
\end{displaymath}
is continuous on $\D^{1,2}$ (in the strong topology).
\end{lemma}
\begin{remark}
It is not necessary that $S''$ is compactly supported in the upcoming
proof (it is sufficient with boundedness/continuity).
\end{remark}
\begin{proof}
Suppose that $V_n \rightarrow V$ in $\D^{1,2}$ as $n \rightarrow \infty$,
and write
\begin{align*}
& \Entropy[(S,Q),\test,V](u)-\Entropy[(S,Q),\test,V_n](u) \\
& \quad = \E{\int_{\R^d} (S(u^0(x)-V)-S(u^0(x)-V_n))\test(0,x) \dx}\\
& \qquad + \E{\iint_{\Pi_T} (S(u-V)-S(u-V_n))\partial_t\test \dxdt} \\
& \qquad + \E{\iint_{\Pi_T}(Q(u,V)-Q(u,V_n))\cdot \nabla \test \dxdt} \\
& \qquad + \E{\iint_{\Pi_T}\int_Z (S''(u-V_n)D_{t,z}V_n-S''(u-V)
D_{t,z}V)\sigma(x,u,z)\test \,d\mu(z)\dxdt} \\
& \qquad +\frac{1}{2}\E{ \iint_{\Pi_T}\int_Z (S''(u-V)-S''(u-V_n))
\sigma(x,u,z)^2\test\,d\mu(z)\dxdt} \\
& \quad =: \mathcal{T}_1 + \mathcal{T}_2 + \mathcal{T}_3 + \mathcal{T}_4 + \mathcal{T}_5.
\end{align*}
We need to show that $\lim_{n \rightarrow \infty}\mathcal{T}_i(n) = 0$
for $1 \leq i \leq 5$.
First, note that $V_n \rightarrow V$ in $L^2(\Omega)$. Next,
\begin{displaymath}
\abs{\mathcal{T}_1} \leq \norm{S}_{\mathrm{Lip}}\E{\abs{V-V_n}}\norm{\test(0)}_{L^1(\R)}.
\end{displaymath}
Similarly,
\begin{displaymath}
\abs{\mathcal{T}_2} \leq \norm{S}_{\mathrm{Lip}}\E{\abs{V-V_n}}\norm{\partial_t\test}_{L^1(\Pi_T)}.
\end{displaymath}
It follows as $V_n \rightarrow V$ in $L^2(\Omega)$ that $\mathcal{T}_1,\mathcal{T}_2 \rightarrow 0$
as $n \rightarrow \infty$.
Concerning $\mathcal{T}_3$, we first observe that for any $\zeta,\xi,\theta \in \R$,
\begin{align*}
&\abs{Q(\zeta,\xi)-Q(\zeta,\theta)} = \abs{\int_\xi^\zeta S'(z-\xi)\Jac f(z)\,dz
- \int_\theta^\zeta S'(z-\theta)\Jac f(z)\,dz} \\
& \qquad \leq \abs{\int_\xi^\zeta (S'(z-\xi)-S'(z-\theta))\Jac f(z)\,dz}
+ \abs{\int_\xi^\theta S'(z-\theta)\Jac f(z)\,dz}.
\end{align*}
Hence,
\begin{align*}
&\abs{\mathcal{T}_3} \leq \E{\iint_{\Pi_T}\abs{\int_V^u (S'(z-V)-S'(z-V_n))
\Jac f(z)\,dz}\abs{\nabla \test} \dxdt} \\
& \qquad\qquad
+ \E{\iint_{\Pi_T}\abs{\int_V^{V_n} S'(z-V_n)\Jac f(z)\,dz}
\abs{\nabla \test} \dxdt} \\
& \qquad =:\mathcal{T}_3^1 + \mathcal{T}_3^2.
\end{align*}
Consider $\mathcal{T}_3^1$. Note that
\begin{displaymath}
\abs{\int_V^u (S'(z-V)-S'(z-V_n))\Jac f(z)\,dz} \leq
\norm{f}_{\mathrm{Lip}}\norm{S''}_\infty\abs{V-V_n}(\abs{u}+ \abs{V}).
\end{displaymath}
Due to H\"{o}lder's inequality it follows that
\begin{multline*}
\mathcal{T}_3^1 \leq \norm{V-V_n}_{L^2(\Omega)}
\norm{f}_{\mathrm{Lip}}\norm{S''}_\infty\norm{\nabla \test}_{L^1(\Pi_T)}^{1/2} \\
\times \E{2\iint_{\Pi_T}(\abs{u}^2 + \abs{V}^2)\abs{\nabla \test}\,dxdt}^{1/2}.
\end{multline*}
Since
\begin{displaymath}
\mathcal{T}_3^2 \leq \norm{S}_{\mathrm{Lip}}\norm{f}_{\mathrm{Lip}}
\E{\abs{V_n-V}}\norm{\nabla \test}_{L^1(\Pi_T)},
\end{displaymath}
it follows that $\lim_{n \rightarrow \infty}\mathcal{T}_3 = 0$.
Concerning the $\mathcal{T}_4$-term, we first split it as follows:
\begin{align*}
\mathcal{T}_4 &= \E{\iint_{\Pi_T}\int_Z S''(u-V_n)(D_{t,z}V_n-D_{t,z}V)\sigma(x,u,z)
\test \,d\mu(z)\dxdt} \\
&+ \E{\iint_{\Pi_T}\int_Z (S''(u-V_n)-S''(u-V))D_{t,z}
V\sigma(x,u,z)\test \,d\mu(z)\dxdt} \\
&= \mathcal{T}_4^1 + \mathcal{T}_4^2.
\end{align*}
By \eqref{eq:EstOnMallTermEntIneq}, $\lim_{n \rightarrow \infty}\mathcal{T}_4^1 = 0$.
Owing to \eqref{eq:EstOnMallTermEntIneq}, the dominated
convergence theorem implies
$\lim_{n \rightarrow \infty}\mathcal{T}_4^2 = 0$.
Finally, by \eqref{eq:EstOnSquareTermEntIneq} and
the dominated convergence theorem,
also $\lim_{n \rightarrow \infty}\mathcal{T}_5 = 0$.
\end{proof}
For the existence proof, it will be convenient to introduce
a weaker notion of entropy solution based on Young
measures (see, e.g., \cite{Bauzet:2012kx,DiPerna1985,EymardGallouetHerbin1995,Panov1996}).
The reason beeing the application of Young measures as generalized
limits in the sense of Theorem~\ref{theorem:YoungMeasureLimitOfComposedFunc}.
Denote by $\Young{\Pi_T \times \Omega;\R}$ the set of all Young
measures from $\Pi_T \times \Omega$ into $\R$,
cf.~Section~\ref{sec:YoungMeasures}. Instead of representing the
solution/limit as an element in $\Young{\Pi_T \times \Omega;\R}$ we
use the notion of entropy process proposed in \cite{EymardGallouetHerbin1995}
or equivalently the strong measure-valued solution proposed in \cite{Panov1996}.
Any probability measure $\nu$ on the real line may be represented by a
measurable function $u:[0,1] \rightarrow \R \cup \seq{\infty}$
such that $\nu$ is the image of the Lebesgue
measure $\Lebesgue$ on $[0,1]$ by $u$.
In fact, we may take (see \cite[\S~2.2.2]{Villani2003})
\begin{equation}\label{eq:ReprOfProcessByYoung}
u(\alpha) = \inf \left\{ \xi \in \R \,:\, \nu((-\infty,\xi]) > \alpha \right\}.
\end{equation}
A Young measure $\nu \in \Young{\Pi_T \times \Omega;\R}$ is thus
represented by a (higher dimensional) function $u:\Pi_T \times [0,1]
\times \Omega \rightarrow \R$, such that
$\nu_{t,x,\omega}(B) = \Lebesgue(u(t,x,\cdot,\omega)^{-1}(B))$
for any measurable $B \subset \R$. The extension to Young
measure-valued solutions is obtained through the embedding defined by
\begin{equation}\label{eq:EmbeddingL2L2times01}
\Phi(u)(t,x,\alpha,\omega) =
u(t,x,\omega).
\end{equation}
Given a functional $F$ we define the extension
\begin{displaymath}
\Y{F}(u) = \int_0^1 F(u(\alpha))\,d\alpha,
\end{displaymath}
so that $\Y{F} \circ \Phi = F$. For $1 \leq p < \infty$ we let
\begin{displaymath}
\norm{u}_{p,\phi \otimes 1} = \left(\int_0^1
\int_{\R^d} \abs{u(x,\alpha)}^p\phi(x)\,dx\,d\alpha\right)^{1/p}.
\end{displaymath}
The associated space is denoted by $L^p(\R^d \times [0,1],\phi)$.
\begin{definition}[Young measure-valued entropy solution]\label{Def:YoungEntropySolution}
A Young measure-valued entropy solution $u = u(t,x,\alpha;\omega)$
of \eqref{eq:StochasticBalanceLaw}, with initial condition
$u^0$ belonging to $L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$, is
a function satisfying:
\begin{itemize}
\item[(i)] $u$ is a predictable process in
$L^2([0,T] \times \Omega;L^2(\R^d \times [0,1],\phi))$.
\item[(ii)] For any random variable $V\in \D^{1,2}$,
any entropy/entropy-flux pair $(S,Q)$ in $\mathscr{E}$,
and all nonnegative test
functions $\test \in C^\infty_c([0,T) \times \R^d)$,
\begin{equation}\label{eq:YoungEntropyCondition}
\Y{\Entropy[(S,Q),\test,V]}(u) \geq 0.
\end{equation}
\end{itemize}
\end{definition}
The next result is concerned with the essential continuity of the
solutions at $t = 0$. A similar argument can be
found in \cite{CancesClementGallouet2011}.
\begin{lemma}[Initial condition]\label{lemma:InitialCondition}
Suppose \eqref{assumption:LipOnf} and \eqref{assumption:LipOnSigma}
are satisfied, and that $u^0$ belongs to $L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Let $u$ be a Young measure-valued entropy
solution of \eqref{eq:StochasticBalanceLaw}
in the sense of Definition~\ref{Def:YoungEntropySolution}.
Let $S:\R \rightarrow [0,\infty)$ be Lipschitz continuous and
satisfy $S(0) = 0$. For any $\psi \in C^\infty_c(\R^d)$,
\begin{displaymath}
\mathcal{T}_{r_0} := \E{\iint_{\Pi_T}\int_{[0,1]}
S(u(t,x,\alpha)-u^0(x))\psi(x)J_{r_0}^+(t)\,d\alpha dxdt}
\rightarrow 0 \mbox{ as } r_0 \downarrow 0.
\end{displaymath}
\end{lemma}
\begin{remark}
The proof does not depend on the differentiability of $J_{r_0}^+$.
Hence the above limit may be replaced by
\begin{displaymath}
\lim_{\tau \downarrow 0} \E{\frac{1}{\tau}\int_0^\tau\int_{\R^d} \int_{[0,1]}
S(u(t,x,\alpha)-u^0(x))\psi(x)\,d\alpha dx dt} = 0.
\end{displaymath}
\end{remark}
\begin{proof}
Let $S \in C^\infty(\R)$ with bounded derivatives. Take
\begin{displaymath}
\test(t,x,y) = \xi_{r_0}(t)\psi(x)J_r(x-y)\, \mbox{ where }\,\xi_{r_0}(t) = 1 - \int_0^t J_{r_0}^+(s)\ds.
\end{displaymath}
Then let $V = u^0(y)$ in \eqref{eq:YoungEntropyCondition} and
integrate in $y$. This implies
\begin{equation}\label{eq:IneqTimeContAtZero}
\begin{split}
I &:= \E{\int_{\R^d}\iint_{\Pi_T}\int_{[0,1]} S(u(t,x,\alpha)-u^0(y))\psi(x)
J_r(x-y)J_{r_0}^+(t) d\alpha\dxdt dy} \\
& \leq \E{\int_{\R^d}\iint_{\Pi_T}\int_{[0,1]}Q(u(t,x,\alpha),u^0(y))\cdot
\nabla_x \test d\alpha\dxdt dy} \\
& \qquad +\E{\int_{\R^d}\int_{\R^d} S(u^0(x)-u^0(y))\test(0,x,y)\dx dy} \\
& \qquad +\frac{1}{2} \E{\int_{\R^d}\iint_{\Pi_T}\int_{[0,1]}\int_Z
S''(u(t,x,\alpha)-u^0(y))\sigma(x,u,z)^2\test \,d\mu(z) d\alpha\dxdt dy} \\
& =: \mathcal{T}^1 + \mathcal{T}^2 + \mathcal{T}^3.
\end{split}
\end{equation}
Let us first observe that
\begin{multline*}
I = \mathcal{T}_{r_0} + E\bigg[\int_{\R^d}\iint_{\Pi_T}
\int_{[0,1]} (S(u(t,x,\alpha)-u^0(y))-S(u(t,x,\alpha)-u^0(x))) \\
\times \psi(x)J_r(x-y)J_{r_0}^+(t) d\alpha\dxdt dy\bigg] =: \mathcal{T}_{r_0} + I^1.
\end{multline*}
We want to take the limit $r_0 \downarrow 0$.
First observe that we have the bound
\begin{displaymath}
\abs{I^1} \leq \norm{S}_{\mathrm{Lip}}\E{\int_{\R^d}\int_{\R^d} \abs{u^0(x)-u^0(y)}
\psi(x)J_r(x-y)dxdy} =: R,
\end{displaymath}
which is independent of $r_0$. Similarly, $\abs{\mathcal{T}^2} \leq R$.
Note that $\xi_{r_0} \rightarrow 0$ a.e.~as $r_0 \downarrow 0$, so due to
assumptions~\eqref{assumption:LipOnf}
and \eqref{assumption:LipOnSigma}, one may conclude by the dominated
convergence theorem and estimates similar
to those in \eqref{eq:EstOnFluxTermEntIneq}
and \eqref{eq:EstOnSquareTermEntIneq} that
\begin{displaymath}
\lim_{r_0 \downarrow 0} \mathcal{T}^1 = \lim_{r_0 \downarrow 0} \mathcal{T}^3 = 0.
\end{displaymath}
Thus, it follows by \eqref{eq:IneqTimeContAtZero} that
\begin{displaymath}
\lim_{r_0 \downarrow 0} \mathcal{T}_{r_0} \leq 2R.
\end{displaymath}
Since $r > 0$ was arbitrary, and $\lim_{r \downarrow 0}R = 0$, we
have arrived at $\lim_{r_0 \downarrow 0} \mathcal{T}_{r_0} \leq 0$.
The desired result follows, since we can approximate
any Lipschitz function uniformly by
smooth functions with bounded derivatives.
\end{proof}
\section{The viscous approximation}\label{sec:ViscousApprox}
For each fixed $\varepsilon>0$, we denote by $\ue$ the
solution of the regularized problem
\begin{equation}\label{eq:ViscousApprox}
\left\{
\begin{aligned}
d\ue + \nabla \cdot f(\ue)dt &= \int_Z \sigma(x,\ue,z)W(dt,dz)
+ \varepsilon \Delta \ue dt, &(t,x) \in \Pi_T, \\
\ue(0,x) &= u^0(x), & x \in \R^d.
\end{aligned}
\right.
\end{equation}
As in the deterministic case, the idea is to let $\varepsilon\to 0$ and
obtain a solution to the stochastic
conservation law \eqref{eq:StochasticBalanceLaw}.
The entropy condition is meant to single out this limit as the
only proper (weak) solution; the entropy solution.
To show that this limit exists, a type of compactness
argument is needed \cite{Bauzet:2012kx, FengNualart2008, ChenKarlsen2012}.
The existence of a unique solution to \eqref{eq:ViscousApprox} may be
found several places \cite{Bauzet:2012kx, FengNualart2008}.
In particular, the semi-group approach presented in \cite[ch.~9]{PeszatZabczyk2007}
may be applied. The functional setting of \cite{PeszatZabczyk2007} is
that of a Hilbert space, and so the natural choice here
is $L^2(\R^d,\phi)$ where $\phi \in \mathfrak{N}$.
Due to the new functional setting, we have chosen to include proofs
for some of the results relating to \eqref{eq:ViscousApprox}.
\subsection{A priori estimates and well-posedness}
Let $S_\varepsilon$ be the semi-group generated by the heat kernel.
That is $S_\varepsilon(t) u = \Phi_\varepsilon(t) \star u$ where
\begin{displaymath}
\Phi_\varepsilon(t,x) := \frac{1}{(4\varepsilon\pi t)^{d/2}}
\exp\left(-\frac{\abs{x}^2}{4 \varepsilon t}\right).
\end{displaymath}
Let $F(u) = \nabla \cdot f(u)$ and $G$ be defined by \eqref{eq:GDef}.
In this setting the key conditions \cite[p.~142]{PeszatZabczyk2007} for
well-posedness of \eqref{eq:ViscousApprox} are:
\begin{itemize}
\item[(F)] $D(F)$ is dense in $L^2(\R^d,\phi)$ and there is a
function $a:(0,\infty) \rightarrow (0,\infty)$ satisfying $\int_0^T a(t)\,dt < \infty$ for all $T < \infty$
such that, for all $t > 0$ and $u,v \in D(F)$,
\begin{align*}
\norm{S_\varepsilon(t)F(u)}_{2,\phi} &\leq a(t)\left(1 + \norm{u}_{2,\phi}\right), \\
\norm{S_\varepsilon(t)(F(u)-F(v))}_{2,\phi} &\leq a(t)\norm{u-v}_{2,\phi}.
\end{align*}
\item[(G)] $D(G)$ is dense in $L^2(\R^d,\phi)$ and there is a
function $b : (0,\infty) \rightarrow (0, \infty)$ satisfying $\int_0^T b^2(t) dt < \infty$
for all $T < \infty$ such that, for all $t > 0$ and $u,v \in D(G)$,
\begin{align*}
\norm{S_\varepsilon(t)G(u)}_{\Lin_2(L^2(Z);L^2(\R^d,\phi))}
& \leq b(t)\left(1 + \norm{u}_{2,\phi}\right), \\
\norm{S_\varepsilon(t)(G(u)-G(v))}_{\Lin_2(L^2(Z);L^2(\R^d,\phi))}
& \leq b(t)\norm{u-v}_{2,\phi}.
\end{align*}
\end{itemize}
Suppose $u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Under assumptions (F) and (G) we may conclude
by \cite[Theorem~9.15, Theorem~9.29]{PeszatZabczyk2007} that
there exists a unique predictable
process $\ue:[0,T] \times \Omega \rightarrow L^2(\R^d,\phi)$ such that
\begin{itemize}
\item[(i)]
\begin{equation}\label{eq:BoundednessOfViscApprox}
\sup_{0 \leq t \leq T}\E{\norm{\ue(t)}_{2,\phi}^2} < \infty.
\end{equation}
\item[(ii)] For all $0 \leq t \leq T$
\begin{equation}\label{eq:MildViscSol}
\begin{split}
\qquad \quad \ue(t,x) &= \int_{\R^d}\Phi_\varepsilon(t,x-y)u^0(y)\,dy \\
& \qquad - \int_{0}^{t}\int_{\R^d} \nabla_x
\Phi_{\varepsilon}(t-s,x-y) \cdot f(\ue(s,y)) \,dyds \\
& \qquad + \int_{0}^{t}\int_Z\int_{\R^d}
\Phi_\varepsilon(t-s,x-y)\sigma(y,\ue(s,y),z)\,dy \,W(ds,dz).
\end{split}
\end{equation}
\item[(iii)] $\ue$ is a weak solution of $\eqref{eq:ViscousApprox}$, i.e., for any
test function $\test \in C^\infty_c(\R^d)$ and any pair of
times $t_0,t$ with $0 \leq t_0 \leq t \leq T$,
\begin{multline}\label{eq:WeakSolutionVisc}
\qquad \int_{\R^d} \ue(t)\test \dx = \int_{\R^d} \ue(t_0)\test\dx
- \int_{t_0}^t\int_{\R^d} f(\ue(s)) \cdot \nabla\test\dx ds \\
+ \int_{t_0}^t\int_{\R^d} \int_Z \sigma(x,\ue(s,x),z)\test W(ds,dz) \dx
+ \varepsilon \int_{t_0}^t\int_{\R^d} \ue \Delta \test\dx ds,
\end{multline}
\qquad \qquad \qquad $dP$-almost surely.
\end{itemize}
To see that conditions (F) and (G) are satisfied
we prove the following estimate:
\begin{lemma}\label{lemma:HeatKernelYoungIneq}
Fix $\phi \in \mathfrak{N}$ and $1 \leq p < \infty$.
Let $v \in W^{1,p}(\R^d,\phi;\R^d)$, $u \in L^p(\R^d,\phi)$. Whenever
$C_\phi\sqrt{4\varepsilon t} \leq 1$,
\begin{align}
&\norm{\Phi_\varepsilon(t) \star u}_{p,\phi}
\leq \kappa_{1,d} \norm{u}_{p,\phi}, \tag{i} \label{eq:estOnHeat}\\
&\norm{\Phi_\varepsilon(t) \star \nabla \cdot v}_{p,\phi}
\leq \frac{\kappa_{2,d}}{\sqrt{\varepsilon t}}\norm{v}_{p,\phi},
\tag{ii}\label{eq:estOnHeatDiv}
\end{align}
where $\kappa_{1,d} =c_{d-1} \frac{d\alpha(d)}{\pi^{d/2}},
\kappa_{2,d} =c_d\frac{d\alpha(d)}{\pi^{d/2}}$, and
\begin{displaymath}
c_d = \int_0^\infty \zeta^d(1 + \zeta)^2\exp(\zeta-\zeta^2)\,d\zeta.
\end{displaymath}
The volume of the unit ball in $\R^d$ is denoted by $\alpha(d)$.
\end{lemma}
Before we give a proof let us see why (F) and (G) follow.
Recall that we may assume $f(0) = 0$ without any loss of generality.
By Lemma~\ref{lemma:HeatKernelYoungIneq}
and \eqref{assumption:LipOnf},
\begin{displaymath}
\norm{S_\varepsilon(t)F(u)}_{2,\phi} = \norm{\Phi_\varepsilon(t)
\star \nabla \cdot f(u)}_{2,\phi}
\leq\underbrace{\frac{\kappa_{2,d}}{\sqrt{\varepsilon t}}
\norm{f}_{\mathrm{Lip}} }_{a(t)}\norm{u}_{2,\phi}.
\end{displaymath}
It remains to observe that $\int_0^T \frac{1}{\sqrt{t}}\,dt = 2\sqrt{T} < \infty$.
The second part of $(F)$ follows similarly.
Let us consider (G). First observe that
\begin{displaymath}
S_\varepsilon(t)G(u)h(x) = \int_Z \left(\int_{\R^d}
\Phi_\varepsilon(t,x-y)\sigma(y,u(y),z)\,dy\right)h(z)\,d \mu(z).
\end{displaymath}
Recall that $HS = \Lin_2(L^2(Z);L^2(\R^d,\phi))$. By
Lemma~\ref{lemma:HeatKernelYoungIneq} and \eqref{assumption:LipOnSigma}
\begin{align*}
\norm{S_\varepsilon(t)G(u)}_{HS}^2
&= \int_Z\int_{\R^d} \left(\int_{\R^d}\Phi_\varepsilon(t,x-y)\sigma(y,u(y),z)\,dy\right)^2
\phi(x) \,dx d\mu(z) \\
&= \int_Z\norm{\Phi_\varepsilon(t) \star \sigma(\cdot,u,z)}_{2,\phi}^2 d\mu(z) \\
&\leq \kappa_{1,d}^2\norm{M}_{L^2(Z)}^2 (\norm{\phi}_{L^1(\R^d)}
+ \norm{u}_{2,\phi})^2.
\end{align*}
This yields the first part of condition (G). The second part follows similarly, in view of
the Lipschitz assumption on $\sigma$.
\begin{proof}[Proof of Lemma~\ref{lemma:HeatKernelYoungIneq}]
Consider \eqref{eq:estOnHeat}. By Proposition~\ref{prop:YoungsForLocalized},
\begin{displaymath}
\norm{\Phi_\varepsilon(t) \star u}_{p,\phi} \leq \underbrace{\left(\int_{\R^d}
\abs{\Phi_\varepsilon(t,x)}(1 + w_{p,\phi}(\abs{x}))\,dx\right)}_{\norm{\Phi(t)}}
\norm{u}_{p,\phi}.
\end{displaymath}
where
\begin{displaymath}
w_{p,\phi}(r) = \frac{C_\phi}{p}r\left(1
+ \frac{C_\phi}{p}r\right)\exp\left(\frac{C_\phi}{p} r\right).
\end{displaymath}
We apply polar coordinates to compute $\norm{\Phi(t)}$. This yields
\begin{align*}
\norm{\Phi(t)} &= \int_0^\infty \int_{\partial B(0,r)}
\abs{\Phi_\varepsilon(t)}(r)(1 + w_{p,\phi}(r))\,dS(r)\,dr \\
&= \frac{d\alpha(d)}{(4\varepsilon\pi t)^{d/2}}\int_0^\infty r^{d-1}
\exp\left(\frac{C_\phi}{p} r-\frac{r^2}{4\varepsilon t}\right)
\\ & \qquad \qquad\qquad\quad
\times \left(\exp\left(-\frac{C_\phi}{p}r\right)
+ \frac{C_\phi}{p}r\left(1 + \frac{C_\phi}{p}r\right)\right)\,dr.
\end{align*}
To simplify, we note that
\begin{displaymath}
\exp\left(-\frac{C_\phi}{p} r\right) + \frac{C_\phi}{p}r\left(1 + \frac{C_\phi}{p}r\right)
\leq \left(1 + \left(\frac{C_\phi}{p}r\right)\right)^2.
\end{displaymath}
Let $\zeta = r/\sqrt{4\varepsilon t}$. Provided
$C_\phi\sqrt{4\varepsilon t} \leq 1$, it follows that
\begin{displaymath}
\frac{C_\phi}{p}r = \frac{C_\phi}{p}\sqrt{4\varepsilon t}\zeta \leq \zeta.
\end{displaymath}
Inserting this we obtain
\begin{displaymath}
\norm{\Phi(t)} \leq \frac{d\alpha(d)}{\pi^{d/2}}\int_0^\infty \zeta^{d-1}\left(1
+ \zeta\right)^2\exp\left(\zeta-\zeta^2\right)\,d\zeta.
\end{displaymath}
Estimate \eqref{eq:estOnHeatDiv} follows
along the same lines. Integration by parts yields
\begin{displaymath}
\int_{\R^d}\Phi_\varepsilon(t,x-y)\nabla \cdot v(y)\,dy
= \int_{\R^d}\nabla_x\Phi_\varepsilon(t,x-y)\cdot v(y)\,dy.
\end{displaymath}
Hence,
\begin{displaymath}
\abs{\Phi_\varepsilon(t) \star \nabla \cdot v} \leq \abs{\nabla \Phi(t)} \star \abs{v}.
\end{displaymath}
By Proposition~\ref{prop:YoungsForLocalized},
\begin{displaymath}
\norm{\nabla_x \Phi_\varepsilon(t) \star v}_{L^p(\R^d)}
\leq \underbrace{\left(\int_{\R^d} \abs{\nabla \Phi_\varepsilon(t,x)}(1
+ w_{p,\phi}(\abs{x}))\,dx\right)}_{\norm{\nabla \Phi(t)}}
\norm{v}_{p,\phi}.
\end{displaymath}
Let $r = \abs{x}$. Then
\begin{displaymath}
\norm{\nabla \Phi(t)} = \int_0^\infty \underbrace{\int_{\partial B(0,r)}
\abs{\nabla \Phi_\varepsilon(t)}(r)(1 + w_{p,\phi}(r))\,dS(r)}_{\Psi(r)}\,dr.
\end{displaymath}
Now,
\begin{displaymath}
\nabla\Phi_\varepsilon(t,x) = -\frac{2\pi x}{(4\pi\varepsilon t)^{d/2 + 1}}
\exp\left(-\frac{\abs{x}^2}{4\varepsilon t}\right),
\end{displaymath}
and so
\begin{displaymath}
\Psi(r) = \frac{d\alpha(d)}{2\varepsilon t\pi^{d/2}}
\left(\frac{r}{\sqrt{4\varepsilon t}}\right)^d
\exp\left(\frac{C_\phi}{p} r-\frac{r^2}{4\varepsilon t}\right)
\left(\exp\left(-\frac{C_\phi}{p} r\right) + \frac{C_\phi}{p}r
\left(1 + \frac{C_\phi}{p}r\right)\right).
\end{displaymath}
Let $\zeta(r) = r/\sqrt{4\varepsilon t}$ and
suppose $C_\phi \sqrt{4\varepsilon t} \leq 1$. Then
\begin{multline*}
\int_0^\infty \left(\frac{r}{\sqrt{4\varepsilon t}}\right)^d\exp
\left(\frac{C_\phi}{p} r-\frac{r^2}{4\varepsilon t}\right)
\left(1 + \left(\frac{C_\phi}{p}r\right)\right)^2\,dr \\
\leq \sqrt{4\varepsilon t} \int_0^\infty
\zeta^d(1 + \zeta)^2\exp(\zeta-\zeta^2)\,d\zeta.
\end{multline*}
This concludes the proof of the lemma.
\end{proof}
The following two lemmas constitute the reason why
Lemma~\ref{lemma:HeatKernelYoungIneq} is the key to
the well-posedness of \eqref{eq:ViscousApprox}.
As we will see, the relevant properties of $\ue$ follow rather
easily with these estimates at hand.
\begin{lemma}\label{lemma:FGenEst}
Let $1 \leq p \leq \infty$ and $\phi \in \mathfrak{N}$. Suppose
$v \in C([0,T];W^{1,p}(\R^d,\phi;\R^d))$.
Set
\begin{displaymath}
\mathcal{T}[v](t,x) = \int_0^t\int_{\R^d}
\Phi_\varepsilon(t-s,x-y)(\nabla \cdot v(s,y)) \,dyds.
\end{displaymath}
Then, for any $1 \leq q < \infty$,
\begin{displaymath}
\norm{\mathcal{T}[v](t)}_{p,\phi}^q \leq
\kappa_{2,d}^q\left(2 \sqrt{\frac{t}{\varepsilon}}\right)^{q-1}
\int_0^t \frac{1}{\sqrt{\varepsilon(t-s)}}\norm{v(s)}_{p,\phi}^q \,ds,
\end{displaymath}
where $\kappa_{2,d} = c_d\frac{d\alpha(d)}{\pi^{d/2}}$ and $c_d$ is
defined in Lemma~\ref{lemma:HeatKernelYoungIneq}.
\end{lemma}
\begin{proof}
By Minkowski's integral inequality \cite[p.271]{Stein1970}
and Lemma~\ref{lemma:HeatKernelYoungIneq},
\begin{align*}
\norm{\mathcal{T}[v](t)}_{p,\phi}^q
&\leq \left(\int_0^t\norm{\Phi_\varepsilon(t-s)
\star (\nabla \cdot v(s,y))}_{p,\phi} \,ds\right)^q \\
&\leq \left(\int_0^t\frac{\kappa_{2,d}}{\sqrt{\varepsilon(t-s)}}
\norm{v(s)}_{p,\phi} \,ds\right)^q.
\end{align*}
If $q = 1$ we are done, so we may assume $1 < q < \infty$. Let $r$
satisfy $1 = r^{-1} + q^{-1}$ and take
\begin{displaymath}
h(s) := \left(\frac{1}{\sqrt{\varepsilon(t-s)}}\right)^{1/r} \quad \mbox{and} \quad
g(s) := \left(\frac{1}{\sqrt{\varepsilon(t-s)}}\right)^{1-1/r}\norm{v(s)}_{p,\phi}.
\end{displaymath}
By H\"older's inequality, $\norm{hg}_{L^1([0,t])}^q \leq
\norm{g}_{L^q([0,t])}^q\norm{h}_{L^r([0,t])}^q$,
and so
\begin{displaymath}
\norm{\mathcal{T}[v](t)}_{p,\phi}^q \leq
(\kappa_{2,d}\norm{h}_{L^r([0,t])})^q \int_0^t
\frac{1}{\sqrt{\varepsilon(t-s)}}\norm{v(s)}_{p,\phi}^q \,ds.
\end{displaymath}
A simple computation yields
\begin{displaymath}
\norm{h}_{L^r([0,t])}^q = \left(\int_0^t
\frac{1}{\sqrt{\varepsilon(t-s)}} \,ds\right)^{q/r}
= \left(2 \sqrt{\frac{t}{\varepsilon}}\right)^{q-1}.
\end{displaymath}
The result follows.
\end{proof}
\begin{lemma}\label{lemma:GGenEst}
Let $2 \leq p < \infty$ and $\phi \in \mathfrak{N}$. Suppose
$v:\Omega \times [0,T] \times Z \times \R^d \rightarrow \R$
is a predictable process satisfying
\begin{displaymath}
\abs{v(s,x,z)} \leq K(s,x)M(z),
\end{displaymath}
for $M \in L^2(Z)$ and a process
$K \in L^2([0,T];L^p(\Omega;L^p(\R^d,\phi)))$. Define
\begin{displaymath}
\mathcal{T}[v](t,x) = \int_0^t \int_Z \int_{\R^d}
\Phi_\varepsilon(t-s,x-y)v(s,y,z)\,dy W(dz,ds).
\end{displaymath}
Then
\begin{displaymath}
\E{\norm{\mathcal{T}[v](t)}_{p,\phi}^p}^{1/p} \leq
c_p^{1/p}\kappa_{1,d}\norm{M}_{L^2(Z)}
\left(\int_0^t \E{\norm{K(s)}_{p,\phi}^p}^{2/p}\,ds\right)^{1/2},
\end{displaymath}
where $c_p$ is the constant appearing in
the Burkholder-Davis-Gundy inequality
and $\kappa_{1,d} = c_{d-1}\frac{d \alpha(d)}{\pi^{d/2}}$, with
$c_d$ defined in Lemma~\ref{lemma:HeatKernelYoungIneq}.
\end{lemma}
\begin{remark}
To prove this result we use the Burkholder-Davis-Gundy
inequality for real-valued processes. Using Banach space valued
versions \cite{CoxVeraar2012, NeervenVeraarWeis2007}, one can derive
more general estimates.
\end{remark}
\begin{proof}
First note that
\begin{displaymath}
M(t,x) = \int_0^t \int_Z \int_{\R^d} \Phi_\varepsilon(\tau-s,x-y)v(s,y,z)\,dy W(dz,ds)
\end{displaymath}
is a martingale on $[0,\tau]$, and so by the
Burkholder-Davis-Gundy inequality \cite{Khoshnevisan2009},
\begin{equation*}
\E{\abs{\mathcal{T}[v](t,x)}^p} \leq c_p\E{\left(\int_0^t\int_Z
\abs{\Phi_\varepsilon(t-s) \star v(s,\cdot,z)(x)}^2\,d\mu(z)\,ds \right)^{p/2}}.
\end{equation*}
Upon integrating in space and applying Minkowski's inequality, it follows that
\begin{align*}
&\E{\norm{\mathcal{T}[v](t)}_{p,\phi}^p}^{2/p} \\
& \hphantom{XXX}\leq c_p^{2/p}\E{\int_{\R^d}\left(\int_0^t
\int_Z\abs{\Phi_\varepsilon(t-s) \star v(s,\cdot,z)(x)}^2
\,d\mu(z) ds \right)^{p/2}\phi(x)\,dx}^{2/p} \\
&\hphantom{XXX}\leq c_p^{2/p}\int_0^t\int_Z\E{\int_{\R^d}
\abs{\Phi_\varepsilon(t-s) \star v(s,\cdot,z)(x)}^p
\phi(x)\,dx}^{2/p}\,d\mu(z) ds.
\end{align*}
By Lemma~\ref{lemma:HeatKernelYoungIneq},
\begin{displaymath}
\E{\norm{\mathcal{T}[v](t)}_{p,\phi}^p}^{2/p} \leq c_p^{2/p}\kappa_{1,d}^2\int_0^t
\int_Z\E{\norm{v(s,\cdot,z)}_{p,\phi}^p}^{2/p}d\mu(z)\,ds.
\end{displaymath}
By assumption,
\begin{displaymath}
\int_0^t \int_Z\E{\norm{v(s,\cdot,z)}_{p,\phi}^p}^{2/p}d\mu(z)\,ds
\leq \norm{M}_{L^2(Z)}^2 \int_0^t \E{\norm{K(s)}_{p,\phi}^p}^{2/p}\,ds.
\end{displaymath}
\end{proof}
For a Banach space $E$ we denote by $\mathcal{X}_{\beta,q,E}$ the space
of pathwise continuous predictable processes
$u:[0,T] \times \Omega \rightarrow E$ normed by
\begin{equation}\label{eq:betanorm}
\norm{u}_{\beta,q,E} :=
\left(\sup_{t \in [0,T]} e^{-\beta t}\E{\norm{u(t)}_E^q}\right)^{1/q}.
\end{equation}
The existence of a solution to \eqref{eq:ViscousApprox} is obtained
by the Banach fixed-point theorem, applied to the operator
\begin{align*}
\mathcal{S}(u)(t,x) &:= \int_{\R^d}\Phi_\varepsilon(t,x-y)u^0(y)\,dy \\
&\quad - \int_{0}^{t}\int_{\R^d}
\nabla_x\Phi_\varepsilon(t-s,x-y) \cdot f(u(s,y)) \,dyds \\
&\quad + \int_{0}^{t}\int_Z \int_{\R^d}
\Phi_\varepsilon(t-s,x-y)\sigma(y,u(s,y),z) \,dy\,W(ds,dz),
\end{align*}
in the space $\mathcal{X}_{\beta,2,L^2(\R^d,\phi)}$, with $\beta \in \R$
sufficiently large. It follows that the sequence $\seq{u^n}_{n \geq 1}$
defined inductively by $u^0 = 0$ and $u^{n+1} = \mathcal{S}(u^n)$
converges to $\ue$ in $\mathcal{X}_{\beta,2,L^2(\R^d,\phi)}$ as $n \rightarrow \infty$.
By Lemmas~\ref{lemma:FGenEst} and \ref{lemma:GGenEst} we are free to use
the space $\mathcal{X}_{\beta,p,L^p(\R^d,\phi)}$ for any $2 \leq p < \infty$ in the
fixed-point argument \cite{FengNualart2008}.
We can use Lemmas~\ref{lemma:FGenEst} and \ref{lemma:GGenEst} to
deduce a continuous dependence result. To do this, we need a measure
of the distance between the coefficients. For the flux function
$f$, the Lipschitz norm is a reasonable choice. Concerning the
noise function $\sigma$, we introduce
the norm $\norm{\sigma}_{\mathrm{Lip}}
= \norm{M_\sigma}_{L^2(Z)}$, where
\begin{displaymath}
M_\sigma(z) = \sup_{x \in \R^d}\left\{\sup_{u \in \R}
\frac{\abs{\sigma(x,u,z)}}{1 + \abs{u}}\right\} + \sup_{x \in \R^d}\left\{\sup_{u \neq v}
\frac{\abs{\sigma(x,u,z)-\sigma(x,v,z)}}{\abs{u-v}}\right\}.
\end{displaymath}
Note that for any $\sigma$ satisfying \eqref{assumption:LipOnSigma}, we
have $\norm{\sigma}_{\text{Lip}} < \infty$.
\begin{proposition}[Continuous dependence]\label{proposition:ContDependVisc}
Let $2 \leq p < \infty$ and $\phi \in \mathfrak{N}$. Let $f_1,f_2$
satisfy \eqref{assumption:LipOnf} and $\sigma_1,\sigma_2$
satisfy \eqref{assumption:LipOnSigma}.
Suppose $u_1^0,u_2^0 \in L^p(\Omega,\F_0,P;L^p(\R^d,\phi))$.
Let $\ue_1$ and $\ue_2$ denote the weak solutions of the
corresponding problems \eqref{eq:ViscousApprox}
with $f = f_i, \sigma = \sigma_i$, and $u^0 = u^0_i$, for $i = 1,2$.
Then, for $\beta > 0$ sufficiently large there exists a
constant $C = C(\beta,\varepsilon,T,f_1,\sigma_1)$ such that
\begin{align*}
\norm{\ue_1 - \ue_2}_{\beta,p,L^p(\R^d,\phi)}
\leq C\bigg(&\E{\norm{u^0_1-u^0_2}_{p,\phi}^p}
+\norm{f_1-f_2}_{\mathrm{Lip}}\norm{\ue_1}_{\beta,p,L^p(\R^d,\phi)} \\
&\,+\norm{\sigma_1-\sigma_2}_{\mathrm{Lip}}
\left(\norm{\phi}_{L^1(\R^d)}
+ \norm{\ue_1}_{\beta,p,L^p(\R^d,\phi)}\right)\bigg),
\end{align*}
where the norm $\norm{\cdot}_{\beta,p,L^p(\R^d,\phi)}$
is defined in \eqref{eq:betanorm}.
\end{proposition}
\begin{proof}
By \eqref{eq:MildViscSol},
\begin{align*}
\ue_1&(t,x)-\ue_2(t,x) = \int_{\R^d}
\Phi_\varepsilon(t,x-y)(u^0_1(y)-u^0_2(y))\,dy \\
&- \int_{0}^{t}\int_{\R^d} \nabla_x\Phi_{\varepsilon}(t-s,x-y)
\cdot (f_1(\ue_1(s,y))-f_2(\ue_2(s,y))) \,dyds \\
&+ \int_{0}^{t}\int_Z\int_{\R^d}
\Phi_\varepsilon(t-s,x-y)(\sigma_1(y,\ue_1(s,y),z)
-\sigma_2(y,\ue_2(s,y),z))\,dy \,W(ds,dz) \\
&= \mathcal{T}_1 + \mathcal{T}_2 + \mathcal{T}_3.
\end{align*}
By Lemma~\ref{lemma:HeatKernelYoungIneq},
\begin{displaymath}
\E{\norm{\mathcal{T}_1(t)}_{p,\phi}^p}
\leq \kappa_{1,d}^p\E{\norm{u^0_1-u^0_2}_{p,\phi}^p}.
\end{displaymath}
where $\kappa_{1,d}$ is defined
in Lemma~\ref{lemma:GGenEst}. Hence
\begin{equation}\label{eq:ContDepEstT1}
\norm{\mathcal{T}_1}_{\beta,p,L^p(\R^d,\phi)}
\leq \kappa_{1,d}\E{\norm{u^0_1-u^0_2}_{p,\phi}^p}^{1/p}.
\end{equation}
Consider $\mathcal{T}_2$. Note that
\begin{multline*}
\E{\norm{f_1(\ue_1(s))-f_2(\ue_2(s))}_{p,\phi}^p}^{1/p} \leq
\norm{f_1 - f_2}_{\mathrm{Lip}}\E{\norm{\ue_1(s)}_{p,\phi}^p}^{1/p} \\
+ \norm{f_2}_{\mathrm{Lip}}\E{\norm{\ue_1(s)-\ue_2(s)}_{p,\phi}^p}^{1/p}.
\end{multline*}
By Lemma~\ref{lemma:FGenEst},
\begin{align*}
&\E{\norm{\mathcal{T}_2(t)}_{p,\phi}^p} \\
& \quad \leq \kappa_{2,d}^p\left(2\sqrt{\frac{t}{\varepsilon}}\right)^{p-1}
\norm{f_1 - f_2}_{\mathrm{Lip}}^p\int_0^t \frac{1}{\sqrt{\varepsilon(t-s)}}
\E{\norm{\ue_1(s)}_{p,\phi}^p} \,ds \\
&\quad\quad + \kappa_{2,d}^p\left(2\sqrt{\frac{t}{\varepsilon}}\right)^{p-1}
\norm{f_2}_{\mathrm{Lip}}^p\int_0^t \frac{1}{\sqrt{\varepsilon(t-s)}}
\E{\norm{\ue_1(s)-\ue_2(s)}_{p,\phi}^p} \,ds.
\end{align*}
Multiplying by $e^{-\beta t}$ and taking the supremum yields
\begin{multline}\label{eq:ContDepEstT2}
\norm{\mathcal{T}_2}_{\beta,p,L^p(\R^d,\phi)}
\leq \delta_{\beta,1}\norm{f_1 - f_2}_{\mathrm{Lip}} \norm{\ue_1}_{\beta,p,L^p(\R^d,\phi)} \\
+ \delta_{\beta,1}\norm{f_2}_{\mathrm{Lip}}\norm{\ue_1-\ue_2}_{\beta,p,L^p(\R^d,\phi)},
\end{multline}
where
\begin{displaymath}
\delta_{\beta,1} = \kappa_{2,d}\sup_{t \in [0,T]}\left(2\sqrt{\frac{t}{\varepsilon}}\right)^{1-1/p}
\left(\int_0^t \frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}} \,ds\right)^{1/p}.
\end{displaymath}
Consider $\mathcal{T}_3$. First, observe that
\begin{displaymath}
\abs{\sigma_1(y,\ue_1,z)-\sigma_2(y,\ue_2,z)} \leq M_{\sigma_1}(z)
\abs{\ue_1 - \ue_2} + M_{\sigma_1-\sigma_2}(z)(1 + \abs{\ue_1}).
\end{displaymath}
Due to a simple extension of Lemma~\ref{lemma:GGenEst},
\begin{align*}
\E{\norm{\mathcal{T}_3(t)}_{p,\phi}^p}^{1/p}
&\leq c_p^{1/p}\kappa_{1,d}\norm{\sigma_1}_{\mathrm{Lip}}
\left(\int_0^t \E{\norm{\ue_1(s) - \ue_2(s)}_{p,\phi}^p}^{2/p}\,ds\right)^{1/2} \\
&+c_p^{1/p}\kappa_{1,d}\norm{\sigma_1-\sigma_2}_{\mathrm{Lip}}
\left(\int_0^t \E{\norm{1 + \abs{\ue_1(s)}}_{p,\phi}^p}^{2/p}\,ds\right)^{1/2}.
\end{align*}
Multiplication by $e^{-\beta t/p}$ and then taking the supremum yields
\begin{multline}\label{eq:ContDepEstT3}
\norm{\mathcal{T}_3}_{\beta,p,L^p(\R^d,\phi)}
\leq \delta_{\beta,2}\norm{\sigma_1}_{\mathrm{Lip}}
\norm{\ue_1 - \ue_2}_{\beta,p,L^p(\R^d,\phi)} \\
+\delta_{\beta,2}\norm{\sigma_1-\sigma_2}_{\mathrm{Lip}}
(\norm{\phi}_{L^1(\R^d)} + \norm{\ue_1}_{\beta,p,L^p(\R^d,\phi)}),
\end{multline}
where
\begin{displaymath}
\delta_{\beta,2} = c_p^{1/p}\kappa_{1,d}\sup_{t \in [0,T]}\left(\int_0^t
e^{-\beta 2(t-s)/p}\,ds\right)^{1/2} \leq c_p^{1/p}\kappa_{1,d}\sqrt{\frac{p}{2\beta}}.
\end{displaymath}
Here we used that $\norm{1}_{\beta,p,L^p(\R^d,\phi)}
= \norm{\phi}_{L^1(\R^d)}$. Combine \eqref{eq:ContDepEstT1}, \eqref{eq:ContDepEstT2},
and \eqref{eq:ContDepEstT3}, and note that
$\delta_{\beta,i} \rightarrow 0$ as $\beta \rightarrow \infty$ for $i = 1,2$.
This concludes the proof.
\end{proof}
In order to apply It\^o's formula to the process $t \mapsto \ue(t,x)$ we
need to know that the weak (mild) solution
$\ue$ of \eqref{eq:ViscousApprox} is in fact a strong solution.
The following result provides the existence of weak derivatives.
\begin{proposition}\label{proposition:SobolevBoundsOnViscApprox}
Fix $\phi \in \mathfrak{N}$ and a multiindex $\tilde{\alpha}$.
Make the following assumptions:
\begin{itemize}
\item[(i)] The flux-function $f$ belongs to $C^{\abs{\tilde{\alpha}}}(\R;\R^d)$
with all derivatives bounded.
\item[(ii)] For each fixed $z \in Z$, $(x,u) \mapsto \sigma(x,u,z)$
belongs to $C^{\abs{\tilde{\alpha}}}(\R^d \times \R)$ and
for each $0 < \alpha \leq \tilde{\alpha}$ and $0 \leq n \leq \abs{\tilde{\alpha}}$ there
exists $M_{\alpha,n} \in L^2(Z)$ such that
\begin{displaymath}
\begin{cases}
\partial_1^\alpha \partial_2^n \sigma(x,u,z)
\leq M_{\alpha,n}(z), & 1 \leq n \leq \abs{\tilde{\alpha}}, \\
\partial_1^\alpha \sigma(x,u,z) \leq M_{\alpha,0}(z)(1 + \abs{u}).
\end{cases}
\end{displaymath}
\item[(iii)] The initial function $u^0$ satisfies for all $\alpha \leq \tilde{\alpha}$,
\begin{displaymath}
\E{\norm{\partial^\alpha u^0}_{p,\phi}^p} < \infty \qquad (2 \leq p < \infty).
\end{displaymath}
\end{itemize}
Let $\ue$ be the weak solution of \eqref{eq:ViscousApprox}.
For any $\alpha \leq \tilde{\alpha}$, there exists a predictable process
\begin{displaymath}
(t,x,\omega) \mapsto \partial^\alpha_x\ue(t,x,\omega)
\mbox{ in }L^p([0,T] \times \Omega;L^p(\R^d,\phi))
\end{displaymath}
such that for all $\test \in C^\infty_c(\Pi_T)$,
\begin{displaymath}
\iint_{\Pi_T} \partial^\alpha_x \ue \test \,dxdt
= (-1)^{\abs{\alpha}}
\iint_{\Pi_T} \ue \partial^\alpha_x \test \,dxdt,
\qquad \text{$dP$-almost surely.}
\end{displaymath}
\end{proposition}
To prove Proposition~\ref{proposition:SobolevBoundsOnViscApprox} we apply
\begin{lemma}\label{lemma:IteratedChainRule}
Let $\sigma \in C^\infty(\R^d \times \R)$ and suppose $u \in C^\infty(\R^d)$.
For any multiindex $\alpha$, let
$\partial^\alpha_x := \prod_k \partial_{x_k}^{\alpha_k}$. Then
\begin{displaymath}
\partial^\alpha_x \sigma(x,u(x)) = \sum_{\zeta \leq \alpha}\sum_{\gamma \in \pi(\zeta)}
C_{\gamma,\alpha}\partial_1^{\alpha-\zeta}\partial_2^{\abs{\gamma}}\sigma(x,u(x))
\prod_{i = 1}^{\abs{\gamma}} \partial_x^{\gamma^i}u(x).
\end{displaymath}
Here $\pi(\zeta)$ denotes all partitions of $\zeta$, i.e., all
multiindices $\gamma = \seq{\gamma^i}_{i \geq 1}$
such that $\sum \gamma^i = \zeta$.
Furthermore, $\abs{\gamma}$ denotes the number of
terms in the partition $\gamma$.
\end{lemma}
\begin{remark}
Whenever $\zeta \neq 0$ we assume that the terms $\gamma^i$ in
the partition $\gamma$ satisfies $\gamma^i \neq 0$.
If $\zeta = 0$ we let $\gamma = \gamma^1 = 0$
and by convention let $\abs{\gamma} = 0$.
\end{remark}
\begin{proof}
One may prove by induction and the chain rule that
\begin{displaymath}
\partial^\alpha_x \sigma(x,u(x)) =
\left[(\partial_z + \partial_y)^\alpha \sigma(y,u(z))\right]_{y = x,z = x}.
\end{displaymath}
By the binomial theorem,
\begin{displaymath}
(\partial_z + \partial_y)^\alpha \sigma(y,u(z)) =
\sum_{\zeta \leq \alpha} \binom{\alpha}{\zeta}
\partial^{\alpha-\zeta}_y\partial^\zeta_z\sigma(y,u(z)).
\end{displaymath}
Thanks to \cite[Propositions 1 and 2]{Hardy2006}, it follows that
\begin{displaymath}
\partial^\zeta_z\sigma(y,u(z)) = \sum_{\gamma \in \pi(\zeta)}
M_{\gamma}\partial_u^{\abs{\gamma}}\sigma(y,u(z))
\prod_i\partial^{\gamma_i}_zu(z),
\end{displaymath}
where $M_{\gamma}$ is a constant.
The result follows by combining the above identities.
\end{proof}
\begin{proof}[Proof of Proposition \ref{proposition:SobolevBoundsOnViscApprox}]
We divide the proof into two steps.
\emph{Step~1 (uniform estimates on $\seq{u^n}_{n \geq 1}$)}.
For all $\zeta < \alpha$, suppose
\begin{equation}\label{eq:InductionHypOnMultider}
\sup_{0 \leq s \leq T}
\E{\norm{\partial^\zeta u^n(s)}_{p,\phi}^p}
\leq C_{\zeta,p} \qquad (2 \leq p < \infty).
\end{equation}
We claim that there exists
a constant $C \geq 0$, independent of $\beta$
and $n$, and a number $\delta_\beta \geq 0$ such that
\begin{equation}\label{eq:MainIndIneqDerEst}
\norm{\partial^\alpha u^{n+1}}_{\beta,p,L^p(\R^d,\phi)}
\leq C + \delta_\beta\norm{\partial^\alpha u^{n}}_{\beta,p,L^p(\R^d,\phi)},
\end{equation}
where $\delta_\beta < 1$ for some $\beta > 0$, and
$\norm{\cdot}_{\beta,p,L^p(\R^d,\phi)}$ is defined in \eqref{eq:betanorm}.
Given \eqref{eq:MainIndIneqDerEst}, it follows that
\begin{displaymath}
\norm{\partial^\alpha u^n}_{\beta,p,L^p(\R^d,\phi)}
\leq C \sum_{k = 0}^{n-1}
\delta_\beta^k \leq \frac{C}{1 - \delta_\beta},
\end{displaymath}
and we are done.
To establish \eqref{eq:MainIndIneqDerEst}, observe that
the weak derivative satisfies
\begin{align*}
\partial^\alpha_xu^{n+1}(t,x)
&= \int_{\R^d}\Phi_\varepsilon(t,x-y) \partial_y^\alpha u^0(y)\,dy \\
&\quad - \int_{0}^{t}\int_{\R^d} \nabla_x\Phi_\varepsilon(t-s,x-y)
\cdot \partial^\alpha_y f(u^n(s,y)) \,dyds \\
&\quad + \int_{0}^{t}\int_Z \int_{\R^d} \Phi_\varepsilon(t-s,x-y)
\partial^\alpha_y\sigma(y,u^n(s,y),z) \,dy\,W(ds,dz) \\
&=: \mathcal{T}_1(t,x) + \mathcal{T}_2(t,x) + \mathcal{T}_3(t,x).
\end{align*}
To justify this, multiply by a test function and
apply the Fubini theorem \cite[p.297]{Walsh1984}.
By the triangle inequality we may estimate each term separately.
Consider $\mathcal{T}_1$. By Lemma~\ref{lemma:HeatKernelYoungIneq},
\begin{displaymath}
\E{\norm{\mathcal{T}_1(t)}_{p,\phi}^p}
\leq \kappa_{1,d}^p\E{\norm{\partial^\alpha u^0}_{p,\phi}^p},
\end{displaymath}
where $\kappa_{1,d}$ is defined in Lemma~\ref{lemma:GGenEst}.
By assumption (iii) it follows that there exists a constant $C$ such that
\begin{equation}\label{eq:DerEstT1}
\norm{\mathcal{T}_1}_{\beta,p,L^p(\R^d,\phi)} \leq C.
\end{equation}
Consider $\mathcal{T}_2$. By Lemma~\ref{lemma:FGenEst},
\begin{align*}
\norm{\mathcal{T}_2}&_{\beta,p,L^p(\R^d,\phi)}^p
= \sup_{t \in [0,T]} e^{-\beta t} \E{\norm{\mathcal{T}_2(t)}_{p,\phi}^p} \\
&\qquad \leq \kappa_{2,d}^p\left(2\sqrt{\frac{T}{\varepsilon}}\right)^{p-1}\sup_{t \in [0,T]}
\int_0^t \frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}}e^{-\beta s}
\E{\norm{\partial^\alpha f(u^n(s))}_{p,\phi}^p}\,ds \\
&\qquad \leq \kappa_{2,d}^p\left(2\sqrt{\frac{T}{\varepsilon}}\right)^{p-1}
\left(\int_0^T \frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}}\,ds\right)
\norm{\partial^\alpha f(u^n(s))}_{\beta,p,L^p(\R^d,\phi)}^p.
\end{align*}
By Lemma~\ref{lemma:IteratedChainRule}, the triangle inequality, and
the generalized H\"older inequality,
\begin{align*}
\E{\norm{\partial^\alpha f(u^n(s))}_{p,\phi}^p}^{1/p}
&\leq \sum_{\gamma \in \pi(\alpha)}
C_{\gamma,\alpha}\norm{\partial^{\abs{\gamma}}f}_{\infty}
\E{\norm{\prod_{i=1}^{\abs{\gamma}} \partial^{\gamma^i}u^n(s)}_{p,\phi}^p}^{1/p} \\
&\leq \sum_{\gamma \in \pi(\alpha)}
C_{\gamma,\alpha}\norm{\partial^{\abs{\gamma}}f}_{\infty}
\prod_{i=1}^{\abs{\gamma}}\E{\norm{ \partial^{\gamma^i}u^n(s)}_{q_i,\phi}^{q_i}}^{1/q_i},
\end{align*}
whenever $\sum_{i = 1}^{\abs{\gamma}}\frac{1}{q_i} = \frac{1}{p}$.
Since $\gamma$ is a partition of $\alpha$, $\sum_{i = 1}^{\abs{\gamma}}\abs{\gamma^i}
= \abs{\alpha}$, and we may take $q_i = \abs{\alpha}p/\abs{\gamma^i}$.
By assumption there exists a
constant $C$, independent of $n$, such that
\begin{displaymath}
\E{\norm{ \partial^{\gamma^i}u^n(s)}_{q_i,\phi}^{q_i}} \leq C,
\end{displaymath}
for all terms where $\gamma^i < \alpha$. Since $C_{\alpha,\alpha} = 1$,
there is another constant $C$ such that
\begin{equation*}
\E{\norm{\partial^\alpha f(u^n(s))}_{p,\phi}^p}^{1/p} \leq C
+ \norm{f'}_\infty \E{\norm{\partial^\alpha u^n(s)}_{p,\phi}^p}^{1/p},
\end{equation*}
for all $n \geq 1$. Multiply by $e^{-\beta t/p}$ and
take the supremum to obtain
\begin{displaymath}
\norm{\partial^\alpha f(u^n)}_{\beta,p,L^p(\R^d,\phi)} \leq C
+ \norm{f'}_\infty \norm{\partial^\alpha u^n}_{\beta,p,L^p(\R^d,\phi)}.
\end{displaymath}
It follows that
\begin{equation}\label{eq:DerEstT2}
\begin{split}
\norm{\mathcal{T}_2}_{\beta,p,L^p(\R^d,\phi)} &
\leq c_d\left(2\sqrt{\frac{T}{\varepsilon}}\right)^{1-1/p}\left(\int_0^T
\frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}}\,ds\right)^{1/p} \\
& \hphantom{XXXXXXXXX} \times \left(C + \norm{f'}_\infty
\norm{\partial^\alpha u^n}_{\beta,p,L^p(\R^d,\phi)}\right).
\end{split}
\end{equation}
Consider $\mathcal{T}_3$. By Lemma~\ref{lemma:IteratedChainRule}
\begin{displaymath}
\begin{split}
\mathcal{T}_3(t,x) &= \sum_{\zeta \leq \alpha}\sum_{\gamma \in \pi(\zeta)}
C_{\gamma,\alpha}\int_{0}^{t}\int_Z \int_{\R^d} \Phi_\varepsilon(t-s,x-y) \\
&\hphantom{XXXXX} \times
\partial_1^{\alpha-\zeta}\partial_2^{\abs{\gamma}}\sigma(y,u^n(s,y),z)
\left(\prod_{i=1}^{\abs{\gamma}} \partial_y^{\gamma^i}u^n(s,y)\right) \,dy\,W(ds,dz) \\
&= \mathcal{T}_3^0(t,x) + \mathcal{T}_3^1(t,x),
\end{split}
\end{displaymath}
where $\mathcal{T}_3^0$ contains the term with $\zeta = 0$.
By Lemma~\ref{lemma:GGenEst}, assumption~(ii), and the
generalised H\"older inequality,
\begin{align*}
\E{\norm{\mathcal{T}_3^1(t)}_{p,\phi}^p}^{1/p}
&\leq \sum_{0 < \zeta \leq \alpha}\sum_{\gamma \in \pi(\zeta)}
C_{\gamma,\alpha}c_p^{1/p}\kappa_{1,d}
\norm{M_{\alpha-\zeta,\abs{\gamma}}}_{L^2(Z)} \\
& \hphantom{XXXXXX} \times \left(\int_0^t\prod_{i=1}^{\abs{\gamma}}
\E{\norm{ \partial^{\gamma^i}u^n(s)}_{q_i,\phi}^{q_i}}^{2/q_i}\,ds\right)^{1/2},
\end{align*}
where $q_i = \abs{\zeta}p/\abs{\gamma^i}$. The term $\mathcal{T}_3^0$ is
estimated similarly by applying the second case of assumption (ii).
It follows from \eqref{eq:InductionHypOnMultider}
that there exists a constant $C$ such that
\begin{displaymath}
\E{\norm{\mathcal{T}_3(t)}_{p,\phi}^p}^{1/p}
\leq C + c_p^{1/p}\kappa_{1,d}\norm{M_{0,1}}_{L^2(Z)}
\left(\int_0^t \E{\norm{\partial^\alpha
u^n(s)}_{p,\phi}^p}^{2/p}\,ds\right)^{1/2}.
\end{displaymath}
Multiplying by $(e^{-\beta t})^{1/p}$ and
taking the supremum yields
\begin{equation}\label{eq:DerEstT3}
\begin{split}
\norm{\mathcal{T}_3}_{\beta,p,L^p(\R^d,\phi)}
&\leq C + c_p^{1/p}\kappa_{1,d}\norm{M_{0,1}}_{L^2(Z)} \\
& \qquad \times \sup_{t \in [0,T]}\left(\int_0^t e^{-2\beta(t-s)/p}
\left(e^{-\beta s}\E{\norm{\partial^\alpha u^n(s)}_{p,\phi}^p}\right)^{2/p}\,ds\right)^{1/2} \\
&\leq C + c_p^{1/p}\kappa_{1,d}\norm{M_{0,1}}_{L^2(Z)}\sqrt{\frac{p}{2\beta}}
\norm{\partial^\alpha u^n}_{\beta,p,L^p(\R^d,\phi)}.
\end{split}
\end{equation}
Combining \eqref{eq:DerEstT1}, \eqref{eq:DerEstT2}, and \eqref{eq:DerEstT3}
we obtain inequality \eqref{eq:MainIndIneqDerEst}, where
\begin{align*}
\delta_\beta &= \kappa_{2,d}\left(2\sqrt{\frac{T}{\varepsilon}}\right)^{1-1/p}
\left(\int_0^T \frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}}\,ds\right)^{1/p}\norm{f'}_\infty \\
& \hphantom{XXXXXXXXXXXXXXXXXX}
+ c_p^{1/p}\kappa_{1,d}\norm{M_{0,1}}_{L^2(Z)}\sqrt{\frac{p}{2\beta}}.
\end{align*}
It is clear that $\delta_\beta \rightarrow 0$ as
$\beta \rightarrow \infty$ and so \eqref{eq:MainIndIneqDerEst} follows.
By induction, estimate \eqref{eq:InductionHypOnMultider} holds for all $\zeta \leq \alpha$.
\emph{Step~2 (convergence of $u^n$)}.
Fix $ \alpha \leq \tilde{\alpha}$. We apply
Theorem~\ref{theorem:YoungMeasureLimitOfComposedFunc}
to the familiy $\seq{\partial^\alpha u^n}_{n \geq 1}$ on the space
\begin{displaymath}
(X,\mathscr{A},\mu) =
(\Omega \times \Pi_T, \Pred \otimes \Borel{\R^d}, dP \otimes dt \otimes \phi(x)dx).
\end{displaymath}
By means of \eqref{eq:InductionHypOnMultider},
\begin{displaymath}
\sup_{n \geq 1}\left\{\E{\iint_{\Pi_T}
\abs{\partial^\alpha u^n(t,x)}^2 \phi(x)\,dxdt}\right\} < \infty.
\end{displaymath}
Hence, $\seq{\partial^\alpha u^n}_{n \geq 1}$ has a
Young measure limit $\nu^\alpha \in \Young{\Omega \times \Pi_T}$.
Next, define $\partial^\alpha \ue(t,x,\omega)
:= \int_\R \,d\nu^\alpha_{t,x,\omega}$.
By definition, the limit has a $\Pred \otimes \Borel{\R^d}$ measurable version.
Furthermore, $\partial^\alpha \ue \in L^p(\Omega \times [0,T];L^p(\R^d,\phi))$,
cf.~proof of Theorem~\ref{theorem:ExistenceOfSolution}
and Lemma~\ref{lemma:LebBochRepr}. Let us show
that $\partial^\alpha \ue$ is the
weak derivative of $\ue$. To this end, observe that
\begin{displaymath}
\iint_{\Pi_T}u^n(t,x)\partial^\alpha \test \,dxdt =
(-1)^{\abs{\alpha}} \iint_{\Pi_T} \partial^\alpha u^n(t,x) \test \,dxdt,
\end{displaymath}
for any $\test \in C^\infty_c(\R^d)$. By Lemma~\ref{lemma:UniformIntCriteria}(ii)
and Theorem~\ref{theorem:DunfordPettis}, there is
a subsequence $\seq{n(j)}_{j \geq 1}$ such that for any $A \in \F$,
\begin{align*}
\lim_{j \rightarrow \infty}\E{\car{A}\iint_{\Pi_T} \partial^\alpha u^{n(j)} \test \,dxdt}
&= \E{\iint_{\Pi_T}\int_\R \test(t,x)\car{A}(\omega)\,d\nu^\alpha_{t,x,\omega}(\xi)\,dxdt} \\
&= \E{\car{A}\iint_{\Pi_T} \partial^\alpha \ue \test\,dxdt}.
\end{align*}
As $u^n \rightarrow \ue$ in
$\mathcal{X}_{\beta,2,L^2(\R^d,\phi)}$, it follows that
\begin{displaymath}
\lim_{n \rightarrow \infty} \E{\car{A}\iint_{\Pi_T}u^n
\partial^\alpha \test \,dxdt}
= \E{\car{A}\iint_{\Pi_T}\ue \partial^\alpha \test \,dxdt}.
\end{displaymath}
This concludes the proof.
\end{proof}
\subsection{Malliavin differentiability}
We will establish the Malliavin differentiability
of the viscous approximations. Furthermore, we will observe that the Malliavin
derivative satisfies a linear parabolic equation. This equation is then applied
to show that $D_{r,z}\ue(t,x) \rightarrow \sigma(x,\ue(r,x),z)$
as $t \downarrow r$ in a weak
sense (Lemma~\ref{lemma:MalliavinDerivativeWeakTimeCont}); a
property that is crucial in the proof of uniqueness.
\begin{proposition}[Malliavin derivative of viscous
approximation]\label{proposition:MalliavinDiffOfViscApprox}
Suppose \eqref{assumption:LipOnf} and \eqref{assumption:LipOnSigma}
are satisfied. Fix $\phi \in \mathfrak{N}$ and $u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Let $\ue$ be the solution of \eqref{eq:ViscousApprox}.
Then $\ue$ belongs to $\D^{1,2}(L^2([0,T];L^2(\R^d,\phi)))$ and
\begin{equation}\label{eq:PointwiseMalliavinBoundOnVisc}
\esssup_{0 \leq t \leq T} \norm{\ue(t)}_{\D^{1,2}(L^2(\R^d,\phi))} < \infty.
\end{equation}
Furthermore, for $dr \otimes d\mu$-a.a.~$(r,z)$, the $L^2(\R^d,\phi)$-valued
process $\seq{D_{r,z}\ue(t)}_{t>r}$ is a predictable weak solution of
\begin{equation}\label{eq:MallEqSat}
\left\{\begin{split}
dw + \nabla \cdot (f'(\ue)w)\dt
&= \int_Z \partial_2\sigma(x,\ue,z')w\, W(dt,dz')
+ \varepsilon\Delta w\dt, \quad t \in [r,T], \\
w(r,x,z) &= \sigma(x,\ue(r,x),z),
\end{split}\right.
\end{equation}
while $D_{r,z}\ue(t) = 0$ if $r > t$. Furthermore
\begin{equation}\label{eq:supInrMallEst}
\esssup_{r \in [0,T]} \left\{\sup_{t \in [0,T]}
E{\norm{D_r\ue(t)}_{L^2(Z;L^2(\R^d,\phi))}^2}\right\} < \infty.
\end{equation}
\end{proposition}
\begin{remark}
Let $w_{r,z}(t,x) = D_{r,z}\ue(t,x)$, $t > r$.
Estimate \eqref{eq:supInrMallEst} may be seen
as a consequence of the Gr\"onwall-type estimate
\begin{displaymath}
\E{\norm{w_{r,z}(t)}_{2,\phi}^2} \leq
\left(1 + Ce^{C(t-r)}\right)
\E{\norm{w_{r,z}(r)}_{2,\phi}^2},
\end{displaymath}
for $t \geq r$. From the perspective of a uniqueness result (see
Lemma~\ref{lemma:MalliavinDerivativeWeakTimeCont}), it is of
interest to know whether one can derive such estimates
independent of $\varepsilon$.
\end{remark}
\begin{proof}
We divide the proof into two steps.
\emph{Step~1 (uniform bounds)}. Consider the Picard
approximation $\seq{u^n}_{n \geq 1}$ of $\ue$. We want to prove that
\begin{equation}\label{eq:UniformBoundOnMallDer}
\sup_{0 \leq t \leq T} \norm{u^n(t)}_{\D^{1,2}(L^2(\R^d,\phi))}
\leq C, \quad \mbox{for all $n \geq 1$}.
\end{equation}
Recall that
\begin{displaymath}
\norm{u^n(t)}_{\D^{1,2}(L^2(\R^d,\phi))}^2 = \E{\norm{u^n(t)}_{2,\phi}^2}
+ \E{\norm{Du^n(t)}_{H \otimes L^2(\R^d,\phi)}^2},
\end{displaymath}
where $H$ is the space $L^2([0,T] \times Z)$.
Note that there is a constant $C$ such that
\begin{equation}\label{eq:UniformBoundOnun}
\sup_{0 \leq t \leq T}\E{\norm{u^n(t)}_{2,\phi}^2} < C,
\end{equation}
by the proof of Proposition~\ref{proposition:SobolevBoundsOnViscApprox},
Step~1 with $\alpha = 0$. Next, we claim that
there exists a constant $C$ such that
\begin{equation}\label{eq:StepwiseBoundForMallDiff}
\norm{Du^{n+1}}_{\beta,2,H \otimes L^2(\R^d,\phi)} \leq C
+\delta_\beta \norm{Du^n}_{\beta,2,H \otimes L^2(\R^d,\phi)},
\quad \mbox{where $\delta_\beta < 1$},
\end{equation}
for some $\beta > 0$. We then conclude that
\begin{displaymath}
\norm{Du^{n}}_{\beta,2,H \otimes L^2(\R^d,\phi)} \leq C \sum_{k = 0}^{n-1}
\delta_\beta^k \leq \frac{C}{1-\delta_\beta},
\end{displaymath}
and \eqref{eq:UniformBoundOnMallDer} follows.
Let us establish \eqref{eq:StepwiseBoundForMallDiff}.
By \cite[Propositions 1.2.4, 1.3.8, and 1.2.8]{NualartMalliavinCalc2006},
\begin{equation}\label{eq:MallDiffOfUn}
\begin{split}
&D_{r,z}u^{n+1}(t,x) \\
& \quad = \int_{\R^d} \Phi_\varepsilon(t-r,x-y) \sigma(y,u^n(r,y),z) \,dy\\
&\qquad - \int_{r}^{t}\int_{\R^d} \nabla_x\Phi_\varepsilon(t-s,x-y)
\cdot f'(u^n(s,y))D_{r,z}u^n(s,y) \,dyds \\
&\qquad + \int_{r}^{t}\int_Z \int_{\R^d} \Phi_\varepsilon(t-s,x-y)
\partial_2\sigma(y,u^n(s,y),z')D_{r,z}u^n(s,y) \,dy\,W(ds,dz') \\
&\quad =: \mathcal{T}_1^n + \mathcal{T}_2^n + \mathcal{T}_3^n,
\end{split}
\end{equation}
for all $r \in (0,t]$. Whenever $r > t$, $D_ru^{n+1}(t) = 0$
since $u^{n+1}$ is adapted, see \cite[Corollary~1.2.1]{NualartMalliavinCalc2006}.
We proceed by estimating each term of \eqref{eq:MallDiffOfUn} separately.
Consider $\mathcal{T}_1^n$. By Lemma~\ref{lemma:HeatKernelYoungIneq}
and assumption \eqref{assumption:LipOnSigma},
\begin{align*}
\norm{\mathcal{T}_1^n(r,t)}_{L^2(Z;L^2(\R^d,\phi))}^2
&= \int_Z \norm{\Phi_\varepsilon(t-r) \star \sigma(\cdot,u^n(r),z)}_{2,\phi}^2\,d\mu(z) \\
&\leq \kappa_{1,d}^2\norm{M}_{L^2(Z)}^2
\left(\norm{\phi}_{L^1(\R^d)} + \norm{u^n(r)}_{2,\phi}\right)^2,
\end{align*}
for each $0 \leq r < t$.
It follows from \eqref{eq:UniformBoundOnun} that
\begin{equation}\label{eq:MallDiffEstT1}
\begin{split}
&\norm{\mathcal{T}_1^n}_{\beta,2,H \otimes L^2(\R^d,\phi)} \\
& \hphantom{XX}\leq \kappa_{1,d}\norm{M}_{L^2(Z)}
\left(\sup_{0 \leq t \leq T}e^{-\beta t}\E{\int_0^t
(\norm{\phi}_{L^1(\R^d)} + \norm{u^n(r)}_{2,\phi})^2\,dr}\right)^{1/2} \leq C.
\end{split}
\end{equation}
Consider $\mathcal{T}_2^n$. By Lemma~\ref{lemma:FGenEst},
\begin{align*}
&\norm{\mathcal{T}_2^n(r,t)}_{L^2(Z;L^2(\R^d,\phi))}^2
= \int_Z \norm{\int_{r}^{t} \nabla\Phi_\varepsilon(t-s)
\star f'(u^n(s))D_{r,z}u^n(s) ds}_{2,\phi}^2\, d\mu(z) \\
& \hphantom{XXXXXX}\leq 2\kappa_{2,d}^2\norm{f'}_{\infty}^2
\sqrt{\frac{t-r}{\varepsilon}}\int_{r}^{t} \frac{1}{\sqrt{\varepsilon(t-s)}}
\int_Z\norm{D_{r,z}u^n(s)}_{2,\phi}^2d\mu(z)\,ds.
\end{align*}
Multiplication by $e^{-\beta t}$ and integration in $r$ yields
\begin{align*}
&e^{-\beta t}\E{\norm{\mathcal{T}_2^n(t)}_{H \otimes L^2(\R^d,\phi)}^2} \\
& \hphantom{XXX}\leq 2\kappa_{2,d}^2\norm{f'}_{\infty}^2
\sqrt{\frac{t}{\varepsilon}}\int_{0}^{t} \frac{e^{-\beta(t-s)}}{\sqrt{\varepsilon(t-s)}}
e^{-\beta s}\E{\norm{Du^n(s)}_{H \otimes L^2(\R^d,\phi)}^2}\,ds.
\end{align*}
It follows that
\begin{multline}\label{eq:MallDiffEstT2}
\norm{\mathcal{T}_2^n}_{\beta,2,H \otimes L^2(\R^d,\phi)}
\leq \kappa_{2,d}\norm{f'}_{\infty}\left(2\sqrt{\frac{T}{\varepsilon}}\int_{0}^{T}
\frac{e^{-\beta(T-s)}}{\sqrt{\varepsilon(T-s)}}\,ds\right)^{1/2} \\
\times \norm{Du^n(s)}_{\beta,2,H \otimes L^2(\R^d,\phi)}.
\end{multline}
Consider $\mathcal{T}_3^n$. Due to \eqref{assumption:LipOnSigma},
\begin{displaymath}
\abs{\partial_2 \sigma(x,u,z')D_{r,z}u^n(s,y)}
\leq M(z')\abs{D_{r,z}u^n(s,y)}.
\end{displaymath}
By Lemma~\ref{lemma:GGenEst},
\begin{multline*}
\E{\norm{\mathcal{T}_3^n(r,t)}_{L^2(Z;L^2(\R^d,\phi))}^2} \\
\leq c_2 \kappa_{1,d}^2\norm{M}_{L^2(Z)}^2
\int_r^t \E{\norm{D_{r}u^n(s)}_{L^2(Z;L^2(\R^d,\phi))}^2}\,ds.
\end{multline*}
Integrate in $r$ and multiply by $e^{-\beta t}$ to obtain
\begin{multline*}
e^{-\beta t}\E{\norm{\mathcal{T}_3^n(t)}_{H \otimes L^2(\R^d,\phi)}^2} \\
\leq c_2 \kappa_{1,d}^2\norm{M}_{L^2(Z)}^2 \int_0^te^{-\beta(t-s)}e^{-\beta s}
\E{\norm{Du^n(s)}_{H \otimes L^2(\R^d,\phi)}^2}\,ds.
\end{multline*}
Hence,
\begin{equation}\label{eq:MallDiffEstT3}
\norm{\mathcal{T}_3^n}_{\beta,2,H \otimes L^2(\R^d,\phi)}
\leq c_2^{1/2}\kappa_{1,d}\norm{M}_{L^2(Z)}
\frac{1}{\sqrt{\beta}}\norm{Du^n}_{\beta,2,H \otimes L^2(\R^d,\phi)}.
\end{equation}
Combining \eqref{eq:MallDiffEstT1}, \eqref{eq:MallDiffEstT2},
and \eqref{eq:MallDiffEstT3}
yields \eqref{eq:StepwiseBoundForMallDiff} with
\begin{displaymath}
\delta_\beta = \kappa_{2,d}\norm{f'}_{\infty}
\left(2\sqrt{\frac{T}{\varepsilon}}\int_{0}^{T}
\frac{e^{-\beta(T-s)}}{\sqrt{\varepsilon(T-s)}}\,ds\right)^{1/2}
+ c_2^{1/2}\kappa_{1,d}\norm{M}_{L^2(Z)} \frac{1}{\sqrt{\beta}},
\end{displaymath}
where $\kappa_{2,d}$ is the constant from Lemma~\ref{lemma:FGenEst},
while $c_2$ and $\kappa_{1,d}$ are the
constants from Lemma~\ref{lemma:GGenEst}.
Note that $\delta_\beta \downarrow 0$ as
$\beta \rightarrow \infty$. Leaving out the integration in $r$
throughout Step~1, we deduce the estimate
\begin{equation}\label{eq:rBoundOnMallApprox}
\norm{D_ru^{n}}_{\beta,2,L^2(Z) \otimes
L^2(\R^d,\phi)} \leq \frac{C}{1-\delta_\beta}.
\end{equation}
\emph{Step~2 (convergence)}.
Let $E$ denote the space $L^2([0,T];L^2(\R^d,\phi))$
and recall that $H = L^2(Z \times [0,T])$.
Consider $\seq{u^n}_{n \geq 1}$ as a sequence in $\D^{1,2}(E)$.
By \eqref{eq:UniformBoundOnMallDer} and the Hilbert
space valued version of \cite[Lemma~1.2.3]{NualartMalliavinCalc2006}
(see \cite[Lemma~5.2]{CarmonaTehranchi2006}), it follows that $\ue$
belongs to $\D^{1,2}(E)$ and that $Du^n \rightharpoonup D\ue$ (weakly)
in $L^2(\Omega;H \otimes E)$, i.e., for any $h \in H, \test \in E$,
and $V \in L^2(\Omega)$,
\begin{displaymath}
\E{\inner{Du^n}{h \otimes \test}_{H \otimes E}V}
\rightarrow \E{\inner{D\ue}{h \otimes \test}_{H \otimes E}V}.
\end{displaymath}
It follows that the map
\begin{displaymath}
(t,\omega) \mapsto D\ue(t,\omega) \in L^2(H \otimes L^2(\R^d,\phi))
\end{displaymath}
is $\Pred^*$-measurable. Note that Lemma~\ref{lemma:LebBochRepr}
extends to this case, so that $(t,\omega) \mapsto D_{r,z}\ue(t,\omega)$
is $\Pred^*$-measurable for $dr \otimes d\mu$ almost all $(r,z) \in [0,T] \times Z$.
For each fixed $t \in [0,T]$, we conclude by \eqref{eq:UniformBoundOnMallDer}
that $\ue(t) \in \D^{1,2}(L^2(\R^d,\phi))$, where $Du^n(t) \rightharpoonup D\ue(t)$
(weakly) along some subsequence. Besides, this limit agrees $dt$-almost everywhere
with the evaluation of the limit taken in $\D^{1,2}(E)$. This follows by definition for
smooth Hilbert space valued random variables
and may be extended to the general case by approximation.
The weak lower semicontinuity of the norm yields
\eqref{eq:PointwiseMalliavinBoundOnVisc}.
Similarly, we may apply \eqref{eq:rBoundOnMallApprox} and
the Banach-Alaoglu theorem to extract a weakly convergent subsequence
in the space $L^\infty([0,T];\mathcal{X}_{\beta,2,L^2(Z) \otimes L^2(\R^d,\phi)})$.
This yields the bound \eqref{eq:supInrMallEst}.
As above, $\ue(t,x) \in \D^{1,2}$ for $dt \otimes dx$-almost all $(t,x)$, and for
such $(t,x)$ we have $D_{r,z}\ue(t,x) = D\ue(t,x,r,z)$ for $d\mu \otimes dr$ almost
all $(r,z)$ where $D\ue(t,x,r,z)$ denotes the evaluation of the limit taken in $\D^{1,2}(E)$.
Taking the Malliavin derivative of \eqref{eq:MildViscSol} (as above on $u^{n+1}$)
it follows that $t \mapsto D_{r,z}\ue(t)$ is a mild solution
of \eqref{eq:MallEqSat} for $dr \otimes d\mu$ almost all $(r,z)$.
To conclude by \cite[Theorem~9.15]{PeszatZabczyk2007} that it
is a weak solution, we verify conditions (F)
and (G), with $F(w,t) = \nabla \cdot (f'(\ue(t))w)$ and
\begin{displaymath}
G(w,t)h(x) = \int_Z \partial_2\sigma(x,\ue(t,x),z)w(x)h(z)\,d\mu(z).
\end{displaymath}
\end{proof}
The next result concerns the limit of $D_{r,z}\ue(t,x)$ as $t \downarrow r$.
In view of Lemma~\ref{proposition:MalliavinDiffOfViscApprox}, this is a
question about the satisfaction of the initial condition for \eqref{eq:MallEqSat}.
\begin{lemma}\label{lemma:MalliavinDerivativeWeakTimeCont}
Let $\phi \in C^\infty_c(\R^d)$ be non-negative. In the setting of
Proposition~\ref{proposition:MalliavinDiffOfViscApprox}, for
$\Psi \in L^2(\Omega \times Z;L^2(\R^d,\phi))$, set
\begin{displaymath}
\mathcal{T}_{r_0}(\Psi) :=
\E{\iiint\limits_{\quad Z \times \Pi_T} \left(D_{r,z}\ue(t,x)
-\sigma(x,\ue(r,x),z)\right)J^+_{r_0}(t-r) \Psi\phi\,dtdxd\mu(z)}.
\end{displaymath}
Then there exists a constant $C$ independent of $r_0$ such that
\begin{equation}\label{eq:UniformBoundOnTr0}
\abs{\mathcal{T}_{r_0}(\Psi)} \leq C \E{\norm{\Psi}_{L^2(Z;L^2(\R^d,\phi))}^2}^{1/2}.
\end{equation}
and $\lim_{r_0 \downarrow 0}\mathcal{T}_{r_0}(\Psi) = 0$
for $dr$-almost all $r \in [0,T]$.
\end{lemma}
\begin{proof}
Note that $\mathcal{T}_{r_0} = \mathcal{T}_{r_0}^1-\mathcal{T}_{r_0}^2$, and
consider each term separately. By H\"older's inequality,
\begin{align*}
\abs{\mathcal{T}_{r_0}^1} &= \abs{\int_0^T\E{\int_Z\int_{\R^d}
\Psi(x,z)D_{r,z}\ue(t,x)\phi(x)\,dxd\mu(z)} J^+_{r_0}(t-r) \,dt} \\
&\leq \esssup_{t \in [0,T]}\E{\int_Z\int_{\R^d}\Psi(x,z)D_{r,z}\ue(t,x)\phi(x)\,dxd\mu(z)} \\
&\leq\E{\norm{\Psi}_{L^2(Z;L^2(\R^d,\phi))}^2}^{1/2}
\esssup_{t \in [0,T]}\seq{\E{\norm{D_{r}\ue(t)}_{L^2(Z;L^2(\R^d,\phi)}^2}^{1/2}}.
\end{align*}
Furthermore, due to \eqref{assumption:LipOnSigma},
\begin{align*}
\mathcal{T}_{r_0}^2 &= E\bigg[\int_Z\int_{\R^d}
\Psi(x,z)\sigma(x,\ue(r,x),z)\phi(x)\,dxd\mu(z)\,\bigg] \\
&\leq \norm{M}_{L^2(Z)}\E{\norm{\Psi}_{L^2(Z;L^2(\R^d,\phi))}^2}^{1/2}
\E{\norm{1 + \abs{\ue(r)}}_{L^2(\R^d,\phi)}^2}^{1/2}.
\end{align*}
The uniform bound \eqref{eq:UniformBoundOnTr0} follows
by \eqref{eq:supInrMallEst} and \eqref{eq:BoundednessOfViscApprox}.
Note that $\Psi\mapsto \mathcal{T}_{r_0}(\Psi)$ is a linear functional
on $L^2(\Omega \times Z;L^2(\R^d,\phi))$, for each $r_0 > 0$.
By \eqref{eq:UniformBoundOnTr0} the family $\seq{\mathcal{T}_{r_0}}_{r_0 > 0}$ is
uniformly continuous. Hence, by approximation, it suffices to prove
the lemma for $\Psi$ smooth in $x$ with bounded derivatives. Let
\begin{displaymath}
\test(t,x,z) = \Psi(x,z)\phi(x)\xi_{r_0,r}(t), \quad
\xi_{r_0,r}(t) =1-\int_0^t J_{r_0}^+(\sigma-r)\,d\sigma.
\end{displaymath}
By Proposition \ref{proposition:MalliavinDiffOfViscApprox},
\begin{align*}
0 &= \int_{\R^d} \sigma(x,\ue(r,x),z)\test(r,x,z)\,dx \\
&\quad + \int_r^T\int_{\R^d} D_{r,z}\ue(t,x)\partial_t\test(t,x,z)\,dxdt \\
&\quad + \int_r^T\int_{\R^d} f'(\ue(t,x))D_{r,z}\ue(t,x) \cdot \nabla \test(t,x,z)\,dxdt \\
&\quad +\varepsilon\int_r^T\int_{\R^d}D_{r,z}\ue(t,x)\Delta \test(t,x,z)\,dxdt \\
&\quad + \int_r^T\int_Z \int_{\R^d}
\partial_2\sigma(x,\ue(t,x),z')D_{r,z}\ue(t,x)\test(t,x,z)\,dxW(dz',dt),
\end{align*}
$dr \otimes d\mu \otimes dP$-almost all $(r,z,\omega)$. Note that
\begin{displaymath}
\partial_t\test(t,x,z)= -\Psi(x,z)\phi(x)J_{r_0}^+(t-r).
\end{displaymath}
Taking expectations and integrating in $z$ we obtain
\begin{align*}
\mathcal{T}_{r_0}(\Psi) &=\E{\int_Z\int_r^T\int_{\R^d}
f'(\ue(t,x))D_{r,z}\ue(t,x) \cdot \nabla(\Psi \phi)\xi_{r_0,r}(t)\,dxdtd\mu(z)} \\
& \qquad +\varepsilon \E{\int_Z\int_r^T
\int_{\R^d}D_{r,z}\ue(t,x)\Delta (\Psi\phi)\xi_{r_0,r}(t)\,dxdtd\mu(z)}.
\end{align*}
As $\lim_{r_0 \downarrow 0}\xi_{r_0,r}(t) = 0$ for all $t > r$, it follows by the
dominated convergence theorem
that $\lim_{r_0 \downarrow 0}\mathcal{T}_{r_0}(\Psi) = 0$.
\end{proof}
\section{Existence of entropy solutions}\label{sec:Existence}
We will now prove the existence entropy
solutions, as defined in Section~\ref{sec:Entropy_Formulation}.
\begin{theorem}\label{theorem:ExistenceOfSolution}
Fix $\phi \in \mathfrak{N}$ and $2 \leq p < \infty$. Suppose
$u^0 \in L^p(\Omega,\F_0,P;L^p(\R^d,\phi))$, and \eqref{assumption:LipOnf}
and \eqref{assumption:LipOnSigma} hold. Then the generalized limit
$u = \lim_{\varepsilon \downarrow 0} \ue$ of
the viscous approximations \eqref{eq:ViscousApprox}
is a Young measure-valued entropy solution
of \eqref{eq:StochasticBalanceLaw} in the sense of
Definition~\ref{Def:YoungEntropySolution}. Moreover,
$u\in L^p(\Omega \times [0,T];L^p(\R^d \times [0,1],\phi))$.
\end{theorem}
The bounds in Proposition~\ref{proposition:SobolevBoundsOnViscApprox}
blow up as $\varepsilon \downarrow 0$.
Below we establish bounds that are independent of
the regularization parameter $\varepsilon > 0$.
\begin{lemma}[Uniform bounds]\label{lemma:UniformBoundsOnVisc}
Suppose \eqref{assumption:LipOnf} and \eqref{assumption:LipOnSigma} hold,
and $u^0$ belongs to $L^p(\Omega,\F_0,P;L^p(\R^d,\phi))$ for some
even number $p\ge 2$ and $\phi \in \mathfrak{N}$.
Then there exists a constant $C$, depending on
$u^0,f,\sigma,p,T, \phi$ but not
on $\varepsilon$, such that
\begin{equation}\label{eq:UniformBoundLpPhi}
\E{\norm{\ue(t)}_{p,\phi}^p} \leq C, \qquad t \in [0,T].
\end{equation}
\end{lemma}
\begin{proof}
Suppose $u^0,f,\sigma$ satisfy the assumptions of
Proposition~\ref{proposition:SobolevBoundsOnViscApprox}
for $\abs{\alpha} \leq 2$. In view of
Proposition~\ref{proposition:ContDependVisc}, the
general result follows by approximation.
Set $\phi_\delta = \phi \star J_\delta$.
By Lemma~\ref{lemma:ContMollWeightedNorm} there
is a constant $C_{\delta}$ satisfying
$\abs{\Delta \phi_\delta} \leq C_{\delta}\phi_\delta$.
By Proposition \ref{proposition:SobolevBoundsOnViscApprox}, $\ue$ is
a strong solution of \eqref{eq:ViscousApprox}.
Hence we may apply It\^o's formula to the
function $S(u) = \abs{u}^p$, cf.~Step 2 in the upcoming proof
of Theorem~\ref{theorem:ExistenceOfSolution}.
After multplying by $\phi_\delta$ and integrating the result in $x$,
\begin{equation*}\label{eq:ItoFormFroSquareAppliedToVisc}
\begin{split}
&\norm{\ue(t)}_{p,\phi_\delta}^p = \norm{u^0}_{p,\phi_\delta}^p \\
& \quad-p\int_0^t\int_{\R^d} \abs{\ue(s,x)}^{p-1}
\sign{\ue(s,x)}\left(\nabla \cdot f(\ue(s,x))
- \varepsilon \Delta \ue(s,x) \right)\phi_\delta(x) \,dxds \\
& \quad +p\int_0^t\int_Z\int_{\R^d} \abs{\ue(s,x)}^{p-1}\sign{\ue(s,x)}
\sigma(x,\ue(s,x),z)\phi_\delta(x)\,dx W(ds,dz) \\
& \quad +\frac{1}{2}p(p-1)\int_0^t\int_{\R^d} \abs{\ue(s,x)}^{p-2}
\int_Z \sigma^2(x,\ue(s,x),z)\phi_\delta(x) \, d\mu(z) dxds.
\end{split}
\end{equation*}
Let $q(u) = p\int_0^u \abs{z}^{p-1}\sign{z}\partial f(z)\dz$ and
note that by \eqref{assumption:LipOnf},
\begin{equation}\label{eq:EntFluxEst}
\abs{q(u)} = \abs{\int_0^{u} p\abs{z}^{p-1}\sign{z}
\partial f(z)\,dz} \leq \norm{f}_{\mathrm{Lip}} \abs{u}^p.
\end{equation}
It follows that $q(\ue(t))\phi_\delta \in L^1(\Omega;L^1(\R^d;\R^d))$ for $0 \leq t \leq T$.
By the chain rule and integration by parts,
\begin{displaymath}
\begin{split}
\mathcal{T}_1 :=& \int_0^t\int_{\R^d} p\abs{\ue(s,x)}^{p-1}
\sign{\ue(s,x)} \nabla \cdot f(\ue(s,x))\phi_\delta(x)\, dx ds\\
=& \int_0^t\int_{\R^d} \partial q(\ue(s,x)) \cdot \nabla \ue(s,x)\phi_\delta(x)\, dx ds \\
=& -\int_0^t\int_{\R^d} q(\ue(s,x))\cdot \nabla \phi_\delta(x)\, dx ds.
\end{split}
\end{displaymath}
By \eqref{eq:EntFluxEst} and the fact that $\phi_\delta \in \mathfrak{N}$,
\begin{displaymath}
\abs{\mathcal{T}_1} \leq C_\phi
\norm{f}_{\mathrm{Lip}}\int_0^t \norm{\ue(s)}_{p,\phi_\delta}^p\,ds.
\end{displaymath}
Again by the chain rule and integration by parts,
\begin{align*}
\mathcal{T}_2 :=& \varepsilon\int_0^t\int_{\R^d} p\abs{\ue(s,x)}^{p-1}\sign{\ue(s,x)}
\Delta \ue(s,x)\phi_\delta(x)\,dxds \\
= & -\varepsilon p(p-1)\int_0^t\int_{\R^d} \abs{\ue(s,x)}^{p-2}
\abs{\nabla \ue(s,x)}^2\phi_\delta(x)\,dxds \\
& \quad -\varepsilon\int_0^t\int_{\R^d} p\abs{\ue(s,x)}^{p-1}
\sign{\ue(s,x)} \nabla \ue(s,x) \cdot \nabla \phi_\delta(x)\,dxds \\
=& -\varepsilon p(p-1)\int_0^t\int_{\R^d} \abs{\ue(s,x)}^{p-2}
\abs{\nabla \ue(s,x)}^2\phi_\delta(x)\,dxds \\
& \quad +\varepsilon\int_0^t\int_{\R^d} \abs{\ue(s,x)}^p
\Delta \phi_\delta(x)\,dxds.
\end{align*}
Hence,
$$
\abs{\mathcal{T}_2} \leq C_{\delta}
\varepsilon \int_0^t \norm{\ue(s)}_{p,\phi_\delta}^p \,ds.
$$
Finally, by assumption~\eqref{assumption:LipOnSigma},
\begin{align*}
\mathcal{T}_3 & := \frac{1}{2}p(p-1)\int_0^t\int_{\R^d}
\abs{\ue(s,x)}^{p-2} \int_Z \sigma^2(x,\ue(s,x),z)\phi_\delta(x) \, d\mu(z) dxds \\
& \leq \frac{1}{2}p(p-1)\norm{M}_{L^2(Z)}^2\int_0^t\int_{\R^d}
\abs{\ue(s,x)}^{p-2}(1 + \abs{\ue(s,x)})^2\phi_\delta(x) \,dxds \\
& \leq p(p-1)\norm{M}_{L^2(Z)}^2
\left(\, \int_0^t \norm{\ue(s)}^{p-2}_{p-2,\phi_\delta}ds
+ \int_0^t\norm{\ue(s)}^p_{p,\phi_\delta}ds\right).
\end{align*}
After taking expectations and summarizing our findings, we arrive at
\begin{equation*}
\begin{split}
\E{\norm{\ue(t)}_{p,\phi_\delta}^p} \leq \underbrace{\E{\norm{u^0}_{p,\phi_\delta}^p}
+p(p-1)\norm{M}_{L^2(Z)}^2\int_0^t\E{\norm{\ue(s)}^{p-2}_{p-2,\phi_\delta}}ds}_{C_2} \\
+ \underbrace{\left(C_\phi\norm{f}_{\mathrm{Lip}}
+ \varepsilon C_{\delta} + p(p-1)
\norm{M}_{L^2(Z)}^2\right)}_{C_1}
\int_0^t \E{\norm{\ue(s)}_{p,\phi_\delta}^p} \ds,
\end{split}
\end{equation*}
and hence, appealing to Gr\"onwall's inequality,
\begin{displaymath}
\E{\norm{\ue(t)}_{p,\phi_\delta}^p}
\leq C_2\left(1 + C_1t e^{C_1t}\right).
\end{displaymath}
Observe that the result holds for $p$ if it also holds
for $p-2$, as long as $u^0$ belongs to $L^p(\Omega;L^p(\R^d,\phi_\delta))$.
For this reason, \eqref{eq:UniformBoundLpPhi} follows by induction for $\phi = \phi_\delta$.
The bound \eqref{eq:UniformBoundLpPhi} follows (for $\phi = \phi$) by Lemma~\ref{lemma:ContMollWeightedNorm}~(ii).
\end{proof}
\begin{remark}
In the forgoing proof, it is certainly possible to apply the
Burkholder-Davis-Gundy inequality, resulting
in the improvement
$$
\E{\, \sup_{0 \leq t \leq T}
\norm{\ue(t)}_{p,\phi}^p} \leq C,
$$
for some constant $C$ independent of $\varepsilon$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{theorem:ExistenceOfSolution}]
We divide the proof into two main steps.
\emph{Step~1 (convergence)}. We apply
Theorem~\ref{theorem:YoungMeasureLimitOfComposedFunc}
to the viscous approximation $\seq{\ue}_{\varepsilon > 0}$
on the measure space
\begin{displaymath}
(X,\mathscr{A},\mu)
= (\Omega \times \Pi_T, \Pred \otimes \Borel{\R^d},
dP \otimes dt \otimes \phi(x)dx).
\end{displaymath}
By Lemma~\ref{lemma:UniformBoundsOnVisc},
\begin{displaymath}
\sup_{\varepsilon > 0}\left\{\E{\iint_{\Pi_T}
\abs{\ue(t,x)}^2 \phi(x)\,dxdt}\right\} < \infty,
\end{displaymath}
so we may take $\zeta(\xi) = \xi^2$. It follows that there
exists a subsequence $\varepsilon_k \downarrow 0$ and a
Young measure $\nu = \nu_{t,x,\omega}$ such that for
any Carath\'eodory function $\psi = \psi(u,t,x,\omega)$ satisfying
$$
\psi(u^{\varepsilon_k}(\cdot),\cdot) \rightharpoonup
\overline{\psi}(\cdot) \,\mbox{ weakly in } L^1(\Omega \times \Pi_T),
$$
we have
\begin{equation}\label{eq:RepOfLimit}
\overline{\psi}(t,x,\omega) = \int_\R \psi(\xi,t,x,\omega)\,d\nu_{t,x,\omega}(\xi)
= \int_0^1 \psi(u(t,x,\alpha,\omega),t,x,\omega)\,d\alpha.
\end{equation}
Here $u(t,x,\cdot,\omega)$ is defined
through \eqref{eq:ReprOfProcessByYoung}, i.e.,
$$
u(t,x,\alpha,\omega) =
\inf \left\{ \xi \in \R \,:\, \nu_{t,x,\omega}((-\infty,\xi]) > \alpha \right\}.
$$
We want to show that the limit $u$ is measurable, i.e., that it has
a version $\tilde{u}$ such that for any $\beta \in \R$,
$$
B_\beta = \seq{(t,x,\alpha,\omega): \tilde{u}(t,x,\alpha,\omega)
\geq \beta} \in \Pred \otimes \Borel{\R^d} \otimes \Borel{[0,1]} .
$$
Note that
$$
\tilde{u}(t,x,\alpha,\omega) \geq \beta
\Leftrightarrow \inf_{\xi \in \R} \seq{\nu_{t,x,\omega}((-\infty,\xi]) > \alpha} \geq \beta
\Leftrightarrow \nu_{t,x,\omega}((-\infty,\beta]) \leq \alpha.
$$
By definition of the Young measure we pick a version (not relabeled)
such that, the mapping $(t,x,\omega) \mapsto \nu_{t,x,\omega}((-\infty,\beta])$
is $\Pred \otimes \Borel{\R^d}$-measurable.
Furthermore, if it were finitely valued it would be clear that $B_\beta$ is
in the product topology, i.e., $B_\beta \in \F \otimes \Borel{\Pi_T} \otimes \Borel{[0,1]}$.
Hence, the result follows upon approximation by simple
functions \cite[Example~5.3.1]{Cohn2013}.
Let us show that $u \in L^p([0,T] \times \Omega;L^p(\R^d \times [0,1],\phi))$. That is,
\begin{equation}\label{eq:BundOnYoungLimit}
\E{\iint_{\Pi_T}\int_0^1 \abs{u(t,x,\alpha)}^p\phi(x)\,d\alpha dxdt} < \infty.
\end{equation}
Let $\psi \in C_c^\infty(\R)$ be supported in $(-1,1)$ and
satisfy $0 \leq \psi \leq 1, \psi(0) = 1$. Take $\psi_R(u) = \psi(u/R)$.
It follows that $\lim_{R \rightarrow \pm \infty}\psi_R(u) = 1$ for each $u \in \R$.
Moreover, with $S_R(u) = \abs{u}^p \psi_R(u)$, note that
$S_R(u) \uparrow \abs{u}^p$ for all $u \in \R$.
Since $S_R$ is compactly supported it follows that
$\seq{S_R(\ue)}_{\varepsilon > 0}$ is uniformly integrable on $X$.
By Theorem \ref{theorem:DunfordPettis}, there is
a subsequence $\varepsilon_{k(j)} \downarrow 0$ (denoted by $\varepsilon_j$)
and a limit $\overline{\psi}$ such that
$S_R(u^{\varepsilon_j}) \rightharpoonup \overline{\psi}$
(weakly) in $L^1(\Omega \times \Pi_T)$, where the weak
limit $\overline{\psi}$ can be expressed in terms of the Young
measure, cf.~\eqref{eq:RepOfLimit}. For this reason,
\begin{align*}
&\E{\iint_{\Pi_T}\int_0^1 S_R(u(t,x,\alpha)) \phi(x)\,d\alpha dxdt}
\\ & \qquad = \lim_{j \rightarrow \infty}
\E{\iint_{\Pi_T} S_R(u^{\varepsilon_j}(t,x)) \phi(x)\,dxdt} \\
& \qquad \leq \limsup_{\varepsilon \downarrow 0}
\E{\iint_{\Pi_T} \abs{\ue(t,x)}^p \phi(x)\,dxdt}
\le C \quad \text{(by Lemma~\ref{lemma:UniformBoundsOnVisc})},
\end{align*}
for some constant $C$ independent of $R$.
The claim \eqref{eq:BundOnYoungLimit} follows
upon sending $R \rightarrow \infty$, applying the
monotone convergence theorem.
\emph{Step~2 (entropy condition)}.
Let us for the moment assume that $f, \sigma, u^0$ satisfy
the assumptions of Proposition \ref{proposition:SobolevBoundsOnViscApprox} for all
multiindices $\abs{\alpha} \leq 2$. Fix an entropy/entropy-flux pair $(S,Q)$ in
$\mathscr{E}$, a nonnegative test function $\test \in C^\infty_c([0,T) \times \R)$, and a
random variable $V \in \Sm$. The goal is to show that the
limit $u$ from Step~1 satisfies $\Y{\Entropy[(S,Q),\test,V]}(u) \geq 0$.
By Proposition~\ref{proposition:SobolevBoundsOnViscApprox}, $\ue$
is a strong solution of \eqref{eq:ViscousApprox}. Indeed, consider
the weak form~\eqref{eq:WeakSolutionVisc}, integrate by parts
(cf.~Proposition~\ref{proposition:SobolevBoundsOnViscApprox}), and
use a (separating) countable subset $\seq{\test_n}_{n \geq 1}
\subset C_c^\infty(\R^d)$ of test functions, to arrive at
\begin{align*}
\ue(t,x) = u^0(x) & + \int_0^t\varepsilon
\Delta \ue(s,x)-\nabla \cdot f(\ue(s,x))\,ds \\
&+\int_0^t \int_Z\sigma(x,\ue(s,x),z) W(ds,dz),
\quad \text{$dx \otimes dP$-almost surely.}
\end{align*}
Next, we apply the anticipating It\^{o} formula (Theorem ~\ref{theorem:AntIto}), for
fixed $x \in \R^d$, to $X_t = \ue(t,x)$ and $F(X,V,t) = S(X-V)\test(t,x)$.
This yields, after taking expectations and integrating in $x$,
\begin{equation}\label{eq:AppliedItoFormToViscApprox}
\begin{split}
0 &= \E{\int_{\R^d} S(u^0-V)\test(0)\dx} \\
& \qquad + \E{\iint_{\Pi_T} S(\ue(t)-V)\partial_t\test(t)\dx dt} \\
&\qquad - \E{\iint_{\Pi_T} \nabla \cdot f(\ue(t))S'(\ue(t)-V)\test(t) \dx dt} \\
&\qquad + \E{\varepsilon\iint_{\Pi_T} \Delta \ue(t)S'(\ue(t)-V)\test(t)\dx dt} \\
&\qquad - \E{\iint_{\Pi_T}\int_Z S''(\ue(t)-V)\test(t)\sigma(x,\ue(t),z)D_{t,z}V\,d\mu(z)\dx dt} \\
&\qquad + \frac{1}{2} \E{\iint_{\Pi_T}\int_Z S''(\ue(t)-V)\test(t)\sigma(x,\ue(t),z)^2\,d\mu(z)\dx dt},
\end{split}
\end{equation}
where $D_{t,z}V$ is the Malliavin derivative of $V$ at $(t,z)$.
By the chain rule and integration by parts,
\begin{align*}
\varepsilon\iint_{\Pi_T} \Delta \ue(t)S'(\ue(t)-V)\test(t)\,dxdt
&= \varepsilon\iint_{\Pi_T}S(\ue(t)-V)\Delta\test(t)\,dxdt \\
&\quad \underbrace{- \varepsilon\iint_{\Pi_T}
S''(\ue(t)-V)\abs{\nabla\ue(t)}^2\test(t)\,dxdt}_{\leq 0}.
\end{align*}
It follows from \eqref{eq:AppliedItoFormToViscApprox} that
\begin{equation}\label{eq:EntIneqApprox}
\begin{split}
&E\Bigg[\int_{\R^d} S(u^0(x)-V)\test(0,x)\dx\Bigg] \\
&\quad +E\Bigg[\iint_{\Pi_T} \underbrace{S(\ue(t,x)-V)
\partial_t\test(t,x)}_{\psi_1(\ue,\cdot)}
+ \underbrace{Q(\ue(t,x),V)\cdot
\nabla \test(t,x)}_{\psi_2(\ue,\cdot)}\dxdt\Bigg] \\
&\quad - E\Bigg[\iint_{\Pi_T} \underbrace{S''(\ue(t,x)-V)
\int_Z\sigma(x,\ue(t,x),z)D_{t,z}V\,d\mu(z)\test(t,x)}_{\psi_3(\ue,\cdot)}\dxdt\Bigg]\\
&\quad + \frac{1}{2}E\Bigg[\iint_{\Pi_T}
\underbrace{S''(\ue(t,x)-V)\int_Z\sigma^2(x,\ue(t,x),z)\,d\mu(z)
\test(t,x)}_{\psi_4(\ue,\cdot)}\dxdt\Bigg] \\
&\quad + \varepsilon E\Bigg[\iint_{\Pi_T} S(\ue(t,x)-V)
\Delta \test(t,x)\dxdt \Bigg] \geq 0.
\end{split}
\end{equation}
At this point we may apply Proposition~\ref{proposition:ContDependVisc} to
relax the assumptions on $f,\sigma, u^0$ to the ones listed
in Theorem~\ref{theorem:ExistenceOfSolution}, leaving
the details to the reader.
Next, we wish to send $\varepsilon \downarrow 0$
in \eqref{eq:EntIneqApprox}; expressing the limits in
terms of the function $u$ obtained in Step~1. Obviously,
\begin{displaymath}
\lim_{\varepsilon \downarrow 0}
\E{\varepsilon \iint_{\Pi_T} S(\ue(t,x)-V)\Delta \test(t,x)\dxdt} = 0.
\end{displaymath}
For the remaining terms, it suffices by Step 1
and the upcoming Theorem~\ref{theorem:DunfordPettis} to show
that $\seq{\psi_i(\ue,\cdot)\phi^{-1}}_{\varepsilon > 0}$
is uniformly integrable ($i=1,2,3,4$). In view of
Lemma~\ref{lemma:UniformIntCriteria}(ii), we must show that
\begin{equation}\label{eq:UniformIntpsii}
\sup_{\varepsilon > 0}\E{\iint_{\Pi_T}
\abs{\psi_i(\ue(t,x),t,x)\phi^{-1}(x)}^2 \phi(x)\,dxdt} < \infty,
\quad i=1,2,3,4.
\end{equation}
As $S$ is in $\mathscr{E}$ and $\test \in C^\infty_c(\Pi_T)$,
\begin{align*}
\abs{\psi_1(\ue(t,x),t,x)\phi^{-1}(x)} &
= \abs{S(\ue(t,x)-V)\partial_t\test(t,x)\phi^{-1}(x)}^2 \\
& \leq 2\norm{S}_{\mathrm{Lip}}^2
\norm{\partial_t \test(t)}_{\infty,\phi^{-1}}(\abs{\ue(t,x)}^2 + \abs{V}^2).
\end{align*}
So \eqref{eq:UniformIntpsii}, with $i=1$,
follows from Lemma~\ref{lemma:UniformBoundsOnVisc}.
The term in \eqref{eq:EntIneqApprox} involving
$\psi_2$ is treated in the same way.
Consider the term involving the Malliavin derivative, namely $\psi_3$.
By \eqref{assumption:LipOnSigma},
$$
\abs{\int_Z\sigma(x,\ue(t,x),z)D_{t,z}V\,d\mu(z)}^2 \leq
\norm{M}_{L^2(Z)}^2 \norm{D_tV}_{L^2(Z)}^2(1 + \abs{\ue(t,x)})^2.
$$
Recall that $V$ is uniformly bounded and
also that $\mathrm{supp}\,(S'') \subset (-R,R)$
for some $R < \infty$. Hence,
$$
S''(\ue-V)(1 + \abs{\ue}) \leq \norm{S''}_\infty(1 + R + \norm{V}_\infty).
$$
Consequently,
\begin{align*}
&\E{\iint_{\Pi_T} \abs{\psi_3(\ue(t,x),t,x)\phi^{-1}(x)}^2 \phi(x)\,dxdt}
\\ &\,
\leq \norm{S''}_\infty^2\norm{M}_{L^2(Z)}^2(1 + R
+ \norm{V}_\infty)^2\norm{\test}_{\infty,\phi^{-1}}^2
\E{\int_0^T\norm{D_tV}_{L^2(Z)}^2\,dt}\norm{\phi}_{L^1(\R^d)},
\end{align*}
and \eqref{eq:UniformIntpsii} holds with $i=3$.
Consider the $\psi_4$-term. By \eqref{assumption:LipOnSigma},
\begin{displaymath}
S''(\ue-V)\int_Z\sigma^2(x,\ue,z)\,d\mu(z)
\leq \norm{S''}_\infty\norm{M}_{L^2(Z)}^2(1 + R + \norm{V}_\infty)^2.
\end{displaymath}
Hence
\begin{multline*}
\E{\iint_{\Pi_T} \abs{\psi_4(\ue(t,x),t,x)\phi^{-1}(x)}^2 \phi(x)\,dxdt} \\
\leq \norm{S''}_\infty^2\norm{M}_{L^2(Z)}^4(1 + R
+ \norm{V}_\infty)^4\norm{\test}_{\infty,\phi^{-1}}^2 \iint_{\Pi_T} \phi(x)\,dxdt.
\end{multline*}
Summarizing, upon sending $\varepsilon \downarrow 0$
along a subsequence, it follows that
$$
\Y{\Entropy[(S,Q),\test,V]}(u) \geq 0,
$$
where $u$ is the process defined in Step 1.
Finally, the result follows for general $V \in \D^{1,2}$ by
the density of $\Sm \subset \D^{1,2}$
and Lemma~\ref{lemma:ContinuityOfEntWRTV}.
\end{proof}
\section{Uniqueness of entropy solutions}\label{sec:Uniqueness}
To prove the uniqueness of Young measure-valued entropy solutions, we
need an additional assumption on $\sigma$: there exists
$M \in L^2(Z)$ and $0 < \kappa \leq 1/2$ such that
\begin{equation}\label{assumption:SigmaRegularity}
\abs{\sigma(x,u,z)-\sigma(y,u,z)} \leq
M(z)\abs{x-y}^{\kappa + 1/2}(1 + \abs{u}) \tag{$\mathcal{A}_{\sigma,1}$},
\end{equation}
for $x,y \in \R^d$ and $u \in \R$. Actually, it suffices
that the criterion is satisfied locally, i.e., for each compact
$K \subset \R^d \times \R^d$ there exists $M = M_K$ such
that \eqref{assumption:SigmaRegularity} is satisfied for all $(x,y) \in K$.
\begin{theorem}\label{theorem:UniquenessOfEntSol}
Fix $\phi \in \mathfrak{N}$, and suppose
$u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Assume that assumptions~\eqref{assumption:LipOnf},
\eqref{assumption:LipOnSigma}, \eqref{assumption:SigmaRegularity}
are satisfied. Let $u$ be the Young measure-valued entropy solution
to \eqref{eq:StochasticBalanceLaw} with initial condition $u^0$
obtained in Theorem~\ref{theorem:ExistenceOfSolution}, and
let $v$ be any Young measure-valued entropy
solution with initial condition $u^0$ in the sense
of Definition~\ref{Def:YoungEntropySolution}. Then
$$
u(t,x,\alpha) = v(t,x,\beta),
\qquad (t,x,\alpha,\beta,\omega) \mbox{-almost everywhere.}
$$
Consequently, $\hat{u} := \int_0^1 u \,d\alpha$ is the unique
entropy solution to \eqref{eq:StochasticBalanceLaw} in
the sense of Definition~\ref{Def:EntropySolution}.
\end{theorem}
The proof is found at the end of this section.
As discussed in the introduction, due
to the lack of Malliavin differentiability at the hyperbolic level, the
uniqueness argument will invoke the viscous approximations and
their limit taken in the weak sense of Young measures.
Retracing the proof of Theorem~\ref{theorem:UniquenessOfEntSol}, making
some small modifications, we obtain the following spatial regularity result:
\begin{proposition}[Spatial regularity]\label{proposition:FracBounds}
Fix $\phi \in \mathfrak{N}$, and suppose $u^0$ belongs to
$L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Under assumptions \eqref{assumption:LipOnf}, \eqref{assumption:LipOnSigma},
and \eqref{assumption:SigmaRegularity} the
entropy solution $u$ to \eqref{eq:StochasticBalanceLaw} satisfies
\begin{multline*}
E\Bigg[\,\,\iint\limits_{\,\, \R^d \times \R^d}
\abs{u(t,x + z)-u(t,x-z)}\phi(x)J_r(z)\,dxdz\Bigg] \\
\leq CE\Bigg[\,\, \iint\limits_{\,\,\R^d \times \R^d}
\abs{u^0(x + z)-u^0(x-z)}\phi(x)J_r(z)\,dxdz\Bigg] + \mathcal{O}(r^\kappa),
\end{multline*}
where the constant $C$ depends only on $C_\phi, \norm{f}_{\mathrm{Lip}}, T$,
and $\kappa$ is the exponent from assumption \eqref{assumption:SigmaRegularity}.
If $\sigma$ is independent of $x$, i.e., $\sigma(x,u,z) = \sigma(u,z)$, then the
last term on the right vanishes, i.e., $\mathcal{O}(\cdot) \equiv 0$.
\end{proposition}
See \cite{DebusscheVovelle2010,ChenKarlsen2012} for
similar results, and how to turn this result into a fractional
$BV$ estimate. The proof of Proposition \ref{proposition:FracBounds} is
found at the very end of this section.
The next lemma contains the ``entropy condition'' at the
parabolic level, which is utilized later in the uniqueness proof.
\begin{lemma}\label{lemma:EntIneqViscForAdapted}
For each fixed $\varepsilon>0$, let $\ue$ be the solution of \eqref{eq:ViscousApprox}.
Suppose $V \in L^2(\Omega)$ is $\F_s$-measurable for some $s \in (0,T)$,
and $0\le \test \in C^\infty_c([0,T) \times \R^d)$ with
$\mathrm{supp}\,(\test)\subset (s,T) \times \R^d$. Then
\begin{equation*}
\begin{split}
& \qquad \E{\iint_{\Pi_T} S(\ue-V)\partial_t\test + Q(\ue,V)\cdot \nabla \test \dxdt}\\
& \qquad \qquad \geq-\frac{1}{2}\E{\iint_{\Pi_T}\int_Z S''(\ue-V)
\sigma(x,\ue,z)^2\test(t,x)\,d\mu(z)\dxdt} \\
& \qquad \qquad \qquad -\varepsilon \E{\iint_{\Pi_T} S(\ue(t)-V)\Delta\test(t)\dx dt},
\end{split}
\end{equation*}
for any entropy/entropy-flux pair $(S,Q)$ in $\mathscr{E}$.
\end{lemma}
\begin{proof}
Consider \eqref{eq:EntIneqApprox}.
Note that for any $V \in \D^{1,2}$ that is $\F_s$-measurable,
\begin{displaymath}
\E{\iint_{\Pi_T}\int_Z S''(\ue(t)-V)\sigma(x,\ue(t),z)D_{t,z}V\test(t)
\,d\mu(z)\dx dt} = 0,
\end{displaymath}
thanks to \cite[Proposition~1.2.8]{NualartMalliavinCalc2006}.
The general result follows by approximation
as in Lemma~\ref{lemma:ContinuityOfEntWRTV}.
\end{proof}
The following ``doubling of variables" lemma is at the heart of the matter.
To some extent it may be instructive to compare its proof with the rather involved
computations in \cite[Lemma~3.2]{FengNualart2008} and \cite[Section~4.1]{Bauzet:2012kx}.
\begin{lemma}\label{lemma:DoubelingWithoutLimits}
Suppose \eqref{assumption:LipOnf}, \eqref{assumption:LipOnSigma} hold.
Fix $\phi \in \mathfrak{N}$, and let $\seq{\ue}_{\varepsilon > 0}$ be a sequence of viscous
approximations with initial condition $u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Let $v$ be a Young measure-valued entropy solution in the sense
of Definition~\ref{Def:YoungEntropySolution} with initial condition
$v^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
For any $0< \gamma <\frac{1}{2}T$ take $t_0 \in (0,T-2\gamma]$ and define
$$
\xi_{\gamma,t_0}(t) := 1 - \int_0^t J_\gamma^+(s-t_0)\ds.
$$
Let $\psi \in C^\infty_c(\R^d)$ be non-negative and define
$$
\test(t,x,s,y) = \frac{1}{2^d}\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)
\xi_{\gamma,t_0}(t)J_{r_0}^+(t-s).
$$
Let $\Sd$ be a function satisfying
$$
\Sd'(\sigma) = 2\int_0^\sigma J_\delta(z)\dz,
\qquad \Sd(0) = 0.
$$
Furthermore, define
$$
Q_\delta(u,c) = \int_c^u \Sd'(z-c)f'(z)\dz,
$$
and note that the pair $(\Sd,Q_\delta)$ belongs
to $\mathscr{E}$.
Then
\begin{equation}\label{eq:DoublingLemmaIneq}
L \geq R + F + \mathcal{T}_1 + \mathcal{T}_2 + \mathcal{T}_3,
\end{equation}
where
\begin{align*}
\begin{split}
&L = \E{\iint_{\Pi_T}\int_{\R^d} \Sd(v^0(y)-\ue(t,x))\test(t,x,0,y) \,dydxdt},
\end{split} \\
&R = -\E{\iiiint_{\Pi_L^2}\int_{[0,1]}\Sd(v-\ue)(\partial_s + \partial_t)\test\,d\beta dX}, \\
&F = -\E{\iiiint_{\Pi_L^2}\int_{[0,1]} Q_\delta(\ue,v)\cdot \nabla_x \test
+ Q_\delta(v,\ue)\cdot \nabla_y\test\,d\beta dX}, \\
\begin{split}
&\mathcal{T}_1 = -\frac{1}{2}\E{ \iiiint_{\Pi_L^2}\int_{[0,1]}\int_Z \Sd''(v-\ue)
\left(\sigma(y,v,z)-\sigma(x,\ue,z)\right)^2\test\,d\mu(z)d\beta dX},
\end{split} \\
&\mathcal{T}_2 = \E{ \iiiint_{\Pi_L^2}\int_{[0,1]}\int_Z \Sd''(v-\ue)\left(D_{s,z}\ue
-\sigma(x,\ue,z)\right)\sigma(y,v,z)\test\,d\mu(z)d\beta dX}, \\
&\mathcal{T}_3 = -\varepsilon \E{\iiiint_{\Pi_L^2}\int_{[0,1]} \Sd(\ue-v)\Delta_x\test\,d\beta dX},
\end{align*}
where $dX = dxdtdyds$.
\end{lemma}
\begin{remark}
In \cite[Section~4.6]{FengNualart2008} the authors prove existence of a
\textit{strong} entropy solution. The additional condition attached to the notion of
strong solution stems from the difficulties in sending
$\varepsilon \downarrow 0$ before $r_0 \downarrow 0$.
In our setting, the existence of a strong entropy solution amounts to
showing that we can send $r_0 \downarrow 0$
and $\varepsilon \downarrow 0$ simultaneously in such a way that
$\lim_{(\varepsilon,t) \downarrow (0,s)}\mathcal{T}_2 = 0$.
This requires a careful study of how the continuity
properties of \eqref{eq:MallEqSat} depends on
$\varepsilon$, cf.~Lemma~\ref{lemma:MalliavinDerivativeWeakTimeCont}.
We do not proceed along this path in this paper, instead we let $r_0 \downarrow 0$
before $\varepsilon \downarrow 0$ as in \cite{Bauzet:2012kx}.
\end{remark}
\begin{proof}
Recall that $\mathrm{supp}(J_{r_0}^+) \subset (0,2r_0)$, so $J_{r_0}^+(t-s)$ is zero
whenever $s \geq t$. Applying Lemma \ref{lemma:EntIneqViscForAdapted}
with $V = v(s,y,\beta)$ and integrating in $y,s,\beta$, we obtain
\begin{equation}\label{eq:EntIneqForViscConstv}
\begin{split}
&\E{\iiiint_{\Pi_T^2}\int_{[0,1]} \Sd(\ue-v)\partial_t\test
+ Q_\delta(\ue,v)\cdot \nabla \test \,d\beta dX}\\
&\geq-\frac{1}{2}\E{\iiiint_{\Pi_T^2}\int_{[0,1]}
\int_Z \Sd''(\ue-v)\sigma(x,\ue,z)^2\test\,d\mu(z)\, d\beta dX}\\
&\qquad -\varepsilon \E{\iiiint_{\Pi_T^2}\int_{[0,1]} \Sd(\ue-v)\Delta_x\test \,d\beta dX}.
\end{split}
\end{equation}
Similarly, in the entropy inequality for $v = v(s,y,\beta)$ we take $V = \ue(t,x)$
and integrate in $t,x$, resulting in
\begin{equation}\label{eq:EntIneqForMeasureValuedvConstue}
\begin{split}
& \E{\iiiint_{\Pi_T^2}\int_{[0,1]} \Sd(v-\ue)\partial_s\test
+ Q_\delta(v,\ue)\cdot \nabla_y \test \,d\beta dX} \\
& +\E{\iint_{\Pi_T}\int_{\R^d} \Sd(v^0(y)-\ue(t,x))\test(t,x,0,y) \,dydxdt}\\
& \geq \E{\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z \Sd''(v-\ue)
D_{s,z}\ue \sigma(y,v,z)\test \,d\mu(z) \,d\beta dX} \\
& \qquad -\frac{1}{2}\E{ \iiiint_{\Pi_T^2}\int_{[0,1]}
\int_Z \Sd''(v-\ue)\sigma(y,v,z)^2\test\,d\mu(z)d\beta dX}.
\end{split}
\end{equation}
The result follows by adding \eqref{eq:EntIneqForViscConstv}
and \eqref{eq:EntIneqForMeasureValuedvConstue}.
\end{proof}
\begin{proposition}[Kato inequality]\label{proposition:KatoInequalityEntSol}
Fix $\phi \in \mathfrak{N}$. Suppose \eqref{assumption:LipOnf},
\eqref{assumption:LipOnSigma}, and \eqref{assumption:SigmaRegularity} hold.
Let $u$ be the Young measure-valued limit of the viscous
approximations $\seq{\ue}_{\varepsilon > 0}$ with initial
condition $u^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$,
constructed in Theorem \ref{theorem:ExistenceOfSolution}.
Let $v$ be a Young measure-valued entropy solution in the
sense of Definition~\ref{Def:YoungEntropySolution} with
initial condition $v^0 \in L^2(\Omega,\F_0,P;L^2(\R^d,\phi))$.
Then, for almost all $t_0 \in (0,T)$ and
any non-negative $\psi \in C^\infty_c(\R^d)$,
\begin{equation}\label{eq:KatoInequalityEntSol}
\begin{split}
& E\bigg[\int_{\R^d}\iint_{[0,1]^2}\abs{u(t_0,x,\alpha)-v(t_0,x,\beta)}
\psi(x)\,d\alpha d\beta dx \bigg] \\
& \quad \leq \E{\int_{\R^d} \abs{u^0(x)-v^0(x)}\psi(x)\dx} \\
& \quad \qquad +E\bigg[\int_0^{t_0}\int_{\R^d}\iint_{[0,1]^2}
\sign{u(t,x,\alpha)-v(t,x,\beta)} \\
&\quad\qquad\qquad\qquad
\times (f(u(t,x,\alpha))-f(v(t,x,\beta)))\cdot
\nabla \psi(x)\,d\beta d\alpha dxdt \bigg].
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
Starting off from \eqref{eq:DoublingLemmaIneq}, we send $r_0$
and $\varepsilon$ to zero (in that order).
Next, we send $(\delta,r)$ to $(0,0)$ simultaneously.
In view of Limits \ref{limit:F} and \ref{limit:T1}, we let $\delta(r) = r^{1 + \eta}$
with $0 < \eta <2\kappa -1$. Finally, we send $\gamma \downarrow 0$.
We arrive at the Kato inequality \eqref{eq:KatoInequalityEntSol}
thanks to the upcoming Limits~\ref{limit:L}--\ref{limit:T3}.
\end{proof}
\begin{remark}\label{remark:testFuncProp}
Later we will make repeated use of two elementary
identities. Set
$$
\xi_r(x):= \frac{1}{2^d}\int_{\R^d}
\psi \left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,dy.
$$
Then note that $\xi_r = \psi \star J_r$.
Indeed, making the change of variable $z = (x+y)/2$,
it follows that $(x-y)/2 = x-z$ and $dy=2^d\,dz$.
Next, consider the change of variables
\begin{displaymath}
\Phi(x,y) = \left(\frac{x + y}{2},\frac{x-y}{2}\right) = (\tilde{x},z).
\end{displaymath}
By the change of variables formula
\begin{displaymath}
\iint_{\R^d \times \R^d} g(\tilde{x},z)\,d\tilde{x}dz =
\iint_{\R^d \times \R^d} g(\Phi(x,y))\abs{\mathrm{det}(\Jac\Phi(x,y))}\,dxdy,
\end{displaymath}
for any measurable function $g(\cdot,\cdot)$. A computation
yields $\abs{\mathrm{det}(\Jac\Phi(x,y))} = 1/2^d$. It follows that
\begin{align*}
\frac{1}{2^d}\iint_{\R^d \times \R^d} &\underbrace{h(x,y)
\psi \left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)}_{g(\Phi(x,y))}\,dxdy \\
&= \iint_{\R^d \times \R^d}
\underbrace{h(\tilde{x} + z,\tilde{x} - z)\psi(\tilde{x})J_r(z)}_{g(\tilde{x},z)}\,d\tilde{x}dz,
\end{align*}
for any measurable function $h(\cdot,\cdot)$. Most of the
time we drop the tilde and write $x$ instead of $\tilde{x}$.
\end{remark}
\begin{limit}\label{limit:L}
With $L$ defined in Lemma~\ref{lemma:DoubelingWithoutLimits},
$$
\lim_{r_0 \downarrow 0}L
= \E{\iint_{\R^d \times \R^d} \Sd(v^0(x - z)-u^0(x + z))\psi(x)J_r(z)\,dxdz}.
$$
If $\delta = \delta(r)$ is a nondecreasing function
satisfying $\delta(r) \downarrow 0$
as $r \downarrow 0$, then
$$
\lim_{\gamma,(\delta,r),\varepsilon,r_0 \downarrow 0} L
= \E{\norm{v^0-u^0}_{1,\psi}}.
$$
\end{limit}
\begin{proof}
Note that
\begin{equation}\label{eq:SdAbsEst}
\abs{\Sd(b)-\Sd(a)} = \abs{\int_a^b\Sd'(z)\dz} \leq \abs{b-a}.
\end{equation}
Furthermore, observe that $\xi_{\gamma,t_0}(t) = 1$ whenever $t \leq t_0$.
Hence, due to Remark~\ref{remark:testFuncProp},
\begin{align*}
&\abs{L -\E{\int_{\R^d}\int_{\R^d} \Sd(v^0(y)-u^0(x))
\frac{1}{2^d}\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,dxdy}} \\
& \qquad \leq \E{\iint_{\Pi_T} \abs{u^0(x)-\ue(t,x)}J_{r_0}^+(t)(\psi \star J_r)(x)\,dxdt},
\end{align*}
whenever $2r_0 < t_0$. Arguing as in Lemma~\ref{lemma:InitialCondition} for the
viscous approximation, it follows that
\begin{align*}
\lim_{r_0 \downarrow 0}L &= \E{\frac{1}{2^d}\int_{\R^d}\int_{\R^d}
\Sd(v^0(y)-u^0(x))\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,dxdy} \\
& = \E{\iint_{\R^d \times \R^d} \Sd(v^0(x - z)-u^0(x + z))\psi(x)J_r(z) \,dxdz}.
\end{align*}
This proves the first limit. The second limit follows by the
dominated convergence theorem
and Lemma~\ref{lemma:DeltaRconvergence}.
\end{proof}
\begin{remark}\label{remark:YoungLimitInDoubling}
To establish Limits~\ref{limit:R} and \ref{limit:F} we
need to send $\varepsilon \downarrow 0$ in terms of the form
$$
\E{\,\,\, \iiiint\limits_{\,\,\Pi_T \times \R^d \times [0,1]}
\Psi(\ue(t,x,\omega),t,x,y,\beta,\omega)\underbrace{\frac{1}{2^d}\phi\left(\frac{x+y}{2}\right)
J_r\left(\frac{x-y}{2}\right)\,d\beta dydxdt}_{d\eta_{\phi,r}}},
$$
where $\Psi$ is continuous in the first variable. Essentially we
proceed as in the proof of Theorem~\ref{theorem:ExistenceOfSolution}, but
now the underlying measure space is $\Pi_T \times \R^d \times [0,1] \times \Omega$
instead of $\Pi_T \times \Omega$.
By Lemma~\ref{lemma:UniformBoundsOnVisc}
and Remark~\ref{remark:testFuncProp},
\begin{multline*}
\sup_{\varepsilon > 0}\left\{\E{\iiiint_{\Pi_T \times \R^d \times [0,1]}
\abs{\ue(t,x)}^2\,d\eta_{\phi,r}}\right\} \\
= \sup_{\varepsilon > 0}
\left\{\E{\int_0^T \norm{\ue(t)}_{2,\phi \star J_r}^2
\,dt}\right\} < \infty.
\end{multline*}
By Theorem~\ref{theorem:YoungMeasureLimitOfComposedFunc}, there
exists $\nu \in \Young{\Pi_T \times \R^d \times [0,1] \times \Omega}$
such that whenever $\Psi(\ue,\cdot) \rightharpoonup \overline{\Psi}$ (weakly)
along some subsequence in
$L^1(\Pi_T \times \R^d \times [0,1] \times \Omega,d\eta_{\phi,r} \otimes dP)$,
\begin{equation*}
\overline{\Psi}
= \int_\R \Psi(\xi,t,x,y,\beta,\omega)\,d\nu_{t,x,\omega}(\xi)
= \int_0^1 \Psi(u(t,x,\alpha,\omega),t,x,y,\beta,\omega)\,d\alpha,
\end{equation*}
where $u$ is defined through \eqref{eq:ReprOfProcessByYoung}.
The fact that $\nu_{t,x,y,\beta,\omega} = \nu_{t,x,\omega}$ comes out since the
limit is independent of $y,\beta$ when $\Psi$ is independent of $y,\beta$.
For measurability considerations, see Step 1 in proof
of Theorem~\ref{theorem:ExistenceOfSolution}.
\end{remark}
\begin{limit}\label{limit:R} With $R$ defined in
Lemma \ref{lemma:DoubelingWithoutLimits},
$$
\lim_{\gamma,\varepsilon, r_0 \downarrow 0} R
= E\Bigg[\, \, \iiiint\limits_{\,\,\, \R^d \times \R^d \times [0,1]^2}
\Sd(v(t_0,x-z,\beta)-u(t_0,x+z,\alpha)) \psi(x)J_r(z)\,d\alpha d\beta dxdz\Bigg],
$$
for $dt$-a.a.~$t_0 \in [0,T]$.
If $\delta = \delta(r)$ is a nondecreasing
function satisfying $\delta(r) \downarrow 0$
as $r \downarrow 0$, then
$$
\lim_{\gamma,(\delta,r),\varepsilon,r_0 \downarrow 0}
R = \E{\int_{\R^d}\iint_{[0,1]^2}
\abs{v(t_0,x,\beta)-u(t_0,x,\alpha)}\psi(x)\,d\alpha d\beta dx},
$$
for $dt$-a.a.~$t_0 \in [0,T]$.
\end{limit}
\begin{proof}
Since $\partial_tJ_{r_0}^+(t-s) = -\partial_sJ_{r_0}^+(t-s)$ and
$\partial_t\xi_{\gamma,t_0}(t) = -J_\gamma^+(t-t_0)$,
$$
(\partial_s + \partial_t)\test(t,x,s,y) = -\frac{1}{2^d}\psi\left(\frac{x+y}{2}\right)
J_r\left(\frac{x-y}{2}\right)J_\gamma^+(t-t_0)J_{r_0}^+(t-s).
$$
It follows that
\begin{align*}
R = E\Bigg[\frac{1}{2^d}\iiiint_{\Pi_T^2}\int_{[0,1]}&
\Sd(v-\ue)\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right) \\
&\qquad \qquad \qquad
\times J_\gamma^+(t-t_0)J_{r_0}^+(t-s)\,d\beta\dX\Bigg].
\end{align*}
Thanks to
$$
\abs{\Sd(v-\ue)} \leq \abs{v} + \abs{\ue},
$$
we can apply the dominated convergence theorem
and Lemma~\ref{lemma:TimeTraceMollLimit}, resulting in
\begin{multline*}
\lim_{r_0 \downarrow 0} R = E\Bigg[\iiiint\limits_{\quad \Pi_T \times \R^d \times [0,1]}
\underbrace{\Sd(v(t,y,\beta)-\ue(t,x))
J_\gamma^+(t-t_0)(\psi\phi^{-1})\left(\frac{x+y}{2}\right)}_{\Psi(\ue,\cdot)} \\
\hphantom{XXXXXXXXXXXXXXXX}\times
\frac{1}{2^d}\phi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,d\beta dy dx dt\Bigg].
\end{multline*}
By Lemma~\ref{lemma:UniformIntCriteria}(ii), $\seq{\Psi_\varepsilon(\ue,\cdot)}$
is uniformly integrable, and so, cf.~Theorem~\ref{theorem:DunfordPettis},
we can extract a weakly convergent subsequence.
By Remarks~\ref{remark:YoungLimitInDoubling} and \ref{remark:testFuncProp},
\begin{align*}
\lim_{\varepsilon, r_0 \downarrow 0} R
&= E\Bigg[\iint_{\Pi_T}\int_{\R^d} \iint_{[0,1]^2}
\Sd(v(t,y,\beta)-u(t,x,\alpha))J_\gamma^+(t-t_0) \\
&\hphantom{XXXXXXXXXXx} \times \frac{1}{2^d}
\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,d\alpha d\beta dydxdt \Bigg] \\
&= E\Bigg[\iint_{\Pi_T}\int_{\R^d} \iint_{[0,1]^2}\Sd(v(t,x-z,\beta)-u(t,x+z,\alpha)) \\
&\hphantom{XXXXXXXXXXXXXx}
\times J_\gamma^+(t-t_0)\psi(x)J_r(z)\,d\alpha d\beta dzdxdt \Bigg].
\end{align*}
Note that
$$
\abs{\Sd(a-b)-\Sd(c-d)} \leq \abs{b-d} + \abs{a-c},
\qquad a,b,c,d \in \R.
$$
Applying this inequality and Lemma~\ref{lemma:TimeTraceMollLimit}, we
can send $\gamma \downarrow 0$ to obtain the first inequality.
To send $(\delta,r) \downarrow (0,0)$ we apply
the dominated convergence theorem
and Lemma~\ref{lemma:DeltaRconvergence}, yielding
$$
\lim_{r,\varepsilon,r_0 \downarrow 0}R
= \E{\iint_{\Pi_T}\iint_{[0,1]^2}
\abs{v(t,x,\beta)-u(t,x,\alpha)}
\psi(x)J_\gamma^+(t-t_0)\,d\alpha d\beta\dxdt}.
$$
To send $\gamma \downarrow 0$ we apply
Lemma~\ref{lemma:TimeTraceMollLimit}.
This provides the second limit.
\end{proof}
\begin{limit}\label{limit:F}
With $F$ defined in Lemma~\ref{lemma:DoubelingWithoutLimits},
\begin{multline}\label{eq:FluxFracBVLimit}
\lim_{\gamma,\varepsilon,r_0 \downarrow 0}F
= E\Bigg[\int_0^{t_0}\iiiint\limits_{\R^d \times \R^d \times [0,1]^2}
\Sd'(u(t,x+z,\alpha)-v(t,x-z,\beta)) \\
\times (f(u(t,x + z,\alpha))-f(v(t,x-z,\beta)))\cdot \nabla \psi(x)J_r(z)
\,d\alpha d\beta dxdzdt\Bigg] \\
+ \mathcal{O}\left(\delta + \frac{\delta}{r}\right).
\end{multline}
If $\delta:[0,\infty) \rightarrow [0,\infty)$ satisfy
$\lim_{r \rightarrow 0}\frac{\delta(r)}{r} = 0$, then
\begin{multline}\label{eq:FluxLimit}
\lim_{\gamma,r,\varepsilon,r_0 \downarrow 0}F
= -E\bigg[\int_0^{t_0}\int_{\R^d}\iint_{[0,1]^2} \sign{u(t,x,\alpha)-v(t,x,\beta)} \\
\times (f(u(t,x,\alpha))-f(v(t,x,\beta)))\cdot \nabla \psi(x)\,d\alpha d\beta dxdt\bigg].
\end{multline}
\end{limit}
\begin{proof}
Using integration by parts,
\begin{displaymath}
Q_\delta(\ue,v) = \Sd'(\ue-v)(f(\ue)-f(v)) - \int_v^{\ue} \Sd''(z-v)(f(z)-f(v))\dz
\end{displaymath}
and
\begin{displaymath}
Q_\delta(v,\ue) = \Sd'(v-\ue)(f(v)-f(\ue)) - \int_{\ue}^v \Sd''(z-\ue)(f(z)-f(\ue))\dz.
\end{displaymath}
Due to the symmetry of $\Sd$,
\begin{align*}
F &= -\E{\iiiint_{\Pi_T^2}\int_{[0,1]} Q_\delta(\ue,v)\cdot \nabla_x \test
+ Q_\delta(v,\ue)\cdot \nabla_y \test\,d\beta dX} \\
&=-\E{\iiiint_{\Pi_T^2}\int_{[0,1]} \Sd'(\ue-v)(f(\ue)-f(v))\cdot(\nabla_x
+ \nabla_y)\test \,d\beta dX} \\
&\quad + \E{\iiiint_{\Pi_T^2}\int_{[0,1]}\left(\int_v^{\ue} \Sd''(z-v)(f(z)-f(v))\dz\right)
\cdot \nabla_x \test\,d\beta dX} \\
& \quad + \E{\iiiint_{\Pi_T^2}\int_{[0,1]}\left(\int_{\ue}^v
\Sd''(z-\ue)(f(z)-f(\ue))\dz\right)\cdot \nabla_y \test\,d\beta dX} \\
&= -F_1 + F_2 + F_3,
\end{align*}
where $dX = dxdtdyds$ as in Lemma~\ref{lemma:DoubelingWithoutLimits}.
Note that
\begin{equation}\label{est:SingTimesLipInt}
\abs{\int_v^u \Sd''(z-v)(f(z)-f(v))\dz}
\leq \norm{f}_{\mathrm{Lip}}\delta,
\qquad u,v\in \R.
\end{equation}
To see this, recall that
$\Sd''(\sigma) = 2J_\delta(\sigma)$.
By \eqref{assumption:LipOnf},
$$
\abs{\int_v^u \Sd''(z-v)(f(z)-f(v))\dz}
\leq 2\norm{f}_{\mathrm{Lip}}\sign{u-v}
\int_v^u J_\delta(z-v)\abs{z-v}\dz,
$$
and letting $\xi = \abs{z-v}/\delta$,
$$
\sign{u-v}\int_v^u J_\delta(z-v)\abs{z-v}\dz
= \delta \int_0^{\delta^{-1}\abs{u-v}}J(\xi)\xi\,d\xi
\leq \frac{\delta}{2}.
$$
In view of \eqref{est:SingTimesLipInt}, it is clear that
$$
F_2 \leq \norm{f}_{\mathrm{Lip}}\delta
\iiiint_{\Pi_T^2}\abs{\nabla_x \test} \dX.
$$
A computation shows $\norm{\nabla \test}_{L^1(\Pi_T^2)} \leq C(1 + r^{-1})$,
for some constant $C$ depending only on $J,T,\psi$.
Consequently,
$$
F_2 \leq C\norm{f}_{\mathrm{Lip}}
\delta\left(1 + \frac{1}{r}\right).
$$
The same type of estimate applies to $F_3$.
Let us consider $F_1$. Observe that
$$
(\nabla_x + \nabla_y)\test(t,x,s,y)
= \frac{1}{2^d}\nabla \psi\left(\frac{x+y}{2}\right)
J_r\left(\frac{x-y}{2}\right)\xi_{\gamma,t_0}(t)J_{r_0}^+(t-s).
$$
For $\delta>0$, define
$$
\mathcal{F}_\delta(a,b) := \Sd'(a-b)(f(a)-f(b)), \qquad a,b \in \R,
$$
and note that $(t,b) \mapsto \mathcal{F}_\delta(\ue(t,x),b)$
obeys the hypotheses of Lemma \ref{lemma:TimeTraceMollLimit}.
By the dominated convergence theorem
and Lemma \ref{lemma:TimeTraceMollLimit},
\begin{multline*}
\lim_{r_0 \downarrow 0}F_1
= E\Bigg[\, \, \iint_{\Pi_T}\int_{\R^d}\int_{[0,1]}
\underbrace{\mathcal{F}_\delta(\ue(t,x),v(t,y,\beta))\cdot \zeta
\left(\frac{x+y}{2}\right)\xi_{\gamma,t_0}(t)}_{\Psi(\ue,\cdot)} \\
\times \frac{1}{2^d}\phi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)
\,d\beta dydxdt\Bigg],
\end{multline*}
where $\zeta(x) = \phi^{-1}(x)\nabla \psi(x)$.
The uniform integrability of $\seq{\Psi(\ue,\cdot)}_{\varepsilon > 0}$
follows thanks to Lemma~\ref{lemma:UniformIntCriteria}(ii).
Indeed, $\abs{\zeta} \leq C_\phi$
and $\abs{\mathcal{F}_\delta(\ue,v)} \leq
\norm{f}_{\mathrm{Lip}}\abs{\ue-v}$, so
$$
\abs{\Psi(\ue,\cdot)}^2 \leq 2C_\phi^2\norm{f}_{\mathrm{Lip}}^2
(\abs{\ue}^2+ \abs{v}^2).
$$
By Theorem~\ref{theorem:DunfordPettis} and Remark~\ref{remark:YoungLimitInDoubling}
\begin{multline*}
\lim_{\varepsilon,r_0 \downarrow 0}F_1
= E\Bigg[\iint_{\Pi_T}\int_{\R^d}\iint_{[0,1]^2}
\mathcal{F}_\delta(u(t,x,\alpha),v(t,y,\beta)) \\
\cdot \frac{1}{2^d}\nabla \psi\left(\frac{x+y}{2}\right)
J_r\left(\frac{x-y}{2}\right)\xi_{\gamma,t_0}(t)
\, d\alpha d\beta dydxdt \Bigg],
\end{multline*}
along a subsequence. Sending $\gamma \downarrow 0$,
applying Remark \ref{remark:testFuncProp}, yields \eqref{eq:FluxFracBVLimit}.
Next we want to prove \eqref{eq:FluxLimit}. To send $r \downarrow 0$ we apply
Lemma \ref{lemma:DeltaRconvergence}. It is easily
verified that condition (i) and (iii) are satisfied with $F_\delta = \mathcal{F}_\delta$.
Consider condition (ii). Since $\Sd'(-\sigma) = -\Sd'(\sigma)$ for all $\sigma \in \R$, it follows that
\begin{align*}
\mathcal{F}_\delta(a,b) - \mathcal{F}_\delta(a,c)
&= \Sd'(b-a)(f(b)-f(a)) - \Sd'(c-a)(f(c)-f(a))\\
&= \int_b^c \partial_z(\Sd'(z-a)(f(z)-f(a)))\dz \\
&= \int_b^c \Sd''(z-a)(f(z)-f(a)))\dz + \int_b^c \Sd'(z-a)f'(z)\dz,
\end{align*}
for $a,b,c \in \R$. By \eqref{est:SingTimesLipInt},
\begin{align*}
\abs{\mathcal{F}_\delta(a,b) - \mathcal{F}_\delta(a,c)} &\leq
\underbrace{\abs{\int_b^c \Sd''(z-a)(f(z)-f(a)))\dz}}_{\leq \norm{f}_{\mathrm{Lip}}2\delta}
+ \underbrace{\abs{ \int_b^c \Sd'(z-a)f'(z)\dz}}_{\leq \norm{f}_{\mathrm{Lip}}\abs{b-c}}.
\end{align*}
This and the symmetry of $\mathcal{F}_\delta$, i.e., $\mathcal{F}_\delta(a,b)
= \mathcal{F}_\delta(b,a)$ for $a,b \in \R$, yields condition (ii).
Hence, by Lemma~\ref{lemma:DeltaRconvergence},
\begin{align*}
\lim_{(\delta,r),\varepsilon,r_0 \downarrow 0} F_1
& = E\bigg[\,\,
\iint_{\Pi_T}\iint_{[0,1]^2}
\sign{u(t,x,\alpha)-v(t,x,\beta)}
\\ & \qquad \qquad
\times (f(u(t,x,\alpha))-f(v(t,x,\beta)))
\cdot \nabla\psi(x)\xi_{\gamma,t_0}(t)
\,d\beta d\alpha dxdt \bigg].
\end{align*}
At long last, Limit \eqref{eq:FluxLimit} follows
by sending $\gamma \downarrow 0$.
\end{proof}
\begin{limit}\label{limit:T1}
Suppose assumptions \eqref{assumption:SigmaRegularity}
and \eqref{assumption:LipOnSigma} hold.
With $\mathcal{T}_1$ defined in Lemma~\ref{lemma:DoubelingWithoutLimits},
$$
\mathcal{T}_1 = \mathcal{O}
\left(\frac{r^{2\kappa + 1}}{\delta} + \delta\right).
$$
If $\sigma$ is independent of $x$, i.e., $\sigma(x,u,z) = \sigma(u,z)$, then
$\mathcal{T}_1 = \mathcal{O}(\delta)$.
\end{limit}
\begin{proof}
By assumption \eqref{assumption:SigmaRegularity}
and \eqref{assumption:LipOnSigma},
$$
\abs{\sigma(y,v,z)-\sigma(x,\ue,z)}
\leq M(z)\abs{y-x}^\kappa(1 + \abs{\ue}) + M(z)\abs{v-\ue}.
$$
and thus
\begin{align*}
\abs{\mathcal{T}_1}
&= \frac{1}{2}\E{ \iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z
\Sd''(v-\ue)\left(\sigma(y,v,z)-\sigma(x,\ue,z)\right)^2\test\,d\mu(z)d\beta\dX} \\
& \leq \norm{M}_{L^2(Z)}^2\E{\iiiint_{\Pi_T^2}\int_{[0,1]}
\Sd''(v-\ue)\abs{x-y}^{2\kappa + 1}(1 +\abs{\ue})^2\test\,d\beta\dX} \\
&\qquad +\norm{M}_{L^2(Z)}^2\E{\iiiint_{\Pi_T^2}\int_{[0,1]}
\Sd''(v-\ue)\abs{v-\ue}^2\test\,d\beta\dX} \\
&=:\mathcal{T}_1^1 + \mathcal{T}_1^2.
\end{align*}
Since $J_r(\frac{x-y}{2}) = 0$ whenever $\abs{x-y} \geq 2r$,
$$
\mathcal{T}_1^1 \leq 4\norm{M_K}_{L^2(Z)}^2
\norm{J}_\infty\frac{r^{2\kappa + 1}}{\delta}
\E{\iiiint_{\Pi_T^2}(1 + \abs{\ue})^2\test \dX}.
$$
Moreover, as
$$
\E{\iiiint_{\Pi_T^2}(1 + \abs{\ue})^2\test \dX}
\leq \int_0^T \E{\norm{1 + \ue(t)}_{2,\psi \star J_r}^2}\,dt,
$$
there is a constant $C > 0$, independent of $r_0, \varepsilon, \delta, \gamma, r$,
such that $\mathcal{T}_1^1 \leq Cr^{2\kappa + 1}\delta^{-1}$.
Regarding the second term $\mathcal{T}_1^2$, observe that
$$
\Sd''(v-\ue)\abs{v-\ue}^2 = J_\delta(v-\ue)\abs{v-\ue}^2
\leq 2\norm{J}_\infty \delta.
$$
Hence, $\mathcal{T}_1^2 \leq \delta 2
\norm{J}_\infty\norm{M}_{L^2(Z)}^2\norm{\test}_{L^1(\Pi_T^2)}$.
Regarding the case $\sigma(x,u,z) = \sigma(u,z)$, observe that $\mathcal{T}_1^1 = 0$.
\end{proof}
Let us consider the term involving the Malliavin derivative.
\begin{limit}\label{limit:T2}
With $\mathcal{T}_2$ defined in
Lemma~\ref{lemma:DoubelingWithoutLimits},
\begin{displaymath}
\lim_{r_0 \downarrow 0}\mathcal{T}_2 = 0.
\end{displaymath}
\end{limit}
\begin{proof}
Let us split $\mathcal{T}_2$ as follows:
\begin{align*}
\mathcal{T}_2 & = E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}
\int_Z \Sd''(v-\ue(s,x))\bigg(D_{s,z}\ue(t,x)-\sigma(x,\ue(s,x),z)\bigg) \\
& \qquad\qquad\qquad \qquad \qquad \qquad\qquad\qquad\qquad
\times \sigma(y,v,z)\test\,d\mu(z)d\beta dX\Bigg] \\
&\qquad
+ E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z \bigg(\Sd''(v-\ue(t,x))-\Sd''(v-\ue(s,x))\bigg) \\
& \qquad\qquad\qquad \qquad \qquad \qquad\qquad\qquad\qquad
\times D_{s,z}\ue(t,x)\sigma(y,v,z)\test\,d\mu(z)d\beta dX\Bigg] \\
&\qquad
+ E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z \Sd''(v-\ue(s,x))
\bigg(\sigma(x,\ue(s,x),z)-\sigma(x,\ue(t,x),z)\bigg)\\
& \qquad\qquad\qquad \qquad \qquad \qquad\qquad\qquad\qquad
\times \sigma(y,v,z)\test\,d\mu(z)d\beta dX\Bigg] \\
&\qquad + E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z \bigg(\Sd''(v-\ue(s,x))-\Sd''(v-\ue(t,x))\\
& \qquad\qquad\qquad \qquad \qquad \qquad\qquad\qquad\qquad
\times \sigma(x,\ue,z)\sigma(y,v,z)\test\,d\mu(z)d\beta dX\Bigg] \\
&=: \mathcal{T}_2^1 + \mathcal{T}_2^2 + \mathcal{T}_2^3 + \mathcal{T}_2^4.
\end{align*}
Consider $\mathcal{T}_2^1$. We want to apply
Lemma~\ref{lemma:MalliavinDerivativeWeakTimeCont}
for fixed $(s,y,\beta)$ with
\begin{align*}
&\Psi_{s,y,\beta}(x,z) = \Sd''(v-\ue(s,x))\sigma(y,v,z), \\
&\phi_y(x) = \frac{1}{2^d}\psi\left(\frac{x+y}{2}\right)
J_r\left(\frac{x-y}{2}\right).
\end{align*}
Then
\begin{displaymath}
\mathcal{T}_2^1 = \iint_{\Pi_T}\int_0^1
\mathcal{T}_{r_0}(\Psi_{s,y,\beta}) \,d\beta dsdy.
\end{displaymath}
By means of Lemma \ref{lemma:MalliavinDerivativeWeakTimeCont},
$\lim_{r_0 \downarrow 0} \mathcal{T}_{r_0}(\Psi_{s,y,\beta}) = 0$
$dsdyd\beta$-a.e., and so
$\lim_{r_0 \downarrow 0}\mathcal{T}_2^1=0$
by the dominated convergence theorem. To this end, in view of
\eqref{eq:UniformBoundOnTr0}, there
exists a constant $C$, not depending on $r_0$, such that
\begin{align*}
\abs{\mathcal{T}_{r_0}(\Psi_{s,y,\beta})}^2
&\leq C^2\E{\iint_{Z \times \R^d}
\abs{\Psi_{s,y,\beta}(x,z)}^2\phi_y(x)\,dx\,d\mu(z)} \\
&\leq C^2\norm{\Sd''}_\infty^2
\E{\int_{Z}\abs{\sigma(y,v,z)}^2(\psi \star J_r)(y)\,d\mu(z)} \\
& \leq C^2\norm{\Sd''}_\infty^2
\norm{M}_{L^2(Z)}^2\E{(1 + \abs{v})^2(\psi \star J_r)(y)}.
\end{align*}
Due to the compact support of $\psi \star J_r$, we see that
$\abs{\mathcal{T}_{r_0}(\Psi_{s,y,\beta})}$ is dominated by an integrable function.
Let us consider $\mathcal{T}_2^2$. Note that
\begin{align*}
& \abs{\Sd''(v-\ue(t,x))-\Sd''(v-\ue(s,x))} \\
& \qquad
\leq \underbrace{\max \seq{2\norm{\Sd''}_\infty,\norm{\Sd''}_{\mathrm{Lip}}
\abs{\ue(t,x)-\ue(s,x)}}}_{\Psi}.
\end{align*}
By H\"older's inequality,
\begin{align*}
\mathcal{T}_2^2 &\leq E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}
\int_Z \Psi^2(s,t,x)\abs{\sigma(y,v,z)}^2
\test\,d\mu(z)d\beta dX\Bigg]^{1/2} \\
& \qquad \quad
\times E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z
\abs{D_{s,z}\ue(t,x)}^2\test\,d\mu(z)d\beta dX\Bigg]^{1/2} \\
& =: F_1 \times F_2.
\end{align*}
By the uniform boundedness of $\Psi$ we can
apply the dominated convergence theorem and
Lemma~\ref{lemma:TimeTraceMollLimit}, to conclude
that $\lim_{r_0 \downarrow 0}F_1 = 0$.
It remains to show that $\abs{F_2} \leq C$, with
$C$ independent of $r_0 > 0$. We deduce easily
\begin{align*}
F_2^2 &= \int_0^T\int_0^T\E{\norm{D_{s}\ue(t)}_{L^2(Z;L^2(\R^d,\psi \star J_r))}^2}
J_{r_0}^+(t-s)\xi_{\gamma,t_0}(t) dsdt \\
& \leq \int_0^T \sup_{t \in [0,T]}
\seq{\E{\norm{D_{s}\ue(t)}_{L^2(Z;L^2(\R^d,\psi \star J_r))}^2}}\,ds,
\end{align*}
and so $\abs{F_2}$ is uniformly bounded by \eqref{eq:supInrMallEst}.
Consider $\mathcal{T}_2^3$. By H\"older's
inequality and \eqref{assumption:LipOnSigma},
\begin{align*}
\abs{\mathcal{T}_2^3} &\leq \norm{\Sd''}_\infty \norm{M}_{L^2(Z)}
E\Bigg[\iiiint_{\Pi_T^2}\int_{[0,1]}\int_Z
\abs{\sigma(y,v,z)}^2\test\,d\mu(z)d\beta dX\Bigg]^{1/2} \\
&\quad\qquad \times
E\Bigg[\iiint\limits_{\quad \R^d \times [0,T]^2}
\abs{\ue(s,x)-\ue(t,x)}^2(\psi \star J_r)(x)J_{r_0}^+(t-s)\,dxdtds\Bigg]^{1/2}.
\end{align*}
By the dominated convergence theorem and Lemma~\ref{lemma:TimeTraceMollLimit},
$\lim_{r_0 \downarrow 0}\mathcal{T}_2^3 = 0$.
The term $\mathcal{T}_2^4$ is treated in the same manner as $\mathcal{T}_2^2$, resulting
in $\lim_{r_0 \downarrow 0}\mathcal{T}_2^4 = 0$.
\end{proof}
\begin{limit}\label{limit:T3}
With $\mathcal{T}_3$ defined in Lemma~\ref{lemma:DoubelingWithoutLimits},
$$
\mathcal{T}_3 = \mathcal{O}(\varepsilon).
$$
\end{limit}
\begin{proof}
Note that
$$
\abs{\Sd(\ue-v)\Delta_x\test} \leq (\abs{\ue} + \abs{v})\abs{\Delta_x\test}.
$$
Using this inequality, it follows from
Lemma~\ref{lemma:UniformBoundsOnVisc} that
$$
\E{\iiiint_{\Pi_T^2}\int_{[0,1]}
\Sd(\ue-v)\Delta_x\test \,d\beta dX} \leq C,
$$
for some constant $C > 0$ independent of $\varepsilon$ and $r_0$.
\end{proof}
Having established Proposition \ref{proposition:KatoInequalityEntSol}, the proof
of Theorem \ref{theorem:UniquenessOfEntSol} follows easily.
\begin{proof}[Proof of Theorem~\ref{theorem:UniquenessOfEntSol}]
In the setting of Proposition~\ref{proposition:KatoInequalityEntSol},
suppose $u^0 = v^0$. Let $\seq{\phi_R}_{R > 1}$ be as in
Lemma~\ref{lemma:NSmCompApprox} and take
$\psi = \phi_R$ in \eqref{eq:KatoInequalityEntSol}.
Exploiting that $\phi$ belongs to $\mathfrak{N}$,
sending $R \rightarrow \infty$ yields
$$
\eta(t_0) \leq
C_\phi\norm{f}_{\mathrm{Lip}}\int_0^{t_0}\eta(t)\,dt,
$$
where
$$
\eta(t) =E\Bigg[\,\,\iiint\limits_{\;\; \R^d \times [0,1]^2}
\abs{u(t,x,\alpha)-v(t,x,\beta)}\phi(x)\,d\beta d\alpha dx\Bigg].
$$
An application of Gr\"onwall's inequality gives $\eta(t) = 0$ for a.a.~$t \in [0,T]$.
Hence $u(t,x,\alpha) = v(t,x,\beta)$ $(t,x,\alpha,\beta,\omega)$-almost everywhere.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{proposition:FracBounds}]
Let $\seq{\phi_R}_{R > 1}$ be as in Lemma~\ref{lemma:NSmCompApprox}, and start off from
Lemma \ref{lemma:DoubelingWithoutLimits} with $\psi = \phi_R$ and $v^0 = u^0$.
We then compute the limits $r_0 \downarrow 0$, $\varepsilon \downarrow 0$,
and $\gamma \downarrow 0$ (in that order). Recall that by
Theorem \ref{theorem:UniquenessOfEntSol}, $v = u$ with
$u =\lim_{\varepsilon \downarrow 0}\ue$.
Furthermore, $u$ is a solution according to Definition~\ref{Def:EntropySolution}.
Due to Limits~\ref{limit:L}--\ref{limit:T3} we arrive at the inequality
\begin{equation}\label{eq:FracIneq}
\begin{split}
&E\Bigg[\,\,\iint\limits_{\,\,\,\R^d \times \R^d}
\Sd(u^0(x-z)-u^0(x + z))\phi_R(x)J_r(z)\,dxdz\Bigg] \\
&\quad
\geq E\Bigg[\,\, \iint\limits_{\,\,\, \R^d \times \R^d}
\Sd(u(t_0,x-z)-u(t_0,x+z)) \phi_R(x)J_r(z)\,dxdz\Bigg] \\
& \quad\qquad
+ E\Bigg[\,\, \int_0^{t_0}
\iint\limits_{\R^d \times \R^d}\Sd'(u(t,x+z)-u(t,x-z)) \\
&\quad\quad \qquad\qquad\qquad
\times (f(u(t,x + z))-f(u(t,x-z)))\cdot \nabla \phi_R(x)J_r(z)\,dxdzdt\Bigg] \\
&\quad\qquad\qquad
+ \mathcal{O}\left(\delta + \frac{\delta}{r}
+ \frac{r^{2\kappa + 1}}{\delta}\right),
\end{split}
\end{equation}
where $\mathcal{O}(\cdot)$ is independent
of $R$, cf.~Limits \ref{limit:F} and \ref{limit:T1}
and Lemmas \ref{lemma:ContMollWeightedNorm}
and \ref{lemma:NSmCompApprox}.
Note that
$$
\abs{\Sd(\sigma)-\abs{\sigma}}
\leq \delta, \qquad
\forall \sigma \in \R,
$$
and $\abs{\nabla \phi} \leq C_\phi\phi$. With the help of
Lemma~\ref{lemma:NSmCompApprox}, we can
now send $R \rightarrow \infty$ in \eqref{eq:FracIneq}, obtaining
$$
\eta(t_0) \leq \eta(0) + C_\phi \norm{f}_{\mathrm{Lip}}\int_0^{t_0}\eta(t)\,dt
+ \mathcal{O}\left(\delta + \frac{\delta}{r} + \frac{r^{2\kappa + 1}}{\delta}\right),
$$
where
$$
\eta(t) = E\Bigg[\,\, \iint\limits_{\,\,\,\R^d \times \R^d}
\abs{u(t,x-z)-u(t,x + z)}\phi(x)J_r(z)\,dxdz\Bigg].
$$
By Gr\"onwall's inequality,
$$
\eta(t) \leq \left(1 + C_\phi
\norm{f}_{\mathrm{Lip}}te^{C_\phi \norm{f}_{\mathrm{Lip}}t}\right)\left(\eta(0) +
\mathcal{O}\left(\delta + \frac{\delta}{r} + \frac{r^{2\kappa+1}}{\delta}\right)\right).
$$
Prescribing $\delta = r^{\kappa + 1}$ concludes the proof.
Regarding the case $\sigma(x,u,z) = \sigma(u,z)$, observe that by Limit~\ref{limit:T1} we
may replace $\mathcal{O}\left(\delta + \frac{\delta}{r} + \frac{r^{2\kappa+1}}{\delta}\right)$
by $\mathcal{O}\left(\delta + \frac{\delta}{r}\right)$ in the
above argument. The result follows by letting $\delta \downarrow 0$.
\end{proof}
\section{Appendix}\label{sec:Appendix}
\subsection{Some ``doubling of variables" tools}
\begin{lemma}\label{lemma:DeltaRconvergence}
Suppose $u,v \in L^1_{\mathrm{loc}}(\R^d)$ and $\seq{F_\delta}_{\delta > 0}$ satisfy:
\begin{itemize}
\item[(i)] There is $F: \R^2 \rightarrow \R$ such
that $F_\delta \rightarrow F$ pointwise as $\delta \downarrow 0$.
\item[(ii)] There exists a constant $C > 0$ such that
$$
\abs{F_\delta(a,b)-F_\delta(c,d)} \leq C(\abs{a-c} + \abs{b-d} + \delta),
$$
for all $a,b,c,d \in \R$ and all $\delta > 0$.
\item[(iii)] There is a constant $C > 0$ such that
$$
\abs{F_\delta(a,a)} \leq C(1 + \abs{a}) \mbox{ for all $\delta > 0$.}
$$
\end{itemize}
Fix $\psi \in C_c(\R^d)$. Suppose
$\delta:[0,\infty) \rightarrow [0,\infty)$
satisfies $\delta(r) \downarrow 0$ as
$r \downarrow 0$. Set
\begin{align*}
\mathcal{T}_r &:= \int_{\R^d}\int_{\R^d} F_{\delta(r)}(u(x),v(y))
\frac{1}{2^d}\psi\left(\frac{x+y}{2}\right)J_r\left(\frac{x-y}{2}\right)\,dydx \\
& \qquad\qquad \qquad
-\int_{\R^d} F(u(x),v(x))\psi(x)\dx.
\end{align*}
Then $\mathcal{T}_r \rightarrow 0$ as $r \downarrow 0$.
\end{lemma}
\begin{proof}
Due to Remark~\ref{remark:testFuncProp},
\begin{align*}
\mathcal{T}_r & = \int_{\R^d}\underbrace{\int_{\R^d}
F_{\delta(r)}(u(x+z),v(x-z))\psi(x)\,dx}_{g_\delta(z)}J_r(z)\,dz
\\ & \qquad\qquad
-\underbrace{\int_{\R^d} F(u(x),v(x))\psi(x)\,dx}_{g(0)}.
\end{align*}
Suppose for the moment that given a number $\varepsilon > 0$,
there exists two numbers $\eta=\eta(\varepsilon) > 0$ and
$\delta=\delta_0(\varepsilon) > 0$ such that
\begin{equation}\label{eq:gEquicontLimAt0}
\abs{g_\delta(z)-g(0)} \leq \varepsilon,
\mbox{ whenever } \abs{z} \leq \eta \mbox{ and } \delta < \delta_0.
\end{equation}
The change of variables $z = r\zeta$ yields
$$
\abs{\mathcal{T}_r} \leq \int_{\R^d}\abs{g_\delta(z)-g(0)}J_r(z)\,dz
= \int_{\R^d}\abs{g_\delta(r\zeta)-g(0)}J(\zeta)\,d\zeta.
$$
Fix $\varepsilon > 0$, and pick $\eta, \delta_0$ as
dictated by \eqref{eq:gEquicontLimAt0}. Let $r_0 > 0$ satisfy
$r_0 \leq \eta$ and $\delta(r_0) \leq \delta_0$.
It follows by \eqref{eq:gEquicontLimAt0} that
$\abs{\mathcal{T}_{r_0}} \leq \varepsilon$.
Hence, $\mathcal{T}_r \downarrow 0$ as $r \downarrow 0$.
Let us now prove \eqref{eq:gEquicontLimAt0}.
By assumption (ii),
\begin{align*}
\abs{g_\delta(z)-g(0)}
& \leq C\int_{\R^d} \abs{u(x+z)-u(x)}\psi(x)\,dx
+C\int_{\R^d} \abs{v(x-z)-v(x)}\psi(x)\,dx \\
& \quad\quad
+\int_{\R^d} \abs{F_\delta(u(x),v(x))-F(u(x),v(x))}\psi(x)\,dx
+ C\delta \norm{\psi}_{L^1(\R^d)}.
\end{align*}
Because of assumptions (i) and (iii), we can apply the
dominated convergence theorem to conclude that
$$
\lim_{\delta \downarrow 0}\int_{\R^d}
\abs{F_\delta(u(x),v(x))-F(u(x),v(x))}\psi(x)\,dx = 0.
$$
It remains to show that
\begin{equation}\label{eq:ContAt0IntTrans}
\lim_{z \rightarrow 0}\int_{\R^d}
\abs{u(x+z)-u(x)}\psi(x)\,dx = 0.
\end{equation}
The term involving $v$ follows by the same argument.
Pick a compact $K \subset \R^d$ such that $\bigcup_{\abs{z} \leq 1}
\mathrm{supp}\,(\psi(\cdot + z)) \subset K$.
Fix $\varepsilon > 0$. By the density of continuous functions in $L^1(K)$,
we can find $w \in C(K)$ such that
$\norm{w-u}_{L^1(K)} \leq \varepsilon$.
Then
$$
\int_{\R^d} \abs{u(x+z)-u(x)}\psi(x)\,dx \leq
2\norm{\psi}_\infty\varepsilon
+ \int_{\R^d}\abs{w(x+z)-w(x)}\psi(x)\,dx,
$$
for any $\abs{z} \leq 1$.
Next we send $z \rightarrow 0$.
The claim \eqref{eq:ContAt0IntTrans} follows
by the dominated convergence theorem and the
arbitrariness of $\varepsilon > 0$.
\end{proof}
\begin{lemma}\label{lemma:TimeTraceMollLimit}
Let $v \in L^p([0,T])$, $1 \leq p < \infty$. Moreover,
Let $F:[0,T] \times \R \rightarrow \R$ be measurable
in the first variable and Lipschitz in the second variable,
$$
\abs{F(s,a)-F(s,b)}
\leq C\abs{a-b}, \qquad \forall a,b \in \R, \forall s \in [0,T],
$$
for some constant $C > 0$. Set
$$
\mathcal{T}_{r_0}(s) = \left(\int_0^T
\abs{F(s,v(t))-F(s,v(s))}^pJ_{r_0}^+(t-s)\,dt\right)^{1/p}.
$$
Then $\mathcal{T}_{r_0}(s) \rightarrow 0$
$ds$-a.e.~as $r_0 \downarrow 0$.
\end{lemma}
\begin{proof}
We can write $v = v_1^n + v_2^n$ with $v_1^n$
continuous and $\norm{v_2^n}_{L^p([0,T])} \leq 1/n$.
This is possible since the continuous functions are dense in $L^p([0,T])$.
Assuming $s\in [0, T-2r_0]$, an application of
the triangle inequality gives
\begin{align*}
\abs{\mathcal{T}_{r_0}(s)} & \leq
C\left(\int_0^T \abs{v(t)- v(s)}^pJ_{r_0}^+(t-s)\,dt\right)^{1/p} \\
& \leq C\left(\int_0^T \abs{v_1^n(t)-v_1^n(s)}^pJ_{r_0}^+(t-s)\,dt\right)^{1/p} \\
& \qquad\qquad
+ (\abs{v_2^n}^p \star J_{r_0}(s))^{1/p} + \abs{v_2^n(s)}.
\end{align*}
Sending $r_0 \downarrow 0$, it follows that
$\lim_{r_0 \downarrow 0}\abs{\mathcal{T}_{r_0}(s)}
\leq 2\abs{v_2^n(s)}$ for $ds$-a.a.~$s \in [0,T)$. Since $v_2^n \rightarrow 0$
in $L^p([0,T])$, it has a subsequence that
converges $ds$-a.e., and this concludes the proof.
\end{proof}
\subsection{Weighted $L^p$ spaces.}
First we make some elementary observations regarding
functions in $\mathfrak{N}$ (see Section~\ref{sec:Entropy_Formulation}
for the definition of $\mathfrak{N}$).
\begin{lemma}\label{lemma:PhiProp}
Suppose $\phi \in \mathfrak{N}$ and $0 < p < \infty$.
Then, for $x,z\in \R^d$,
$$
\abs{\phi^{1/p}(x+z)-\phi^{1/p}(x)}
\leq w_{p,\phi}(\abs{z})\phi^{1/p}(x),
$$
where
$$
w_{p,\phi}(r) = \frac{C_\phi}{p}r\left(1 + \frac{C_\phi}{p}r e^{C_\phi r/p}\right),
$$
which is defined for all $r \geq 0$. As a consequence it follows that if
$\phi(x_0) = 0$ for some $x_0 \in \R^d$, then $\phi \equiv 0$ (and by
definition $\phi \notin \mathfrak{N}$).
\end{lemma}
\begin{proof}
Set $g(\lambda) = \phi^{1/p}(x + \lambda z)$. Then
$$
g'(\lambda) = \frac{1}{p}\phi^{1/p-1}(x + \lambda z)
(\nabla \phi(x + \lambda z) \cdot z).
$$
Since $\phi \in \mathfrak{N}$, it follows that
$\abs{g'(\lambda)} \leq \frac{C_\phi}{p}g(\lambda)\abs{z}$. Hence
$$
g(\lambda) \leq g(0) + \frac{C_\phi}{p}
\abs{z}\int_0^\lambda g(\xi)\,d\xi.
$$
By Gr\"onwall's inequality,
$$
g(\lambda) \leq g(0)\left(1 + \frac{C_\phi}{p}\abs{z}
\lambda e^{C_\phi\abs{z}\lambda/p}\right).
$$
Hence,
$$
\abs{g(1)-g(0)} \leq \frac{C_\phi}{p}\abs{z}g(0)
\left(1 + \frac{C_\phi}{p}\abs{z} e^{C_\phi\abs{z}/p}\right).
$$
This concludes the proof.
\end{proof}
Next, we consider an adaption of Young's inequality for convolutions.
\begin{proposition}\label{prop:YoungsForLocalized}
Fix $\phi \in \mathfrak{N}$.
Suppose $f \in C_c(\R^d)$, and
$g \in L^p(\R^d,\phi)$ for some finite $p\ge 1$. Then
$$
\norm{f \star g}_{L^p(\R^d,\phi)} \leq
\left(\int_{\R^d}\abs{f(x)}(1 + w_{p,\phi}(\abs{x}))\,dx \right)
\norm{g}_{L^p(\R^d,\phi)}.
$$
where $w_{p,\phi}$ is defined in Lemma \ref{lemma:PhiProp}.
\end{proposition}
\begin{proof}
First observe that
\begin{align*}
\norm{f \star g}_{L^p(\R^d,\phi)}^p
&= \int_{\R^d}\abs{\int_{\R^d} f(x-y)g(y)\,dy}^p \phi(x)\,dx \\
&\leq \int_{\R^d}\left(\int_{\R^d} \abs{f(x-y)}
\left(\frac{\phi(x)}{\phi(y)}\right)^{1/p}\abs{g(y)}\phi^{1/p}(y)\,dy\right)^p \,dx.
\end{align*}
By Lemma~\ref{lemma:PhiProp}(iii),
$$
\left(\frac{\phi(x)}{\phi(y)}\right)^{1/p} \leq
\frac{1}{\phi^{1/p}(y)}\left(\phi^{1/p}(y) + \abs{\phi^{1/p}(x)-\phi^{1/p}(y)}\right)
\leq \left(1 + w_{p,\phi}(\abs{x-y})\right).
$$
Set
$$
\zeta(x) := \abs{f(x)}(1 + w_{p,\phi}(\abs{x})),
\qquad
\xi(x) := \abs{g(x)}\phi^{1/p}(x).
$$
Then, by Young's inequality for convolutions,
$$
\norm{f \star g}_{L^p(\R^d,\phi)} \leq \norm{\zeta \star \xi}_{L^p(\R^d)}
\leq \norm{\zeta}_{L^1(\R^d)}\norm{\xi}_{L^p(\R^d)}.
$$
\end{proof}
\begin{lemma}\label{lemma:ContMollWeightedNorm}
Fix $\phi \in \mathfrak{N}$, and let $w_{p,\phi}$ be defined in
Lemma \ref{lemma:PhiProp}. Let $J$ be a mollifier as defined
in Section~\ref{sec:Entropy_Formulation} and
take $\phi_\delta = \phi \star J_\delta$ for $\delta > 0$. Then
\begin{itemize}
\item[(i)] $\phi_\delta \in \mathfrak{N}$ with $C_{\phi_\delta} = C_\phi$.
\item[(ii)] For any $u \in L^p(\R^d,\phi)$,
$$
\abs{\norm{u}_{p,\phi}^p-\norm{u}_{p,\phi_\delta}^p}
\leq w_{1,\phi}(\delta)\min\seq{\norm{u}_{p,\phi}^p,\norm{u}_{p,\phi_\delta}^p}.
$$
\item[(iii)]
\begin{displaymath}
\abs{\Delta \phi_\delta(x)} \leq \frac{1}{\delta}
C_\phi\norm{\nabla J}_{L^1(\R^d)}(1 + w_{1,\phi}(\delta))^2\phi_\delta(x).
\end{displaymath}
\end{itemize}
\end{lemma}
\begin{proof}
Consider (i). Young's inequality for convolutions
yields $\phi_\delta \in L^1(\R^d)$. Furthermore,
$$
\abs{\nabla(\phi \star J_\delta)(x)}
= \abs{\int_{\R^d} J_\delta(y)\nabla\phi(x-y)\,dy}
\leq C_\phi (\phi \star J_\delta)(x).
$$
Consider (ii). By Lemma \ref{lemma:PhiProp},
\begin{align*}
\abs{\norm{u}_{p,\phi}^p -\norm{u}_{p,\phi_\delta}^p}
&= \abs{\int_{\R^d}\int_{\R^d} \abs{u(x)}^p(\phi(x-z)-\phi(x))
J_\delta(z)\,dzdx} \\
&\leq \min\seq{\norm{u}_{p,\phi}^p,\norm{u}_{p,\phi_\delta}^p}
\int_{\R^d}w_{1,\phi}(\abs{z})J_\delta(z)\,dz.
\end{align*}
This proves (ii). Consider(iii). Integration by parts yields
\begin{displaymath}
\abs{\Delta (\phi_\delta)(x)} =
\abs{\int_{\R^d}\nabla J_\delta(x-y) \cdot \nabla \phi (y)\,dy}
\leq C_\phi\int_{\R^d}\abs{\nabla J_\delta(x-y)}\phi (y)\,dy.
\end{displaymath}
By Lemma~\ref{lemma:PhiProp},
\begin{align*}
\int_{\R^d}\abs{\nabla J_\delta(x-y)}\phi (y)\,dy
&\leq \left(\int_{\R^d}\abs{\nabla J_\delta(x-y)}(1 + w_{1,\phi}(\abs{x-y}))\,dy\right)\phi(x) \\
&\leq \frac{1}{\delta}\norm{\nabla J}_{L^1(\R^d)}(1 + w_{1,\phi}(\delta))\phi(x).
\end{align*}
Again, by Lemma~\ref{lemma:PhiProp}
\begin{displaymath}
\phi(x) \leq \abs{\phi(x)-\phi_\delta(x)} + \phi_\delta(x)
\leq (1 + w_{1,\phi}(\delta))\phi_\delta(x).
\end{displaymath}
The result follows.
\end{proof}
\begin{lemma}\label{lemma:NSmCompApprox}
Let $\phi \in \mathfrak{N}$. Then there exists
$\seq{\phi_R}_{R > 1} \subset C^\infty_c(\R^d)$ such that
\begin{itemize}
\item[(i)] $\phi_R \rightarrow \phi$ and $\nabla \phi_R \rightarrow \nabla \phi$
pointwise in $\R^d$ as $R \rightarrow \infty$,
\item[(ii)] $\exists$ a constant $C$ independent of $R > 1$ such that
$$
\max \seq{\norm{\phi_R}_{\infty,\phi^{-1}},
\norm{\nabla \phi_R}_{\infty,\phi^{-1}}} \leq C.
$$
\end{itemize}
\end{lemma}
\begin{proof}
Modulo a mollification step, we may assume $\phi \in C^\infty$.
Let $\zeta \in C_c^\infty(\R^d)$ satisfy $0 \leq \zeta \leq 1$, $\zeta(0) =1$.
Let $\phi_R(x) := \phi(x)\zeta(R^{-1}x)$. Then
$$
\nabla \phi_R(x) = \nabla \phi(x)
\zeta(R^{-1}x) + R^{-1}\phi(x)\nabla \zeta(R^{-1}x).
$$
Hence (i) follows. Clearly, $\norm{\phi_R}_{\infty,\phi^{-1}}
= \sup_x \seq{\abs{\phi_R(x)}\phi^{-1}(x)} = \norm{\zeta}_\infty$.
Furthermore,
$$
\abs{\nabla \phi_R(x)} \leq
\left(C_\phi\zeta(R^{-1}x) + R^{-1}\abs{\nabla \zeta(R^{-1}x)}\right)\phi(x).
$$
Hence, $\norm{\nabla \phi_R}_{\infty,\phi^{-1}}
\leq C_\phi + R^{-1}\norm{\nabla \zeta}_\infty$.
\end{proof}
\subsection{A version of It\^{o}'s formula}
Here we establish the particular
anticipating It\^{o} formula applied in
the proof of Theorem \ref{theorem:ExistenceOfSolution}.
\begin{theorem}\label{theorem:AntIto}
Let
$$
X(t) = X_0 + \int_0^t\int_Z u(s,z)\,W(dz,ds)
+ \int_0^t v(s)\,ds,
$$
where $u:[0,T] \times Z \times \Omega \rightarrow \R$ and
$v:[0,T] \times \Omega \rightarrow \R$ are jointly measurable
and $\seq{\F_t}$-adapted processes, satisfying
\begin{equation}\label{eq:AssumptionItoProcess}
\E{\bigg(\int_0^T\int_Z u^2(s,z)\,d\mu(z)ds\bigg)^2} < \infty,
\qquad \E{\int_0^T v^2(s)\,ds} < \infty.
\end{equation}
Let $F:\R^2 \times [0,T] \rightarrow \R$ be twice continuously
differentiable. Suppose there exists a
constant $C > 0$ such that for all
$(\zeta,\lambda,t) \in \R^2 \times [0,T]$,
\begin{align*}
& \abs{F(\zeta,\lambda,t)},\abs{\partial_3F(\zeta,\lambda,t)}
\leq C(1 + \abs{\zeta} + \abs{\lambda}), \\
& \abs{\partial_1F(\zeta,\lambda,t)},
\abs{\partial_{1,2}^2F(\zeta,\lambda,t)},
\abs{\partial_1^2F(\zeta,\lambda,t)} \leq C.
\end{align*}
Let $V \in \Sm$. Then $s \mapsto \partial_1F(X(s),V,s)u(s)$ is
Skorohod integrable, and
\begin{align*}
F(X(t),V,t) &= F(X_0,V,0) \\
&\qquad +\int_0^t\partial_3F(X(s),V,s)\,ds \\
&\qquad +\int_0^t\int_Z\partial_1F(X(s),V,s)u(s,z)\,W(dz,ds) \\
&\qquad +\int_0^t\partial_1F(X(s),V,s)v(s)\,ds \\
&\qquad +\int_0^t\int_Z \partial_{1,2}^2F(X(s),V,s)D_{s,z}Vu(s,z)\,d\mu(z)ds \\
&\qquad +\frac{1}{2}\int_0^t\int_Z \partial_1^2F(X(s),V,s)u^2(s,z)\,d\mu(z)ds,
\quad \text{$dP$-almost surely}.
\end{align*}
\end{theorem}
\begin{proof}
The proof follows \cite[Theorem~3.2.2 and Proposition~1.2.5]{NualartMalliavinCalc2006}.
We give an outline and some details where there are
considerable differences. Furthermore, we assume that $F$ is
independent of $t$ as this is a standard modification.
Set $t_i^n = \frac{it}{2^n}$, $0 \leq i \leq 2^n$.
By Taylor's formula,
\begin{multline*}
F(X(t),V) = F(X_0,V)
+ \underbrace{\sum_{i=0}^{2^n-1}
\partial_1F(X(t_i^n),V)(X(t_{i+1}^n)-X(t_i^n))}_{\mathcal{T}^1_n} \\
+ \underbrace{\frac{1}{2}\sum_{i=0}^{2^n-1}
\partial_1^2F(\overline{X}_i,V)(X(t_{i+1}^n)-X(t_i^n))^2}_{\mathcal{T}^2_n},
\end{multline*}
where $\overline{X}_i$ denotes a random intermediate point
between $X(t_i^n)$ and $X(t_{i+1}^n)$.
As in the proof of \cite[Proposition~1.2.5]{NualartMalliavinCalc2006},
$$
\mathcal{T}^2_n \rightarrow \frac{1}{2}\int_0^t\int_Z
\partial_1^2F(X(s),V)u^2(s,z)\,d\mu(z)ds,
\quad
\text{in $L^1(\Omega)$ as $n \rightarrow \infty$.}
$$
Note that
\begin{multline*}
\mathcal{T}^1_n = \underbrace{\sum_{i=0}^{2^n-1}
\partial_1F(X(t_i^n),V)\int_{t_i^n}^{t_{i+1}^n}
\int_Z u(s,z)\,W(dz,ds)}_{\mathcal{T}^{1,1}_n} \\
+ \underbrace{\sum_{i=0}^{2^n-1} \partial_1F(X(t_i^n),V)
\int_{t_i^n}^{t_{i+1}^n}v(s)\,ds}_{\mathcal{T}^{1,2}_n}.
\end{multline*}
Clearly,
$$
\mathcal{T}^{1,2}_n \rightarrow \int_0^t\partial_1F(X(s),V)v(s)\,ds,
\quad
\text{in $L^1(\Omega)$ as $n \rightarrow \infty$.}
$$
Consider $\mathcal{T}^{1,1}_n$.
By \cite[Proposition~1.3.5]{NualartMalliavinCalc2006},
$s \mapsto \partial_1F(X(t_i^n),V)u(s)$
is Skorohod integrable on $[t_i^n,t_{i+1}^n]$ and
\begin{multline*}
\mathcal{T}^{1,1}_n =
\underbrace{\sum_{i=0}^{2^n-1}\int_{t_i^n}^{t_{i+1}^n}
\int_Z \partial_1F(X(t_i^n),V)u(s,z)\,W(dz,ds)}_{\mathcal{T}^{1,1,1}_n} \\
+ \underbrace{\sum_{i=0}^{2^n-1} \int_{t_i^n}^{t_{i+1}^n}
\int_Z\partial_{1,2}^2F(X(t_i^n),V)
D_{s,z}Vu(s,z)\,d\mu(z)ds}_{\mathcal{T}^{1,1,2}_n}.
\end{multline*}
As before
$$
\mathcal{T}^{1,1,2}_n
\rightarrow \int_0^t\int_Z \partial_{1,2}^2F(X(s),V)D_{s,z}Vu(s,z)\,d\mu(z)ds,
\quad
\text{in $L^1(\Omega)$ as $n \rightarrow \infty$.}
$$
Consider $\mathcal{T}^{1,1,1}_n$. Let
$$
\zeta_n(s,z) = \sum_{i=0}^{2^n-1}\partial_{1,2}^2F(X(t_i^n),V)
\car{[t_i^n,t_{i+1}^n)}(s)D_{s,z}Vu(s,z),
$$
and note that $\zeta_n$ is Skorohod integrable on $[0,t]$.
We need to show the following:
\begin{itemize}
\item[(i)] There exists $\zeta \in L^2(\Omega;H)$
such that $\zeta_n \rightarrow \zeta$ in $L^2(\Omega;H)$.
\item[(ii)] There exists a $G \in L^2(\Omega)$
such that for each $U \in \Sm$
$$
\E{\int_0^t\int_Z \zeta_n(s,z)W(dz,ds)U} \rightarrow \E{GU}.
$$
\end{itemize}
Then we may conclude by \cite[Proposition~1.3.6]{NualartMalliavinCalc2006}
that $\zeta$ is Skorohod integrable and $\int_0^t \zeta(s)\,dW(s) = G$.
The result then follows. Consider (i). Let
$$
\zeta(s,z) = \partial_{1,2}^2F(X(s),V)D_{s,z}Vu(s,z).
$$
Then
$$
\E{\int_0^t\int_Z \abs{\zeta_n(s)-\zeta(s)}^2\,d\mu(z)ds}
\leq \E{H_n\int_0^t\int_Z \abs{D_{s,z}Vu(s,z)}^2\,d\mu(z)ds},
$$
where
$$
H_n = \sup_{\abs{t_i^n-s} \leq t2^{-n}}
\seq{\abs{\partial_{1,2}^2F(X(t_i^n),V)-\partial_{1,2}^2F(X(s),V)}^2}.
$$
Hence, (i) follows by the dominated convergence theorem. Consider (ii).
The existence of a random variable $G$ follows by the convergence of the other terms.
This also yields the weak convergence. It remains to check that
$G \in L^2(\Omega)$. This is a consequence
of assumptions \eqref{eq:AssumptionItoProcess}.
\end{proof}
\subsection{The Lebesgue-Bochner space}\label{sec:LebBoch}
Let $(X,\mathscr{A},\mu)$ be a $\sigma$-finite measure space
and $E$ a Banach space. In the previous sections
$X = [0,T] \times \Omega$, $\mu = dt \otimes dP$, $E$ is
typically $L^p(\R^d,\phi)$ for some $1 \leq p < \infty$, and $\mathscr{A}$ is the
predictable $\sigma$-algebra $\Pred$. A function $u:X \rightarrow E$
is \emph{strongly $\mu$-measurable} if there exists a sequence
of $\mu$-simple functions $\seq{u_n}_{n \geq 1}$ such that
$u_n \rightarrow u$ $\mu$-almost everywhere.
By a $\mu$-\emph{simple function} $s:X \rightarrow E$
we mean a function of the form
\begin{displaymath}
s(\zeta) = \sum_{k = 1}^N \car{A_k}(\zeta)x_k, \qquad \zeta \in X,
\end{displaymath}
where $x_k \in E$ and $A_k \in \mathscr{A}$ satisfy
$\mu(A_k)< \infty$ for all $1 \leq k \leq N$.
The Lebesgue-Bochner space $L^p(X,\mathscr{A},\mu;E)$ is the
linear space of $\mu$-equivalence classes of strongly
measurable functions $u:X \rightarrow E$ satisfying
\begin{displaymath}
\int_X \norm{u(\xi)}_E^p \,d\mu(\xi) < \infty.
\end{displaymath}
A map $u:X \rightarrow E$ is \emph{weakly $\mu$-measurable} if
the map $\xi \mapsto \inner{u(\xi)}{\test^*}$ has a $\mu$-version which
is $\mathscr{A}$-measurable for each $\test^*$ in the dual space $E^*$.
By the Pettis measurability theorem \cite[Theorem~1.11]{Neerven2007}, strong
$\mu$-measurability is equivalent to weak
$\mu$-measurability, whenever $E$ is separable.
For $u \in L^1(X,\mathscr{A},\mu;L^1(\R^d,\phi))$, it is convenient
to know that $\zeta \mapsto u(\zeta)(x)$ has a $\mu$-version which
is $\mathscr{A}$-measurable for almost all $x$. In fact this is crucial to
the manipulations performed in the previous sections.
The following result verifies that this is indeed the case.
\begin{lemma}\label{lemma:LebBochRepr}
Let $(X,\mathscr{A},\mu)$ be a $\sigma$-finite measure space
and $\phi \in \mathfrak{N}$. Let
\begin{displaymath}
\Psi:L^1(X \times \R^d,\mathscr{A} \otimes \Borel{\R^d},d\mu \otimes d\phi)
\rightarrow L^1(X,\mathscr{A},\mu;L^1(\R^d,\phi))
\end{displaymath}
be defined by $\Psi(u)(\xi)=u(\xi,\cdot)$. Then $\Psi$ is an isometric isomorphism.
\end{lemma}
\begin{remark}
The measure space $(X \times \R^d,\mathscr{A} \otimes \Borel{\R^d},d\mu \otimes d\phi)$
is not necessarily complete. Strictly speaking we should rather consider its completion.
What this ensures is that every representative is measurable
with respect to the complete $\sigma$-algebra. A remedy is to define
$L^1(X \times \R^d,\mathscr{A} \otimes \Borel{\R^d},d\mu \otimes d\phi)$ by
asking that any element $u$ has a $d\mu \otimes d\phi$-version $\tilde{u}$
which is $\mathscr{A} \otimes \Borel{\R^d}$-measurable. Now, $\tilde{u}(\cdot,x)$ is
$\mathscr{A}$ measurable, and so for $d\phi$-almost all $x$, $u(\cdot,x)$
has a $\mu$-version which is $\mathscr{A}$-measurable.
\end{remark}
\begin{proof}
Let us first check that $\Psi(u) \in L^1(X;L^1(\R^d,\phi))$.
By the Pettis measurability theorem \cite[Theorem~1.11]{Neerven2007}, strong
$\mu$-measurability follows due to the separability of $L^1(\R^d,\phi)$ if $\Psi(u)$ is
weakly $\mu$-measurable. That is, for any $\test \in L^\infty(\R^d,\phi)$, the map
\begin{displaymath}
\xi \mapsto \int_{\R^d} \test(x)u(\xi,x)\phi(x)\,dx,
\end{displaymath}
has a $\mu$-version which is $\mathscr{A}$ measurable. This is a consequence
of Fubini's theorem \cite[Proposition~5.2.2]{Cohn2013}. The fact that $\Psi$ is an
isometry is obvious. It remains to prove that $\Psi$ is surjective.
Let $v \in L^1(X;L^1(\R^d,\phi))$. By definition there exists
a sequence $\seq{v_n}_{n \geq 1}$ of simple
functions such that $v_n \rightarrow v$ $\mu$-almost everywhere. Set
\begin{displaymath}
v_n(\xi) = \sum_{k=1}^{N_n} \car{A_{k,n}}(\xi)f_{k,n}, \quad u_n(\xi,x) = v_n(\xi)(x),
\end{displaymath}
where $A_{k,n} \in \mathscr{A}$, $f_{k,n} \in L^1(\R^d,\phi)$. Note that
$u_n$ is $\mathscr{A} \otimes \Borel{\R^d}$ measurable, and $\Psi(u_n) = v_n$.
By the Lebesgue dominated convergence theorem, $v_n \rightarrow v$
in $L^1(X,L^1(\R^d,\phi))$ \cite[Proposition~1.16]{Neerven2007}.
By the isometry property, $\seq{u_n}_{n \geq 1}$ is Cauchy, and so by
completeness there exists $u$ such that
\begin{displaymath}
u_n \rightarrow u \mbox{ in } L^1(X \times \R^d,\mathscr{A}
\otimes \Borel{\R^d},d\mu \otimes d\phi).
\end{displaymath}
Since
\begin{align*}
\int_X \norm{v-\Psi(u)}_{1,\phi}\,d\mu
&= \lim_{n \rightarrow \infty}\int_X \norm{v_n-\Psi(u)}_{1,\phi}\,d\mu \\
&= \lim_{n \rightarrow \infty}\iint_{X \times \R^d}
\abs{u_n(\xi,x)-u(\xi,x)}\, d\mu \otimes d\phi(\xi,x) = 0,
\end{align*}
it follows that $\Psi(u) = v$.
\end{proof}
\subsection{Young measures}\label{sec:YoungMeasures}
The purpose of this subsection is to provide a
reference for some results concerning Young measures
and their application as generalized limits.
Let $(X,\mathscr{A},\mu)$ be a $\sigma$-finite measure space,
and $\mathscr{P}(\R)$ denote the set of probability measures on $\R$. In the
previous sections $X$ is typically $\Pi_T \times \Omega$.
A \emph{Young measure} from $X$ into $\R$ is a function
$\nu:X \rightarrow \mathscr{P}(\R)$ such that $x \mapsto \nu_x(B)$ is
$\mathscr{A}$-measurable for every Borel measurable set $B \subset \R$.
We denote by $\Young{X,\mathscr{A},\mu;\R}$, or simply $\Young{X;\R}$ if the
measure space is understood, the set of all Young measures from $X$ into $\R$.
The following theorem is proved in \cite[Theorem~6.2]{Pedregal1997} in the
case that $X \subset \R^n$ and $\mu$ is the Lebesgue measure:
\begin{theorem}\label{theorem:YoungMeasureLimitOfComposedFunc}
Let $(X,\mathscr{A},\mu)$ be a $\sigma$-finite measure space.
Let $\zeta:[0,\infty) \rightarrow [0,\infty]$ be a continuous, nondecreasing
function satisfying $\lim_{\xi \rightarrow \infty}\zeta(\xi) = \infty$ and
$\seq{u^n}_{n \geq 1}$ a sequence of measurable functions such that
$$
\sup_{n} \int_X \zeta(\abs{u^n})d\mu(x) < \infty.
$$
Then there exist a subsequence $\seq{u^{n_j}}_{j \geq 1}$
and $\nu \in \Young{X,\mathscr{A},\mu;\R}$ such that for
any Carath\'eodory function $\psi:\R \times X \rightarrow \R$
with $\psi(u^{n_j}(\cdot),\cdot) \rightharpoonup \overline{\psi}$ (weakly)
in $L^1(X)$, we have
$$
\overline{\psi}(x) = \int_{\R} \psi(\xi,x)\,d\nu_x(\xi).
$$
\end{theorem}
Recall that $\psi:\R \times X \rightarrow \R$ is a Carath\'eodory function
if $\psi(\cdot,x):\R \rightarrow \R$ is continuous for all $x \in X$
and $\psi(u,\cdot):X \rightarrow \R$
is measurable for all $u \in \R$. The proof is based on the embedding
of $\Young{X;\R}$ into $L^\infty_{w*}(X,\Rad{\R})$.
Here $\Rad{\R}$ denotes the space of Radon
measures on $\R$ and $L^\infty_{w*}(X,\Rad{\R})$ denotes the
space of weak$*$-measurable bounded maps $\nu:X \rightarrow \Rad{\R}$.
The crucial observation is that $(L^1(X,C_0(\R)))^*$ is isometrically isomorphic
to $L^\infty_{w*}(X,\Rad{\R})$ also in the case that
$(X,\mathscr{A},\mu)$ is an abstract $\sigma$-finite measure space.
It is relatively straightforward to go through the proof and
extend to this more general case \cite[Theorem~2.11]{Malek1996}.
Note however that the use of weighted $L^p$ spaces
allows us to stick with the version
for finite measure spaces.
\subsection{Weak compactness in $L^1$.}
To apply Theorem~\ref{theorem:YoungMeasureLimitOfComposedFunc} one
must first be able to extract from
$\seq{\psi(u^n(\cdot),\cdot)}_{n \geq 1}$ a weakly
convergent subsequence in $L^1(X)$.
The key result is the Dunford-Pettis Theorem.
\begin{definition}\label{def:equiintegrability}
Let $\mathcal{K} \subset L^1(X,\mathscr{A},\mu)$.
\begin{itemize}
\item[(i)] $\mathcal{K}$ is \emph{uniformly integrable}
if for any $\varepsilon > 0$ there
exists $c_0(\varepsilon)$ such that
$$
\sup_{f \in \mathcal{K}} \int_{\abs{f} \geq c} \abs{f} \,d\mu
< \varepsilon,
\quad \mbox{whenever $c > c_0(\varepsilon)$.}
$$
\item[(ii)] $\mathcal{K}$ has \emph{uniform tail} if for
any $\varepsilon > 0$ there exists
$E \in \mathscr{A}$ with $\mu(E) < \infty$ such that
$$
\sup_{f \in \mathcal{K}}\int_{X \setminus E} \abs{f} \,d\mu
<\varepsilon.
$$
\end{itemize}
If $\mathcal{K}$ satisfies both (i) and (ii) it is
said to be \emph{equiintegrable}.
\end{definition}
\begin{remark}
Note that (ii) is void when $\mu$ is finite. As a consequence
uniform integrability and equiintegrability are equivalent for finite measure spaces.
\end{remark}
\begin{theorem}[Dunford-Pettis]\label{theorem:DunfordPettis}
Let $(X,\mathscr{A},\mu)$ be a $\sigma$-finite measure space.
A subset $\mathcal{K}$ of $L^1(X)$ is
relatively weakly sequentially compact
if and only if it is equiintegrable.
\end{theorem}
By the Eberlain-\v{S}mulian theorem \cite{Whitley1967}, in the
weak topology of a Banach space, relative weak compactness
is equivalent with relative sequentially weak compactness.
There are a couple of well known reformulations of uniform integrability.
\begin{lemma}\label{lemma:UniformIntCriteria}
Suppose $\mathcal{K} \subset L^1(X)$ is bounded.
Then $\mathcal{K}$ is uniformly integrable if and only if:
\begin{itemize}
\item[(i)] For any $\varepsilon > 0$ there exists
$\delta(\varepsilon) > 0$ such that
$$
\sup_{f \in \mathcal{K}}\int_E \abs{f} \,d\mu <
\varepsilon, \quad
\mbox{whenever $\mu(E) <\delta(\varepsilon)$.}
$$
\item[(ii)] There is an increasing function
$\Psi:[0,\infty) \rightarrow [0,\infty)$
such that $\Psi(\zeta)/\zeta \rightarrow \infty$
as $\zeta \rightarrow \infty$ and
$$
\sup_{f \in \mathcal{K}}\int_X \Psi(\abs{f(x)}) \,d\mu(x) < \infty.
$$
\end{itemize}
\end{lemma}
\begin{remark}\label{remark:DominatedFamilyUniformInt}
Suppose there exists $g \in L^1(X)$ such
that $\abs{f} \leq g$ for all $f \in \mathcal{K}$. Then
$$
\sup_{f \in \mathcal{K}}\int_E \abs{f} \,d\mu \leq \int_E g \,d\mu,
\qquad \forall E \in \mathscr{A}.
$$
Since $\seq{g} \subset L^1(X)$ is uniformly
integrable, it follows by Lemma~\ref{lemma:UniformIntCriteria}(i)
that $\mathcal{K}$ is uniformly integrable.
\end{remark}
|
1,108,101,565,417 | arxiv | \section{Introduction}
High resolution spectroscopy has been employed to infer the chemistry and history of populations across the Milky Way. Advances in infrared detector technology and instrumentation have recently enabled large scale studies directed toward the central parts of the Galaxy - the bulge and nuclear star cluster- where extinction and crowding are both high. We refer in particular to the inner $\pm2^\circ$ of the Galactic plane that has escaped many detailed investigations due to extreme optical extinction caused by dust in the line of sight. It is easy to forget that for some of the red giants in the central cluster, SgrA* is closer than Proxima Centauri is to the Sun. How does such a unique environment affect chemical evolution?
We are engaged in a long-term project employing high resolution infrared spectroscopy to lift the reddening veil and to explore the Galactic bulge and center in the near-IR. For an accurate abundance determination, we need high resolution, high S/N spectra of red giants, which we can achieve using K-band spectroscopy observed at KECK with the NIRSPEC spectrometer \citep{nirspec} and the VLT with the CRIRES spectrometer \citep{crires1}. We also need a line list at least as good in the near-IR as is available in the optical. It is now possible to derive abundances for the light elements as well as iron, for red giants in the Nuclear Star Cluster at the center of the Galaxy \citep[see, e.g. Figure \ref{fig1} and][]{ryde:16}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[angle=-90,width=5.5in]{spectra.pdf}
\caption{Example of a spectrum of a Galactic Center giant observed with KECK/NIRSPEC at a spectral resolution of $R=23\,000$ for an integration time of 40 minutes, achieving SNR = 100. Some atomic lines are marked. The brown bands show the regions with good CN and CO lines for the simultaneous fit of the C and N abundances. The yellow bands show good continuum bands for the normalization of the spectra. This M giant ($K_S = 11.0$) has determined stellar parameters of $T_\mathrm{eff}=3900\,\mathrm{K}$, $\log(g)=1.5$, and [Fe/H]$=+0.1$ (for details see Rich et al. 2018.).
}
\label{fig1}
\end{center}
\end{figure}
\section{The stellar metallicity distribution and $\alpha$-element trends in the Bulge}
The metallicity distribution of the stars in the bulge are shown to be wide, from approximately [Fe/H]$=-1.5$ to $+0.5$ or more \citep[see e.g.][]{johnson:13,schultheis:15,rojas:17}. The question of the bimodality of this distribution is still debated; although gaussian mixture models applied to the abundance distribution support bimodality, corollary support for bimodality from other physical parameters e.g. kinematics and composition is not robust \citep[e.g.][]{johnson:14,rojas:17}. The composition trends appear similar to that of the thick disk, but significant trends differ \citep[]{johnson:14} and of course, the thick disk never reaches [Fe/H]=+0.5 as seen in the bulge. A vertical metallicity gradient, with increasing metallicity with decreasing Galactic latitude is generally found down to approximately 500 pc. At distances of 140-400 pc from the nucleus, \citet{rich:12} find no abundance gradient with $\langle \rm [Fe/H] \rangle=-0.2$. The alpha-element trends are, however, found to be similar in all fields all the way into the center \citep{johnson:11,ryde:16}.
The Nuclear Star Cluster, is of special interest as it is projected within the sphere of influence of Sgr A$^*$, surrounded by the Bulge. While the inner bulge is predominantly old, the Cluster also has ongoing star-formation. It can therefore not be assumed that these populations should be of the same origin. There are several formation routes which can be tested by determining the metallicities of the stars in the Cluster. One of our goals is to connect old stars in Galactic Center and in the Bulge, which can be done with a homogeneous data set, using same analysis technique.
\section{Near-IR Spectroscopy - Challenges}
Since stars in the inner Bulge and in the Nuclear Cluster are faint and cool, the spectra are difficult to analyze. The cooler the star, the greater the impact of molecular absorption on the entirety of the spectrum and the risk of unaccounted blending of molecular and atomic lines, even at high dispersion. Further, the K-band is a relatively new wavelength region being used for stellar abundance analysis, meaning that work has to be put into developing the methods to analyze the spectra; a final challenge is that many lines of the light elements of interest are present but too strong for abundance analysis. A careful analysis is required and the observations have to be optimized to minimize systematics. High spectral resolution is needed for accurately determining the composition of these bulge giants.
Elements that have spectral lines appropriate for an abundance analysis are Fe, C, N, F, Sc, Na, Al, and the alpha elements Mg, Si, S, Ca, and Ti. A new atomic line list is needed (Thorsbro et al., these proceedings) but the important CN lines are well described with the list provided in \citet{sneden:14}.
In the near-IR, spectral lines saturate at a smaller line depth than at shorter wavelengths. Strong lines should therefore be avoided since they get increasingly abundance insensitive, and are increasingly sensitive to the largely unknown microturbulence parameter. Furthermore, the cores can be sensitive to non-LTE effects, such as scattering, and they might probe the outer layers of the atmosphere where the physics is uncertain. In low-resolution spectra, and especially at low signal-to-noise levels, one might be tempted to use these strong lines.
\section{Results}
After analysing VLT/CRIRES spectra of fields at latitudes $b=0$, $-1$, and $-2^\circ$ in \citet{ryde_schultheis:15,ryde:16}, a total of $\simeq 50$ stars in the corresponding fields at Northern latitudes are under analysis, Ca.\ 25 giants in the Nuclear Star Cluster have been observed with Keck/NIRSPEC \citep[][Rich et al.2018, and Thorsbro et al. 2018]{ryde:16}. Figure \ref{fig2} shows stellar spectra of two such giants in the Nuclear Star Cluster. We see a broad metallicity distribution with few stars having [Fe/H]$<-1$ (less than $\sim5\%$) and we do not detect extremely metal-rich stars. A full metallicity distribution will be presented in Rich et al. (2018, in prep.) and the $\alpha$-element trends in Thorsbro et al. (2018 in prep.).
\begin{figure}[!bht]
\begin{center}
\includegraphics[angle=90,width=6.3in]{two_spectra_new.pdf}
\caption{Here we present analysis of two of the observed Galactic Center stars, one metal-poor
and one metal-rich, shown with the black spectra. The red line is the synthetic spectra. The green spectra show the telluric features. The Fe lines in this wavelength range used for the metallicity determination, are marked with blue vertical lines.
We use effective temperatures determined by low resolution observations of the CO-bandhead
at $2.3\,\mu$m and $\log(g)$ values are based on de-reddened photometry (for details see Rich et al. 2018, in prep.).
}
\label{fig2}
\end{center}
\end{figure}
\bibliographystyle{aa}
|
1,108,101,565,418 | arxiv | \subsection{Contractive shadowing}
We prove here a {\em shadowing lemma} saying that under some conditions a loose pseudo-orbit of a chain of contracting maps is shadowed by a true orbit of the mapping sequence. In particular, a closed pseudo-orbit is shadowed by a periodic orbit of the mapping chain.
Given a metric space $(X,d)$, denote the closed $r$-ball around $x\in X$ by
$$B(x,\varepsilon):=\{\, z\in X\,\colon\, d(z,x)\leq \varepsilon \,\}\;. $$
Given an open set $X^0\subset X$, define
$$ X^0(\varepsilon):=\{\, x\in X^0\,\colon\, d(x,\partial X^0)\geq \varepsilon\,\}\;,$$
where $\partial X^0$ denotes the topological boundary of $X^0$ in $(X,d)$.
\begin{lemma}[shadowing lemma] \label{shadow:lemma}
Consider $\varepsilon>0$ and
$0<\delta<\kappa<1$ such that $\delta/(1-\kappa)<\varepsilon<1/2$.
Given a family $\{ (X_j,d_j)\}_{0\leq j\leq n}$ of compact metric spaces with diameter $1$, a chain of continuous mappings $\{g_j:X_j^0 \to X_{j+1}\}_{0\leq j\leq n-1}$ defined on open sets $X^0_j\subset X_j$,
and a sequence of points
$x_j\in X_j$, assume that for every $0\leq j\leq n-1$:
\begin{enumerate}
\item[(a)] $x_j\in X^0_j$ and $d(x_j,\partial X^0_j)=1$,
\item[(b)] $g_j$ has Lipschitz constant
$\leq \kappa$ on $X^0_j(\varepsilon)$,
\item[(c)] $g_j(x_j)\in X^0_{j+1}(2\,\varepsilon)$,
\item[(d)] $g_j(X^0_j(\varepsilon))\subset B(g_j(x_j),\delta)$.
\end{enumerate}
Then, setting $g^{(n)} :=g_{n-1}\circ \ldots\circ g_1\circ g_0$, the following hold:
\begin{enumerate}
\item[(1)] the composition $g^{(n)}$ is defined on $B(x_0,\varepsilon)$ and ${\rm Lip}(g^{(n)}\vert_{B(x_0,\varepsilon)})\leq \kappa^n$,
\item[(2)] $d (\, g_{n-1}(x_{n-1}), \, g^{(n)}(x_0)\, )\leq
\frac{\delta}{1-\kappa}$,
\item[(3)] if $x_0=g_{n-1}(x_{n-1})$ then
$g^{(n)}(B(x_0,\varepsilon))\subset B(x_0,\varepsilon)$ and there is
a point $x^\ast\in B(x_0,\varepsilon)$
such that $g^{(n)}(x^\ast)=x^\ast$ and
$d\left( x_0, x^\ast \right)\leq
\frac{\delta}{(1-\kappa)(1-\kappa^n)}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof's inductive scheme is better understood
with the help of figure~\ref{chain:orbits},
where we set $z^i_j:= (g_{j-1}\circ \ldots \circ g_{i+1}\circ g_i)(x_i)$ for $i\leq j\leq n$. Of course we have to prove that all points $z^i_j$ are well-defined.
\begin{figure}[h]
$$\begin{array}{cccccccccccccc}
X_0 & & X_1 & & X_2 & & X_3 & & \ldots & & X_{n-1} & & X_n\\
\hline \\
z^0_0 & \stackrel{g_0}{\longrightarrow} & z^0_1
& \stackrel{g_1}{\longrightarrow} & z^0_2
& \stackrel{g_2}{\longrightarrow} & z^0_3
& \stackrel{g_3}{\longrightarrow} & \; \ldots \; & \stackrel{g_{n-2}}{\longrightarrow} & z^0_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^0_n \\
& & & & \boxed{\delta} & & \boxed{\kappa\,\delta} & & & & \boxed{\kappa^{n-3}\delta} & & \boxed{\kappa^{n-2}\delta}\\
& & z^1_1
& \stackrel{g_1}{\longrightarrow} & z^1_2
& \stackrel{g_2}{\longrightarrow} & z^1_3
& \stackrel{g_3}{\longrightarrow} & \; \ldots \; & \stackrel{g_{n-2}}{\longrightarrow} & z^1_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^1_n \\
& & & & & & \boxed{\delta} & & & & \boxed{\kappa^{n-4}\delta} & & \boxed{\kappa^{n-3}\delta}\\
& & & & z^2_2
& \stackrel{g_2}{\longrightarrow} & z^2_3
& \stackrel{g_3}{\longrightarrow} & \; \ldots \; & \stackrel{g_{n-2}}{\longrightarrow} & z^2_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^2_n \\
& & & & & & & & & & \boxed{\kappa^{n-5}\delta} & & \boxed{\kappa^{n-4}\delta} \\
& & & & & & z^3_3
& \stackrel{g_3}{\longrightarrow} & \; \ldots \; & \stackrel{g_{n-2}}{\longrightarrow} & z^3_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^3_n \\
& & & & & & & \ddots & & & \vdots & & \vdots\\
& & & & & & & & z^{n-2}_{n-2} & \stackrel{g_{n-2}}{\longrightarrow} & z^{n-2}_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^{n-2}_n \\
& & & & & & & & & & & & \boxed{\delta} \\
& & & & & & & & & & z^{n-1}_{n-1} & \stackrel{g_{n-1}}{\longrightarrow} & z^{n-1}_n \\
& & & & & & & & & & & & \\
& & & & & & & & & & & & z^{n}_n \\
\hline
\end{array} $$
\caption{Family of orbits for the chain of mappings $\{g_j:X^0_j\to X_{j+1}\}_j$.
}
\label{chain:orbits}
\end{figure}
The boxed expressions represent upper bounds on the distance between the points respectively above and below the box.
The $i$-th row represents the orbit of $x_i\in X_i$ by the chain of mappings $\{g_j\}_{j\geq i}$. All points in the $j$-th column belong to the space $X_j$.
To explain the last upper bound at the bottom of each column,
first notice that $z^i_i=x_i$.
By (a), $z^{i-1}_{i}=g_{i-1}(x_{i-1})$ is well-defined,
and by (c), $z^{i-1}_{i}\in X^0_{i}(2\,\varepsilon)\subset X^0_{i}(\varepsilon)$.
Likewise $z^{i-2}_{i-1}\in X^0_{i-1}(\varepsilon)$,
and $z_i^{i-2}=g_{i-1}(g_{i-2}(x_{i-2}))$ is well-defined.
Then by (d) we have
\begin{equation}\label{delta:dist}
d(z^{i-1}_i, z_i^{i-2}) = d(g_{i-1}(x_{i-1}),g_{i-1}(g_{i-2}(x_{i-2}))) \leq \delta\;.
\end{equation}
All other bounds are obtained applying (b) inductively.
More precisely, we prove by induction in the column index $j$
that
\begin{enumerate}
\item[(i)] all points $z^i_j$ in the $j$-th column are well-defined and belong to $ X^0_j(\varepsilon)$,
\item[(ii)] distances between between consecutive points in the column $j$ are bounded by the expressions in figure~\ref{chain:orbits}, i.e.,
for all $1\leq i\leq j-1$,
\begin{equation}\label{zij:claim}
d(z_j^{i-1},z_j^{i})\leq \kappa^{j-i-1}\,\delta \;.
\end{equation}
\end{enumerate}
The initial inductive steps, $j=0,1,2$, follow from (a), (c) and ~\eqref{delta:dist}.
Assume now that the points $z_j^i$ in $j$-th column satisfy (i) and (ii). Then their images $z^i_{j+1}=g_{j}(z_j^i)$ are well-defined.
By (b) we have for all $1\leq i\leq j-1$,
$$d(z_{j+1}^{i-1},z_{j+1}^{i}) =
d(g_j(z_j^{i-1}), g_j(z_j^{i})) \leq
\kappa\,d(z_j^{i-1},z_j^{i})\leq \kappa^{j-i}\,\delta \;. $$
Together with ~\eqref{delta:dist} this proves (ii) for the column $j+1$. To prove (i) consider any $1\leq i\leq j$.
By (c) and the triangle inequality,
\begin{align*}
d( z^i_{j+1}, \partial X^0_{j+1}(\varepsilon)) &\geq
d( z^j_{j+1}, \partial X^0_{j+1}(\varepsilon)) - d( z^i_{j+1}, z^j_{j+1})\\
&\geq
d( g_j(x_j), \partial X^0_{j+1}(\varepsilon)) - \sum_{l=i+1}^{j} d( z^{l-1}_{j+1}, z^{l}_{j+1})\\
&\geq
2\,\varepsilon - \sum_{l=i+1}^{j} \kappa^{j-l}\,\delta\geq
2\,\varepsilon-\frac{\delta}{1-\kappa} \geq \varepsilon\;.
\end{align*}
This proves (i) for the column $j+1$, and concludes the induction.
\smallskip
Conclusion (1) follows from (b) and the following claim, to be proved by induction in $i$.
For every $i=0,1,\ldots, n-1$, \,
$g^{(i)}( B(x_0,\varepsilon))\subset X^0_{i}(\varepsilon)$, where $g^{(i)}=g_{i-1}\circ \ldots \circ g_0$.
Consider first the case $i=0$.
Given $x\in B(x_0,\varepsilon)$, \,
$$d(x,\partial X^0_0)\geq d(x_0,\partial X^0_0)-d(x,x_0)\geq 1-\varepsilon>\varepsilon\;.$$
This implies that $d(g_0(x),g_0(x_0))\leq\kappa\,d(x,x_0)$.
Thus
\begin{align*}
d( g_0(x), \partial X^0_1) &\geq
d( g_0(x_0), \partial X^0_1) - d(g_0(x_0), g_0(x)) \geq 2\,\varepsilon - d(g_0(x_0), g_0(x))\\
&\geq 2\,\varepsilon - \kappa\, d(x_0,x)\geq
2\,\varepsilon - \kappa\, \varepsilon >\varepsilon\;
\end{align*}
which proves that $g_0( B(x_0,\varepsilon) )\subset X^0_1(\varepsilon)$.
Assume now that for every $l\leq i-1$,
$$(g_l\circ \ldots\circ g_0)( B(x_0,\varepsilon))\subset X^0_{l+1}(\varepsilon)\;. $$
By (b), $g^{(i)}$ acts
as a $\kappa^i$ contraction on $B(x_0,\varepsilon)$
and $g^{(i)}( B(x_0,\varepsilon) )\subset X^0_{i}(\varepsilon)$.
Thus for every $x\in B(x_0,\varepsilon)$,
\begin{align*}
d( g^{(i+1)}(x), \partial X^0_{i+1}) &\geq
d( g_i(x_i), \partial X^0_{i+1}) - d(g_i(x_i), g^{(i+1)}(x))\\
&\geq 2\,\varepsilon - d(z_{i+1}^0, z_{i+1}^i) -
d( z_{i+1}^0, g^{(i+1)}(x))\\
&\geq 2\,\varepsilon - \sum_{l=0}^{i-1} d(z_{i+1}^l, z_{i+1}^{l+1}) -
d( g^{(i+1)}(x_0), g^{(i+1)}(x))\\
&\geq 2\,\varepsilon - (\delta+\kappa\, \delta +\ldots+\kappa^{i-1}\, \delta)
- \kappa^{i}\, d(x_0, x) \\
&\geq 2\,\varepsilon - (\delta+ \kappa\, \delta + \ldots+\kappa^{i-1}\, \delta) - \kappa^{i}\, \varepsilon \\
&\geq 2\,\varepsilon - (1-\kappa)\,\varepsilon\,(1 + \kappa + \ldots+\kappa^{i-1} ) - \kappa^{i}\, \varepsilon
= \varepsilon\;
\end{align*}
which proves that $g^{(i+1)}( B(x_0,\varepsilon))\subset X^0_{i+1}(\varepsilon)$, and establishes the claim above.
Thus $g^{(n)}$ is well-defined on $B(x_0,\varepsilon)$,
and, because of assumption (b), $g^{(n)}$ is a $\kappa^n$ Lipschitz contraction on this ball.
This proves (1).
Item (2) follows by~\eqref{zij:claim}. In fact
$$
d(g_{n-1}(x_{n-1}), g^{(n)}(x_0) ) =
d(z^{n-1}_n , z^0_n ) \leq \sum_{l=1}^{n-1} d(z^{l}_n , z^{l-1}_n ) \leq \sum_{l=1}^{n-1} \kappa^{n-l-1}\,\delta \leq \frac{\delta}{1-\kappa}\;.
$$
Finally we prove (3). Assume $x_0=g_{n-1}(x_{n-1})$.
It is enough to see that
$g^{(n)}( B(x_0,\varepsilon))\subset B(x_0,\varepsilon)$,
because by (1) $g^{(n)}$ acts as a $\kappa^n$-contraction in the closed ball $B(x_0,\varepsilon)$. The conclusion on the existence of a fixed point, as well as the proximity bound, follow from the classical fixed point theorem for Lipschitz contractions.
Given $x\in B(x_0,\varepsilon)$,
we know from the previous calculation that \,
$$d(x_0, g^{(n)}(x_0))<\delta + \kappa\,\delta + \ldots + \kappa^{n-2}\,\delta \;.$$
Hence
\begin{align*}
d( g^{(n)}(x), x_0) &\leq
d(g^{(n)}(x), g^{(n)}(x_0)) + d(g^{(n)}(x_0), x_0)\\
&\leq \kappa^{n-1}\, d(x, x_0) + \delta + \kappa\,\delta + \cdots + \kappa^{n-2}\,\delta\\
&\leq \delta + \kappa\,\delta + \cdots + \kappa^{n-2}\,\delta +
\kappa^{n-1}\,\varepsilon \\
&\leq (1-\kappa)\,\varepsilon\,(1 + \kappa + \cdots + \kappa^{n-2} ) +
\kappa^{n-1}\,\varepsilon \\
& =(1-\kappa)\,\varepsilon\, \frac{1-\kappa^{n-1}}{1-\kappa} +
\kappa^{n-1}\,\varepsilon = \varepsilon\;.
\end{align*}
Thus $g^{(n)}(x)\in B(x_0,\varepsilon)$, which proves that
$g^{(n)}(B(x_0,\varepsilon))\subset B(x_0,\varepsilon)$.
\end{proof}
\begin{figure}
\begin{center}
\includegraphics*[scale={.6}]{shadowing.png}
\end{center}
\caption{Shadowing property for a chain of contractive mappings}
\end{figure}
\subsection{Statement and proof of the AP}
In the AP's statement and proof we will use the notation introduced in subsection~\ref{subsection angles}.
Given a chain of linear mappings
$\{g_j:V_{j}\to V_{j+1}\}_{0\leq j\leq n-1}$ we denote the composition of the first $i$ maps by
$g^{(i)}:= g_{i-1}\, \ldots \, g_1\, g_0$.
\begin{theorem}[Avalanche Principle] \label{Theorem:AP} There exists a constant
$c>0$ such that
given $0<\varepsilon<1$, \;$0<\kappa\leq c\,\varepsilon^ 2$
and a chain of linear mappings
$\{g_j:V_{j}\to V_{j+1}\}_{0\leq j\leq n-1}$ between Euclidean spaces $V_j$, \, if
\begin{enumerate}
\item[(a)] $\sigma(g_i)\leq \kappa$,\,
for $0\leq i\leq n-1$, and
\item[(b)] $\alpha(g_{i-1}, g_{i})\geq \varepsilon$,\;
for $1\leq i\leq n-1$,
\end{enumerate}
then
\begin{enumerate}
\item[(1)] $d(\overline{\mathfrak{v}}(g^{(n)}), \overline{\mathfrak{v}}(g_{0}))
\lesssim \kappa\,\varepsilon^{-1}$\,,
\smallskip
\item[(2)] $d(\overline{\mathfrak{v}}(g^{(n)\ast}), \overline{\mathfrak{v}}(g_{n-1}^\ast))
\lesssim \kappa\,\varepsilon^{-1}$\,,
\smallskip
\item[(3)] $ \sigma(g^{(n)}) \leq \left(\frac{\kappa\,(4+2\,\varepsilon)}{\varepsilon^{2}}\right)^n$\,,
\smallskip
\item[(4)] $\displaystyle
\abs{\log \norm{g^{(n)}} +
\sum_{i=1}^{n-2} \log \norm{g_i} -
\sum_{i=1}^{n-1}
\log \norm{g_i\,g_{i-1}} } \lesssim n\,\frac{\kappa}{\varepsilon^2}
\; .$
\end{enumerate}
\end{theorem}
\begin{remark}[On the assumptions]
\label{rmk on the assumptions}
Assumption (a) says that the (first) gap ratio of each $g_j$ is large,
${\rm gr}(g_j)\geq \kappa^{-1}$.
By propositions~\ref{prod:2:lemma} and ~\ref{prop angle rift}, assumption (b) is equivalent to a condition on the rift,
$\rho(g_{j-1},g_j)\geq\varepsilon$\, for all $j=1,\ldots,n-1$.
\end{remark}
\begin{remark}[On the conclusions]
\label{rmk on the conclusions}
Conclusions (1) and (2) say that the most expanding direction $\overline{\mathfrak{v}}(g^{(n)})$ of the product $g^{(n)}$, and its image
$\varphi_{g^{(n)}} \overline{\mathfrak{v}}(g^{(n)})$, are respectively $\kappa/\varepsilon$-close to the most expanding direction $\overline{\mathfrak{v}}(g_0)$ of $g_0$, and to the image $\varphi_{g_{n-1}} \overline{\mathfrak{v}}(g_{n-1})$ of the most expanding direction of $g_{n-1}$.
Conclusion (3) says that the composition map $g^{(n)}$ has a large gap ratio.
Finally, conclusion (4) is equivalent to
$$ e^{-n\, C\,\kappa\,\varepsilon^{-2}}\leq \frac{\norm{g_{n-1}\, \ldots \, g_1 \,g_0} \, \norm{g_1}\, \ldots\, \norm{g_{n-2}} }{ \norm{g_1\,g_0}\, \ldots\, \norm{g_{n-1}\,g_{n-2}} }\leq e^{n\, C\,\kappa\,\varepsilon^{-2}} \;, $$
for some universal constant $C>0$.
These inequalities describe the asymptotic almost multiplicative behavior of the rifts
$$ \rho(g_0,g_1,\ldots, g_{n-1})\asp{ C\,\kappa/\varepsilon^{2}} \rho(g_0,g_1)\,\rho(g_1,g_2)\,\ldots\, \rho(g_{n-2},g_{n-1}) \;. $$
\end{remark}
\begin{proof}
The strategy of the proof is to look at the contracting action of linear mappings $g_j$ on the projective space.
For each $j=0,1,\ldots, n$ consider the compact metric space
$X_j=\mathbb{P}(V_j)$ with the normalized Riemannian distance,
$d(\hat{u},\hat{v})=\frac{2}{\pi}\,\rho(\hat{u},\hat{v})$, and define for
$0\leq j <n$
\begin{align*}
X^0_j &:=\{\, \hat{v}\in X_j\,\colon\,
\alpha(\hat{v},\overline{\mathfrak{v}}(g_j))>0\,\}\;,\\
Y^0_j &:=\{\, \hat{v}\in X_j\,\colon\,
\alpha(\hat{v},\overline{\mathfrak{v}}(g_{j-1}^\ast))>0\,\}\;.
\end{align*}
The domain of the projective map $\varphi_{g_j}:\mathbb{P}(g_j)\subset X_j\to X_{j+1}$ clearly contains the open set $X^0_j$.
Analogously, the domain of $\varphi_{g_{j-1}^\ast}:\mathbb{P}(g_j^\ast)\subset X_{j} \to X_{j-1}$ contains $Y^0_j$.
We will apply lemma~\ref{shadow:lemma} to chains
of projective maps formed out of the mappings $\varphi_{g_j}:X^0_j\to X_{j+1}$ and their adjoints
$\varphi_{g_{j-1}^\ast}:Y^0_j\to X_{j-1}$.
Take positive numbers $\varepsilon$ and $\kappa$ such that
$0<\kappa\ll \varepsilon^2$, let $r := \sqrt{1- \varepsilon^2/4 } $,
and define the following input parameters for the application of lemma~\ref{shadow:lemma},
\begin{align*}
\varepsilon_{{\rm sh}} &:= \frac{1}{\pi}\,\arcsin \varepsilon\;, \\
\kappa_{{\rm sh}} &:= \kappa\,\frac{r+\sqrt{1-r^2}}{1-r^2}
\asymp \frac{ 4\,\kappa }{\varepsilon^2}\;, \\
\delta_{{\rm sh}} &:=\frac{ \kappa\,r }{\sqrt{1-r^2}}
\asymp \frac{ 2\,\kappa }{\varepsilon}\;.
\end{align*}
A simple calculation shows that there exists
$0<c<1$ such that for any $0<\varepsilon<1$ and $0<\kappa\leq c\,\varepsilon^2$, the pre-conditions
$0<\delta_{{\rm sh}}<\kappa_{{\rm sh}}<1$ and $\frac{\delta_{{\rm sh}}}{1-\kappa_{{\rm sh}}}<\varepsilon_{{\rm sh}}<1/2$ of the shadowing lemma are satisfied.
Define $x_j=\overline{\mathfrak{v}}(g_j)$ and
$x_j^\ast=\overline{\mathfrak{v}}(g_{j-1}^\ast)$. This lemma is going to be applied to the following chains of maps and sequences of points
\begin{align*}
\text{(A)} \quad &\varphi_{g_0}, \ldots,
\varphi_{g_{n-1}}, \varphi_{g_{n-1}^\ast}, \ldots, \varphi_{g_0^\ast}\;, \quad
x_0, \ldots, x_{n-1}, x_n^\ast,
\ldots, x_1^\ast \;,\\
\text{(B)} \quad &\varphi_{g_{n-1}^\ast}, \ldots, \varphi_{g_0^\ast}, \varphi_{g_0}, \ldots,
\varphi_{g_{n-1}} \;, \quad
x_n^\ast,
\ldots, x_1^\ast,x_0,\ldots, x_{n-1} \;,
\end{align*}
from which we will infer the conclusions (1) and (2).
Let us check now that assumptions (a)-(d) of
lemma~\ref{shadow:lemma} hold in both cases (A) and (B).
By definition
$\partial X^0_j:=\{\, \hat{v}\in X_j\,\colon\, \alpha(\hat{v}, x_j)=0\,\}=\{\, \hat{v}\in X_j\,\colon\, \hat{v}\perp x_j \,\}$.
Hence, if $\hat{v}\in \partial X^0_j$ then $d(x_j,\hat{v})=1$, which proves that $d(x_j,\partial X^0_j)=1$.
Analogously, $\partial Y^0_j =\{\, \hat{v}\in X_j\,\colon\, \hat{v}\perp x_j^\ast \,\}$ and $d(x_j^\ast,\partial Y^0_j)=1$. Therefore assumption (a) holds.
By definition of $ X^0_j(\varepsilon)$,
\begin{align*}
\hat{v}\in X^0_j(\varepsilon) \; &\Leftrightarrow \;
d(\hat{v}, \partial X^0_j) \geq \varepsilon
\; \Leftrightarrow \;
\rho(\hat{v}, \partial X^0_j) \geq \frac{\pi}{2}\, \varepsilon \\
&\Leftrightarrow \; \delta(\hat{v}, \partial X^0_j) =\alpha(\hat{v}, x_j) \geq \sin\left(\frac{\pi}{2}\,\varepsilon\right) \\
& \Leftrightarrow \; \delta(\hat{v}, x_j) \leq \cos\left(\frac{\pi}{2}\,\varepsilon\right) \;.
\end{align*}
Similarly, by definition of $Y^0_j(\varepsilon)$,
$$ \hat{v}\in Y^0_j(\varepsilon) \; \Leftrightarrow \; \delta(\hat{v}, x_j^\ast) \leq \cos\left(\frac{\pi}{2}\,\varepsilon\right)\;. $$
Thus, because
$$ \cos\left( \frac{\pi}{2}\,\varepsilon_{{\rm sh}}\right)
=\cos\left(\frac{1}{2}\,\arcsin \varepsilon \right) \leq \sqrt{1-\frac{\varepsilon^2}{4}}= r\;,$$
we have \,
$X^0_j(\varepsilon_{{\rm sh}})\subset B^{(\delta)}(x_j,r)$
\, and \,
$Y^0_j(\varepsilon_{{\rm sh}})\subset B^{(\delta)}(x_j^\ast,r)$, and assumption (b) holds by proposition~\ref{proj:contr} (3).
By the gap assumption,
\begin{align*}
\alpha(\varphi_{g_j}(x_j), x_{j+1})
= \alpha( \overline{\mathfrak{v}}(g_j^\ast), \overline{\mathfrak{v}}(g_{j+1}))
= \alpha(g_j,g_{j+1})\geq \varepsilon \;.
\end{align*}
Therefore
\begin{align*}
d(\varphi_{g_j}(x_j), \partial X^0_{j+1})
&= \frac{2}{\pi}\,\arcsin \delta(\varphi_{g_j}(x_j), \partial X^0_{j+1}) = \frac{2}{\pi}\,\arcsin \alpha(\varphi_{g_j}(x_j), x_{j+1})\\
&\geq \frac{2}{\pi}\,\arcsin \varepsilon = 2\,\varepsilon_{{\rm sh}}\;.
\end{align*}
Similarly, by the gap assumption,
\begin{align*}
\alpha(\varphi_{g_{j-1}^\ast}(x_j^\ast), x_{j-1}^\ast)
= \alpha( \overline{\mathfrak{v}}(g_{j-1} ), \overline{\mathfrak{v}}(g_{j-1}^\ast))
= \alpha(g_{j+1}^\ast, g_j^\ast) = \alpha(g_j,g_{j+1})\geq \varepsilon \;,
\end{align*}
and in the same way we infer that
\begin{align*}
d(\varphi_{g_{j-1}^\ast}(x_j^\ast), \partial Y^0_{j-1})\geq \frac{2}{\pi}\,\arcsin \varepsilon = 2\,\varepsilon_{{\rm sh}}\;.
\end{align*}
This proves that (c) of the shadowing lemma holds.
Notice that in both cases (A) and (B), the assumption (c) holds trivially for the middle points, because
$\varphi_{g_{n-1}}(x_{n-1})=x_n^\ast\in Y_n^0(2\,\varepsilon_{{\rm sh}})$ and
$\varphi_{g_{0}^\ast}(x_{1}^\ast)=x_0\in X_0^0(2\,\varepsilon_{{\rm sh}})$.
It was proved above that $X^0_j(\varepsilon_{{\rm sh}})\subset B^{(\delta)}(x_j,r)$
\, and \,
$Y^0_j(\varepsilon_{{\rm sh}})\subset B^{(\delta)}(x_j^\ast,r)$.
By~\eqref{metric equivalence} we have
$d(\hat{u},\hat{v})\leq \delta(\hat{u},\hat{v})$.
Thus by proposition~\ref{proj:contr} (1),
$$ \varphi_{g_j}( X_j^0(\varepsilon_{{\rm sh}}))\subset B^{(\delta)}(x_j^\ast,\delta_{{\rm sh}})\subset B^{(d)}(x_j^\ast,\delta_{{\rm sh}})
\; \text{ with }\;
x_j^\ast = \varphi_{g_j}(x_j)\;,$$
and analogously,
$$ \varphi_{g_{j-1}^\ast}( Y_j^0(\varepsilon_{{\rm sh}}))\subset B^{(\delta)}(x_{j-1},\delta_{{\rm sh}})\subset B^{(d)}(x_{j-1},\delta_{{\rm sh}}) \; \text{ with }\;
x_{j-1} = \varphi_{g_{j-1}^\ast}(x_j^\ast)\;.$$
Hence, (d) of lemma~\ref{shadow:lemma} holds.
Therefore,
because $\varphi_{g_0^\ast}(x_1^\ast)=x_0$
and $\varphi_{g_{n-1}}(x_{n-1})=x_n^\ast$,
conclusion (2) of lemma~\ref{shadow:lemma} holds
for both chains (A) and (B).
The projective points $\overline{\mathfrak{v}}(g^{(n)})$
and $\overline{\mathfrak{v}}(g^{(n)\ast})$ are the unique fixed points of the chains of mappings (A) and (B), respectively. Hence, by the shadowing lemma both distances $d(x_0, \overline{\mathfrak{v}}(g^{(n)}))$ and
$d(x_n^\ast, \overline{\mathfrak{v}}(g^{(n)\ast}))$ are bounded above by
$$\frac{\delta_{{\rm sh}}}{(1-\kappa_{{\rm sh}})\,(1-\kappa_{{\rm sh}}^{2n})}
\asymp \delta_{{\rm sh}} \asymp\frac{\kappa}{\varepsilon}\;.$$
This proves conclusions (1) and (2) of the AP.
\medskip
From proposition~\ref{derivative varphig}
we infer that for any $g\in \mathcal{L}(V)$,
$$ \norm{ (D\varphi_g)_{\overline{\mathfrak{v}}(g)} } =\frac{s_2(g)}{\norm{g}} = \sigma(g)\;.$$
Hence, by (1) of the shadowing lemma,
\begin{align*}
\sigma(g^{(n)}) &= \norm{ (D\varphi_{g^{(n)}})_{\overline{\mathfrak{v}}(g^{(n)})} } \leq {\rm Lip}(\varphi_{g^{(n)}}\vert_{ B(\overline{\mathfrak{v}}(g_0),\varepsilon_{{\rm sh}})} ) \\
&\leq (\kappa_{{\rm sh}})^n \leq
\left( \frac{\kappa\,(4+ 2\,\varepsilon)}{\varepsilon^2}\right)^n\;.
\end{align*}
This proves conclusion (3) of the AP.
Before proving (4), notice that applying
(3) to the chain of
linear maps $g_0,\ldots, g_{i-1}$ we get that $g^{(i)}:=g_{i-1}\,\ldots \, g_0$ has a first gap ratio for all $i=1,\ldots, n$.
We claim that
\begin{equation}\label{claim}
\abs{\alpha(g^{(i)},g_i)-\alpha(g_{i-1},g_i)} \lesssim \kappa\,\varepsilon^{-1} \;.
\end{equation}
By (2) of the AP, on the chain of
linear maps $g_0,\ldots, g_{i-1}$,
$$ d(\overline{\mathfrak{v}}(g^{(i)\ast}), \overline{\mathfrak{v}}(g_{i-1}^\ast) )\leq \frac{\delta_{{\rm sh}}}{(1-\kappa_{{\rm sh}})(1- \kappa_{{\rm sh}}^{2i})} \lesssim \kappa\,\varepsilon^{-1} \;. $$
Hence, by proposition~\ref{prop aangle continuity}
\begin{align*}
\abs{ \alpha(g^{(i)},g_i)-\alpha(g_{i-1},g_i) } &=
\abs{ \alpha(\overline{\mathfrak{v}}(g^{(i)\ast}), \overline{\mathfrak{v}}(g_i) )-\alpha(\overline{\mathfrak{v}}(g_{i-1}^\ast), \overline{\mathfrak{v}}(g_i ) ) }\\
& \leq d(\overline{\mathfrak{v}}(g^{(i)\ast}),\overline{\mathfrak{v}}(g_{i-1}^\ast))
\lesssim \kappa\,\varepsilon^{-1} \;.
\end{align*}
For any $i$, the logarithm of any ratio between the four factors
$\alpha( g^{(i)}, g_i)$, $\beta( g^{(i)}, g_i)$,
$\alpha( g_{i-1}, g_i)$ and $\beta( g_{i-1}, g_i)$
is of order $\kappa\,\varepsilon^{-2}$.
In fact, by~(\ref{claim})
\begin{align*}
\abs{\log\frac{ \alpha(g^{(i)},g_i) }{ \alpha(g_{i-1},g_i) } }
&\leq \frac{1}{\varepsilon}\,\abs{ \alpha(g^{(i)},g_i) -\alpha(g_{i-1},g_i) }
\lesssim \kappa\,\varepsilon^{-2} \;.
\end{align*}
By Lemma~\ref{alpha:beta:bound},
and since $\sigma_{\tau_j}(g_i)\leq \kappa$,
the other ratios are of the same magnitude as this one.
Thus, for some universal constant $C>0$, each of these four ratios is inside the interval
$[e^{-C\,\kappa\,\varepsilon^ {-2}},e^{C\,\kappa\,\varepsilon^ {-2}}]$.
Finally, applying proposition~\ref{svp:lemma:norm}
to the rifts $\rho(g_0,\ldots, g_{n-1})$,
$\rho(g_0,g_1)$, $\rho(g_1,g_2)$, etc, we have
$$ e^{-n\, C\,\kappa\,\varepsilon^{-2}} \leq \prod_{i=1}^{n-1} \frac{\alpha(g^{(i)},g_i)}{\beta(g_{i-1},g_i)} \leq
\frac{\rho(g_0,\ldots, g_{n-1})}{\prod_{i=1}^{n-1}\rho(g_{i-1},g_i)} \leq
\prod_{i=1}^{n-1} \frac{\beta(g^{(i)},g_i)}{\alpha(g_{i-1},g_i)} \leq
e^{ n\, C\,\kappa\,\varepsilon^{-2}}\;, $$
which by remark~\ref{rmk on the conclusions} is equivalent to (4).
\end{proof}
\bigskip
Next proposition is a practical reformulation of
the Avalanche Principle.
\begin{proposition} \label{AP-practical}
There exists $c>0$ such that
given $0<\epsilon<1$, $0<\kappa\leq c\,\epsilon^ 2$
and \, $g_0, g_1,\ldots, g_{n-1}\in{\rm Mat}(m,\mathbb{R})$, \,
if
\begin{align*}
\rm{(gaps)} \ & {\rm gr} (g_i) > \frac{1}{\ka} & \text{for all } & \ \ 0 \le i \le n-1
\\
\rm{(angles)} \ & \frac{\norm{ g_i \cdot g_{i-1} }}{\norm{g_i} \, \norm{ g_{i-1}}} > \ep & \
\text{for all } & \ \ 1 \le i \le n-1
\end{align*}
then
\begin{align*}
& \max\left\{ \, d(\overline{\mathfrak{v}}(g^{(n)\ast}), \overline{\mathfrak{v}}(g_{n-1}^\ast)),\,
d(\overline{\mathfrak{v}}(g^{(n)}), \overline{\mathfrak{v}}(g_{0})) \, \right\}
\lesssim \kappa\,\ep^{-1} \\
& \sabs{ \log \norm{ g^{(n)} } + \sum_{i=1}^{n-2} \log \norm{g_i} - \sum_{i=1}^{n-1} \log \norm{ g_i \cdot g_{i-1}} } \lesssim n \cdot \frac{\ka}{\ep^2} \;.
\end{align*}
\end{proposition}
\begin{proof}
Consider the constant $c>0$ in theorem~\ref{Theorem:AP},
let $c':=c\,(1-2\,c^2)$ and assume $0<\kappa \le c'\,\epsilon^2$.
Assumption (gaps) here is equivalent to assumption (a) of
theorem~\ref{Theorem:AP}. By proposition~\ref{prop angle rift}, the assumption (angles) here implies
\begin{align*}
\alpha(g_{i-1}, g_i) & \geq \rho(g_{i-1}, g_i)\,
\sqrt{1-\frac{2\,\kappa^2}{\rho(g_{i-1}, g_i)^2} } \\
& \geq \epsilon\,
\sqrt{1-\frac{2\,\kappa^2}{\epsilon^2} } \geq \epsilon\,
\sqrt{1- 2\,c^2\,\epsilon^2 } =:\epsilon'\;,
\end{align*}
Since $0<\kappa\leq c'\,\epsilon^2$, \, and \,
$ c'\,\epsilon^2 \leq c\,(1-2\,c^2\,\epsilon^2)\,\epsilon^2
= c\,(\epsilon')^2$ we have
$0<\kappa\leq c \,(\epsilon')^2$.
Thus, because $\epsilon\asymp \epsilon'$, this proposition follows from conclusions (1), (2) and (4) of
theorem~\ref{Theorem:AP}.
\end{proof}
\subsection{Consequences of the AP}
Given a chain of linear maps
$\{g_j:V_j\to V_{j+1}\}_{0\leq j\leq n-1}$ between Euclidean spaces $V_j$, and integers $0\leq i<j\leq n$ we
define
$$ g^{(j,i)}:= g_{j-1}\circ \ldots \circ g_{i+1}\circ g_i\;. $$
With this notation the following relation holds for
$0\leq i <k <j\leq n$,
$$ g^{(j,i)} = g^{(j,k)}\circ g^{(k,i)} \;.$$
Next proposition states, in a quantified way, that the most expanding directions $\overline{\mathfrak{v}}(g^{n,i)})\in\mathbb{P}(V_i)$ are almost invariant under the adjoints of the chain mappings.
\begin{proposition} \label{prop::almost invariance}
Under the assumptions of theorem~\ref{Theorem:AP}, where $0<\kappa\ll \varepsilon^2$,
$$ d( \varphi_{g_i^\ast}\,\overline{\mathfrak{v}}(g^{(n,i+1)}), \overline{\mathfrak{v}}(g^{(n,i)})) \lesssim \frac{\kappa}{\varepsilon}\, (\frac{\kappa\,(4+2\,\varepsilon)}{\varepsilon^2})^{n-i}\;.$$
\end{proposition}
\begin{proof}
Consider $\kappa$, $\varepsilon$, $\kappa_{{\rm sh}}$ and $\varepsilon_{{\rm sh}}$
as in theorem~\ref{Theorem:AP}.
From the proof of item (3) of the AP, applied to the chain of mappings $g_{n-1}^\ast,\ldots, g_{i}^\ast$,
we conclude that the composition
$g^{(n,i)}=g_{i}^\ast\circ \ldots \,\circ g_{n-1}^\ast$
is a $(\kappa_{{\rm sh}})^{n-i}$-Lipschitz contraction on the ball
$B(\overline{\mathfrak{v}}(g_{n-1}^\ast),\varepsilon_{{\rm sh}})$.
On the other hand, by (2) of the AP we have
$d( \,\overline{\mathfrak{v}}(g^{(n,i+1)\ast} , \overline{\mathfrak{v}}(g_{n-1}^\ast) \,)\lesssim\kappa\,\varepsilon^{-1}$ and \, $d( \,\overline{\mathfrak{v}}(g_{n-1}^\ast) , \overline{\mathfrak{v}}(g^{(n,i)\ast} \,)\lesssim\kappa\,\varepsilon^{-1}$.
Since $\kappa\,\varepsilon^{-1}\ll \varepsilon \asymp\varepsilon_{{\rm sh}}$,
both projective points $\overline{\mathfrak{v}}(g^{(n,i)\ast})$ and $\overline{\mathfrak{v}}(g^{(n,i+1)\ast})$ belong to the ball $B(\overline{\mathfrak{v}}(g_{n-1}^\ast),\varepsilon_{{\rm sh}})$.
Thus,
\begin{align*}
&d( \varphi_{g_i^\ast}\, \overline{\mathfrak{v}}(g^{(n,i+1)}), \overline{\mathfrak{v}}(g^{(n,i)}) ) =\\
&\qquad = d(\, \varphi_{g_i^\ast}\circ\varphi_{g^{(n,i+1)\ast} } \,\overline{\mathfrak{v}}(g^{(n,i+1)\ast} ),
\, \varphi_{g^{(n,i)\ast} }\, \overline{\mathfrak{v}}(g^{(n,i)\ast} )\, ) \\
&\qquad = d(\, \varphi_{g^{(n,i)\ast} } \,\overline{\mathfrak{v}}(g^{(n,i+1)\ast} ),
\, \varphi_{g^{(n,i)\ast} }\, \overline{\mathfrak{v}}(g^{(n,i)\ast} )\, ) \\
&\qquad \leq (\kappa_{{\rm sh}})^{n-i} \, d( \,\overline{\mathfrak{v}}(g^{(n,i+1)\ast} , \overline{\mathfrak{v}}(g^{(n,i)\ast} \,) \\
&\qquad \leq (\frac{\kappa\, (4+2\,\varepsilon)}{\varepsilon^2})^{n-i} \, \left( d( \,\overline{\mathfrak{v}}(g^{(n,i+1)\ast} , \overline{\mathfrak{v}}(g_{n-1}^\ast) \,) + d( \,\overline{\mathfrak{v}}(g_{n-1}^\ast) , \overline{\mathfrak{v}}(g^{(n,i)\ast} \,) \right)\\
&\qquad \lesssim \frac{2\,\kappa}{\varepsilon}\, (\frac{\kappa\, (4+2\,\varepsilon)}{\varepsilon^2})^{n-i} \;.
\end{align*}
which proves the proposition.
\end{proof}
Most expanding directions and norms of products of chains matrices under an application of the AP
admit the following modulus of continuity.
\begin{proposition}
Let $c>0$ be the universal constant in theorem~\ref{Theorem:AP}.
Given numbers $0<\varepsilon<1$ and $0<\kappa<c\,\varepsilon^2$,
and given two chains of matrices $g_0,\ldots, g_{n-1}$ and $g_0',\ldots, g_{n-1}'$ in ${\rm Mat}(m,\mathbb{R})$, both satisfying the assumptions of the AP for the given parameters $\kappa$ and $\varepsilon$, \, if \,
$d_{{\rm rel}} (g_i,g_i')<\delta$ \, for all $i=0,1,\ldots, n-1$,
then
\begin{enumerate}
\item[(a)] $d( \, \overline{\mathfrak{v}}(g_{n-1}\,\ldots\, g_0), \,
\overline{\mathfrak{v}}(g_{n-1}'\,\ldots\, g_0')\, ) \lesssim \frac{ \kappa}{\varepsilon} + 8\,\delta $,
\item[(b)]\;
$\displaystyle \abs{\log \frac{\norm{g_{n-1}\,\ldots\, g_0}}{\norm{g_{n-1}'\,\ldots\, g_0'}} }\lesssim n\,\left( \frac{\kappa}{\varepsilon^2} + \frac{ \delta }{\varepsilon} \right) $.
\end{enumerate}
\end{proposition}
\begin{proof}
Item (a) follows from conclusion (1) of theorem~\ref{Theorem:AP}, and
proposition~\ref{lipschitz:eigendir},
\begin{align*}
d(\, \overline{\mathfrak{v}}( g_{n-1}\,\ldots\, g_0),\,
\overline{\mathfrak{v}}( g_{n-1}'\,\ldots\, g_0')\,) &\leq
d(\, \overline{\mathfrak{v}}( g_{n-1}\,\ldots\, g_0),\,
\overline{\mathfrak{v}}(g_0)\,) \\
&\qquad + d(\overline{\mathfrak{v}}(g_0),\overline{\mathfrak{v}}(g_0')) +
d(\, \overline{\mathfrak{v}}(g_0'),\,
\overline{\mathfrak{v}}( g_{n-1}'\,\ldots\, g_0')\,)\\
&\lesssim 2\,\frac{\kappa}{\varepsilon} +
\frac{16\,\delta}{1-\kappa^2} \lesssim \frac{ \kappa}{\varepsilon} + 8\,\delta \;.
\end{align*}
Assuming $\norm{g_i}\geq \norm{g_i'}$, we have
$$ \frac{\norm{g_i}}{\norm{g_i'}}
\leq 1+ \frac{\norm{g_i-g_i'}}{\norm{g_i'}}
\leq 1+ \frac{\norm{g_i}}{\norm{g_i'}}\, d_{{\rm rel}} (g_i,g_i')
\leq 1+ \delta\,\frac{\norm{g_i}}{\norm{g_i'}}
$$
which implies
$$ \frac{\norm{g_i}}{\norm{g_i'}}\leq \frac{1}{1-\delta} \;. $$
Because the case $\norm{g_i}\leq \norm{g_i'}$ is analogous,
we conclude that
$$ \abs{\log \frac{\norm{g_i}}{\norm{g_i'}} }
\leq \log\left( \frac{1}{1-\delta}\right)
\leq \frac{\delta}{1-\delta} \asymp \delta \;. $$
Since the two chains of matrices satisfy the assumptions of the AP we have
$$ \frac{\norm{g_i\,g_{i-1}}}{\norm{g_i}\,\norm{g_{i-1}}}
\geq \alpha(g_{i-1},g_i)\geq \varepsilon \quad \text{ and }\quad
\frac{\norm{g_i'\,g_{i-1}'}}{\norm{g_i'}\,\norm{g_{i-1}'}}
\geq \alpha(g_{i-1}',g_i')\geq \varepsilon \;. $$
A simple calculation gives
$$ d_{{\rm rel}} (\,g_i\,g_{i-1}, \, g_i'\,g_{i-1}' \,)\leq
\frac{2}{(1-\delta)^2}\, \frac{\delta}{\varepsilon}
\asymp \frac{\delta}{\varepsilon} \;.$$
Therefore
$$ \abs{\log \frac{\norm{g_i\,g_{i-1}}}{\norm{g_i'\,g_{i-1}'}} }
\lesssim \frac{\delta}{\varepsilon} \;.$$
Hence, by conclusion (4) of the AP we have
\begin{align*}
\abs{\log \frac{\norm{g_{n-1}\,\ldots \,g_{0}}}{\norm{g_{n-1}'\,\ldots \, g_{0}'}} } &\leq
\abs{\log \frac{\norm{g_{n-1}\,\ldots \,g_{0}}\,\norm{g_1}\,\ldots\, \norm{g_{n-2}}}{ \norm{g_1\,g_0}\,\ldots\,\norm{g_{n-1}\,g_{n-2}} } } \\
&\qquad + \abs{\log \frac{ \norm{g_1'\,g_0'}\,\ldots\,\norm{g_{n-1}'\,g_{n-2}'} }{\norm{g_{n-1}'\,\ldots \,g_{0}'}\,\norm{g_1'}\,\ldots\, \norm{g_{n-2}'}} } \\
&\qquad + \sum_{i=1}^{n-2} \abs{ \log \frac{\norm{g_i'} }{\norm{g_i} } }
+ \sum_{i=1}^{n-1} \abs{ \log \frac{\norm{g_i\,g_{i-1} } }{\norm{g_i'\,g_{i-1}' }} }\\
&\lesssim 2\,n\,\frac{\kappa}{\varepsilon^2}
+(n-2)\,\delta + (n-1)\,\frac{\delta}{\varepsilon} \\
&\lesssim n\,\left( \frac{\kappa}{\varepsilon^2} + \frac{ \delta }{\varepsilon} \right)\;,
\end{align*}
which proves (b).
\end{proof}
\smallskip
Next proposition is a flag version of the AP.
Let $\tau=(\tau_1,\ldots, \tau_k)$ be a signature
with $0<\tau_1<\tau_2 <\ldots <\tau_k <m $.
We call $\tau$-block product to any of the functions
$\pi_{\tau,j}:{\rm Mat}(m,\mathbb{R})\to \mathbb{R}$,
$$
\pi_{\tau,j}(g):= s_{\tau_{j-1}+1}(g) \, \ldots \, s_{\tau_j}(g)\;,\qquad 1\leq j\leq k \;,
$$
where by convention $\tau_0=0$.
A $\tau$-singular value product, abbreviated $\tau$-s.v.p., is any product of distinct $\tau$-block products.
By definition, $\tau$-block products are $\tau$-singular value products. Other examples of $\tau$-singular value products are the functions
$$p_{\tau_j} (g) = s_1 (g) \, \ldots \, s_{\tau_j} (g)
= \norm{\wedge_{\tau_j} g} \;. $$
Note that for every $1 \le j \le k$ we have:
$$\pi_{\tau,j}(g) = \frac{p_{\tau_j} (g)}{p_{\tau_{j-1}} (g)}\;,$$
and
$$p_{\tau_j} (g) = \pi_{\tau,1}(g) \, \ldots \, \pi_{\tau,j}(g)\;. $$
\begin{proposition}
\label{Flag:AP}
Let $c>0$ be the universal constant in theorem~\ref{Theorem:AP}.
Given numbers $0<\varepsilon<1$, $0<\kappa\leq c\,\varepsilon^ 2$
and a chain of matrices
$g_j\in {\rm Mat}(m,\mathbb{R})$, with $j=0,1,\ldots, n-1$, \, if
\begin{enumerate}
\item[(a)] $\sigma_\tau(g_i)\leq \kappa$,\,
for $0\leq i\leq n-1$, and
\item[(b)] $\alpha_\tau(g_{i-1}, g_{i})\geq \varepsilon$,\;
for $1\leq i\leq n-1$,
\end{enumerate}
then
\begin{enumerate}
\item[(1)] $d(\overline{\mathfrak{v}}_\tau(g^{(n)\ast}), \overline{\mathfrak{v}}_\tau(g_{n-1}^\ast))
\lesssim \kappa\,\varepsilon^{-1}$
\item[(2)] $d(\overline{\mathfrak{v}}_\tau(g^{(n)}), \overline{\mathfrak{v}}_\tau(g_{0}))
\lesssim \kappa\,\varepsilon^{-1}$
\item[(3)]
$ \sigma_\tau(g^{(n)}) \leq \left(\frac{\kappa\,(4+ 2\,\varepsilon)}{\varepsilon^{2}}\right)^n$
\item[(4)] for any $\tau$-s.v.p. function $\pi$,
$$
\abs{\log \pi(g^{(n)}) +
\sum_{i=1}^{n-2} \log \pi(g_i) -
\sum_{i=1}^{n-1}
\log \pi(g_i\,g_{i-1}) } \lesssim n\,\frac{\kappa}{\varepsilon^2}
\; . $$
\end{enumerate}
\end{proposition}
\begin{proof}
For each $j=1,\ldots, k$, consider the chain of matrices
$\wedge_{\tau_j}g_0, \wedge_{\tau_j}g_1, \ldots, \wedge_{\tau_j} g_{n-1}$.
Assumptions (a) and (b) here imply the corresponding assumptions of theorem~\ref{Theorem:AP} for all these chains of exterior power matrices.
Hence, by (1) of the AP
\begin{align*}
d(\overline{\mathfrak{v}}_{\tau_j}(g^{(n)\ast}), \overline{\mathfrak{v}}_{\tau_j}(g_{n-1}^\ast)) &=
d(\Psi(\overline{\mathfrak{v}}_{\tau_j}( g^{(n)\ast})), \Psi(\overline{\mathfrak{v}}_{\tau_j}( g_{n-1}^\ast)))\\
&=d( \overline{\mathfrak{v}}(\wedge_{\tau_j} g^{(n)\ast}), \overline{\mathfrak{v}}(\wedge_{\tau_j} g_{n-1}^\ast)) \lesssim
\kappa\,\varepsilon^{-1}\;.
\end{align*}
Thus, taking the maximum in $j$ we get
$d(\overline{\mathfrak{v}}_{\tau}(g^{(n)\ast}), \overline{\mathfrak{v}}_{\tau}(g_{n-1}^\ast))\lesssim
\kappa\,\varepsilon^{-1}$, which proves (1).
Conclusion (2) follows in the same way.
Similarly, from (3) of theorem~\ref{Theorem:AP},
we infer the corresponding conclusion here
$$
\sigma_{\tau}(g^{(n)}) =
\max_{1\leq j\leq k} \sigma_{\tau_j}(g^{(n)})
= \max_{1\leq j\leq k} \sigma(\wedge_{\tau_j} g^{(n)}) \leq \left(\frac{\kappa\,(4+ 2\,\varepsilon)}{\varepsilon^{2}}\right)^n\;.
$$
Let us now prove (4).
For the $\tau$-s.v.p. $\pi(g)=p_{\tau,j}(g)=\norm{\wedge_{\tau_j} g}$ conclusion (4) is a consequence of the corresponding conclusion of theorem~\ref{Theorem:AP}.
For the $\tau$-block product
$\pi=\pi_{\tau,j}$, since
$$ \log \pi(g) = \log \norm{\wedge_{\tau_j} g }
- \log \norm{\wedge_{\tau_{j-1}} g } \;, $$
conclusion (4) follows again from theorem~\ref{Theorem:AP} (4).
Finally, since any $\tau$-s.v.p. is a finite product of $\tau$-block products we can reduce (4) to the previous case.
\end{proof}
We finish this section with a version of the AP for complex matrices.
The {\em singular values} of a complex matrix $g\in{\rm Mat}(m,\mathbb{C})$ are defined to be the eigenvalues of the positive semi-definite hermitian matrix
$g^\ast\,g$, where $g^\ast$ stands for the transjugate of $g$, i.e., the conjugate transpose of $g$. Similarly, the
{\em singular vectors} of $g$
are defined as the eigenvectors of $g^\ast\,g$.
The sorted singular values of $g\in {\rm Mat}(m,\mathbb{C})$ are denoted by
\, $ s_1(g)\geq s_2(g) \geq \ldots \geq s_m(g) $.
The top singular value of $g$ coincides with its norm,
$s_1(g)=\norm{g}$.
The (first) gap ratio of $g$ is the quotient
$\sigma(g):=s_2(g)/s_1(g)\leq 1$. We say that $g\in{\rm Mat}(m,\mathbb{C})$ has a (first) gap ratio when
$\sigma(g)<1$. When this happens the complex eigenspace
$$ \{\, v\in\mathbb{C}^m\,\colon\, g^\ast\,g\,v=\norm{g}\,v\,\}
= \{\, v\in\mathbb{C}^m\,\colon\, \norm{g\,v}=\norm{g}\,\norm{v} \,\} $$
has complex dimension one, and determines a point
in $\mathbb{P}(\mathbb{C}^m)$, denoted by $\overline{\mathfrak{v}}(g)$ and referred as the $g$-most expanding direction.
Given points $\hat{v},\hat{u}\in\mathbb{P}(\mathbb{C}^m)$, we set
\begin{equation}\label{complex aangle def}
\alpha(\hat{v},\hat{u}):= \frac{\abs{\langle v,u \rangle}}{\norm{v}\,\norm{u}} \qquad \text{ where } \quad
v\in\hat{v},\; u\in\hat{u}\;.
\end{equation}
Given $g,g'\in {\rm Mat}(m,\mathbb{C})$, both with (first) gap ratios,
we define the {\em angle between} $g$ and $g'$ to be
$$ \alpha(g,g'):= \alpha(\overline{\mathfrak{v}}(g^\ast),\overline{\mathfrak{v}}(g'))\;. $$
With these definitions, the real version of the AP leads in a straightforward manner to a slightly weaker complex version, stated and proved below.
However, adapting the original proof to the complex case,
replacing each real concept by its complex analog,
would lead to the same stronger estimates as in theorem~\ref{Theorem:AP}.
\begin{proposition}[Complex AP]
\label{complex:AP}
Let $c>0$ be the universal constant in theorem~\ref{Theorem:AP}.
Given numbers $0<\varepsilon<1$, $0<\kappa\leq c\,\varepsilon^ 4$
and a chain of matrices
$g_j\in {\rm Mat}(m,\mathbb{C})$, with $j=0,1,\ldots, n-1$, \, if
\begin{enumerate}
\item[(a)] $\sigma(g_i)\leq \kappa$,\,
for $0\leq i\leq n-1$, and
\item[(b)] $\alpha(g_{i-1}, g_{i})\geq \varepsilon$,\;
for $1\leq i\leq n-1$,
\end{enumerate}
then
\begin{enumerate}
\item[(1)] $d(\overline{\mathfrak{v}}(g^{(n)\ast}), \overline{\mathfrak{v}}(g_{n-1}^\ast))
\lesssim \kappa\,\varepsilon^{-2}$
\item[(2)] $d(\overline{\mathfrak{v}}(g^{(n)}), \overline{\mathfrak{v}}(g_{0}))
\lesssim \kappa\,\varepsilon^{-2}$
\item[(3)]
$ \sigma(g^{(n)}) \leq \left(\frac{\kappa\,(4+ 2\,\varepsilon^2)}{\varepsilon^{4}}\right)^n$
\item[(4)]
$\displaystyle
\abs{\log \norm{ g^{(n)} } +
\sum_{i=1}^{n-2} \log \norm{g_i} -
\sum_{i=1}^{n-1}
\log \norm{ g_i\,g_{i-1} } } \lesssim n\,\frac{\kappa}{\varepsilon^4}
\; . $
\end{enumerate}
\end{proposition}
\begin{proof}
Make the identification $\mathbb{C}^m\equiv \mathbb{R}^{2m}$,
and given $g\in{\rm Mat}(m,\mathbb{C}^m)$
denote by $g^\mathbb{R}\in{\rm Mat}(2m,\mathbb{R})$ the matrix representing the linear operator $g:\mathbb{R}^{2m}\to\mathbb{R}^{2m}$ in the canonical basis.
We make explicit the relationship between gap ratios and angles of the complex matrices and $g,g'\in{\rm Mat}(m,\mathbb{C})$, and the gap ratios and angles of their real analogues $g^\mathbb{R}$ and $(g')^\mathbb{R}$.
Given $g\in{\rm Mat}(m,\mathbb{C})$, for each eigenvalue $\lambda$ of $g$,
the matrix $g^\mathbb{R}$ has a corresponding pair of eigenvalues $\lambda,\overline{\lambda}$.
Since $g\mapsto g^\mathbb{R}$ is a $C^\ast$-algebra homomorphism, we have
$(g^\ast\,g)^\mathbb{R}=(g^\mathbb{R})^\ast\,(g^\mathbb{R})$.
Therefore, for all $i=1,\ldots, m$,
$s_i(g)=s_{2i-1}(g^\mathbb{R}) = s_{2i}(g^\mathbb{R})$.
In particular, considering the signature $\tau=(2)$,
\begin{equation}
\label{sgap real-complex}
\sigma_{(2)}(g^\mathbb{R})
= \frac{s_3(g^\mathbb{R})}{s_1(g^\mathbb{R})}
=\frac{s_2(g)}{s_1(g)} = \sigma(g)
\;.
\end{equation}
The $g$-most expanding direction
$\overline{\mathfrak{v}}(g)\in\mathbb{P}(\mathbb{C}^m)$ is a complex line
which we can identify with the real $2$-plane
$\overline{\mathfrak{v}}_{(2)}(g^\mathbb{R})$.
This identification, $\overline{\mathfrak{v}}(g)\equiv \overline{\mathfrak{v}}_{(2)}(g^\mathbb{R})$, comes from a natural isometric embedding
$\mathbb{P}(\mathbb{C}^m) \hookrightarrow {\rm Gr}_2(\mathbb{R}^{2m})$.
Consider two points $\hat{v},\hat{u}\in\mathbb{P}(\mathbb{C}^m)$
and take unit vectors $v\in \hat{v}$ and $u\in \hat{u}$. Denote by $U,V\subset \mathbb{C}^m$ the complex lines spanned by these vectors, which are planes in ${\rm Gr}_{2}(\mathbb{R}^{2m})$.
Consider the complex orthogonal projection onto the complex line $V$,
$\pi_{u,v}:U\to V$, defined by
$\pi_{u,v}(x):=\langle x,v\rangle\,v$.
By~\eqref{complex aangle def} we have $\alpha(\hat{v},\hat{u})=\norm{\pi_{u,v}}$.
On the other hand, since $\pi_{u,v}\circ \pi_{u,v}=\pi_{u,v}$ and
$\langle x- \pi_{u,v}(x), v\rangle =0$ for all $x\in U$, it follows that $\pi_{u,v}$ is the restriction to $U$ of the (real) orthogonal projection onto the $2$-plane $V$.
Thus, by proposition~\ref{prop: alpha = det Pi(E F)}(b),
$$ \alpha_2(U,V)=\abs{\det{}_\mathbb{R}(\pi_{u,v})} =\abs{\det{}_\mathbb{C}(\pi_{u,v})}^2 = \norm{\pi_{u,v}}^2=\alpha(\hat{v},\hat{u})^2\;. $$
In particular,
\begin{equation}
\label{angle real-complex}
\alpha_{(2)}(g^\mathbb{R}, (g')^\mathbb{R}) =
\alpha_{(2)}(\overline{\mathfrak{v}}((g^\mathbb{R})^\ast) , \overline{\mathfrak{v}}((g')^\mathbb{R})) =
\alpha(\overline{\mathfrak{v}}(g^\ast) , \overline{\mathfrak{v}}(g'))^2 = \alpha(g,g')^2\;.
\end{equation}
Take $\kappa,\varepsilon>0$ such that
$\kappa <c\,\varepsilon^4$, $0<\varepsilon<1$, and consider a chain of matrices $g_j\in{\rm Mat}(m,\mathbb{C})$, $j=0,1,\ldots, n-1$ satisfying the assumptions (a) and (b) of the complex AP.
By~\eqref{sgap real-complex} and~\eqref{angle real-complex},
the assumptions (a) and (b) of proposition~\ref{Flag:AP} hold for the chain of real matrices
$ g_j^\mathbb{R}\in{\rm Mat}(2m,\mathbb{R})$, $j=0,1,\ldots, n-1$, with parameters $\kappa$ and $\varepsilon^2$, and with $\tau=(2)$.
Therefore conclusions (1)-(4) of the complex AP follow from the corresponding conclusions of proposition~\ref{Flag:AP}.
In conclusion (4) we use the $(2)$-singular value product $\pi(g):=\norm{g}^2 = \norm{\wedge_2 g^\mathbb{R}}$.
\end{proof}
\subsection{Projective spaces}
\label{subsection projective spaces}
The projective space is the simplest compact model to study the action of a linear map.
Given a $n$-dimensional Euclidean space $V$, consider the equivalence relation defined on $V\setminus\{0\}$ by
$u\equiv v$ if and only if $u=\lambda\,v$ for some $\lambda\neq 0$.
For $v\in V\setminus\{0\}$, the set
$\hat{v}:=\{\, \lambda\,v\,\colon\, \lambda\in\mathbb{R}\setminus\{0\}\,\}$ is the equivalence class of the vector $v$ by this relation.
The {\em projective space of $V$} is the quotient \, $\mathbb{P}(V):=\{\, \hat{v}\,\colon\, v\in V\setminus\{0\}\, \}$ of $V\setminus\{0\}$ by this equivalence relation.
It is a compact topological space when endowed with the quotient topology.
The unit sphere
$\mathbb{S}(V):=\{\, v\in V\,\colon\, \norm{v}=1\,\}$ is a compact Riemannian manifold of constant curvature $1$ and diameter $\pi$.
The natural projection $\hat \pi:\mathbb{S}(V)\to \mathbb{P}(V)$, $\hat \pi(v)=\hat{v}$,
is a (double) covering map. Hence the projective space $\mathbb{P}(V)$
has a natural smooth Riemannian structure for which the
covering map $\hat \pi$ is a local isometry.
Thus $\mathbb{P}(V)$ is a compact Riemannian manifold with constant curvature $1$ and diameter $\frac{\pi}{2}$.
Given a linear map $g\in \mathcal{L}(V)$ define
$\mathbb{P}(g):=\{\, \hat{v}\in \mathbb{P}(V)\,\colon\, g\,v\neq 0\,\}$.
We refer to the linear map $\varphi_g:\mathbb{P}(g)\subset \mathbb{P}(V)\to\mathbb{P}(V)$, $\varphi_g(\hat{v}) := \hat \pi(\frac{g\,v}{\norm{g\,v}})$, as the {\em projective action of $g$ on } $\mathbb{P}(V)$.
If $g$ is invertible then $\varphi_g: \mathbb{P}(V)\to\mathbb{P}(V)$ is a diffeomorphism with inverse $\varphi_{g^{-1}}: \mathbb{P}(V)\to\mathbb{P}(V)$.
Through these maps, the group ${\rm GL}(V)$, of all linear automorphisms on $V$, acts transitively on the projective space $\mathbb{P}(V)$.
We will consider three different metrics on the projective space $\mathbb{P}(V)$. The Riemannian distance, $\rho$, measures the length of an arc connecting two points in the sphere.
More precisely, given $u,v\in\mathbb{S}(V)$,
\begin{equation}
\label{projective Riemannian metric}
\rho(\hat{u},\hat v):=\min\{\, \angle(u,v), \angle(u,-v)\}\;.
\end{equation}
The second metric, $d$, corresponds to the Euclidean distance measured in the sphere. More precisely, given $u,v\in\mathbb{S}(V)$,
\begin{equation}
\label{projective Euclidean metric}
d(\hat{u},\hat v):=\min\{\, \norm{u-v}, \norm{u+v}\}
\end{equation}
measures the smallest chord of the arcs between $u$ and $v$, and between $u$ and $-v$. The third metric, $\delta$, measures the sine of the arc between two points in the sphere. More precisely, given $u,v\in\mathbb{S}(V)$,
\begin{equation}
\label{projective sine metric}
\delta(\hat{u},\hat{v}):= \frac{\norm{u\wedge v}}{\norm{u}\,\norm{v}} = \sin(\angle(u,v))\;.
\end{equation}
The fact that $\delta$ is a metric on $\mathbb{P}(V)$ follows from the sine addition law, which implies that
$\sin (\theta+\theta') \leq \sin \theta + \sin\theta'$,
for all $\theta,\theta'\in [0,\frac{\pi}{2}]$.
These three distances are equivalent. For all $\hat{u},\hat{v}\in\mathbb{P}(V)$,
\begin{equation}\label{metric relations}
\delta(\hat{u},\hat{v})=\sin \rho (\hat{u},\hat{v})\quad \text{ and }\quad d(\hat{u},\hat{v})={\rm chord} \, \rho (\hat{u},\hat{v})\;.
\end{equation}
The inequalities
$$ \frac{2\,\theta}{\pi} \leq \sin \theta \leq {\rm chord}\,\theta= 2\, \sin ({\theta}/{2}) \leq \theta \qquad \forall\, 0\leq \theta\leq \frac{\pi}{2}$$
imply that
\begin{equation}\label{metric equivalence}
\frac{2}{\pi}\,\rho(\hat{u},\hat{v})\leq \delta(\hat{u},\hat{v}) \leq d(\hat{u},\hat{v}) \leq \rho(\hat{u},\hat{v}) \;.
\end{equation}
Because of ~\eqref{metric relations}, these three metrics determine the same group of isometries on the projective space.
\bigskip
\subsection{Exterior algebra}\label{exterior algebra}
Exterior Algebra was introduced by H. Grassmann in the `Ausdehnungslehre'. We present here an informal description of some of its properties. See the book of Shlomo Stenberg ~\cite{Stenberg-book} for a rigorous treatment of the subject.
Let $V$ be a finite $n$-dimensional Euclidean space.
Given $k$ vectors $v_1,\ldots, v_k\in V$, their $k$-th exterior product is a formal skew-symmetric product
$v_1\wedge \ldots \wedge v_k$, in the sense that
for any permutation $\sigma=(\sigma_1,\ldots,\sigma_k) \in {\rm S}_k$,
$$ v_{\sigma_1} \wedge \ldots \wedge v_{\sigma_k}
= (-1)^{{\rm sgn}(\sigma)} v_1\wedge \ldots \wedge v_k
\;. $$
These formal products are elements of an anti-commutative and associative graded algebra $(\wedge_\ast V, +,\wedge)$, called the {\em exterior algebra} of $V$.
Formal products $v_1\wedge \ldots \wedge v_k$ are called
{\em simple $k$-vectors} of $V$.
The {\em $k$-th exterior power of $V$}, denoted by $\wedge_k V$,
is the linear span of all simple $k$ vectors of $V$.
Elements of $\wedge_k V$ are called {\em $k$-vectors}.
An easy consequence of this formal definition is that
$v_1\wedge \ldots \wedge v_k=0$ if and only if $v_1,\ldots, v_k$
are linearly dependent. Another simple consequence is that given
two bases $\{v_1,\ldots, v_k\}$ and $\{w_1,\ldots, w_k\}$ of the same $k$-dimensional linear subspace of $V$, if for some real matrix
$A=(a_{ij})$ we have $w_i=\sum_{j=1}^k a_{ij}\,v_j$
for all $i=1,\ldots, k$, then
$$ w_1\wedge \ldots \wedge w_k = (\det A)\,v_1\wedge \ldots \wedge v_k\;.$$
More generally, two families
$\{v_1,\ldots, v_k\}$ and $\{w_1,\ldots, w_k\}$ of linearly independent vectors span the same
$k$-di\-men\-sional subspace if and only if for some real number
$\lambda\neq 0$, $ w_1\wedge \ldots \wedge w_k = \lambda \,v_1\wedge \ldots \wedge v_k$.
Hence we identify the line spanned by a simple $k$-vector $v=v_1\wedge \ldots \wedge v_k$,
i.e., the projective point $\hat{v}\in \mathbb{P}(\wedge_k V)$ determined by $v$, with the $k$-di\-men\-sional subspace spanned by the vectors $\{v_1,\ldots, v_k\}$,
denoted hereafter by $\linspan{ v_1\wedge \ldots \wedge v_k }$.
The subspaces $\wedge_k V$ induce the grading structure of the exterior algebra $\wedge_\ast V$, i.e.,
we have the direct sum decomposition $\wedge_\ast V=\oplus_{k=0}^{\dim V} \wedge_k V$
with $(\wedge_k V)\wedge (\wedge_{k'} V)\subset \wedge_{k+k'} V$
for all $0\leq k,k'\leq \dim V$.
Geometrically, the exterior product operation
$\wedge:\wedge_k V\times \wedge_{k'} V \to \wedge_{k+k'} V$
corresponds to the algebraic sum of linear subspaces, in the sense that given families $\{v_1,\ldots, v_k\}$ and $\{w_1,\ldots, w_k\}$ of linearly independent vectors such that
$\linspan{ v_1\wedge \ldots \wedge v_k } \cap
\linspan{ w_1\wedge \ldots \wedge w_{k'} }=0$, then
$$\linspan{ v_1\wedge \ldots \wedge v_k\, \wedge \, w_1\wedge \ldots \wedge w_{k'} } =
\linspan{ v_1\wedge \ldots \wedge v_k } +
\linspan{ w_1\wedge \ldots \wedge w_{k'} } \;. $$
Let $\Lambda_k^n$ be the set of all $k$-subsets
$I=\{i_1,\ldots, i_k\}\subset \{1,\ldots, n\}$,
with $i_1<\ldots <i_k$,
and order it lexicographically.
Given a basis $\{e_1, \ldots, e_n\}$ of $V$,
define for each $I\in\Lambda^n_k$, the $k$-th exterior product $e_I=e_{i_1}\wedge \ldots\wedge e_{i_k}$.
Then the ordered family $\{e_I\,\colon \,I\in\Lambda^n_k\}$ is a basis of $\wedge_k V$. In particular\,
$\dim \wedge_k V=\binom{n}{k}$.
The exterior algebra $\wedge_\ast V$ inherits an Euclidean
structure from $V$. More precisely, there is a unique inner product
on $\wedge_\ast V$ such that for any orthonormal basis $\{e_1, \ldots, e_n\}$ of $V$, the family
$\{\, e_I\,\colon\, I\in \Lambda^n_k, \; 0\leq k\leq n\,\}$
is an orthonormal basis of the exterior algebra $\wedge_\ast V$.
A simple $k$-vector
$v_1\wedge \ldots \wedge v_k$ of norm one will be called a {\em unit $k$-vector}. From the previous considerations
the correspondence
$v_1\wedge \ldots \wedge v_k\mapsto \linspan{v_1\wedge \ldots \wedge v_k}$ is one-to-one, between the set of unit $k$-vectors in $\wedge_k V$ and the set of oriented $k$-dimensional linear subspaces of $V$.
In particular, if $V$ is an oriented Euclidean space then the $1$-dimensional space $\wedge_n V$ has a canonical unit $n$-vector,
denoted by $\omega$, and called the {\em volume element} of $\wedge_n V$. In this case there is a unique operator,
called the {\em Hodge star} operator,
$\ast:\wedge_\ast V\to \wedge_\ast V$ defined by
$$ v\wedge (\ast w) = \langle v, w\rangle\, \omega,\quad \text{ for all }\; v,w\in \wedge_\ast V\;. $$
The Hodge star operator maps
$\wedge_k V$ isomorphically, and isometrically, onto $\wedge_{n-k} V$, for all
$0\leq k\leq n$.
Geometrically it corresponds to the
orthogonal complement operation on linear subspaces, i.e., for any simple $k$-vector,
$$ \linspan{\ast (v_1\wedge \ldots \wedge v_k) }
= \linspan{ v_1\wedge \ldots \wedge v_k }^\perp\;. $$
A dual product operation $\vee :\wedge_\ast V\times \wedge_\ast V\to \wedge_\ast V$ can be defined by
$$ v \vee w := \ast( (\ast v)\wedge (\ast w) ),\quad
\text{ for all }\; v,w\in \wedge_\ast V \;.$$
This operation maps
$\wedge_k V\times \wedge_{k'} V$ to $\wedge_{k+k'-n} V$, and describes the intersection operation on linear subspaces,
in the sense that given families $\{v_1,\ldots, v_k\}$ and $\{w_1,\ldots, w_k\}$ of linearly independent vectors with
$\linspan{ v_1\wedge \ldots \wedge v_k } +
\linspan{ w_1\wedge \ldots \wedge w_{k'} }= V$, then
$$\linspan{ (v_1\wedge \ldots \wedge v_k) \vee (w_1\wedge \ldots \wedge w_{k'}) } =
\linspan{ v_1\wedge \ldots \wedge v_k } \cap
\linspan{ w_1\wedge \ldots \wedge w_{k'} } \;. $$
By duality, this interpretation of the $\vee$-operation reduces to the previous ones on sums ($\wedge$) and complements ($\ast$).
Any linear map $g:V\to V$ induces
a linear map $\wedge_k g:\wedge_k V\to \wedge_k V$,
called the {\em $k$-th exterior power} of $g$,
such that for all $v_1,\ldots, v_k\in V$,
$$\wedge_k g(v_1\wedge \ldots \wedge v_k)=
g(v_1)\wedge \ldots \wedge g(v_k)\;.$$
This construction is functorial
in the sense that for all linear maps $g,g':V\to V$,
$$\wedge_k{\rm id}_V = {\rm id}_{\wedge_k V}, \;\; \wedge_k(g'\circ g)=\wedge_k g'\circ \wedge_k g\quad \;\; \text{ and }\;\;
\wedge_k g^\ast = (\wedge_k g)^\ast \;,$$
where $g^\ast:V\to V$ denotes the {\em adjoint operator}.
A clear consequence of these properties is that if $g:V\to V$ is an {\em orthogonal automorphism}, i.e.,
$g^\ast\circ g={\rm id}_V$, then so is
$\wedge_k g: \wedge_k V\to \wedge_k V$.
Consider a matrix $A\in{\rm Mat}_n(\mathbb{R})$.
Given $I,J\in\Lambda^ n_k$,
we denote by $A_{I\times J}$ the square sub-matrix of $A$
indexed in $I\times J$.
If a linear map $g:V\to V$ is represented by
$A$ relative to a basis $\{e_1, \ldots, e_n\}$,
then the $k$-th exterior power
$\wedge_k g:\wedge_k V\to \wedge_k V$ is represented
by the matrix $\wedge_k A:= (\det A_{I\times J})_{I,J}$ relative to the basis $\{e_I\,:\,I\in\Lambda^n_k\}$. The matrix $\wedge_k A$ is called the {\em $k$-th exterior power} of $A$.
Obviously, matrix exterior powers satisfy the same functorial properties
as linear maps, i.e., for all $A,A'\in{\rm Mat}_n(\mathbb{R})$,
$$\wedge_k{\rm I}_n = {\rm I}_{\binom{n}{k}}, \;\; \wedge_k(A' A)=(\wedge_k A') (\wedge_k g)\quad \;\; \text{ and }\;\;
\wedge_k A^\ast = (\wedge_k A)^\ast \;,$$
where $A^\ast$ denotes the {\em transpose} matrix of $A$.
\bigskip
\subsection{Grassmann manifolds} \label{grassmannians}
Grassmannians, like projective spaces, are compact Riemannian manifolds which stage the action of linear maps.
For each $0\leq k\leq n$, the {\em Grassmannian} ${\rm Gr}_k(V)$ is the space of all $k$-dimensional linear subspaces of $V$.
Notice that the projective space $\mathbb{P}(V)$ and the Grassmannian ${\rm Gr}_1(V)$ are the same object if we identify each point $\hat{v}\in\mathbb{P}(V)$ with the line $\langle v\rangle= \{\, \lambda\,v\,\colon\, \lambda\in\mathbb{R} \,\}$. The full Grassmannian ${\rm Gr}(V)$ is the union of all Grassmannians ${\rm Gr}_k(V)$ with $0\leq k\leq n$. Denote by $\mathcal{L}(V)$ the algebra of linear endomorphisms on $V$, and consider the map
$\pi:{\rm Gr}(V)\to \mathcal{L}(V)$, $E\mapsto \pi_E$, that assigns the orthogonal projection $\pi_E$ onto $E$,
to each subspace $E\in{\rm Gr}(V)$. This map is one-to-one, and we endow ${\rm Gr}(V)$ with the unique topology that makes the map $\pi:{\rm Gr}(V)\to \pi({\rm Gr}(V))$ a homeomorphsim. With it, ${\rm Gr}(V)$ becomes a compact space, and
each Grassmannian ${\rm Gr}_k(V)$ is a closed connected subspace of ${\rm Gr}(V)$.
The group ${\rm GL}(V)$ acts transitively on each Grassmannian.
The action of ${\rm GL}(V)$ on ${\rm Gr}_k(V)$ is given by
$\cdot:{\rm GL}(V)\times {\rm Gr}_k(V)\to {\rm Gr}_k(V)$,
$(g,E)\mapsto g\,E$.
The special orthogonal group ${\rm SO}(V)$,
of orientation preserving orthogonal automorphisms, acts transitively on Grassmannians too. All Grassmannians are compact homogeneous spaces.
For each $0\leq k\leq n$, the {\em Pl\"ucker} embedding is the map
$\psi:{\rm Gr}_k(V)\to \mathbb{P}(\wedge_k V)$ that to each subspace $E$ in ${\rm Gr}_k(V)$
assigns the projective point $\hat{v}\in \mathbb{P}(\wedge_k V)$,
where $v=v_1\wedge \ldots \wedge v_k$ is any simple $k$-vector formed as exterior product of a basis $\{ v_1,\ldots, v_k \}$ of $E$. This map is one-to-one and equivariant, i.e.,
for all $g\in{\rm GL}(V)$ and $E\in{\rm Gr}(V)$,
\begin{equation}\label{Plucker equivariance}
\psi(g\,E) = \varphi_{ \wedge_k g }\psi(E) \;.
\end{equation}
We will consider the metrics
$\rho, d,\delta :{\rm Gr}_k(V) \times {\rm Gr}_k(V)\to [0,+\infty)$ defined for any given $E,F\in{\rm Gr}_k(V)$ by
\begin{align}
\label{Grassmann:arc distance}
\rho(E,F) &:= \rho(\psi(E),\psi(F)) \;,\\
\label{Grassmann:distance}
d(E,F) &:= d(\psi(E),\psi(F)) \;,\\
\label{Grassmann:sine distance}
\delta(E,F) &:= \delta(\psi(E),\psi(F)) \;.
\end{align}
which assign diameter $\frac{\pi}{2}$, $\sqrt{2}$ and $1$, respectively, to the manifold ${\rm Gr}_k(V)$.
These distances are preserved by orthogonal linear maps in ${\rm SO}(V)$.
We also define the {\em minimum distance} between
any two subspaces $E,F\in{\rm Gr}(V)$,
$$ \delta_{{\rm min}}(E,F):= \min_{u\in E\setminus\{0\}, v\in F\setminus\{0\}} \delta(\hat{u},\hat{v})\;, $$
and the {\em Hausdorff distance} between
subspaces $E,F\in {\rm Gr}_k(V)$,
$$ \delta_{{\rm H}}(E,F):= \max\left\{ \,
\max_{u\in E\setminus\{0\}} \delta_{{\rm min}}(\hat{u},F),\,
\max_{v\in F\setminus\{0\}} \delta_{{\rm min}}(\hat{v},E)\,
\right\}\;. $$
\begin{definition}
\label{def orthog proj}
Given $E,F\in{\rm Gr}(V)$, we denote by
$\pi_F:V\to V$ the orthogonal projection onto $F$,
and by $\pi_{E,F}:E\to F$ the restriction of $\pi_F$ to $E$.
\end{definition}
\begin{proposition}
\label{delta, deltamin, deltaH}
Given $E,F\in{\rm Gr}_k(V)$,
\begin{enumerate}
\item[(a)]\; $\displaystyle \delta(E,F) = \sqrt{1-\det(\pi_{E,F})^2} =
\sqrt{1-\det(\pi_{F,E})^2} $,
\item[(b)]\; $\displaystyle \delta_{{\rm H}}(E,F)= \norm{\pi_{E,F^\perp}} = \norm{\pi_{F,E^\perp}} $,
\item[(c)]\; $\displaystyle \delta_{{\rm H}}(E,F) \leq \delta(E,F) $.
\end{enumerate}
\end{proposition}
\begin{proof}
Consider the unit $k$-vectors $e=\Psi(E)$ and $f=\Psi(F)$.
For (a) notice first that
$\delta(E,F)=\delta(e,f)=\sqrt{1-\langle e,f\rangle^2}$.
Since the exterior power $\wedge_k \pi_{F,E}:\wedge_k F\to \wedge_k E$ is also an orthogonal projection we have
$\langle e,f\rangle =
\langle e,\wedge_k \pi_{F,E} (f)\rangle = \norm{\wedge_k \pi_{F,E}} = \abs{\det(\pi_{F,E}) } $.
Given an orthogonal map $g\in{\rm SO}(V)$ such that
$g(F)=E$, we have $g^{-1}(E^\perp)=F^\perp$
and $\pi_{E,F^\perp}= g^{-1}\circ \pi_{F,E^\perp}\circ g$. Therefore $\norm{\pi_{E,F^\perp}} = \norm{\pi_{F,E^\perp}} $.
Item (b) follows because for any unit vector $u\in\hat{u}$, with $\hat{u}\in\mathbb{P}(E)$,
$$ \norm{\pi_{E,F^\perp} (u)} = \min_{v\in F\setminus\{0\}} \delta(\hat{u},\hat{v})\;. $$
Since $\pi_{E,F}$ is an orthogonal projection all its singular values are in the range $[0,1]$.
Hence, for any unit vector $u\in E$,
$\norm{\pi_{E,F}(u)}\geq \mathfrak{m}(\pi_{E,F})\geq \det(\pi_{E,F})$. Thus
\begin{align*}
\norm{\pi_{E,F^\perp} (u)}^2
= 1- \norm{\pi_{E,F} (u)}^2
\leq 1- \det(\pi_{E,F})^2\;,
\end{align*}
and (c) follows taking the maximum over all unit vectors
$u\in E$.
\end{proof}
Given $k,k'\geq 0$ such that $k+k'\geq n=\dim V$, the
intersection of subspaces is an operation
$\cap:{\rm Gr}_{k,k'}(\cap)\subset {\rm Gr}_k(V)\times {\rm Gr}_{k'}(V)\to {\rm Gr}_{k+k'-n}(V)$
where
\begin{definition}
\label{def intersection domain}
the domain is defined by
$$ {\rm Gr}_{k,k'}(\cap):=\{\,(E,E')\in {\rm Gr}_k(V)\times {\rm Gr}_{k'}(V)\,\colon\,
E+E'=V\, \}\;. $$
\end{definition}
Similarly, given $k,k'\geq 0$ such that $k+k'\leq n=\dim V$, the
algebraic sum of subspaces is operation
$+:{\rm Gr}_{k,k'}(+)\subset {\rm Gr}_k(V)\times {\rm Gr}_{k'}(V)\to {\rm Gr}_{k+k'-n}(V)$
where
\begin{definition}
\label{def sum domain}
the domain is defined by
$$ {\rm Gr}_{k,k'}(+):=\{\,(E,E')\in {\rm Gr}_k(V)\times {\rm Gr}_{k'}(V)\,\colon\,
E\cap E' =\{0\}\, \}\;. $$
\end{definition}
The considerations in subsection~ \ref{exterior algebra} show that
the Pl\"ucker embedding satisfies the following relations:
\begin{proposition}
\label{prop }
Given $E\in {\rm Gr}_k(V)$, $E'\in{\rm Gr}_{k'}(V)$,
consider unit vectors $v\in \Psi(E)$ and $v'\in\Psi(E')$.
\begin{enumerate}
\item[(a)]\, If $(E,E')\in {\rm Gr}_{k,k'}(\cap)$ \, then \,
$ \psi(E\cap E')=\widehat{v\vee v'}$.
\item[(b)]\, If $ (E,E')\in {\rm Gr}_{k,k'}(+)$ \, then \,
$\psi(E + E')=\widehat{v\wedge v'} $.
\end{enumerate}
\end{proposition}
A duality between sums and intersections stems from these facts.
\begin{proposition}\label{+cap duality}
The orthogonal complement operation $E\mapsto E^\perp$ is a $d$-isometric involution on ${\rm Gr}(V)$ which maps $ {\rm Gr}_{k,k'}(+)$ to $ {\rm Gr}_{n-k,n-k'}(\cap)$
and satisfies for all $(E,E')\in {\rm Gr}_{k,k'}(+)$,
\begin{equation*}
(E+E')^\perp = (E^\perp)\cap (E')^\perp \;.
\end{equation*}
\end{proposition}
The composition semigroup $\mathcal{L}(V)$ has two partial actions
on Grassmannians, called the {\em push-forward action}
and the {\em pull-back action}. Before introducing them a couple facts is needed.
\begin{definition}
\label{def kernel, range}
Given $g\in \mathcal{L}(V)$, we denote by
$\rm K g:=\{\, v\in V\,\colon\, g\,v=0\,\}$
the the {\em kernel of $g$}, and by
$\rm R g:=\{\, g\,v\,\colon\, v\in V\,\}$
the {\em range of $g$}.
\end{definition}
\begin{lemma}\label{push-forwards,pull-backs}
Given $g\in\mathcal{L}(V)$ and $E\in{\rm Gr}(V)$,
\begin{enumerate}
\item if $E\cap (\rm K g)=\{0\}$ then the linear map $g\vert_E:E\to g(E)$ is an isomorphism,
and in particular\, $\dim g(E)=\dim E$.
\item if $E + (\rm R g)= V$ then the linear map $g^ \ast\vert_{E^ \perp}:E^ \perp\to g^{-1}(E)^ \perp$ is an isomorphism,
and in particular\, $\dim g^{-1}(E)=\dim E$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first statement is obvious because
if $E\cap (\rm K g)=\{0\}$ then $\rm K (g\vert_E)=\{0\}$.
If $E + (\rm R g)=V$ then, since $\rm K g^\ast=(\rm R g)^\perp$,
we have $E^ \perp\cap (\rm K {g^ \ast})=E^ \perp\cap (\rm R {g})^ \perp=(E+\rm R g)^ \perp = \{0\}$.
Hence by 1, the linear map $g^ \ast\vert_{E^ \perp}:{E^ \perp}\to g^ \ast(E^ \perp)$ is an isomorphism.
It is now enough to remark that $ g^ \ast(E^ \perp) = g^{-1}(E)^ \perp$.
In fact, the inclusion $ g^ \ast(E^ \perp) \subset g^{-1}(E)^ \perp$ is clear.
Since $g^\ast\vert{E^\perp}$ is injective,
$\dim g^ \ast(E^ \perp) = \dim (E^\perp)$. On the other hand, by the transversality condition,
$g^{-1}(E)$ has dimension
\begin{align*}
\dim g^{-1}(E) &= \dim \left( (g\vert_{(\rm K g)^ \perp})^{-1}(E\cap \rm R g)\right) +\dim (\rm K g)\\
&= \dim (E\cap \rm R g) +\dim (\rm K g)\\
&= \dim (E)+\dim(\rm R g)-n +\dim (\rm K g) = \dim(E)\;.
\end{align*}
Hence both $ g^ \ast(E^ \perp) $ and $ g^{-1}(E)^ \perp$ have dimension equal to $\dim( E^ \perp)$, and the equality follows.
\end{proof}
Given $g\in \mathcal{L}(V)$ and $k\geq 0$ such that $k+\dim (\rm K g)\leq n=\dim V$,
the {\em push-forward by $g$} is the map
$ \varphi_g: {\rm Gr}_{k}(g)\subset {\rm Gr}_k(V) \to {\rm Gr}_{k}(V)$, $E\mapsto g E$,
where
\begin{definition}
\label{def push forward domain}
the domain is defined by
$$ {\rm Gr}_{k}(g):=\{\, E \in {\rm Gr}_k(V) \,\colon\,
E\cap (\rm K g)=\{0\}\, \}\;. $$
\end{definition}
Similarly, given $k\geq 0$ such that $k+\dim(\rm R g)\geq n=\dim V$, the
{\em pull-back by $g$} is the map
$ \varphi_{g^{-1}}: {\rm Gr}_{k}(g^{-1})\subset {\rm Gr}_k(V) \to {\rm Gr}_{k}(V)$, $E\mapsto g^{-1} E$,
where
\begin{definition}
the domain is defined by
\label{def pull back domain}
$$ {\rm Gr}_{k}(g^{-1}):=\{\, E \in {\rm Gr}_k(V) \,\colon\,
E + (\rm R g) =V \, \}\;. $$
\end{definition}
From the proof of proposition~\ref{push-forwards,pull-backs} we obtain a duality between push-forwards and pull-backs which can be expressed as follows.
\begin{proposition} \label{push-pull duality}
Given $g\in \mathcal{L}(V)$ and $k\geq 0$ such that $k+\dim (\rm R g)\geq n=\dim V$, we have
${\rm Gr}_k(g^{-1})={\rm Gr}_{n-k}(g^\ast)^\perp$ and for all $E\in {\rm Gr}_k(g^{-1})$,
\begin{equation*}
(g^{-1} E)^\perp = g^\ast(E^\perp) \;.
\end{equation*}
\end{proposition}
In section~\ref{le} we will derive modulus of Lipschitz continuity, w.r.t. the metric $\delta$, for the sum, intersection, push-forward and pull-back operations.
\subsection{Flag manifolds}
Let $V$ be a finite $n$-dimensional Euclidean space.
Any strictly increasing sequence of linear subspaces
$F_1\subset F_2\subset \ldots \subset F_{k}\subset V$ is called a {\em flag} in the Euclidean space $V$.
Formally, flags are denoted as lists $F=(F_1,\ldots, F_{k})$.
The sequence $\tau=(\tau_1,\ldots, \tau_k)$ of dimensions $\tau_j=\dim F_j$ is called the {\em signature} of the flag $F$.
The integer $k$ is called the {\em length} of the flag $F$, and the {\em length} of the signature $\tau$.
Let $\mathscr{F}(V)$ be the set of all flags in $V$, and define $ \mathscr{F}_\tau(V)$ to be the space of flags with a given signature $\tau$.
Two special cases of flag spaces are the projective
space $\mathbb{P}(V)=\mathscr{F}_\tau(V)$, when $\tau=(1)$,
and the Grassmannian ${\rm Gr}_k(V)=\mathscr{F}_\tau(V)$, when $\tau=(k)$.
The general linear group ${\rm GL}(V)$ acts naturally on $\mathscr{F}(V)$.
Given $g\in{\rm GL}(V)$ the action of $g$ on $\mathscr{F}_\tau(V)$ is given by the
map $\varphi_g:\mathscr{F}_\tau(V) \to \mathscr{F}_\tau(V)$, $\varphi_g F=(g F_1,\ldots, g F_{k})$. The special orthogonal subgroup ${\rm SO}(V)\subset {\rm GL}(V)$
acts transitively on $\mathscr{F}_\tau(V)$. Hence,
all flag manifolds $\mathscr{F}_\tau(V)$ are compact homogeneous spaces.
Each of them is a compact connected Riemannian manifold where the
group ${\rm SO}(V)$ acts by isometries.
Since $\mathscr{F}_\tau(V)\subset {\rm Gr}_{\tau_1}(V)\times {\rm Gr}_{\tau_2}(V)\times \ldots
\times {\rm Gr}_{\tau_k}(V)$, the product distances
\begin{align}
\label{rho:tau}
\rho_\tau(F,F') &=\max_{1\leq j\leq k} \rho(F_j,F_j') \\
\label{d:tau}
d_\tau(F,F') &=\max_{1\leq j\leq k} d(F_j,F_j') \\
\label{delta:tau}
\delta_\tau(F,F') &=\max_{1\leq j\leq k} \delta(F_j,F_j')
\end{align}
are equivalent to the Riemannian distance on $\mathscr{F}_\tau(V)$.
With these metrics, the flag manifold $\mathscr{F}_\tau(V)$ has diameter $\frac{\pi}{2}$, $\sqrt{2}$ and $1$, respectively.
The group ${\rm SO}(V)$ acts isometrically on $\mathscr{F}_\tau(V)$ with respect to these distances.
Given a signature $\tau=(\tau_1,\ldots, \tau_k)$,
if $n=\dim V$, we define
$$\tau^\perp :=(n-\tau_k, \ldots, n-\tau_1)\;. $$
When $\tau=(\tau_1,\ldots, \tau_k)$ we will write
$\tau^\perp=(\tau_1^\perp,\ldots, \tau_k^\perp)$,
where $\tau_j^\perp= n-\tau_{k+1-i}$.
\begin{definition}
\label{def flag orth complement}
Given a flag $F=(F_1,\ldots, F_k)\in\mathscr{F}_\tau(V)$, its
{\em orthogonal complement } is the ${\tau^\perp}$-flag\,
$F^\perp : =(F_k^\perp,\ldots, F_1^\perp)$.
\end{definition}
The map $\cdot^\perp:\mathscr{F}(V)\to \mathscr{F}(V)$ is an isometric involution on $\mathscr{F}(V)$,
mapping $\mathscr{F}_{\tau}(V)$ onto $\mathscr{F}_{{\tau^\perp}}(V)$.
The involution character, $(F^\perp)^\perp = F$
for all $F\in\mathscr{F}(V)$, is clear.
As explained in section~\ref{exterior algebra}, the Hodge star operator $\ast:\wedge_k V \to \wedge_{n-k} V$ is an isometry between these Euclidean spaces.
By choice of metrics on the Grassmannians, see~\eqref{Grassmann:distance}, the Pl\"ucker embeddings are isometries. Finally, the Pl\"ucker embedding conjugates the
orthogonal complement map $\cdot^\perp:{\rm Gr}_k(V)\to{\rm Gr}_{n-k}(V)$ with the Hodge star operator. Hence for each $0\leq k\leq n$, the map $\cdot^\perp:{\rm Gr}_k(V)\to{\rm Gr}_{n-k}(V)$ is
an isometry. The analogous conclusion for flags follows from the defintion of distance $d_\tau$.
Given $g\in \mathcal{L}(V)$ and a signature $\tau$ such that $\tau_i+\dim (\rm K g)\leq n$ for all $i$,
the {\em push-forward by $g$} on flags is the map
$ \varphi_g: \mathscr{F}_{\tau}(g)\subset \mathscr{F}_\tau(V) \to \mathscr{F}_{\tau}(V)$, $\varphi_g F := (g \,F_1,\ldots, g\,F_k)$,
where
\begin{definition}
\label{def push forward flag domain}
the domain is defined by
\begin{align*}
\mathscr{F}_{\tau}(g) &:=\{\, F \in \mathscr{F}_\tau(V) \,\colon\,
F_k\cap (\rm K g)=\{0\}\, \}\;.
\end{align*}
\end{definition}
Similarly, given a signature $\tau$ such that $\tau_i+\dim(\rm R g)\geq n$ for all $i$, the
{\em pull-back by $g$} on flags is the map
$ \varphi_{g^{-1}}: \mathscr{F}_{\tau}(g^{-1})\subset \mathscr{F}_\tau(V) \to \mathscr{F}_{\tau}(V)$, $\varphi_{g^{-1}} F := (g^{-1} F_1,\ldots, g^{-1} F_k)$, where
\begin{definition}
\label{def pull-back flag domain}
the domain is defined by
\begin{align*}
\mathscr{F}_{\tau}(g^{-1}) &:=\{\, F \in \mathscr{F}_\tau(V) \,\colon\,
F_1 + (\rm R g) =V \, \}\;.
\end{align*}
\end{definition}
The duality between duality between push-forwards and pull-backs is expressed as follows.
\begin{proposition} \label{push-pull duality: flags}
Given $g\in \mathcal{L}(V)$,\,
$\mathscr{F}_\tau(g^{-1})=\mathscr{F}_{\tau^\perp}(g^\ast)^\perp$ and for all $F\in \mathscr{F}_\tau(g^{-1})$,
\begin{equation*}
(\varphi_{g^{-1}} F)^\perp = \varphi_{g^\ast}(F^\perp) \;.
\end{equation*}
\end{proposition}
\subsection{ Projective action }
\begin{proposition}\label{proj:lip}
Given $p,q\in V\setminus\{0\}$,
$$ \norm{\frac{p}{\norm{p}} - \frac{q}{\norm{q}} } \leq
\max\{ \frac{1}{\norm{p}}, \frac{1}{\norm{q}} \}\,
\norm{p-q} \;.$$
\end{proposition}
\begin{proof}
Given to vectors $u,v\in V$ with $\norm{u}\geq \norm{v}=1$
we have
$$\norm{ \frac{u}{\norm{u}} - \frac{v}{\norm{v}} }\leq
\norm{u-v} \;. $$
Assume for instance that $\norm{p}\geq \norm{q}$,
so that
$$\max\{ \norm{p}^{-1}, \norm{q}^{-1} \} = \norm{q}^{-1}\;.$$
Applying the previous inequality with
$u= \frac{p}{\norm{q}}$ and
$v = \frac{q}{\norm{q}}$, we get
\begin{align*}
\norm{ \frac{p}{\norm{p}} -
\frac{q}{\norm{q}} } & = \norm{ \frac{u}{\norm{u}} - \frac{v}{\norm{v}} }
\leq \norm{u-v} = \norm{\frac{p}{\norm{q}}-
\frac{q}{\norm{q}}} \\
& =
\norm{q}^{-1} \,
\norm{p-q} =
\max\{ \norm{p}^{-1}, \norm{q}^{-1} \}\,
\norm{p-q}\;.
\end{align*}
\end{proof}
Given a linear map $g\in\mathcal{L}(V)$, the projective action of $g$ is given by the map
$\varphi_g:\mathbb{P}(g)\to\mathbb{P}(g^\ast)$,
$\varphi_g(\hat{p}):=\widehat{g\,p}$.
For any non collinear vectors $p,q\in V$ with $\norm{p}=\norm{q}=1$,
define
$$v_{p}(q):= \frac{ q- \langle p, q\rangle \,p}{\norm{q- \langle p, q\rangle \,p}} $$
to be the versor of the orthogonal projection
of $q$ onto $p^\perp$.
\begin{proposition} \label{Lip:proj:action}
Given $g\in\mathcal{L}(V)$, and points $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$,
$$ \frac{\delta(\varphi_g(\hat{p}), \varphi_g(\hat{q}))}{\delta(\hat{p},\hat{q})} = \frac{\norm{g p \wedge g v_p(q)}}{\norm{g\,p}\,\norm{g\,q}}\;. $$
\end{proposition}
\begin{proof}
Let $p\in \hat{p}$ and $q\in\hat{q}$ be unit vectors such that
$\theta=\angle(p,q)\in [0,\frac{\pi}{2}]$.
We can write $q= (\cos\theta)\,p + (\sin\theta)\,v_p(q)$.
Hence
$$ \delta(\hat{p},\hat{q})=\norm{p\wedge q}= (\sin\theta)\,\norm{p\wedge v_p(q)}= \sin\theta\;,$$
and
$$ \delta(\varphi_g(\hat{p}), \varphi_g(\hat{q}))=
\frac{\norm{ g\,p \wedge g\,q} }{\norm{g\,p}\,\norm{g\,q} } =
(\sin\theta)\,\frac{\norm{g p\wedge g v_p(q)}}{\norm{g\,p}\,\norm{g\,q} } \;.$$
\end{proof}
Given a unit vector $v\in V$, $\norm{v}=1$,
denote by $\pi_v,\pi_v^\perp:V\to V$
the orthogonal projections $\pi_v(x):=\langle v,x\rangle\, v$,
respectively $\pi_v^ \perp(x):=x -\langle v,x\rangle\, v$.
\begin{lemma}\label{diff:proj}
Given $u,v\in V$ non-collinear with $\norm{u}=\norm{v}=1$,
denote by $P$ the plane spanned by $u$ and $v$. Then
\begin{enumerate}
\item[(a)] is $\pi_v-\pi_u$ a self-adjoint endomorphism,
\item[(b)] $\rm K (\pi_v-\pi_u)= P^ \perp$,
\item[(c)] the restriction $\pi_v-\pi_u:P\to P$ is anti-conformal
with similarity factor $\abs{\sin\angle (u,v)}$,
\item[(d)] $\norm{\pi^\perp_v-\pi^\perp_u}=\norm{\pi_v-\pi_u}\leq \norm{v-u}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Item (a) follows because orthogonal projections are self-adjoint operators.
Given $w\in P^\perp$, we have $\pi_u(w)=\pi_v(w)=0$, which implies
$w\in \rm K (\pi_u-\pi_v)$. Hence $P^\perp \subset \rm K (\pi_u-\pi_v)$. Since $u$ and $v$ are non-collinear,
$\pi_u-\pi_v$ has rank $2$. Thus $\rm K (\pi_u-\pi_v) = P^\perp$, which proves (b).
For (c) we may assume that
$V=\mathbb{R}^2$ and consider $u=(u_1,u_2)$, $v=(v_1,v_2)$, with $u_1^ 2+u_2^ 2=v_1^ 2+v_2^ 2=1$.
The projections $\pi_u$ and $\pi_v$ are represented by the matrices
$$ U= \left(\begin{array}{cc}
u_1^2 & u_1 u_2 \\
u_1 u_2 & u_2^ 2
\end{array} \right)\quad \text{ and }\quad
V = \left(\begin{array}{cc}
v_1^2 & v_1 v_2 \\
v_1 v_2 & v_2^ 2
\end{array} \right)$$
w.r.t. the canonical basis. Hence $\pi_v-\pi_u$ is given by
$$ V-U= \left(\begin{array}{cc}
v_1^2-u_1^2 & v_1 v_2 - u_1 u_2 \\
v_1 v_2 - u_1 u_2 & v_2^ 2 - u_2^ 2
\end{array} \right) = \left(\begin{array}{cc}
\beta & \alpha \\
\alpha & -\beta
\end{array} \right)$$
where $\alpha=v_1 v_2 - u_1 u_2$ and $\beta= v_1^2-u_1^2 = -(v_2^ 2 - u_2^ 2)$.
This proves that the restriction of $\pi_v-\pi_u$ to the plane $P$ is
anti-conformal. The similarity factor of this map is
$$ \norm{\pi_v-\pi_u} = \norm{\pi_v(u)-u}=\norm{\pi_v^\perp(u) }
=\abs{\sin \angle(u,v)}$$
Finally, since $u - \langle v,u\rangle\,v\perp v$,
\begin{align*}
\norm{\pi^\perp_v-\pi^\perp_u}^2 &=\norm{\pi_v-\pi_u}^2 = \norm{\pi_v^\perp(u) }^2\\
&= \norm{ u - \langle v,u\rangle\,v}^2\\
&= \norm{u-v}^2- \norm{ v - \langle v,u\rangle\,v}^2\\
&\leq \norm{u-v}^2\;.
\end{align*}
\end{proof}
Given a point $\hat{p}\in\mathbb{P}(V)$, we identify the tangent to the projective space at $\hat{p}$ as $T_{\hat{p}}\mathbb{P}(V)=p^\perp$,
for any representative $p\in\hat{p}$.
\begin{proposition}\label{derivative varphig}
Given $g\in \mathcal{L}(V)$, $\hat{x} \in\mathbb{P}(g)$,
and a representative $x\in\hat{x}$,
the derivative of the map
$\varphi_g:\mathbb{P}(g)\to \mathbb{P}(g^\ast)$ at $\hat{x}$ is given by
$$ (D\varphi_g)_{\hat{x}}\, v =
\frac{g\,v-\langle \frac{g\,x}{\norm{g\,x}}, \, g\,v\rangle\, \frac{g\,x}{\norm{g\,x}} }{\norm{g\,x}} =
\frac{1}{\norm{g\,x}}\, \pi_{ g x/\norm{g x} }^ \perp (g\,v) $$
\end{proposition}
\begin{proof}
The sphere $\mathbb{S}(V):=\{\, v\in V\,\colon\,
\norm{v}=1\,\}$ is a double covering space of $\mathbb{P}(V)$, whose covering map is the canonical projection $\hat \pi:\mathbb{S}(V)\to\mathbb{P}(V)$.
With the identification $T_{\hat{p}}\mathbb{P}(V)=p^\perp$, the derivative of $\hat \pi$, $D{\hat \pi}_x:T_x\mathbb{S}(V)\to T_{\hat{x}}\mathbb{P}(V)$, is the identity linear map.
The map $\varphi_g$ lifts
to the map defined on the sphere by
$\widetilde{\varphi}_g(x):=\frac{g\,x}{\norm{g\,x}}$.
Hence we can identify the derivatives $(D\varphi_g)_{\hat{x}}$ and
$(D\widetilde{\varphi}_g)_x$. The explicit expression for
$(D\widetilde{\varphi}_g)_x v$ follows by a simple calculation.
\end{proof}
We will use the following closed ball notation
$$B^{(d)}(\hat{p},r):=\{\, \hat{x}\in\mathbb{P}(V)\,\colon\,
d(\hat{x},\hat{p})\leq r\,\} \;, $$
where the superscript emphasizes the distance in matter.
Given a projective map $f:X\subset \mathbb{P}(V)\to\mathbb{P}(V)$,
we denote by ${\rm Lip}_d(f)$ the least Lipschitz
constant of $f$ with respect to the distance $d$.
Next proposition refers to the projective metrics
$\delta$ and $\rho$ defined in subsection~\ref{subsection projective spaces}.
\begin{proposition} \label{proj:contr} Given $0<\kappa<1$ and $g\in\mathcal{L}(V)$ such that ${\rm gr}(g)\geq \kappa^{-1}$,
\begin{enumerate}
\item[(1)] $\varphi_g\left( B^{(\delta)}(\overline{\mathfrak{v}}(g),r) \right)\subset B^{(\delta)}(\overline{\mathfrak{v}}(g^\ast), \kappa\,r/\sqrt{1-r^ 2})$, \, for any $0<r <1$,
\item[(2)] $\varphi_g\left( B^{(\rho)}(\overline{\mathfrak{v}}(g),a) \right)\subset B^{(\rho)}(\overline{\mathfrak{v}}(g^\ast), \kappa\,\tan a)$, \, for any $0<a <\frac{\pi}{2}$,
\item[(3)] ${\rm Lip}_\rho( \varphi_g\vert_{ B^{(\delta)}(\overline{\mathfrak{v}}(g),r)} ) \leq
\kappa\, \frac{ r+\sqrt{1-r^2} }{1-r^2}$, \, for any $0<r <1$.
\end{enumerate}
\end{proposition}
\begin{proof}
Item (1) of this proposition follows from
proposition~\ref{prop expansion aangle} (b), because
$$ \delta(\hat{w},\overline{\mathfrak{v}}(g))<r \quad \text{ implies} \quad
\alpha(\hat{w},\overline{\mathfrak{v}}(g))=\sqrt{1-\delta(\hat{w},\overline{\mathfrak{v}}(g))^2}
\geq \sqrt{1-r^2}\;.$$
Item (2) reduces to (1), because we have
$\delta(\hat{u},\hat{v}) =\sin \rho(\hat{u},\hat{v})$, which implies that $B^{(\rho)}(\hat{v}, a)= B^{(\delta)}(\hat{v}, \sin a)$.
To prove (3), take unit vectors $v\in \overline{\mathfrak{v}}(g)$ and $v^\ast\in \overline{\mathfrak{v}}(g^\ast)$
such that $g\,v=\norm{g}\,v^\ast$.
Because $v$ is a $g$-most expanding vector,
$\norm{\pi^\perp_{v^\ast}\circ g} =\norm{g\circ \pi_v^\perp}\leq s_2(g)\leq \kappa\,\norm{g}$.
Given $\hat x$ such that $\delta(\hat x, \overline{\mathfrak{v}}(g))<r$,
and a unit vector $x\in\hat x$, by proposition~\ref{prop expansion aangle} (a)
$$ \frac{\norm{g}}{\norm{g x}}\leq \frac{1}{\alpha(\hat x,\overline{\mathfrak{v}}(g))} \leq \frac{1}{\sqrt{1-r^2}}\;. $$
Using item (b) of the same proposition we get
$$ \norm{ \widetilde{\varphi}_g(x) -v^\ast} \leq
\delta(\varphi_g(\hat x), \overline{\mathfrak{v}}(g^\ast))\leq\frac{\sigma(g)}{\alpha( \hat x, \overline{\mathfrak{v}}(g))}\,\delta( \hat x, \overline{\mathfrak{v}}(g))
\leq \frac{\kappa\, r}{\sqrt{1-r^2}}
$$
By proposition~\ref{derivative varphig} we have
$$ (D\varphi_g)_x\, v=\frac{1}{\norm{g x}}\,
\pi_{v^\ast}^\perp(g\,v) + \frac{1}{\norm{g x}}\left( \pi_{\widetilde{\varphi}_g(x)}^\perp - \pi_{v^\ast}^\perp\right)(g\,v)\;.
$$
Thus, by lemma~\ref{diff:proj} (d),
\begin{align*}
\norm{ (D\varphi_g)_x } &\leq
\frac{\kappa\,\norm{g}}{\norm{g x}}
+ \frac{\norm{ \widetilde{\varphi}_g(x) - v^\ast }\,\norm{g}}{\norm{g x}} \\
&\leq
\frac{\kappa}{\sqrt{1-r^2}}
+ \frac{\kappa\,r }{1-r^2}
= \frac{\kappa\,(r+\sqrt{1-r^2}) }{1-r^2} \;.
\end{align*}
Since $B^{(\delta)}(\overline{\mathfrak{v}}(g),r)$ is a convex Riemannian disk,
by the mean value theorem $\varphi_g\vert_ {B^{(\delta)}(\overline{\mathfrak{v}}(g),r)}$ has Lipschitz constant
$\leq \frac{\kappa\,(r+\sqrt{1-r^2}) }{1-r^2}$ with respect to distance $\rho$.
\end{proof}
\bigskip
\subsection{ Operations on flag manifolds }
As before let $V$ be a $n$-dimensional Euclidean space.
Recall that the Grassmann manifold ${\rm Gr}_k(V)$ identifies through the Pl\"ucker embedding with a submanifold of $\mathbb{P}(\wedge_k V)$. Up to a sign, $E\in {\rm Gr}_k(V)$
is identified with the unit $k$-vector $e = e_1\wedge \ldots \wedge e_k$
associated to any orthonormal basis $\{e_1,\ldots, e_k\}$ of $E$.
Recall that the Grassmann distance~\eqref{Grassmann:distance} on ${\rm Gr}_k(V)$ can be characterized by
$$ d(E_1, E_2):= \min\{ \norm{e_1-e_2}, \norm{e_1+e_2}\}\; , $$
where $e_j$ is a unit $k$-vector of $E_j$, for $j=1,2$.
\begin{definition}
Given $E,F\in{\rm Gr}(V)$, we say that $E$ and $F$ are
$(\cap)$ transversal \, iff\, $E+F=V$.
Analogously, we say that $E$ and $F$ are
$(+)$ transversal \, iff\, $E\cap F =\{0\}$.
\end{definition}
The following numbers quantify the transversality of two linear subspaces.
\begin{definition} Given $E\in {\rm Gr}_r(V)$ and $F\in {\rm Gr}_s(V)$,
consider a unit $r$-vector $e$ of $E$,
a unit $s$-vector $f$ of $F$, a unit $(n-r)$-vector $e^\perp$ of $E^\perp$,
and a unit $(n-s)$-vector $f^\perp$ of $F^\perp$.
We define
\begin{align*}
\theta_+(E,F) &:= \norm{e \wedge f}\;, \\
\theta_\cap(E,F) &:= \norm{e^\perp \wedge f^\perp } \;.
\end{align*}
Since the chosen unit vectors are unique up to a sign, these quantities are well-defined.
\end{definition}
\begin{remark}
\label{rmk dim constraint}
If $r + s>n$ then $\theta_+(E,F)=0$.
Similarly, if $r + s< n$ then $\theta_\cap(E,F)=0$.
\end{remark}
\begin{remark}
\label{rmk rel between transversalities}
Given $E,F\in{\rm Gr}(V)$,\,
$ \theta_\cap(E,F) = \theta_+(E^\perp,F^\perp)$.
\end{remark}
Next proposition establishes a Lispchitz modulus of continuity
for the sum and intersection operations on Grassmannians in terms of
the previous quantities.
\begin{proposition}
\label{sum:inters:modulus cont}
Given $r,s\in\mathbb{N}$ and $E,E'\in{\rm Gr}_r(V)$, $F,F'\in{\rm Gr}_s(V)$,
\begin{enumerate}
\item[(1)]\; $\displaystyle d(E + F, E' + F') \leq \max\left\{ \frac{1}{\theta_+ (E,F)},
\frac{1}{\theta_+ (E',F')} \right\} \,( d(E,E') + d(F,F') ) \;,$
\smallskip
\item[(2)]\; $\displaystyle d(E \cap F, E' \cap F') \leq \max\left\{ \frac{1}{\theta_\cap (E,F)},
\frac{1}{\theta_\cap (E',F')} \right\} \,( d(E,E') + d(F,F') ) \;.$
\end{enumerate}
\end{proposition}
\begin{proof} (1)\; Consider unit $r$-vectors $e$ and $e'$ representing the
subspaces $E$ and $E'$ respectively. Consider also
unit $s$-vectors $f$ and $f'$ representing the
subspaces $F$ and $F'$ respectively.
By Proposition~\ref{proj:lip}
\begin{align*}
d(E+F,E'+F') &= \norm{ \frac{e\wedge f}{\norm{e\wedge f}}
- \frac{e'\wedge f'}{\norm{e'\wedge f'}} } \\
&\leq K \, \norm{ e\wedge f - e'\wedge f' }\\
&\leq K \, ( \norm{ e\wedge (f-f') } +
\norm{ (e - e')\wedge f' } )\\
&\leq K \, ( \norm{ e- e' } +
\norm{f - f' } )\\
\end{align*}
where
$K = \max\{\norm{e\wedge f}^{-1},
\norm{e'\wedge f'}^{-1} \} =
\max\{\theta_{+}(E,F)^{-1},
\max\{\theta_{+}(E',F')^{-1} \}$.
\noindent
(2) reduces to (1) by duality (see Proposition~\ref{+cap duality}).
\end{proof}
Next proposition gives an alternative characterization of the transversality measurements $\theta_+(E,F)$ and $\theta_\cap(E,F)$.
Let, as before, $\pi_E:V\to E$ denote the orthogonal projection onto a subspace $E\subset V$,
and define the restriction $\pi_{E,F}:= \pi_F\vert_E:E\to F$.
\begin{proposition}
\label{prop theta continuity modulus}
Given $E\in{\rm Gr}_r(V)$ and $F\in{\rm Gr}_s(V)$,
\begin{enumerate}
\item[(1)] \; $\theta_+(E,F) = \abs{\det (\pi_{E,F^\perp})} = \abs{\det (\pi_{F,E^\perp})}$.
\item[(2)] \; $\theta_\cap(E,F) = \abs{\det (\pi_{E^\perp,F})} = \abs{\det (\pi_{F^\perp,E})}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Notice that $E\cap F = \rm K (\pi_{E,F^\perp})= \rm K(\pi_{F,E^\perp})$.
If $E\cap F\neq \emptyset$ then the three terms in (1) vanish. Otherwise $\pi_{E,F^\perp}$ and $\pi_{F,E^\perp}$ are isomorphisms.
Take an orthonormal basis $\{f_1,\ldots, f_s, f_{s+1},\ldots, f_{s+r},\ldots, f_n\}$
such that $\{ f_1,\ldots, f_s\}$ spans $F$ and the family of vectors
$\{f_1,\ldots, f_r, f_{s+1},\ldots, f_{s+r}\}$ spans $E+F$.
Consider the unit $s$-vector
$f=f_1\wedge\ldots \wedge f_s$ of $F$,
and a unit $r$-vector $e=e_1\wedge \ldots \wedge e_r$ of $E$. Then
\begin{align*}
\theta_+(E,F) &= \norm{(e_1\wedge \ldots \wedge e_r)\wedge (f_1\wedge \ldots \wedge f_s)}\\
&= \norm{\pi_{E, F^\perp}(e_1)\wedge \ldots \wedge \pi_{E, F^\perp}(e_r)\wedge f_1\wedge \ldots \wedge f_s }\\
&= \abs{\det (\pi_{E, F^\perp})}\, \norm{f_{s+1}\wedge \ldots \wedge f_{s+r}\wedge f_1\wedge \ldots \wedge f_s }
= \abs{\det (\pi_{E, F^\perp})}\;.
\end{align*}
Reversing the roles of $E$ and $F$, and because $\norm{e\wedge f}$ is symmetric in $e$ and $f$,
we obtain $\theta_+(E,F) = \abs{\det \pi_{F, E^\perp}}$, which proves (1).
By duality and remark~\ref{rmk rel between transversalities}, item (2) reduces to (1).
\end{proof}
The measurement on the $(\cap)$ transversality admits the following lower bound in terms of the angle in definition~\ref{grassmann rho delta alpha}.
\begin{proposition}
Given $E\in {\rm Gr}_r(V)$ and $F\in {\rm Gr}_s(V)$,
if $E+F=V$ then
$$ \theta_{\cap}(E,F)\geq \alpha_r(E, E\cap F + F^\perp)\;.$$
\end{proposition}
\begin{proof}
Combining lemmas~\ref{theta:monot} and ~\ref{theta aangle} we have
\begin{align*}
\theta_{\cap}(E,F) & \geq \theta_{\cap}(E,F\cap(E\cap F)^\perp) = \alpha_r(E, (F\cap(E\cap F)^\perp)^\perp)\\
&= \alpha_r(E, (E\cap F) + F^\perp )\;.
\end{align*}
\end{proof}
\begin{lemma}\label{theta:monot}
Given $E\in {\rm Gr}_r(V)$, $E'\in {\rm Gr}_{r'}(V)$ and $F\in {\rm Gr}_s(V)$
such that $r+s\geq n$ and $E\subseteq E'$ then
$\displaystyle \theta_{\cap}(E',F)\geq \theta_{\cap}(E,F)$.
\end{lemma}
\begin{proof} Because $E\subset E'$, we have
$\pi_{F^ \perp, E} = \pi_{E', E}\circ \pi_{F^ \perp, E'}$. Hence
\begin{align*}
\theta_{\cap}(E,F) &= \abs{\det(\pi_{F^ \perp, E}) } = \abs{\det(\pi_{\pi_{E'}(F^ \perp), E})}\,\abs{\det( \pi_{F^ \perp, E'})} \\
&\leq \abs{\det( \pi_{F^ \perp, E'})} = \theta_{\wedge}(E',F) \;,
\end{align*}
where $\abs{\det(\pi_{\pi_{E'}(F^ \perp), E})} \leq 1$ because $\norm{ \pi_{E} } \leq 1$.
\end{proof}
\begin{lemma}\label{theta aangle}
Given $E, E'\in {\rm Gr}_r(V)$, \;
$\displaystyle \theta_{\cap}(E',E^ \perp) = \alpha_r(E',E)$.
\end{lemma}
\begin{proof}
Take orthonormal basis $\{v_1,\ldots, v_r\}$ of $E$, and
$\{v_1',\ldots, v_r'\}$ of $E'$. Then
\begin{align*}
\theta_{\cap}(E',E^\perp) &= \abs{ \det (\pi_{E',E} ) } \\
&= \abs{\langle \wedge_r \pi_{E,E'} (v_1 \wedge \ldots \wedge v_r), v_1'\wedge \ldots \wedge v_r'\rangle } \\
&= \abs{\langle \pi_{E'}(v_1)\wedge \ldots \wedge \pi_{E'}(v_r), v_1'\wedge \ldots \wedge v_r'\rangle } \\
&= \abs{\langle v_1\wedge \ldots \wedge v_r, v_1'\wedge \ldots \wedge v_r'\rangle }
=\alpha_r(E,E')\;.
\end{align*}
\end{proof}
Next proposition gives a modulus of lower semi-continuity for the transversality measurement $\theta_{\cap}$.
\begin{proposition}\label{theta cap: E->E0}
Given $E, E_0\in {\rm Gr}_r(V)$ and $F, F_0\in {\rm Gr}_s(V)$,
$$\theta_{\cap}(E,F)\geq \theta_{\cap}(E_0,F_0)
- d(E,E_0)-d(F,F_0)\;.$$
\end{proposition}
\begin{proof}
Consider unit vectors
$e\in\Psi(E^\perp)$, $f\in\Psi(F^\perp)$,
$e_0\in\Psi(E_0^\perp)$ and $f_0\in\Psi(F_0^\perp)$,
chosen so that
\begin{align*}
d(E,E_0) &=d(E^\perp,E_0^\perp)=\norm{e-e_0}\;,\\
d(F,F_0) &=d(F^\perp,F_0^\perp)=\norm{f-f_0}\;.
\end{align*}
Hence
\begin{align*}
\theta_{\cap}(E,F) &= \norm{e\wedge f} \geq
\norm{e_0\wedge f_0} -\norm{e\wedge f - e_0\wedge f_0}\\
&\geq \theta_{\cap}(E_0,F_0) -
\norm{ e\wedge (f-f_0)} -\norm{(e-e_0)\wedge f_0}\\
&\geq \theta_{\cap}(E_0,F_0) -
\norm{f-f_0} -\norm{e-e_0}\\
&\geq \theta_{\cap}(E_0,F_0) -
d(F,F_0) -d(E,E_0)\;.
\end{align*}
\end{proof}
The exterior product is a continuous operation.
A lower bound on its modulus of continuity can be expressed in terms of the angle between the arguments.
\begin{proposition}\label{prop norm{u wedge v} >= norm{u} norm{v}}
Given $E,F\in{\rm Gr}_k(V)$, and families of vectors
$\{u_1,\ldots, u_k\}\subset E$ and $\{u_{k+1},\ldots, u_{k+i}\}\subset F^\perp$ with $1\leq i\leq m-k$,
\begin{enumerate}
\item[(a)] $\displaystyle \norm{u_1\wedge \ldots \wedge u_k\wedge u_{k+1}\wedge \ldots \wedge u_{k+i}} \leq \norm{u_1\wedge \ldots \wedge u_k}\,\norm{u_{k+1}\wedge \ldots \wedge u_{k+i}}$,
\item[(b)] $\displaystyle \norm{u_1\wedge \ldots \wedge u_k\wedge u_{k+1}\wedge \ldots \wedge u_{k+i}} \geq \alpha(E,F)\,\norm{u_1\wedge \ldots \wedge u_k}\,\norm{u_{k+1}\wedge \ldots \wedge u_{k+i}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Since $\pi_{F^\perp,E^\perp}$ is an orthogonal projection, all its singular values are in $[0,1]$.
Thus, because $\abs{\det (\pi_{F^\perp,E^\perp})}$ is the product of all singular values, while $\mathfrak{m}(\wedge_i \, \pi_{F^\perp,E^\perp})$ is the product of the
$i$ smallest singular values, we have
$$\abs{\det (\pi_{F^\perp,E^\perp})}
\leq \mathfrak{m}(\wedge_i \, \pi_{F^\perp,E^\perp})
\leq \norm{ \wedge_i \, \pi_{F^\perp,E^\perp} }\leq 1\;.$$
Hence
\begin{align*}
\norm{u_1\wedge \ldots \wedge u_k\wedge u_{k+1}\wedge \ldots \wedge u_{k+i}}
&= \norm{u_1\wedge \ldots \wedge u_k\wedge \pi_{F^\perp,E^\perp}(u_{k+1})\wedge \ldots \wedge \pi_{F^\perp,E^\perp}(u_{k+i})} \\
&= \norm{ u_1 \wedge \ldots u_k}\,
\norm{ \pi_{F^\perp,E^\perp}(u_{k+1})\wedge \ldots \pi_{F^\perp,E^\perp}(\wedge u_{k+i})} \\
&\leq \norm{\wedge_i \, \pi_{F^\perp,E^\perp}}\, \norm{ u_1\wedge \ldots \wedge u_k}\,\norm{ u_{k+1}\wedge \ldots \wedge u_{k+i}} \\
&\leq \norm{ u_1\wedge \ldots \wedge u_k}\,\norm{ u_{k+1}\wedge \ldots \wedge u_{k+i}} \;,
\end{align*}
which proves (a).
By proposition~\ref{prop: alpha = det Pi(E F)} we have
$$\alpha(E,F)=\alpha(F^\perp,E^\perp)=\abs{\det (\pi_{F^\perp,E^\perp})}
\leq \mathfrak{m}( \wedge_i (\pi_{F^\perp,E^\perp}))\;.
$$
Thus
\begin{align*}
\norm{u_1\wedge \ldots \wedge u_k\wedge u_{k+1}\wedge \ldots \wedge u_{k+i}}
&= \norm{u_1\wedge \ldots \wedge u_k\wedge \pi_{F^\perp,E^\perp}(u_{k+1})\wedge \ldots \wedge \pi_{F^\perp,E^\perp}(u_{k+i})} \\
&= \norm{ u_1 \wedge \ldots u_k}\,
\norm{ \pi_{F^\perp,E^\perp}(u_{k+1})\wedge \ldots \pi_{F^\perp,E^\perp}(\wedge u_{k+i})} \\
&\geq \mathfrak{m}(\wedge_i \, \pi_{F^\perp,E^\perp})\, \norm{ u_1\wedge \ldots \wedge u_k}\,\norm{ u_{k+1}\wedge \ldots \wedge u_{k+i}} \\
&\geq \alpha(E,F)\, \norm{ u_1\wedge \ldots \wedge u_k}\,\norm{ u_{k+1}\wedge \ldots \wedge u_{k+i}} \;,
\end{align*}
which proves (b).
\end{proof}
Of course the angle function $\alpha$ is Lipschitz continuous.
\begin{proposition}
\label{prop aangle continuity}
Given $u,u',v,v'\in \mathbb{P}(V)$,
$$ \abs{\alpha(u,v) -\alpha(u',v')} \leq d(u,u') + d(v,v') \;. $$
\end{proposition}
\begin{proof}
Exercise.
\end{proof}
\bigskip
The intersection of complementary flags satisfying the appropriate transversality conditions determines a decomposition of the Euclidean space $V$.
We end this subsection proving a modulus of continuity for this intersection operation.
Consider a signature $\tau=(\tau_1,\ldots, \tau_k)$ of length $k$ with $\tau_k<\dim V$.
We make the convention that
$\tau_0=0$ and $\tau_{k+1}=\dim V$.
\begin{definition}\label{def decomposition}
A $\tau$-decomposition is a family of linear subspaces $E_{\cdot}=\{E_i\}_{1\leq i\leq k+1}$ in ${\rm Gr}(V)$ such that $V=\oplus_{i=1}^{k+1} E_i$ and
$\dim E_i=\tau_i-\tau_{i-1}$ for all
$1\le i\leq k+1$.
\end{definition}
Let $\mathscr{D}_\tau(V)$ denote
the space of all $\tau$-decompositions, which is a metric space with the distance
$$ d_\tau(E_{\cdot}, E_{\cdot}')=
\max_{1\leq i \leq k+1} d_{\tau_i-\tau_{i-1}}(E_i,E_i')\;, $$
and where $d_{\tau_i-\tau_{i-1}}$ stands for the distance~\eqref{Grassmann:distance} in ${\rm Gr}_{\tau_i-\tau_{i-1}}(V)$.
Given two flags $F \in\mathscr{F}_\tau(V)$ and $F' \in\mathscr{F}_{\tau^\perp}(V)$, we will define a decomposition, denoted by $F\sqcap F'$, formed out of intersecting the components of these flags.
For that we introduce the following a measurement.
\begin{definition}
\label{def sqcap tranvsersality}
Given two flags $F \in\mathscr{F}_\tau(V)$ and $F' \in\mathscr{F}_{\tau^\perp}(V)$, let
$$ \theta_{\sqcap}(F,F'):=\min_{1\leq i\leq k}
\theta_{\cap}(F_i,F_{k-i+1}')\;.$$
\end{definition}
Notice that $\dim F_i=\tau_i$ and
$\dim F_{k-i+1}'=\tau_{k-i+1}^\perp=\dim V -\tau_i$, i.e., the subspaces $F_i$ and $F_{k-i+1}'$ have complementary dimensions. We will refer to this quantity as the measurement of the transversality between the flags $F$ and $F'$.
In the next proposition we complete $F$ and $F'$ to full flags of length $k+1$ setting $F_{k+1}=F_{k+1}'=V$.
Assume also that $\tau_0=0$ and $\tau_{k+1}=\dim V$.
\begin{proposition}
\label{label theta sqcap >0 decomp well-def}
If $\theta_{\sqcap}(F,F')>0$ then the following is a direct sum decomposition in the space $\mathscr{D}_\tau(V)$,
$$ V=\bigoplus_{i=1}^{k+1} F_i\cap F_{k-i+2}' \;, $$
with $\dim (F_i\cap F_{k-i+2}') = \tau_{i}-\tau_{i-1}$ for all $1\leq i\leq k+1$.
\end{proposition}
\begin{proof}
Since the subspaces $F_i$ and $F_{k-i+1}'$ have complementary dimensions, the relation $\theta_{\cap}(F_i,F_{k-i+1}')>0$ implies that
\begin{equation}\label{V=Fi+Fk-i+1'}
V=F_i\oplus F_{k-i+1}'\;.
\end{equation}
By lemma~\ref{theta:monot},
$\theta_{\cap}(F_i,F_{k-i+2}')\geq \theta_{\cap}(F_i,F_{k-i+1}')>0$. Therefore
$F_i + F_{k-i+2}'=V$ and
\begin{align*}
\dim(F_i\cap F_{k-i+2}') &=
\tau_i + \tau_{k-i+2}^\perp-\dim V\\
&=
\tau_i + (\dim V - \tau_{i-1})-\dim V = \tau_i-\tau_{i-1}\;.
\end{align*}
We prove by finite induction in $i=1,\ldots, k+1$
that
\begin{equation}\label{Fi oplus}
F_i= \bigoplus_{j\leq i} F_j\cap F_{k-j+2}'\;.
\end{equation}
Since $F_{k+1}=V$ the proposition will follow from this relation at $i=k+1$.
For $i=1$, ~\eqref{Fi oplus} reduces to $F_1=F_1\cap V$. The induction step follows
from
$$ F_{i+1}=F_{i}\oplus \left( F_{i+1}\cap F_{k-i+1}'\right) \;.$$
Since the following dimensions add up
\begin{align*}
\dim F_{i+1}=\tau_{i+1} & = \tau_{i} + (\tau_{i+1}-\tau_{i}) \\
& = \dim F_{i} +\dim (F_{i+1}\cap F_{k-i+1}') \;,
\end{align*}
it is enough to see that
$$ F_{i}\cap \left( F_{i+1}\cap F_{k-i+1}'\right) = F_{i}\cap F_{k-i+1}' =\{0\}\;, $$
which holds because of~\eqref{V=Fi+Fk-i+1'}.
\end{proof}
Hence, by the previous proposition we can define
\begin{definition}
\label{def sqcap decomp operation}
Given flags $F\in \mathscr{F}_\tau(V)$ and $F'\in\mathscr{F}_{\tau^\perp}(V)$ such that $\theta_{\sqcap}(F,F')>0$ we define
$F\sqcap F':= \{ F_i\cap F_{k-i+2}'\}_{1\leq i\leq k+1}$ and call it the intersection decomposition of the flags $F$ and $F'$.
\end{definition}
Next proposition gives a modulus of lower semi-continuity for the transversality measurement $\theta_{\sqcap}$.
\begin{proposition}
\label{theta cap: F,F'->F0,F0'}.
Given $F, F_0\in \mathscr{F}_\tau(V)$ and $F', F_0'\in \mathscr{F}_{\tau^\perp}(V)$,
$$\theta_{\sqcap}(F,F')\geq \theta_{\sqcap}(F_0,F_0')
- d_{\tau}(F,F_0)-d_{\tau^\perp}(F',F_0')\;.$$
\end{proposition}
\begin{proof}
Apply proposition~\ref{theta cap: E->E0}.
\end{proof}
The modulus of continuity for the intersection map
$\sqcap:\mathscr{F}_\tau(V)\times \mathscr{F}_{\tau^\perp}(V)\to \mathscr{D}_\tau(V)$ is established below.
\begin{proposition}
\label{decomp:modulus cont}
Given flags $F_1, F_2\in\mathscr{F}_\tau(V)$ and $F_1', F_2'\in\mathscr{F}_{\tau^\perp}(V)$,
$$ d_\tau(F_1\sqcap F_1', F_2\sqcap F_2') \leq \max\left\{ \frac{1}{\theta_\sqcap (F_1,F_1')},
\frac{1}{\theta_\sqcap (F_2,F_2')} \right\} \,( d_{\tau}(F_1,F_2) + d_{\tau^\perp}(F_1',F_2') ) \;.$$
\end{proposition}
\begin{proof}
The proof reduces to apply proposition~\ref{sum:inters:modulus cont}.
\end{proof}
Any two linear maps $g_0,g_1\in\mathcal{L}(V)$
having $\tau$-gap ratios, and such that $\alpha_\tau(g_0,g_1)>0$, determine a $\tau$-decomposition of $V$ as intersection of the image by $\varphi_g$ of the $g_0$ most expanding $\tau$-flag with the $g_1$ least expanding $\tau^\perp$-flag. Recall definitions~\ref{def most expanding flag} and~\ref{def least expanding flag}.
The corresponding intersection measurement is bounded below by the angle $\alpha_\tau(g_0,g_1)$.
\begin{proposition}
\label{aangle bound}
Given $g_0,g_1\in\mathcal{L}(V)$, if ${\rm gr}_\tau(g_0)>1$ and ${\rm gr}_\tau(g_1)>1$ then
$$\theta_{\sqcap}(\underline{\mathfrak{v}}_{\tau^\perp}(g_1),
\overline{\mathfrak{v}}_\tau(g_0^\ast) )\geq \alpha_\tau(g_0,g_1)\;.$$
In particular, if $\alpha_\tau(g_0,g_1)>0$ the flags $\overline{\mathfrak{v}}_\tau(g_0^\ast)$ and $\underline{\mathfrak{v}}_{\tau^\perp}(g_1)$ determine the decomposition $\overline{\mathfrak{v}}_\tau(g_0^\ast)\sqcap \underline{\mathfrak{v}}_{\tau^\perp}(g_1) \in\mathscr{D}_\tau(V)$.
\end{proposition}
\begin{proof} Let $n=\dim V$.
Consider the flags $F= \overline{\mathfrak{v}}_{\tau}(g_0^\ast)$ and $F' = \underline{\mathfrak{v}}_{{\tau}^\perp}(g_1)$.
We have $F_i= \overline{\mathfrak{v}}_{\tau_i}(g_0^\ast)$ \, and \,
$F_{k-i+1} = \underline{\mathfrak{v}}_{\tau_{k-i+1}^\perp}(g_1) = \underline{\mathfrak{v}}_{n-\tau_{i}}(g_1) = \overline{\mathfrak{v}}_{\tau_i}(g_1)^\perp$. Hence by lemma~\ref{theta aangle},
$$ \theta_{\cap}(F_i,F_{k-i+1}') =
\theta_{\cap}(\overline{\mathfrak{v}}_{\tau_i}(g_0^\ast),\overline{\mathfrak{v}}_{\tau_i}(g_1)^\perp) = \alpha_{\tau_i }(\overline{\mathfrak{v}}_{\tau_i}(g_0^\ast),\overline{\mathfrak{v}}_{\tau_i}(g_1)) = \alpha_{\tau_i }(g_0,g_1)\;, $$
and taking the minimum,\,
$\theta_{\sqcap}(F,F')\geq \alpha_{\tau}(g_0,g_1)$.
\end{proof}
\subsection{ Dependence on the linear map}
We establish a modulus of Lipschitz continuity for the most expanding direction of a linear endomorphism with a gap between its first and second singular values. For any $0<\kappa<1$, consider the set
$\mathcal{L}_\kappa :=\{\, g\in \mathcal{L}(V)\,:\,
{\rm gr}(g)\geq \frac{1}{\kappa} \,\}$. We denote by $\overline{\mathfrak{v}}:\mathcal{L}_\kappa\to\mathbb{P}(V)$ the map that assigns the $g$-most expanding direction to each $g\in \mathcal{L}_\kappa$.
The {\em relative distance} between
linear maps $g,g'\in \mathcal{L}(V)\setminus\{0\}$ is defined as
$$ d_{{\rm rel}} (g,g'):=\frac{ \norm{g-g'} }{ \max\{ \norm{g}, \norm{g'} \}}\;. $$
Notice that this relative distance is not a metric.
It does not satisfy the triangle inequality. We introduce it just to lighten the notation.
\begin{proposition}\label{lipschitz:eigendir}
The map $\overline{\mathfrak{v}}:\mathcal{L}_\kappa\to\mathbb{P}(V)$ is locally Lipschitz.
More precisely, given $0<\kappa<1$ there exists $\varepsilon_0 >0$ such that for any $g_1,g_2\in \mathcal{L}_\kappa$ satisfying $d_{{\rm rel}} (g_1,g_2)\leq \varepsilon_0 $,
$$d(\overline{\mathfrak{v}}(g_1), \overline{\mathfrak{v}}(g_2))\leq \frac{16}{1-\kappa^2}\, d_{{\rm rel}} (g_1,g_2) \;. $$
\end{proposition}
\begin{proof}
Let $g\in \mathcal{L}_\kappa$ and $\lambda>0$.
The singular values (resp. singular vectors) of $g$ are the eigenvalues (resp. eigenvectors) of $\sqrt{g^\ast\, g}$.
Hence $s_j(\lambda\,g)= \lambda\, s_j(g)$ , for all $j$. We also have
$\overline{\mathfrak{v}}(\lambda g)=\overline{\mathfrak{v}}(g)$ and ${\rm gr}(\lambda\, g)={\rm gr}(g)$.
Consider the subspace
$\mathcal{L}_\kappa(1):=\{\, g\in \mathcal{L}_\kappa\, \colon\, \norm{g}=1\,\}$.
The projection $g\mapsto g/\norm{g}$ takes
$\mathcal{L}_\kappa$ to $\mathcal{L}_\kappa(1)$.
It also satisfies $\overline{\mathfrak{v}}(g/\norm{g})=\overline{\mathfrak{v}}(g)$
and
$$ \norm{ \frac{g_1}{\norm{g_1}} - \frac{g_2}{\norm{g_2}} }\leq
2\, d_{{\rm rel}} (g_1,g_2) \;. $$
Hence we can focus our attention on the
restricted map
$\overline{\mathfrak{v}}:\mathcal{L}_\kappa(1)\to\mathbb{P}(V)$.
Let $\mathcal{L}^+_\kappa(1)$ denote the subspace of $g\in \mathcal{L}_\kappa(1)$ such that
$g=g^ \ast\geq 0$, i.e., $g$ is positive semi-definite.
Given $g \in\mathcal{L}_\kappa(1)$,
we have $\norm{g^\ast\, g}=1=\norm{g}$, ${\rm gr}(g^\ast g)={\rm gr}(g)^2$ and $\overline{\mathfrak{v}}(g^\ast g)=\overline{\mathfrak{v}}(g)$.
Also, for all $g_1, g_2\in\mathcal{L}_\kappa(1)$,
\begin{align*}
\norm{g_1^\ast\,g_1 - g_2^\ast\,g_2 } &\leq
\norm{g_1^\ast}\,\norm{g_1 - g_2 } +
\norm{g_1^\ast - g_2^\ast } \,\norm{g_2}\\
&= ( \norm{g_1^\ast} + \norm{g_2})\,\norm{g_1 - g_2 }
\leq 2\,\norm{g_1-g_2}\;.
\end{align*}
Hence, the mapping $g\mapsto g^\ast\,g$ takes $\mathcal{L}_\kappa(1)$ to $\mathcal{L}^+_{\kappa^2}(1)$ and has Lispschitz constant $2$.
Therefore, it is enough to prove that the
restricted map
$\overline{\mathfrak{v}}:\mathcal{L}^+_{\kappa^ 2}(1)\to\mathbb{P}(V)$
has (locally) Lipschitz constant $4\,(1-\kappa^2)^{-1}$.
Let $\delta_0$ be a small positive number and take
$0<\varepsilon_0\ll \frac{ \delta_0}{4}$.
The size of $\delta_0$ will be fixed throughout the rest of the proof
according to necessity.
Take $h_1,h_2\in\mathcal{L}_{\kappa^2}^+(1)$
such that $\norm{h_1-h_2}<\varepsilon_0$ and set
$\hat{p}_0:=\overline{\mathfrak{v}}(h_1)$. By Proposition~\ref{proj:contr} we have
$$\varphi_{h_1}\left(B(\hat{p}_0,\delta_0) \right) \subset B \left(\hat{p}_0, \frac{\kappa^2\delta_0}{\sqrt{1-\delta_0^2}} \right)
\subset B(\hat{p}_0, \delta_0 ) \;, $$
where all balls refer to the projective sine-metric $\delta$
defined in~\eqref{projective sine metric}.
The second inclusion holds if $\delta_0$ is chosen small enough.
Take any $\hat{p}\in B(\hat{p}_0, \delta_0)$ and choose unit vectors
$p\in\hat{p}$ and $p_0\in\hat{p}_0$ such that $\langle p,p_0\rangle>0$.
Then $p = \langle p,p_0\rangle\,p_0 + w$, with $w\in p_0^\perp$,
$h_1(p_0)= p_0$ and
$h_1(w)\in p_0^\perp$. Hence
\begin{align*}
\norm{h_1(p)} & = \norm{\langle p,p_0\rangle\,p_0 +
h_1(w) } \geq \langle p,p_0\rangle \\
& = \sqrt{1-\norm{p\wedge p_0}^2}\geq \sqrt{1-\delta_0^2} \geq 1/2 \;,
\end{align*}
and again, assuming $\delta_0$ is small,
$$
\norm{h_2(p)} \geq \norm{h_1(p)} -\norm{h_1-h_2} \geq\sqrt{1-\delta_0^2} -\varepsilon_0
\geq 1/2 \;.
$$
Thus, by Lemma~\ref{varphi:gi} below, for all $\hat{p}\in B(\hat{p}_0,\delta_0)$,
$$ d(\varphi_{h_1}(\hat{p}), \varphi_{h_2}(\hat{p})) \leq 2\, \norm{h_1-h_2} \;. $$
Choosing $\varepsilon_0$ small enough, $\frac{\kappa^2\,\delta_0}{\sqrt{1-\delta_0^2}} + 2\,\varepsilon_0 < \delta_0$. This implies that
$$\varphi_{h_2}\left(B(\hat{p}_0,\delta_0) \right) \subset B(\hat{p}_0,\delta_0) \;. $$
By Proposition~\ref{proj:contr} we know that
$T_1=\varphi_{h_1}\vert_{B(\hat{p}_0,\delta_0)}$ has Lispchitz
constant $\kappa'= \kappa^2\,\frac{\delta_0+\sqrt{1-\delta_0^2}}{1-\delta_0^2}\approx \kappa^2$, and assuming $\delta_0$ is small enough we have $\frac{1}{1-\kappa'}\leq \frac{2}{1-\kappa^2}$.
Notice that although the Lispchitz constant in this proposition refers to the Riemannian metric $\rho$, since the ratio ${\rm Lip}_\delta(T_1)/{\rm Lip}_\rho(T_1)$ approaches $1$ as $\delta_0$ tends to $0$, we can assume that
${\rm Lip}_\delta(T_1)\leq \kappa'$.
Thus, by Lemma~\ref{Ti} below applied to $T_1$ and $T_2=\varphi_{h_2}\vert_{B(\hat{p}_0,\delta_0)}$,
we have $d(T_1,T_2)\leq 2\,\norm{h_1-h_2}$ and
$$
d(\overline{\mathfrak{v}}(h_1),\overline{\mathfrak{v}}(h_2)) \leq \frac{1}{1-\kappa'}\, d(T_1,T_2)\leq \frac{4}{1-\kappa^2}\,
\norm{h_1-h_2} \;.
$$
\end{proof}
\begin{lemma} \label{Ti}
Let $(X,d)$ be a complete metric space,
$T_1:X\to X$ a Lipschitz contraction
with ${\rm Lip}(T_1)<\kappa<1$,
$x_1^\ast=T_1 (x_1^\ast)$ a fixed point,
and $T_2:X\to X$ any other map with a fixed point $x_2^\ast=T_2 (x_2^\ast)$. Then
$$ d(x_1^\ast, x_2^\ast)\leq \frac{1}{1-\kappa}\, d(T_1,T_2) \;,$$
where $d(T_1,T_2):=\sup_{x\in X} d(T_1(x), T_2 (x))$.
\end{lemma}
\begin{proof}
\begin{align*}
d(x_1^\ast, x_2^\ast) & =
d(T_1(x_1^\ast), T_2(x_2^\ast))\\
&\leq
d(T_1(x_1^\ast), T_1(x_2^\ast)) + d(T_1(x_2^\ast), T_2(x_2^\ast))\\
&\leq
\kappa\, d(x_1^\ast, x_2^\ast) + d(T_1, T_2)\;,
\end{align*}
which implies that
$$ d(x_1^\ast, x_2^\ast) \leq \frac{1}{1-\kappa}\, d(T_1, T_2)\;. $$
\end{proof}
\begin{lemma}
\label{lemma wedge i g1 minus wedge i g2}
Given $g_1,g_2\in\mathcal{L}(V)$, for any $1\leq i \leq \dim V$,
$$ \norm{ \wedge_i g_1 - \wedge_i g_2 }
\leq i\, \max\{ 1, \norm{g_1}, \norm{g_2} \}^{i-1}\,\norm{g_1-g_2}\;. $$
\end{lemma}
\begin{proof}
Given any unit $i$-vector $v_1\wedge \ldots \wedge v_i\in \wedge_i V$, determined by an orthonormal family of vectors $\{v_1,\ldots, v_i\}$,
\begin{align*}
& \norm{(\wedge_i g_1)(v_1\wedge \ldots \wedge v_i)
- (\wedge_i g_2)(v_1\wedge \ldots \wedge v_i) }
= \\
& \qquad = \norm{ (g_1 v_1)\wedge \ldots \wedge (g_1 v_i)
- (g_2 v_1)\wedge \ldots \wedge (g_2 v_i) }\\
& \qquad \leq \sum_{j=1}^i
\norm{ (g_1 v_1)\wedge \ldots \wedge (g_1 v_{j-1}) \wedge (g_1 v_j
- g_2 v_j)\wedge (g_2 v_{j+1}) \wedge \ldots \wedge (g_2 v_i) } \\
& \qquad \leq \sum_{j=1}^i
\norm{g_1}^{j-1}\,\norm{g_2}^{i-j} \,
\norm{ g_1 v_j - g_2 v_j} \\
&\qquad \leq i\, \max\{1, \norm{g_1}, \norm{g_2} \}^{i-1}\,\norm{g_1-g_2}\;.
\end{align*}
\end{proof}
Given a dimension $1\leq l \leq \dim V$ and $0<\kappa<1$, consider the set
$$ \mathcal{L}_{l, \kappa} :=\{\, g\in \mathcal{L}(V)\,:\,
{\rm gr}_l(g)\geq \kappa^{-1} \,\}\;,$$
and define
$$ C_l(g_1,g_2):= \frac{l\,\max\{ 1, \norm {g_1},\norm{g_2}\}^{l-1}}{\max\{\norm{ 1, \wedge_l g_1}, \norm{\wedge_l g_2} \}}\;. $$
\begin{corollary}
\label{coro lipschitz:medir}
The map $\overline{\mathfrak{v}}:\mathcal{L}_{l,\kappa}\to{\rm Gr}_l(V)$ is locally Lipschitz.
More precisely, given $0<\kappa<1$ there exists $\varepsilon_0 >0$ such that for any $g_1,g_2\in \mathcal{L}_{l,\kappa}$ such that $\norm{g_1-g_2}\leq \varepsilon_0 \,C_l(g_1,g_2)^{-1}$, we have
$$d(\overline{\mathfrak{v}}_l(g_1), \overline{\mathfrak{v}}_l(g_2))\leq \frac{16}{1-\kappa^2}\,C_l(g_1,g_2)\,
\norm{g_1-g_2}\;. $$
\end{corollary}
\begin{proof}
By lemma~\ref{lemma wedge i g1 minus wedge i g2},
$d_{{\rm rel}} (\wedge_l g_1, \wedge_l g_2) \leq C_l(g_1,g_2)\,\norm{g_1-g_2}$.
Apply proposition~\ref{lipschitz:eigendir}
to the linear maps $\wedge_l g_j:\wedge_l V \to \wedge_l V$, $j=1,2$.
\end{proof}
Given $g\in\mathcal{L}(V)$ having $k$ and $k+r$ gap ratios, if a subspace $E\in{\rm Gr}_k(V)$ close to the $g$ most expanding subspace $\overline{\mathfrak{v}}_k(g)$ then the restriction $g\vert_{E^\perp}$ has a $r$-gap ratio and the most expanding $r$-dimensional subspace of $g\vert_{E^\perp}$ is close to the intersection of
$\overline{\mathfrak{v}}_{k+r}(g)$ with $E^\perp$. Next proposition expresses this fact in a quantitative way.
\begin{proposition}\label{prop dist med g | E perp}
Given $\varkappa>0$ small enough, and integers $1\leq k < k+r \leq \dim V$, there exists $\delta_0>0$ such that for all $g\in\mathcal{L}(V)$
and $E\in{\rm Gr}_k(V)$, if
\begin{enumerate}
\item[(a)]\; $\sigma_k(g)<\varkappa$ \, and \, $\sigma_{k+r}(g)<\varkappa$,
\item[(b)]\; $\delta(E,\overline{\mathfrak{v}}_k(g))<\delta_0$
\end{enumerate}
then
\begin{enumerate}
\item[(1)]\; $\sigma_r(g\vert_{E^\perp})\leq 2\,\varkappa$,
\item[(2)]\; $\displaystyle \delta\left(\, \overline{\mathfrak{v}}_r(g\vert_{E^\perp}), \,
\overline{\mathfrak{v}}_{k+r}(g)\cap E^\perp\,
\right) \leq \frac{20}{1-4\,\varkappa^2}\,\delta(E,\overline{\mathfrak{v}}_k(g))$.
\end{enumerate}
\end{proposition}
\begin{proof}
Consider the compact space
$$ \mathcal{K}_r=\{\, h\in \mathcal{L}(V)\,\colon\,
\norm{h}\leq 1 \; \text{ and }\; \sigma_r(h)\leq \varkappa\,\}\;. $$
By uniform continuity of $\sigma_r$ on $\mathcal{K}_r$ there exists $\delta_0>0$ such that for all
$h\in \mathcal{L}(V)$ if there exists $h_0\in \mathcal{K}_r$
with $\norm{h-h_0}<\delta_0$ then $\sigma_r(h)\leq 2\,\varkappa$.
Recall that $\pi_F$ denotes the orthogonal projection onto a linear subspace $F\subset V$.
Given $g \in \mathcal{L}(V)$ such that (a) holds,
consider the map $h=\frac{g}{\norm{g}}\circ \pi_{\overline{\mathfrak{v}}_k(g)^\perp}$. We have $h\in \mathcal{K}_r$ because
$\sigma_r(h)= \sigma_r(g \circ \pi_{\overline{\mathfrak{v}}_k(g)^\perp})
= \sigma_{k+r}(g)<\varkappa$.
Given $E\in {\rm Gr}_k(V)$ such that (b) holds,
we define $h_E=\frac{g}{\norm{g}}\circ \pi_{E^\perp}$.
Then
$$ \norm{h-h_E}\leq \norm{ \pi_{\overline{\mathfrak{v}}_k(g)^\perp} - \pi_{E^\perp} } \lesssim \delta(\overline{\mathfrak{v}}_k(g)^\perp, E^\perp) = \delta(E, \overline{\mathfrak{v}}_k(g)) <\delta_0 \;, $$
which implies that
$\sigma_r(g\vert_{E^\perp})=\sigma_r(h_E) \leq 2\,\varkappa$,
and hence proves (1).
For (2) we use the following triangle inequality
\begin{align*}
\delta( \overline{\mathfrak{v}}_r(g\vert_{E^\perp}), \overline{\mathfrak{v}}_{k+r}(g)\cap E^\perp ) &\leq \; \delta( \overline{\mathfrak{v}}_r( h_E), \overline{\mathfrak{v}}_r(h) ) \\
&\quad + \delta( \overline{\mathfrak{v}}_r(h),
\overline{\mathfrak{v}}_{k+r}(g)\cap \overline{\mathfrak{v}}_k(g)^\perp ) \\
&\quad + \delta( \overline{\mathfrak{v}}_{k+r}(g)\cap \overline{\mathfrak{v}}_k(g)^\perp,\,
\overline{\mathfrak{v}}_{k+r}(g)\cap E^\perp ) \\
&\leq \left( \frac{16\,r}{1-4\,\varkappa^2} + 0 + 1\right)\,\delta(E,\overline{\mathfrak{v}}_k(g))\\
&\leq \frac{20}{1-4\,\varkappa^2} \,\delta(E,\overline{\mathfrak{v}}_k(g))\; .
\end{align*}
The bound on the first distance is obtained through
corollary~\ref{coro lipschitz:medir}, with $C_r(h_E,h)=r$.
The second distance is zero.
Finally the bound on the third distance comes from
proposition~\ref{sum:inters:modulus cont} (2), using that
$\theta_\cap( \overline{\mathfrak{v}}_{k+r}(g), \overline{\mathfrak{v}}_{k}(g)^\perp )=1$,
because $\overline{\mathfrak{v}}_{k}(g)\subset \overline{\mathfrak{v}}_{k+r}(g)$.
\end{proof}
\begin{lemma}\label{varphi:gi}
Given $g_1, g_2\in \mathcal{L}(V)$, $\hat{p}\in\mathbb{P}(g_1)\cap \mathbb{P}(g_2)$ and any unit vector $p\in\hat p$,
$$ d(\varphi_{g_1}(\hat{p}), \varphi_{g_2}(\hat{p})) \leq
\max\{ \frac{1}{\norm{g_1\,p}}, \frac{1}{\norm{g_2\,p}} \}\,
\norm{g_1-g_2} \;.$$
\end{lemma}
\begin{proof} Assume $p\in V$ is a unit vector such that $\hat{p}\in \mathbb{P}(g_1)\cap \mathbb{P}(g_2)$.
Applying proposition~\ref{proj:lip} to the non-zero vectors
$g_1 \,p$ and $g_2\,p$, we get
\begin{align*}
d(\varphi_{g_1}(\hat{p}), \varphi_{g_2}(\hat{p})) &\leq
\norm{ \frac{g_1 \,p}{\norm{g_1 \,p}} -
\frac{g_2 \,p}{\norm{g_2 \,p}} } \\
&\leq \max\{ \norm{g_1\,p}^{-1}, \norm{g_2\,p}^{-1} \}\,
\norm{g_1\, p-g_2\, p} \\
&\leq \max\{ \norm{g_1\,p}^{-1}, \norm{g_2\,p}^{-1} \}\,
\norm{g_1 -g_2} \;.
\end{align*}
\end{proof}
The final four lemmas of this subsection apply to invertible linear maps in ${\rm GL}(V)$. They express the continuity of the map $g\mapsto \varphi_g$ with values in the space of Lipschitz or H\"older continuous maps on the projective space. These facts will be useful in ~\cite{LEbook}.
\begin{lemma}\label{delta:1}
Given $g_1, g_2\in {\rm GL}(V)$, and $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$,
$$ \abs{ \frac{\delta(\varphi_{g_1}(\hat{p}), \varphi_{g_1}(\hat{q}))}{\delta(\hat{p},\hat{q})} - \frac{\delta(\varphi_{g_2}(\hat{p}), \varphi_{g_2}(\hat{q}))}{\delta(\hat{p},\hat{q})} }
\leq C(g_1,g_2)\, \norm{g_1 - g_2}\;, $$
where \, $C(g_1,g_2):= (\norm{g_1^{-1}}^ 2 + \norm{g_2}^2\,\norm{g_1^{-1}}^2\,\norm{g_2^{-1}}^2)\,(\norm{g_1}+ \norm{g_2})$.
\end{lemma}
\begin{proof}
Given $p\in\hat p$ and $q\in\hat q$, by proposition~\ref{Lip:proj:action}
\begin{align*}
& \abs{ \frac{\delta(\varphi_{g_1}(\hat{p}), \varphi_{g_1}(\hat{q}))}{\delta(\hat{p},\hat{q})} - \frac{\delta(\varphi_{g_2}(\hat{p}), \varphi_{g_2}(\hat{q}))}{\delta(\hat{p},\hat{q})} } =
\abs{ \frac{ \norm{g_1 p \wedge g_1 v_p(q) } }{ \norm{g_1 p} \norm{g_1 q} } -
\frac{ \norm{g_2 p \wedge g_2 v_p(q) } }{ \norm{g_2 p} \norm{g_2 q} } }\\
&\qquad \leq \frac{ \norm{g_1 p \wedge g_1 v_p(q) - g_2 p \wedge g_2 v_p(q) } }{ \norm{g_1 p} \norm{g_1 q} } \\
&\qquad \quad + \, \abs{ \frac{1}{\norm{g_1 p} \norm{g_1 q}} - \frac{1}{\norm{g_2 p} \norm{g_2 q}} } \, \norm{ g_2 p \wedge g_2 v_p(q) } \\
&\qquad \leq \norm{g_1^ {-1}}^2\, \norm{g_1 p \wedge (g_1 v_p(q) - g_2 v_p(q)) } +
\norm{g_1^{-1}}^2\, \norm{(g_1 p - g_2 p) \wedge g_2 v_p(q) } \\
&\qquad \quad + \, \norm{g_1^{-1}}^2\,\norm{g_2^{-1}}^2\,( \norm{g_1 p}\,\abs{\norm{g_1 q}- \norm{g_2 q}} +
\norm{g_2 q}\,\abs{\norm{g_1 p} - \norm{g_2 p}} )\,\norm{g_2}^2 \\
&\qquad \leq \norm{g_1^ {-1}}^2\, (\norm{g_1} + \norm{g_2})\, \norm{g_1-g_2} \\
&\qquad \quad + \, \norm{g_2}^2\, \norm{g_1^{-1}}^2\,\norm{g_2^{-1}}^2\,(\norm{g_1}+ \norm{g_2})\, \norm{g_1-g_2} \\
&\qquad = (\norm{g_1^{-1}}^ 2 + \norm{g_2}^2\, \norm{g_1^{-1}}^2\,\norm{g_2^{-1}}^2)\,(\norm{g_1}+ \norm{g_2})\,\norm{g_1-g_2} \;.
\end{align*}
\end{proof}
\begin{lemma}\label{delta:3} Given $g\in{\rm GL}(V)$ and $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$,
$$ \frac{1}{\norm{g}^{2}\,\norm{g^{-1}}^{2}} \leq \frac{\delta(\varphi_g(\hat{p}), \varphi_g(\hat{q}))}{\delta(\hat{p},\hat{q})} \leq \norm{g}^2\,\norm{g^{-1}}^2 \;.$$
\end{lemma}
\begin{proof}
Given $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$ consider unit vectors
$p\in\hat{p}$, $q\in \hat{q}$ and set $v=v_p(q)$.
We have $\norm{p}=\norm{q}=\norm{v}=1$ and $\langle p,v\rangle=0$. This last relation implies $\norm{p\wedge v}=1$. Hence
$$ \norm{g p \wedge g v}= \norm{(\wedge_2 g)(p\wedge v)}\geq \norm{(\wedge_2 g)^{-1}}^{-1}\geq \norm{g^{-1}}^ {-2} \;. $$
Analogously
$$ \norm{g p \wedge g v}= \norm{(\wedge_2 g)(p\wedge v)}\leq \norm{\wedge_2 g}\leq \norm{g}^ 2 \;. $$
We also have
$$ \norm{g^{-1}}^{-2} \leq \norm{g\,p}\,\norm{g\,q} \leq \norm{g}^2\;. $$
To finish the proof combine these inequalities with proposition~\ref{Lip:proj:action}.
\end{proof}
\bigskip
Given $g\in{\rm GL}(V)$, we define
\begin{equation} \label{ell}
\ell(g):= \max\{ \log \norm{g},\log \norm{g^{-1}} \}\;.
\end{equation}
\begin{lemma}\label{Lipschitz:Proj}
For every $g\in{\rm GL}(V)$ and $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$,
$$ -4\,\ell(g) \leq \log \left[\frac{\delta(\varphi_g(\hat{p}), \varphi_g(\hat{q}))}{\delta(\hat{p},\hat{q})}\right]\leq 4\,\ell(g)\;.$$
\end{lemma}
\begin{proof}
Follows from ~lemma~\ref{delta:3}. \end{proof}
\begin{lemma}\label{delta:2}
Given $g_1, g_2\in{\rm GL}(V)$, $0<\alpha\leq 1$ and $\hat{p}\neq \hat{q}$ in $\mathbb{P}(V)$,
$$ \abs{ \left(\frac{\delta(\varphi_{g_1}(\hat{p}), \varphi_{g_1}(\hat{q}))}{\delta(\hat{p},\hat{q})}\right)^ \alpha -
\left(\frac{\delta(\varphi_{g_2}(\hat{p}), \varphi_{g_2}(\hat{q}))}{\delta(\hat{p},\hat{q})} \right)^ \alpha }
\leq C_1(g_1,g_2)\, \norm{g_1 - g_2}\;, $$
where $C_1(g_1,g_2)= \alpha\, \max\{\norm{g_1}\,\norm{g_1^{-1}}, \norm{g_2}\,\norm{g_2^{-1}}\}^{2(1-\alpha)}\, C(g_1,g_2)$,
and $C(g_1,g_2)$ stands for the constant in lemma~\ref{delta:1}.
\end{lemma}
\begin{proof}
Setting $\Delta_1:=\frac{\delta(\varphi_{g_1} \hat{p}, \varphi_{g_1} \hat{q})}{\delta(\hat{p},\hat{q})}$ and $\Delta_2:=\frac{\delta(\varphi_{g_2} \hat{p}, \varphi_{g_2} \hat{q})}{\delta(\hat{p},\hat{q})}$,
from lemmas~\ref{delta:1} and~\ref{delta:3} we get
\begin{align*}
\abs{ \Delta_1^\alpha - \Delta_2^\alpha } &\leq \alpha\, \max\{\Delta_1^{\alpha-1}, \Delta_2^{\alpha-1}\} \,\abs{\Delta_1-\Delta_2}\\
&\leq \alpha\, \max\{ \norm{g_1}\,\norm{g_1^{-1}}, \norm{g_2}\,\norm{g_2^{-1}} \}^{2(1-\alpha)} \,\abs{\Delta_1-\Delta_2}\\
&\leq \alpha\, \max\{ \norm{g_1}\,\norm{g_1^{-1}}, \norm{g_2}\,\norm{g_2^{-1}} \}^{2(1-\alpha)} \, C(g_1,g_2) \,\norm{ g_1- g_2}\;.
\end{align*}
\end{proof}
\section{Grassmann Geometry}
\label{grassmann}
\input{grassmann.tex}
\bigskip
\section{Singular Value Geometry}
\label{svg}
\input{svg.tex}
\bigskip
\section{Lipschitz Estimates}
\label{le}
\input{le.tex}
\bigskip
\section{Avalanche Principle}
\label{ap}
\input{ap.tex}
\subsection*{Acknowledgments}
\bigskip
The first author was supported by
Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia,
UID/MAT/04561/2013.
The second author was supported by the Norwegian Research Council project no. 213638, "Discrete Models in Mathematical Analysis".
\bigskip
\bibliographystyle{amsplain}
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\subsection{Singular value decomposition}
Let $V$ be an Euclidean space of dimension $n$.
\begin{definition}
\label{singular values}
Given $g\in\mathcal{L}(V)$, the {\em singular values} of $g$ are the square roots of the eigenvalues of the
quadratic form $Q_g:V\to\mathbb{R}$, $Q_g(v)=\norm{g\,v}^2=\langle g v, gv\rangle$, i.e., the eigenvalues of the
positive semi-definite self-adjoint operator $\sqrt{g^\ast g}$.
\end{definition}
Given $g\in \mathcal{L}(V)$, let
$$ s_1(g) \geq s_2(g)\geq \ldots \geq s_n(g)\geq 0\;,$$
denote the sorted singular values of $g$. The adjoint $g^ \ast$ has the same singular values as $g$
because the operators $\sqrt{g^\ast g}$ and $\sqrt{g\, g^\ast}$ are conjugate.
The largest singular value, $s_1(g)$, is the square root of the maximum value of $Q_g$ over the unit sphere,
i.e., $s_1(g)=\max_{\norm{v}=1} \norm{g\,v}=\norm{g}$ is the operator norm of $g$.
Likewise, the least singular value, $s_n(g)$, is the square root of the minimum value of $Q_g$ over the unit sphere,
i.e., $s_n(g)=\min_{\norm{v}=1} \norm{g\,v}$. This number also denoted by $\mathfrak{m}(g)$ is called the {\em least expansion} of $g$.
If $g$ is invertible $\mathfrak{m}(g)=\norm{g^{-1}}^{-1}$,
while otherwise $\mathfrak{m}(g)=0$.
\begin{definition}
\label{singular vectors}
The eigenvectors of the quadratic form $Q_g$, i.e., of the positive semi-definite self-adjoint operator $\sqrt{g^\ast g}$,
are called the {\em singular vectors} of $g$.
\end{definition}
By the spectral theory of self-adjoint operators, for any $g\in\mathcal{L}(V)$
there exists an orthonormal basis consisting of singular vector of $g$.
\begin{proposition}
Given $g\in\mathcal{L}(V)$ and $v\in V$ be a unit singular vector of $g$ such that $g^\ast g\,v=\lambda^2\,v$, there exists a unit vector
$w\in V$ such that
\begin{enumerate}
\item[(a)] $g\,v=\lambda\, w$,
\item[(b)] $g\,g^\ast\,w=\lambda^2\,w$, i.e., $w$ is a singular vector of $g^\ast$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $v\in V$ be a unit singular vector of $g$.
Then $g^\ast g\,v=\lambda^2\,v$ and
$\lambda^2=\langle\lambda^2\,v,v\rangle=\langle g^\ast g\,v,v\rangle=\norm{g\,v}^ 2$,
which implies that $\lambda=\norm{g\,v}$.
Since $(g\,g^\ast)\,(g\,v)=g\,(g^\ast g)\,v=\lambda^2\,g\,v$, if $\lambda\neq 0$ then
setting $w= g\,v/\norm{g\,v}=\lambda^{-1}\,g\,v$, we have $(g\,g^\ast)\,w= \lambda^2\,w$, which proves that
$w$ is a singular vector of $g^\ast$. By definition $g\,v=\lambda\,w$.
When $\lambda=0$,
take $w$ to be any unit vector in $\rm K g^\ast$. Notice that $\dim(\rm K g)=\dim(\rm K g^ \ast)$.
In this case $v$ and $w$ are singular vectors of $g$ and $g^\ast$, respectively, such that $g\,v=0= \lambda\,w$.
\end{proof}
By the previous proposition, given $g\in\mathcal{L}(V)$ there exist two orthonormal singular vector basis of $V$,
$\{v_1(g),\ldots, v_n(g)\}$ and $\{v_1(g^\ast),\ldots, v_n(g^\ast)\}$
for $g$ and $g^\ast$, respectively, such that
\begin{equation*}
g\, v_j(g)= s_j(g)\, v_j(g^\ast) \quad \text{ for all }\; 1\leq j \leq n\;.
\end{equation*}
Denote by $D_g$ the diagonal matrix with diagonal entries $s_j(g)$, $1\leq j \leq n$, seen as an operator $D_g\in\mathcal{L}(\mathbb{R}^ n)$. Define the linear maps $U_g, U_{g^\ast}:\mathbb{R}^n\to V$ by $U_g(e_j)=v_ j(g)$ and $U_{g^\ast}(e_j)=v_ j(g^\ast)$, for all $1\leq j \leq n$, where the $e_j$ are the vectors of the canonical basis in $\mathbb{R}^n$. Then the following decomposition holds
$$ g= U_{g^\ast}\,D_g\, (U_g)^\ast\;,$$
known as the {\em singular value decomposition} (SVD) of $g$.
We say that $g$ has a {\em simple singular spectrum}
if its $n$ singular values are all distinct. When $g$ has simple singular spectrum, the singular vectors
$v_j(g)$ and $v_j(g^\ast)$ above are uniquely determined up to a sign, and in particular they determine well-defined projective points
$\overline{\mathfrak{v}}_j(g), \overline{\mathfrak{v}}_j(g^\ast)\in\mathbb{P}(V)$.
\begin{definition}\label{def singular basis}
Given $g\in\mathcal{L}(V)$, we call {\em singular basis} of $g$ to any orthonormal basis
$\{v_1,\ldots, v_m\}$ of $V$ ordered such that
$\norm{g\,v_i } = s_i(g)$,
for all $i=1,\ldots, m$.
\end{definition}
\subsection{Gaps and most expanding directions}
Consider a linear map $g\in\mathcal{L}(V)$ and a number $0\leq k \leq \dim V$.
\begin{definition}
\label{gap ratio}
The {\em $k$-th gap ratio } of $g$ is defined to be
\begin{equation*}
{\rm gr}_k(g):=\frac{s_k(g)}{s_{k+1}(g)}\geq 1 \;.
\end{equation*}
\end{definition}
We will write ${\rm gr}(g)$ instead of
${\rm gr}_1(g)$.
\begin{definition}
\label{def has a gap ratio}
We say that $g$ {\em has a first singular gap} when
${\rm gr}(g)>1$. More generally, we say that $g$ {\em has a $k$ singular gap} when ${\rm gr}_k(g)>1$.
\end{definition}
In some occasions it is convenient to work with the inverse quantity, denoted by
\begin{equation}
\label{inverse gap ratio}
\sigma_k(g):={\rm gr}_k(g)^{-1}\leq 1 \;.
\end{equation}
\begin{proposition}\label{sing vals and ext pws}
For any $1\leq k \leq \dim V$,
$\norm{\wedge_k g} = s_1(g)\,\ldots\, s_k(g)$.
\end{proposition}
\begin{proof}
Let $n=\dim V$. Consider orthonormal singular vector basis
$\{v_1,\ldots, v_n\}$ and $\{v_1^\ast,\ldots, v_n^\ast\}$
for $g$ and $g^\ast$, respectively, such that
$$ g\, v_j = s_j \, v_j^\ast,\; \text{ where }\; s_j=s_j(g) \quad \text{ for all }\; 1\leq j \leq n\;.$$
Given $I=\{i_1,\ldots, i_k\}\in \Lambda^n_k$, with
$i_1<\ldots < i_k$, we have
$$(\wedge_k g)(v_{i_1}\wedge \ldots \wedge v_{i_k})
= (s_{i_1} \ldots s_{i_k} )\, (v_{i_1}^\ast\wedge \ldots \wedge v_{i_k}^\ast)\;. $$
Therefore, the $k$-vectors $v_I=v_{i_1}\wedge \ldots \wedge v_{i_k}$ and $v_I^\ast=v_{i_1}^\ast\wedge \ldots \wedge v_{i_k}^\ast$ form two orthonormal singular vector basis for $\wedge_k g$ and $\wedge_k g^\ast$, respectively,
while the products $s_I= s_{i_1} \ldots s_{i_k}$ are the singular values of both $\wedge_k g$ and $\wedge_k g^\ast$.
Since the largest singular value is attained with $I=\{1,\ldots, k\}$, $\norm{\wedge_k g} = s_1 \,\ldots\, s_k$.
\end{proof}
\begin{corollary}
For any $1\leq k <\dim V$, \;
$${\rm gr}_k(g) = \frac{\norm{\wedge_k g}^2}{\norm{\wedge_{k-1} g}\, \norm{\wedge_{k+1} g}}\;.$$
\end{corollary}
Given $g\in\mathcal{L}(V)$, if ${\rm gr}(g)>1$ then
the singular value $s_1(g)=\norm{g}$ is simple.
\begin{definition}
\label{def medir}
We denote by $\overline{\mathfrak{v}}(g)\in\mathbb{P}(V)$ the associated singular direction, and refer to it as the {\em $g$-most expanding direction}.
\end{definition}
By definition we have
\begin{equation}\label{varphi g:mostexp proj}
\varphi_g \overline{\mathfrak{v}}(g)= \overline{\mathfrak{v}}(g^\ast)\;.
\end{equation}
More generally, given $1\leq k\leq \dim V$,
if ${\rm gr}_k(g)>1$
\begin{definition}
\label{def medir k}
we define the
{\em $g$-most expanding $k$-subspace} to be
$$ \overline{\mathfrak{v}}_k(g):=\Psi^{-1}\left( \overline{\mathfrak{v}}(\wedge_k g) \right)\; ,$$
where $\Psi$ stands for the Pl\"ucker embedding
defined in subsection~\ref{grassmannians}.
\end{definition}
The subspace $\overline{\mathfrak{v}}_k(g)$ is the direct sum of all singular directions associated with the singular values
$s_1(g),\ldots, s_k(g)$.
We have
\begin{equation}\label{varphi g:mostexp grassmann}
\varphi_g \overline{\mathfrak{v}}_k(g)= \overline{\mathfrak{v}}_k(g^\ast)\;.
\end{equation}
Analogously, let $n=\dim V$ and assume ${\rm gr}_{n-k}(g)>1$.
\begin{definition}
\label{def ledir}
We define the {\em $g$-least expanding $k$-subspace} as
$$ \underline{\mathfrak{v}}_k(g) := \overline{\mathfrak{v}}_{n-k}(g)^\perp \;.$$
\end{definition}
The subspace $\underline{\mathfrak{v}}_k(g)$ is the direct sum of all singular directions associated with the singular values
$s_{n-k+1}(g),\ldots, s_n(g)$.
Again we have
\begin{equation}\label{varphi g:leastexp grassmann}
\varphi_g \underline{\mathfrak{v}}_k(g)= \underline{\mathfrak{v}}_k(g^\ast)\;.
\end{equation}
\bigskip
Let $\tau=(\tau_1,\ldots,\tau_k)$ be a signature with $1\leq \tau_1<\ldots <\tau_k\leq \dim V$.
\begin{definition}
\label{def tau gap ratio}
We define the {\em $\tau$-gap ratio} of $g$ to be
$$ {\rm gr}_\tau(g):= \min_{1\leq j\leq k} {\rm gr}_{\tau_j}(g)\;. $$
When ${\rm gr}_\tau(g)>1$ we say that $g$ {\em has a $\tau$-gap pattern}.
\end{definition}
Note that ${\rm gr}_\tau(g)>1$ means that $g$ has a
$\tau_j$ singular gap for $1\leq j\leq k$.
Recall that $\mathscr{F}_\tau(V)$ denotes the space of all $\tau$-flags, i.e.,
flags $F=(F_1,\ldots, F_k)$ such that $\dim (F_j)=\tau_j$ for $j=1,\ldots, k$.
\begin{definition}
\label{def most expanding flag}
If ${\rm gr}_\tau(g)>1$ the {\em most expanding $\tau$-flag} is defined to be
$$ \overline{\mathfrak{v}}_\tau(g):=( \overline{\mathfrak{v}}_{\tau_1}(g),\ldots, \overline{\mathfrak{v}}_{\tau_k}(g))\in \mathscr{F}_\tau(V) \;. $$
\end{definition}
Given $g\in\mathcal{L}(V)$ the domain of its push-forward action on $\mathscr{F}_\tau(V)$ is
\begin{definition}
\label{def flag push-forward domain}\quad
$\displaystyle \mathscr{F}_\tau(g):=\{\, F\in\mathscr{F}_\tau(V)\,\colon\,
F_k\cap \rm K_g=0\,\}\;.$
\end{definition}
The {\em push-forward} of a flag $F\in \mathscr{F}_\tau(g)$ by $g$ is defined to be
$$ \varphi_g F = g\, F :=
(g\,F_1,\ldots, g\,F_k)\;. $$
\begin{proposition} Given $g\in\mathcal{L}(V)$ such that ${\rm gr}_\tau(g)>1$, the push-forward induces a map
$\varphi_g: \mathscr{F}_\tau(g)\to\mathscr{F}_\tau(g^\ast)$ such that
$ \varphi_g \overline{\mathfrak{v}}_\tau(g)= \overline{\mathfrak{v}}_\tau(g^\ast)$.
\end{proposition}
\begin{proof}
Given $F\in\mathscr{F}_\tau(g)$, we have
$F_j\cap \rm K_g=0$ for all $j=1,\ldots, k$.
Hence $\dim g F_j= \dim F_j=\tau_j$ for all $j$,
which proves that $\varphi_g F \in \mathscr{F}_\tau(V)$. To check that
$\varphi_g F \in \mathscr{F}_\tau(g^\ast)$ we need to show that $g F_k\cap \rm K_{g^\ast}=0$.
Assume $g\,v\in \rm K_{g^\ast}$, with $v\in F_k$,
and let us see that $g\,v=0$. By assumption $g^\ast g\, v=0$, which implies $(g\,g^\ast)\,g\,v=0$.
Since the self-adjoint map $g\,g^\ast$ induces an automorphism on $\rm R_g$, we conclude that $g\,v=0$.
The second statement follows from~\eqref{varphi g:mostexp grassmann}.
\end{proof}
Given $g\in\mathcal{L}(V)$, the domain of its pull-back action on $\mathscr{F}_\tau(V)$ is
\begin{definition}
\label{def flag pull-back domain}
\quad
$\displaystyle \mathscr{F}_\tau^{-1}(g):=\{\, F\in\mathscr{F}_\tau(V)\,\colon\,
F_1 + \rm R_g=V\,\}\;.$
\end{definition}
The {\em pull-back} of a flag $F\in \mathscr{F}_\tau(g)$ by $g$ is defined to be
$$ \varphi_g F = g^{-1} F :=
(g^{-1} F_1,\ldots, g^{-1} F_k)\;. $$
\begin{definition}
\label{def least expanding flag}
If ${\rm gr}_{\tau^\perp}(g)>1$ the {\em least expanding $\tau$-flag} is defined as
$$ \underline{\mathfrak{v}}_\tau(g):=( \underline{\mathfrak{v}}_{\tau_1}(g),\ldots, \underline{\mathfrak{v}}_{\tau_k}(g))\in \mathscr{F}_\tau(V) \;. $$
\end{definition}
\begin{proposition}
\label{prop relation least most expanding subspaces}
If ${\rm gr}_{\tau}(g)>1$ then \,
$\underline{\mathfrak{v}}_{\tau^\perp} (g) =
\overline{\mathfrak{v}}_{\tau}(g)^\perp$.
\end{proposition}
\begin{proof}
Let $\{v_1,\ldots, v_n\}$ be a singular basis of $g$.
Since this basis is orthonormal,
\begin{align*}
\underline{\mathfrak{v}}_{n-k}(g) =\langle v_{k+1},\ldots, v_n \rangle = \langle v_1,\ldots, v_k\rangle^\perp = \overline{\mathfrak{v}}_k(g)^\perp\;.
\end{align*}
Hence
$$ \underline{\mathfrak{v}}_{\tau^\perp}(g) =
( \underline{\mathfrak{v}}_{n-\tau_k}(g),\ldots, \underline{\mathfrak{v}}_{n-\tau_1}(g) ) = ( \overline{\mathfrak{v}}_{\tau_1}(g),\ldots,
\overline{\mathfrak{v}}_{\tau_k}(g) )^\perp = \overline{\mathfrak{v}}_\tau(g)^\perp\;.$$
\end{proof}
\begin{proposition} Given $g\in\mathcal{L}(V)$ such that ${\rm gr}_{\tau^\perp}(g)>1$, the pull-back induces a map
$\varphi_g^{-1}:\mathscr{F}_\tau^{-1}(g)\to \mathscr{F}_\tau^{-1}(g^\ast)$ such that $ \varphi_g^{-1} \underline{\mathfrak{v}}_\tau(g)= \underline{\mathfrak{v}}_\tau(g^\ast)$.
\end{proposition}
\begin{proof}
Given $F\in\mathscr{F}_\tau^{-1}(g)$, we have
$F_j + \rm R_g=V$ for all $j=1,\ldots, k$.
Hence $\dim g^{-1}F_j= \dim F_j=\tau_j$ for all $j$,
which proves that $\varphi_g^{-1} F \in \mathscr{F}_\tau(V)$. To check that
$\varphi_g^{-1} F \in \mathscr{F}_\tau^{-1}(g^\ast)$ just notice that $ g^{-1} F_1+ \rm R_{g^\ast} \supseteq
\rm K_g + \rm K_g^\perp = V$.
The second statement follows from~\eqref{varphi g:leastexp grassmann}
and proposition~\ref{prop relation least most expanding subspaces}.
\end{proof}
We end this subsection proving that the orthogonal complement involution conjugates the push-forward action by
$g\in\mathcal{L}(V)$ with the pull-back action by the adjoint map $g^\ast$.
\begin{proposition} \label{pfwd-pbck:conjugation}
Given $g\in\mathcal{L}(V)$ such that ${\rm gr}_{\tau^\perp}(g)>1$,
the action of $\varphi_{g}^{-1}$ on $\mathscr{F}_\tau(V)$ is
conjugate to the action of $\varphi_{g^\ast}$ on
$\mathscr{F}_{\tau^\perp}(V)$ by the orthogonal complement involution.
More precisely, we have
$\mathscr{F}_{\tau}^{-1}(g)=\mathscr{F}_{\tau^\perp}(g^\ast)^\perp$
and $\mathscr{F}_{\tau}^{-1}(g^\ast)=\mathscr{F}_{\tau^\perp}(g)^\perp$, and the following diagram commutes
$$ \begin{CD}
\mathscr{F}_{\tau^\perp}(g^\ast) @>\varphi_{g^\ast} >> \mathscr{F}_{\tau^\perp}(g) \\
@V \cdot^\perp VV @VV \cdot^\perp V \\
\mathscr{F}_{\tau}^{-1}(g) @>>\varphi_{g}^{-1}> \mathscr{F}_{\tau}^{-1}(g^\ast)
\end{CD} \;. $$
\end{proposition}
\begin{proof}
To see that $\mathscr{F}_{\tau}^{-1}(g)=\mathscr{F}_{\tau^\perp}(g^\ast)^\perp$, notice that the following equivalences hold:
\begin{align*}
F\in \mathscr{F}_{\tau}^{-1}(g)\quad & \Leftrightarrow \quad F_1+\rm R_g=V \\
& \Leftrightarrow \quad F_1^\perp\cap \rm K_{g^\ast} = 0 \quad \Leftrightarrow \quad
F^\perp\in \mathscr{F}_{\tau^\perp}(g^\ast)\;.
\end{align*}
Exchanging the roles of $g$ and $g^\ast$ we obtain the relation
$\mathscr{F}_{\tau}^{-1}(g^\ast)=\mathscr{F}_{\tau^\perp}(g)^\perp$.
Finally, notice it is enough to prove the diagram's commutativity at the Grassmannian level. For that we use proposition~\ref{push-pull duality}.
\end{proof}
\subsection{Angles and expansion }
\label{subsection angles}
Throughout this subsection let $\hat{p},\hat{q}\in \mathbb{P}(V)$, and $p\in\hat{p}$, $q\in \hat{q}$ denote representative vectors.
The projective distance $\delta(\hat{p},\hat{q})$ was defined by
$$ \delta(\hat{p},\hat{q}):=\sqrt{1-\frac{\langle p,q\rangle^2}{\norm{p}^2\norm{q}^2}} = \frac{\norm{p\wedge q}}{\norm{p}\,\norm{q}}
=\sin \rho(\hat{p},\hat{q}) \;. $$
The complementary quantity plays a special role in the sequel.
\begin{definition}
\label{def aangle(p,q)}
The $\alpha$-angle between $\hat p$ and $\hat q$ is defined to be
$$ \alpha(\hat{p},\hat{q} ):= \frac{\vert \langle p,q\rangle\vert }{\norm{p}\,\norm{q} } =\cos \rho(\hat{p},\hat{q})\;. $$
\end{definition}
In order to give a geometric meaning to this angle we define the {\em projective orthogonal complement} of $\hat{p}\in\mathbb{P}(V)$ as
$$ \orthC{\hat{p}} :=\{\, \hat{x}\in\mathbb{P}(V) \,\colon\, \langle x, p\rangle=0\quad \text{ for }\; x\in\hat x \,\}\;. $$
The number $\alpha(\hat{p},\hat{q} )$ is the sine of the minimum angle between
$\hat{p}$ and $\orthC{\hat{q}}$.
\begin{proposition}
For any $\hat p, \hat q\in \mathbb{P}(V)$,
\begin{align}\label{aangle:sine rho}
&\alpha(\hat{p},\hat{q}) =\sin \rho_{{\rm min}}(\hat{p},\orthC{\hat{q}}) =\delta_{{\rm min}}(\hat{p},\orthC{\hat{q}})\\
\label{aangle:delta:orth}
& \alpha(\hat{p},\hat{q})=0\; \Leftrightarrow \;
\delta(\hat{p},\hat{q})=1 \; \Leftrightarrow \; p\perp q\;.
\end{align}
\end{proposition}
These concepts extend naturally to Grassmannians and flag manifolds.
\begin{definition}
\label{grassmann rho delta alpha}
Given $E,F\in{\rm Gr}_k(V)$, we define the $\alpha$-angle between them
\begin{align*}
\alpha(E,F)= \alpha_k(E,F):=\alpha(\Psi(E),\Psi(F))\;,
\end{align*}
where $\Psi:{\rm Gr}_k(V)\to\mathbb{P}(\wedge_k V)$ denotes the Pl\"ucker embedding (see subsection ~\ref{grassmannians}).
\end{definition}
\begin{definition}
\label{def Grassmann orth complement}
We say that two $k$-subspaces $E,F\in{\rm Gr}_k(V)$ {\em are orthogonal}, and we write $E\perp F$,\,
iff \, $\alpha(E,F)=0$.
\end{definition}
The {\em Grassmannian orthogonal complement} of $F$ is defined as
$$ \orthC{F}:=\{\, E\in {\rm Gr}_k(V)\,\colon\, \alpha(E,F)=0\,\}\;. $$
As before, the number $\alpha(E,F)$ measures the sine of the minimum angle between
$E$ and $\orthC{F}$.
\begin{proposition}
\label{aangle:sine rho:grassmann}
For any $E, F\in {\rm Gr}_k(V)$,
\begin{equation*}
\alpha(E,F)=\sin \rho_{{\rm min}}(E,\orthC{F}) =\delta_{{\rm min}}(E,\orthC{F})\;.
\end{equation*}
\end{proposition}
Next we characterize the angle $\alpha(E,F)$.
Consider the notation of definition~\ref{def orthog proj}.
\begin{proposition}\label{prop: alpha = det Pi(E F)}
Given $E,F\in{\rm Gr}_k(V)$,
\begin{enumerate}
\item[(a)] $\alpha(E,F)= \alpha(E^\perp, F^\perp)$,
\item[(b)] $\alpha(E,F)= \abs{\det(\pi_{E,F})} = \abs{\det(\pi_{F,E})}$,
\item[(c)] $E\perp F$\, iff \,
there exists a pair $(e,f)$ of unit vectors
such that $e\in E\cap F^\perp$ and $f\in F\cap E^\perp$,
\item[(d)] $\delta_{{\rm min}}(E,F^\perp)\geq \alpha(E,F)$.
\end{enumerate}
\end{proposition}
\begin{proof} Given $E,F\in{\rm Gr}_k(V)$, take orthonormal
basis $\{u_1,\ldots, u_k\}$ and $\{v_1,\ldots, v_k\}$ of $E$ and $F$, respectively, and consider the associated unit $k$-vectors $u= u_1\wedge \ldots \wedge u_k$ and
$v= v_1\wedge \ldots \wedge v_k$, so that $u\in \Psi(E)$ and $v\in\Psi(F)$.
Using the Hodge star operator we obtain unit vectors $\ast u\in \Psi(E^\perp)$ and $\ast v\in \Psi(F^\perp)$.
Hence
$$\alpha(E^\perp,F^\perp)=\abs{\langle\ast u, \ast v \rangle}
=\abs{\langle u, v \rangle} =\alpha(E,F)\;, $$
which proves (a). Also
\begin{align*}
\alpha(E,F) & :=
\abs{ \langle \, u_1\wedge \ldots \wedge u_k, \,
v_1\wedge \ldots \wedge v_k\, \rangle}\\
&=\abs{\det \left( \begin{array}{cccc}
\langle u_1,v_1\rangle & \langle u_1,v_2\rangle & \ldots & \langle u_1,v_k\rangle\\
\langle u_2,v_1\rangle & \langle u_2,v_2\rangle & \ldots & \langle u_2,v_k\rangle\\
\vdots & \vdots & \ddots & \vdots \\
\langle u_k,v_1\rangle & \langle u_k,v_2\rangle & \ldots & \langle u_k,v_k\rangle
\end{array}\right)}\\
&= \abs{\det (\pi_{E,F})}
\;.
\end{align*}
For the second equality write $u_i=w_i +\sum_{j=1}^k \langle u_i,v_j\rangle\,v_j$ with $w_i\in F^\perp$ and use the anti-symmetry of the exterior product. For the third equality remark that the matrix with entries $\langle u_i,v_j\rangle$ represents $\pi_{E,F}$ w.r.t. the given orthonormal basis.
By symmetry, $\alpha(E,F)= \abs{\det(\pi_{F,E})}$. This proves (b).
From these relations,
$\alpha(E,F)=0$ \, $\Leftrightarrow$\, $\rm K (\pi_{E,F})\neq \{0\}$ \, $\Leftrightarrow$\, $\rm K (\pi_{F,E})\neq \{0\}$,
which explains (c).
By proposition~\ref{delta, deltamin, deltaH} (b), and because all singular values of $\pi_{E,F}$ are in $[0,1]$,
$$ \delta_{{\rm min}}(E,F^\perp) = \norm{\pi_{E,F}}
\geq \abs{\det( \pi_{E,F} ) } =
\alpha_k(E,F) \;,$$
which proves (d).
\end{proof}
Finally, we extend $\alpha$-angle to flags.
Consider a signature $\tau$ of length $k$.
\begin{definition}
\label{def tau aangle}
Given flags $F,G\in
\mathscr{F}_\tau(V)$, define
\begin{align*}
\alpha(F,G) = \alpha_\tau(F,G) &:=
\min_{1\leq j\leq k} \alpha(F_j,G_j)\;.
\end{align*}
\end{definition}
\begin{definition}
\label{def Flag orth complement}
We say that two $\tau$-flags $F,G\in\mathscr{F}_\tau(V)$ {\em are orthogonal}, and we write $F\perp G$,\,
iff \, $F_j\perp G_j$ for some $j=1,\ldots, k$ .
\end{definition}
Comparing the two definitions, for all $F, G\in \mathscr{F}_\tau(V)$
$$ \alpha_\tau(F,G)=0\quad \Leftrightarrow\quad G\perp F \;. $$
Hence, the {\em orthogonal
flag hyperplane} of $F$ is defined as
$$ \orthC{F}:=\{\, \Sigma(F):=\{\, G\in\mathscr{F}_\tau(V)\,:\,
\alpha(G,F)=0\,\}\;. $$
As in the previous cases, the number $\alpha_\tau(F,G)$ measures the sine of the minimum angle between
$F$ and $\orthC{G}$.
\begin{proposition}
\label{aangle:sine rho:grassmann}
For any $F, G\in \mathscr{F}_\tau(V)$,
\begin{equation*}
\alpha(E,F)=\sin \rho_{{\rm min}}(F,\orthC{G}) =\delta_{{\rm min}}(F,\orthC{G})\;.
\end{equation*}
\end{proposition}
\medskip
Consider a sequence of linear maps $g_0, g_1, \ldots, g_{n-1} \in \mathcal{L}(V)$. The following quantities, called {\em expansion rifts}, measure
the break of expansion in the composition
$g_{n-1}\,\ldots\, g_1\, g_0$ of the maps $g_j$.
\begin{definition}
\label{def rift}
The first expansion rift of the sequence above is the number
$$\rho (g_0, g_1, \ldots, g_{n-1}) := \frac{\norm{g_{n-1} \ldots g_1 g_0}}{ \norm{g_{n-1}} \ldots \norm{g_1} \norm{g_0}}\in [1,+\infty)\,. $$
Given $1\leq k\leq \dim V$, the $k$-th expansion rift is
$$\rho_k (g_0, g_1, \ldots, g_{n-1}) := \rho (\wedge_k g_0, \wedge_k g_1, \ldots, \wedge_k g_{n-1})\,. $$
Given a signature $\tau=(\tau_1,\ldots, \tau_k)$,
the $\tau$-expansion rift is defined as
$$\rho_\tau (g_0, g_1, \ldots, g_{n-1}) := \min_{1\leq j\leq k}
\rho_{\tau_j} (g_0, g_1, \ldots, g_{n-1})\,. $$
\end{definition}
\medskip
The key concept of this section is that of angle between linear maps. The quantity $\alpha(g,g')$, for instance,
is the sine of the angle between
$\varphi_g(\overline{\mathfrak{v}}(g))=\overline{\mathfrak{v}}(g^\ast)$ and $\orthC{\overline{\mathfrak{v}}(g')}$.
As we will see, this angle is a lower bound on the expansion rift of two linear maps $g$ and $g'$.
\begin {definition}\label{alpha:def}
Given $g,g'\in\mathcal{L}(V)$, we define
\begin{align*}
\alpha (g,g')&:=\alpha (\overline{\mathfrak{v}}(g^\ast),\overline{\mathfrak{v}}(g'))
\quad &\text{ if }\; g \; \text{ and }\; g' \; \text{ have a first gap ratio} \\
\alpha_k(g,g') &:=\alpha (\overline{\mathfrak{v}}_{k}(g^\ast),\overline{\mathfrak{v}}_{k}(g'))\quad &\text{ if }\; g \; \text{ and }\; g' \; \text{ have a }\; k \; \text{ gap ratio } \\
\alpha_\tau(g,g') &:=\alpha (\overline{\mathfrak{v}}_{\tau}(g^\ast),\overline{\mathfrak{v}}_{\tau}(g')) \quad &\text{ if }\; g \; \text{ and }\; g' \; \text{ have a }\; \tau \; \text{ gap pattern}.
\end{align*}
\end {definition}
\medskip
The following exotic operation is introduced to obtain an upper bound
on the expansion rift $\rho(g,g')$.
Consider the algebraic operation $ a\oplus b := a+b -a\,b $ on the set $[0,1]$.
Clearly
$([0,1],\oplus)$ is a commutative semigroup isomorphic to $([0,1],\cdot)$. In fact, the transformation $\Phi:([0,1],\oplus)\to ([0,1],\cdot)$, $\Phi(x):= 1-x$,
is a semigroup isomorphism. We summarize some properties of this
operation.
\begin{proposition} \label{oplus:prop}
For any $a,b,c\in [0,1]$,
\begin{enumerate}
\item[(1)] $0\oplus a = a$,
\item[(2)] $1\oplus a = 1$,
\item[(3)] $a\oplus b = (1-b)\,a+b = (1-a)\,b+a $,
\item[(4)] $a\oplus b <1$\; $\Leftrightarrow$\; $a<1$ and $b<1$,
\item[(5)] $a\leq b$ \; $\Rightarrow$\; $a\oplus c\leq b\oplus c$,
\item[(6)] $b>0$\; $\Rightarrow$\;
$({a} {b}^{-1}\oplus c)\,b\leq a\oplus c$,
\item[(7)] $a\,c + b\,\sqrt{1-a^2}\,\,\sqrt{1-c^ 2} \leq \sqrt{a^2 \oplus b^2}$.
\end{enumerate}
\end{proposition}
\begin{proof} Items (1)-(6) are left as exercises.
For the last item consider the function
$f:[0,1]\to [0,1]$
defined by $f(c):= a\,c + b\,\sqrt{1-a^2}\,\,\sqrt{1-c^ 2}$.
A simple computation shows that
$$ f'(c)=a-\frac{b\,c\,\sqrt{1-a^2}}{\sqrt{1-c^2}}$$
The derivative $f'$ has a zero at $c= a/\sqrt{a\oplus b}$,
and one can check that this zero is a global maximum of $f$.
Since
$f( a/\sqrt{a\oplus b} )=\sqrt{a^2\oplus b^2}$,
item (7) follows.
\end{proof}
\begin {definition}\label{beta:def}
Given $g,g'\in\mathcal{L}(V)$ with $\tau$-gap patterns,
the upper $\tau$-angle between $g$ and $g'$ is defined to be
\begin{equation*}
\beta_\tau(g,g'):=\sqrt{ {\rm gr}_\tau(g)^{-2} \oplus \alpha_\tau(g,g')^ 2 \oplus {\rm gr}_\tau(g')^{-2}}\;.
\end{equation*}
We will write $\beta_k(g,g')$ when $\tau=(k)$, and
$\beta(g,g')$ when $\tau=(1)$.
\end{definition}
Next proposition relates norm expansion by $g$ and distance contraction by $\varphi_g$ with angles and gap ratios.
\begin{proposition}\label{prop expansion aangle}
Given $g\in\mathcal{L}(V)$ with
$\sigma(g)<1$,
a point $\hat{w}\in \mathbb{P}(V)$ and a unit vector $w\in\hat{w}$,
\begin{enumerate}
\item[(a)]\quad $\displaystyle \alpha(\hat{w},\overline{\mathfrak{v}}(g))\,\norm{g} \leq \norm{g\,w} \leq \norm{g} \, \sqrt{ \alpha(\hat{w},\overline{\mathfrak{v}}(g))^2 \oplus \sigma(g)^{2}} $,
\smallskip
\item[(b)]\quad $\displaystyle \delta( \varphi_g(\hat{w}), \overline{\mathfrak{v}}(g^\ast) )\leq \frac{\sigma(g)}{\alpha(\hat{w},\overline{\mathfrak{v}}(g))}\,\delta(\hat{w}, \overline{\mathfrak{v}}(g)) \, $.
\end{enumerate}
\end{proposition}
\begin{proof}
Let us write $\alpha= \alpha(\hat{w},\overline{\mathfrak{v}}(g))$ and $\sigma= \sigma(g)$.
Take a unit vector $v\in \overline{\mathfrak{v}}(g)$
such that $\angle(v,w)$ is non obtuse. Then $w = \alpha \, v + u$ with $u\perp v$ and $\norm{u}=\sqrt{1-\alpha^2}$. Choosing a unit vector
$v^\ast\in\overline{\mathfrak{v}}(g^\ast)$, we have $g w = \alpha\,\norm{g}\,v^\ast + g u$ with $g u\perp v^\ast$ and $\norm{g u}\leq \sqrt{1-\alpha^2} \,s_2(g)= \sqrt{1-\alpha^2} \,\sigma\,\norm{g}$.
We define the number $0\leq \kappa\leq \sigma$ so that
$\norm{g u} = \sqrt{1-\alpha^2} \,\kappa\,\norm{g}$.
Hence
\begin{align*}
\alpha^2\,\norm{g}^2 &\leq \alpha^2\,\norm{g}^2 +
\norm{g u}^2 = \norm{g w}^2\;,
\end{align*}
and also
\begin{align*}
\norm{g w}^2 & = \alpha^2\,\norm{g}^2 +
\norm{g u}^2 =
\norm{g}^2\,\left(\alpha^2 + (1-\alpha^2) \kappa^{2} \right)\\
&= \norm{g}^2 \, \left( \alpha^2 \oplus \kappa^{2}\right)
\leq
\norm{g}^2 \, \left( \alpha^2 \oplus \sigma^{2}\right) \;,
\end{align*}
which proves (a).
Item (b) follows from
\begin{align*}
\delta\left( \varphi_g(\hat{w}), \overline{\mathfrak{v}}(g^\ast) \right)
&= \frac{\norm{ g\,v \wedge g w}}{\norm{g v}\,\norm{g w}}
= \frac{\norm{ g\,v \wedge g u}}{\norm{g}\,\norm{g w}} = \frac{\norm{ v^\ast \wedge g u}}{ \norm{g w}} \\
& = \frac{\norm{ g u}}{ \norm{g w}}
\leq \frac{\sigma\,\sqrt{1-\alpha^2}\,\norm{g}}{\alpha\,\norm{g} } = \frac{\sigma\,\delta(\hat{w}, \overline{\mathfrak{v}}(g))}{\alpha }\;.
\end{align*}
\end{proof}
Next proposition relates the expansion rift
$\rho(g,g')$ with the angle $\alpha(g,g')$
and the upper angle $\beta(g,g')$.
\begin{proposition}\label{prod:2:lemma}
Given $g, g' \in\mathcal{L}(V)$ with a $(1)$-gap pattern,
$$ \alpha(g,g') \leq \frac{\norm{g'\,g}}{\norm{g'}\,\norm{g}}
\leq \beta(g,g') $$
\end{proposition}
\begin{proof}
Let $\alpha:=\alpha(g,g')=\alpha(\overline{\mathfrak{v}}(g^\ast),\overline{\mathfrak{v}}(g'))$ and take unit vectors $v\in\overline{\mathfrak{v}}(g)$, $v^\ast\in\overline{\mathfrak{v}}(g^\ast)$
and $v'\in\overline{\mathfrak{v}}(g')$ such that
$\langle v^\ast, v'\rangle= \alpha >0$ and
$g\,v=\norm{g}\,v^\ast$.
Since $\varphi_g(\overline{\mathfrak{v}}(g))=\overline{\mathfrak{v}}(g^\ast)$,
$w=\frac{g\,v}{\norm{g\,v}}$ is a unit vector in $\hat{w} = \overline{\mathfrak{v}}(g^\ast)$. Hence, applying proposition~\ref{prop expansion aangle} (a) to $g'$ and $\hat{w}$, we get
$$ \alpha(g,g')\,\norm{g'}
= \alpha(\hat{w}, \overline{\mathfrak{v}}(g')) \,\norm{g'}
\leq \norm{ \frac{g'\,g\,v}{\norm{g\,v}} } \leq \frac{\norm{g'\,g}}{\norm{g}}\;, $$
which proves the first inequality.
For the second, consider $\hat{w}\in\mathbb{P}(g)$ and a unit vector $w\in \hat{w}$ such that
$a:=\langle w, v\rangle = \alpha(\hat{w}, \overline{\mathfrak{v}}(g)) \geq 0$.
Then $w = a\, v+ \sqrt{1-a^2}\, u$,
where $u$ is a unit vector orthogonal to $v$. It follows that
$g\, w = a\, \norm{g}\,v^\ast + \sqrt{1-a^2}\, g\,u$ with $g\,u\perp v^\ast$, and $\norm{g\, u} = \kappa\,\norm{g}$ for some $0\leq \kappa\leq \sigma(g)$. Therefore
$$\frac{\norm{g\, w}^2}{\norm{g}^2}
= a^2 + (1-a^2)\,\kappa^2 = a^2\oplus \kappa^2 \;. $$
and
$$ \frac{g\, w}{\norm{g\,w}}
= \frac{a}{\sqrt{a^2\oplus \kappa^2}}\, v^\ast
+ \frac{\sqrt{1-a^2}}{\sqrt{a^2\oplus \kappa^2}}\,\frac{g\,u}{\norm{g}}\;. $$
The vector $v'$ can be written as
$v'= \alpha\, v^\ast + w'$ with $w'\perp v^\ast$ and $\norm{w'}=\sqrt{1-\alpha^2}$.
Set now $b:= \alpha( \varphi_g(\hat{w}), \overline{\mathfrak{v}}(g'))$. Then
\begin{align*}
b = \abs{\langle
\frac{ g\,w}{\norm{g\, w}}, v' \rangle} &\leq \frac{\alpha\,a}{\sqrt{a^2\oplus\kappa^2}}
+ \frac{\sqrt{1-a^2} }{\sqrt{a^2\oplus\kappa^2}}\,\frac{\abs{\langle g\,u, v' \rangle}}{\norm{g}}
\\
&\leq \frac{\alpha\,a}{\sqrt{a^2\oplus\kappa^2}}
+ \frac{\kappa \, \sqrt{1-a^2}}{\sqrt{a^2\oplus\kappa^2}}\,
\abs{ \langle \frac{g\,u}{\norm{g\,u}}, w' \rangle }\\
&\leq \frac{\alpha\,a}{\sqrt{a^2\oplus\kappa^2}}
+ \frac{\kappa \, \sqrt{1-a^2}}{\sqrt{a^2\oplus\kappa^2}}\,
\norm{w'}\\
&\leq \frac{\alpha\,a}{\sqrt{a^2\oplus\kappa^2}}
+ \frac{\kappa \, \sqrt{1-a^2}\, \sqrt{1-\alpha^2} }{\sqrt{a^2\oplus\kappa^2}} \leq \frac{\sqrt{\alpha^2\oplus\kappa^2}}{\sqrt{a^2\oplus\kappa^2}}
\;.
\end{align*}
We use Lemma~\ref{oplus:prop} (7) on the last inequality.
Finally, by proposition~\ref{prop expansion aangle} (a)
\begin{align*}
\norm{ g'\,g\,w} &\leq \norm{g'}\,\sqrt{b^2\oplus \sigma(g')^2}\,\norm{g\,w}\\
&\leq \norm{g'}\,\norm{g}\,\sqrt{b^2\oplus \sigma(g')^2}\,\sqrt{a^2\oplus \kappa^2}\\
&\leq \norm{g'}\,\norm{g}\,\sqrt{\kappa^2\oplus \alpha^2\oplus \sigma(g')^2 }
\leq \beta(g,g')\,\norm{g'}\,\norm{g} \;,
\end{align*}
where on the two last inequalities use items (6) and (5) of lemma~\ref{oplus:prop}.
\end{proof}
\begin{corollary}
Given $g, g' \in\mathcal{L}(V)$ with a $(k)$-gap pattern,
$$ \alpha_k(g,g') \leq \frac{\norm{ \wedge_k (g'\, g) }}{\norm{\wedge_k g'}\,\norm{\wedge_k g}}
\leq \beta_k(g,g') $$
\end{corollary}
\begin{proof}
Apply proposition~\ref{prod:2:lemma} to the composition
$(\wedge_k g')\,(\wedge_k g)$. Notice that by definition~\ref{def medir k}, the Pl\"ucker embedding satisfies $\Psi(\overline{\mathfrak{v}}_{k}(g) )= \overline{\mathfrak{v}}(\wedge_k g)$. Hence
$$ \alpha_k(g,g') =\alpha(\overline{\mathfrak{v}}_{k}(g^\ast),\overline{\mathfrak{v}}_{k}(g'))
= \abs{\langle \overline{\mathfrak{v}}(\wedge_k g), \overline{\mathfrak{v}}(\wedge_k g') \rangle} =\alpha (\wedge_k g, \wedge_kg')\;.
$$
\end{proof}
Next lemmas show how close the bounds
$\alpha(g,g')$ and $\beta(g,g')$ are, to each other, and to the rift $\rho(g,g')$.
\begin{lemma} \label{alpha:beta:bound}
Given $g,g'\in\mathcal{L}(V)$ with $(1)$-gap patterns,
$$ 1\leq \frac{\beta (g,g')}{\alpha (g,g')}\leq \sqrt{ 1+
\frac{{\rm gr} (g)^{-2}\oplus {\rm gr} (g')^{-2}}{ \alpha (g,g')^2} }\;. $$
\end{lemma}
\begin{proof}
Just notice that
$$ \frac{\sqrt{\kappa^2\oplus \alpha^2\oplus (\kappa')^2}}{\alpha} \leq
\sqrt{\frac{\alpha^2 + (\kappa^2\oplus (\kappa')^2) }{\alpha^2}}
= \sqrt{1 + \frac{ \kappa^2\oplus (\kappa')^2 }{\alpha^2}} \;.$$
\end{proof}
\begin{proposition}
\label{prop angle rift}
Given $g, g' \in \mathcal{L}(V)$ with a $(1)$-gap pattern
$$ \alpha(g,g')\geq \rho(g,g')\,\sqrt{1-\frac{{\rm gr}(g)^{-2} + {\rm gr}(g')^{-2} }{\rho(g,g')^2} } \;.$$
\end{proposition}
\begin{proof}
By proposition~\ref{prod:2:lemma}
$$\rho(g,g')^2\leq \beta(g,g')^2\leq
\alpha(g,g')^2 + \sigma(g)^2 + \sigma(g')^2 \;,$$
which implies the claimed inequality.
\end{proof}
These inequalities then imply the following more general fact.
\begin{proposition} \label{svp:lemma:norm}
Given $g_0, g_1,\ldots, g_{n-1}\in\mathcal{L}(V)$, if for all $1\leq i\leq n-1$ the linear maps
$g_i$ and $g^{(i)}= g_{i-1}\ldots g_0$
have $(1)$-gap patterns, then
$$ \prod_{i=1}^{n-1}
\alpha (g^{(i)},g_i)
\leq \frac{\norm{ g_{n-1}\ldots g_1 g_0} }{
\norm{g_{n-1}}\ldots \norm{g_{1}}
\norm{g_{0}}} \leq \prod_{i=1}^{n-1}
\beta (g^{(i)},g_i) $$
\end{proposition}
\begin{proof}
By definition
$g^{(n-1)}=g_{n-1}\ldots g_1 g_0$, and by convention $g^{(0)}={\rm id}_V$.
Hence $\norm{ g_{n-1}\ldots g_1 g_0} = \prod_{i=0}^{n-1}\frac{\norm{g^{(i+1)}}}{\norm{g^{(i)}}}$. This implies that
\begin{align*}
\frac{\norm{ g_{n-1}\ldots g_1 g_0}}{\norm{g_{n-1}}\ldots \norm{g_{1}}}
&=\left(\prod_{i=0}^{n-1}\frac{1}{\norm{g_i}}\right)\,
\left(\prod_{i=0}^{n-1}\frac{\norm{g^{(i+1)}}}{\norm{g^{(i)}}}\right)\\
&= \prod_{i=0}^{n-1}\frac{\norm{g_i\,g^{(i)}}}{\norm{g_i}\,\norm{g^{(i)}}}\;.
\end{align*}
It is now enough to apply proposition~\ref{prod:2:lemma} to each factor.
\end{proof}
|
1,108,101,565,419 | arxiv | \section{Introduction}\label{sec:intro}
Structured fluids composed of discrete particles, bubbles or droplets are abundant in industry and nature. The importance of these materials is highlighted by the century long continued scientific attention which their flow behavior has received. At sufficient volume fraction of particulate matter, the flow behavior of such structured fluids is generally viewed as consisting of two regimes. In the slow flow limit, interactions are contact-based and the shear stress is rate-independent. At higher driving rates, the material becomes more fluid-like: inertia, collisions or the viscosity of the interstitial fluid~\citep{courrech2003} starts to play a role; the driving stress is then well described by a power law originating mostly from collisional or viscous energy losses. These regimes are often phenomenologically combined by the Herschel-Bulkley (HB) model~\cite{herschel1926}:
\begin{equation}
\displaystyle \tau=\tau_0 + k\dot{\gamma}^n.
\label{floweq:1}
\end{equation}
In this equation, $\tau$ denotes the shear stress, $\tau_0$ the yield stress, $\dot{\gamma}$ the shear rate, $k$ a proportionality constant and $n$ a power law index. The HB model effectively captures the macroscopic flow response of dense granular materials~\cite{jop2006}, emulsions and foams~\cite{paredes2013,dinkgreve2015}, as well as suspensions~\cite{dijksman2010}. Note that for all these systems, the volume fraction $\phi$ has to be high enough in order for the dispersed phase to ``jam'' and resist flow in the slow flow limit~\cite{siemens2010}. The HB constitutive equation also serves as input for flow modeling of amorphous materials deep into the regime where these material seem to be solid-like, in particular as local flow rule in the very successful ``fluidity'' based kinetic elasto-plastic flow modeling \cite{goyon2008,jop2012,kamrin2012}. Even so, although the HB model is applied in a wide variety of materials, exactly how microscopic interactions affect the HB ingredients is an area of active study. There are many microscopic features relevant for the macroscopic flow behavior; the surprising role of roughness, charges, lubrication, adhesion and friction~\cite{johnson2000,zhou2001,lootens2005,becu2006,seto2013, wyart2014, clavaud2017,comtet2017,pons2017,Coulomb2017} have already been suggested especially in faster ``inertial'' flows, where local flow properties can ``turn on'' frictional effects~\cite{seto2013} giving even strong deviations from HB behavior. It is suggested that fluctuations affect $n$ in various regimes~\cite{tighefluct}, but also that microscopic friction coefficients do not significantly affect the HB model~\cite{kamrin2014}.\\
\begin{figure}[t!]
\centering
\includegraphics[width=14cm]{Fig1_new2}
\caption{\label{friction}(a) Schematic phase diagram. Emulsions are located in the limit of zero microscopic friction constant $\mu_m$. Granular materials and suspensions of solid particles exist at finite $\mu_m$. To obtain HB behavior, the solid fraction $\phi$ of these materials must be above some finite $\phi_{rcp}$ limit, which generally depends on $\mu$~\cite{silbert2010,Hsu2018}, where steric hindrance becomes important. Using soft particles, one can obtain volume fractions above this steric limit for finite $\mu_m$. (b) Schematic of the Couette geometry used in our experiments. $\Omega_i$ is the applied rotation rate, $M$ is the measured torque. Particles are confined to a constant volume environment.}
\end{figure}
Here we show experimentally that microscopic frictional interactions between suspended particles in a dense ``granular emulsion'' have a significant influence even in the \emph{slow} flow limit. We perform experiments using dense suspensions of soft particles confined in fixed volume and sheared in a Couette geometry. Their softness allows us to suspend the particles at high volume fraction ($\phi > \phi_{rcp}$, the random close packing density) while still being able to make them flow. The soft particle suspension we use can therefore be made similarly dense as an emulsion, yet the interactions between the particles are frictional as in a granular material. In our perspective, the granular emulsions we employ exist in the top right corner of the schematic phase diagram sketched in Fig.~\ref{friction}a. Our granular emulsions are therefore rather different from discontinuous shear thickening fluids, as particles are densely packed at all shear rates, at a finite pressure, and therefore always feature semi-permanent contacts among particles.\\
\indent We find that granular emulsions have a well defined effective friction coefficient with two peculiar properties: the effective friction coefficient can either be similar, \emph{or} much higher than that of the microscopic coefficient, depending on the magnitude of the microscopic friction coefficient. Furthermore, the effective friction coefficient of the suspension can be rate dependent such that it gets \emph{smaller} at higher shear rates. Even though weak flow instabilities in flowing suspensions have been observed before~\cite{lu2007, dijksman2011}, we find ``yield stress'' reductions of up to a factor two. Our results highlight the importance of understanding the coupling between microscopic interactions and macroscopic flow behavior and their integration in numerical and theoretical modeling approaches for dense particulate media.\\
The paper is set up as follows: in Sec.~\ref{sec:mnm} we first present the flow geometry in which we perform all rheological measurements and a characterization of the custom made hydrogel particles in terms of their size, hardness and frictional properties. In Sec.~\ref{sec:stress} we present an overview of experimental results for shear and confining stress dynamics for various suspension types, at various experimental settings. We make a cross comparison of these results and briefly discuss how pressure controlled experiments compare to our volume controlled experimental data. To gain further insight into the rheology of granular emulsions, we discuss their flow behavior and fluctuations in Sec.~\ref{sec:flow}. An overall discussion and conclusion section follows the presented results.
\section{Materials \& Methods}\label{sec:mnm}
\subsection{Flow setup}\label{subsec:flowsetup}
We use a custom, 3D-printed (Stratasys Objet 30) Couette cell to perform the flow experiments, see Fig.~\ref{friction}b and Ref.~\cite{workamp2017}. The inner cylinder has radius $r_i$ = 25~mm, while the outer cylinder has radius $r_o$ = 45~mm, such that the gap $r_o-r_i$ = 20~mm $\approx$ 10$d$ with $d$ the particle diameter. We drive the inner cylinder using a rheometer (Anton-Paar MCR301 or MCR501). Both inner and outer cylinder are made rough with teeth of approximately 2.5~mm to minimize wall slip. The height of the shear cell $L$ = 20~mm~$\approx 10d$; the rheometer measures/provides a torque $M$. There is a top cover on the cell that confines only the particles; the fluid can freely move in and out of the cell. Unless otherwise mentioned, we thus confine the particles to a constant volume in all experiments, while we let the particle pressure adjust to the shear rate and amount of particles added to the volume. We thus measure the pressure exerted by the particles only. We measure the particle pressure $P^p$ on a separate lid embedded in the cover. The lid is attached to a load cell. Solvent can freely flow in and out of the cell through the gap around the pressure-sensing lid and cover, and through the gap between the rotating inner cylinder and the cover. The benefit of this approach is that in typical emulsion rheology experiments, performing constant (particle) volume experiments is impossible due to the size of the droplets involved, and confining stresses are at least partly induced by surface tension at the free boundary, which can assume any shape. Our constant volume experiments allow an effective control and characterization of confining pressure. Experimental protocols and results are discussed in the next section.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{Fig2_MnM}
\caption{\label{fig:MnM} (a) Typical results of uniaxial compression tests of hydrogel particles. Normal force $F_n$ as a function of overlap $\delta$, for a particle of 15\% ($\triangledown$), 10\% ($\diamond$) and 5\% gelatin ($\triangle$), as well as for PAAm ($\circ$). Using Hertzian contact theory, we find the elastic modulus $E$ of the particles as $F_n=\frac{4}{3}\frac{E}{1-\nu^2}R^{1/2}\delta^{3/2}$, where we assume Poisson's ratio $\nu = 0.5$ (incompressible material) and R is the particle radius. (b) Schematic of our tribology setup. (c) Material friction coefficient $\mu_m$ as a function of sliding velocity $v$ for gelatin-gelatin contact (with 15 ($\triangledown$), 10 ($\diamond$), or 5 wt\% gelatin ($\triangle$)) and PAAm-PAAm contacts ($\circ$). Error bars denote the 95\% confidence interval of $\mu_m$, based on linear regression of $F_f(F_n)$.}
\end{figure}
\subsection{Particle hardness}\label{subsec:parthardchar}
We aim to perform experiments on suspensions of macroscopic particles in which we only vary the friction coefficient, while keeping \emph{all} other experimental setting the same. We therefore need to make particles manually, from materials with different surface properties. The materials we choose are hydrogels, because they are soft, can be made through custom synthesis methods~\cite{workamp2016} and are known to have tunable frictional behavior~\cite{gong1998,gong2006,baumberger2006}. In particular, we use low friction polyacrylamide (PAAm) and chemically cross-linked gelatin. We produce bidisperse mixtures with mean diameters $d$ around 2~mm. We make the PAAm particles using a monomer solution that contains 20 wt\% acrylamide and 1 wt\% N,N'-methylenebis(acrylamide) as a cross-linker. We prepare gelatin particles of 5, 10 and 15~wt\% gelatin. To ensure the gelatin particles remain stable to dissolution, we cross-link them with glutaraldehyde~\cite{damink1995glutaraldehyde}. We can keep the composition and the stiffness of the PAAm and gelatin suspensions the same by choosing the right gelatin concentration. To show this, we use uniaxial compression to measure the elastic moduli of the particles. We find the Youngs moduli to be approximately $8.1\times 10^1$ kPa (5\% gelatin), $3.2\times 10^2$ kPa (10\% gelatin), $9.1\times 10^2$ kPa (15\% gelatin) and $3.1\times 10^2$ kPa (PAAm); see Fig.~\ref{fig:MnM}a. Note that the PAAm particles and the 10\% gelatin particles have the same modulus.
\subsection{Hydrogel friction characterization}\label{subsec:tribo}
We measure the frictional behavior of the hydrogels using a modified version of the ``pin-on-disk'' method. Usually, the pin is held stationary while the disk rotates, see e.g. Ref.~\cite{pitenis2014}. Instead of driving the disk, we drive the pin, a hemispherical gel head (radius 7 mm) using a rheometer (Anton-Paar MCR501). The gel head is securely held on a 3D-printed arm (length $l$ = 3~cm) connected to the rheometer axis, and rubs over a flat gel slab of the same material. The hydrogel samples used for measurement of the friction coefficient have the same chemistry as the particles and are molded using petri-dishes to create flat disks, and silicone rubber (Smooth-On Oomoo) to prepare a hemispherical probe. To ensure a smooth surface of the hemispherical cap, the rubber mold is cast using a ball produced for ball bearing purposes, which have superior smoothness and roundness.\\
\indent We drive the arm at rotation rates ranging from $10^{-3}$ to $10^{-1}$~rps, corresponding to sliding velocities $v$ from $1.9 \times 10^{-4}$ to $1.9 \times 10^{-2}$~m~s$^{-1}$. We measure the torque $M$ and normal force $F_n$ at different heights of the hemispherical probe, to get a range of $F_n$. As the hydrogel surfaces as well as the arm are submersed in water, we correct $F_n$ for buoyancy and $M$ for the viscous contribution of the water. We calculate the frictional force as $F_f = M/l$. We use only the data where 0.02~mN $<F_n<$ 20~mN; in this regime $F_f$($F_n$) is linear and regression yields $\mu_m$. At higher loads, $F_f$ depends more weakly on $F_n$. In our rheology measurements, the particle pressure $P^p$ is around 1~kPa; an estimate of the load on each particle is $P^pd^2$, yielding normal forces in the same range as in our friction measurements. The setup is schematically depicted in Fig.~\ref{fig:MnM}b. Although there is an error associated to measuring a friction coefficient on a circular sliding path rather than in a straight line~\cite{krick2010}, this error is negligible here, since the arm $l$ is much larger than the maximum radius of the contact area ($a_{max}\sim$~1~mm).\\
\indent In Fig.~\ref{fig:MnM}c, we plot $\mu_m$ as a function of the sliding velocity $v$, for the different materials. The errorbars denote the 95\% confidence interval for $\mu_m$. For the polyacrylamide surfaces, the friction coefficient is on the order of 10$^{-2}$ and little effect of $v$ is observed, in agreement with Ref.~\cite{uruena2015}. Only at the highest rate a small decrease can be seen. However, the error bar on this data point is relatively large as the frictional force is small while viscous contributions from the fluid in which the arm is rotating are significant at this rate. Although their results only concern PAAm hydrogels, Ref.~\cite{uruena2015} also helps interpret the polymer concentration dependence of $\mu_m$ for the cross-linked gelatin. The authors show that decreasing the mesh size (i.e. increasing the polymer concentration) of the gel increases its friction coefficient, in agreement with our findings. The friction coefficient of all cross-linked gelatin decreases with $v$. Note that since the modulus and all other particle parameters are the same for PAAm and 10 wt\% gelatin suspensions that we will make, the only difference between them is their frictional behavior.
\section{Stress dynamics}\label{sec:stress}
\subsection{Rate dependence of shear stress and confining pressure}
To explore the effect of contact friction in suspensions, we measure the shear stress and confining pressure for suspensions made with PAAm and gelatin particles. Composing particles of three different concentrations of gelatin provide us a range of $\mu_m \in \{0.01 \ldots 0.6\}$ as outlined in Sec~\ref{subsec:tribo}. We can confine the suspensions by simply adding more particles in the same volume and measure the resultant confining pressure as a function of shear rate, as is typical for dry granular materials and suspensions~\cite{dacruz2005,boyer2011}. We thus perform measurements at different constant volume fractions. The volume fractions are not known, but we can characterize the density through the pressure $P^p$ at the lowest shear rate $\dot{\gamma}_0 = 1.2 \times 10^{-2}$~s$^{-1}$. Since $P^p$ is finite even at zero shear, we know that $\phi>\phi_{rcp}$, the random close packing density. Due to the size of the particle used, pore fluid flow effects are negligible. It is challenging to match the density of the particles with the solvent as the hydrogel particles are porous to their swelling solvent and their swelling depends on environmental conditions; we therefore use water as the solvent. The maximum hydrostatic pressure can be estimated to be $P_g = \Delta\rho g L \approx$~20~Pa, with $\Delta\rho$ the density difference. As $P^p >> P_g$, we expect no influence of $P_g$. Note that regardless of driving form, $P^p$ in all our experiments on both PAAm and gelatin suspensions is never more than 1.5\% of the modulus: the particles are very weakly compressed and hence remain spherical at all times, so multiple contact effects~\cite{brodu2015,hohler2017} can be neglected. Before measuring the flow curve, we pre-shear the sample at our maximum shear rate ($1.2 \times 10^{2}$~s$^{-1}$) for 10 seconds. After this, we decrease the rate, measuring the required stress for one full rotation of the tool for 41 logarithmically spaced shear rates. Both the top cover and bottom of the cell we use are made of smooth acrylic, so we can perform flow profile measurements via transmission-based particle image velocimetry (see Ref.~\cite{workamp2016} for details). Note that the numerical value used for the shear strese and shear rate in a Couette geometry is subject to some arbitrary choices, due to the inhomogeneity of stress and flow field even at fixed $\Omega_i$; following~\cite{rheochem,chatte2018} we use $\dot{\gamma} \equiv \langle\dot{\gamma}\rangle = \Omega_i\frac{r_o^2+r_i^2}{r_o^2-r_i^2}$ and $\tau \equiv \langle\tau\rangle = M\frac{r_o^2+r_i^2}{4\pi Lr_o^2r_i^2}$; the geometric correction coefficients are all of order one.\\
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Fig3_SM_Flowcurves_pagew}
\caption{\label{fig:flowcurves} Shear stress $\tau$ as a function of shear rate $\dot{\gamma}$ at different volume fraction for PAAm (a), 5\% (b), 10\% (c) and 15\% gelatin (d). We characterize the volume fraction by the pressure $P^p$ at the lowest measured $\dot{\gamma}$. Solid lines represent HB fits according to Eq.~\ref{floweq:1}. From a to d, $\mu_m$ for the particles used increases from 0.01 to 0.6.}
\end{figure}
The results of the rate dependent shear stress measurements for all four material types are shown in Fig.~\ref{fig:flowcurves}. Our hydrogel friction measurements indicate that the material friction coefficient $\mu_m$ of PAAm to be approximately 0.01. This means the PAAm particles resemble emulsion droplets: they are deformable and have negligible friction. The rheology of the PAAm suspension is indeed what one may expect for an emulsion: it is well fitted with the HB model (solid lines in Fig.~\ref{fig:flowcurves}a). We find exponents $n$ of about 0.5$-$0.6, similar to what one finds in dense emulsions~\cite{dinkgreve2015}, owing to the deformability of the particles~\cite{barnes1989, brown2014}. The gelatin suspension flow curves are different in character. All gelatin suspension flow curves display non-monotonic behavior, with a distinct minimum or ``dip'' around 5 s$^{-1}$. The dip location seems to be independent of the overall modest pressure variation, but gets more pronounced the higher $\mu_m$ is.\\
At the same time during the same experiments, we measure the confining pressure; results are shown in Fig.~\ref{fig:Pp_rate}. Again we find that the PAAm suspension displays a monotonic increase of the confining pressure with the shear rate. For the gelatin suspension, the pressure dynamics is more subtle: at low $\mu_m$ and high pressure, the confining pressure is also monotonically increasing with shear rate. However, as $\mu_m$ increases, a non-monotonicity becomes apparent at low pressure; at the largest $\mu_m$, all pressure dynamics displays this dip. Additionally, at the largest $\mu_m$, the low shear rate dynamics of $P^p$ is weakly rate dependent. Note that the vertical axes in Fig~\ref{fig:Pp_rate} are linear and not logarithmic as those in Fig~\ref{fig:flowcurves}; the dips in $P^p(\dot{\gamma})$ are less pronounced than those in $\tau(\dot{\gamma})$. The pressure measurements are robust; we performed additional experiments in which we measured the confining pressure from the cylinder wall and found the pressure dynamics in the radial direction had the same rate dependence as $P^p$ (not shown).
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Fig4_SM_Pp_vs_rate_pagew}
\caption{\label{fig:Pp_rate} Confining pressure $P^p$ as a function of shear rate $\dot{\gamma}$ at different volume fraction for PAAm (a), 5\% (b), 10\% (c) and 15\% gelatin (d). Same colors/symbols as in Fig.~\ref{fig:flowcurves}.}
\end{figure}
\subsection{Pressure rescaling}
To interpret the flow curves shown in the previous section, we borrow the ideas from dry granular materials~\cite{dacruz2005} and suspensions~\cite{boyer2011}: we investigate the rheology by computing the \emph{macroscopic} friction coefficient $\mu=\tau(\dot{\gamma})/P^p(\dot{\gamma})$. We combine the shear stress data and the measured confining pressure $P^p$ for all points in the flow curve. We plot $\mu(\dot{\gamma})$ in Fig.~\ref{fig:MuRate}. For all suspensions, the shear stress scales with the confining pressure exerted on the particles and hence we obtain a good collapse of the data obtained at different $P^p$. It is immediately obvious that the PAAm suspension flow behavior in Fig.~\ref{fig:MuRate}a is different from the gelatin suspensions in Fig.~\ref{fig:MuRate}b-d in several ways. We find that for the PAAm suspension, the quasistatic suspension friction coefficient at low shear rates is constant and approximately 0.16. This value is much higher than the material friction coefficient $\mu_m \sim 0.01$. While it has been observed before that even at $\mu_m = 0$, $\mu > 0$ (see for example Refs~\cite{chialvo2012,trulsson2017,peyneau2008}), we find this result counter-intuitive, as it suggests that contact friction is indeed not the main source of dissipation in PAAm suspensions, despite the pressure rescaling. This observation is perhaps related to how the friction coefficient of a rough solid depends on the height distribution of the asperities but only weakly on the pressure~\cite{zhuravlev2007,gw1966}. Furthermore, for the PAAm suspension the effective suspension friction coefficient is a monotonically increasing function of the shear rate.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Fig5_SM_Mu_vs_rate_pagew}
\caption{\label{fig:MuRate} Effective friction coefficient $\mu$ as a function of shear rate $\dot{\gamma}$ at different pressures for PAAm (a) and gelatin: 5\% (b), 10\% (c) and 15\% (d). Same symbols as in Fig.~\ref{fig:flowcurves}. The inset in c shows the 10\% data as a function of the shear rate rescaled by the pressure.}
\end{figure}
\newpage
Gelatin particles suspensions always have a significant effective friction coefficient in the limit of zero shear rate. The suspension friction coefficient also seems to be of the same order as the microscopic friction coefficient. Upon increasing the shear rate, the suspension friction coefficient however initially decreases before entering the more commonly observed rate dependent regime; the larger $\mu_m$, the stronger the decrease. Initially, the decrease seems logarithmic, yet there is always a pronounced minimum in $\mu_m(\dot{\gamma})$. Note that the location of the minimum is at constant $\dot{\gamma}$ for each material, rather than at a constant inertial number~\cite{dacruz2005} $I = \dot{\gamma}d\sqrt{\rho/P^p}$ or viscous number~\cite{boyer2011} $J = \frac{\eta_f \dot{\gamma}}{P^p}$, where $\rho$ is the particle density and $\eta_f$ the viscosity of the suspending fluid. To highlight this fact, we plot $\mu$ as a function of $\dot{\gamma}/P^p$ in the inset of Fig.~\ref{fig:MuRate}c. The collapse of the data is certainly not as good as in the main panel, especially in the slow flow limit. The shear rate at which the minimum occurs thus seems to change little with $P^p(\dot{\gamma})$. At higher gelatin concentration, the particles also change in stiffness by a factor 10 as documented in Sec.~\ref{sec:mnm}, whereas the location of the minimum does not appear to systematically change in panel Fig.~\ref{fig:MuRate}b-d. The role of particle stiffness is perhaps not always crucial in slow flows~\cite{Coulomb2017}, but the absence of good rescaling with either $P^p$ or $E$ suggests that another, perhaps contact-based, time scale is causing the instability.
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\linewidth]{Fig6_summary}
\caption{\label{fig:comparison} (a) Shear stress $\tau$ normalized with the yield stress $\tau_0$ as function of shear rate $\dot{\gamma}$ for suspensions at fixed volume. We estimate $\tau_0$ as $\tau$ at the lowest $\dot{\gamma}$ considered here. The PAAm suspension ($\circ$, $P^p = 0.24$~kPa) is fitted well by the HB model (see Fig.~\ref{fig:flowcurves}a), while gelatin suspensions (with 15 ($\triangledown$, $P^p = 0.79$~kPa), 10 ($\diamond$, $P^p = 0.87$~kPa), or 5 wt\% gelatin ($\triangle$, $P^p = 0.30$~kPa)) display a flow instability. (b) For the same data as in (a), a comparison of $\mu(\dot{\gamma})$ from PAAm and the three gelatin suspension types. (c) Torque $M$ as a function of shear rate $\dot{\gamma}$ for gravitational suspensions of 10 wt\% gelatin ($\diamond$), PAAm ($\circ$) and glass beads ($*$). All measurements performed in our Couette cell but now filled to a height of $\approx 3/4h$, i.e. there is no pressure on the lid and the particles are jammed by hydrostatic pressure $P_g$ only. Due to the larger density of the glass beads, their yield stress is also larger.}
\end{figure}
\subsection{Comparison of flow curves}
We can go a step further and directly compare the flow curves of different hydrogel suspensions in one figure. We would like to stress that while we change the hydrogel chemistry, all other particle and suspension characteristics such as hardness, size and polydispersity, system volume, boundary conditions et cetera are the same between a PAAm particle suspension and a gelatin particle suspension composed with particles made from a 10\% gelatin solution. We first compare the shear stress behavior in Fig.~\ref{fig:comparison}a; to make a good comparison, we normalize the data on the zero-shear stress value. This allows us to even more directly compare the role of particle hardness: there is no observable trend with the particle modulus in the location of the minimum in the flow curve for the three gelatin-based suspensions, so the flow curves do not seem to be affected by this pressure scale. The depth of the minimum however increases with increasing polymer concentration and thus seems to depend on $\mu_m$.
When we compare the suspension friction coefficients in Fig.~\ref{fig:comparison}b, we see that the four different suspensions behave similar in the high flow rate regime; at low flow rates, the observed minimum in the flow curve for gelatin coincides with the effective friction coefficient for the PAAm suspension. Indeed, recent numerical work suggests that $\mu(\dot{\gamma})$ should depend on $\mu_m$~\cite{trulsson2017}, contradicting earlier results that suggest that $\mu$ is a universal function~\cite{gallier2014}. Here, we observe that at sufficiently high $\dot{\gamma}$, the gelatin data is clearly similar to that of the PAAm data. Note that the exponent $n$ seems to be much less than $1$, contrary to expectations for inertial or viscously damped granular flows, but in agreement with dense emulsion flows~\cite{dinkgreve2015}.
\subsection{Volume control versus pressure control}\label{subsec:vvspcontrol}
In the experiments discussed above, we exclusively focused on volume controlled experiments, in which the volume fraction is fixed and the granular pressure and shear stress depend on the shear rate. This is potentially problematic, as $\phi$ is considered the slaved variable in most flow modeling efforts~\cite{boyer2011,henann2013}. However, the flow behavior we have observed is not limited to controlled volume contexts. We can drive our granular emulsions also without the presence of the confining lid and observe the same qualitative behavior. Without lid, the confining pressure scale is then the hydrostatic pressure generated by the density mismatch of the particles and the water. This constant pressure environment also allows us to compare our granular emulsions with a suspension of glass beads in water. The results are shown in Fig.~\ref{fig:comparison}c. Clearly, in the pressure controlled environment of the open Couette cell, we observe that \textit{i} the PAAm suspension has a monotonically increasing flow curve, \textit{ii} the gelatin suspension has a minimum in the flow curve. \textit{iii} the glass bead suspension shows a modest shear weakening behavior conform to other work~\citep{dijksman2011}. Note that the nonmonotonicity of gelatin suspensions even reproduces in unconfined split bottom geometry~\cite{2010_sm_dijksman} driven flows (not shown). We thus conclude that volume control in our experiments did not significantly change the coupling between microscopic contact mechanics and macroscopic flow phenomenology, again suggesting some other contact based time scale is determining the instability and/or the difference between the PAAm and gelatin suspensions.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.9\linewidth]{Fig7_abc}
\caption{\label{fig:flowprofiles} (a) Normalized angular velocity $\Omega(r)/\Omega_i$ as a function of the distance from the inner cylinder for gelatin ($\diamond$) and PAAm ($\circ$) suspensions at similar $P^p \approx$ 0.2~kPa. Yellow and blue datapoints represent the suspension at $\dot{\gamma} = 4.8 \times 10^{-2}$ s$^{-1}$, while purple and red datapoints are at $\dot{\gamma} = 1.2$ s$^{-1}$. Adapted from Ref.~\cite{workamp2017}. Inset: normalized standard deviation $SD$ of the pixel intensity time series, as a function of the distance from the inner cylinder. Same color coding as in the main panel. (b) Still image from supplementary video showing the difference in particle flow fluctuations in PAAm and 10\% gelatin suspensions (Multimedia View). (c) Material friction coefficient $\mu_m$ as a function of the effective friction coefficient of the suspension $\mu$ for all materials, sliding velocities and volume fractions. The dashed line represents $\mu_m=\frac{3}{2}\mu$, dash-dotted line denotes the critical value $\mu_0$. Same symbols as in Fig.~\ref{fig:comparison}.}
\end{figure}
\section{Flow behavior}\label{sec:flow}
\subsection{Flow profiles}
The Herschel-Bulkley behavior observed for the PAAm suspension, and the flow instability observed for gelatin suggest that our granular emulsions will show shear banding~\cite{schall2010, bonn2017review}. To determine the flow profiles in the gap of our Couette cell, we perform particle image velocimetry (PIV), using a method described in more detail elsewhere~\cite{workamp2017}. In short: imaging the flowing suspension in transmission provides sufficient contrast to elucidate local velocities using standard PIV methods. In Fig.~\ref{fig:flowprofiles}a we plot the angular velocity $\Omega(r)$ normalized with the angular velocity of the inner cylinder $\Omega_i$, for suspensions of PAAm and 10\% gelatin at similar pressure ($P^p \approx$ 0.2~kPa), for two different driving rates: $\dot{\gamma} = 4.8 \times 10^{-2}$ s$^{-1}$ and $\dot{\gamma} = 1.2~$s$^{-1}$, both below $\dot{\gamma}$ of the minimum. A more extensive dataset can be found in Ref.~\cite{workamp2017}. For the PAAm suspensions, the shear bands are relatively wide and insensitive to the driving rate, whereas for gelatin the shear bands are narrow and slightly rate-dependent. However, for both materials, $\Omega(r)$ decays to less than 10\% of $\Omega_i$ within a few particle diameters $d$.
\subsection{Flow Fluctuations}
Even though the decay of the velocity profiles shows that flow ceases entirely beyond a couple of particle diameters from the rotating cylinder, we observe that the suspended particles still fluctuate in their position even in the static zone. That PAAm and gelatin suspensions display different fluctuations can be observed visually when running the experiment. Particles outside of the shear band are clearly much more ``agitated'' in a PAAm suspension compared to the gelatin case. These velocity fluctuations are a crucial element in dispersion based flow modeling~\cite{jop2012,kamrin2017}. Measuring the actually relevant particle-level velocity fluctuations is not possible in our experiment, but we can qualitatively measure the extent of such fluctuations. We estimate particle position fluctuations by calculating the standard deviation $SD$ of the time series of the intensity fluctuations in the forward scattered light passing through the suspension. Values are averaged over the azimuthal direction and normalized with the mean intensity of the image. $SD$ signifies both flow and uncorrelated particle motion, and is plotted in the inset of Fig.~\ref{fig:flowprofiles}a, for the same experiments represented in the main panel. From this analysis, two observations stand out: it is clear that particle fluctuations extend the entire gap ($r-r_i\approx 10d$) for the PAAm suspension, even though the flow is localized to a shear band of only a few particle diameters. By contrast, in the case of gelatin, $SD$ decays to zero within 1 or 2 particle diameters away from the shear band. Second, the extent of particle motion fluctuations for the PAAm suspensions are rate dependent, whereas the normalized flow profiles are not. These observations can be qualitatively assessed with multimedia video~\ref{fig:flowprofiles}b. We conclude that the microscopic frictional interaction mechanisms also significantly affect the local flow behavior: friction enhances shear banding and suppresses particle-level fluctuations.\\
\subsection{Microscopic interpretation}
The measured flow profiles allow us to estimate the relative velocities of the particles in the suspension. Since the microscopic friction is weakly rate dependent for the gelatin particles we can attempt to see how $\mu_m(v)$ and $\mu(\dot{\gamma})$ are connected. Since $\Omega(r)$ decays to less than 10\% of $\Omega_i$ within a few particle diameters $d$, we estimate the particle relative sliding velocities as $\Omega_i r_i$. The sliding velocities in our friction measurements are then corresponding to values of $\dot{\gamma}$ just below the observed minima in $\mu(\dot{\gamma})$. We can therefore speculate that a microscopic timescale in $\mu_m(v)$ plays a role in the observed flow instability. The observed irrelevance of pressure versus volume control in Sec.~\ref{subsec:vvspcontrol} points towards a more microscopic underpinning of the observed minimum. Our data is however inconclusive: the strongest instability is observed for the 15\% gelatin particle, which shows the least amount of rate dependence in $\mu_m(v)$. Due to experimental limitations, we cannot extend the range of $\mu_m(v)$ to higher $v$, covering the entire $\dot{\gamma}$ range. We can nevertheless directly compare our $\mu_m(v)$ and $\mu(\dot{\gamma})$ by plotting $\mu_m(v)$ for each $\mu(\dot{\gamma})$ with a similar sliding velocity. We plot $\mu_m$ as a function of $\mu$ in Fig.~\ref{fig:flowprofiles}c, for all materials and $P^p$. The dashed line serves as a reference to indicate what a linear relation between the two variables would look like on this log-log scaling; specifically, it represents $\mu_m=\frac{3}{2}\mu$. For the gelatin, all data points lie close to this line: the collapse is certainly not perfect, but the deviations in $\mu$ are all smaller than 0.15. This is important, as for the PAAm suspensions, at much lower $\mu_m = 0.01$, $\mu$ differs distinctly from $\mu_m$, by as much as $0.15$. This deviation suggests that a different dissipation mechanism must contribute to the shear resistance of the PAAm suspension, that sets a minimum $\mu$, which we call $\mu_0$. We find $\mu_0 \approx 0.16$. In simulations of slow flows of frictionless suspensions~\cite{chialvo2012,trulsson2017} and frictionless dry granular materials~\cite{peyneau2008}, values of approximately 0.1 are found, suggesting a bigger contribution of fabric and force anisotropy~\cite{rothenburg1989,majmudar2005,peyneau2008,peyneau2008b,azema2014,azema2015} in our experiments. Unexpectedly, for frictional particles like the gelatin particles used here, the correction due to anisotropy/geometry seems to disappear, and the suspension friction coefficient is set exclusively by the material friction coefficient. Knowing that even in the PAAm suspension a finite $\mu_0$ is observed, even though tangential contact force components are absent makes it all the more surprising that, approximately, $\mu_m=\frac{3}{2}\mu$ for the gelatin suspensions: the contributions of friction and geometric effects to the shear stress do not seem to be simply additive; it seems that $\mu = \mu_m + C(\mu_m)$ in which constant $C(\mu_m) \sim \mu_0$ for $\mu_m \ll 0.1$, but $C(\mu_m) \rightarrow 0$ for $\mu_m > 0.1$ Adding to the confusion, numerical simulations of comparable systems have found contradicting relationships between $\mu_m$ and $\mu$: see Refs.~\cite{chialvo2012,gallier2014,trulsson2017}. Finding how anisotropy emerges from grain-scale friction, velocity and perhaps other microscopic contact and force correlations hence seems to be an important next step to understand suspension rheology.
What is the microscopic source of the instability? We would like to note that the instability observed in Fig.~\ref{fig:comparison}c for the glass beads and gelatin suspension is of different character. The glass bead suspension has a logarithmic negative rate dependence, that beyond a certain flow rate gets overtaken by inertial dynamics. The source of this rate dependence is perhaps related to self-weakening due to mechanical agitations present in the material~\cite{wortel2016} that propagate fast enough due to the hardness of the particles and the limited damping of the low viscosity solvent (water). In contrast, the gelatin suspension has a broad and deep minimum in the flow curve. The time scale responsible for this minimum is not clear. One option is that it is related to a hydrodynamic particle contact effect. Due to the composition of the hydrogel particles used in this study, probing the role of the fluid viscosity was not possible, yet this remains a promising avenue for future work. \\
\section{Conclusions}
We probe the role of microscopic friction in slow dispersed media flows by synthesizing soft particles that allow us to perform experiments above the random close packing limit. By measuring both shear and confining stresses during flow, we find that friction plays an outsize role in all aspects of the flow: rheology, flow profiles and particle-level fluctuations of such suspensions are significantly affected by the microscopic friction coefficient. In the ``emulsion'' limit, where the material friction friction $\mu_m$ is smaller than a critical value $\mu_0$, the macroscopic friction $\mu$ remains finite. This suggests that dissipation in dispersed media can emerge from non-frictional, perhaps geometric sources or velocity fluctuations. Upon increasing the material friction coefficient $\mu_m > \mu_0$, we find that the flow behavior of the granular emulsion becomes unstable, while the effective friction coefficient of the suspension approaches that of the microscopic value; we find that the suspension friction coefficient $\mu$ is set by $\frac{2}{3}\mu_m$. Our results show that the ``granular emulsion'' phase yields a wide range of different, unexpected and potentially useful flow behaviors. The observations provide new benchmarks for modeling approaches and could serve as input to get more insight in the microscopic underpinning of fluidity and anisotropy based modeling of dispersed media.
\begin{acknowledgments}
We thank Pieter de Visser, Raisa Rudge and Jonathan Bar\'{e}s for their help and useful discussions concerning the tribology measurements, and Sepideh Alaie for her help with making the particles and measuring the flow profiles.
\end{acknowledgments}
|
1,108,101,565,420 | arxiv | \section{Introduction}
The Jordan type of a graded Artinian algebra $A$ and linear form $\ell$ is a partition determining the Jordan block decomposition for the (nilpotent) multiplication map by $\ell$ on $A$ which is denoted by $P_{\ell,A}=P_\ell$. Jordan type determines the weak and strong Lefschetz properties of Artinian algebras. A graded Artinian algebra $A$ is said to satisfy the weak Lefschetz property (WLP) if multiplication map by a linear form on $A$ has maximal rank in every degree. If this holds for all powers of a linear form the algebra $A$ is said to have the strong Lefschetz property (SLP). It is known that an Artinian algebra $A$ has the WLP if there is a linear form $\ell$ where the number of parts in $P_\ell$ is equal to the Sperner number of $A$, the maximum value of the Hilbert function $h_A$. Also $A$ has the SLP if there is a linear form $\ell$ such that $P_\ell = h^\vee_A$ the conjugate partition of $h_A$ see \cite{IMM}.
Jordan type of a linear form for an Artinian algebra captures more information than the weak and Strong Lefschetz properties. Recently, there has been studies about Jordan types of Artinian algebras also in more general settings, see \cite{IMM, IMM2, IKVZ} and their references.
Studying Artinian Gorenstein algebras is of great interest among the researchers in the area. Gorenstein algebras are commutative Poincar\'e duality algebras \cite{MW} and thus natural algebraic objects to cohomology rings of smooth complex projective varieties.
There has been many studies in the Lefschetz properties and Jordan types of Artinian Gorenstein algebras \cite{CG, HW, GZ, Gondim, MH}.
Gorenstein algebras of codimension two are complete intersections and they all satisfy the SLP. The list of all possible Jordan types of linear forms, not necessarily generic linear forms, for complete intersection algebras of codimension two is provided in \cite{CIJT}.
In this article, we study the ranks of multiplication maps by linear forms on graded Artinian Gorenstein algebras that are quotients of polynomial ring $S = \K[x_1,\dots ,x_n]$ where $\K$ is a field of characteristic zero. In Section \ref{section-rksec}, we study such algebras with arbitrary codimension in terms of their Jordan types. We present an approach to determine the Jordan types of Artinian Gorenstein algebras using Macaulay duality. We assign a natural invariant to an Artinian Gorenstein algebra $A$ providing the ranks of multiplication maps by a linear form $\ell$ in different degrees, called \emph{rank matrix}, $M_{\ell,A}$, Definition \ref{rkmatrix-def}. There is a 1-1 correspondence between rank matrices and so called \emph{Jordan degree types} in Proposition \ref{rkmatrix-1-1-JDT-prop}. We provide necessary conditions for a rank matrix in Lemmas \ref{diffO-seq} and \ref{additiveRank}.
We use this approach in Section \ref{codim3section} for Artinian Gorenstein algebras in polynomial rings with three variables. We give a complete list of rank matrices that occur for some Artinian Gorenstein algebra $A$ and linear form $\ell$ where $\ell^3=0$ and $\ell^2\neq 0$, see Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd} for algebras with even and odd socle degrees respectively. As an immediate consequence in Corollary \ref{2linescorollary} we list rank matrices for linear forms where $\ell^2=0$.
In Theorem \ref{JT-theorem} we prove that the Jordan types of Artinian Gorenstein algebras with codimension three and linear forms $\ell$ where $\ell^4=0$ is uniquely determined by the ranks of at most three multiplication maps, or equivalently, three mixed Hessians.
\section{Preliminaries}
Let $S = \K[x_1,\dots ,x_n]$ be a polynomial ring equipped with standard grading over a field $\K$ of characteristic zero. Let $A=S/I$ be a graded Artinian ( its Krull dimension is zero) algebra where $I$ is an homogeneous ideal.
The \emph{Hilbert function} of a graded Artinian algebra $A=S/I$ is a vector of non-negative integers and we denote it by $h_A=(1,h_1,\dots ,h_d)$ where $h_A(i)=h_i=\dim_{\K}(A_i)$. The integer $d$ is called the \emph{socle degree} of $A$, that is the largest integer $i$ such that $h_A(i)>0$. A graded Artinian algebra $A$ is \emph{ Gorenstein} if $h_d=1$ and its Hilbert function is symmetric, i.e. $h_A(i)=h_A(d-i)$ for $0\leq i\leq d$.\par
A famous result of F. H. S. Macaulay \cite{Macaulay} provides a bound on the growth of Hilbert functions of graded Artinian algebras. F. H. S. Macaulay characterizes all vectors of non-negative integers that occur as Hilbert functions of standard graded algebras. Such a sequence is called an \emph{O-sequence}. \par
Let $R=\K[X_1,\dots , X_n ]$ be the Macualay dual ring of $S$. Given a homogeneous ideal $I\subset S$ the \emph{inverse system} of $I$ is defined to be a graded $S$-module $M\subset R$ such that $S$ acts on $R$ by differentiation. For more details of Macaulay's inverse system see \cite{Geramita} and \cite{IK}.
For graded Artinian Gorenstein algebras the inverse system is generated by only one form.
\begin{theorem}\cite{MW}\label{dualGen}
Let $A=S/I$ be a graded Artinian algebra. Then $A$ is Gorenstein if and only if there exists a polynomial $F\in R =\K[X_1,\dots ,X_n]$ such that $I=\ann_S(F)$.
\end{theorem}
From a result by F. H. S. Macaulay \cite{F.H.S} it is known that an Artinian standard graded $\mathsf{k}$-algebra $A=S/I$ is Gorenstein if and only if there exists $F\in R_d$, such that $I=\ann_S(F)$. T. Maeno and J. Watanabe \cite{MW} described higher Hessians of dual generator $F$ and provided a criterion for Artinian Gorenstein algebras having the SLP or WLP.
\begin{definition}\cite[Definition 3.1]{MW}
Let $F$ be a polynomial in $R$ and $A= S/\ann_S(F)$ be its associated Artinian Gorenstein algebra. Let $\mathcal{B}_{j} = \lbrace \alpha^{(j)}_i+\ann_S(F) \rbrace_i$ be a $\mathsf{k}$-basis of $A_j$. The entries of the $j$-th Hessian matrix of $F$ with respect to $\mathcal{B}_j$ are given by
$$
(\Hess^j(F))_{u,v}=(\alpha^{(j)}_u\alpha^{(j)}_v \circ F).
$$
We note that when $j=1$ the form $\Hess^1(F)$ coincides with the usual Hessian. Up to a non-zero constant multiple $\det \Hess^j(F)$ is independent of the basis $\mathcal{B}_j$. By abusing notation we will write $\mathcal{B}_{j} = \lbrace \alpha^{(j)}_i \rbrace_i$ for a basis of $A_j$.
\end{definition}
R. Gondim and G. Zappal\`a \cite{GZ} introduced a generalization of Hessians which provides the rank of multiplication maps by powers a linear form which are not necessarily symmetric.
\begin{definition}\cite[Definition 2.1]{GZ}
Let $F$ be a polynomial in $R$ and $A= S/\ann_S(F)$ be its associated Gorenstein algebra. Let $\mathcal{B}_{j} = \lbrace \alpha^{(j)}_i\rbrace_i$ and $\mathcal{B}_{k} = \lbrace \beta^{(k)}_i\rbrace_i$ be $\mathsf{k}$-bases of $A_j$ and $A_k$ respectively. The \emph{Hessian matrix of order $(j,k)$} of $F$ with respect to $\mathcal{B}_j$ and $\mathcal{B}_k$ is
$$
(\Hess^{(j,k)}(F))_{u,v}=(\alpha^{(j)}_u\beta^{(k)}_v \circ F).
$$
When $j=k$, $\Hess^{(j,j)}(F)=\Hess^{j}(F)$.
\end{definition}
\begin{definition}
Let $A= S/\ann(F)$ where $F\in R_d$. Pick bases $\B_j = \lbrace \alpha^{(j)}_u\rbrace_u$ and $\B_{d-j} = \lbrace \beta^{(d-j)}_u\rbrace_u$ be $\K$-bases of $A_j$ and $A_{d-j}$ respectively. The \emph{catalecticant matrix of $F$} with respect to $\B_j$ and $\B_{d-j}$ is
$$
\Cat^j_F=(\alpha^{(j)}_u\beta^{(d-j)}_v F)_{u,v=1}.
$$
\end{definition}
The rank of the $j$-th catalecticant matrix of $F$ is equal to the Hilbert function of $A$ in degree $j$, see \cite[Definition 1.11]{IK}.
We recall the definition of the Jordan degree type for a graded Artinian algebra and linear form.
\begin{definition}{\cite[Definition 2.28]{IMM}}\label{JDT}
Let $A$ be a graded Artinian algebra and $\ell\in A_1$. Suppose that $P_{\ell,A}=(p_1,\dots ,p_t)$ is the Jordan type for $\ell$ and $A$, then there exist elements $z_1, \dots z_t\in A$, which depend on $\ell$, such that $\{\ell^iz_k\mid 1\leq k\leq t, 0\leq i\leq p_k-1\}$ is a $\mathsf{k}$-basis for $A$. The Jordan blocks of the multiplication map by $ \ell$ is determined by the strings $\mathsf{s}_k=\{z_k, \ell z_k, \dots , \ell^{p_k-1}z_k\}$, and $A$ is the direct sum $A=\langle \mathsf{s}_1\rangle \oplus\dots \oplus \langle \mathsf{s}_t\rangle$. Denote by $d_k$ the degree of $z_k$. Then the \emph{Jordan degree type}, is defined to be the indexed partition $\mathcal{S}_{\ell,A}=({p_1}_{d_1}, \dots ,{p_t}_{d_t})$.
\end{definition}
\section{Rank matrices for Artinian Gorenstein algebras of linear forms}\label{section-rksec}
Throughout this section let $S=\K[x_1,\dots, x_n]$ be a polynomial ring with $n\geq 2$ variables equipped with standard grading over a filed $\K$ of characteristic zero. We let $A=S/\ann(F)$ be a graded Artinian Gorenstein algebra with dual generator $F\in R=\K[X_1,\dots , X_n ]$ that is a homogeneous polynomial of degree $d\geq 2$.
\begin{definition}\label{rkmatrix-def}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra with socle degree $d$. For linear form $\ell\in A$ define the \textit{rank matrix}, $M_{\ell,A}$, of $A$ and $\ell$ to be the upper triangular square matrix of size $d+1$ with the following $i,j$-th entry
$$(M_{\ell,A})_{i,j} = \rk\left(\times \ell^{j-i} : A_i\longrightarrow A_j\right),$$
for every $i\leq j$. For $i>j$ we set $ (M_{\ell,A})_{i,j}=0$.
\end{definition}
\begin{definition}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra with socle degree $d$ and linear form $\ell$. For each $0\leq i\leq d$ define the Artinian Gorenstein algebra, $A^{(i)}$, with the dual generator $\ell^i\circ F$
$$
A^{(i)} := S/\ann(\ell^i\circ F).
$$
\end{definition}
\noindent We note that when $i=0$ the algebra $A^{(0)}$ coincides with $A$. \par
\noindent
\begin{remark}\label{r_ij-remark}
By the definition of higher and mixed Hessians for every $0\leq i<j$ we have that
\begin{equation}
\rk \Hess^{(i,d-j)}_\ell (F) = (M_{\ell,A})_{i,j}.
\end{equation}
\end{remark}
For each $0\leq i\leq d$ denote the $i$-the diagonal vector of $M_{\ell,A}$ by $\diag(i,M_{\ell,A})$,
$$\diag(i,M_{\ell,A}):=((M_{\ell,A})_{0,i},(M_{\ell,A})_{1,i+1},\dots ,(M_{\ell,A})_{d-i,d}).$$
We show that for every $0\leq i\leq d$ the vector $\diag(i,M_{\ell,A})$ is the Hilbert function of some Artinian Gorenstein algebra.
We denote the Macaulay inverse system module of $A=S/\ann(F)$ by $\langle F\rangle$.
\begin{proposition}\label{diagprop}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra with socle degree $d\geq 2$ and $\ell$ be a linear form. Then
$$
\diag(i,M_{\ell,A}) = h_{A^{(i)}},
$$
for every $0\leq i\leq d$.
\end{proposition}
\begin{proof}
By the definition of rank matrix $M_{\ell,A}$ we have that the entries on the $i$-th diagonal of $M_{\ell,A}$ are exactly the ranks of multiplication map by $\ell^i$ on $A$ in various degrees.
Using Macaulay duality for every $0\leq j\leq \lfloor\frac{d-i}{2}\rfloor$ we get the following
\begin{align*}
rk\left(\times \ell^{i} : A_j\longrightarrow A_{i+j}\right)&=rk\left(\circ \ell^{i} : \langle F\rangle_{i+j}\longrightarrow \langle F\rangle_{j}\right)\\&=\dim_{\K} \langle \ell^i\circ F \rangle_j\\
&=\dim_{\K}(S/\ann(\ell^i\circ F))_{j}.
\end{align*
Note that the socle degree of $A^{(i)}$ is equal to $d-i$. The proof is complete since $h_{A^{(i)}}$ is symmetric about $\lfloor\frac{d-i}{2}\rfloor$.
\end{proof}
\begin{example}\label{firstEx}
Let $A=\K[x_1,x_2,x_3]/\ann(F)$ be Artinian Gorenstein algebra where $F=X_1^2X_2^2X_3^2$. We have that
$h_A=\left(1,3,6,7,6,3,1\right)$. Consider $\ell=x_1$, then
$$h_{A^{(1)}}=h_{S/\ann(x_1\circ F)}=\left(1,3,5,5,3,1\right),\quad
h_{A^{(2)}}=h_{S/\ann(x_1^2\circ F)}=\left(1,2,3,2,1\right),
$$
and $x_1^i\circ F=0$ for $i\geq 3$. Then the rank matrix is as follows
$$M_{x_1,A}= \begin{pmatrix}
1&1&1&0&0&0&0\\
0&3&3&2&0&0&0\\
0&0&6&5&3&0&0\\
0&0&0&7&5&2&0\\
0&0&0&0&6&3&1\\
0&0&0&0&0&3&1\\
0&0&0&0&0&0&1\\
\end{pmatrix}.
$$
By Remark \ref{r_ij-remark} we have that
\begin{align*}
&\rk \Hess_{x_1}^{(0,5)}=\rk \Hess_{x_1}^{(0,4)}=1, \rk \Hess_{x_1}^{(1,4)}=3,\rk \Hess_{x_1}^{(1,3)}=2,\\
& \rk \Hess_{x_1}^{(2,3)}=5,\hspace*{2mm}\text{and}\hspace*{2mm} \rk \Hess_{x_1}^{(2,2)}=3.
\end{align*}
\end{example}
Te following two lemmas provide conditions on every rank matrix $M_{\ell,A}$.
First we set a notation. For a vector $\mathbf{v}$ of positive integers of length $l$ denote by $\mathbf{v}_+$ the vector of length $l+1$ obtained by adding zero to vector $\mathbf{v}$, that is $\mathbf{v}_+ = (0,\mathbf{v})$.
\begin{lemma}\label{diffO-seq}
For every $0\leq i\leq d-1$, the difference vector $ h_{A^{(i)}}-(h_{A^{(i+1)}})_+$ is an O-sequence.
\end{lemma}
\begin{proof
Using Macaulay duality, for every $j\geq 1$ we have
\begin{align*}
& h_{A^{(i)}}(j)-h_{A^{(i+1)}}(j-1) = \dim_{\K}\langle \ell^i\circ F\rangle_j -\dim_{\K}\langle \ell^{i+1}\circ F\rangle_{j-1}=\dim_{\K}\left(\langle \ell^i\circ F\rangle/\langle \ell^{i+1}\circ F\rangle\right)_{j}.
\end{align*}
For $j=0$ we have that $\dim_{\K}\left(\langle \ell^i\circ F\rangle/\langle \ell^{i+1}\circ F\rangle\right)_{0}=1$ if ${A^{(i)}}\neq 0$. If ${A^{(i)}}=0$ then clearly ${A^{(i+1)}}= 0$ and so $ h_{A^{(i)}}-(h_{A^{(i+1)}})_+$ is the zero vector.\par
\noindent We conclude that $h_{A^{(i)}}-(h_{A^{(i+1)}})_+$ is the Hilbert function of $\left(\langle \ell^i\circ F\rangle/\langle \ell^{i+1}\circ F\rangle\right)$, and hence it is an O-sequence.
\end{proof}
\begin{lemma}\label{additiveRank}
For every $i,j\geq 1$, the following inequality holds
$$
h_{A^{(i-1)}}(j)+h_{A^{(i+1)}}(j-1)\geq h_{A^{(i)}}(j)+h_{A^{(i)}}(j-1).
$$
\end{lemma}
\begin{proof}
The inclusion map $\langle \ell^{i+1}\circ F\rangle\hookrightarrow \langle \ell^{i}\circ F\rangle$ for every $i\geq 0$ induces the following commutative diagram
$$
\xymatrix{
0\ar[r]& \langle \ell^{i+1}\circ F\rangle \ar[d]\ar[r]&\langle \ell^{i}\circ F\rangle \ar[d]\ar[r]& \langle \ell^{i}\circ F\rangle/ \langle \ell^{i+1}\circ F\rangle \ar[d]^{\varphi}\ar[r]&0\\
0\ar[r]& \langle \ell^i\circ F\rangle\ar[r]&\langle \ell^{i-1}\circ F\rangle\ar[r]& \langle \ell^{i-1}\circ F\rangle/ \langle \ell^i\circ F\rangle\ar[r]&0\\
}
$$
which shows that $\varphi$ is also injective.
Using Lemma \ref{diffO-seq} we get that $h_{A^{(i)}}(j)-h_{A^{(i+1)}}(j-1) = \dim_{\K}\left(\langle \ell^i\circ F\rangle/\langle \ell^{i+1}\circ F\rangle\right)_{j},$ for every $i,j\geq 1$ that implies the desired inequality.
\end{proof}
\begin{remark}
The above lemma shows that for every $i,j\geq 1$ the following inequality holds
$$
\rk \Hess_\ell^{(j,d-i-j+1)}+\rk \Hess_\ell^{(j-1,d-i-j)}\geq \rk \Hess_\ell^{(j,d-i-j)}+\rk \Hess_\ell^{(j-1,d-i-j+1)}.
$$
\end{remark}
As a consequence of the above lemmas, we provide necessary conditions for an upper triangular square matrix of size $d+1$ with non-negative integers to occur for an Artinian Gorenstein algebra $A$ and linear form $\ell\in A_1$.
\begin{corollary}\label{cor-rkmatrix}
Let $M$ be an upper triangular matrix of size $d+1$ with non-negative entries. Then $M$ is the rank matrix of some Artinian Gorenstein algebra $A$ and linear form $\ell\in A_1$, only if the following conditions are satisfied.
\begin{itemize}
\item[$(i)$] For every $0\leq i\leq d$, $\diag(i,M)$ is an O-sequences, and $h_A=\diag(0,M)$;
\item[$(ii)$] for every $0\leq i\leq d-1$, the difference vector $\diag(i,M)-\left(\diag(i+1,M)\right)_+$ is an O-sequences;
\item[$(iii)$] for any $2\times 2$ square submatrix of successive entries on and above the diagonal of $M$ of the form $\begin{pmatrix}
u&v\\
w&z\\
\end{pmatrix}$ we have that $w+v\geq u+z$.
\end{itemize}
\end{corollary}
\begin{proof}
It is an immediate consequence of Proposition \ref{diagprop} and Lemmas \ref{diffO-seq} and \ref{additiveRank}.
\end{proof}
\begin{example}
The following matrix does not occur as the rank matrix of some Artinian Gorenstein algebra and linear form $\ell$.
$$ M=\begin{pmatrix}
1&1&1&0&0&0\\
0&3&2&2&0&0\\
0&0&3&3&2&0\\
0&0&0&3&2&1\\
0&0&0&0&3&1\\
0&0&0&0&0&1\\
\end{pmatrix}.
$$
Since condition $(ii)$ in Corollary \ref{cor-rkmatrix} is not satisfied; in fact $\diag(0,M)-(\diag(1,M))_+=(1,3,3,3,3,1)-(0,1,2,3,2,1)=(1,2,1,0,1,0)$ is not an O-sequence.\par
Corollary \ref{cor-rkmatrix} also implies that the following matrix is not a possible rank matrix for some $A$ and $\ell$.
$$ N=\begin{pmatrix}
1&1&1&0&0&0\\
0&3&3&1&0&0\\
0&0&5&4&1&0\\
0&0&0&5&3&1\\
0&0&0&0&3&1\\
0&0&0&0&0&1\\
\end{pmatrix}.
$$
In fact, for submatrix $\begin{pmatrix} 3&1\\5&4\end{pmatrix}$ the condition $(iii)$ is not satisfied.
\end{example}
\begin{definition}[Jordan degree type matrix] Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra and $\ell\in A$ a linear form. Assume that $M_{\ell,A}$ is the rank matrix of $A$ and $\ell$. We define
the \emph{Jordan degree type matrix}, $J_{\ell,A}$, of $A$ and $\ell$ to be the upper triangular matrix with the following non-negative entries
\begin{align}\label{J(A,l)definition}
(J_{\ell,A})_{i,j} :
=& (M_{\ell,A})_{i,j}+(M_{\ell,A})_{i-1,j+1}-(M_{\ell,A})_{i-1,j}-(M_{\ell,A})_{i,j+1},
\end{align}
where we set $(M_{\ell,A})_{i,j}$ if either $i< 0$ or $j< 0$.
\begin{equation}\label{JDT_ij}
(J_{\ell,A})_{i,j} = h_{A^{(k)}}(i)+h_{A^{(k+2)}}(i-1)-h_{A^{(k+1)}}(i-1)-h_{A^{(k+1)}}(i),
\end{equation}
such that for every $k$, $h_{A^{(k)}}(-1):=0$.
\end{definition}
Recall from Lemma \ref{additiveRank} that for each $0\leq i\leq j$, $(J_{\ell,A})_{ij}$ is non-negative.
\begin{proposition}\label{rkmatrix-1-1-JDT-prop}
There is a 1-1 correspondence between the two matrices $M_{\ell,A}$ and $J_{\ell,A}$ associated to a pair $(A,\ell)$.
\end{proposition}
\begin{proof}
We use Equation (\ref{J(A,l)definition}) to provide an algorithm to obtain $J_{\ell,A}$ from $M_{\ell,A}$. For each $0\leq i\leq j$ define matrix $J^\prime_{\ell,A}$ as the following
\begin{equation}\label{J'matrixdef}
(J^\prime_{\ell,A})_{i,j} := (M_{\ell,A})_{i,j}-(M_{\ell,A})_{i,j+1},
\end{equation}
where we set $(M_{\ell,A})_{i,j}=0$ if either $i< 0$ or $j< 0$.
Then define the upper triangular matrix $J_{\ell,A}$ where its entry $i,j$ for every $0\leq i\leq j$ is equal to
\begin{equation}\label{JfromJ'def}
(J_{\ell,A})_{i,j} = (J^\prime_{\ell,A})_{i,j}-(J^\prime_{\ell,A})_{i-1,j},
\end{equation}
where we set $(J^\prime_{\ell,A})_{i,j}=0$ if either $i< 0$ or $j< 0$.
We obtain $M_{\ell,A}$ from $J^\prime_{\ell,A}$ in two steps.
First we get the matrix $J^\prime_{\ell,A}$ from $J_{\ell,A}$. For each $0\leq i\leq j$, we have the following
\begin{equation}
(J^\prime_{\ell,A})_{i,j} = (J_{\ell,A})_{i,j}+(J_{\ell,A})_{i-1,j},
\end{equation}
where we set $(J_{\ell,A})_{i,j}=0$ if either $i< 0$ or $j< 0$.
Then for each $0\leq i\leq j$,
\begin{equation}
(M_{\ell,A})_{i,j}=(J^\prime_{\ell,A})_{i,j}+(J^\prime_{\ell,A})_{i,j+1},
\end{equation}
where we set $(J^\prime_{\ell,A})_{i,j}=0$ if either $i< 0$ or $j< 0$.
\end{proof}
\begin{example}
We illustrate the procedure provided in Proposition \ref{rkmatrix-1-1-JDT-prop} for the Artinian Gorenstein algebra given in Example \ref{firstEx} with the rank matrix $M_{\ell,A}$. Using Equations (\ref{J'matrixdef}) and (\ref{JfromJ'def}) we get the following matrices.
$$M_{\ell,A}= \begin{pmatrix}
1&1&1&0&0&0&0\\
0&3&3&2&0&0&0\\
0&0&6&5&3&0&0\\
0&0&0&7&5&2&0\\
0&0&0&0&6&3&1\\
0&0&0&0&0&3&1\\
0&0&0&0&0&0&1\\
\end{pmatrix},
\hspace*{0mm} J^\prime_{\ell,A} = \begin{pmatrix}
0&0&1&0&0&0&0\\
0&0&1&2&0&0&0\\
0&0&1&2&3&0&0\\
0&0&0&2&3&2&0\\
0&0&0&0&3&2&1\\
0&0&0&0&0&2&1\\
0&0&0&0&0&0&1\\
\end{pmatrix}, \hspace*{0mm} J_{\ell,A} = \begin{pmatrix}
0&0&1&0&0&0&0\\
0&0&0&2&0&0&0\\
0&0&0&0&3&0&0\\
0&0&0&0&0&2&0\\
0&0&0&0&0&0&1\\
0&0&0&0&0&0&0\\
0&0&0&0&0&0&0\\
\end{pmatrix}. $$
\end{example}
Define the decreasing sequence $\mathbf{d}:=(\dim_{\mathsf{k}}A^{(0)}, \dim_{\mathsf{k}}A^{(1)}, \dots , \dim_{\mathsf{k}}A^{(d)})$, and recall that the second difference sequence of $\mathbf{d}$ is denoted by $\Delta^2 \mathbf{d}$ and its $i$-th entry is given by $$\Delta^2 \mathbf{d}(i)=\dim_{\mathsf{k}}A^{(i)}+\dim_{\mathsf{k}}A^{(i+2)}-2\dim_{\mathsf{k}}A^{(i+1)},$$
where we set $\dim_{\mathsf{k}}A^{(i)}=0$ for $i>d$.
\begin{proposition}\label{JT}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra with socle degree $d\geq 2$ and let $\ell\in A$ be a linear form. Then the Jordan type partition of $\ell$ for $A$ is given by
$$
P_{\ell,A} = \big(\underbrace{d+1,\dots ,d+1}_{n_{d}} ,\underbrace{d,\dots ,d}_{n_{d-1}} ,\dots ,\underbrace{2,\dots ,2}_{n_1},\underbrace{1,\dots ,1}_{n_0}\big),
$$
such that $\mathbf{n}=(n_0,n_{1}, \dots , n_d)=\Delta^2 \mathbf{d}.$
\end{proposition}
\begin{proof}
The Jordan type partition of $\ell$ for $A$ is equal to the dual partition of the following partition
\begin{equation}\label{jordantype}
\Big(\rk (\times \ell^0)-\rk (\times \ell^1),\rk (\times \ell^1)-\rk (\times \ell^2),\dots , \rk (\times \ell^{d-1})-\rk (\times \ell^d),\rk (\times \ell^d)\Big).
\end{equation}
Since for each $0\leq i\leq d$ the rank of the multiplication map $ \times \ell^{i}:A_j\longrightarrow A_{j+i}$ is equal to the rank of differentiation map $\circ \ell^i :\langle F \rangle_{i+j}\longrightarrow\langle F\rangle_{j}$, where $\langle F\rangle$ is the dual algebra to $A$. Thus the rank of $ \times \ell^i:A_j\longrightarrow A_{j+i}$ is equal to $\dim_{\K}\left( S/\ann(\ell^i\circ F)\right)_j$ and therefore we have
$$\rk \left( \times \ell^i: A\longrightarrow A\right) = \sum^{d-i}_{j=0} \dim_{\K}\left( S/\ann(\ell^i\circ F)\right)_j=\dim_{\mathrm{k}}A^{(i)}.$$
So (\ref{jordantype}) is equal to the following partition
$$
\big(\dim_{\mathrm{k}}A^{(0)}-\dim_{\mathrm{k}}A^{(1)},\dim_{\mathrm{k}}A^{(1)}-\dim_{\mathrm{k}}A^{(2)},\dots ,\dim_{\mathrm{k}}A^{({d-1})}-\dim_{\mathrm{k}}A^{(d)},\dim_{\mathrm{k}}A^{(d)}\big).
$$
The dual partition to the above partition is the Jordan type partition of $A$ and $\ell$ as we claimed.
\end{proof}
\begin{example}
Consider the Artinian Gorenstein algebra given in Example \ref{firstEx} and linear form $\ell=x_1$. The Jordan degree type matrix of $A$ and $\ell$ is equal to the following matrix
$$J_{\ell,A} = \begin{pmatrix}
0&0&1&0&0&0&0\\
0&0&0&2&0&0&0\\
0&0&0&0&3&0&0\\
0&0&0&0&0&2&0\\
0&0&0&0&0&0&1\\
0&0&0&0&0&0&0\\
0&0&0&0&0&0&0\\
\end{pmatrix}.
$$
We have that $P_{\ell,A} = (\underbrace{3, \dots ,3}_9)$. In order to obtain the Jordan degree type $\mathcal{S}_{\ell,A}$ we recall the Definition \ref{JDT} and note that the degree of each part in $P_{\ell,A}$ is equal to the row index of the corresponding entry in $J_{\ell,A}$, so $\mathcal{S}_{\ell,A}=(3_0,3_1,3_1,3_2,3_2,3_2,3_3,3_3,3_4)$.
\end{example}
\begin{remark}
Equation (\ref{JDT_ij}) may be expressed in terms of the mixed Hessians.
\begin{small}
\begin{equation}
(J_{\ell,A})_{i,j} = \rk\Hess_\ell^{(i,d-i-k)}(F)+\rk\Hess_\ell^{(i-1,d-i-k-1)}(F)-\rk\Hess_\ell^{(i-1,d-i-k)}(F)-\rk\Hess_\ell^{(i,d-i-k-1)}(F).
\end{equation}
\end{small}
This recovers a result by R. Gondim and B. Costa \cite[Theorem 4.7]{CG} determining Jordan types of Artinian Gorenstein algebras and linear forms using the ranks of mixed Hessians.
\end{remark}
\section{Jordan types of Artinian Gorenstein algebras of codimension three}\label{codim3section}
In this section we consider graded Artinian Gorenstein quotients of $S=\K[x,y,z]$ where $\mathrm{char}(\K)=0$. For an Artinian Gorenstein algebra $A=S/\ann(F)$ with dual generator $F\in R=\mathsf{k}[X,Y,Z]$ of degree $d\geq 2$ and a linear form $\ell$ we explain how we find the rank matrix $M_{\ell,A}$, and as a consequence the Jordan type $P_{\ell,A}$.
Let $L_1, L_2,L_3$ be linear forms in the dual ring $R=\mathsf{k}[X,Y,Z]$ such that $\ell\circ L_1\neq 0$ and $\ell\circ L_2=\ell\circ L_3 = 0$.
By linear change of coordinates we may assume that $L_1=X$, $L_2=Y$ and $L_3=Z$. Then $F$ can be written in the following form
$$
F = \sum_{i=0}^dX^iG_{d-i},
$$
where for each $0\leq i\leq d$, $G_{d-i}$ is a homogeneous polynomial of degree $d-i$ in the variables $Y$ and $Z$. In general $G_{d-i}$ could be a zero polynomial for some $i$.
\subsection{Jordan types with parts of length at most four}\label{length3}
We will provide the list of all possible rank matrices $M_{\ell,A}$ such that $A$ is an Artinian Gorenstein algebra and $\ell$ is a linear form in $A$ where $\ell^3=0$.
Assuming $\ell^3=0$ implies that $M_{\ell,A}$ has at most three non-zero diagonals. Consequently, we provide a formula to compute the Jordan type partitions for Artinian Gorenstein algebras and linear forms $\ell$ such that $\ell^4=0$, which are Jordan types with parts of length at most four.
Consider Artinian Gorenstein algebra $A=S/\ann(F)$ with socle degree $d\geq 2$ and linear form $\ell$ such that $\ell^3=0$. Without loss of generality we assume that $\ell=x$ and that $F$ is in the following form
\begin{equation}\label{F}
F = X^2G_{d-2}+XG_{d-1}+G_d,
\end{equation}
where
$$G_{d}=\sum_{j=0}^{d}\frac{a_{j}}{j!(d-j)!}{Y^{d-j}Z^{j}}, \quad G_{d-1}=\sum_{j=0}^{d-1}\frac{b_{j}}{j!(d-j-1)!}{Y^{d-j-1}Z^{j}},$$
and
$$ G_{d-2}=\frac{1}{2}\sum_{j=0}^{d-2}\frac{c_{j}}{j!(d-j-2)!}{Y^{d-j-2}Z^{j}}.$$
In order to make the computations simpler, we choose the coefficients of the terms in $F$ in a way that the entries of the catalecticant matrices of $F$ are either zero or one.
We first consider the case when $\ell^3=0$ but $\ell^2\neq 0$. Therefore, we assume that $G_{d-2}\neq 0$ since otherwise we get $\ell^2=0$. Recall that $A^{(0)}=A$, $A^{(1)}=S/\ann(\ell\circ F)$, $A^{(2)}=S/\ann(\ell^2\circ F)$ and $A^{(i)}=S/\ann(\ell^i\circ F)=0$, for every $i\geq 3$.
We determine all rank matrices that occur for such algebras and linear forms $\ell$ where $\ell^3=0$. Equivalently, we determine all possible Hilbert functions for $A, A^{(1)}$ and $A^{(2)}$. The rank matrices are slightly different for even and odd socle degrees, as excepted, thus we treat these cases separately. We first prove our result for Artinian Gorenstein algebras with even socle degree $d\geq 2$. Later, in similar cases for odd socle degrees we refer to the relevant proof given for even socle degrees.
We will show in the theorems bellow that the rank matrix, $M_{\ell,A}$, for $\ell^3=0$ and $A$ is determined by three of its entries. These entries are exactly the maximum values in non-zero diagonals of $M_{\ell,A}$, that are maximum values of $h_A$, $h_{A^{(1)}}$ and $h_{A^{(2)}}$. The maximum value of the Hilbert function of an Artinian Gorenstein algebra is obtained in the middle degree. We denote by $r,s$ and $t$ the maximum value for the Hilbert function of $h_{A^{(2)}}, h_{A^{(1)}}$ and $h_A$ respectively. We first provide all possible triples $(r,s,t)$.
\begin{lemma}[Even socle degree]\label{maxvaluesevenLemma}
There exists an Artinian Gorenstein algebra $A$ with even socle degree $d\geq 2$ and linear form $\ell\in A_1$ where $\ell^2\neq 0$ but $\ell^3=0$, such that $$(r,s,t)=(h_{A^{(2)}}(\frac{d}{2}-1) ,h_{A^{(1)}}(\frac{d}{2}-1),h_A(\frac{d}{2}))$$ if and only if
\begin{itemize}
\item [$(1)$] $r\in [1,\frac{d}{2}-1]$, $s\in[2r,\frac{d}{2}+r]$ and $t\in[2s-r,\frac{d}{2}+s+1]$, for $d\geq 4$; or
\item [$(2)$] $r=\frac{d}{2}$, $s=d-1$ and $t\in[\frac{3d}{2}-2,\frac{3d}{2}]$, for $d\geq 2$.
\end{itemize}
\end{lemma}
\begin{proof}
We prove the statement by analysing the catalecticant matrices in the desired degrees. In each case we first determine all possible ranks for each catalecticant matrix and then for each possible value we provide polynomials $G_{d-2}, G_{d-1}$ and $G_d$ as in (\ref{F}) giving the certain ranks. \par
The maximum value of the Hilbert function of $A$ occurs in degree $\frac{d}{2}$ and it is equal to $\rk\Cat_F(\frac{d}{2})$. Pick the following monomial basis for $A_{\frac{d}{2}}$
$$
\B_{\frac{d}{2}} = \{x^{\frac{d}{2}},x^{\frac{d}{2}-1}y,x^{\frac{d}{2}-1}z,x^{\frac{d}{2}-2}y^2,x^{\frac{d}{2}-2}yz,x^{\frac{d}{2}-2}z^2,\dots , y^{\frac{d}{2}},y^{\frac{d}{2}-1}z,\dots ,z^{\frac{d}{2}}\}.
$$
Then the catalecticant matrix of $F$ with respect to $\B_{\frac{d}{2}}$ is equal to
\begin{equation}
\Cat_F(\frac{d}{2})=\left[\begin{array}{@{}c|c|c@{}}
\mathbf{0}& \mathbf{0}&{\Cat_{G_{d-2}}{(\frac{d}{2}-2)}}
\\\hline
\mathbf{0}&\Cat_{G_{d-2}}{(\frac{d}{2}-1)}&\Cat_{G_{d-1}}{(\frac{d}{2}-1)}\\\hline
\Cat_{G_{d-2}}{(\frac{d}{2})}&\Cat_{G_{d-1}}{(\frac{d}{2})}&\Cat_{G_{d}}{(\frac{d}{2})}
\end{array}
\right].\\
\end{equation}
Which is equal to
\begin{equation}\label{catmatrix3lines}
\Cat_F(\frac{d}{2}) = \left[\begin{array}{@{}cccc|cccc|cccc@{}}
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{\frac{d}{2}} \\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{1} & c_{2} &\cdots & c_{\frac{d}{2}+1} \\
\vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{\frac{d}{2}-2} & c_{\frac{d}{2}-1} &\cdots & c_{d-2} \\\hline
0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{\frac{d}{2}-1} & b_{0} & b_{1}&\cdots & b_{\frac{d}{2}} \\
0 & 0 & \cdots & 0 & c_{1} & c_{2} &\cdots & c_{\frac{d}{2}} & b_{1} & b_{2} &\cdots & b_{\frac{d}{2}+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & c_{\frac{d}{2}-1} & c_{\frac{d}{2}} &\cdots & c_{d-2} & b_{\frac{d}{2}-1} & b_{\frac{d}{2}} &\cdots & b_{d-1} \\\hline
c_{0} & c_{1}&\cdots & c_{\frac{d}{2}-2} & b_{0} & b_{1}&\cdots & b_{\frac{d}{2}-1} &a_{0} & a_{1}&\cdots & a_{\frac{d}{2}} \\
c_{1} & c_{2} &\cdots & c_{\frac{d}{2}-1} & b_{1} & b_{2} &\cdots & b_{\frac{d}{2}}& a_{1} & a_{2} &\cdots & a_{\frac{d}{2}+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
c_{\frac{d}{2}} & c_{\frac{d}{2}+1} &\cdots & c_{d-2} & b_{\frac{d}{2}} & b_{\frac{d}{2}+1} &\cdots & b_{d-1}& a_{\frac{d}{2}} & a_{\frac{d}{2}+1} &\cdots & a_{d} \\
\end{array}\right].
\end{equation}
Since any Artinian algebra of codimension two has the SLP the rank of the $j$-th Hessian matrices of polynomials $G_{d-2},G_{d-1}$ and $G_d$ are equal to the ranks of their $j$-th catalecticant matrices. By linear change of coordinates, we may assume that $z$ is the strong Lefschetz element for Artinian Gorenstein algebra $\mathsf{k}[y,z]/\ann(G_{d-2})$. This implies that the lower right square submatrices of the catalecticant matrices of $G_{d-2}$ in all degrees have maximal rank. Likewise, we may assume that $y$ is the strong Lefschetz element for the Artinian Gorenstein algebra $\mathsf{k}[y,z]/\ann(G_{d-1})$ which means that the upper left square submatrices of the catalecticant matrices of $G_{d-1}$ in different degrees are all full rank. \par
Observe that $r=h_{A^{(2)}}(\frac{d}{2}-1)\in [1,\frac{d}{2}]$. To show $(1)$ we assume $r=h_{A^{(2)}}(\frac{d}{2}-1)\in [1,\frac{d}{2}-1]$ which implies that $
h_{A^{(2)}}(\frac{d}{2}-2)= h_{A^{(2)}}(\frac{d}{2}-1)= h_{A^{(2)}}(\frac{d}{2})=r.
$ We assume that the ranks of the lower right submatrices of $\Cat_{G_{d-2}}(\frac{d}{2}-2), \Cat_{G_{d-2}}(\frac{d}{2}-1)$ and $\Cat_{G_{d-2}}(\frac{d}{2})$ are equal to $r$ and setting $c_{d-r-1}=1$ and $c_{i}=0$ for every $i\neq d-r-1 $ provides the desired property. So
\begin{equation}\label{G_(d-2)even(1)}
G_{d-2}= \frac{Y^{r-1}Z^{d-r-1}}{(r-1)!(d-r-1)!}, \quad \text{for all}\quad r\in[1,\frac{d}{2}-1].
\end{equation}
Now in order to obtain possible values for $s=h_{A^{(1)}}(\frac{d}{2}-1)$, we notice that $s\in [2r, 2r+\rk \mathbf{B}]$ where $\mathbf{B}$ is the following matrix
$$
\mathbf{B}=\left(\begin{array}{@{}ccccccc@{}}
b_{0} &\cdots & b_{\frac{d}{2}-r} \\
\vdots & \reflectbox{$\ddots$} &\vdots\\
b_{\frac{d}{2}-1-r}&\cdots & b_{d-1-2r} \end{array}\right).
$$
Since the socle degree of $A^{(1)}$ is equal to $d-1$ that is an odd integer, we get that $ h_{A^{(1)}}(\frac{d}{2}-1)= h_{A^{(1)}}(\frac{d}{2}) = s$. For every $s\in [2r, 2r+\rk\mathbf{B}]$, we have $\rk\mathbf{B}=s-2r$. We may assume that the upper left submatrix of $\mathbf{B}$ has rank $s-2r$. Setting $G_{d-1}=0$ provides that $\rk\mathbf{B}=s-2r=0$. And setting $b_{s-2r-1}=1$ and $b_i=0$ for every $i\neq s-2r-1$ implies that $\rk\mathbf{B}=s-2r\neq 0$. Equivalently, we set
\begin{equation}\label{G_(d-1)even(1)}
G_{d-1}=\left\{
\begin{array}{ll}
0 & \text{if $s-2r=0$},\\
\frac{Y^{d-s+2r}Z^{s-2r-1}}{(d-s+2r)!(s-2r-1)!} & \text{if $1\leq s-2r\leq \frac{d}{2}-r.$}\\
\end{array}
\right.
\end{equation}
This implies that, there exists $A$ such that $h_{A^{(1)}}(\frac{d}{2}-1)=s $ if and only if $s\in [2r,\frac{d}{2}+r].$\par
To obtain possible values for $t=h_A(\frac{d}{2})$, first notice that $t\in [2s-r, 2s-r+\rk\mathbf{A}]$, for
$$
\mathbf{A}=\left(\begin{array}{@{}ccccccc@{}}
a_{2s-4r} &\cdots & a_{\frac{d}{2}-3r+s} \\
\vdots & \reflectbox{$\ddots$} &\vdots\\
a_{\frac{d}{2}-3r+s}&\cdots & a_{d-2r} \end{array}\right).
$$
For every $t\in [2s-r, 2s-r+\rk\mathbf{A}]$, we have that $\rk\mathbf{A}=t-2s+r$. We may assume that the rank of the upper left submatrix of $\mathbf{A}$ is equal to $t-2s+r$.
For $G_d=0$ we get $\rk\mathbf{A}=t-2s+r=0$. Setting $a_{t-3r-1}=1$ and $a_i=0$ for every $i\neq t-3r-1$ provides that $\rk\mathbf{A}=t-2s+r\neq 0$. In other words, we choose $G_d$ as the following
\begin{equation}\label{G_(d)even(1)}
G_{d}=\left\{
\begin{array}{ll}
0 & \text{if $t-2s+r=0$},\\
\frac{Y^{d-t+3r+1}Z^{t-3r-1}}{(d-t+3r+1)!(t-3r-1)!} & \text{if $1\leq t-2s+r\leq \frac{d}{2}+r-s+1.$}\\
\end{array}
\right.
\end{equation}
So there exists $A$ such that $t=h_A(\frac{d}{2})$ if and only if
$t\in [2s-r,\frac{d}{2}+s+1]$.
To prove $(2)$ assume $h_{A^{(2)}}(\frac{d}{2}-1)=\frac{d}{2}$. This implies that the Hilbert function of $h_{A^{(2)}}$ has the maximum possible value up to degree $\frac{d}{2}-1$ and since the socle degree of $A^{(2)}$ is even and is equal to $d-2$ we have $$h_{A^{(2)}}(\frac{d}{2}-2)=h_{A^{(2)}}(\frac{d}{2})=\frac{d}{2}-1.$$ So setting $c_{\frac{d}{2}-1}=1$ and $c_i=0$ for every $i\neq \frac{d}{2}-1$, or equivalently,
\begin{equation}\label{Gd-2maxeven}
G_{d-2} = \frac{Y^{\frac{d}{2}-1}Z^{\frac{d}{2}-1}}{(\frac{d}{2}-1)!(\frac{d}{2}-1)!}
\end{equation}
provides the desired ranks for the catalecticant matrices $\Cat_{G_{d-2}}(\frac{d}{2}-2)$, $\Cat_{G_{d-2}}(\frac{d}{2}-1)$ and $\Cat_{G_{d-2}}(\frac{d}{2})$. \\
We have that $h_{{A^{(1)}}}(\frac{d}{2}-1)=\rk\Cat_{x\circ F}(\frac{d}{2}-1)$, and
\begin{equation}
\Cat_{x\circ F}(\frac{d}{2}-1) = \left[\begin{array}{@{}c|c@{}}
\mathbf{0}&{\Cat_{G_{d-2}}{(\frac{d}{2}-2)}}
\\\hline
\Cat_{G_{d-2}}{(\frac{d}{2}-1)}&\Cat_{G_{d-1}}{(\frac{d}{2}-1)}\\
\end{array}
\right].\\
\end{equation}
Since $\rk {\Cat_{G_{d-2}}{(\frac{d}{2}-2)}}=\frac{d}{2}-1$ and $\rk\Cat_{G_{d-2}}{(\frac{d}{2}-1)}=\frac{d}{2}$, the rank of the above matrix is maximum possible and is equal to $d-1$. This means that for every choice of polynomial $G_{d-1}$ in this case we have
$$
h_{A^{(1)}}(\frac{d}{2}-1)=d-1.
$$
In order to find possbile values for $h_A(\frac{d}{2})$, note that the rank of $\Cat_F(\frac{d}{2})$ is at most equal to $\frac{3d}{2}$. Also
$$\frac{3d}{2}-2=\rk\Cat_{G_{d-2}}(\frac{d}{2}-2)+\rk\Cat_{G_{d-2}}(\frac{d}{2}-1)+\rk\Cat_{G_{d-2}}(\frac{d}{2})\leq \rk\Cat_F(\frac{d}{2})\leq \frac{3d}{2}.$$
Note that setting $G_{d-2}$ as (\ref{Gd-2maxeven}), $G_{d-1}=0$ and $G_d$ equal to the following
\begin{equation}\label{Gdmaxeven}
G_{d}=\left\{
\begin{array}{ll}
0 & \text{for $t=\frac{3d}{2}-2$},\\
\frac{Y^{d}}{(d)!} & \text{for $t=\frac{3d}{2}-1,$}\\
\frac{Y^{d}}{(d)!}+ \frac{Z^{d}}{(d)!}& \text{for $t=\frac{3d}{2}$}.\\
\end{array}
\right.
\end{equation}
provides the desired ranks for the catalecticant matrix $\Cat_F(\frac{d}{2})$ in (\ref{catmatrix3lines}).
\end{proof}
We now prove that the rank matrix of $A$, or equivalently, Hilbert functions of $A$, $A^{(1)}$ and $A^{(2)}$ are completely determined by the maximum values of $h_{A^{(2)}}$, $h_{A^{(1)}}$ and $h_{A}$. We then provide all rank matrices for each possible combination of integers $(r,s,t)$ listed in Lemma \ref{maxvaluesevenLemma}.
\begin{theorem}[Even socle degree]\label{3linesHFtheorem-even}
Let $A$ be an Artinian Gorenstein algebra with even socle degree $d\geq 2$ and $\ell\in A_1$ such that $\ell^2\neq 0$ and $\ell^3=0$. Then Hilbert functions of $A$, $A^{(1)}$ and $A^{(2)}$ are completely determined by $(r,s,t)=(h_{A^{(2)}}(\frac{d}{2}-1) ,h_{A^{(1)}}(\frac{d}{2}-1),h_A(\frac{d}{2}))$. More precisely,
\begin{itemize}
\item[$(1)$] if $d\geq 4$, $r\in [1,\frac{d}{2}-1]$, $s\in[2r,\frac{d}{2}+r]$ and $t\in [2s-r,\frac{d}{2}+s+1]$, then
\begin{equation}\label{HFeven(1)}
h_{A^{(2)}}(i)=\left\{
\begin{array}{ll}
i+1 & 0\leq i\leq r-1,\\
r & r\leq i\leq \frac{d}{2}-1,\\
\end{array}
\right.\quad
h_{A^{(1)}}(i)=\left\{
\begin{array}{ll}
2i+1 & 0\leq i\leq r-1,\\
i+r+1 & r\leq i\leq s-r-1,\\
s & s-r\leq i\leq \frac{d}{2}-1.
\end{array}
\right.
\end{equation}
\begin{itemize}
\item If $t=3r$ then there are two possible Hilbert functions for $A$
\begin{equation}\label{HFevenA(1,1)}
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r-1,\\
3r & r\leq i\leq\frac{d}{2},\\
\end{array}
\right.
\text{and} \quad
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r-1,\\
3r-1& i=r,\\
3r & r+1\leq i\leq \frac{d}{2},\\
\end{array}
\right.
\end{equation}
\item otherwise, i.e., $t>3r$ we have
\begin{equation}\label{HFevenA(1)o.w.}
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r,\\
2i+r+1 & r+1\leq i\leq s-r-1,\\
2i+r+1& i=s-r, \hspace*{2mm}\text{if}\hspace*{2mm}t>2s-r\hspace*{2mm}\text{and}\hspace*{2mm}s>2r,\\
2i+r& i=s-r, \hspace*{2mm}\text{if}\hspace*{2mm}t>2s-r \hspace*{2mm}\text{and}\hspace*{2mm}s=2r,\\
i+s+1 &s-r+1\leq i\leq t-s-1,\\
t & t-s \leq i\leq \frac{d}{2}.
\end{array}
\right.
\end{equation}
\end{itemize}
\item[$(2)$]
If $d\geq 2$, $r=\frac{d}{2}$, $s=d-1$ and $t\in[\frac{3d}{2}-2,\frac{3d}{2}]$,
then for every $0\leq i\leq \frac{d}{2}-1$
\begin{align}\label{HFeven(2)}
h_{A^{(2)}}(i)=i+1,\quad h_{A^{(1)}}(i)=2i+1, \hspace*{2mm}\text{and}
\end{align}
\begin{align}\label{HFevenA(2.1)}
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq \frac{d}{2}-1,\\
t & i=\frac{d}{2}.\\
\end{array}
\right.
\end{align}
\end{itemize}
\end{theorem}
\begin{proof}
We first show $(1)$. Since the Hilbert function of Artinian Gorenstein algebras are symmetric it is enough to determine it up to the middle degree.
We have that
$$
A^{(2)} = S/\ann(\ell^2\circ F) = S/\ann(G_{d-2}).
$$
So $A^{(2)}$ is an Artinian Gorenstein algebra with codimension at most two and the maximum value of $h_{A^{(2)}}$ is equal to $r$. The Hilbert function of $A^{(2)}$ increases by exactly one until it reaches $r$ and it stays $r$ up to the middle degree, $\frac{d}{2}-1$. So we get $h_{A^{(2)}}$ as we claimed.\par
The assumption on $r$ implies that $h_{A^{(2)}}(\frac{d}{2}-2)=h_{A^{(2)}}(\frac{d}{2}-1)=r$. So
$$
(h_{A^{(1)}}-(h_{A^{(2)}})_+)(\frac{d}{2}-1) = h_{A^{(1)}}(\frac{d}{2}-1)-h_{A^{(2)}}(\frac{d}{2}-2) = s-r.
$$
Since $(h_{A^{(1)}}-(h_{A^{(2)}})_+)(1)\leq 2$, Lemma \ref{diffO-seq} implies that for every $0\leq i\leq s-r-1,$
$$(h_{A^{(1)}}-(h_{A^{(2)}})_+)(i)=i+1.$$
So since $0\leq r-1\leq s-r-1$, for every $0\leq i\leq r-1$ we have that
$$
h_{A^{(1)}}(i) = i+1+h_{A^{(2)}}(i-1)= i+1+i = 2i+1.
$$
If $r-1< s-r-1$, then for $r\leq i\leq s-r-1$ we have
$$
h_{A^{(1)}}(i) = i+1+h_{A^{(2)}}(i-1)= i+1+r.
$$
We have that $r\leq s-r$, so $h_{A^{(2)}}(i)=r$ for every $s-r-1\leq i\leq \frac{d}{2}-1$ which implies that $h_{A^{(1)}}(i) = s$, for every $s-r\leq i\leq \frac{d}{2}-1$.
We now determine the Hilbert function of $A$. By assumption we have $(h_A-(h_{A^{(1)}})_+)({\frac{d}{2}})= h_{A}(\frac{d}{2})-h_{A^{(1)}}(\frac{d}{2}-1)=t-s$. On the other hand, $(h_{A}-(h_{A^{(1)}})_+)(1)\leq 2$ and by Lemma \ref{diffO-seq} we conclude that $h_{A}-(h_{A^{(1)}})_+$ is the Hilbert function of some algebra with codimension at most two. So for every $0\leq i\leq t-s-1$
$$
(h_A-(h_{A^{(1)}})_+)({i}) = i+1.
$$
By assumption we have $0\leq r-1\leq s-r-1\leq t-s-1$, so for every $1\leq i\leq r-1$
$$
h_A(i) =i+1+h_{A^{(1)}}(i-1)= i+1+2(i-1)+1=3i.
$$
\begin{itemize}
\item Suppose that $r=t-s$, then $s=2r$ and $t=3r$. Since we have $r\leq \frac{d}{2}-1$ and the Hilbert function of an algebra with codimension two is unimodal, we get
$$
\left(h_A-(h_{A^{(1)}})_+\right)({r})\geq \left(h_A-(h_{A^{(1)}})_+\right)({r-1})
$$ and thus
$$h_A(r)\geq r+h_{A^{(1)}}(r-1) = r+2(r-1)+1=3r-1.
$$
Thus we have two possible values for $h_A(r)$, that is either equal to $3r-1$ or $3r$. Clearly, $h_A(i)=3r$ for every $r+1\leq i\leq \frac{d}{2}$.
\item Now suppose that $r<t-s$. Then $$h_A(r) = r+1+h_{A^{(1)}}(r-1)=r+1+2(r-1)+1=3r.$$
If $r<s-r-1$, then for every $r+1\leq i\leq s-r-1$ we get
$$
h_A(i)=i+1+h_{A^{(1)}}(i-1) = i+1+r+i=2i+r+1.
$$
If $s-r-1<t-s-1$, then
\begin{equation*}
h_A(s-r) =s-r+1+h_{A^{(1)}}(s-r-1) = \left\{
\begin{array}{ll}
s-r+1+s & \text{if}\hspace*{2mm} s>2r,\\
s-r+1+s-1 & \text{if}\hspace*{2mm} s=2r.\\
\end{array}
\right.
\end{equation*}
If $s-r<t-s-1$, then for every $s-r+1\leq i\leq t-s-1$ we get
$$
h_A(i)=i+1+h_{A^{(1)}}(i-1)=i+1+s.
$$
Since the Hilbert function of an Artinian algebra with codimension two is unimodal and $t-s\leq \frac{d}{2}+1$ we have that
$$
\left(h_A-(h_{A^{(1)}})_+\right)({t-s})\geq \left(h_A-(h_{A^{(1)}})_+\right)({t-s-1})=t-s.
$$
Therefore,
\begin{equation}\label{h(t-s)}
h_A(t-s)\geq t-s+h_{A^{(1)}}(t-s-1).
\end{equation}
If $s-r-1<t-s-1$, then $h_A(t-s) \geq t-s+s = t$. Therefore, for every $t-s\leq i\leq \frac{d}{2}$ we have that $h_A(i)=t$.
\noindent If $s-r=t-s$, assuming $r= s-r$ implies that $s=2r$ and $t=3r$ which contradicts the assumption that $r<t-s$. So we have $r\leq s-r-1$. Using (\ref{h(t-s)}) we get that
$$
h_A(t-s)\geq t-s+h_{A^{(1)}}(t-s-1)= t-s+s=t .
$$ We conclude that $h_A(i)=t$, for every $t-s\leq i\leq \frac{d}{2}$.
\end{itemize}
We now prove $(2)$. Notice that
$$\frac{d}{2}-1=\frac{3d}{2}-2 -(d-1)\leq \left(h_A-(h_{A^{(1)}})_+\right)({\frac{d}{2}})\leq \frac{3d}{2} -(d-1)=\frac{d}{2}+1.$$
If $ \frac{d}{2}\leq \left(h_A-(h_{A^{(1)}})_+\right)({\frac{d}{2}})\leq \frac{d}{2}+1$, then for every $1\leq i\leq \frac{d}{2}-1$ we have that $\left(h_A-(h_{A^{(1)}})_+\right)(i)=i+1$ which implies that
$$
h_A(i)=i+1+2(i-1)+1=3i.
$$
If $\left(h_A-(h_{A^{(1)}})_+\right)({\frac{d}{2}})= \frac{d}{2}-1$, then for $1\leq i\leq \frac{d}{2}-2$ we have $h_A(i)=i+1+2(i-1)+1=3i$. On the other hand, for every $d\geq 6$ we have that
\begin{equation*}
\Cat_F(\frac{d}{2}-1)=\left[\begin{array}{@{}c|c|c@{}}
\mathbf{0}& \mathbf{0}&{\Cat_{G_{d-2}}{(\frac{d}{2}-3)}}
\\\hline
\mathbf{0}&\Cat_{G_{d-2}}{(\frac{d}{2}-2)}&\Cat_{G_{d-1}}{(\frac{d}{2}-2)}\\\hline
\Cat_{G_{d-2}}{(\frac{d}{2}-1)}&\Cat_{G_{d-1}}{(\frac{d}{2}-1)}&\Cat_{G_{d}}{(\frac{d}{2}-1)}
\end{array}
\right].\\
\end{equation*}
Which implies that
$$
\frac{3d}{2}-3= h_{A^{(2)}}(\frac{d}{2}-3)+h_{A^{(2)}}(\frac{d}{2}-2)+h_{A^{(2)}}(\frac{d}{2}-1)\leq h_A(\frac{d}{2}-1),
$$
and since $\Cat_F(\frac{d}{2}-1)$ is a square matrix of size $\frac{3d}{2}-3$ we get $h_A(\frac{d}{2}-1)=\frac{3d}{2}-3$. For $d=4$ similar argument implies that
$$
3= h_{A^{(2)}}(0)+h_{A^{(2)}}(1)\leq h_A(1).
$$
For $d=2$ there is noting to show.
\end{proof}
Now we state and prove the analogues statements to Lemma \ref{maxvaluesevenLemma} and Theorem \ref{3linesHFtheorem-even} for Artinian Gorenstein algebras with odd socle degrees.
\begin{lemma}[Odd socle degree]\label{maxvaluesoddLemma}
There exists an Artinian Gorenstein algebra $A$ with odd socle degree $d\geq 3$ and linear form $\ell\in A_1$ where $\ell^2\neq 0$ and $\ell^3=0$, such that
$$
(r,s,t)=\left( h_{A^{(2)}}(\frac{d-1}{2}), h_{A^{(1)}}(\frac{d-1}{2}), h_{A}(\frac{d-1}{2})\right)
$$
if and only if
\begin{itemize}
\item[$(1)$] $r\in [1,\frac{d-1}{2}-1], s\in[2r,\frac{d-1}{2}+r]$ and $t\in[2s-r,\frac{d-1}{2}+s+1]$, for $d\geq 5$; or
\item[$(2)$] $r\in [1,\frac{d-1}{2}-1], s=\frac{d-1}{2}+r+1$ and $t=d+r$, for $d\geq 5$; or
\item[$(3)$] $r=\frac{d-1}{2}, s\in [d-1,d]$ and $t\in [\frac{d-1}{2}+s-1,3\frac{d-1}{2}]$, for $d\geq 3$.
\end{itemize}
\end{lemma}
\begin{proof}
The maximum value of the Hilbert function of $A$ occurs in degree $\frac{d-1}{2}$ and it is equal to the rank of the following catalecticant matrix
\begin{equation}
\Cat_F(\frac{d-1}{2})=\left[\begin{array}{@{}c|c|c@{}}
\mathbf{0}& \mathbf{0}&{\Cat_{G_{d-2}}{(\frac{d-1}{2}-2)}}
\\\hline
\mathbf{0}&\Cat_{G_{d-2}}{(\frac{d-1}{2}-1)}&\Cat_{G_{d-1}}{(\frac{d-1}{2}-1)}\\\hline
\Cat_{G_{d-2}}{(\frac{d-1}{2})}&\Cat_{G_{d-1}}{(\frac{d-1}{2})}&\Cat_{G_{d}}{(\frac{d-1}{2})}
\end{array}
\right]\\
\end{equation}
which is equal to
\begin{Small}
\begin{equation}\label{catmatrix3linesodd}
\Cat_F(\frac{d-1}{2}) = \left[\begin{array}{@{}cccc|cccc|cccc@{}}
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{\frac{d+1}{2}} \\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{1} & c_{2} &\cdots & c_{\frac{d+1}{2}+1} \\
\vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{\frac{d-1}{2}-2} & c_{\frac{d-1}{2}-1} &\cdots & c_{d-2} \\\hline
0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{\frac{d-1}{2}} & b_{0} & b_{1}&\cdots & b_{\frac{d+1}{2}} \\
0 & 0 & \cdots & 0 & c_{1} & c_{2} &\cdots & c_{\frac{d-1}{2}+1} & b_{1} & b_{2} &\cdots & b_{\frac{d+1}{2}+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & c_{\frac{d-1}{2}-1} & c_{\frac{d-1}{2}} &\cdots & c_{d-2} & b_{\frac{d-1}{2}-1} & b_{\frac{d-1}{2}} &\cdots & b_{d-1} \\\hline
c_{0} & c_{1}&\cdots & c_{\frac{d-1}{2}-1} & b_{0} & b_{1}&\cdots & b_{\frac{d-1}{2}} &a_{0} & a_{1}&\cdots & a_{\frac{d+1}{2}} \\
c_{1} & c_{2} &\cdots & c_{\frac{d-1}{2}} & b_{1} & b_{2} &\cdots & b_{\frac{d-1}{2}+1}& a_{1} & a_{2} &\cdots & a_{\frac{d+1}{2}+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
c_{\frac{d-1}{2}} & c_{\frac{d-1}{2}+1} &\cdots & c_{d-2} & b_{\frac{d-1}{2}} & b_{\frac{d-1}{2}+1} &\cdots & b_{d-1}& a_{\frac{d-1}{2}} & a_{\frac{d-1}{2}+1} &\cdots & a_{d} \\
\end{array}\right].
\end{equation}
\end{Small}
We note that $r=h_{A^{(2)}}(\frac{d-1}{2})\in [1,\frac{d-1}{2}]$. First assume that $r\in [1,\frac{d-1}{2}-1]$ and note that the socle degree of $A^{(2)}$ is odd, then we have that $
h_{A^{(2)}}(\frac{d-1}{2}-2)= h_{A^{(2)}}(\frac{d-1}{2}-1)= \linebreak h_{A^{(2)}}(\frac{d-1}{2})=r.
$
We may assume that the ranks of the lower right submatrices of \linebreak$\Cat_{G_{d-2}}(\frac{d-1}{2}-2), \Cat_{G_{d-2}}(\frac{d-1}{2}-1)$ and $\Cat_{G_{d-2}}(\frac{d-1}{2})$ are equal to $r$. Setting $c_{d-r-1}=1$ and $c_i=0$ for every $i\neq d-r-1$, or equivalently setting $G_{d-2}$ as the following provides the desired property
\begin{equation}\label{Gd-2}
G_{d-2}= \frac{Y^{r-1}Z^{d-r-1}}{(r-1)!(d-r-1)!}, \quad \text{for each}\quad r\in[1,\frac{d-1}{2}-1].
\end{equation}
The Hilbert function of $A^{(1)}$ in degree $\frac{d-1}{2}$ is equal to $2r+\rk\mathbf{B}$ where
\begin{equation}\label{matrixBodd}
\mathbf{B}=\left(\begin{array}{@{}ccccccc@{}}
b_{0} &\cdots & b_{\frac{d-1}{2}-r} \\
\vdots & \reflectbox{$\ddots$} &\vdots\\
b_{\frac{d-1}{2}-r}&\cdots & b_{d-1-2r} \end{array}\right).
\end{equation}
So $s=h_{A^{(1)}}(\frac{d-1}{2})\in [2r, \frac{d-1}{2}+r+1]$. Suppose that $s\in [2r, \frac{d-1}{2}+r]$. This implies that $ h_{A^{(1)}}(\frac{d-1}{2}-1)= h_{A^{(1)}}(\frac{d-1}{2}) = s$. To prove $(1)$, we use the same argument that we used to prove Lemma \ref{maxvaluesevenLemma} part $(1)$. Therefore, the following choice of $G_{d-1}$ and $G_d$ completes the proof of part $(1)$.
\begin{equation*}
G_{d-1}=\left\{
\begin{array}{ll}
0 & \text{if $s-2r=0$},\\
\frac{Y^{d-s+2r}Z^{s-2r-1}}{(d-s+2r)!(s-2r-1)!} & \text{if $1\leq s-2r\leq \frac{d-1}{2}-r,$}\\
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
G_{d}=\left\{
\begin{array}{ll}
0 & \text{if $t-2s+r=0$},\\
\frac{Y^{d-t+3r+1}Z^{t-3r-1}}{(d-t+3r+1)!(t-3r-1)!} & \text{if $1\leq t-2s+r\leq \frac{d-1}{2}+r-s+1.$}\\
\end{array}
\right.
\end{equation*}
Now assume that $s=h_{A^{(1)}}(\frac{d-1}{2})=\frac{d-1}{2}+r+1$, which is the maximum possible for $r\in [1,\frac{d-1}{2}-1]$. The following submatrices of $\Cat_{G_{d-1}}(\frac{d-1}{2})$ having maximal rank, that is equal to $\frac{d-1}{2}-r+1$, implies that $\rk\Cat_{x\circ F}(\frac{d-1}{2})=\frac{d-1}{2}+r+1$.
\begin{equation*}
\mathbf{B}=\left[\begin{array}{@{}ccccccc@{}}
b_{0} &\cdots & b_{\frac{d-1}{2}-r} \\
\vdots & \reflectbox{$\ddots$} &\vdots\\
b_{\frac{d-1}{2}-r}&\cdots & b_{d-1-2r} \end{array}\right].
\end{equation*}
This forces the following submatrix of $\Cat_{G_{d-1}}(\frac{d-1}{2}-1)$ to have maximal rank, that is equal to $\frac{d-1}{2}-r$.
\begin{equation*}
\mathbf{B^\prime}=\left[\begin{array}{@{}ccccccc@{}}
b_{0} &\cdots & b_{\frac{d+1}{2}-r} \\
\vdots & \reflectbox{$\ddots$} &\vdots\\
b_{\frac{d-1}{2}-1-r}&\cdots & b_{d-1-2r} \end{array}\right].
\end{equation*}
Setting $b_{\frac{d-1}{2}+r}=1$ and $b_{i}=0$ for every $i\neq \frac{d-1}{2}+r$, or equivalently,
$$
G_{d-1} = \frac{Y^{\frac{d-1}{2}-r}Z^{\frac{d-1}{2}+r}}{(\frac{d-1}{2}-r)!(\frac{d-1}{2}+r)!}
$$
provides that
$$h_{A^{(1)}}(\frac{d-1}{2})=\frac{d-1}{2}+r+1, \hspace*{2mm}\text{and} \hspace*{2mm} h_{A^{(1)}}(\frac{d-1}{2}-1)=\frac{d-1}{2}+r.$$
Since $\mathbf{B}$ and $\mathbf{B^\prime}$ both have maximal ranks for every choice of $G_d$, we conclude
$$
h_A(\frac{d-1}{2})=\rk\Cat_{F}(\frac{d-1}{2}) = 3r+\rk\mathbf{B}+\rk\mathbf{B^\prime} = d+r.
$$
Now we assume that $r=\frac{d-1}{2}$ as in $(3)$. Since $d$ is an odd integer $h_{A^{(2)}}(\frac{d-1}{2}-1)=h_{A^{(2)}}(\frac{d-1}{2})=\frac{d-1}{2}$ and $h_{A^{(2)}}(\frac{d-1}{2}-2) = \frac{d-1}{2}-1$. Setting $c_{\frac{d-1}{2}-1}=1$ and $c_i=0$ for every $i\neq \frac{d-1}{2}-1$, that is
\begin{equation}\label{random}
G_{d-2} = \frac{Y^{\frac{d-1}{2}}Z^{\frac{d-1}{2}-1}}{(\frac{d-1}{2})!(\frac{d-1}{2}-1)!},
\end{equation}
implies that $h_{A^{(2)}}(\frac{d-1}{2})=\frac{d-1}{2}$.
In order to find possible values for $h_{A^{(1)}}(\frac{d-1}{2})$, note that
\begin{equation*}
h_{A^{(1)}}(\frac{d-1}{2}) = \rk\left[\begin{array}{@{}c|c@{}}
\mathbf{0}&{\Cat_{G_{d-2}}{(\frac{d-1}{2}-1)}}
\\\hline
\Cat_{G_{d-2}}{(\frac{d-1}{2})}&\Cat_{G_{d-1}}{(\frac{d-1}{2})}\\
\end{array}
\right],\\
\end{equation*}
is a square matrix of size $d$. On the other hand $$d-1=\rk\Cat_{G_{d-2}}(\frac{d-1}{2}-1)+\rk\Cat_{G_{d-2}}(\frac{d-1}{2}-1)\leq h_{A^{(1)}}(\frac{d-1}{2}).$$
For the polynomial $G_{d-2}$ as in (\ref{random}) we get that the last column of the above matrix is zero. So setting $G_{d-1}=0$ gives $h_{A^{(1)}}(\frac{d-1}{2})=d-1$ and setting $G_{d-1}=\frac{Z^{d-1}}{(d-1)!}$ gives that $h_{A^{(1)}}(\frac{d-1}{2})=d$.
To find possible values for $h_A(\frac{d-1}{2})$ we note that the number of rows in the catalecticant matrix (\ref{catmatrix3linesodd}) is equal to $3\frac{d-1}{2}$.
If $h_{A^{(1)}}(\frac{d-1}{2})=d$, then independent of the choice of $G_d$,the Hilbert function $h_A(\frac{d-1}{2})$ is equal to the maximum possible. So,
$$\rk\Cat_F(\frac{d-1}{2})\geq h_{A^{(1)}}(\frac{d-1}{2})+h_{A^{(2)}}(\frac{d-1}{2}-2)=d+\frac{d-1}{2}-1=3\frac{d-1}{2}.$$
If $h_{A^{(1)}}(\frac{d-1}{2})=d-1$, then
$$
\rk\Cat_F(\frac{d-1}{2})\geq h_{A^{(1)}}(\frac{d-1}{2})+h_{A^{(2)}}(\frac{d-1}{2}-2)=d-1+\frac{d-1}{2}-1=3\frac{d-1}{2}-1.
$$ Setting $G_{d-1}=0$ and $G_{d}=0$ provides that
$ h_{A}(\frac{d-1}{2})=3\frac{d-1}{2}-1$. And setting $G_{d-1}=0$ and $G_{d}=\frac{Z^d}{d!}$ provides that $ h_{A}(\frac{d-1}{2})=3\frac{d-1}{2}$. In fact, with this choice the last column of $\Cat_F(\frac{d-1}{2})$ becomes non-zero and linearly independent from the previous columns.
\end{proof}
\begin{theorem}[Odd socle degree]\label{3linesHFtheorem-odd}
Let $A$ be an Artinian Gorenstein algebra with odd socle degree $d\geq 3$ and $\ell\in A_1$ such that $\ell^2\neq 0$ and $\ell^3=0$. Then Hilbert functions of $A$, $A^{(1)}$ and $A^{(2)}$ are completely determined by $(r,s,t)=( h_{A^{(2)}}(\frac{d-1}{2}), h_{A^{(1)}}(\frac{d-1}{2}), h_{A}(\frac{d-1}{2}))$. More precisely,
\begin{itemize}
\item[$(1)$] if $d\geq 5$, $r\in [1,\frac{d-1}{2}-1], s\in[2r,\frac{d-1}{2}+r]$ and $t\in [2s-r,\frac{d-1}{2}+s+1]$, then
\begin{equation}\label{HFodd(1)}
h_{A^{(2)}}(i)=\left\{
\begin{array}{ll}
i+1 & 0\leq i\leq r-1,\\
r & r\leq i\leq\frac{d-1}{2},\\
\end{array}
\right.\quad
h_{A^{(1)}}(i)=\left\{
\begin{array}{ll}
2i+1 & 0\leq i\leq r-1,\\
i+r+1 & r\leq i\leq s-r-1,\\
s & s-r\leq i\leq \frac{d-1}{2}.\\
\end{array}
\right.
\end{equation}
\begin{itemize}
\item If $t=3r$ then there are two possible Hilbert functions for $A$
\begin{equation}\label{HFoddA(1,1)}
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r-1,\\
3r & r\leq i\leq\frac{d-1}{2}.\\
\end{array}
\right.
\text{and} \quad
h_{A}(i)=\left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r-1,\\
3r-1& i=r,\\
3r & r+1\leq i\leq \frac{d-1}{2},\\
\end{array}
\right.
\end{equation}
\item otherwise
\begin{equation}\label{HFoddA(1,2)}
h_A(i)= \left\{
\begin{array}{ll}
1 & i=0,\\
3i & 1\leq i\leq r,\\
2i+r+1& r+1\leq i\leq s-r-1,\\
2i+r+1& i=s-r, \hspace*{2mm}\text{if}\hspace*{2mm}t>2s-r\hspace*{2mm}\text{and}\hspace*{2mm}s>2r,\\
2i+r& i=s-r, \hspace*{2mm}\text{if}\hspace*{2mm}t>2s-r \hspace*{2mm}\text{and}\hspace*{2mm}s=2r,\\
i+s+1 & s-r+1\leq i\leq t-s-1,\\
t & t-s\leq i\leq \frac{d-1}{2}.\\
\end{array}
\right.
\end{equation}
\end{itemize}
\item[$(2)$] If $d\geq 3$, $r\in[1,\frac{d-1}{2}-1],s=\frac{d-1}{2}+r+1 $ and $t=d+r$, then
\begin{equation}\label{HFodd(2)}
h_{A^{(1)}}(i)= \left\{
\begin{array}{ll}
2i+1 & 0\leq i\leq r,\\
i+1+r & r+1\leq i\leq\frac{d-1}{2},\\
\end{array}
\right. \text{and}\quad h_{A}(i)= \left\{
\begin{array}{ll}
1& i=0,\\
3i & 1\leq i\leq r+1,\\
2i+1+r & r+2\leq i\leq\frac{d-1}{2}.\\
\end{array}
\right.
\end{equation}
\item[$(3)$] For $d\geq 3$ if $r=\frac{d-1}{2}, s\in [d-1,d]$ and $t\in [\frac{d-1}{2}+s-1,3\frac{d-1}{2}]$, then for every $i\in [0,\frac{d-1}{2}-1]$
\begin{equation}\label{HFodd(3)}
h_{A^{(2)}}(i) = i+1, \quad h_{A^{(1)}}(i) = 2i+1\hspace*{2mm}\text{and} \hspace*{2mm} h_A(0)=1, h_A(i) =3i.
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof}
We first prove part $(1)$. The assumptions on $r$ and $s$ imply that
$$ h_{A^{(1)}}(\frac{d-1}{2}-1)=h_{A^{(1)}}(\frac{d-1}{2})=h_{A^{(1)}}(\frac{d-1}{2}+1)=s,
$$
and
$$ h_{A^{(2)}}(\frac{d-1}{2}-1)=h_{A^{(2)}}(\frac{d-1}{2})=r.
$$
Therefore, applying Theorem \ref{3linesHFtheorem-even} part $(1)$ for $d-1$ completes the proof of $(1)$.\par
\noindent Now we show $(2)$. First note that $h_{A^{(2)}}$ is the same as the previous case. By the assumption we have that
\begin{align*}
(h_{A^{(1)}}-(h_{A^{(2)}})_+)(\frac{d-1}{2}) &= h_{A^{(1)}}(\frac{d-1}{2})-h_{A^{(2)}}(\frac{d-1}{2}-1)\\
&= h_{A^{(1)}}(\frac{d-1}{2})-h_{A^{(2)}}(\frac{d-1}{2})\\
&=\frac{d-1}{2}+r+1-r= \frac{d-1}{2}+1.
\end{align*}
Using Lemma \ref{diffO-seq}, we get that
$(h_{A^{(1)}}-(h_{A^{(2)}})_+)(i) = i+1
$ for every $i\in[0,\frac{d-1}{2}]$.
Therefore, $h_{A^{(1)}}$ is what we claimed.
To obtain $h_A$, we note that
\begin{align*}
(h_{A}-(h_{A^{(1)}})_+)(\frac{d-1}{2}) = h_{A}(\frac{d-1}{2})-h_{A^{(1)}}(\frac{d-1}{2}-1)= d+r- h_{A^{(1)}}(\frac{d-1}{2}-1).
\end{align*}
If $r<\frac{d-1}{2}-1$, then we have $h_{A^{(1)}}(\frac{d-1}{2}-1)=\frac{d-1}{2}+r$ and if $r=\frac{d-1}{2}-1$, then $h_{A^{(1)}}(\frac{d-1}{2}-1)=2(\frac{d-1}{2}-1)+1=d-2.$ In both cases we get that
\begin{align*}
(h_{A}-(h_{A^{(1)}})_+)(\frac{d-1}{2}) = d+r-(d-2) = r+2 = \frac{d-1}{2}+1.
\end{align*}
So for every $i\in[0,\frac{d-1}{2}]$ we have $(h_{A}-(h_{A^{(1)}})_+)(i) = i+1,
$ and therefore
$
h_A(i)=i+1+h_{A^{(1)}}(i-1)
$
which implies the desired Hilbert function for $A$.
To prove $(3)$ we get $h_{A^{(2)}}$ by replacing $r$ by $\frac{d-1}{2}$ in the previous case. By the assumption we have that
$$
\frac{d-1}{2}\leq (h_{A^{(1)}}-(h_{A^{(2)}})_+)(\frac{d-1}{2})\leq \frac{d-1}{2}+1.
$$
So for every $i\in [0,\frac{d-1}{2}-1]$ $$h_{A^{(1)}}(i)=i+1+h_{A^{(2)}}(i-1)=2i+1.$$
To obtain $h_A$ we observe that
\begin{align*}
(h_{A}-(h_{A^{(1)}})_+)(\frac{d-1}{2}) \geq \frac{d-1}{2}+s-1-(d-2)
=s-\frac{d-1}{2}\geq \frac{d-1}{2}.
\end{align*}
Therefore,($h_{A}-(h_{A^{(1)}})_+)(i)=i+1$ for every $i\in [0,\frac{d-1}{2}-1]$, and equivalently, we have $h_A(0)=1$ and $h_A(i)=i+1+2(i-1)+1=3i$ for every $i\in [1,\frac{d-1}{2}-1]$.
\end{proof}
We prove that the lists of rank matrices given in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd} are exhaustive lists.
\begin{theorem}\label{allHF´sPossible-even}
A vector of non-negative integers $h$ is the Hilbert function of some Artinian Gorenstein algebra $A=S/\ann(F)$ such that there exists a linear form $\ell$ satisfying $ \ell^2\neq 0$ and $\ell^3=0$ if and only if $h$ is equal to one of the Hilbert functions provided in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd}.
\end{theorem}
\begin{proof}
In Lemmas \ref{maxvaluesevenLemma} and \ref{maxvaluesoddLemma} we provide the complete list of possible values for the maximum of the Hilbert function of any Artinian Gorenstein algebras $A$ where $\ell^3=0$. In fact, for each maximum value we produce a dual generator $F$ for $A$. On the other hand, in Theorems \ref{3linesHFtheorem-even} and \ref{maxvaluesoddLemma} we prove that for each possible maximum value the Hilbert function of $A$ is uniquely determined by the maximum value in all the cases except when $r\in [1,\lfloor\frac{d}{2}\rfloor-1]$ and $t=3r$ for every $d\geq 4$, in which we have two possibilities for $h_A$. We show that both Hilbert functions provided for $h_A$ occur for some Artinian Gorenstein algebra $A$.\par
First assume that $d\geq 6$, $r\in[2,\lfloor\frac{d}{2}\rfloor-1]$ and $t=3r$. This implies that $s=2r$. Fixing $h_A(r)$ to be either $3r-1$ or $3r$ we provide a degree $d$ polynomial satisfying (\ref{F}) as the dual generator for Artinian Gorenstein algebra $A$ such that $h_A(\lfloor\frac{d}{2}\rfloor)=3r$.
Pick the following monomial basis for $A_r$
$$\B_r=\{x^r,x^{r-1}y,x^{r-1}z,x^{r-2}y^2,\dots ,y^r,y^{r-1},z,\dots , z^r\}$$
so
\begin{equation}
\Cat_F(r)=\left[\begin{array}{@{}c|c|c@{}}
\mathbf{0}& \mathbf{0}&{\Cat_{G_{d-2}}{(r-2)}}
\\\hline
\mathbf{0}&\Cat_{G_{d-2}}{(r-1)}&\Cat_{G_{d-1}}{(r-1)}\\\hline
\Cat_{G_{d-2}}{(r)}&\Cat_{G_{d-1}}{(r)}&\Cat_{G_{d}}{(r)}
\end{array}
\right].\\
\end{equation}
This is equal to
\begin{equation}\label{catmatrix3lines-r}
\Cat_F(r) = \left[\begin{array}{@{}cccc|cccc|cccc@{}}
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{d-r} \\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{1} & c_{2} &\cdots & c_{d-r+1} \\
\vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots & \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0 & c_{r-2} & c_{r-1} &\cdots & c_{d-2} \\\hline
0 & 0 & \cdots & 0 & c_{0} & c_{1}&\cdots & c_{d-r-1} & b_{0} & b_{1}&\cdots & b_{d-r} \\
0 & 0 & \cdots & 0 & c_{1} & c_{2} &\cdots & c_{d-r} & b_{1} & b_{2} &\cdots & b_{d-+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
0 & 0 & \cdots & 0 & c_{r-1} & c_{r} &\cdots & c_{d-2} & b_{r-1} & b_{r} &\cdots & b_{d-1} \\\hline
c_{0} & c_{1}&\cdots & c_{d-r-2} & b_{0} & b_{1}&\cdots & b_{d-r-1} &a_{0} & a_{1}&\cdots & a_{d-r} \\
c_{1} & c_{2} &\cdots & c_{d-r-1} & b_{1} & b_{2} &\cdots & b_{d-r}& a_{1} & a_{2} &\cdots & a_{d-r+1} \\
\vdots &\vdots & \reflectbox{$\ddots$} & \vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots& \vdots & \vdots & \reflectbox{$\ddots$} &\vdots\\
c_{r} & c_{r+1} &\cdots & c_{d-2} & b_{r} & b_{r+1} &\cdots & b_{d-1}& a_{r} & a_{r+1} &\cdots & a_{d} \\
\end{array}\right].
\end{equation}
Using what we have shown in Lemmas \ref{maxvaluesevenLemma} and \ref{maxvaluesoddLemma} part $(1)$, setting $c_{d-r-1}=1$ and all other coefficients in the polynomial $F$ to be zero, or equivalently,
\begin{equation}\label{3rFeven}
F = \frac{X^2 Y^{r-1}Z^{d-r-1}}{2(r-1)!(d-r-1)!}
\end{equation}
provides that $h_A(\lfloor\frac{d}{2}\rfloor)=3r$.
Therefore, since $G_{d-1}=G_d=0$ we get that
$$
h_A(r)=\rk\Cat_F(r)=\rk\Cat_{F}(r-2)+ \rk\Cat_{F}(r-1)+\rk\Cat_{F}(r)=3r-1,
$$
where the ranks of $\Cat_{F}(r-2), \Cat_{F}(r-1)$ and $\Cat_{F}(r)$, or equally, $h_{A^{(2)}}(r-2),h_{A^{(2)}}(r-1)$ and $h_{A^{(2)}}(r)$ are given in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd}.
In order to provide a polynomial $F$ as the dual generator of $A$ where $h_A(\lfloor\frac{d}{2}\rfloor)=3r=h_A(r)=3r$, we set $c_{d-r-1}=a_{d-r}=1$ and all other coefficients to be zero, so
\begin{equation}\label{3rFeven}
F = \frac{X^2 Y^{r-1}Z^{d-r-1}}{2(r-1)!(d-r-1)!} + \frac{Y^rZ^{d-r}}{r!(d-r)!}.
\end{equation}
We observe that setting $a_{d-r}=1$ in the matrices (\ref{catmatrix3lines}) and (\ref{catmatrix3linesodd}), the number of linearly independent columns does not increase and is equal to $3r$. On the other hand, setting $a_{d-r}=1$ increases the number of linearly independent columns of $\Cat_F(r)$ in (\ref{catmatrix3lines-r}) by one. In fact, by setting $a_{d-r}=1$ the last column of (\ref{catmatrix3lines-r}) becomes non-zero and not included in the span of the previous columns. Thus, the number of linearly independent columns in (\ref{catmatrix3lines-r}) is equal to $3r$, so $h_A(r)=3r$.
\noindent Now assume that $d\geq 4$ and $r=1$. We use the same argument as the previous case for the following matrix.
\begin{equation}\label{catmatrix3lines(r=1)}
\Cat_F(r)=\left[\begin{array}{@{}c|c|c@{}}
\mathbf{0}&{\Cat_{G_{d-2}}{(r-1)}}&{\Cat_{G_{d-1}}{(r-1)}}
\\\hline
\Cat_{G_{d-2}}{(r)}&\Cat_{G_{d-1}}{(r)}&\Cat_{G_{d}}{(r)}\\
\end{array},
\right]\\
\end{equation}
similarly setting $F = \frac{X^2 Z^{d-2}}{2(d-2)!}$ provides that $\rk\Cat_F(1)=2$, and setting $F = \frac{X^2 Z^{d-2}}{2(d-2)!} + \frac{Y Z^{d-1}}{(d-1)!}$ provides $\rk\Cat_F(1)=3$. We notice that in both cases $h_A(\lfloor\frac{d}{2}\rfloor)=3$.
\end{proof}
As an immediate consequence of above results we get the complete list of possible rank matrices for Artinian Gorenstein algebras and linear forms such that $\ell^2=0$. We may assume that $\ell\neq 0$, since otherwise multiplication map by $\ell$ is trivial. So we have $A^{(1)}\neq 0$ and $A^{(i)}=0$, for all $i\geq 2$. We denote by $r$ and $s$ the maximum values for $h_{A^{(1)}}$ and $h_A$ respectively.
\begin{corollary}\label{2linescorollary}
There exists an Artinian Gorenstein algebra $A$ with socle degree $d\geq 2$ and $\ell\in A_1$ where $\ell\neq 0$ and $\ell^2=0$, such that
$$
(r,s)=\left(h_{A^{(1)}}(\lfloor\frac{d}{2}\rfloor), h_{A}(\lfloor\frac{d}{2}\rfloor)\right)
$$
if and only if
\begin{itemize}
\item $r\in[1,\lceil\frac{d}{2}\rceil-1]$ and $s\in [2r,\lceil\frac{d}{2}\rceil+r ]$, for $d\geq 3$; or
\item $r= \lceil\frac{d}{2}\rceil$ and $s=d$ if $d\geq 3$ is odd; $s=d,d+1$ if $d\geq 2$ is even.
\end{itemize}
Moreover, the Hilbert functions of $A$ and $A^{(1)}$ are completely determined by $(r,s)$ as the following
\begin{equation}\label{HF2lines}
h_{A^{(1)}}(i)=\left\{
\begin{array}{ll}
i+1 & 0\leq i\leq r-1,\\
r & r\leq i\leq\lfloor\frac{d}{2}\rfloor.\\
\end{array}
\right.\quad
h_{A}(i)=\left\{
\begin{array}{ll}
2i+1 & 0\leq i\leq r-1,\\
i+r+1 & r\leq i\leq s-r-1,\\
s & s-r\leq i\leq \lfloor\frac{d}{2}\rfloor.\\
\end{array}
\right.
\end{equation}
\end{corollary}
\begin{proof}
The proof is immediate by considering rank matrices with two non-zero diagonals given by $h_{A^{(1)}}$ and $h_{A^{(2)}}$ provided in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd}.
\end{proof}
\begin{remark}
The above threorems provide complete lists of rank matrices, $M_{\ell,A}$, for Artinian Gorenstein algebras $A$ of codimension two and three satisfying $\ell^3=0$. In fact, there might exists a linear form $\ell^\prime\neq \ell$ such that $\ell^\prime= 0$.
\end{remark}
We are now able to formulate our last result which provides a formula to compute Jordan types of Artinian Gorenstein algebras with parts of length at most four in terms of at most three parameters $(r,s,t)$ in the above theorems.
Using Remark \ref{r_ij-remark}, we provide the formulas in terms of the ranks of mixed Hessians in certain degrees.
\begin{theorem}\label{JT-theorem}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra with socle degree $d\geq 2$ and $\ell\neq 0$ be a linear form such that $\ell^4=0$. The Jordan type $P_{\ell,A}$ is one of the followings.
\begin{itemize}
\item If $\ell^3\neq 0$ then the Jordan type partition of $A$ for $\ell$ is given by
\begin{equation}\label{JT4}
P_{\ell,A}=(\underbrace{4,\dots ,4}_{\Delta^2\mathbf{d}(3)},\underbrace{3,\dots ,3}_{\Delta^2\mathbf{d}(2)}, \underbrace{2,\dots ,2}_{\Delta^2\mathbf{d}(1)},\underbrace{1,\dots ,1}_{\Delta^2\mathbf{d}(0)}),
\end{equation}
where $\mathbf{d}=(\dim_\mathsf{k} A,\dim_\mathsf{k} A^{(1)},\dim_\mathsf{k} A^{(2)}, \dim_\mathsf{k} A^{(3)})$ and the Hilbert functions of $A^{(1)}$, $A^{(2)}$ and $A^{(3)}$ are given in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd} for parameters
$$(r,s,t)=\left(\rk\Hess_\ell^{(\lfloor\frac{d}{2}\rfloor-1, \lceil\frac{d}{2}\rceil-2)}(F), \rk\Hess_\ell^{(\lfloor\frac{d}{2}\rfloor-1, \lceil\frac{d}{2}\rceil-1)}(F),\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor)}(F)\right).$$
\noindent Moreover, if $t\neq 3r$, then $P_{\ell,A}$ is uniquely determined by non-zero integers $(r,s,t)$. Otherwise, if $t=3r$, then $P_{\ell,A}$ is uniquely determined by non-zero integers \linebreak$(r,\rk\Hess_\ell^{(r,d-r-1)}(F)).$
\item If $\ell^3=0$ and $\ell^2\neq 0$, then
\begin{equation}\label{JT3}
P_{\ell,A}=(\underbrace{3,\dots ,3}_{\Delta^2\mathbf{d}(2)},\underbrace{2,\dots ,2}_{\Delta^2\mathbf{d}(1)}, \underbrace{1,\dots ,1}_{\Delta^2\mathbf{d}(0)}),
\end{equation}
where $\mathbf{d}=(\dim_\mathsf{k} A,\dim_\mathsf{k} A^{(1)},\dim_\mathsf{k} A^{(2)})$ and the Hilbert functions of $A^{(1)}$ and $A^{(2)}$ are given in Corollary \ref{2linescorollary} for parameters
$$(r,s)=\left(\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor)}(F),\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor-1)}(F)\right).$$
Moreover, $P_{\ell,A}$ is uniquely determined by non-zero integers $(r,s).$
\item If $\ell^2=0$ and $\ell\neq 0$, then
\begin{equation}\label{JT2}
P_{\ell,A}=(\underbrace{2,\dots ,2}_{\Delta^2\mathbf{d}(1)},\underbrace{1,\dots ,1}_{\Delta^2\mathbf{d}(0)}),
\end{equation}
where $\mathbf{d}=(\dim_\mathsf{k} A,\dim_\mathsf{k} A^{(1)})$
where the Hilbert function of $A^{(1)}$ is given in Corollary \ref{2linescorollary} for parameter $$r=\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor)}(F).$$
Moreover, $P_{\ell,A}$ is uniquely determined by the non-zero integer $r.$
\end{itemize}
\end{theorem}
\begin{proof}
First assume that $\ell^3\neq 0$ and notice that the socle degree of $A^{(1)}=S/\ann(\ell\circ F)$ equals to $d-1$. Recall from Remark \ref{r_ij-remark} that
$$r=\rk\Hess_\ell^{(\lfloor\frac{d}{2}\rfloor-1,\lceil\frac{d}{2}\rceil-2 )}(F)= h_{A^{(3)}}(\lfloor\frac{d}{2}\rfloor-1), $$
$$s=\rk\Hess_\ell^{(\lfloor\frac{d}{2}\rfloor-1,\lceil\frac{d}{2}\rceil-1 )}(F)= h_{A^{(2)}}(\lfloor\frac{d}{2}\rfloor-1), $$
and
$$t=\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor,\lfloor\frac{d}{2}\rfloor )}(F)= h_{A^{(1)}}(\lfloor\frac{d-1}{2}\rfloor).$$
Then using Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd}, we get the ranks of multiplication maps by $\ell$, $\ell^2$ and $\ell^3$ on $A$ in various degrees from the rank matrix of $A^{(1)}$ in terms of $r,s,t$. Then using Proposition \ref{JT}, we get $P_{\ell,A}$, as we claimed in (\ref{JT4}). Moreover, we proved in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd} that the rank matrix of $A^{(1)}$ is uniquely determined in terms of $r,s$ and $t$ except when $t=3r$. In this case, there are two possible rank matrices for $A^{(1)}$ that is determined uniquely in terms of $r$ and $\Hess_\ell^{(r,d-r-1)}(F)$.
Now suppose that $\ell^3=0$ and $\ell^2\neq 0$. Ranks of multiplication maps on $A$ by $\ell$ and $\ell^2$ are uniquely determined by $r$ and $s$ in Corollary \ref{2linescorollary}, where
$$r=\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor)}(F)=h_{A^{(2)}}(\lfloor\frac{d-1}{2}\rfloor),\hspace*{2mm} \text{and} \hspace*{2mm} s=\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor-1)}(F)=h_{A^{(1)}}(\lfloor\frac{d-1}{2}\rfloor).$$ Therefore, Proposition \ref{JT} implies that $P_{\ell,A}$ is equal to (\ref{JT3}).
Assume that $\ell^2=0$ and $\ell\neq 0$ and that $r=\rk\Hess_\ell^{(\lfloor\frac{d-1}{2}\rfloor, \lfloor\frac{d}{2}\rfloor)}(F)=h_{A^{(1)}}(\lfloor\frac{d-1}{2}\rfloor)$. Then, Corollary \ref{2linescorollary} provides the rank of multiplication map by $\ell$, by providing $h_{A^{(1)}}$ which implies the desired Jordan type in this case.
\end{proof}
\begin{remark}
One may use \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd} to get the Jordan degree types with parts of length at most four, similar to the above theorem. Thus, such Jordan degree type is also determined uniquely by at most the ranks of three mixed Hessians.\par
More precise formulas for $P_{\ell,A}$ could be obtained directly from rank matrices provided in Theorems \ref{3linesHFtheorem-even} and \ref{3linesHFtheorem-odd}.
\end{remark}
\begin{example}
Let $A=S/\ann(F)$ be an Artinian Gorenstein algebra where $$F=X^3Y^4+X^3Z^4+X^2YZ^4+Y^3Z^4.$$ We have that
$$\rk\Hess^{(2,2)}_x(F)=2, \quad \rk\Hess^{(2,3)}_x(F)=4, \hspace*{2mm}\text{and}\hspace*{2mm}\rk\Hess^{(3,3)}_x(F)=7.$$
So using Theorem \ref{3linesHFtheorem-even} for $(r,s,t)=(2,4,7)$ we get the rank matrix for $A^{(1)}=S/\ann(x\circ F)$ and $\ell=x$ that is equal to
$$M_{x,A^{(1)}}= \begin{pmatrix}
1&1&1&0&0&0&0\\
0&3&3&2&0&0&0\\
0&0&6&4&2&0&0\\
0&0&0&7&4&2&0\\
0&0&0&0&6&3&1\\
0&0&0&0&0&3&1\\
0&0&0&0&0&0&1\\
\end{pmatrix}.
$$
Therefore, using Equation \ref{JT4} we get the Jordan type of $A$ and $x$, which is equal to
$$
P_{x,A}=(\underbrace{4,\dots ,4}_8,\underbrace{2,\dots ,2}_3, \underbrace{1,\dots ,1}_{\dim_{\mathsf{k}}A-38})= (\underbrace{4,\dots ,4}_8,\underbrace{2,\dots ,2}_3)
$$
sicne we have $h_A=(1,3,6,9,9,6,3,1)$.
Moreover, using the correspondence to the Jordan degree type matrix in Proposition \ref{rkmatrix-1-1-JDT-prop} we get that the Jordan degree type partition of $A$ for $x$ is equal to $\mathcal{S}_{x,A}=(4_0,4_1,4_1,4_2,4_2,4_3,4_3,4_4,2_2, 2_3,2_4).$
\end{example}
Based on computations in Macaulay2, for large number of cases up to socle degree nine, we have no example of Artinian Gorenstein algebras over polynomial rings with three variables that the necessary conditions given in Lemmas \ref{diffO-seq} and \ref{additiveRank} are not sufficient. So we pose the following conjecture.
\begin{conjecture}
Let $M$ be an upper triangular matrix of size $d+1$ with non-negative entries. Then $M$ is the rank matrix of some Artinian Gorenstein algebra $A$ of codimension three and linear form $\ell\in A_1$, if and only if the following conditions are satisfied.
\begin{itemize}
\item[$(i)$] For every $0\leq i\leq d$, $\diag(i,M)$ is an O-sequences, and $h_A=\diag(0,M)$;
\item[$(ii)$] for every $0\leq i\leq d-1$, the difference vector $\diag(i,M)-\left(\diag(i+1,M)\right)_+$ is an O-sequences;
\item[$(iii)$] for any $2\times 2$ square submatrix of successive entries on and above the diagonal of $M$ of the form $\begin{pmatrix}
u&v\\
w&z\\
\end{pmatrix}$ we have that $w+v\geq u+z$.
\end{itemize}
\end{conjecture}
\section{Acknowledgment}
The author would like to thank Mats Boij for many helpful discussions and Anthony Iarrobino for his comments on the first draft of this paper. Experiments using the algebra software Macaulay2 \cite{Mac2} were essential to get the ideas behind some of the proofs. This work was supported by the grant VR2013-4545.
|
1,108,101,565,421 | arxiv | \section{Introduction}
The celebrated \emph{Cahn--Hilliard equation},
\begin{equation*}
u_t=\Delta\left(-\varepsilon^2\Delta u+F'(u)\right),
\end{equation*}
where $\varepsilon$ is a positive constant and $F:\mathbb{R}\rightarrow\mathbb{R}$ is a double well potential with wells of equal depth,
was originally proposed in \cite{Cahn-Hilliard} to model phase separation in a binary system at a fixed temperature,
with constant total density and where $u$ stands for the concentration of one of the two components.
Among the phase transformations involved in phase separation, a peculiar one is named ``spinodal decomposition",
which indicates the stage during which the mixture quickly becomes inhomogeneous, forming a fine-grained structure (cfr. \cite{Cahn,Grant2,Mai-Wan}).
In order to model the early stages of spinodal decomposition in certain glasses,
some physicists \cite{Galenko,GalenkoJou,LeZaGa} proposed the following \emph{hyperbolic relaxation of the Cahn--Hilliard equation}
\begin{equation}\label{eq:hyp-CH-multiD}
\tau u_{tt}+u_t=\Delta\left(-\varepsilon^2\Delta u+F'(u)\right),
\end{equation}
where $\tau$ is a positive constant.
In particular, the \emph{hyperbolic version} \eqref{eq:hyp-CH-multiD} has been firstly proposed by Galenko in \cite{Galenko},
following the classical Maxwell--Cattaneo modification of the Fick's diffusion law \cite{Cat}.
Many papers have been devoted to the study of the dynamics of the solutions to \eqref{eq:hyp-CH-multiD}.
Without claiming to be complete, we list the following papers:
for the long-time behavior of the solutions and the limiting behavior as $\tau\to0$ in the one-dimensional case
see \cite{Debussche,ZhengMilani2004,ZhengMilani2005,BonGraMir} and references therein;
for the multidimensional-case among others, we mention \cite{GraSchZel,GraSchZel2}.
In this paper, we are interested in studying the metastable dynamics of the solutions to the one-dimensional version of \eqref{eq:hyp-CH-multiD}
\begin{equation}\label{eq:hyp-CH}
\tau u_{tt}+u_t=\left(-\varepsilon^2u_{xx}+F'(u)\right)_{xx}, \qquad \qquad x\in(0,1),\, t>0,
\end{equation}
subject to homogeneous Neumann boundary conditions
\begin{equation}\label{eq:Neumann}
u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0, \qquad \forall\,t\geq0,
\end{equation}
and initial data
\begin{equation}\label{eq:initial}
u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), \qquad x\in[0,1].
\end{equation}
Precisely, we are interested in describing the behavior of the solutions to the initial boundary value problem \eqref{eq:hyp-CH}-\eqref{eq:Neumann}-\eqref{eq:initial}
when the parameter $\varepsilon$ is very small and the function $F\in C^4(\mathbb{R})$ satisfies
\begin{equation}\label{eq:ass-F}
F(\pm1)=F'(\pm1)=0, \qquad F''(\pm1)>0 \qquad \mbox{ and } \qquad F(u)>0, \quad \mbox{ for } \, u\neq\pm1.
\end{equation}
The simplest example of function satisfying \eqref{eq:ass-F} is $F(u)=\frac14(u^2-1)^2$.
The existence and persistence for an exponentially large time of metastable states with $N$ transitions between $-1$ and $+1$
for the IBVP \eqref{eq:hyp-CH}-\eqref{eq:initial} has been proved in \cite{FLM17pre}
by using an energy approach firstly introduced in \cite{Bron-Kohn} to study the \emph{Allen--Cahn equation}
\begin{equation}\label{eq:AC}
u_t=\varepsilon^2u_{xx}-F'(u),
\end{equation}
and subsequently used in \cite{Bron-Hilh} to prove existence of metastable states for the classic Cahn--Hilliard equation
\begin{equation}\label{eq:CH}
u_t=\left(-\varepsilon^2u_{xx}+F'(u)\right)_{xx}, \qquad \qquad x\in(0,1),\, t>0.
\end{equation}
Here, we investigate the metastable properties of the solutions to \eqref{eq:hyp-CH} by using a different approach,
the dynamical approach proposed by Carr and Pego in \cite{Carr-Pego} and Fusco and Hale \cite{Fusco-Hale} to study the Allen--Cahn equation \eqref{eq:AC}
and used for the Cahn--Hilliard equation \eqref{eq:CH} in \cite{AlikBateFusc91} and \cite{Bates-Xun1,Bates-Xun2}.
The dynamical approach gives a more precise description of the dynamics of the solution to the IBVP \eqref{eq:hyp-CH}-\eqref{eq:initial}
and allows us to derive a system of ODEs which describes the evolution of such solution.
Before presenting our results, let us briefly describe the dynamics of the solutions to the classic Cahn--Hilliard equation \eqref{eq:CH}
with homogeneous Neumann boundary conditions \eqref{eq:Neumann} and recall some previous results on the metastable behavior of the solutions.
First of all, notice that any constant function is a equilibrium solution to \eqref{eq:CH}-\eqref{eq:Neumann}
and that, by integrating the equation \eqref{eq:CH} and using the boundary conditions \eqref{eq:Neumann} one finds out that the total mass $\displaystyle\int_a^b u(x,t)\,dx$ is conserved.
A linear analysis of the equation \eqref{eq:CH} about a constant solution
shows that spatially homogeneous equilibria in the \emph{spinodal region} (where $F''<0)$ are unstable \cite{Grant2}.
Moreover, it is sufficient to take an initial datum which is a small perturbation of a fixed constant in the spinodal region and the corresponding solution exhibits the phenomenon of \emph{spinodal decomposition}:
after a relatively short time, the solution to \eqref{eq:CH}-\eqref{eq:Neumann} is approximately close to $+1$ or $-1$ (the positions of the global minimum of $F$) except near a finite number $N$ of transition layers.
The first mathematical treatment and rigorous verification of such phenomenon is performed in \cite{Grant2}.
After the spinodal decomposition, the solution, which has a $N$-\emph{transition layers structure}, evolves so slow that the profile \emph{appears} to be stable.
On the other hand, it is well-known that the Cahn--Hilliard equation \eqref{eq:CH} possesses the Lyapunov functional
\begin{equation}\label{eq:energy}
E_\varepsilon[u]=\int_a^b\left[\frac{\varepsilon^2}2u_x^2+F(u)\right]\,dx,
\end{equation}
and the solutions converge as $t\to+\infty$ to a stationary solution \cite{Zheng1986}.
The problem to minimize the energy functional \eqref{eq:energy} among all the functions satisfying $\displaystyle\int_a^b u\,dx=M$ (the total mass being conserved),
has been investigated in \cite{CarrGurtSlem} for the one-dimensional case and in \cite{Modica} for the multi-dimensional case.
In particular, in \cite{CarrGurtSlem} it has been proved that if $\varepsilon$ is small enough and $M\in(-1,1)$, then
all the minimizers are strictly monotone functions.
Therefore, the solution to \eqref{eq:CH}-\eqref{eq:Neumann} converges to a limit with a single transition and, as a consequence, we have an example of \emph{metastable dynamics}:
the solution maintains the (unstable) $N$-transitions layer structure for a very long time $T_\varepsilon$ and then converges to the asymptotic limit with a single transition.
Precisely, the evolution of the solutions depends only from the interactions between the layers, which move with an exponentially small velocity as $\varepsilon\to0$;
it follows that the lifetime $T_\varepsilon$ of a metastable state with $N$ transitions is exponentially large as $\varepsilon\to0$,
namely $T_\varepsilon=\mathcal O\left(e^{C/\varepsilon}\right)$ where $C>0$ depends only on $F$ and on the distance between the layers.
As it was previously mentioned, there are at least two different approaches to study the metastable dynamics of the solutions,
which have been proposed in the study of the Allen--Cahn equation \eqref{eq:AC}.
The energy approach of \cite{Bron-Kohn} is based on $\Gamma$-convergence properties of the functional \eqref{eq:energy} and
it has been applied to the Cahn--Hilliard equation \eqref{eq:CH} in \cite{Bron-Hilh};
it permits to handle both Neumann \eqref{eq:Neumann}
and Dirichlet boundary conditions of the type
\begin{equation}\label{eq:Dirichlet}
u(0,t)=\pm1, \quad u(1,t)=\pm1 \qquad \mbox{ and } \qquad u_{xx}(0,t)=u_{xx}(1,t)=0, \qquad \forall\,t\geq0.
\end{equation}
On the other hand, the dynamical approach of \cite{Carr-Pego,Fusco-Hale} is performed in \cite{AlikBateFusc91},
where the authors consider the case of an initial datum with a $2$-transition layer structure and in \cite{Bates-Xun1,Bates-Xun2},
where the general case of $N+1$ layers ($N\geq1$) is considered.
This approach permits to describe in details the movement of the layers.
In the two layer case, for the conservation of the mass, we have that the layers move in an almost rigid way (they move in the same direction at approximately the same \emph{exponentially small} velocity);
when the layers are more than $2$ the situation is more complicated and we will describe their dynamics in Section \ref{sec:layers}.
Each of the previous approaches has its advantages and drawbacks.
The dynamical approach gives more precise results: it gives the exact order of the speed of the slow motion, and allows us to accurately describe the movement of the layers,
but permits to study only the case of homogeneous boundary conditions and the proofs are complicated and lengthy.
The energy approach is fairly simple, it provides a rather clear and intuitive explanation for the slow motion and permits to handle both Neumann \eqref{eq:Neumann}
and Dirichlet \eqref{eq:Dirichlet} boundary conditions, but it gives only an upper bound for the velocity of the layers.
We also recall that the energy approach permits to study the vector-valued version of \eqref{eq:CH}, that is
when $u$ takes value in $\mathbb{R}^m$ and the potential $F$ vanishes only in a finite number of points (for details, see \cite{Grant}).
Finally, we mention that both the dynamical and the energy approach can be applied to study the metastability for the following hyperbolic variations of the Allen--Cahn equation
\begin{equation}\label{eq:hypAC}
\tau u_{tt}+ g(u)u_t=\varepsilon^2u_{xx}-F'(u),
\end{equation}
for any positive function $g\in C^1(\mathbb{R})$ (cfr. \cite{Folino,Folinoproc,Folino2,FLM17}).
In this paper, we apply the dynamical approach to the IBVP \eqref{eq:hyp-CH}-\eqref{eq:Neumann}-\eqref{eq:initial}.
The well-posedness and the asymptotic behavior as $t\to+\infty$ of the solutions to such IBVP are investigated in \cite{Debussche}.
A fundamental difference with respect to the classic Cahn--Hilliard equation \eqref{eq:CH} is that the homogeneous Neumann boundary conditions \eqref{eq:Neumann}
do not imply conservation of the mass;
as we will see in Section \ref{sec:prelimin} the solution to the IBVP \eqref{eq:hyp-CH}-\eqref{eq:Neumann}-\eqref{eq:initial}
conserves the mass if and only if the initial velocity $u_1$ is of zero mean.
Therefore, to apply the dynamical approach we need a further assumption on $u_1$;
however, by using the energy approach, it is possible to prove the metastable dynamics of the solutions without the assumption of zero-mean for $u_1$
(for details see \cite[Remark 2.7]{FLM17pre}).
The main idea of the dynamical approach introduced by Carr and Pego in \cite{Carr-Pego} is to construct a family of functions $u^{\bm h}$, which approximates a metastable states with $N+1$ transitions
located at $\bm h=(h_1,h_2,\dots,h_{N+1})$, consider the decomposition
\begin{equation}\label{eq:decom}
u(x,t)=u^{\bm h(t)}(x)+w(x,t),
\end{equation}
for the solution $u$ and study the evolution of the remainder function $w$ and of the transition points $h_1,h_2,\dots,h_{N+1}$.
By inserting the decomposition \eqref{eq:decom} in the equation \eqref{eq:AC} and imposing an orthogonality condition on $w$,
it is possible to derive an ODE-PDE coupled system for $(\bm h,w)$ and prove that the solution $u$ is well-approximated by $u^{\bm h}$ as $\varepsilon\to0$ and evolves very slowly
until either two transition points are close enough or a transition point is close enough to the boundary points of the interval $(0,1)$.
In other words, with the dynamical approach it is possible to prove the existence of an \emph{approximately invariant manifold} $\mathcal{M}$,
consisting of functions with $N+1$ transitions between $-1$ and $+1$:
if the initial datum $u_0$ is in a particular tubular neighborhood of $\mathcal{M}$, then the transition points move with an exponentially small velocity
and the solution remains in such neighborhood for an exponentially long time.
Then, since the remainder $w$ is very small as $\varepsilon\to0$, by using the approximation $w\approx0$ in \eqref{eq:decom}
one can derive a system of ODEs for $\bm h$, which accurately describes the movement of the layers, and so the evolution of the solution $u$,
until the transition points are well-separated and far away from $0$ and $1$.
This strategy has been applied to the integrated version of \eqref{eq:hyp-CH}-\eqref{eq:Neumann} in \cite{Bates-Xun1,Bates-Xun2}
and gives a precise description of the metastable dynamics of the solutions.
In the following, we will show how to adapt this strategy to the hyperbolic version \eqref{eq:hyp-CH} and we will analyze the differences with respect to \eqref{eq:CH}.
In particular, we will prove the existence of an approximately invariant manifold $\mathcal{M}_0$ contained in a narrow channel for the initial boundary problem \eqref{eq:hyp-CH}-\eqref{eq:Neumann}-\eqref{eq:initial}:
if the initial datum \eqref{eq:initial} is in the channel, then the solution $u$ remains in the channel for an exponentially long time.
Moreover, in the channel the following estimates hold:
\begin{equation}\label{eq:estimates-intro}
\|u-u^{\bm h}\|_{{}_{L^\infty}}\leq C\varepsilon^{-5/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right), \qquad \qquad |\bm h'|_{{}_\infty}\leq C\varepsilon^{-2}\tau^{-1/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right),
\end{equation}
where $A:=\sqrt{\min\{F''(-1),F''(+1)\}}$, $\ell^{\bm h}:=\min\{h_j-h_{j-1}\}$ and $|\cdot|_{{}_{\infty}}$ denotes the maximum norm in $\mathbb{R}^N$.
Furthermore, we will derive the following system of ODEs
\begin{equation}\label{eq:ODE-intro}
\tau \bm h''+\bm h'+\tau\bm{\mathcal{Q}}(\bm h,\bm h')=\bm{\mathcal P}(\bm h),
\end{equation}
which describes the movement of the transition layers and has to be compared with the system $\bm h'=\bm{\mathcal P}(\bm h)$,
which describes the dynamics of the solutions to the classic Cahn-Hilliard equation;
for the formulas of $\bm{\mathcal P}$ and $\bm{\mathcal{Q}}$ see Section \ref{sec:layers}.
The rest of the paper is organized as follows.
In Section \ref{sec:prelimin} we give all the definitions, the preliminaries and we construct the approximate invariant manifold $\mathcal{M}_0$
following the ideas of \cite{Carr-Pego}, \cite{Bates-Xun1}.
Section \ref{sec:slow} contains the main result of the paper, Theorem \ref{thm:main}, where we prove that the manifold $\mathcal{M}_0$
is approximately invariant for \eqref{eq:hyp-CH}-\eqref{eq:Neumann}, by constructing a slow channel which contains $\mathcal{M}_0$ and where
the solutions stay for an exponentially long time and satisfy \eqref{eq:estimates-intro}.
Finally, Section \ref{sec:layers} is devoted to the description of the movement of the layers.
We will derive the system of ODEs \eqref{eq:ODE-intro} and we will analyze the differences between the classic Cahn--Hilliard equation \eqref{eq:CH} and its hyperbolic relaxation \eqref{eq:hyp-CH}.
\section{Preliminaries}\label{sec:prelimin}
In this section we collect some results of \cite{Carr-Pego}, \cite{Bates-Xun1,Bates-Xun2} we will use later
and we introduce the \emph{extended base manifold} $\mathcal{M}_0$, which is, as we shall prove in Section \ref{sec:slow},
approximately invariant for the boundary value problem \eqref{eq:hyp-CH}-\eqref{eq:Neumann}.
\subsection{Approximate metastable states}
The aim of this subsection is to construct a family of functions with $N+1$ transitions between $-1$ and $+1$, approximating metastable states for \eqref{eq:hyp-CH}.
Such construction was firstly introduced by Carr and Pego \cite{Carr-Pego} to describe the metastable dynamics of the solutions to the Allen--Cahn equation \eqref{eq:AC},
and then it has also been used to study the metastability for the Cahn--Hilliard equation \cite{Bates-Xun1,Bates-Xun2}
and for hyperbolic variants of the Allen--Cahn equation \cite{FLM17}.
Here, we briefly recall the construction of the family and some useful properties we will use later, for details see \cite{Carr-Pego}.
For fixed $\rho>0$, we introduce the set
\begin{equation*}
\Omega_\rho:=\bigl\{{\bm h}\in\mathbb{R}^{N+1}\, :\,0<h_1<\cdots<h_{N+1}<1,\quad
h_j-h_{j-1}>\varepsilon/\rho\mbox{ for } j=1,\dots,N+2\bigr\},
\end{equation*}
where we define $h_0:=-h_1$ and $h_{N+2}:=2-h_{N+1}$, because of the homogeneous Neumann boundary conditions \eqref{eq:Neumann}.
In what follows, we fix a minimal distance $\delta\in(0,1/(N+1))$ and we consider the parameters $\varepsilon$ and $\rho$ such that
\begin{equation}\label{eq:triangle}
0<\varepsilon<\varepsilon_0\qquad\textrm{and}\qquad \delta<\frac{\varepsilon}{\rho}<\frac{1}{N+1},
\end{equation}
for some $\varepsilon_0>0$ to be chosen appropriately small.
We associate to any $\bm h\in\Omega_\rho$ a function $u^{\bm h}=u^{\bm h}(x)$ which approximates a metastable
state with $N+1$ transition points located at $h_1,\dots,h_{N+1}$.
To do this, we make use of the solutions to the following boundary value problem:
given $\ell>0$, let $\phi(\cdot,\ell,+1)$ be the solution to
\begin{equation}\label{eq:fi}
\mathcal{L}^{AC}(\phi):=-\varepsilon^2\phi_{xx}+F'(\phi)=0, \qquad \quad
\phi\bigl(-\tfrac12\ell\bigr)=\phi\bigl(\tfrac12\ell\bigr)=0,
\end{equation}
with $\phi>0$ in $(-\tfrac12\ell,\tfrac12\ell)$,
and $\phi(\cdot,\ell,-1)$ the solution to \eqref{eq:fi} with $\phi<0$ in $(-\tfrac12\ell,\tfrac12\ell)$.
The functions $\phi(\cdot,\ell,\pm1)$ are well-defined if $\ell/\varepsilon$ is sufficiently large, and they depend on $\varepsilon$
and $\ell$ only through the ratio $\varepsilon/\ell$.
Moreover, we have
\begin{equation*}
\max_x|\phi(\cdot,\ell,\pm1)|=|\phi(0,\ell,\pm1)|=M_\pm(\ell/\varepsilon)
\qquad \quad \textrm{and} \qquad \quad
\max_x|\phi_x(\cdot,\ell,\pm1)|\leq C\varepsilon^{-1},
\end{equation*}
where $C>0$ is a constant depending only on the function $F$.
In particular, $M_\pm$ tends to $+1$ as $\varepsilon/\ell\to 0$ (more details in Proposition \ref{prop:alfa,beta}).
The family of the approximate metastable states is constructed by matching together
the functions $\phi(\cdot,\ell,\pm1)$, using smooth cut-off functions:
given $\chi:\mathbb{R}\rightarrow[0,1]$ a $C^\infty$-function with $\chi(x)=0$ for $x\leq-1$ and $\chi(x)=1$ for $x\geq1$, set
\begin{equation*}
\chi^j(x):=\chi\left(\frac{x-h_j}\varepsilon\right) \qquad\textrm{and}\qquad
\phi^j(x):=\phi\left(x-h_{j-1/2},h_j-h_{j-1},(-1)^j\right),
\end{equation*}
where
\begin{equation*}
h_{j+1/2}:=\tfrac12(h_j+h_{j+1})\qquad j=0,\dots,N+1,
\end{equation*}
(note that $h_{1/2}=0$, $h_{N+3/2}=1$).
Then, we define the function $u^{\bm h}$ as
\begin{equation}\label{eq:uh}
u^{\bm h}:=\left(1-\chi^j\right)\phi^j+\chi^j\phi^{j+1} \qquad \textrm{in}\quad I_j:=[h_{j-1/2},h_{j+1/2}],
\end{equation}
for $j=1,\dots,N+1$, and the manifold
\begin{equation*}
\mathcal{M}^{AC}:=\{u^{\bm h} :\bm h\in\Omega_\rho\}.
\end{equation*}
In \cite{Carr-Pego}, the authors show that the manifold $\mathcal{M}^{AC}$ is approximately invariant for the Allen--Cahn equation \eqref{eq:AC}.
On the other hand, the \emph{extended} manifold
\begin{equation*}
\mathcal{M}^{AC}_{{}_0}:=\mathcal{M}^{AC}\times\{0\}=\{(u^{\bm h},0) :u^{\bm h}\in\mathcal{M}^{AC}\}
\end{equation*}
is approximately invariant for the hyperbolic variant \eqref{eq:hypAC}, see \cite{FLM17}.
To get an idea of the structure of the function $u^{\bm h}$ defined in \eqref{eq:uh},
we recall that, if $\rho>0$ is sufficiently small and $\bm h\in\Omega_\rho$, then $u^{\bm h}\approx\pm1$ away from $h_j$ for $j=1,\dots,N+1$,
and $u^{\bm h}(x)\approx\Phi\left((x-h_j)(-1)^{j-1}\right)$ for $x$ near $h_j$, where $\Phi$ is the unique solution to the problem
\begin{equation*}
\mathcal{L}^{AC}(\Phi):=-\varepsilon^2\Phi_{xx}+F'(\Phi)=0, \qquad \qquad \lim_{x\rightarrow\pm\infty} \Phi(x)=\pm\infty, \qquad\qquad \Phi(0)=0.
\end{equation*}
For instance, in the case $F(u)=\frac14(u^2-1)^2$, the unique solution is $\Phi(x)=\tanh(x/\sqrt2\varepsilon)$.
In conclusion, we say that $u^{\bm h}$ is a smooth function of $\bm h$ and $x$,
which is approximately $\pm1$ except near $N+1$ transition points located at $h_1,\cdots,h_{N+1}$;
moreover, $\mathcal{L}^{AC}(u^{\bm h})=0$ except in an $\varepsilon$--neighborhood of the transition points $h_j$.
Precisely, we have
\begin{equation}\label{eq:prop-uh}
\begin{aligned}
u^{\bm h}(0)&=\phi(0,2h_1,-1)<0,
&\qquad u^{\bm h}(h_{j+1/2})&=\phi\left(0,h_{j+1}-h_j,(-1)^{j+1}\right),\\
u^{\bm h}(h_j)&=0,
&\qquad \mathcal{L}^{AC}(u^{\bm h}(x))&=0\quad \textrm{for }|x-h_j|\geq\varepsilon,
\end{aligned}
\end{equation}
for any $j=1,\dots,N+1$.
Central to the study of the metastable dynamics of the solutions to both the Allen--Cahn and the Cahn--Hilliard equation is an accurate characterization of the
quantities $u^{\bm h}(h_{j+1/2})=\phi\left(0,h_{j+1}-h_j,(-1)^{j+1}\right)$ and $F\left(u^{\bm h}(h_{j+1/2})\right)$,
because the motion of the transition points $h_1,\dots,h_{N+1}$ depend essentially on these quantities.
Since $\phi(0,\ell,\pm1)$ depends only on the ratio $r=\varepsilon/\ell$, we can define
\begin{equation*}
\alpha_\pm(r):=F(\phi(0,\ell,\pm1)), \qquad \quad \beta_\pm(r):=1\mp\phi(0,\ell,\pm1).
\end{equation*}
By definition, $\phi(0,\ell,\pm1)$ is close to $+1$ or $-1$ and so, $\alpha_\pm(r), \beta_\pm(r)$ are close to $0$.
The next result characterizes the leading terms in $\alpha_\pm$ and $\beta_\pm$ as $r\to 0$.
\begin{prop} [Carr--Pego \cite{Carr-Pego}] \label{prop:alfa,beta}
Let $F$ be such that \eqref{eq:ass-F} holds and set
\begin{equation*}
A_\pm^2:=F''(\pm1), \qquad K_{\pm}=2\exp\left\{\int_0^1\left(\frac{A_\pm}{(2F(\pm t))^{1/2}}-\frac{1}{1-t}\right)\,dt\right\}.
\end{equation*}
There exists $r_0>0$ such that if $0<r<r_0$, then
\begin{equation*}
\begin{aligned}
\alpha_\pm(r)&=\tfrac12K^2_\pm A^2_\pm\,\exp(-{A_\pm}/r\bigr)\bigl\{1+O\left(r^{-1} \exp(-{A_\pm}/2r)\right)\bigr\},\\
\beta_\pm(r)&=K_\pm\,\exp\bigl(-{A_\pm}/2r\bigr)\bigl\{1+O\left(r^{-1} \exp(-{A_\pm}/2r)\right)\bigr\},
\end{aligned}
\end{equation*}
with corresponding asymptotic formulae for the derivatives of $\alpha_\pm$ and $\beta_\pm$.
\end{prop}
For $j=1,\dots,N+1$, we set
\begin{equation*}
l_j:=h_{j+1}-h_{j}, \qquad \qquad r_{j}:=\frac{\varepsilon}{l_j},
\end{equation*}
and
\begin{equation*}
\alpha^{j}:=\left\{\begin{aligned}
&\alpha_+(r_{j}) &j \textrm{ odd},\\
&\alpha_-(r_{j}) &j \textrm{ even},\\
\end{aligned}\right.
\qquad
\beta^{j}:=\left\{\begin{aligned}
&\beta_+(r_{j}) &j \textrm{ odd},\\
&\beta_-(r_{j}) &j \textrm{ even}.\\
\end{aligned}\right.
\end{equation*}
\begin{rem}\label{rem:alfa}
Let $\bm h\in\Omega_\rho$ with $\varepsilon,\rho$ satisfying \eqref{eq:triangle} and let $l^{\bm h}:=\min\{h_j-h_{j-1}\}$.
Then, the quantities $\alpha^j$ and $\beta^j$ are exponentially small in $\varepsilon$, namely there exists $C>0$ (independent of $\varepsilon$) such that
\begin{align}
0<\alpha^j\leq C\exp\left(-\frac{Al_j}{\varepsilon}\right)\leq C\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right), \label{eq:alfaj}\\
0<\beta^j\leq C\exp\left(-\frac{Al_j}{2\varepsilon}\right)\leq C\exp\left(-\frac{Al^{\bm h}}{2\varepsilon}\right), \label{eq:betaj}
\end{align}
where $A:=\sqrt{\min\{F''(-1),F''(+1)\}}$.
Moreover, assuming that $F$ is an even function and so that $\alpha_+\equiv\alpha_-$,
from Proposition \ref{prop:alfa,beta} we get
\begin{equation*}
\frac{\alpha^j}{\alpha^i}\leq C\exp\left(-\frac{A}{\varepsilon}(l_j-l_i)\right),
\end{equation*}
for some $C>0$.
Hence, if $l_j-l_i\geq \kappa$ for some $\kappa>0$, we deduce
\begin{equation}\label{eq:alfa^j-alfa^i}
\alpha^j\leq C\exp\left(-\frac{A\kappa}{\varepsilon}\right)\alpha^i.
\end{equation}
Therefore, if $l_j>l_i$ then $\alpha^j<\alpha^i$, and for $\varepsilon/\kappa\ll1$, $\alpha^j$ is \emph{exponentially small} with respect to $\alpha^i$.
\end{rem}
Now, let us introduce the \emph{barrier function}
\begin{equation}\label{eq:barrier}
\Psi(\bm h):=\sum_{j=1}^{N+1}{\langle\mathcal{L}^{AC}\bigl(u^{\bm h}\bigr),k^{\bm h}_j\rangle}^2=\sum_{j=1}^{N+1}\bigl(\alpha^{j+1}-\alpha^{j}\bigr)^2,
\end{equation}
where $\mathcal{L}^{AC}$ is the Allen--Cahn differential operator introduced above and the functions $k^{\bm h}_j$ are defined by
\begin{equation*}
k^{\bm h}_j(x):=-\gamma^j(x)u^{\bm h}_x(x), \qquad \mbox{ with } \;
\gamma^j(x):=\chi\left(\frac{x-h_{j}-\varepsilon}\varepsilon\right)\left[1-\chi\left(\frac{x-h_{j+1}+\varepsilon}\varepsilon\right)\right].
\end{equation*}
By construction, $k^{\bm h}_j$ are smooth functions of $x$ and $\bm h$ and are such that
\begin{equation*}
\begin{aligned}
k^{\bm h}_j(x)&=0 &\quad \textrm{for}\quad &x\notin[h_{j-1/2},h_{j+1/2}],\\
k^{\bm h}_j(x)&=-u^{\bm h}_x(x) &\quad \textrm{for}\quad &x\in[h_{j-1/2}+2\varepsilon,h_{j+1/2}-2\varepsilon].
\end{aligned}
\end{equation*}
Such functions are fundamental in the study of the metastability for the Allen--Cahn equation \eqref{eq:AC} and for the hyperbolic Allen--Cahn equation \eqref{eq:hypAC} (see \cite{Carr-Pego} and \cite{FLM17}, respectively),
and play a crucial role in the study of the metastability for the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH}.
We recall that there exists $C>0$ independent of $\varepsilon$ such that
\begin{equation}\label{eq:ineq-k}
\|k_j^{\bm h}\|+\varepsilon\|k^{\bm h}_{ij}\|\leq C\varepsilon^{-1/2}, \qquad \qquad \mbox{where } \quad
k^{\bm h}_{ji}:=\partial_{h_i} k^{\bm h}_j.
\end{equation}
For the proof of \eqref{eq:ineq-k} see \cite[Proposition 2.3]{Carr-Pego}.
In conclusion, we collect some useful properties of the derivative of $u^{\bm h}$ with respect to $h_j$ we will use later;
we will use the notation
\begin{equation*}
u^{\bm h}_j:=\frac{\partial u^{\bm h}}{\partial h_j},
\qquad \quad u^{\bm h}_{ji}:=\frac{\partial^2 u^{\bm h}}{\partial h_j\partial h_i}.
\end{equation*}
\begin{lem}\label{lem:u^h_j}
The interval $[h_{j-1}-\varepsilon,h_{j+1}+\varepsilon]$ contains the support of $u^{\bm h}_j$ and
\begin{equation*}
u^{\bm h}_j(x)=\begin{cases}
\mathcal{O}\left(\varepsilon^{-1}\beta^{j-1}\right), & x\in[h_{j-1}+\varepsilon, h_{j-1/2}], \\
-u^{\bm h}_x(x)+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^{j})\right), \qquad & x\in I_j,\\
\mathcal{O}\left(\varepsilon^{-1}\beta^{j}\right), & x\in [h_{j+1/2},h_{j+1}+\varepsilon], \\
0, & \mbox{otherwise},
\end{cases}
\end{equation*}
for $j=1,\dots,N+1$.
Moreover, there exists $C>0$ such that
\begin{equation}\label{eq:uh_j}
\varepsilon\|u^{\bm h}_j\|_{{}_{L^\infty}}+\varepsilon^{1/2} \|u_j^{\bm h}\|\leq C, \qquad \qquad j=1,\dots,N+1.
\end{equation}
\end{lem}
For the precise formula for $u^{\bm h}_j$ and the proof of Lemma \ref{lem:u^h_j} see \cite[Sections 7-8]{Carr-Pego}.
Thanks to Lemma \ref{lem:u^h_j}, we can state that if we neglect the exponentially small terms, then $u^{\bm h}_j$ is equal to $-u^{\bm h}_x$ in $I_j$ and it is zero for $x\notin I_j$.
We will use such approximation in Section \ref{sec:layers} to derive the ODE describing the motion of the transition layers $h_1,\dots,h_{N+1}$.
\subsection{Base Manifold}
In this subsection we define the base manifold for the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH}.
Integrating the equation \eqref{eq:hyp-CH} in $[0,1]$ and using the homogeneous Neumann boundary conditions \eqref{eq:Neumann},
we obtain that the total mass $m(t):=\displaystyle\int_0^1u(y,t)\,dy$ satisfies the ODE
\begin{equation*}
\tau m''(t)+m'(t)=0, \qquad m(0)=\int_0^1 u_0(y)\,dy, \quad m'(0)=\int_0^1u_1(y)\,dy.
\end{equation*}
Then, as a trivial consequence, $m(t)=m(0)+\tau m'(0)(1-e^{-t/\tau})$ and the total mass $m$ is conserved if and only if
\begin{equation}\label{eq:ass-u1}
\int_0^1 u_1(y)\,dy=0.
\end{equation}
From now on, we will assume that the initial velocity satisfies \eqref{eq:ass-u1} in order to have conservation of the mass,
and we also assume that the initial profile $u_0$ has mass equal to $M$, for some $M\in(-1,1)$.
It follows that
\begin{equation}\label{eq:conservation}
m(t)=\int_0^1 u_0(y)\,dy=M\in(-1,1), \qquad \qquad \mbox{ for any } t\geq0.
\end{equation}
Since the total mass is conserved, we introduce the manifold
\begin{equation*}
\mathcal{M}^{CH}:=\left\{u^{\bm h}\in \mathcal{M}^{AC}: \displaystyle\int_0^1u^{\bm h}(x)\,dx=M\right\}.
\end{equation*}
In \cite{Bates-Xun1,Bates-Xun2}, the authors study the dynamics of the solutions to the Cahn--Hilliard equation \eqref{eq:CH} in a neighborhood of $\mathcal{M}^{CH}$
and show that such manifold is approximately invariant for \eqref{eq:CH}.
In this paper, we will show that the \emph{extended base manifold}
\begin{equation*}
\mathcal{M}^{CH}_{{}_{0}}:=\mathcal{M}^{CH}\times\{0\}=\left\{(u^{\bm h},0)\,: \, u^{\bm h}\in\mathcal{M}^{CH} \right\}
\end{equation*}
is approximately invariant for the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH}.
From now on, we drop the superscript $CH$ and we use the notation $\mathcal{M}^{CH}=\mathcal{M}$ and $\mathcal{M}^{CH}_{{}_{0}}=\mathcal{M}_{{}_{0}}$.
The following lemma of \cite{Bates-Xun1} is crucial in the study of the metastable dynamics of \eqref{eq:hyp-CH}
in a neighborhood of $\mathcal{M}_{{}_{0}}$. For reader's convenience, we report here below its proof.
\begin{lem}\label{lem:M(h)}
Let $M(\bm h):=\displaystyle\int_0^1u^{\bm h}(x)\,dx$, for $\bm h\in\Omega_\rho$.
Then, $M(\bm h)$ is a smooth function of $\bm h$ and
\begin{equation*}
\frac{\partial M}{\partial h_j}=2(-1)^{j}+ \mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^j)\right).
\end{equation*}
\end{lem}
\begin{proof}
By differentiating the function $M(\bm h)$ with respect to $h_j$, and by using Lemma \ref{lem:u^h_j}, we infer
\begin{align*}
\frac{\partial M}{\partial h_j}&=\int_0^1u^{\bm h}_j(x)\,dx=-\int_{I_j}u^{\bm h}_x(x)\,dx+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^{j})\right)\\
&=u^{\bm h}(h_{j-1/2})-u^{\bm h}(h_{j+1/2})+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^{j})\right).
\end{align*}
for $j=1,\dots,N+1$.
From \eqref{eq:prop-uh} and the definition of $\beta^j$, it follows that
\begin{equation}\label{eq:uh(h_1/2)}
u^{\bm h}(h_{j+1/2})=(-1)^{j+1}+(-1)^{j}\beta^j, \qquad \qquad j=0,\dots,N+1.
\end{equation}
Therefore, we can conclude that
\begin{equation*}
\frac{\partial M}{\partial h_j}=2(-1)^j+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^{j})\right),
\end{equation*}
and the proof is complete.
\end{proof}
The previous lemma shows that the manifold $\mathcal{M}$ can be parameterized by the first $N$ components $(h_1,\dots, h_N)$ of $\bm h$.
Indeed, if $u^{\bm h}\in\mathcal{M}$, applying Lemma \ref{lem:M(h)} and the implicit function theorem,
we can think $h_{N+1}$ as a function of $(h_1,\dots, h_N)$, namely there exists $g:\mathbb{R}^N\rightarrow\mathbb R$ such that
\begin{equation*}
h_{N+1}=g(h_1,\dots,h_N),
\end{equation*}
and we have
\begin{equation}\label{eq:N+1-der}
\begin{aligned}
\frac{\partial h_{N+1}}{\partial h_j}&=-\frac{\partial M/\partial h_j}{\partial M/\partial h_{N+1}}=
-\frac{2(-1)^{j}+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{j-1},\beta^j)\right)}{2(-1)^{N+1}+\mathcal{O}\left(\varepsilon^{-1}\max(\beta^{N},\beta^{N+1})\right)}\\
&=(-1)^{N-j}+\mathcal{O}\left(\varepsilon^{-1}\exp\left(-\frac{Al^{\bm h}}\varepsilon\right)\right),
\end{aligned}
\end{equation}
where we used \eqref{eq:betaj}.
Hence, we introduce the new variable $\bm\xi$, consisting of the first $N$ components of $\bm h$, and we will denote $u^{\bm h}\in\mathcal{M}$ by $u^{\bm\xi}$.
Moreover, we denote by $\mathbf{G}:\mathbb{R}^N\rightarrow\mathbb{R}^{N+1}$ the function $\mathbf{G}(\bm\xi)=(\xi_1,\dots,\xi_N,g(\xi_1,\dots,\xi_N))$,
and in the following we will interchangeably use $\bm\xi$ and $\bm h$, meaning $\bm h=\mathbf{G}(\bm\xi)$.
Finally, we have that
\begin{equation*}
u^{\bm\xi}_j:=\frac{\partial u^{\bm\xi}}{\partial\xi_j}=\frac{\partial u^{\bm h}}{\partial h_j}+\frac{\partial u^{\bm h}}{\partial h_{N+1}}\frac{\partial h_{N+1}}{\partial h_j},
\end{equation*}
for $j=1,\dots,N$, and using \eqref{eq:uh_j} we get
\begin{equation}\label{eq:uxi_j}
\varepsilon\|u^{\bm\xi}_j\|_{{}_{L^\infty}}+\varepsilon^{1/2} \|u_j^{\bm\xi}\|\leq C, \qquad \qquad j=1,\dots,N.
\end{equation}
Following the previous works \cite{AlikBateFusc91, Bates-Xun1,Bates-Xun2} on the metastability for the classic Cahn--Hilliard equation \eqref{eq:CH},
we will consider an integrated version of \eqref{eq:hyp-CH}.
If $u$ is a solution to \eqref{eq:hyp-CH} with homogeneous Neumann boundary conditions \eqref{eq:Neumann} and initial data \eqref{eq:initial}, with $u_1$ satisfying \eqref{eq:ass-u1},
then $\tilde{u}(x,t):=\displaystyle\int_0^x u(y,t)\,dy$ satisfies the integrated hyperbolic Cahn--Hilliard equation
\begin{equation}\label{eq:integrated-CH}
\tau \tilde{u}_{tt}+\tilde{u}_t=-\varepsilon^2\tilde{u}_{xxxx}+F'(\tilde{u}_x)_x, \qquad \qquad x\in(0,1), \; t>0,
\end{equation}
with initial data
\begin{equation*}
\tilde{u}(x,0)=\tilde{u}_0(x), \qquad \tilde{u}_t(x,0)=\tilde{u}_1(x), \qquad \qquad x\in[0,1],
\end{equation*}
and Dirichlet boundary conditions
\begin{equation}\label{eq:bound-ut}
\tilde{u}(0,t)=0, \quad \tilde{u}(1,t)=M, \qquad \tilde{u}_{xx}(0,t)=\tilde{u}_{xx}(1,t)=0, \qquad \forall\,t\geq0,
\end{equation}
where $M\in(-1,1)$ is the total mass of the solution.
Here and in all the paper we use the following notation:
given a function $u:[0,1]\rightarrow\mathbb{R}$, we denote by $\tilde{u}:[0,1]\rightarrow\mathbb{R}$ the function
\begin{equation*}
\tilde{u}(x):=\int_0^x u(y)\,dy.
\end{equation*}
Rewrite \eqref{eq:integrated-CH} as the system
\begin{equation}\label{eq:integrated-CH-sys}
\begin{cases}
\tilde{u}_t=\tilde{v},\\
\tau \tilde{v}_t=\mathcal{L}(\tilde{u})-\tilde{v},
\end{cases}
\end{equation}
where we introduced the integrated Cahn--Hilliard differential operator
$$
\mathcal{L}(\tilde{u}):=-\varepsilon^2 \tilde{u}_{xxxx}+\left(F'(\tilde{u}_x)\right)_x.
$$
Observe that
\begin{equation}\label{eq:AC-CH}
\mathcal{L}(\tilde{u})=-\frac{d}{dx}\mathcal{L}^{AC}(u),
\end{equation}
where $\mathcal{L}^{AC}(u):=\varepsilon^2 u_{xx}-F'(u)$ is the Allen--Cahn differential operator introduced in the previous subsection.
Since $\mathcal{L}^{AC}(u^{\bm h})=0$ except in an $\varepsilon$--neighborhood of the transition points $h_j$ (see \eqref{eq:prop-uh}),
we have that the same property holds for $\mathcal{L}(\tilde{u}^{\bm\xi})$, namely
\begin{equation*}
\mathcal{L}(\tilde{u}^{\bm\xi}(x))=0, \qquad \qquad \mbox{ for } |x-h_j|>\varepsilon, \qquad j=1,\dots,N+1.
\end{equation*}
More precisely, one can prove that (see \cite[Lemma 5.1]{Bates-Xun1})
\begin{equation}\label{eq:Lut^xi}
\|\mathcal{L}(\tilde{u}^{\bm\xi})\|\leq C\varepsilon^{-1}\sum_{j=1}^{N+1}|\alpha^{j+1}-\alpha^j|\leq C\varepsilon^{-1}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right),
\end{equation}
for some positive constant $C$.
Hence, the $L^2$--norm of $\mathcal{L}(\tilde{u}^{\bm\xi})$ is exponentially small in $\varepsilon$.
We will study the dynamics of the solutions to \eqref{eq:hyp-CH}-\eqref{eq:Neumann} in a neighborhood of $\mathcal{M}_{{}_{0}}$ by considering the integrated version \eqref{eq:integrated-CH-sys}
and using the decomposition $\tilde{u}=\tilde{u}^{\bm\xi}+\tilde{w}$, where $u^{\bm\xi}\in\mathcal{M}^{CH}$ is defined in \eqref{eq:uh} and $\tilde{w}\in H$ for
\begin{equation}\label{eq:H}
H:=\left\{\tilde{w}\in H^4(0,1)\, : \, \tilde{w}=\tilde{w}_{xx}=0 \, \mbox{ at } x=0,1 \mbox{ and } \; \langle\tilde{w},E_j^{\bm\xi}\rangle=0, \; \mbox{ for } j=1,\dots,N \right\},
\end{equation}
where $E_j^{\bm\xi}$ are linear functions of $\tilde{u}^{\bm h}_j$ and $\tilde{u}^{\bm h}_{j+1}$ to be determined later.
By using the formula of Lemma \ref{lem:u^h_j}, Proposition \ref{prop:alfa,beta} and Remark \ref{rem:alfa}, we obtain
\begin{equation*}
\tilde{u}_j^{\bm h}(x):=\int_0^xu^{\bm h}_j(y)\,dy=\begin{cases}
0, & x\leq h_{j-3/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{j-1},\\
-u^{\bm h}(x)+u^{\bm h}(h_{j-1/2})+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_j,\\
-u^{\bm h}(h_{j+1/2})+u^{\bm h}(h_{j-1/2})+\mathcal{O}(e^{-c/\varepsilon}), & x\geq h_{j+1/2},
\end{cases}
\end{equation*}
for $j=1,\dots, N$, and
\begin{equation*}
\tilde{u}_{N+1}^{\bm h}(x):=\int_0^xu^{\bm h}_{N+1}(y)\,dy=\begin{cases}
0, & x\leq h_{N-1/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{N},\\
-u^{\bm h}(x)+u^{\bm h}(h_{N+1/2})+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_{N+1}.
\end{cases}
\end{equation*}
Here and in what follows $c$ is a generic positive constant independent on $\varepsilon$.
Using \eqref{eq:uh(h_1/2)}, we deduce
\begin{equation}\label{eq:ut-j-h}
\tilde{u}_j^{\bm h}(x)=\begin{cases}
0, & x\leq h_{j-3/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{j-1},\\
-u^{\bm h}(x)+(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_j,\\
2(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), & x\geq h_{j+1/2},
\end{cases}
\end{equation}
for $j=1,\dots, N$, and
\begin{equation}\label{eq:ut-N+1-h}
\tilde{u}_{N+1}^{\bm h}(x)=\begin{cases}
0, & x\leq h_{N-1/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{N},\\
-u^{\bm h}(x)+(-1)^{N+1}+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_{N+1}.
\end{cases}
\end{equation}
Since $\tilde{u}^{\bm\xi}_j=\tilde{u}_j^{\bm h}+\tilde{u}_{N+1}^{\bm h}\frac{\partial h_{N+1}}{\partial h_j}$, for \eqref{eq:N+1-der}, \eqref{eq:ut-j-h} and \eqref{eq:ut-N+1-h}, we get
\begin{equation}\label{eq:ut-j-xi}
\tilde{u}_j^{\bm \xi}(x)=\begin{cases}
0, & x\leq h_{j-3/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{j-1},\\
-u^{\bm\xi}(x)+(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), & x\in I_j,\\
2(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), & x\in[h_{j+1/2},h_{N+1/2}],\\
-u^{\bm\xi}(x)(-1)^{N-j}+(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), \qquad & x\in I_{N+1},
\end{cases}
\end{equation}
for $j=1,\dots,N$.
Let $\omega_j:=\tilde{u}_j^{\bm h}+\tilde{u}_{j+1}^{\bm h}$, $j=1,\dots,N$; one has
\begin{equation}\label{eq:omega_j}
\omega_j(x)=\begin{cases}
0, & x\leq h_{j-3/2},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\in I_{j-1},\\
-u^{\bm\xi}(x)+(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_j\cup I_{j+1},\\
\mathcal{O}(e^{-c/\varepsilon}), & x\geq h_{j+3/2}.
\end{cases}
\end{equation}
Then, the functions $\omega_j$ are either zero or exponentially small outside of $I_j\cup I_{j+1}$.
Now, we can define the functions $E^{\bm\xi}_j$ introduced above:
\begin{equation}\label{eq:Ej-Qj}
E^{\bm\xi}_j(x):=\omega_j(x)-Q_j(x),
\end{equation}
where
\begin{equation*}
Q_j(x):=\left(-\tfrac{1}{6}x^3+\tfrac12x^2-\tfrac13x\right)\omega''_j(0)+\tfrac16(x^3-x)\omega''_j(1)+x\omega_j(1).
\end{equation*}
As it was shown in \cite[Section 3, formula (54)]{Bates-Xun1}, the terms $\omega''_j(0)$, $\omega''_j(1)$ and $\omega_j(1)$ are exponentially small as $\varepsilon\to0^+$.
Hence, $Q_j$ are exponentially small functions introduced so that $E_j^{\bm\xi}$ satisfies
\begin{equation*}
E^{\bm\xi}_j(x)=E^{\bm\xi}_{jxx}(x)=0, \qquad \mbox{ for } x=0,1, \qquad j=1,\dots,N.
\end{equation*}
The functions $E_j^{\bm\xi}$ are good approximations of the first $N$ eigenfunctions of the eigenvalue problem
\begin{align*}
L^{\bm\xi} u:=&-\varepsilon^2u_{xxxx}+\left(F''(u^{\bm\xi})u_x\right)_x=\lambda u, \qquad & \mbox{ in } (0,1),\\
&u(x)=u''(x)=0, \qquad & \mbox{ for } x=0,1,
\end{align*}
where $L^{\bm\xi}$ is the linearized operator of $\mathcal{L}$ about $\tilde{u}^{\bm\xi}$.
Indeed, in \cite{Bates-Xun1} it is proved that $L^{\bm\xi}$ has $N$ exponentially small eigenvalues and that all the others are bounded away from zero uniformly in $\varepsilon$ (see \cite[Theorem A]{Bates-Xun1}).
From \eqref{eq:omega_j} and the fact that $Q_j$ are exponentially small functions, we obtain
\begin{equation}\label{eq:Ej}
E^{\bm\xi}_j(x)=\begin{cases}
-u^{\bm\xi}(x)+(-1)^j+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad & x\in I_j\cup I_{j+1},\\
\mathcal{O}(e^{-c/\varepsilon}), & \mbox{otherwise},
\end{cases}
\end{equation}
for $i=j,\dots,N$.
We conclude this section recalling that the existence of the coordinate system $\tilde{u}=\tilde{u}^{\bm\xi}+\tilde{w}$ with $\tilde{w}\in H$ in a neighborhood of $\mathcal{M}$
has been proved in \cite[Theorem A.7]{Bates-Xun2}.
For Lemma \ref{lem:M(h)} and the subsequent comments, in the following result we can use $\bm\xi$ and $\bm h$ interchangeably.
\begin{thm}\label{thm:existence-coord}
There exists $\rho_0>0$ such that if $\rho\in(0,\rho_0)$ and $\tilde{u}$ satisfies
\begin{equation*}
\tilde{u}(0)=\tilde{u}_{xx}(0)=\tilde{u}_{xx}(1)=0, \quad \tilde{u}(1)=M, \qquad \mbox{and } \qquad \|\tilde{u}-\tilde{u}^{\bm k}\|_{{}_{L^\infty}}\leq \varepsilon^2
\end{equation*}
for some $\bm k\in\Omega_\rho$, then there is a unique $\bar{\bm{h}}\in\Omega_\rho$ such that
\begin{equation*}
\tilde{u}=\tilde{u}^{\bar{\bm{h}}}+\tilde{w}, \qquad \mbox{ with } \qquad \langle\tilde{w},E_j^{\bm\xi}\rangle=0, \;\; j=1,\dots,N.
\end{equation*}
Moreover, if $\|\tilde{u}-\tilde{u}^{\bm{h}^*}\|_{{}_{L^\infty}}=\inf\{ \|\tilde{u}-\tilde{u}^{\bm k}\|_{{}_{L^\infty}}\, :\, \bm k\in\Omega_\rho\}$ for some $\bm{h}^*\in\Omega_\rho$,
then there exists a positive constant $C$ such that
\begin{equation*}
|\bar{\bm{h}}-\bm{h}^*|\leq C\|\tilde{u}-\tilde{u}^{\bm{h}^*}\|_{{}_{L^\infty}}, \qquad \mbox{ and } \qquad \|\tilde{u}-\tilde{u}^{\bar{\bm{h}}}\|_{{}_{L^\infty}}\leq C\|\tilde{u}-\tilde{u}^{\bm{h}^*}\|_{{}_{L^\infty}}.
\end{equation*}
\end{thm}
\section{Slow dynamics in a neighborhood of the base manifold}\label{sec:slow}
The aim of this section is to study the dynamics of the solutions to \eqref{eq:hyp-CH}-\eqref{eq:Neumann} in a neighborhood of the manifold $\mathcal{M}_{{}_0}$
and to prove that $\mathcal{M}_{{}_0}$ is approximately invariant for \eqref{eq:hyp-CH}-\eqref{eq:Neumann}.
To do this, we will consider the integrated version \eqref{eq:integrated-CH}-\eqref{eq:bound-ut}.
Since
\begin{equation}\label{eq:u-ut}
\|\tilde{u}\|_{{}_{L^\infty}}\leq\|u\|_{{}_{L^\infty}},
\end{equation}
if $\|u-u^{\bm h}\|_{{}_{L^\infty}}$ is sufficiently small for some $\bm h\in\Omega_\rho$, we can use Theorem \ref{thm:existence-coord} and the decomposition $\tilde{u}=\tilde{u}^{\bm h}+\tilde{w}$ introduced in Section \ref{sec:prelimin}.
\subsection{Equations of motion and slow channel}
Let $(\tilde{u},\tilde{v})$ be a solution to \eqref{eq:integrated-CH-sys} with $\tilde{u}=\tilde{u}^{\bm\xi}+\tilde{w}$ and $\tilde{w}\in H$, where $H$ is the space defined in \eqref{eq:H};
it follows that the variables $(\tilde{w},\tilde{v})$ satisfy
\begin{equation*}
\begin{cases}
\tilde{w}_t=\tilde{v}-\displaystyle\sum_{j=1}^N\tilde{u}_j^{\bm\xi}\xi'_j,\\
\tau\tilde{v}_t=\mathcal{L}(\tilde{u}^{\bm\xi}+\tilde{w})-\tilde{v}.
\end{cases}
\end{equation*}
Expanding we get
\begin{equation*}
\mathcal{L}(u^{\bm\xi}+\tilde{w})=\mathcal{L}(\tilde{u}^{\bm\xi})+L^{\bm\xi}\tilde{w}+(f_2\tilde{w}_x^2)_x,
\qquad\textrm{ where }\quad
f_2:=\int_0^1(1-s)F'''(\tilde{u}^{\bm\xi}_x+s\tilde{w}_x)\,ds,
\end{equation*}
and $L^{\bm\xi}$ is the linearized operator of $\mathcal{L}$ about $\tilde{u}^{\bm\xi}$, that is $L^{\bm\xi}\tilde{w}:=-\varepsilon^2\tilde{w}_{xxxx}+\left(F''(u^{\bm\xi})\tilde{w}_x\right)_x$.
Hence, we obtain the following system for $(\tilde{w},\tilde{v})$:
\begin{equation}\label{eq:w-v}
\begin{cases}
\tilde{w}_t=\tilde{v}-\displaystyle\sum_{j=1}^N\tilde{u}_j^{\bm\xi}\xi'_j,\\
\tau\tilde{v}_t=\mathcal{L}(\tilde{u}^{\bm\xi})+L^{\bm\xi}\tilde{w}+(f_2\tilde{w}_x^2)_x-\tilde{v}.
\end{cases}
\end{equation}
In order to obtain the equation for $\bm\xi=\bm\xi(t)$, we make use of the orthogonality condition
\begin{equation}\label{eq:orthogonality}
\langle \tilde{w},E^{\bm\xi}_j\rangle=0, \qquad \mbox{ for } j=1,\dots,N,
\end{equation}
where the functions $E_j^{\bm\xi}$ are defined in Section \ref{sec:prelimin} and satisfy \eqref{eq:Ej}.
By differentiating with respect to $t$ the conditions \eqref{eq:orthogonality} and by using the first equation of \eqref{eq:w-v}, we infer
\begin{equation}\label{eq:xi}
\langle \tilde{v},E^{\bm\xi}_j\rangle-\sum_{i=1}^N\langle\tilde{u}_i^{\bm\xi},E^{\bm\xi}_j\rangle\xi'_i+\sum_{i=1}^N\langle\tilde{w},E^{\bm\xi}_{ji}\rangle\xi'_i=0,
\qquad j=1,\dots,N,
\end{equation}
where we introduced the notation $E^{\bm\xi}_{ji}:=\partial_i E^{\bm\xi}_j$.
Rewrite \eqref{eq:xi} in the compact form
\begin{equation}\label{eq:xi-compact}
D(\bm\xi,\tilde{w})\bm\xi'=Y(\bm\xi,\tilde{v}),
\end{equation}
where
\begin{equation*}
D_{ji}(\bm\xi,\tilde{w}):=\langle\tilde{u}_i^{\bm\xi},E^{\bm\xi}_j\rangle-\langle\tilde{w},E^{\bm\xi}_{ji}\rangle, \qquad \mbox{ and } \qquad Y_j(\bm\xi,\tilde{v}):=\langle \tilde{v},E^{\bm\xi}_j\rangle.
\end{equation*}
Therefore, combining \eqref{eq:w-v} and \eqref{eq:xi-compact} we obtain the ODE-PDE coupled system
\begin{equation}\label{eq:w-v-xi}
\begin{cases}
\tilde{w}_t=\tilde{v}-\displaystyle\sum_{j=1}^N\tilde{u}_j^{\bm\xi}\xi'_j,\\
\tau\tilde{v}_t=\mathcal{L}(\tilde{u}^{\bm\xi})+L^{\bm\xi}\tilde{w}+{(f_2\tilde{w}_x^2)}_x-\tilde{v},\\
D(\bm\xi,\tilde{w})\bm\xi'=Y(\bm\xi,\tilde{v}).
\end{cases}
\end{equation}
Now, let us define the slow channel where we will study the dynamics of \eqref{eq:w-v-xi}.
Let $\bm\xi$ such that $\bm h=\mathbf{G}(\bm\xi)\in\Omega_\rho$ and $\tilde{w}\in C^2(0,1)$ with $\tilde{w}=0$ at $x=0,1$; define
\begin{align*}
A^{\bm\xi}(\tilde{w})&:=-\langle L^{\bm\xi}\tilde{w},\tilde{w}\rangle=\int_0^1\left[\varepsilon^2\tilde{w}_{xx}^2+F''(u^{\bm\xi})\tilde{w}^2_x\right]\,dx,\\
B(\tilde{w})&:=\int_0^1\left[\varepsilon^2\tilde{w}_{xx}^2+\tilde{w}^2_x\right]\,dx.
\end{align*}
We recall the following lemma of \cite{Bates-Xun1}.
\begin{lem}
For any $\tilde{w}\in C^2(0,1)$ with $\tilde{w}=0$ at $x=0,1$, we have
\begin{align}
\|\tilde{w}\|_{{}_{L^\infty}}^2&\leq B(\tilde{w}), \label{eq:wt-B}\\
\varepsilon\|\tilde{w}_x\|_{{}_{L^\infty}}^2&\leq(1+\varepsilon)B(\tilde{w}). \label{eq:wt_x-B}
\end{align}
Moreover, assume that $F$ satisfies \eqref{eq:ass-F}.
There exists $\rho_0>0$ such that if $\rho\in(0,\rho_0)$ and $\bm h=\mathbf{G}(\bm\xi)\in\Omega_\rho$, then for any $\tilde{w}$ as above and satisfying the orthogonality condition \eqref{eq:orthogonality}, we have
\begin{equation}\label{eq:A^xi-B}
CA^{\bm\xi}(\tilde{w})\geq\varepsilon^2B(\tilde{w}),
\end{equation}
for some positive constant $C$ independent of $\varepsilon$ and $\tilde{w}$.
\end{lem}
For the proof of this lemma see \cite[Lemmas 4.1 and 4.2]{Bates-Xun1}.
Let us define the energy functional
\begin{equation*}
E^{\bm\xi}[\tilde{w},\tilde{v}]:=\frac12A^{\bm\xi}(\tilde{w})+\frac\tau2\|\tilde{v}\|^2+\varepsilon^\theta\tau\langle\tilde{w},\tilde{v}\rangle, \qquad \qquad \mbox{ for } \, \theta>0,
\end{equation*}
and the slow channel
\begin{align*}
\mathcal{Z}_{{}_{\Gamma,\rho}}:=\biggl\{(\tilde{u},\tilde{v})\,:\,\tilde{u}=\tilde{u}^{\bm\xi}+\tilde{w},\;\; (\tilde{w},\tilde{v})\in H\times H^2(0,1), \; \bm\xi \, &\mbox{ is such that }\, \bm h=\mathbf{G}(\bm\xi)\in\overline\Omega_\rho,\\
& \qquad \mbox{and } E^{\bm{\xi}}[\tilde{w},\tilde{v}]\leq\Gamma\varepsilon^{-2}\Psi(\bm h)\biggr\},
\end{align*}
for $\Gamma,\rho>0$, where the space $H$ and the barrier function $\Psi$ are defined in \eqref{eq:H} and \eqref{eq:barrier}, respectively.
Studying the dynamics of the solutions to \eqref{eq:integrated-CH-sys} in the slow channel $\mathcal{Z}_{{}_{\Gamma,\rho}}$ is equivalent to study the dynamics of the solutions to \eqref{eq:w-v-xi} in the set
\begin{align*}
\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}:=\biggl\{(\tilde{w},\tilde{v},\bm\xi) \, : \, (\tilde{w},\tilde{v})\in H\times H^2(0,1), \; \bm\xi \, &\mbox{ is such that }\, \bm h=\mathbf{G}(\bm\xi)\in\overline\Omega_\rho,\\
&\qquad \mbox{ and } E^{\bm{\xi}}[\tilde{w},\tilde{v}]\leq\Gamma\varepsilon^{-2}\Psi(\bm h)\biggr\}.
\end{align*}
Hence, we will study the dynamics of \eqref{eq:w-v-xi} in the set $\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$.
The first step is the following proposition, which gives estimates for solutions $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ to \eqref{eq:w-v-xi}.
\begin{prop}\label{prop:estimate-channel}
Let us assume that $F\in C^4(\mathbb{R})$ satisfies conditions \eqref{eq:ass-F}.
Given $N\in\mathbb{N}$ and $\delta\in(0,1/(N+1))$, there exists $\varepsilon_0>0$ such that if $\varepsilon,\rho$ satisfy \eqref{eq:triangle}, $\theta>1$,
and $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$, then
\begin{align}
A^{\bm\xi}(\tilde{w})\leq C E^{\bm\xi}[\tilde{w},\tilde{v}]\leq C\Gamma\varepsilon^{-2}\exp\left(-\frac{2Al^{\bm h}}{\varepsilon}\right), \label{eq:A^xi-E}\\
\varepsilon^2\|\tilde{w}\|^2_{{}_{L^\infty}}+\tau\|\tilde{v}\|^2_{{}_{L^2}}\leq CE^{\bm\xi}[\tilde{w},\tilde{v}]\leq C\Gamma\varepsilon^{-2}\exp\left(-\frac{2Al^{\bm h}}{\varepsilon}\right), \label{eq:wt-vt-E}
\end{align}
for some positive constant $C>0$ (independent of $\varepsilon$, $\tau$ and $\theta$).
Moreover, if $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ is a solution to \eqref{eq:w-v-xi} for $t\in[0,T]$, then
\begin{equation}\label{eq:xi'-estimate}
|\bm\xi'|_{{}_\infty}\leq C\varepsilon^{-2}\tau^{-1/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right).
\end{equation}
\end{prop}
\begin{proof}
Let us prove \eqref{eq:A^xi-E}.
Using Young inequality, we infer
\begin{equation*}
2\varepsilon^\theta|\langle\tilde{w},\tilde{v}\rangle|\leq\varepsilon^{2\theta}\|\tilde{w}\|^2_{{}_{L^{\infty}}}+\|\tilde{v}\|^2,
\end{equation*}
and so, for the definitions of $A^{\bm\xi}$ and $E^{\bm\xi}$, we have
\begin{equation*}
A^{\bm\xi}(\tilde{w})=2E^{\bm\xi}[\tilde{w},\tilde{v}]-\tau\|\tilde{v}\|^2-2\varepsilon^\theta\tau\langle\tilde{w},\tilde{v}\rangle\leq 2E^{\bm\xi}[\tilde{w},\tilde{v}]+\varepsilon^{2\theta}\tau\|\tilde{w}\|^2_{{}_{L^{\infty}}}.
\end{equation*}
Using \eqref{eq:wt-B} and \eqref{eq:A^xi-B}, we deduce
\begin{equation*}
\|\tilde{w}\|^2_{{}_{L^{\infty}}}\leq B(\tilde{w})\leq C\varepsilon^{-2} A^{\bm\xi}(\tilde{w}),
\end{equation*}
and then
\begin{equation*}
A^{\bm\xi}(\tilde{w})\leq 2E^{\bm\xi}[\tilde{w},\tilde{v}]+C\varepsilon^{2(\theta-1)}\tau A^{\bm\xi}(\tilde{w}).
\end{equation*}
Since $\theta>1$, we can choose $\varepsilon_0$ so small that $C\varepsilon^{2(\theta-1)}\tau\leq\nu<1$ for $\varepsilon\in(0,\varepsilon_0)$, and conclude that
\begin{equation*}
A^{\bm\xi}(\tilde{w})\leq CE^{\bm\xi}[\tilde{w},\tilde{v}].
\end{equation*}
The second inequality of \eqref{eq:A^xi-E} follows from the facts that $E^{\bm\xi}[\tilde{w},\tilde{v}]\leq\Gamma\varepsilon^{-2}\Psi(\bm h)$ in $\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ and that
the definition of the barrier function $\Psi$ \eqref{eq:barrier} and \eqref{eq:alfaj} imply that
\begin{equation}\label{eq:Psi<exp}
\Psi(\bm h)\leq C\exp\left(-\frac{2Al^{\bm h}}{\varepsilon}\right),
\end{equation}
for some $C>0$ independent of $\varepsilon$.
The proof of \eqref{eq:wt-vt-E} is similar.
From \eqref{eq:A^xi-B} and Young inequality, one has
\begin{equation*}
E^{\bm\xi}[\tilde{w},\tilde{v}]\geq C\varepsilon^2B(\tilde{w})+\frac\tau2\|\tilde{v}\|^2-\varepsilon^{2\theta}\tau\|\tilde{w}\|^2-\frac\tau4\|\tilde{v}\|^2.
\end{equation*}
Hence, for \eqref{eq:wt-B} we get
\begin{equation*}
E^{\bm\xi}[\tilde{w},\tilde{v}]\geq C\varepsilon^2\|\tilde{w}\|^2_{{}_{L^\infty}}+\frac\tau4\|\tilde{v}\|^2-\varepsilon^{2\theta}\tau\|\tilde{w}\|^2_{{}_{L^\infty}}\geq \left(C-\varepsilon^{2(\theta-1)}\tau\right)\varepsilon^2\|\tilde{w}\|^2_{{}_{L^\infty}}+\frac\tau4\|\tilde{v}\|^2,
\end{equation*}
and we obtain \eqref{eq:wt-vt-E} choosing $\varepsilon$ sufficiently small (again since $\theta>1$) and using \eqref{eq:Psi<exp}.
It remains to prove \eqref{eq:xi'-estimate}.
Let us consider the equation for $\bm\xi$ in \eqref{eq:w-v-xi} and the matrix $D(\bm\xi,\tilde{w})$ of elements $D_{ij}(\bm\xi,\tilde{w}):=\langle\tilde{u}_j^{\bm\xi},E^{\bm\xi}_i\rangle-\langle\tilde{w},E^{\bm\xi}_{ij}\rangle$.
These elements have been already studied in \cite{Bates-Xun1,Bates-Xun2} and one has
\begin{equation}\label{eq:aij}
a_{ij}:=\langle\tilde{u}_j^{\bm\xi},E_i\rangle=\begin{cases}
(-1)^{i+j}4l_{j+1}+\mathcal{O}(\varepsilon), \qquad \qquad & i\geq j, \\
\mathcal{O}(\varepsilon), & i\leq j,
\end{cases}
\end{equation}
where $l_j:=h_j-h_{j-1}$ is the distance between the layers (see formulas (4.27) in \cite{Bates-Xun2}), and $\|E^{\bm\xi}_{ij}\|\leq C\varepsilon^{-1/2}$.
These results can be obtained by using the formulas \eqref{eq:ut-j-xi} and \eqref{eq:Ej} for $\tilde{u}_j^{\bm\xi}$ and $E^{\bm\xi}_j$.
In particular, the bound for $\|E^{\bm\xi}_{ij}\|$ can be easily obtained by differentiating with respect to $\xi_j$ the formula \eqref{eq:Ej}
without the exponentially small terms and by using \eqref{eq:uxi_j}.
From \eqref{eq:wt-vt-E}, it follows that
\begin{equation*}
|\langle\tilde{w},E^{\bm\xi}_{ji}\rangle|\leq\|\tilde{w}\|\|E^{\bm\xi}_{ji}\|\leq C\varepsilon^{-5/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right).
\end{equation*}
Hence, for $\varepsilon$ sufficiently small, we have
\begin{equation*}
D(\bm\xi,\tilde{w}):=\left(\begin{array}{ccccc} 4l_2 & 0 & 0 & \dots & 0\\
-4l_3 & 4l_3 & 0 & \dots & 0\\
4l_4 & -4l_4 & 4l_4 & \dots & 0\\
\dots & \dots & \dots & \dots & \dots\\
(-1)^{N-1}4l_{N+1} & (-1)^{N-2}4l_{N+1} & (-1)^{N-3}4l_{N+1} & \dots & 4l_{N+1}
\end{array}\right)+\mathcal{O}(\varepsilon),
\end{equation*}
and its inverse
\begin{equation*}
D^{-1}(\bm\xi,\tilde{w})=\left(\begin{array}{cccccc} \displaystyle{\frac1{4l_2}} & 0 & 0 & \dots & 0 & 0\\
\displaystyle{\frac1{4l_2}} & \displaystyle{\frac1{4l_3}} & 0 & \dots & 0 & 0\\
0 & \displaystyle\frac1{4l_3} & \displaystyle\frac1{4l_4} & \dots & 0 & 0\\
\dots & \dots & \dots & \dots & \dots\\
0 & 0 & 0 & \dots & \displaystyle\frac1{4l_N} & \displaystyle\frac1{4l_{N+1}}\\
\end{array}\right)+\mathcal{O}(\varepsilon).
\end{equation*}
Let us rewrite the equation for $\bm\xi$ in \eqref{eq:w-v-xi} as
\begin{equation*}
\bm\xi'=D^{-1}(\bm\xi,\tilde{w})Y(\bm\xi,\tilde{v}).
\end{equation*}
Since
\begin{equation*}
|Y_j(\bm\xi,\tilde{v})|=|\langle \tilde{v},E^{\bm\xi}_j\rangle|\leq\|\tilde{v}\|\|E^{\bm\xi}_j\|\leq C\|\tilde{v}\|,
\end{equation*}
where in the last passage we used the formula \eqref{eq:Ej} for $E^{\bm\xi}_j$,
we deduce
\begin{equation*}
|\bm\xi'|_{{}_{\infty}}\leq C\|D^{-1}(\bm\xi,\tilde{w})\|_{{}_{\infty}}\|\tilde{v}\|,
\end{equation*}
where $\|\cdot\|_{{}_{\infty}}$ denotes the matrix norm induced by the vector norm $|\cdot|_{{}_{\infty}}$.
To estimate such matrix norm, we use the assumption $\bm h\in\Omega_\rho$, which implies $l_j>\varepsilon/\rho$ for any $j$.
Therefore, we can conclude that
\begin{equation}\label{eq:xi-vt}
|\bm\xi'|_{{}_{\infty}}\leq C\varepsilon^{-1}\|\tilde{v}\|,
\end{equation}
and the proof of \eqref{eq:xi'-estimate} follows from \eqref{eq:wt-vt-E}.
\end{proof}
\subsection{Main result}
Thanks to the estimates \eqref{eq:A^xi-E}, \eqref{eq:wt-vt-E} and \eqref{eq:xi'-estimate}, we can state that if $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ is a solution to \eqref{eq:w-v-xi} in $[0,T]$,
then the $L^\infty$--norm of $\tilde{w}$, the $L^2$--norm of $\tilde{v}$ and the velocity of $\bm\xi$ are exponentially small in $\varepsilon$.
This implies that if $u=u^{\bm h}+w$ is a solution to \eqref{eq:hyp-CH} such that $(\tilde{u},\tilde{u}_t)\in\mathcal{Z}_{{}_{\Gamma,\rho}}$ for $t\in[0,T]$,
then the $L^\infty$--norm of $w$ and the velocity of the transition points $(h_1,\dots,h_{N+1})$ are exponentially small.
Indeed, to estimate the norm of $w$, we use \eqref{eq:wt_x-B}, \eqref{eq:A^xi-B} and \eqref{eq:A^xi-E} and we get
\begin{equation*}
\|w(\cdot,t)\|_{{}_{L^\infty}}=\|\tilde{w}_x(\cdot,t)\|_{{}_{L^\infty}}\leq C\varepsilon^{-1/2} B(\tilde{w})^{1/2}\leq C\varepsilon^{-3/2} A^{\bm\xi}(\tilde{w})^{1/2}\leq C\varepsilon^{-5/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right),
\end{equation*}
for some $C>0$ independent of $\varepsilon$.
On the other hand, the velocity of the transition points $(h_1,\dots,h_N)$ is exponentially small for \eqref{eq:xi'-estimate} and the fact that $h_i=\xi_i$ for $i=1,\dots,N$;
to estimate the velocity of $h_{N+1}$, we use \eqref{eq:N+1-der} and the relation
\begin{equation}\label{eq:h'_N+1}
h'_{N+1}=\sum_{j=1}^N\frac{\partial h_{N+1}}{\partial h_j}h'_j=\sum_{j=1}^N\left[(-1)^{N-j}+\mathcal{O}\left(\varepsilon^{-1}\exp\left(-\frac{Al^{\bm h}}\varepsilon\right)\right)\right]h'_j.
\end{equation}
From \eqref{eq:h'_N+1} and \eqref{eq:xi'-estimate} it follows that
\begin{equation*}
|h'_{N+1}(t)|\leq C\varepsilon^{-2}\tau^{-1/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right).
\end{equation*}
Therefore, we can state that if $u=u^{\bm h}+w$ is a solution to \eqref{eq:hyp-CH} such that $(\tilde{u},\tilde{u}_t)\in\mathcal{Z}_{{}_{\Gamma,\rho}}$ for $t\in[0,T]$, then
\begin{equation*}
\|u-u^{\bm h}\|_{{}_{L^\infty}}\leq C\varepsilon^{-5/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right), \qquad \qquad |\bm h'|_{{}_\infty}\leq C\varepsilon^{-2}\tau^{-1/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right),
\end{equation*}
for $t\in[0,T]$.
In other words, there exists a neighborhood of the manifold $\mathcal{M}_{{}_0}$ where the solution $u$ to \eqref{eq:hyp-CH}-\eqref{eq:Neumann}
is well approximated by $u^{\bm h}$; thus, $u$ is a function with $N+1$ transitions between $-1$ and $+1$,
and the velocity of the transition points is exponentially small.
Let us focus the attention on a lower bound of the time $T_\varepsilon$ taken for the solution to leave such neighborhood of $\mathcal{M}_{{}_0}$.
To this aim, we observe that a solution can leave the slow channel $\mathcal{Z}_{{}_{\Gamma,\rho}}$ either if $\bm h=\mathbf{G}(\bm\xi)\in\partial\Omega_\rho$,
meaning that two transition points are close enough, namely $h_j-h_{j-1}=\varepsilon/\rho$ for some $j$,
or if the energy functional is large enough, precisely $E^{\bm{\xi}}[\tilde{w},\tilde{v}]=\Gamma\varepsilon^{-2}\Psi(\bm h)$.
We will prove that solutions leave the slow channel only if two transition points are close enough;
then, since the transition points move with exponentially small velocity, the time taken for the solution to leave the slow channel is exponentially large.
Precisely, we will prove the following result.
\begin{thm}\label{thm:main}
Let us assume that $F\in C^4(\mathbb{R})$ satisfies conditions \eqref{eq:ass-F} and consider the IBVP \eqref{eq:hyp-CH}-\eqref{eq:Neumann}-\eqref{eq:initial} with $u_1$ satisfying \eqref{eq:ass-u1}.
Given $N\in\mathbb{N}$ and $\delta\in(0,1/(N+1))$, there exist $\varepsilon_0,\theta_0>0$ and $\Gamma_2>\Gamma_1>0$ such that if $\varepsilon,\rho$ satisfy \eqref{eq:triangle}, $\theta>\theta_0$,
$\Gamma\in[\Gamma_1,\Gamma_2]$, and the initial datum $(u_0,u_1)$ is such that
\begin{equation*}
(\tilde{u}_0,\tilde{u}_1)\in\,\stackrel{\circ}{\mathcal{Z}}_{{}_{\Gamma,\rho}}=\bigl\{(\tilde{u},\tilde{v})\in\mathcal{Z}_{{}_{\Gamma,\rho}}\, : \, {\bm h}=\mathbf{G}(\bm\xi)\in\Omega_\rho
\;\;\textrm{and}\;\; E^{\bm\xi}[w,v]<\Gamma\varepsilon^{-2}\Psi({\bm h})\bigr\},
\end{equation*}
then the solution $(u,u_t)$ is such that $(\tilde{u},\tilde{u}_t)$ remains in $\mathcal{Z}_{{}_{\Gamma,\rho}}$ for a time $T_\varepsilon>0$, and for any $t\in[0,T_\varepsilon]$ one has
\begin{equation}\label{eq:u-uh,h'}
\|u-u^{\bm h}\|_{{}_{L^\infty}}\leq C\varepsilon^{-5/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right), \qquad \qquad |\bm h'|_{{}_\infty}\leq C\varepsilon^{-2}\tau^{-1/2}\exp\left(-\frac{Al^{\bm h}}{\varepsilon}\right),
\end{equation}
where $A:=\sqrt{\min\{F''(-1),F''(+1)\}}$, $\ell^{\bm h}:=\min\{h_j-h_{j-1}\}$ and $|\cdot|_{{}_{\infty}}$ denotes the maximum norm in $\mathbb{R}^N$.
Moreover, there exists $C>0$ such that
\begin{equation*}
T_\varepsilon\geq C\varepsilon^2\tau^{1/2}(\ell^{\bm h(0)}-\varepsilon/\rho)\exp(A\delta /\varepsilon).
\end{equation*}
\end{thm}
The proof of Theorem \ref{thm:main} is based on the following proposition, which gives an estimate on the time derivative of $E^{\bm\xi}[\tilde{w},\tilde{v}]$ along the solutions to the system \eqref{eq:w-v-xi}.
\begin{prop}\label{prop:energy-estimate}
Let us assume that $F\in C^4(\mathbb{R})$ satisfies conditions \eqref{eq:ass-F}.
Given $N\in\mathbb{N}$ and $\delta\in(0,1/(N+1))$, there exist $\varepsilon_0,\theta_0>0$ and $\Gamma_2>\Gamma_1>0$ such that if $\varepsilon,\rho$ satisfy \eqref{eq:triangle}, $\theta>\theta_0$,
$\Gamma\in[\Gamma_1,\Gamma_2]$, and $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ is a solution to \eqref{eq:w-v-xi} for $t\in[0,T]$, then for some $\eta\in(0,1)$ and $\mu>0$, we have
\begin{equation}\label{eq:E-Psi}
\frac d{dt}\bigl\{E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h)\bigr\} \leq
-\eta\varepsilon^\mu\bigl\{E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h)\bigr\}, \qquad \qquad \mbox{for } t\in[0,T].
\end{equation}
\end{prop}
\begin{proof}
In all the proof, symbols $C, c, \eta$ denote generic positive constants, independent on $\varepsilon$, and with $\eta\in(0,1)$.
Let us differentiate with respect to $t$ the three terms of the energy functional $E^{\bm\xi}$.
For the first term, direct differentiation and the first equation of \eqref{eq:w-v-xi} give
\begin{equation*}
\begin{aligned}
\frac{d}{dt}\left\{\frac12 A^{\bm\xi}(\tilde{w})\right\} & =-\frac{d}{dt}\left\{\frac12\langle L^{\bm\xi}\tilde{w},\tilde{w}\rangle\right\}
=-\langle L^{\bm\xi}\tilde{w},\tilde{w}_t\rangle+\frac12\langle{(F''(u^{\bm \xi}))}_t,\tilde{w}_x^2\rangle\\
& =-\langle L^{\bm\xi}\tilde{w},\tilde{v}\rangle+\sum_{j=1}^N\xi'_j\langle L^{\bm\xi}\tilde{w},\tilde{u}_j^{\bm\xi}\rangle
+\frac12\sum_{j=1}^N\xi'_j\langle F'''(u^{\bm \xi})\tilde{u}_j^{\bm\xi},\tilde{w}_x^2\rangle.
\end{aligned}
\end{equation*}
Using the self-adjointness of the operator $L^{\bm\xi}$ and inequality \eqref{eq:xi-vt}, we infer
\begin{equation*}
\sum_{j=1}^N|\xi'_j\langle L^{\bm\xi}\tilde{w},\tilde{u}_j^{\bm\xi}\rangle|= \sum_{j=1}^N|\xi'_j\langle \tilde{w},L^{\bm\xi}\tilde{u}_j^{\bm\xi}\rangle| \leq C\varepsilon^{-1}\|\tilde{v}\|\|\tilde{w}\|\max_j\|L^{\bm\xi}\tilde{u}_j^{\bm\xi}\|.
\end{equation*}
For the last term of the latter inequality, we have that
\begin{align*}
L^{\bm\xi}\tilde{u}_j^{\bm\xi} & =L^{\bm\xi}\tilde{u}_j^{\bm h}+\frac{\partial h_{N+1}}{\partial h_j}L^{\bm\xi}\tilde{u}_{N+1}^{\bm h}
=\frac{\partial}{\partial h_j}\mathcal{L}(\tilde{u}^{\bm h})+\frac{\partial h_{N+1}}{\partial h_j}\frac{\partial}{\partial h_{N+1}}\mathcal{L}(\tilde{u}^{\bm h})\\
& = -\frac{\partial}{\partial h_j}\frac{\partial}{\partial x}\mathcal{L}^{AC}(u^{\bm h})-\frac{\partial h_{N+1}}{\partial h_j}\frac{\partial}{\partial h_{N+1}}\frac{\partial}{\partial x}\mathcal{L}^{AC}(u^{\bm h}),
\end{align*}
and from \cite[Lemma 5.2]{Bates-Xun1}, it follows that
\begin{equation*}
\|L^{\bm\xi}\tilde{u}_j^{\bm\xi}\|\leq C\varepsilon^{-4}\exp\left(-\frac{Al^{\bm h}}{2\varepsilon}\right)\leq C\varepsilon^{-4}\exp\left(-\frac{A\delta}{2\varepsilon}\right),
\end{equation*}
where we used \eqref{eq:triangle}.
On the other hand, the formula \eqref{eq:ut-j-xi} and the inequalities \eqref{eq:wt_x-B}, \eqref{eq:A^xi-B} and \eqref{eq:xi-vt} yield
\begin{align*}
\left|\sum_{j=1}^N\xi'_j\langle F'''(u^{\bm \xi})\tilde{u}_j^{\bm\xi},\tilde{w}_x^2\rangle\right| & \leq C|\bm\xi|_{{}_{\infty}}\|\tilde{w}_x\|_{{}_{L^\infty}}\|\tilde{w}_x\|^2\max_j\|\tilde{u}_j^{\bm\xi}\|\leq C\varepsilon^{-2}B(\tilde{w})\|\tilde{v}\|\\
& \leq C\varepsilon^{-4}A^{\bm\xi}(\tilde{w})\|\tilde{v}\|.
\end{align*}
Therefore, for the first term of the energy we conclude
\begin{equation}\label{eq:first-energy}
\frac{d}{dt}\left\{\frac12 A^{\bm\xi}(\tilde{w})\right\}\leq-\langle L^{\bm\xi}\tilde{w},\tilde{v}\rangle+C\varepsilon^{-5}\exp(-c/\varepsilon)\|\tilde{v}\|\|\tilde{w}\|+C\varepsilon^{-4}A^{\bm\xi}(\tilde{w})\|\tilde{v}\|.
\end{equation}
For what concerns the second term in the energy $E^{\bm\xi}$, the second equation of \eqref{eq:w-v-xi} gives
\begin{equation*}
\begin{aligned}
\frac{d}{dt}\Bigl\{\frac\tau2\|\tilde{v}\|^2\Bigr\}& =\langle \tau v_t,v\rangle
=\langle\mathcal{L}(\tilde{u}^{\bm\xi})+L^{\bm\xi}\tilde{w}+{(f_2\tilde{w}^2_x)}_x-\tilde{v},\tilde{v}\rangle\\
&\leq\langle L^{\bm\xi}\tilde{w},\tilde{v}\rangle+\|\mathcal{L}(\tilde{u}^{\bm\xi})\|\|v\|+\langle{(f_2\tilde{w}^2_x)}_x,\tilde{v}\rangle-\|\tilde{v}\|^2\\
&\leq \langle L^{\bm\xi}\tilde{w},\tilde{v}\rangle-\frac12\|\tilde{v}\|^2+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2+\langle{(f_2\tilde{w}^2_x)}_x,\tilde{v}\rangle.
\end{aligned}
\end{equation*}
By expanding
\begin{align*}
{(f_2\tilde{w}^2_x)}_x & = {(f_2)}_x\tilde{w}^2_x+2f_2\tilde{w}_x\tilde{w}_{xx}\\
& = \tilde{w}^2_x\int_0^1(1-s)F''''(\tilde{u}_x^{\bm\xi}+s\tilde{w}_x)(\tilde{u}^{\bm\xi}_{xx}+s\tilde{w}_{xx})\,ds+2f_2\tilde{w}_x\tilde{w}_{xx},
\end{align*}
we deduce the estimate
\begin{equation}\label{eq:f2wx}
\begin{aligned}
|\langle{(f_2\tilde{w}^2_x)}_x,\tilde{v}\rangle| & \leq\|{(f_2\tilde{w}^2_x)}_x\|\|\tilde{v}\|\leq C\left(\|\tilde{w}_x\|^2_{{}_{L^\infty}}\|\tilde{u}^{\bm\xi}_{xx}+\tilde{w}_{xx}\|+\|\tilde{w}_x\|_{{}_{L^\infty}}\|\tilde{w}_{xx}\|\right)\|\tilde{v}\|\\
& \leq C\left\{\varepsilon^{-1}B(\tilde{w})\left(\varepsilon^{-1}+\varepsilon^{-1}B(\tilde{w})^{1/2}\right)+\varepsilon^{-3/2}B(\tilde{w})\right\}\|\tilde{v}\|,\\
& \leq C\left\{\varepsilon^{-4}+\varepsilon^{-5}A^{\bm\xi}(\tilde{w})^{1/2}\right\}\|\tilde{v}\|A^{\bm\xi}(\tilde{w}),
\end{aligned}
\end{equation}
where the estimates \eqref{eq:wt_x-B}, \eqref{eq:A^xi-B},
\begin{equation*}
\|\tilde{u}^{\bm\xi}_{xx}\|=\|u^{\bm\xi}_x\|\leq C\varepsilon^{-1}, \qquad \mbox{ and } \qquad \|\tilde{w}_{xx}\|^2\leq\varepsilon^{-2}B(\tilde{w}).
\end{equation*}
have been used.
Hence, we obtain
\begin{equation}\label{eq:second-energy}
\frac{d}{dt}\Bigl\{\frac\tau2\|\tilde{v}\|^2\Bigr\}\leq\langle L^{\bm\xi}\tilde{w},\tilde{v}\rangle-\frac12\|\tilde{v}\|^2+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2+C\left\{1+\varepsilon^{-1}A^{\bm\xi}(\tilde{w})^{1/2}\right\}\varepsilon^{-4}\|\tilde{v}\|A^{\bm\xi}(\tilde{w}).
\end{equation}
Finally, the time derivative of the scalar product $\langle w,\tau v\rangle$ can be bounded by
\begin{equation*}
\begin{aligned}
\frac{d}{dt}\langle \tilde{w},\tau\tilde{v}\rangle
&=\tau\|\tilde{v}\|^2-\tau\sum_{j=1}^N\xi'_j\langle\tilde{u}_j^{\bm\xi},\tilde{v}\rangle+\langle\tilde{w},\mathcal{L}(\tilde{u}^{\bm\xi})+L^{\bm\xi}\tilde{w}+{(f_2\tilde{w}_x^2)}_x-\tilde{v}\rangle\\
&\leq\tau\|\tilde{v}\|^2+\tau|\bm\xi'|_{{}_{\infty}}\|\tilde{u}_j^{\bm\xi}\|\|\tilde{v}\|+\|\tilde{w}\|\|\mathcal{L}(\tilde{u}^{\bm\xi})\|-A^{\bm\xi}(\tilde{w})-\langle\tilde{w},\tilde{v}\rangle+\langle\tilde{w},{(f_2\tilde{w}_x^2)}_x\rangle.
\end{aligned}
\end{equation*}
where we used that $A^{\bm\xi}(\tilde{w})=-\langle\tilde{w},L^{\bm\xi}\tilde{w}\rangle$.
By using \eqref{eq:ut-j-xi}, \eqref{eq:xi-vt} and estimating as in \eqref{eq:f2wx}, we infer
\begin{equation*}
\begin{aligned}
\varepsilon^\theta\frac{d}{dt}\langle \tilde{w},\tau\tilde{v}\rangle\leq-\varepsilon^\theta A^{\bm\xi}(\tilde{w})+\varepsilon^\theta\tau(1+C\varepsilon^{-1})&\|\tilde{v}\|^2-\frac{\varepsilon^\theta}\tau\langle \tilde{w},\tau\tilde{v}\rangle+\varepsilon^\theta\|\tilde{w}\|\|\mathcal{L}(\tilde{u}^{\bm\xi})\|\\
&+C\left\{1+\varepsilon^{-1}A^{\bm\xi}(\tilde{w})^{1/2}\right\}\varepsilon^{\theta-4}\|\tilde{w}\|A^{\bm\xi}(\tilde{w}).
\end{aligned}
\end{equation*}
For Young inequality, we have
\begin{equation*}
\varepsilon^\theta\|\tilde{w}\|\|\mathcal{L}(\tilde{u}^{\bm\xi})\|\leq C\varepsilon^{2\theta}\|\tilde{w}\|^2+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2\leq C\varepsilon^{2(\theta-1)}A^{\bm\xi}(\tilde{w})+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2,
\end{equation*}
and we can estimate the third term of $E^{\bm\xi}$ as
\begin{equation}\label{eq:third-energy}
\begin{aligned}
\varepsilon^\theta\frac{d}{dt}\langle \tilde{w},\tau\tilde{v}\rangle\leq&-\varepsilon^\theta A^{\bm\xi}(\tilde{w})-\frac{\varepsilon^\theta}\tau\langle \tilde{w},\tau\tilde{v}\rangle+\varepsilon^\theta\tau(1+C\varepsilon^{-1})\|\tilde{v}\|^2+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2\\
&+C\varepsilon^{2(\theta-1)}A^{\bm\xi}(\tilde{w})+C\left\{1+\varepsilon^{-1}A^{\bm\xi}(\tilde{w})^{1/2}\right\}\varepsilon^{\theta-4}\|\tilde{w}\|A^{\bm\xi}(\tilde{w}).
\end{aligned}
\end{equation}
Collecting \eqref{eq:first-energy}, \eqref{eq:second-energy} and \eqref{eq:third-energy}, we deduce
\begin{equation*}
\begin{aligned}
\frac{d}{dt}E^{\bm\xi}[\tilde{w},\tilde{v}] \leq &-\varepsilon^\theta A^{\bm\xi}(\tilde{w})-\left(\frac12-\varepsilon^\theta\tau(1+C\varepsilon^{-1})\right)\|\tilde{v}\|^2-\frac{\varepsilon^\theta}\tau\langle \tilde{w},\tau\tilde{v}\rangle+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2\\
&+C\varepsilon^{2(\theta-1)}A^{\bm\xi}(\tilde{w})+R^{\bm\xi}[\tilde{w},\tilde{v}],
\end{aligned}
\end{equation*}
where
\begin{align*}
R^{\bm\xi}[\tilde{w},\tilde{v}]:= C\varepsilon^{-5}\exp(-c/\varepsilon)\|\tilde{v}\|&\|\tilde{w}\|+C\varepsilon^{-4}A^{\bm\xi}(\tilde{w})\|\tilde{v}\|\\
&+C\varepsilon^{-4}\left\{1+\varepsilon^{-1}A^{\bm\xi}(\tilde{w})^{1/2}\right\}\left\{\|\tilde{v}\|+\varepsilon^{\theta}\|\tilde{w}\|\right\}A^{\bm\xi}(\tilde{w}).
\end{align*}
Choosing $\theta>2$ and $\varepsilon_0$ so small that
\begin{equation}\label{eq:cond-eps-tau}
C\varepsilon^{\theta-1}\tau\leq\frac12-\eta,
\end{equation}
for any $\varepsilon\in(0,\varepsilon_0)$, we obtain
\begin{equation*}
\frac{d}{dt}E^{\bm\xi}[\tilde{w},\tilde{v}] \leq -\eta\varepsilon^\theta A^{\bm\xi}(\tilde{w})-\eta\|\tilde{v}\|^2-\frac{\varepsilon^\theta}\tau\langle \tilde{w},\tau\tilde{v}\rangle+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2+R^{\bm\xi}[\tilde{w},\tilde{v}].
\end{equation*}
Therefore, we conclude that there exists $\mu>0$ (independent on $\varepsilon$) such that
\begin{equation*}
\frac{d}{dt}E^{\bm\xi}[\tilde{w},\tilde{v}] \leq -\eta\varepsilon^\mu E^{\bm\xi}[\tilde{w},\tilde{v}]-\eta\|\tilde{v}\|^2+C\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2+R^{\bm\xi}[\tilde{w},\tilde{v}],
\end{equation*}
for some $\eta\in(0,1)$.
Indeed, the condition \eqref{eq:cond-eps-tau} implies $\varepsilon^\theta/\tau>C\varepsilon^{2\theta-1}$ and, since $\theta>2$, we can choose $\mu\geq2\theta-1$.
Now, let us use that $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ for $t\in[0,T]$;
from Proposition \ref{prop:estimate-channel} it follows that there exists $\varepsilon_0$ (dependent on $\Gamma$ and $\tau$) such that
\begin{equation*}
R^{\bm\xi}[\tilde{w},\tilde{v}]\leq C\exp(-c/\varepsilon)\Gamma\varepsilon^{-2}\Psi(\bm h),
\end{equation*}
for any $\varepsilon\in(0,\varepsilon_0)$.
Since, for \eqref{eq:barrier} and \eqref{eq:Lut^xi} one has
\begin{equation*}
\|\mathcal{L}(\tilde{u}^{\bm\xi})\|^2\leq C\varepsilon^{-2}\Psi(\bm h),
\end{equation*}
we infer
\begin{equation}\label{eq:d/dtE}
\frac{d}{dt}E^{\bm\xi}[\tilde{w},\tilde{v}]\leq -\eta\varepsilon^\mu E^{\bm\xi}[\tilde{w},\tilde{v}]-\eta\|\tilde{v}\|^2+C\varepsilon^{-2}\Psi(\bm h).
\end{equation}
Now, let us compute the time derivative of the barrier function $\Psi$.
Direct differentiation gives
\begin{equation*}
\frac{d\Psi}{dt}=2\sum_{i,j=1}^{N+1}\langle\mathcal{L}^{AC}(u^{\bm h}),k^{\bm h}_j\rangle
\Bigl\{\langle\mathcal{L}^{AC}(u^{\bm h}),k_{ji}^{\bm h}\rangle+\langle L^{AC}u^{\bm h}_i,k^{\bm h}_j\rangle\Bigr\}h'_i,
\end{equation*}
where $L^{AC}$ is the linearization of $\mathcal{L}^{AC}(u)$ about $u^{\bm h}$, i.e.
\begin{equation*}
L^{AC}w:=\varepsilon^2w_{xx}-F''(u^{\bm h})w.
\end{equation*}
Using the estimates provided by inequalities \eqref{eq:ineq-k} and \eqref{eq:xi-vt}, we deduce
\begin{equation*}
\begin{aligned}
\bigl|\langle\mathcal{L}^{AC}(u^{\bm h}),k_{ji}^{\bm h}\rangle h_i'\bigr|
&\leq|\bm h'|_{{}_\infty}\|\mathcal{L}^{AC}(u^{\bm h})\|\|k^{\bm h}_{ji}\|
\leq C\varepsilon^{-5/2}\|\mathcal{L}^{AC}(u^{\bm h})\|\|v\|,\\
\bigl|\langle L^{AC}u^{\bm h}_i,k^{\bm h}_j\rangle h'_i\bigr|
&\leq|\bm h'|_{{}_\infty}\|k^{\bm h}_j\|\|L^{AC}u^{\bm h}_i\|
\leq C\exp(-c/\varepsilon)\|v\|,
\end{aligned}
\end{equation*}
where in the last passage the inequality $\|L^{AC}u^{\bm h}_i\|\leq C\varepsilon^{-1/2}\exp(-Al^{\bm h}/\varepsilon)$ has been used (see \cite[Proposition 7.2]{Carr-Pego2}).
Thus, since
\begin{equation*}
|\langle\mathcal{L}^{AC}(u^{\bm h}),k^{\bm h}_j\rangle|\leq C\varepsilon^{-1/2} \|\mathcal{L}^{AC}(u^{\bm h})\|,
\end{equation*}
we deduce the bound
\begin{equation*}
\left|\frac{d\Psi}{dt}\right|
\leq C\varepsilon^{-1/2} \left\{\varepsilon^{-2}\|\mathcal{L}^{AC}(u^{\bm h})\|+\exp(-c/\varepsilon)\right\}\|\mathcal{L}^{AC}(u^{\bm h})\|\|v\|.
\end{equation*}
It is well known (see \cite[Proposition 3.5]{Carr-Pego}) that
\begin{equation}
\|\mathcal{L}^{AC}(u^{\bm h})\|^2\leq C\varepsilon\sum_{j=1}^{N+1}|\alpha^{j+1}-\alpha^j|^2\leq C\varepsilon\Psi(\bm h),
\end{equation}
where we used the definition of $\Psi$ \eqref{eq:barrier}.
Therefore, we obtain
\begin{equation*}
\begin{aligned}
\left|\Gamma\frac{d\Psi}{dt}\right|
&\leq C\,\Gamma\bigl\{\varepsilon^{-3/2}\Psi^{1/2}+\exp(-c/\varepsilon)\bigr\}\|v\|\Psi^{1/2}\\
&\leq \eta\|v\|^2+C\,\Gamma^2\bigl\{\varepsilon^{-3/2}\Psi^{1/2}+\exp(-c/\varepsilon)\bigr\}^2\Psi.
\end{aligned}
\end{equation*}
Hence, observing that $\Psi\leq C\exp\bigl(-c/\varepsilon\bigr)$, we end up with
\begin{equation}\label{eq:Psi'}
\left|\Gamma\frac{d\Psi}{dt}\right|\leq \eta\|v\|^2+C\,\Gamma^2\exp(-c/\varepsilon)\Psi.
\end{equation}
Combining \eqref{eq:d/dtE} and \eqref{eq:Psi'}, we obtain that if $(\bm\xi,\tilde{w},\tilde{v})\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ is a solution of \eqref{eq:w-v-xi}, then
\begin{equation*}
\frac d{dt}\bigl\{E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h)\bigr\} \leq
-\eta\varepsilon^\mu E^{\bm\xi}[\tilde{w},\tilde{v}]+C\bigl(\varepsilon^{-2}+\Gamma^2\exp(-c/\varepsilon)\bigr)\Psi,
\end{equation*}
for some $\eta\in(0,1)$.
Therefore the estimate \eqref{eq:E-Psi} follows from
\begin{equation*}
C\exp(-c/\varepsilon)\Gamma^2-\eta\,\varepsilon^{\mu-2}\Gamma +C\varepsilon^{-2}\leq 0,
\end{equation*}
and the latter is verified for $\Gamma\in [\Gamma_1,\Gamma_2]$, provided $\varepsilon\in(0,\varepsilon_0)$ with $\varepsilon_0$ sufficiently small so that
$\eta^2\varepsilon^{2\mu} - 4C^2\varepsilon^{2}\exp(-c/\varepsilon) > 0$.
\end{proof}
\begin{rem}
Regarding the role of the parameter $\tau$ and its possible dependence on $\varepsilon$, we observe that Propositions \ref{prop:estimate-channel} and \ref{prop:energy-estimate}
are valid if the condition \eqref{eq:cond-eps-tau} holds.
Therefore, the parameter $\tau$ can be chosen of the order $\mathcal{O}\left(\varepsilon^{-k}\right)$ for some $k>0$ and the results of this section hold true by choosing $\theta>\max\{2,k+1\}$;
in particular, the estimate \eqref{eq:E-Psi} is valid with $\mu=\theta+k$.
On the other hand, if either $\tau$ is independent on $\varepsilon$ or $\tau\to0^+$ as $\varepsilon\to0^+$, we can choose any $\theta>2$ and the estimate \eqref{eq:E-Psi} is valid with $\mu=\theta$.
In general, if $\tau=\tau(\varepsilon)$ for some function $\tau:\mathbb{R}^+\rightarrow\mathbb{R}^+$,
then we can prove the results of Propositions \ref{prop:estimate-channel} and \ref{prop:energy-estimate} by working with the energy
\begin{equation*}
E^{\bm\xi}[\tilde{w},\tilde{v}]:=\frac12A^{\bm\xi}(\tilde{w})+\frac\tau2\|\tilde{v}\|^2+f(\varepsilon)\tau\langle\tilde{w},\tilde{v}\rangle,
\end{equation*}
where $f:\mathbb{R}^+\to\mathbb{R}^+$ is a function such that $f(\varepsilon)\tau(\varepsilon)/\varepsilon\to0^+$ and $f(\varepsilon)/\varepsilon^2\to0^+$ as $\varepsilon\to0^+$.
\end{rem}
Now, we have all the tools to prove our main result.
\begin{proof}[Proof of Theorem \ref{thm:main}]
Let $(\tilde{u}_0,\tilde{u}_1)\in\,\stackrel{\circ}{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ and let $(\tilde{u},\tilde{v})\in\mathcal{Z}_{{}_{\Gamma,\rho}}$
for $t\in[0,T_\varepsilon]$ be the solution to \eqref{eq:integrated-CH-sys}.
Then, $\tilde{u}=\tilde{u}^{\bm\xi}+\tilde{w}$ and $(\tilde{w},\tilde{v},\bm\xi)\in\hat{\mathcal{Z}}_{{}_{\Gamma,\rho}}$ solves the system \eqref{eq:w-v-xi} for $t\in[0,T_\varepsilon]$.
We have already seen that the property \eqref{eq:u-uh,h'} holds.
Assume that $T_\varepsilon$ is maximal and apply Proposition \ref{prop:energy-estimate}; from \eqref{eq:E-Psi}, it follows that
\begin{equation*}
\frac d{dt}\Bigl\{\exp(\eta\varepsilon^\mu t)(E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h))\Bigr\}\leq0,
\quad \qquad t\in[0,T_\varepsilon]
\end{equation*}
and so,
\begin{equation*}
\exp(\eta\varepsilon^\mu t)\left\{E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h)\right\}(t)\leq\left\{E^{\bm{\xi}}[\tilde{w},\tilde{v}]-\Gamma\varepsilon^{-2}\Psi(\bm h)\right\}(0)<0,
\qquad t\in[0,T_\varepsilon].
\end{equation*}
Therefore, $(\tilde{u},\tilde{v})$ remains in the channel $\mathcal{Z}_{{}_{\Gamma,\rho}}$ while $\bm h=\mathbf{G}(\bm\xi)\in\partial\Omega_\rho\in\Omega_\rho$
and if $T_\varepsilon<+\infty$ is maximal, then $\bm h(T_\varepsilon)\in\partial\Omega_\rho$, that is
\begin{equation}\label{hfrontiera}
h_j(T_\varepsilon)-h_{j-1}(T_\varepsilon)=\varepsilon/\rho \qquad \textrm{for some } j.
\end{equation}
From \eqref{eq:u-uh,h'} it follows that for all $t\in[0,T_\varepsilon]$, one has
\begin{equation}\label{dhmax}
|h_j(t)-h_j(0)|\leq C\varepsilon^{-2}\tau^{-1/2}\exp(-Al^{\bm h(t)}/\varepsilon)t \qquad \textrm{for any } j=1,\dots,N+1,
\end{equation}
where $l^{\bm h(t)}$ is the minimum distance between layers at the time $t$.
Combining \eqref{hfrontiera} and \eqref{dhmax}, we obtain
\begin{equation*}
\varepsilon/\rho\geq l^{\bm h(0)}-2C\varepsilon^{-2}\tau^{-1/2}\exp(-A/\rho)T_\varepsilon.
\end{equation*}
Hence, using \eqref{eq:triangle} we have
\begin{equation*}
T_\varepsilon\geq C\bigl(\ell^{\bm h(0)}-\varepsilon/\rho\bigr)\varepsilon^2\tau^{1/2}\exp(A/\rho)\geq
C\bigl(\ell^{\bm h(0)}-\varepsilon/\rho\bigr)\varepsilon^2\tau^{1/2}\exp(A\delta/\varepsilon),
\end{equation*}
and the proof is complete.
\end{proof}
\section{Layer dynamics}\label{sec:layers}
As we have seen in the previous section, there exist metastable states for the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH},
that are approximately equal to $+1$ or $-1$ except near $N+1$ transition points moving with exponentially small velocity.
The aim of this section is to derive and study a system of ODEs describing the movement of the transition layers.
Precisely, after deriving a system of ODEs from \eqref{eq:xi-compact},
we will compare such system with the one obtained in the case of the classic Cahn--Hilliard equation \eqref{eq:CH} by studying, in particular, the limit as $\tau\to0$.
\subsection{Equations of transition layers}
In order to derive the system of ODEs, we use the approximation $(\tilde{w},\tilde{v})\approx(0,\sum_{j=1}^N\xi'_j\tilde{u}_j^{\xi})$; substituting $\tilde{w}=0$ in \eqref{eq:xi} we get
\begin{equation*}
\sum_{i=1}^N\langle \tilde{u}_i^{\bm\xi},E_j^{\bm\xi}\rangle\xi'_i=\langle v,E^{\bm\xi}_j\rangle, \qquad j=1,\dots,N.
\end{equation*}
In order to eliminate the variable $v$, let us differentiate and multiply by $\tau$ the latter equation:
\begin{align*}
\tau\sum_{i,l=1}^N & \bigl(\langle\tilde{u}^{\bm\xi}_{il},E^{\bm\xi}_j\rangle+\langle\tilde{u}^{\bm\xi}_i,E^{\bm\xi}_{jl}\rangle\bigr)\xi'_l\xi'_i
+\tau\sum_{i=1}^N\langle\tilde{u}^{\bm\xi}_i,E^{\bm\xi}_j\rangle\xi''_i\\
& =-\langle\mathcal{L}(\tilde{u}^{\bm\xi}),E^{\bm\xi}_j\rangle-\langle v,E^{\bm\xi}_j\rangle+
\tau\sum_{l=1}^N\langle v,E^{\bm\xi}_{jl}\rangle\xi'_l, \qquad j=1,\dots,N.
\end{align*}
Using the approximation $\tilde{v}\approx\sum_{j=1}^N\xi'_j\tilde{u}_j^{\xi}$, we obtain
\begin{equation}\label{eq:xi-sum1}
\tau\sum_{i=1}^N\langle\tilde{u}_i^{\bm\xi},E^{\bm\xi}_j\rangle\xi''_i+\sum_{i=1}^N\langle\tilde{u}_i^{\bm\xi},E^{\bm\xi}_j\rangle\xi'_i+\tau\sum_{i,l=1}^N\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_j\rangle\xi'_i\xi'_l=
\langle\mathcal{L}\big(\tilde{u}^{\bm\xi}\big),E^{\bm\xi}_j\rangle,
\end{equation}
for $j=1,\dots,N$.
In order to simplify \eqref{eq:xi-sum1}, let us compute the terms $a_{ij}=\langle\tilde{u}_i^{\bm\xi},E^{\bm\xi}_j\rangle$, $\langle\tilde{u}_{il}^{\bm \xi},E^{\bm\xi}_j\rangle$
and $\langle\mathcal{L}\big(\tilde{u}^{\bm\xi}\big),E^{\bm\xi}_j\rangle$.
The formula for $a_{ij}$ is given in \eqref{eq:aij} and implies that the matrix $(a_{ij})\in\mathbb{R}^{N\times N}$ has the form
\begin{equation*}
(a_{ij})=\left(\begin{array}{ccccc} 4l_2 & 0 & 0 & \dots & 0\\
-4l_3 & 4l_3 & 0 & \dots & 0\\
4l_4 & -4l_4 & 4l_4 & \dots & 0\\
\dots & \dots & \dots & \dots & \dots\\
(-1)^{N-1}4l_{N+1} & (-1)^{N-2}4l_{N+1} & (-1)^{N-3}4l_{N+1} & \dots & 4l_{N+1}
\end{array}\right)+\mathcal{O}(\varepsilon),
\end{equation*}
with inverse
\begin{equation*}
(a_{ij})^{-1}:=\left(\begin{array}{cccccc} \displaystyle{\frac1{4l_2}} & 0 & 0 & \dots & 0 & 0\\
\displaystyle{\frac1{4l_2}} & \displaystyle{\frac1{4l_3}} & 0 & \dots & 0 & 0\\
0 & \displaystyle\frac1{4l_3} & \displaystyle\frac1{4l_4} & \dots & 0 & 0\\
\dots & \dots & \dots & \dots & \dots\\
0 & 0 & 0 & \dots & \displaystyle\frac1{4l_N} & \displaystyle\frac1{4l_{N+1}}\\
\end{array}\right)+\mathcal{O}(\varepsilon).
\end{equation*}
Next, for Lemma \ref{lem:u^h_j}, \eqref{eq:AC-CH} and the definition $E^{\bm\xi}_j$ \eqref{eq:Ej-Qj}, we have
\begin{equation}\label{eq:L,Ei}
\begin{aligned}
\langle\mathcal{L}\big(\tilde{u}^{\bm\xi}\big),E^{\bm\xi}_j\rangle & =\langle\mathcal{L}^{AC}\big(u^{\bm\xi}\big),u^{\bm h}_j+u^{\bm h}_{j+1}-Q'_{j}\rangle
= \langle\mathcal{L}^{AC}\big(u^{\bm\xi}\big),u^{\bm h}_j+u^{\bm h}_{j+1}\rangle+\mathcal{O}(e^{-c/\varepsilon})\\
& =\int_{{I_j}\cup{I_{j+1}}}\left(\varepsilon^2 u^{\bm h}_{xx}(x)-F'(u^{\bm h}(x))\right)u^{\bm h}_x(x)\,dx+\mathcal{O}(e^{-c/\varepsilon})\\
& =\alpha^{j+2}-\alpha^{j}+\mathcal{O}(e^{-c/\varepsilon}),
\end{aligned}
\end{equation}
for $j=1,\dots,N$.
Finally, let us compute the terms $\langle\tilde{u}_{il}^{\bm \xi},E^{\bm\xi}_j\rangle$, by using the formulas \eqref{eq:ut-j-h} and \eqref{eq:Ej} for $u_i^{\bm h}$ and $E^{\bm\xi}_j$, respectively.
In what follows, we omit the tedious, but straightforward computation of the derivatives of the exponentially small terms,
because one can prove (using the bounds in \cite{Bates-Xun1,Bates-Xun2,Carr-Pego,Carr-Pego2}) that they remain exponentially small in $\varepsilon$.
Therefore, differentiating the identities
\begin{equation*}
\tilde{u}^{\bm\xi}_i=\tilde{u}_i^{\bm h}+\tilde{u}_{N+1}^{\bm h}\frac{\partial h_{N+1}}{\partial h_i}, \qquad \mbox{ and } \qquad \frac{\partial h_{N+1}}{\partial h_i}=(-1)^{N-i}+\mathcal{O}(e^{-c/\varepsilon}),
\end{equation*}
we obtain
\begin{equation*}
\tilde{u}^{\bm\xi}_{il}=\tilde{u}_{il}^{\bm h}+(-1)^{N-l}\tilde{u}_{i,N+1}^{\bm h}+(-1)^{N-i}\tilde{u}_{N+1,l}^{\bm h}
+(-1)^{i+l}\tilde{u}_{N+1,N+1}^{\bm h}+\mathcal{O}(e^{-c/\varepsilon}).
\end{equation*}
From \eqref{eq:ut-j-h} and the formula for $u_i^{\bm h}$ of Lemma \ref{lem:u^h_j}, it follows that
\begin{equation*}
\tilde{u}_{ii}^{\bm h}(x)=\begin{cases}
u^{\bm h}_x(x)+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad &x\in I_i,\\
\mathcal{O}(e^{-c/\varepsilon}), & \mbox{otherwise},
\end{cases}
\end{equation*}
and $\tilde{u}_{il}^{\bm h}(x)=e$, $i\neq l$, for $i=1,\dots,N+1$.
Hence, we have
\begin{align}
\tilde{u}^{\bm\xi}_{ii}(x)& =\begin{cases}
u^{\bm h}_x(x)+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad & x\in I_i\cup I_{N+1},\\
\mathcal{O}(e^{-c/\varepsilon}), & \mbox{otherwise},
\end{cases} \qquad & \mbox{ for } i=1,\dots,N,\label{eq:ut-jj-xi}
\\
\tilde{u}^{\bm\xi}_{il}(x)& =\begin{cases}
(-1)^{i+l}u^{\bm h}_x(x)+\mathcal{O}(e^{-c/\varepsilon}), \qquad \qquad & x\in I_{N+1},\\
\mathcal{O}(e^{-c/\varepsilon}), & \mbox{otherwise},
\end{cases} & \mbox{ for } i\neq l. \label{eq:ut-jl-xi}
\end{align}
Thanks to the formulas \eqref{eq:Ej}, \eqref{eq:ut-jj-xi} and \eqref{eq:ut-jl-xi}, we can compute the quantities $\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_j\rangle$.
Let us start with the case $i=l=j\neq N$; for \eqref{eq:Ej} and \eqref{eq:ut-jj-xi}, we deduce
\begin{align*}
\langle\tilde{u}_{ii}^{\bm\xi},E^{\bm\xi}_i\rangle&=\int_0^1\tilde{u}_{ii}^{\bm\xi}(x)E^{\bm\xi}_i(x)\,dx=
\int_{h_{i-1/2}}^{h_{i+1/2}}u^{\bm h}_x(x)\left[(-1)^i-u^{\bm h}(x)\right]dx+\mathcal{O}(e^{-c/\varepsilon})\\
&=-\frac12\left[(-1)^i-u^{\bm h}(x)\right]^2\bigg|_{h_{i-1/2}}^{h_{i+1/2}}+\mathcal{O}(e^{-c/\varepsilon})=-2+\mathcal{O}(e^{-c/\varepsilon}),
\end{align*}
for $i=1,\dots,N-1$.
In the case $i=l=j=N$, we have
\begin{align*}
\langle\tilde{u}_{NN}^{\bm\xi},E^{\bm\xi}_N\rangle&=\int_0^1\tilde{u}_{NN}^{\bm\xi}(x)E^{\bm\xi}_N(x)\,dx=
\int_{h_{N-1/2}}^{1}u^{\bm h}_x(x)\left[(-1)^N-u^{\bm h}(x)\right]dx+\mathcal{O}(e^{-c/\varepsilon})\\
&=-\frac12\left[(-1)^N-u^{\bm h}(x)\right]^2\bigg|_{h_{N-1/2}}^{1}+\mathcal{O}(e^{-c/\varepsilon})= \mathcal{O}(e^{-c/\varepsilon}).
\end{align*}
The latter equality together with the expression for $(a_{ij})^{-1}$ and \eqref{eq:L,Ei} gives the equation for $\xi$ in the case $N=1$ (two layers):
equation \eqref{eq:xi-sum1} in the case $N=1$ becomes
\begin{equation*}
\tau\xi''+\xi'=\frac{1}{4l_2}(\alpha^{3}-\alpha^{1}).
\end{equation*}
Consider now the case $i=l=j+1$, $j\neq N$ with $N>1$; for the formulas \eqref{eq:Ej} and \eqref{eq:ut-jj-xi}, we infer
\begin{align*}
\langle\tilde{u}_{j+1,j+1}^{\bm\xi},E^{\bm\xi}_j\rangle&=\int_0^1\tilde{u}_{j+1,j+1}^{\bm\xi}(x)E^{\bm\xi}_j(x)\,dx=
\int_{h_{j+1/2}}^{h_{j+3/2}}u^{\bm h}_x(x)\left[(-1)^j-u^{\bm h}(x)\right]dx+\mathcal{O}(e^{-c/\varepsilon})\\
&=-\frac12\left[(-1)^j-u^{\bm h}(y)\right]^2\bigg|_{h_{j+1/2}}^{h_{j+3/2}}+\mathcal{O}(e^{-c/\varepsilon})=2+\mathcal{O}(e^{-c/\varepsilon}),
\end{align*}
If $j\neq N$ and either $i=l\neq j, j+1$ or $i\neq l$, then all the terms $\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_j\rangle$ are negligible for \eqref{eq:Ej} and \eqref{eq:ut-jl-xi}.
In conclusion, for $j\neq N$, we have
\begin{equation*}
\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_j\rangle=\mathcal{O}(e^{-c/\varepsilon})+\begin{cases}
-2, \qquad \qquad & i=l=j,\\
2, & i=l=j+1,\\
0, &\mbox{otherwise}.
\end{cases}
\end{equation*}
Hence, the first $N-1$ equations of \eqref{eq:xi-sum1} become
\begin{equation*}
\sum_{i=1}^{N}(\tau\xi''_i+\xi'_i)a_{ij}
+2\tau\left[\left(\xi'_{j+1}\right)^2-\left(\xi'_j\right)^2\right]=\alpha^{j+2}-\alpha^{j},
\end{equation*}
for $j=1,\dots,N-1$.
The last equation of \eqref{eq:xi-sum1} is more difficult because the functions $\tilde{u}_{il}^{\bm\xi}$ and $E^{\bm\xi}_N$ are not negligible in $I_{N+1}$.
We have already seen that $\langle\tilde{u}_{NN}^{\bm\xi},E^{\bm\xi}_N\rangle=e$, for the other terms we have
\begin{align*}
\langle\tilde{u}_{ii}^{\bm\xi},E^{\bm\xi}_N\rangle&=\int_0^1\tilde{u}_{ii}^{\bm\xi}(x)E^{\bm\xi}_N(x)\,dx=
\int_{h_{N+1/2}}^{1}u^{\bm h}_x(x)\left[(-1)^N-u^{\bm h}(x)\right]dx+\mathcal{O}(e^{-c/\varepsilon})\\
&=-\frac12\left[(-1)^N-u^{\bm h}(x)\right]^2\bigg|_{h_{N+1/2}}^{1}+\mathcal{O}(e^{-c/\varepsilon})=2+\mathcal{O}(e^{-c/\varepsilon}),
\end{align*}
for $i=1,\dots, N-1$, and
\begin{align*}
\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_N\rangle&=\int_0^1\tilde{u}_{il}^{\bm\xi}(x)E^{\bm\xi}_N(x)\,dx=
(-1)^{i+l}\int_{h_{N+1/2}}^{1}u^{\bm h}_x(x)\left[(-1)^N-u^{\bm h}(x)\right]dx+\mathcal{O}(e^{-c/\varepsilon})\\
&=-\frac{(-1)^{i+l}}2\left[(-1)^N-u^{\bm h}(x)\right]^2\bigg|_{h_{N+1/2}}^{1}+\mathcal{O}(e^{-c/\varepsilon})=2(-1)^{i+l}+\mathcal{O}(e^{-c/\varepsilon}),
\end{align*}
for $i\neq l$.
Therefore,
\begin{equation*}
\langle\tilde{u}_{il}^{\bm\xi},E^{\bm\xi}_N\rangle=\mathcal{O}(e^{-c/\varepsilon})+\begin{cases}
0, \qquad \qquad \qquad & i=l=N,\\
2, & i=l\neq N,\\
2(-1)^{i+l}, &\mbox{otherwise}.
\end{cases}
\end{equation*}
It follows that the last equation of \eqref{eq:xi-sum1} becomes
\begin{equation*}
\sum_{i=1}^N(\tau\xi''_i+\xi'_i)a_{ij}
+2\tau\left[\sum_{i=1}^{N-1}\left(\xi'_i\right)^2+\sum_{i\neq l}(-1)^{i+l}\xi'_i\xi'_l\right]=\alpha^{N+2}-\alpha^N.
\end{equation*}
Since
\begin{align*}
\sum_{i=1}^{N-1}\left(\xi'_i\right)^2+\sum_{i\neq l}(-1)^{i+l}\xi'_i\xi'_l&=\left(\sum_{i=1}^{N-1}(-1)^{N-j}\xi'_i\right)^2+2\xi'_N\sum_{i=1}^{N-1}(-1)^{N-j}\xi'_i\\
&=\left(\sum_{i=1}^{N}(-1)^{N-i}\xi'_i-\xi'_N\right)\left(\sum_{i=1}^{N}(-1)^{N-i}\xi'_i+\xi'_N\right),
\end{align*}
we can rewrite
\begin{equation*}
\sum_{i=1}^N(\tau\xi''_i+\xi'_i)a_{ij}
+2\tau\left[\left(\sum_{i=1}^{N}(-1)^{N-i}\xi'_i\right)^2-\left(\xi'_N\right)^2\right]=\alpha^{N+2}-\alpha^N.
\end{equation*}
By applying the inverse matrix $(a_{ij})^{-1}$, we obtain the following equation for $\bm\xi$:
\begin{equation*}
\begin{aligned}
\tau\xi_1''+\xi_1'+\frac{\tau}{2l_2}Q(\xi'_2,\xi'_1)&=P_1(\bm h),\\
\tau\xi_i''+\xi_i'+\frac{\tau}{2l_i}Q(\xi'_i,\xi'_{i-1})+\frac{\tau}{2l_{i+1}}Q(\xi'_{i+1},\xi'_i)&=P_{i-1}(\bm h)+P_i(\bm h), \\
&\quad i=2,\dots,N-1, \\
\tau\xi''_N+\xi'_N+\frac\tau{2l_N}Q(\xi'_N,\xi'_{N-1})+\frac\tau{2l_{N+1}}Q\left(\sum_{j=1}^{N}(-1)^{N-j}\xi'_j,\xi'_N\right)&=
P_{N-1}(\bm h)+P_N(\bm h),
\end{aligned}
\end{equation*}
where we introduced the functions
\begin{equation}\label{eq:Q}
Q(x,y):=x^2-y^2, \qquad\mbox{ and}\qquad P_i(\bm h):=\frac{1}{4l_{i+1}}(\alpha^{i+2}-\alpha^{i}), \qquad i=1,\dots,N.
\end{equation}
Therefore, we derived the equation for $\bm\xi=(\xi_1,\dots,\xi_N)$;
recall that the transition points are located at $\bm h=(h_1,\dots,h_N,h_{N+1})$ and that $\xi_i=h_i$ for $i=1,\dots,N$;
the position of the last point $h_{N+1}$ is determined by the other points $(h_1,\dots,h_N)$ for the conservation of the mass.
In order to write the equation for $\bm h=(h_1,\dots,h_N,h_{N+1})$, which is more natural, symmetric and easy to handle,
we use \eqref{eq:h'_N+1} by neglecting the exponentially smallest terms;
thus, we consider the approximations
\begin{equation*}
h'_{N+1}\approx\sum_{j=1}^N(-1)^{N-j}h'_j, \qquad \qquad h''_{N+1}\approx\sum_{j=1}^N(-1)^{N-j}h''_j.
\end{equation*}
We can now write the equation for $\bm h=(h_1,\dots,h_N,h_{N+1})$.
In the case $N=1$ (two layers) we have
\begin{equation}\label{eq:h2layers}
\begin{aligned}
\tau h_1''+h_1'=\frac{1}{4l_2}(\alpha^3-\alpha^1),\\
\tau h_2''+h_2'=\frac{1}{4l_2}(\alpha^3-\alpha^1).
\end{aligned}
\end{equation}
In general, for $N\geq2$ the equations are
\begin{equation}\label{eq:hN+1layers}
\begin{aligned}
\tau h_1''+h_1'+\frac{\tau}{2l_2}Q(h'_2,h'_1)&=P_{1}(\bm h),\\
\tau h_i''+h_i'+\frac{\tau}{2l_{i}}Q(h'_i,h'_{i-1})+\frac\tau{2l_{i+1}}Q(h'_{i+1},h'_i)&=P_{i-1}(\bm h)+P_i(\bm h), \qquad \, i=2,\dots,N, \\
\tau h_{N+1}''+h_{N+1}'+\frac{\tau}{2l_{N+1}}Q(h'_{N+1},h'_N)&=P_{N}(\bm h).
\end{aligned}
\end{equation}
Equations \eqref{eq:h2layers} and \eqref{eq:hN+1layers} imply the following equations for the interval length $l_j$ (remember that $l_1=h_1-h_0=2h_1$ and $l_{N+2}=h_{N+2}-h_{N+1}=2(1-h_{N+1})$).
For $N=1$ one has
\begin{equation}\label{eq:l2layers}
\begin{aligned}
\tau l_1''+l_1'&=\frac{1}{2l_2}(\alpha^3-\alpha^1),\\
\tau l_2''+l'_2&=0,\\
\tau l_3''+l_3'&=-\frac{1}{2l_2}(\alpha^3-\alpha^1).
\end{aligned}
\end{equation}
For $N=2$:
\begin{equation}\label{eq:l3layers}
\begin{aligned}
\tau l_1''+l_1'+\frac{\tau}{l_2}\left[\left(h_2'\right)^2-\left(h_1'\right)^2\right]&=\frac{1}{2l_2}(\alpha^3-\alpha^1),\\
\tau l_2''+l'_2+\frac{\tau}{2l_{3}}\left[\left(h'_3\right)^2-\left(h_2'\right)^2\right]&=\frac{1}{4l_{3}}(\alpha^{4}-\alpha^{2}),\\
\tau l_3''+l_3'-\frac{\tau}{2l_2}\left[\left(h_{2}'\right)^2-\left(h_{1}'\right)^2\right]&=-\frac{1}{4l_2}(\alpha^{3}-\alpha^{1}),\\
\tau l_4''+l_4'-\frac{\tau}{l_{3}}\left[\left(h'_3\right)^2-\left(h_2'\right)^2\right]&=-\frac{1}{2l_{3}}(\alpha^{4}-\alpha^{2}).
\end{aligned}
\end{equation}
In general, for $N\geq3$:
\begin{equation}\label{eq:lN+1layers}
\begin{aligned}
\tau l_1''+l_1'+\frac\tau{l_2} Q(h'_2,h'_1)&=2P_1(\bm h),\\
\tau l_2''+l_2'+\frac\tau{2l_3}Q(h'_3,h_2)&=P_2(\bm h),\\
\tau l_i''+l_i'+\frac\tau{2l_{i+1}}Q(h'_{i+1},h'_i)-\frac{\tau}{2l_{i-1}}Q(h'_{i-1},h'_{i-2})&=P_{i}(\bm h)-P_{i-2}(\bm h), \qquad i=3,\dots,N, \\
\tau l_{N+1}''+l_{N+1}'-\frac{\tau}{2l_{N}}Q(h'_N,h'_{N-1})&=-P_{N-1}(\bm h),\\
\tau l_{N+2}''+l_{N+2}'-\frac\tau{l_{N+1}}Q(h'_{N+1},h'_N)&=-2P_{N}(\bm h).
\end{aligned}
\end{equation}
Observe that $l_1/2$ and $l_{N+2}/2$ are the distances of $h_1$ and $h_{N+1}$ from the boundary points $0$ and $1$, respectively.
Let $L_-$ and $L_+$ be the length of all the intervals where the solution is approximately $-1$ and $+1$, respectively; namely
\begin{align*}
L_-:&=\frac{l_1}2+\sum_{i=1}^{N/2} l_{2i+1}, \qquad \qquad \qquad &L_+&=\sum_{i=1}^{N/2} l_{2i}+\frac{l_{N+2}}{2}, \qquad &\mbox{ if }\, N \mbox{ is even},\\
L_-:&=\frac{l_1}2+\sum_{i=1}^{(N-1)/2} l_{2i+1}+\frac{l_{N+2}}{2}, &L_+&=\sum_{i=1}^{(N+1)/2} l_{2i}, &\mbox{ if }\, N \mbox{ is odd}.
\end{align*}
It follows that these quantities satisfy
\begin{equation}\label{eq:L_pm}
\tau L_{\pm}''+L_{\pm}'=0.
\end{equation}
\subsection{Comparison with the classic Cahn--Hilliard equation}
In this subsection, we study the equations describing the movement of the transition points derived above,
and we analyze the differences with the corresponding equations valid for the classic Cahn--Hilliard equation \eqref{eq:CH}.
Rewrite the equations \eqref{eq:h2layers} and \eqref{eq:hN+1layers} in a compact form:
in the case of two layers ($N=1$), see equations \eqref{eq:h2layers}, we get
\begin{equation}\label{eq:h2layers-compact}
\tau\bm h''+\bm h'=\bm{\mathcal{P}}(\bm h),
\end{equation}
where $\bm h=(h_1,h_2)$ and $\bm{\mathcal{P}}:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined by
$$
\mathcal P_i(h_1,h_2):=\frac{\alpha^3-\alpha^1}{4(h_2-h_1)}, \qquad \qquad i=1,2.
$$
In the case of $N+1$ layers with $N\geq2$, we rewrite \eqref{eq:hN+1layers} as
\begin{equation}\label{eq:hN+1layers-compact}
\tau \bm h''+\bm h'+\tau\bm{\mathcal{Q}}(\bm h,\bm h')=\bm{\mathcal P}(\bm h),
\end{equation}
where $\bm h=(h_1,\dots,h_{N+1})$ and $\bm{\mathcal P}:\mathbb{R}^{N+1}\rightarrow\mathbb{R}^{N+1}$ is defined by
\begin{equation}\label{eq:P(h)}
\bm{\mathcal P}(\bm h):=\left(\begin{array}{c}
\displaystyle\frac{\alpha^3-\alpha^1}{4(h_2-h_1)}\vspace{0.1cm}\\
\displaystyle{\frac{\alpha^3-\alpha^1}{4(h_2-h_1)}+\frac{\alpha^4-\alpha^2}{4(h_3-h_2)}}\\
\vdots \\ \vdots \\
\displaystyle{\frac{\alpha^{N+1}-\alpha^{N-1}}{4(h_N-h_{N-1})}+\frac{\alpha^{N+2}-\alpha^N}{4(h_{N+1}-h_N)}}\vspace{0.1cm}\\
\displaystyle{\frac{\alpha^{N+2}-\alpha^N}{4(h_{N+1}-h_N)}}
\end{array}\right),
\end{equation}
and $\bm{\mathcal{Q}}:\mathbb{R}^{N+1}\times\mathbb{R}^{N+1}\rightarrow\mathbb{R}^{N+1}$ is
\begin{equation}\label{eq:Q(h)}
\bm{\mathcal Q}(\bm h,\bm h'):=\left(\begin{array}{c}
\displaystyle\frac{\left(h_2'\right)^2-\left(h_1'\right)^2}{2(h_2-h_1)},\vspace{0.1cm}\\
\displaystyle\frac{\left(h_2'\right)^2-\left(h_1'\right)^2}{2(h_2-h_1)}+\frac{\left(h_3'\right)^2-\left(h_2'\right)^2}{2(h_3-h_2)}\\
\vdots\\ \vdots \\
\displaystyle\frac{\left(h_N'\right)^2-\left(h_{N-1}'\right)^2}{2(h_N-h_{N-1})}
+\frac{\left(h_{N+1}'\right)^2-\left(h_N'\right)^2}{2(h_{N+1}-h_N)}\vspace{0.1cm}\\
\displaystyle\frac{\left(h_{N+1}'\right)^2-\left(h_N'\right)^2}{2(h_{N+1}-h_N)}
\end{array}\right).
\end{equation}
Both in the case \eqref{eq:h2layers-compact} and in the case \eqref{eq:hN+1layers-compact}, taking formally the limit as $\tau\to0^+$
one obtains the system describing the motion of the transition layers in the classic Cahn--Hilliard equation \eqref{eq:CH} (see equations (4.36) in \cite{Bates-Xun2}).
Indeed, in \cite{Bates-Xun2} the authors derived the following system of ODEs to approximately describe the motion of the transition points $h_1,h_2,\dots,h_{N+1}$ when they are well separated:
\begin{equation}\label{eq:h_i-CahnHilliard}
\begin{aligned}
h_1'&=\frac{1}{4l_2}(\alpha^3-\alpha^1),\\
h_j'&=\frac{1}{4l_j}(\alpha^{j+1}-\alpha^{j-1})+\frac{1}{4l_{j+1}}(\alpha^{j+2}-\alpha^j), \qquad \qquad \qquad j=2,\dots,N\\
h_{N+1}'&=\frac{1}{4l_{N+1}}(\alpha^{N+2}-\alpha^N).
\end{aligned}
\end{equation}
Let us briefly describe the behavior of the solutions to \eqref{eq:h_i-CahnHilliard} when $F$ is an even function
and $h_i$, $h_{i-1}$ are the closest transition points at time $t=0$, namely assume that there exists a unique $i\in\{1,\dots,N+2\}$ such that
\begin{equation*}
l_i(0)<l_j(0), \quad j\neq i, \quad j=1,\dots,N+2.
\end{equation*}
In this case, we can use Remark \ref{rem:alfa} and from the estimate \eqref{eq:alfa^j-alfa^i} it follows that $\alpha^i\gg\alpha^j$ for all $j\neq i$,
and the terms $\alpha^j$ with $j\neq i$ are exponentially small with respect to $\alpha^i$ as $\varepsilon\to0^+$.
As a consequence, we can describe the motion of the transition layers in the case of the Cahn--Hilliard equation \eqref{eq:CH} as follows.
In the case $N=1$, the two transition points $h_1$ and $h_2$ move to the right (respectively left) if $l_3=2(1-h_2)<l_1=2h_1$ (respectively $l_3>l_1$) and we have $h'_1\approx h'_2$;
thus, the points move together in an almost rigid way, they move in the same direction at approximately the same speed.
In the case $N=2$, we have two transitions points moving in the same direction at approximately the same speed $v$,
while the speed of the third one is exponentially small with respect to $v$,
and so, the third point is essentially static.
Finally, consider the case $N\geq3$ with $i\in\{3,\dots,N\}$;
the term $\alpha^i$ appears in the equations for $h_{i-2}$, $h_{i-1}$, $h_{i}$ and $h_{i+1}$,
and so we have four points moving at approximately the same speed,
while all the other layers remain essentially stationary in time.
Precisely, we have
\begin{equation*}
h'_{i-2}>0, \; h'_{i-1}>0, \; h'_{i}<0, \; h'_{i+1}<0, \; h'_j=\mathcal{O}\left(e^{-C/\varepsilon}h'_i\right) \, \mbox{ for } j\notin\{i-2,i-1,i,i+1\}.
\end{equation*}
Roughly speaking, the system \eqref{eq:h_i-CahnHilliard} shows that the shortest distance between layers decreases:
the closest layers move towards each other, each being followed by its nearest transition point from ``behind'', at approximately the same speed,
until the points $h_i$ and $h_{i-1}$ are sufficiently close.
Hence, the loss of the mass due to the annihilation of the transitions at $h_{i-1}$ and $h_i$ is compensated by the movement of the nearest neighbor $h_{i-2}$ and $h_{i+1}$.
This property, due to the conservation of the mass, is a fundamental difference with respect to the Allen--Cahn equation \eqref{eq:AC}.
For such equation, Carr and Pego \cite{Carr-Pego} derived the following equations for the transition points $h_j$:
\begin{equation*}
h'_j=C\varepsilon\left(\alpha^{j+1}-\alpha^j\right), \qquad \qquad j=1,\dots,N+1,
\end{equation*}
where $C$ is a constant depending only on $F$.
Then, in the case of the Allen--Cahn equation, the closest layers move towards each other at approximately the same speed satisfying $|h'_i|\approx\varepsilon|\alpha^{i+1}-\alpha^i|$,
while all the other points remain essentially stationary in time.
As it was already mentioned, system \eqref{eq:h_i-CahnHilliard} was derived in \cite[Section 4]{Bates-Xun2} in order to approximately describe
the movement of the transition layers for the Cahn--Hilliard equation \eqref{eq:CH} until the points are well separated, with distance $l_j>\varepsilon/\rho$.
A detailed analysis of the motion of the layers for \eqref{eq:CH} can be also found in \cite{SunWard},
where the authors studied in details layer collapse events and presented many numerical simulations
confirming that the layer dynamics is closely described by \eqref{eq:h_i-CahnHilliard}.
However, system \eqref{eq:h_i-CahnHilliard} provides an accurate description of the motion of the points corresponding to the annihilating interval and its two nearest neighbors,
but it may be slightly inaccurate for other layers.
For example, in \cite{SunWard} it is showed that if $(h_{i-1},h_i)$ is the annihilating interval for some $i\in\{3,\dots,N\}$,
all the points $h_j$ with $j\notin\{i-2,i-1,i,i+1\}$ move at an algebraic slower speed in $\varepsilon$ than $h_i$.
In contrast, we saw that for \eqref{eq:h_i-CahnHilliard} the points $h_j$ move exponentially slower than the collapsing layers.
Apart from that, system \eqref{eq:h_i-CahnHilliard} provides a good description of the layer dynamics for the classic Cahn--Hilliard equation \eqref{eq:CH}.
In the case of the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH}, the movement of the layer is approximately describe by equations \eqref{eq:h2layers-compact} and \eqref{eq:hN+1layers-compact},
and so, we have to take in account the inertial term $\tau\bm h''$ and the quadratic term $\tau\bm{\mathcal{Q}}(\bm h,\bm h')$ (when $N\geq2$).
In the following, we shall compute some numerical solutions in order to analyze the differences between systems \eqref{eq:hN+1layers} and \eqref{eq:h_i-CahnHilliard}.
To do this, we use Proposition \ref{prop:alfa,beta} choosing $A_+=A_-=\sqrt2$ and $K_+=K_-=4$, which corresponds to the choice $F(u)=\frac14(u^2-1)^2$;
then, we use the approximation
\begin{equation*}
\alpha^j\approx16\exp\left(-\frac{\sqrt2(h_j-h_{j-1})}{\varepsilon}\right).
\end{equation*}
The values of the initial data for the ODEs \eqref{eq:hN+1layers} depend on the choice of the initial datum $(u_0,u_1)$ for the PDE \eqref{eq:hyp-CH};
precisely, we assume that $u_0=u^{\bm{h^0}}$ for some $\bm{h^0}\in\Omega_\rho$, and so $\bm h(0)=\bm{h^0}$ represents the positions of the transition points at time $t=0$,
while the first $N$ components of $\bm h'(0)$ satisfy the third equation of \eqref{eq:w-v-xi}, and $h_{N+1}$ is given by \eqref{eq:h'_N+1} (for the conservation of the mass).
Therefore, we have
\begin{equation}\label{eq:initial-h}
\begin{aligned}
h'_1(0)&=\frac{1}{4l_2(0)}\langle\tilde{u}_1,E_1^{\bm\xi}\rangle+\mathcal{O}(\varepsilon\langle\tilde{u}_1,E_1^{\bm\xi}\rangle),\\
h'_j(0)&=\frac{1}{4l_j(0)}\langle\tilde{u}_1,E_{j-1}^{\bm\xi}\rangle+\frac{1}{4l_{j+1}(0)}\langle\tilde{u}_1,E_j^{\bm\xi}\rangle+\mathcal{O}(\varepsilon\langle\tilde{u}_1,E_j^{\bm\xi}\rangle),
\qquad \qquad j=2,\dots,N\\
h'_{N+1}(0)&=\sum_{j=1}^N\frac{\partial h_{N+1}}{\partial h_j}h'_j(0)=\frac{1}{4l_{N+1}(0)}\langle\tilde{u}_1,E_N^{\bm\xi}\rangle+\mathcal{O}(\varepsilon\|\tilde{u}_1\|).
\end{aligned}
\end{equation}
As we have previously done for the ODEs \eqref{eq:hN+1layers}, we consider equations \eqref{eq:initial-h} without the smallest terms $\mathcal{O}(\varepsilon\langle\tilde{u}_1,E_j^{\bm\xi}\rangle)$.
By reasoning as in the computation of \eqref{eq:L_pm}, we get $L_{\pm}'(0)=0$, and so $L_\pm(t)=0$ for all $t$ and this is consistent with the mass conservation.
In particular, let us stress that in the 2 layers case $(N=1)$ we have $h'_1(0)=h'_2(0)$.
Finally, notice that choosing $\tilde{u}_1=\mathcal L(\tilde{u}_0)$ we deduce that $h'_j(0)$ satisfies \eqref{eq:h_i-CahnHilliard}.
We want to focus the attention on the role of the parameter $\tau$ and we consider the same initial data for \eqref{eq:hN+1layers} and \eqref{eq:h_i-CahnHilliard};
in particular for \eqref{eq:h2layers}-\eqref{eq:hN+1layers}, we choose $h'_j(0)$ satisfying \eqref{eq:h_i-CahnHilliard}, meaning that $h'_j(0)$ satisfy \eqref{eq:initial-h} with $\tilde{u}_1=\mathcal L(\tilde{u}_0)$.
Let us start with the case of $2$ layers.
Observe that $l_2=h_2-h_1$ satisfies \eqref{eq:l2layers} and since $h'_1(0)=h'_2(0)$, we have $l_2(t)=l_2(0)$ for any time $t$.
In the first example, we choose $\varepsilon=0.07$: in Table \ref{table:2layers_eps0.07} we show the numerical computation of the difference $h_1(t)-h_1(0)$ for different times $t$
and for different values of $\tau$ ($\tau=0$ corresponds to system \eqref{eq:h_i-CahnHilliard});
since $l_2$ is constant in time, we get $h_2(t)=s(t)+h_2(0)$; in Figure \ref{fig:h1} we show the graph of $h_1$ for $\tau=0$ and $\tau=50$.
\begin{table}[h!]
\vskip0.2cm
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline TIME $t$ & $s(t)$, $\tau=0$ & $s(t)$, $\tau=5$ & $s(t)$, $\tau=50$ \\
\hline $300$ & $-0.0128$ & $-0.0126$ & $-0.0113$ \\
\hline $600$ & $-0.0534$ & $-0.0497$ & $-0.0364$ \\
\hline $665$ & $-0.1240$ & $-0.0830$ & $-0.0475$ \\ \hline
\end{tabular}
\caption{The numerical computation of $s(t)=h_1(t)-h_1(0)$ for $\varepsilon=0.07$ and different values of $\tau$.
The initial positions of the layers are $h_1(0)=0.31$, $h_2(0)=0.66$.}
\label{table:2layers_eps0.07}
\end{center}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[height=5.5cm, width=6.9cm]{h1_tau0}\quad
\includegraphics[height=5.5cm, width=6.9cm]{h1_tau50}
\caption{The graph of $h_1(t)$ for $\varepsilon=0.07$ in the case of systems \eqref{eq:h_i-CahnHilliard} (left) and \eqref{eq:h2layers} (right) with $\tau=50$.}
\label{fig:h1}
\end{figure}
We see that the greater $\tau$ is, the slower the movement of the layers is.
In particular, in Figure \ref{fig:h1} we see that the behavior of $h_1$ is the same, but the time taken for $h_1$ to reach the position $0.1$ is greater in the case
of system \eqref{eq:h2layers} with $\tau=50$.
Now, we consider an example with $6$ layers.
For $\varepsilon=0.008$, in Tables \ref{table:6layers-tau0} and \ref{table:6layers-tau125} we numerically compute the difference $h_i(t)-h_i(0)$ for $i=1,\dots,6$
in the case $\tau=0$ and $\tau=\varepsilon^{-1}=125$, respectively,
and, in particular, we see that in the case $\tau=125$ the layers move slower than in the case without inertial terms.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline $s_i(t)$ & $t=10^2$ & $t=10^4$ & $t=10^5$ & $t=1.55*10^5$ \\
\hline $s_1(t)$ & $2.99*10^{-7}$ & $3.00*10^{-5}$ & $3.13*10^{-4}$ & $4.96*10^{-4}$ \\
\hline $s_2(t)$ & $2.13*10^{-6}$ & $2.19*10^{-4}$ & $3.27*10^{-3}$ & $1.36*10^{-2}$ \\
\hline $s_3(t)$ & $1.54*10^{-6}$ & $1.60*10^{-4}$ & $2.64*10^{-3}$ & $1.25*10^{-2}$ \\
\hline $s_4(t)$ & $-2.03*10^{-6}$ & $-2.09*10^{-4}$ & $-3.09*10^{-3}$ & $-1.26*10^{-2}$ \\
\hline $s_5(t)$ & $-1.79*10^{-6}$ & $-1.85*10^{-4}$ & $-2.82*10^{-3}$ & $-1.21*10^{-2}$ \\
\hline $s_6(t)$ & $-4.76*10^{-8}$ & $-4.75*10^{-6}$ & $-4.62*10^{-5}$ & $-6.99*10^{-5}$ \\
\hline
\end{tabular}
\caption{The numerical computation of $s_i(t)=h_i(t)-h_i(0)$ in the case of system \eqref{eq:h_i-CahnHilliard} for $\varepsilon=0.008$.
The initial positions of the layers are $0.18, 0.32, 0.45, 0.57, 0.71, 0.86$.}
\label{table:6layers-tau0}
\end{center}
\end{table}
\begin{table}[h!]
\vskip0.2cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline $s_i(t)$ & $t=10^2$ & $t=10^4$ & $t=10^5$ & $t=1.55*10^5$ \\
\hline $s_1(t)$ & $2.99*10^{-7}$ & $3.00*10^{-5}$ & $3.13*10^{-4}$ & $4.94*10^{-4}$ \\
\hline $s_2(t)$ & $2.13*10^{-6}$ & $2.19*10^{-4}$ & $3.26*10^{-3}$ & $1.27*10^{-2}$ \\
\hline $s_3(t)$ & $1.54*10^{-6}$ & $1.60*10^{-4}$ & $2.63*10^{-3}$ & $1.17*10^{-2}$ \\
\hline $s_4(t)$ & $-2.03*10^{-6}$ & $-2.09*10^{-4}$ & $-3.08*10^{-3}$ & $-1.18*10^{-2}$ \\
\hline $s_5(t)$ & $-1.79*10^{-6}$ & $-1.84*10^{-4}$ & $-2.81*10^{-3}$ & $-1.13*10^{-2}$ \\
\hline $s_6(t)$ & $-4.76*10^{-8}$ & $-4.75*10^{-6}$ & $-4.62*10^{-5}$ & $-7.09*10^{-5}$ \\
\hline
\end{tabular}
\caption{The numerical computation of $s_i(t)=h_i(t)-h_i(0)$ in the case of system \eqref{eq:hN+1layers}.
The values of the parameters are $\varepsilon=0.008$ and $\tau=125$; the initial positions of the layers are $0.18, 0.32, 0.45, 0.57, 0.71, 0.86$.}
\label{table:6layers-tau125}
\end{center}
\end{table}
In the previous computations we choose the same initial velocities for \eqref{eq:h_i-CahnHilliard}-\eqref{eq:hN+1layers} and
the only difference is that in the case of system \eqref{eq:hN+1layers} the layers move slower than \eqref{eq:h_i-CahnHilliard}.
On the other hand, choosing different initial velocities, according to \eqref{eq:initial-h}, we can observe different dynamics.
For instance, in the case of system \eqref{eq:hN+1layers} the points can change direction:
in Table \ref{table:change} we consider the same $\varepsilon$, $\tau$ and initial positions of the Table \ref{table:6layers-tau125}, but with opposite initial velocities,
namely, we choose $\tilde{u}_1=-\mathcal L(\tilde{u}_0)$ in \eqref{eq:initial-h}.
We see that the points change direction and after that we have the same behavior of Table \ref{table:6layers-tau125}.
\begin{table}[h!]
\vskip0.2cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline $s_i(t)$ & $t=10^2$ & $t=2*10^2$ & $t=10^4$ & $t=10^5$ & $t=1.55*10^5$ \\
\hline $s_1(t)$ & $-0.11*10^{-6}$ & $1.4*10^{-9}$ & $2.93*10^{-5}$ & $3.12*10^{-4}$ & $4.94*10^{-4}$ \\
\hline $s_2(t)$ & $-0.80*10^{-6}$ & $9.7*10^{-9} $ & $2.14*10^{-4}$ & $3.25*10^{-3}$ & $1.22*10^{-2}$ \\
\hline $s_3(t)$ & $-0.58*10^{-6}$ & $7*10^{-9}$ & $1.56*10^{-4}$ & $2.62*10^{-3}$ & $1.12*10^{-2}$ \\
\hline $s_4(t)$ & $0.76*10^{-6}$ & $-9.3*10^{-9}$ & $-2.04*10^{-4}$ & $-3.07*10^{-3}$ & $-1.14*10^{-2}$ \\
\hline $s_5(t)$ & $0.67*10^{-6}$ & $-8.2*10^{-9}$ & $-1.80*10^{-4}$ & $-2.80*10^{-3}$ & $-1.09*10^{-2}$ \\
\hline $s_6(t)$ & $0.02*10^{-6}$ & $-2.3*10^{-10}$ & $-4.63*10^{-6}$ & $-4.61*10^{-5}$ & $-7.06*10^{-5}$ \\
\hline
\end{tabular}
\caption{In this table we consider the same initial positions and the same values of $\varepsilon$ and $\tau$ of Table \ref{table:6layers-tau125},
but initial velocities with opposite sign respect to Table \ref{table:6layers-tau125}.}
\label{table:change}
\end{center}
\end{table}
We conclude this paper by comparing the solutions to systems \eqref{eq:hN+1layers} and \eqref{eq:h_i-CahnHilliard} as $\tau\to0^+$.
Let us rewrite system \eqref{eq:hN+1layers-compact} in the form
\begin{equation}\label{eq:system-h-h'}
\begin{cases}
\bm h'=\bm\eta,\\
\tau\bm\eta'=\bm{\mathcal P}(\bm h)-\bm\eta-\tau\bm{\mathcal Q}(\bm h,\bm\eta),
\end{cases}
\end{equation}
and system \eqref{eq:h_i-CahnHilliard} in the form
\begin{equation}\label{eq:h-CahnHilliard}
\begin{cases}
\bm h'=\bm\eta,\\
\bm\eta=\bm{\mathcal P}(\bm h),
\end{cases}
\end{equation}
where $\bm{\mathcal P}$ and $\bm{\mathcal Q}$ are defined in \eqref{eq:P(h)} and \eqref{eq:Q(h)}.
Notice that the functions $\bm{\mathcal P}$ and $\bm{\mathcal Q}$ are not well defined when $h_j=h_{j+1}$ for some $j$,
but here we are interested in studying the system \eqref{eq:system-h-h'} when $l_j(t)>\delta$ for any $t\in[0,T]$ and any $j$ for some positive $\delta$ and $T$,
because system \eqref{eq:system-h-h'} describes the movement of the transition points when they are well separated for the hyperbolic Cahn--Hilliard equation \eqref{eq:hyp-CH}.
Therefore, in the following we consider system \eqref{eq:system-h-h'} for $t\in[0,T]$ where $T$ is such that $l_j(t)>\delta>0$ for any $t\in[0,T]$ and any $j\in\{1,N+2\}$.
Denote by $(\bm h,\bm\eta)$ the solutions to \eqref{eq:system-h-h'} and $(\bm h_c,\bm\eta_c)$ the solutions of \eqref{eq:h-CahnHilliard},
and set
\begin{equation*}
\mathcal E_\tau(t):=|\bm h(t)-\bm h_c(t)|+\tau|\bm\eta(t)-\bm\eta_c(t)|.
\end{equation*}
A general theorem of Tihonov on singular perturbations can be applied to systems \eqref{eq:system-h-h'}-\eqref{eq:h-CahnHilliard} to prove that
if $(\bm h,\bm\eta)$ is a bounded solution of \eqref{eq:system-h-h'} for $t\in[0,T]$ and $\mathcal E_\tau(0)\rightarrow0$ as $\tau\rightarrow0$,
then $\bm h\to\bm h_c$ uniformly in $[0,T]$ and $\bm\eta\to\bm\eta_c$ uniformly in $[t_1,T]$ for any $t_1>0$ as $\tau\to0^+$.
\begin{prop}\label{prop:tau0}
Fix $\varepsilon,\rho$ satisfying \eqref{eq:triangle} with $\varepsilon_0$ sufficiently small.
Let $(\bm h,\bm \eta)$ be a solution of \eqref{eq:system-h-h'} and $(\bm h_c,\bm\eta_c)$ a solution of \eqref{eq:h-CahnHilliard},
with $ \bm h(t),\bm h_c(t)\in\Omega_\rho$ for any $t\in[0,T]$.
Then, there exists $C>0$ (independent of $\tau$) such that
\begin{equation}\label{E(t)<}
\mathcal E_\tau(t)\leq C(\mathcal E_\tau(0)+\tau), \qquad \quad \mbox{ for } t\in[0,T].
\end{equation}
Moreover,
\begin{align}
\int_0^T|\bm\eta(t)-\bm\eta_c(t)|dt &\leq C(\mathcal E_\tau(0)+\tau), \label{eta-L1}\\
|\bm\eta(t)-\bm\eta_c(t)| &\leq C(\mathcal E_\tau(0)+\tau), \qquad \quad \mbox{ for } t\in[t_1,T],\label{eta-inf}
\end{align}
for all $t_1\in(0,T)$.
In particular, from \eqref{E(t)<}, \eqref{eta-L1} and \eqref{eta-inf}, it follows that, if $\mathcal E_\tau(0)\rightarrow0$ as $\tau\rightarrow0$, then
\begin{equation*}
\lim_{\tau\rightarrow0}\sup_{t\in[0,T]}|\bm h(t)-\bm h_c(t)|
=\lim_{\tau\rightarrow0} \int_0^T|\bm\eta(t)-\bm\eta_c(t)|dt
=\lim_{\tau\rightarrow0}\sup_{t\in[t_1,T]}|\bm\eta(t)-\bm\eta_c(t)|=0,
\end{equation*}
for any $t_1\in(0,T)$.
\end{prop}
\begin{proof}
For $t\in[0,T]$, define
\begin{equation*}
\bm\delta_{\bm h}(t):=\bm h(t)-\bm h_c(t), \qquad \quad
\bm\delta_{\bm\eta}(t):=\bm\eta(t)-\bm\eta_c(t).
\end{equation*}
Since $\bm h(t),\bm h_c(t)\in\Omega_\rho$ for $t\in[0,T]$, by using Proposition \ref{prop:alfa,beta} and using that $l_j>\delta>0$, we get
\begin{equation}\label{eq:P-Q-stime}
\begin{aligned}
|\mathcal{P}(\bm h_c)|&\leq \frac{C}{\delta}\exp(-A\delta/\varepsilon), \qquad \qquad\quad & |J\mathcal{P}(\bm h_c)|&\leq \frac{C}{\varepsilon^2\delta^2}\exp(-A\delta/\varepsilon), \\
|\mathcal{P}(\bm h_c+\bm\delta_{\bm h})-\mathcal{P}(\bm h_c)|&\leq \frac{C}{\varepsilon^2\delta^2}\exp(-A\delta/\varepsilon)|\bm\delta_{\bm h}|, & |\mathcal Q(\bm h,\bm\eta)|&\leq \frac{C}{\delta}|\bm\eta|^2.
\end{aligned}
\end{equation}
for all $t\in[0,T]$.
Here and in what follows, $C$ is a positive constant independent of $\tau$ whose value may change from line to line.
We have
\begin{equation*}
\bm\delta_{\bm h}'=\bm\eta-\bm\eta_c, \qquad \quad
\tau\bm\delta_{\bm\eta}'=\mathcal{P}(\bm h_c+\bm\delta_c)-\mathcal{P}(\bm h_c)-\bm\delta_{\bm\eta}-\tau\mathcal Q(\bm h,\bm\eta)-\tau J\mathcal{P}(\bm h_c)\mathcal{P}(\bm h_c).
\end{equation*}
Since $\displaystyle\frac d{dt}|\bm\delta|=\frac{\bm\delta'\cdot\bm\delta}{|\bm\delta|}$ for any $\bm\delta(t)\in\mathbb{R}^{N+1}$,
using estimates \eqref{eq:P-Q-stime} and Cauchy--Schwarz inequality, we obtain
\begin{equation*}
\frac d{dt}|\bm\delta_{\bm h}|\leq|\bm\delta_{\bm\eta}|, \qquad
\tau\frac d{dt}|\bm\delta_{\bm\eta}|\leq C|\bm\delta_{\bm h}|-|\bm\delta_{\bm\eta}|+C\tau.
\end{equation*}
Summing, one has
\begin{equation*}
\frac d{dt}\left(|\bm\delta_{\bm h}|+\tau|\bm\delta_{\bm\eta}|\right)\leq C|\bm\delta_{\bm h}|+C\tau,
\end{equation*}
and so,
\begin{equation}\label{d/dt E<}
\frac d{dt}\mathcal E_\tau(t)\leq C\left(\mathcal{E}_\tau(t)+\tau\right), \qquad \quad \mbox{ for } t\in[0,T].
\end{equation}
Integrating \eqref{d/dt E<} and applying Gr\"onwall's Lemma, we obtain \eqref{E(t)<}.
In particular, from \eqref{E(t)<}, it follows that
\begin{equation}\label{delta_h}
|\bm\delta_{\bm h}(t)|\leq C(\mathcal E_\tau(0)+\tau), \qquad \quad\qquad \mbox{ for } t\in[0,T].
\end{equation}
Substituting \eqref{delta_h} into the equation for $\bm\delta_{\bm\eta}$, we obtain
\begin{equation*}
\tau\frac{d}{dt}|\bm\delta_{\bm\eta}|\leq-|\bm\delta_{\bm\eta}|+C(\mathcal E_\tau(0)+\tau),
\end{equation*}
and integrating the latter estimate we infer \eqref{eta-L1}; moreover, we have
\begin{equation*}
\frac{d}{dt}\left(\tau e^{t/\tau}|\bm\delta_{\bm\eta}(t)|\right)\leq C(\mathcal E_\tau(0)+\tau)e^{t/\tau},
\end{equation*}
and so
\begin{equation*}
|\bm\delta_{\bm\eta}(t)| \leq C(\mathcal E_\tau(0)+\tau) +\mathcal E_\tau(0)\frac{e^{-t/\tau}}{\tau},
\end{equation*}
for $t\in[0,T]$.
Therefore, for any fixed $t_1\in(0,T)$, we obtain \eqref{eta-inf}.
\end{proof}
\section*{Acknowledgments}
This is a pre-print of an article published in Journal of Dynamics and Differential Equations.
The final authenticated version is available online at: https://doi.org/10.1007/s10884-019-09806-6.
We thank the anonymous referee for the careful review and for the comments which helped us to improve the paper.
|
1,108,101,565,422 | arxiv | \section{Introduction}
Deep convolutional neural networks (DCNNs) have achieved remarkable success in many fields, such as computer vision, natural language processing, information retrieval, etc. However, training and deploying DCNNs usually require a large amount of computational cost and power consumption, which is greatly challenging the extensive applications in industry. As a result, many recent studies have been focusing on how to accelerate the inference of neural networks by fixed-point quantization on weights or activations~\cite{PACT,PostTraining4bit,GoogleCVPR2018,Whitepaper,LearnInterval,zhou2016dorefa,Apprentice,HAQ,INQ,TSQ,hou2018lossaware}, and design dedicated hardware utilizing the efficient integer arithmetic~\cite{EIE,cambricon,tpu,ascend310}. The successful progress surprisingly shows that the bit-width can be reduced to extremely low such as 4-bit while bringing quite little hurt to the accuracy for inference \cite{dsq,quantization_networks,lsq}.
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{optimization.pdf}
\caption{The fundamental idea of our unified INT8 training. $\mathbf{g_x}$ and $\mathbf{\hat{g}_x}$ represent the original float gradient and the quantized one, respectively. $\alpha$ and $\beta$ represent different direction deviations that quantization brings. The red lines present crash cases when the direction deviation is large. The left subfigure indicates that clipping gradient properly to reduce direction deviation within the convergence boundary can avoid crash. The right subfigure points out that controlling learning rate (step size) could promise a stable parameter updating by counteracting negative effect of deviation. }
\vspace{-0.1in}
\label{fig:optimize}
\end{figure}
Besides inference, low-bit training can also promise considerable acceleration, which further quantizes gradients and utilizes low-bit efficient compute kernel for both the forward and backward propagation. As analyzed in \cite{cnn-benchmark}, the computation of backward propagation occupies more time than that of forward propagation. So accelerating the training utilizing low-bit quantization has greater potential when considering the backward process. There has existed 16-bit floating-point (FP16) training, which proves the feasibility of low-bit training \cite{MPT,DFP,Flexpoint}. But it is restricted to limited advanced GPUs based on Turing or Volta architecture. Compared with FP16, the 8-bit integer (INT8) operation is widely supported by general GPUs based on Turing, Volta and even low-end Pascal architectures. Besides, the 8-bit integer arithmetic is theoretically and practically 2$\times$ faster than FP16 and 4$\times$ faster than FP32. Therefore, INT8 training enjoys better efficiency, lower power consumption and better versatility on off-the-shelf hardware.
Despite the attractive benefits, when quantizing gradients to 8-bit, the normal training tends to become unstable, since the distortion of gradients easily misleads the direction of training and causes crash of optimization. This definitely makes INT8 training very difficult, especially for the deep networks. Currently only a few studies have attempted to solve this problem~\cite{zhou2016dorefa,wage,wageubn,banner2018scalable,fp8training,PerTensorFX}. Unfortunately, all of them just tested limited quantization-friendly networks with high redundancy, and usually require complex structure adjustment or introduce additional operation to reduce quantization error, while significantly increasing the computational complexity. Besides, most of these works lack the theoretical analysis on the ad-hoc tricks, and even worse, none of them reports the practical speedup in the real-world case. All these reasons make the existing INT8 training methods stay far away from the practicality without the universal design.
To build a robust and unified INT8 training framework, we conduct deeper explorations in the challenges of gradient quantization. We empirically find that the distribution of gradients owns four special characteristics: sharp and wide, evolutionary, depth-specific and structure-specific. These unique characteristics make gradient quantization quite different from the naive quantization on weights or activations, and INT8 training more difficult to be stabilized. It is important to understand the behaviors and effects of quantized gradient in the convergence of the training. Therefore, we theoretically establish the convergence bound with respect to the gradient quantization error and the learning rate.
Based on the special characteristics and the theoretical analysis, we propose two universal techniques: Direction Sensitive Gradient Clipping and Deviation Counteractive Learning Rate Scaling to stabilize the INT8 training. The Direction Sensitive Gradient Clipping minimizes the direction deviation by pursuing an appropriate clipping as the training process evolves. Sometimes even if the clipping helps reduce the quantization error, it may still suffer from the accumulated gradient deviations across deep layers. To eliminate this effect, the Deviation Counteractive Learning Rate Scaling is further devised to promise stable parameter updating. The fundamental idea of our method is shown in Figure \ref{fig:optimize}. Extensive experiments on a variety of network structures and tasks prove the superiority and versatility of our method.
Our contribution can be summarized as below:
\begin{itemize}
\item We observe four special characteristics on the gradient distribution: sharp and wide, evolutionary, depth-specific and structure-specific, which cause the larger quantization error of gradients.
\item We theoretically provide the convergence bound of INT8 training, and respectively devise two universal techniques that can stabilize the INT8 training.
\item We are the first to achieve stable INT8 training of various networks such as MobileNetV2/InceptionV3 and various tasks such as object detection, with comparable accuracy to full-precision training.
\item We build a flexible and unified INT8 training framework for various tasks using various networks, which can easily replace the original full-precision training.
\item We are the first to complete practical acceleration of INT8 training on low-end GPUs with Pascal architecture, i.e., NVIDIA GeForce GTX 1080Ti, achieving about 22\% speedup without too much code optimization.
\end{itemize}
\section{Related Work}
Compared to huge amount of studies on accelerating inference by model quantization~\cite{XnorNet,LQNet,BinaryConnect,HAQ,RAD,DiscoveringLowPrecision}, there are few works exploring quantized training including backward propagation comprehensively. DoReFa-Net~\cite{zhou2016dorefa} quantizes gradients to 4 and 6 bits, but only experiments AlexNet with low precision gradient. WAGE~\cite{wage} and WAGEUBN~\cite{wageubn} quantize gradient to 8-bit integer, but they both incur considerable loss of accuracy (greater than $5\%$). RangeBN~\cite{banner2018scalable} and FP8 training \cite{fp8training} achieve accuracy comparable to full-precision models, but they both use floating-point number in gradients, which is not beneficial for hardware optimization to boost the speed.
Besides quantized training, most low-precision training research keeps gradient precision in 16-bit floating-point. Flexpoint~\cite{Flexpoint}, MPT~\cite{MPT} and DFP~\cite{DFP} all use 16-bit floating-point to train DNNs with accuracy comparable to full-precision model. To perform more efficient training of neural networks, INT8 training has more advantages over FP16 training.
\iffalse
For examples, we are the first to quantize gradient of neural networks such as MobileNetV2 \cite{MobileNet} and InceptionV3 \cite{Inception} to 8-bit integer. Also, our framework easily accomplishes the INT8 training of networks for object detection task. Finally, combined with several techniques to reduce extra computation overhead that quantization brings, we pioneer the real speedup of INT8 training containing 8-bit gradient computation, which is tested with ResNet-50 \cite{Resnet} using NVIDIA GeForce GTX 1080Ti.
\fi
\section{Unified INT8 Training}
In this paper, we aim to build a unified INT8 training framework, which utilizes 8-bit integer arithmetic to accelerate the expensive training process of deep neural networks including both the forward and backward propagation.
\subsection{Preliminaries}
Symmetric uniform quantization is the most efficient scheme among existed quantization methods, due to its hardware-friendly computation. Therefore, to guarantee the acceleration performance, we build the INT8 training framework based on it. Given the data $x$ (i.e., weights, activations, and gradients) following in the range $(l, u)$ and a clipping value $c \in (0, \max(|l|, |u|)]$, the symmetric uniform quantization can be formulated as:
\iffalse
\begin{equation}
\label{eq:quant}
q = \min(\max(\mathtt{round}(\frac{x}{s}), -127), 127)
\end{equation}
\fi
\begin{equation}
\label{eq:quant}
q = \mathtt{round}(\frac{\mathtt{clip}(x, c)}{s}),
\end{equation}
where $\mathtt{clip(x, c)=\min(\max(x, -c), c)}$, $s=\frac{c}{2^{8-1}-1}$ indicates the scaling factor to project the floating-point number to fixed-point 8-bit integer, and $q$ represents the quantized fixed-point number. Subsequently, the corresponding dequantized data $\hat{x}$ can be calculated by:
\begin{equation}
\label{eq:dequant}
\hat{x} = q \cdot s.
\end{equation}
Different from most prior studies that mainly focus on speeding up the inference (i.e., the forward propagation), our INT8 training framework attempts to further accelerate the backward propagation during the training stage, by applying quantization to the gradients. Namely, we pursue the quantize-dequantized gradients $\mathbf{\hat{g}}$ from full-precision gradients $\mathbf{g}$ in a proper way.
To ensure the quantized gradients maintain an unbiased expectation compared with the original ones, we adopt the stochastic rounding following \cite{pmlr-v37-gupta15}:
\begin{equation}
\mathtt{round_s(x)} = \begin{cases}
\lfloor x \rfloor, & \mathtt{w.p.} \quad 1-(x-\lfloor x \rfloor) \\
\lfloor x \rfloor + 1, & \mathtt{w.p.} \quad x-\lfloor x \rfloor \\
\end{cases}.
\end{equation}
Unfortunately, although the stochastic rounding technique limits the quantization error to some extent from the statistical view, the perturbation for each training iteration is still inevitable and harmful for convergence, whose reasons will be discussed in the following section.
\begin{figure}[tp!]
\centering
\subfigure[the accuracy curve]{
\includegraphics[width=.47\linewidth]{crash_acc-eps-converted-to.pdf}
}
\subfigure[the loss curve]{
\includegraphics[width=.47\linewidth]{crash_loss-eps-converted-to.pdf}
}
\caption{Crashed training of MobileNetV2 on CIFAR-10 after quantizing gradients to 8-bit.}
\vspace{-0.1in}
\label{fig:crash}
\end{figure}
\subsection{Challenges of Gradient Quantization}
\label{section:challenges}
Gradients determine the direction of optimization and the magnitude of parameter update and thus play a critical role in pursuing high accurate models. In INT8 training, after we apply quantization to gradients, the perturbation introduces deviation to the optimization direction. Once the deviation accumulates to an unacceptable degree, the training process may be unstable and even crash, resulting in severe performance degradation. Figure \ref{fig:crash} shows our empirical observation that for some special network architectures like MobileNetV2, directly quantizing gradients causes a rapid crash of training.
\begin{figure}[tp!]
\subfigure[gradients are different from weights and activations]{
\includegraphics[width=1\linewidth]{awg_wise-eps-converted-to.pdf}
}
\subfigure[gradients keep evolving during training]{
\includegraphics[width=1\linewidth]{step_wise-eps-converted-to.pdf}
}
\subfigure[gradients of different depths have have different patterns]{
\includegraphics[width=1\linewidth]{layer_wise-eps-converted-to.pdf}
}
\subfigure[gradients of different structures have different patterns]{
\includegraphics[width=\linewidth]{conv_wise-eps-converted-to.pdf}
}
\caption{Distributions of activations, weights and gradients with respect to different layers of MobileNetV2 and training iterations.}
\label{fig:data_distritbuion}
\vspace{-0.1in}
\end{figure}
To further investigate the essential reasons behind this phenomenon, we conduct detailed analysis on the distribution of gradients during training without gradient quantization, as shown in Figure \ref{fig:data_distritbuion}. We surprisingly observe that the gradients own the following unique characteristics:
\begin{itemize}
\item[C1:] \textbf{Sharp and Wide.} As shown in Figure \ref{fig:data_distritbuion}(a), compared to weights and activations, gradients follow an unusual distribution that has more values concentrated around zero while a certain number of extreme values also exists. Therefore, the distribution curve is very sharp with small values taking the majority of gradients, but the range is relatively very wide. This makes many gradients quantized to zero and the quantization error significantly large when using uniform quantization.
\item[C2:] \textbf{Evolutionary.} Figure \ref{fig:data_distritbuion}(b) depicts how the gradient distribution of the same layer evolves with respect to the training iterations. We can find that as the training goes on, the shape of gradient distribution becomes much sharper and narrower, which means it is impossible to fix the quantization settings throughout the training process, as we usually do for weights and activations, such as assuming the same clipping range in the whole training.
\item[C3:] \textbf{Depth-Specific.} Figure \ref{fig:data_distritbuion}(c) compares the distribution of gradients in different layers. It is obvious that the distributions in the shallow layers are sharper with larger extreme values than the deeper layers. This means that the preceding layers of the deep neural networks often face more severe quantization loss.
\item[C4:] \textbf{Structure-Specific.} As can be seen in Figure \ref{fig:data_distritbuion}(d), the gradients of layers with different structures present apparently different patterns. For MobileNetV2, the second convolutional layer in each block is of depth-wise structure. Its gradients own larger range and sharper shape even in the deeper block, making MobileNetV2 harder to quantize from the aspect of gradients.
\end{itemize}
Based on the above observations, we can conclude that the gradients differ from weights and activations largely, which inevitably causes an unstable training, when simply adopting the common quantization techniques for weights and activations. This means that we need certain techniques to take care of distinctiveness in gradient quantization, which brings great challenges to the real and unified INT8 training in practice.
Before turning to devise the desired techniques considering the speciality of gradients, we first attempt to understand the gradient's effect on the training stability, by theoretically revealing the connections between training convergence and gradient quantization. This will provide us a reliable clue to build the robust and unified INT8 training framework.
\subsection{Stabilize Training: A Theoretical Perspective}
\label{convergence_analysis}
As commonly used in the analysis of deep learning optimizers \cite{duchi2011adaptive,Adam,AMSGrad,Luo2019AdaBound}, the ability of convergence is usually evaluated by the regret $R(T)$.
\begin{equation}
\label{eq:regret}
R(T) = \sum_{t=1}^{T} (f_t(\mathbf{w}_t)-f_t(\mathbf{w}^*)),
\end{equation}
where $T$ indicates the number of iterations. $\mathbf{w}_t \in \mathbb{S}$ is the parameter at time $t$ in the convex compact set $\mathbb{S}$, and $f_t(\mathbf{w}_t)$ denotes the corresponding loss function. The optimal parameter is represented by $\mathbf{w}^*$. If the average regret $\frac{R(T)}{T}$ approaches zero quickly as $T$ increases, the speed and ability of convergence can be guaranteed.
Due to the complexity of the DCNNs, it is very difficult to directly analyze its behaviors. As the prior studies \cite{QSGD,DeeperUnderstanding,hou2018analysis,BCGD} do, we first make the following assumptions:
\begin{assumption} \label{assumption:1}
$f_t$ is convex;
\end{assumption}
\begin{assumption} \label{assumption:2}
$\forall \mathbf{w}_i, \mathbf{w}_j \in \mathbb{S}, \|\mathbf{w}_i-\mathbf{w}_j \|_\infty \leq D_\infty$.
\end{assumption}
Although the convexity assumption may not hold for deep networks, analysis based on this can provide reasonable and valuable insights for us, which has been proved in previous studies \cite{duchi2011adaptive,Luo2019AdaBound,hou2018analysis,BCGD}.
Taking the standard stochastic gradient descent algorithm into consideration, the optimization based on quantized gradient $\mathbf{\hat{g}}_t$ and learning rate $\eta_t$ can be formulated as:
\begin{equation}
\mathbf{w}_{t+1} = \mathbf{w}_t - \eta_t\mathbf{\hat{g}}_t.
\end{equation}
Then we have the following theoretical finding (see the supplementary materials for detailed proof):
\begin{thm}
\label{thm:1}
If define the error of quantized gradients as $\mathbf{\epsilon}_t = \mathbf{g}_t-\mathbf{\hat{g}}_t$, then with assumption \ref{assumption:1} and \ref{assumption:2}, we have:
\begin{equation}
\label{eq:avg_regret}
\frac{R(T)}{T} \leq \underbrace{\frac{d D_\infty^2}{2T\eta_{T}}\vphantom{\sum_{t=1}^{T}}}_{(1)} + \underbrace{ \frac{D_\infty}{T} \sum_{t=1}^{T} \| \mathbf{\epsilon}_t \|\vphantom{\sum_{t=1}^{T}}}_{(2)} + \underbrace{\frac{1}{T}\sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{\hat{g}}_{t}\|^2}_{(3)}.
\end{equation}
\end{thm}
\iffalse
\begin{proof}
According to assumption \ref{assumption:1},
\begin{equation}
\label{eq:convex_conclusion}
f_t(\mathbf{w}_t) - f_t(\mathbf{w}^*) \leq \mathbf{g}_t^\top(\mathbf{w}_t-\mathbf{w}^*).\nonumber
\end{equation}
Consider one step of update and the quantization error $\mathbf{\epsilon}_t$,
\begin{equation}
\label{eq:rearrange_eq}
\begin{split}
\mathbf{g}_t^\top(\mathbf{w}_t-\mathbf{w}^*) &=
\frac{(\mathbf{w}_t-\mathbf{w}^*)^2-(\mathbf{w}_{t+1}-\mathbf{w}^*)^2}{2\eta_{t}}\\\nonumber
&\quad + \mathbf{\epsilon}_t(\mathbf{w}_t-\mathbf{w}^*) + \frac{\eta_{t}}{2}(\mathbf{g}_t-\mathbf{\epsilon}_t)^2.\nonumber
\end{split}
\end{equation}
Then, with Equation \eqref{eq:regret}, we hav
\begin{equation}
\label{ieq:regret}
\begin{split}
R(T) &\leq [\frac{1}{2\eta_1}(\mathbf{w}_1-\mathbf{w}^{*})^2 - \frac{1}{2\eta_T}(\mathbf{w}_{T+1}-\mathbf{w}^*)^2] \\
&+ \sum_{t=2}^{T}(\frac{1}{2\eta_t}-\frac{1}{2\eta_{t-1}})(\mathbf{w}_{t}-\mathbf{w}^*)^2 \\
&+ \sum_{t=1}^{T}[\mathbf{\epsilon}_{t}(\mathbf{w}_{t}-\mathbf{w}^*) + \frac{\eta_t}{2}(\mathbf{g}_{t}-\mathbf{\epsilon}_{t})^2].\nonumber
\end{split}
\end{equation}
Finally according to assumption \ref{assumption:2} and Cauchy’s inequality,
\begin{align}
R(T) &\leq \frac{d D_\infty^2}{2\eta_{T}} + \sum_{t=1}^{T} \| \mathbf{\epsilon}_t \| \| \mathbf{w}_t-\mathbf{w}^* \| + \sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{\hat{g}}_{t}\|^2 \nonumber \\
&\leq \frac{d D_\infty^2}{2\eta_{T}} + D_\infty \sum_{t=1}^{T} \| \mathbf{\epsilon}_t \| + \sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{\hat{g}}_{t}\|^2.\nonumber
\end{align}
The average regret $\frac{R(T)}{T}$ satisfies Theorem \ref{thm:1}.
\end{proof}
\fi
\iffalse
Taking the standard stochastic gradient descent algorithm into consideration, the optimization based on quantized gradient $\hat{g}_t$ can be formulated as:
\begin{equation}
w_{t+1} = w_t - \eta_t\hat{g_t}.
\end{equation}
Since $f_t$ is convex,
\begin{equation}
f_t(\mathbf{w_t}) - f_t(\mathbf{w^*}) \leq \mathbf{g_t}^\top(\mathbf{w_t}-\mathbf{w^*}) = \sum_{i=1}^{d} g_{t,i}(w_{t,i} - w_i^*).
\end{equation}
Based on the fact that $(w_{t+1,i}-w_i^*)^2 = (w_{t,i} - \eta_{t,i}\ \hat{g}_{t,i}-w_i^*)^2$ and the learning rate $\eta_t$ is non-zero,
\iffalse
\begin{equation}
\begin{split}
&(w_{t+1,i}-w_i^*)^2 = (w_{t,i} - \eta_{t,i}\ \hat{g}_{t,i}-w_i^*)^2 \\
&\quad =(w_{t,i}-w_i^*)^2 - 2(w_{t,i}-w_i^*)\eta_{t,i}\ \hat{g}_{t,i}+\eta_{t,i}^2\ \hat{g}_{t,i}^2
\end{split}
\end{equation}
\fi
we have
\begin{equation}
\begin{split}
\hat{g}_{t,i}(w_{t,i}-w_i^*) &= \frac{1}{2\eta_{t,i}}(w_{t,i}-w_i^*)^2+\frac{\eta_{t,i}}{2}\hat{g}_{t,i}^2 \\
&\quad-\frac{1}{2\eta_{t,i}}(w_{t+1,i}-w_i^*)^2
\end{split}
\end{equation}
The error of quantized gradients is defined as:
\begin{equation}
\epsilon_{t,i} = g_{t,i}-\hat{g}_{t,i}
\end{equation}
Replacing $\hat{g}_{t,i}$ in the \eqref{eq:rearrange} with $g_{t,i}$ and $\epsilon_{t,i}$, we can get that:
\begin{equation}
\begin{split}
&g_{t,i}(w_{t,i}-w_i^*) = \\
&\quad \frac{1}{2\eta_{t,i}}[(w_{t,i}-w_i^*)^2-(w_{t+1,i}-w_i^*)^2)]\\
&\quad + \epsilon_{t,i}(w_{t,i}-w_i^*) + \frac{\eta_{t,i}}{2}(g_{t,i}-\epsilon_{t,i})^2
\end{split}
\end{equation}
So according to the \eqref{ieq:convex_conclusion} and \eqref{ieq:substitude}, the regret
\begin{equation}
\begin{split}
R(T) &\leq \sum_{t=1}^{T} \sum_{i=1}^{d} (\frac{1}{2\eta_{t,i}} [(w_{t,i}-w_i^*)^2 - (w_{t+1,i}-w_i^*)^2] \\
&\quad \quad \quad \quad \quad + \epsilon_{t,i}(w_{t,i}-w_i^*) + \frac{\eta_{t,i}}{2}(g_{t,i}-\epsilon_{t,i})^2) \\
&= \sum_{i=1}^{d} [\frac{1}{2\eta_{1,i}}(w_{1,i}-w_i^*)^2 - \frac{1}{2\eta_{T,i}}(w_{T+1,i}-w_i^*)^2] \\
&\quad + \sum_{t=2}^{T}\sum_{i=1}^{d}(\frac{1}{2\eta_{t,i}}-\frac{1}{2\eta_{t-1,i}})(w_{t,i}-w_i^*)^2 \\
&\quad + \sum_{t=1}^{T} \sum_{i=1}^{d} [\epsilon_{t,i}(w_{t,i}-w_i^*) + \frac{\eta_{t,i}}{2}(g_{t,i}-\epsilon_{t,i})^2]
\end{split}
\end{equation}
Because the solution space is limited, $\forall \mathbf{w_m}, \mathbf{w_n} \in \mathbb{S}$
\begin{equation}
\|\mathbf{w_n}-\mathbf{w_m} \|_\infty \leq D_\infty
\end{equation}
we can further relax the above \eqref{ieq:regret} to:
\begin{equation}
\begin{split}
R(T) &\leq \sum_{i=1}^{d} \frac{D_\infty^2}{2\eta_{1,i}} + \sum_{t=2}^{T}\sum_{i=1}^{d}(\frac{1}{2\eta_{t,i}}-\frac{1}{2\eta_{t-1,i}})D_\infty^2\\
&\quad + \sum_{t=1}^{T} \sum_{i=1}^{d} [\epsilon_{t,i}(w_{t,i}-w_i^*) + \frac{\eta_{t,i}}{2}(g_{t,i}-\epsilon_{t,i})^2]
\end{split}
\end{equation}
Assuming that all layers have the same learning rate, then
\begin{equation}
\begin{split}
R(T) &\leq \frac{d\ D_\infty^2}{2\eta_{T}} + \sum_{t=1}^{T} \mathbf{\epsilon_t}(\mathbf{w_t}-\mathbf{w^*}) + \sum_{t=1}^{T} \frac{\eta_{t}}{2}(\mathbf{g_{t}}-\mathbf{\epsilon_{t}})^2
\end{split}
\end{equation}
Based on Cauchy's inequality, we finally get:
\begin{equation}
\begin{split}
R(T) &\leq \frac{d\ D_\infty^2}{2\eta_{T}} + \sum_{t=1}^{T} \| \mathbf{\epsilon_t \|}\cdot \| \mathbf{w_t}-\mathbf{w^*} \| + \sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{g_{t}}-\mathbf{\epsilon_{t}}\|^2 \\
&\leq \frac{d\ D_\infty^2}{2\eta_{T}} + D_\infty \sum_{t=1}^{T} \| \mathbf{\epsilon_t \|} + \sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{\hat{g}_{t}}\|^2 \\
\end{split}
\end{equation}
Thus the average regret
\begin{equation}
\frac{R(T)}{T} \leq \underbrace{\frac{d\ D_\infty^2}{2T\eta_{T}}\vphantom{\sum_{t=1}^{T}}}_{(1)} + \underbrace{ \frac{D_\infty}{T} \sum_{t=1}^{T} \| \mathbf{\epsilon_t} \|\vphantom{\sum_{t=1}^{T}}}_{(2)} + \underbrace{\frac{1}{T}\sum_{t=1}^{T} \frac{\eta_{t}}{2}\|\mathbf{\hat{g}_{t}}\|^2}_{(3)}
\end{equation}
\fi
We can find that the bound of average regret is dominated by three terms. Term (1) approaches zero as $T$ increases and thus can be ignored in gradient quantization. Term (2) indicates the quantization error of gradients greatly affects the ability to converge, and it is usually large, as analyzed in Section \ref{section:challenges}. For term (3), its magnitude is mainly influenced by the learning rate and l2-norm of quantized gradients. Based on the theoretical analysis, to stabilize INT8 training, we have two basic principles for designing better quantization techniques: (1) reduce the quantization error of gradients; (2) scale down the learning rate. They are also very intuitive since, on the one hand, a lower quantization error means small deviation of optimization direction and thus avoids the training crash, on the other hand, it is a common sense that decreasing the learning rate gradually promises a better solution in the optimization.
Now with the design principles, the question is how to devise the universal techniques for INT8 training, meanwhile take the characteristics of gradients into consideration. We respectively present two novel techniques: Direction Sensitive Gradient Clipping and Deviation Counteractive Learning Rate Scaling, which together lower the average regret bound and guarantee stable INT8 training.
\subsection{Direction Sensitive Gradient Clipping}
Considering the basic operation $\mathbf{z} = \mathbf{W}^\top \mathbf{a}$ in deep neural networks, the gradients of weights $\mathbf{g_W}$ actually can be calculated by $\mathbf{g_z}^\top \mathbf{a}$. From this aspect, the quantization error of $\mathbf{g_W}$ in \eqref{eq:avg_regret} mainly stems from that of activation gradients $\mathbf{g_z}$. Therefore, in our INT8 training we can mainly concern the quantization of $\mathbf{g_z}$, which will help control the error of quantized gradients in \eqref{eq:avg_regret}. For simplicity of notations, in the following discussion we directly use $\mathbf{g}$ to denote $\mathbf{g_z}$.
To minimize quantization error, previous works mainly seek the optimal clipping value $c$ in \eqref{eq:quant} by assuming certain data distribution, e.g. Gaussian distribution \cite{PostTraining4bit,HWGQ,truncated_gaussian,banner2018scalable,hou2018analysis,RAD}. However, according to the gradient characteristics C1 and C2 we discover, it is unpractical to make a common assumption for an evolutionary and unusual gradient distribution. To further prove this point, we do the Kolmogorov–Smirnov test with distribution parameter solved by maximum likelihood estimation, and report the KS-statistics that consistently reject the assumption that gradients obey any common distribution in Table \ref{table:hyp_test}.
\begin{table}[t!]
\caption{KS-statistics of gradient and weight with respect to different layers' conv3 in MobiletNetV2, the last column indicates the maximum value that can accept the hypothesis at significance level of 0.05.}
\label{table:hyp_test}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Data}} & \multicolumn{3}{|c|}{Distribution} & \multirow{2}{*}{Critical value} \\
\cline{3-5}
\multicolumn{2}{|c|}{} & Gaussian & Laplace & Student & \\
\hline
\hline
\multirow{2}{*}{layer0} & $\mathbf{g}$ & 0.1934 & 0.0790 & 0.2005 & 0.0012 \\
& $\mathbf{w}$ & 0.0391 & 0.0721 & 0.1011 & 0.0765 \\
\hline
\multirow{2}{*}{layer8} & $\mathbf{g}$ & 0.2061 & 0.1091 & 0.2303 & 0.0024\\
& $\mathbf{w}$ & 0.0294 & 0.0569 & 0.1084 & 0.0110 \\
\hline
\end{tabular}
\vspace{-0.14in}
\end{table}
To find the optimal clipping value $c$ without any assumption, a straightforward idea is to keep the quantized gradient consistent with the original one by gradient descent algorithm. Usually, one can model the consistency using the popular mean-square error (MSE). Unfortunately, due to characteristics C2 and C3 of gradients with huge discrepancy and fluctuation in their magnitudes, MSE makes the optimization vulnerable and unable to work under the same simple setting across various layers.
Therefore, to pursue the desired clipping values of different layers that promise stable training, we choose cosine distance to guide the learning of clipping values, which not only avoids the negative effect of the varied gradients' magnitudes, but also keeps the network optimization directions consistent:
\begin{equation}
\label{eq:cos_err}
d_c = 1 -\cos(<\mathbf{g}, \mathbf{\hat{g}}>) = 1 - \frac{\mathbf{g} \cdot \mathbf{\hat{g}}}{|\mathbf{g}| \cdot |\mathbf{\hat{g}|}}
\end{equation}
where $\mathbf{g}$ and $\mathbf{\hat{g}}$ denote the original floating-point gradient and its quantize-dequantized counterpart.
The cosine distance measures the direction deviation of quantized gradients. As shown in Figure \ref{fig:CosAcc}, when $d_c$ increases to a certain level, the whole training crashes. There exists strong correlation between $d_c$ and training stability, which proves that cosine distance can effectively reflect the influence of gradient quantization on the convergence. By minimizing the deviation, we subsequently reduce term (2) in \eqref{eq:avg_regret}. Figure \ref{fig:effect}(a) shows the quantization error using different clipping values, where there exists an optimal clipping value that substantially reduces the cosine distance.
\begin{figure}[t!]
\centering
\subfigure[the accuracy curve]{
\includegraphics[width=.47\linewidth]{err_acc-eps-converted-to.pdf}
}
\subfigure[the loss curve]{
\includegraphics[width=.47\linewidth]{err_loss-eps-converted-to.pdf}
}
\caption{Model crashes when $d_c$ exceeds limits.}\label{fig:CosAcc}
\vspace{-0.1in}
\end{figure}
\subsection{Deviation Counteractive Learning Rate Scaling}
The theoretical analysis on convergence ability of quantized training indicates the necessity of scaling down learning rate, since the quantization error of gradients cannot vanish completely. To validate this point, we decrease the learning rate of the original crashed training of MobileNetV2 mentioned in Section \ref{section:challenges} and find that it defers and even eliminates the crash with an extremely low learning rate, although facing a performance degradation (see the red, green and orange lines in Figure \ref{fig:effect}(b)).
\begin{figure}[t!]
\subfigure[effect of clipping]{
\includegraphics[width=0.42\linewidth]{clip_err-eps-converted-to.pdf}
}
\hspace{0.1in}
\subfigure[effect of scaling strategies]{
\includegraphics[width=0.43\linewidth]{acc-eps-converted-to.pdf}
}
\caption{The effect of clipping and learning rates on INT8 training. $\gamma$ in (a) represents optimal clipping value. In (b), $\eta1$ sets initial learning rate as 0.1 with $\phi(d_c)$ scaling, $\eta2$, $\eta3$ and $\eta4$ choose 0.01, 0.05, 0.1 as initial learning rate respectively without scale.}\label{fig:effect}
\end{figure}
Since the gradients are backward propagated layer by layer, the minor gradient deviation will accumulate exponentially after massive multiplication and addition calculation. To address this issue, we further propose the Deviation Counteractive Learning Rate Scaling to balance out the error by exponentially decaying the learning rate according to the degree of direction deviation $d_c$, the scaling function is formulated at:
\begin{equation}
\phi(d_c) = \max(e ^ {-\alpha d_c}, \beta)
\end{equation}
where $\alpha$ controls the decay degree and $\beta$ limits the lower bound of scaling.
This scaling function generates a factor to scale down the original full-precision learning rate. We empirically find that the self-adapting scaling function performs well in a layer-wise way, adaptively adjusting the learning rate according to the direction deviations in different layers. This counteracts the undesired effects of the gradient deviations across layers, and exactly addresses the challenges of the depth-specific and structure-specific patterns as observed in characteristics C3 and C4 in Section \ref{section:challenges}. The blue line in Figure \ref{fig:effect}(b) demonstrates that the training equipped with $\phi(d_c)$ scaling achieves higher accuracy than the manually adjusted ones (tested with MobileNetV2 on CIFAR-10).
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{replace.png}
\caption{Flexible INT8 convolutional layer replacement.}
\label{fig:replace}
\end{figure}
\subsection{General Purpose Training Framework}
\begin{table}[t!]
\caption{Overhead reduced with Periodic Update (on ResNet-50).}
\label{table:period}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
Period & 1 & 10 & 100 & 1000 \\
\hline
\hline
Average time(s/iter)& 1.006 & 0.364 & 0.301 & 0.297 \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
In addition to ensuring the stable and accurate convergence, in practice our unified INT8 training framework should also satisfy the following three features:
\noindent (1) \textbf{Easy to plug into any DCNN architecture.} To realize this, we implement an automatic match and replacement mechanism in PyTorch \cite{pytorch} that correspondingly substitutes convolutional and fully-connected layers with 8-bit counterpart. The whole workflow including both forward and backward passes is shown in Figure \ref{fig:replace}.
\noindent (2) \textbf{No excessive extra computational overhead.} To avoid the extra time cost of calculating clipping value, we design a Periodic Update method to optimize the clipping value periodically. As we can see in Table \ref{table:period}, the Periodic Update method dramatically reduces the computational overhead of optimizing the clipping value.
\noindent (3) \textbf{Easy to implement on off-the-shelf hardware.} To validate the potential of that, we utilizes the DP4A instruction (8-bit integer 4-element vector dot product) on low-end NVIDIA Pascal GPUs to implement efficient 8-bit kernels for calculating gradients. To the best of our knowledge, we are the first to achieve practical acceleration of INT8 training including the backward propagation. The detailed speedup will be reported and discussed in Section \ref{section:speed}.
\iffalse
Our proposed general techniques is flexible to integrate into any CNN architecture, which enables the unified and simple replacement to realize INT8 training, as shown in Figure \ref{fig:replace}. To avoid the extra time cost of calculating clipping value, we design a Periodic Update method to optimize the clipping value periodically. As we can see in Table \ref{table:period}, the Periodic Update method dramatically reduces the computational overhead of optimizing the clipping value.
The 8-bit fixed-point operation has been widely used for inference, most existed hardware have the ability to speed up the training including backward propagation such as Ascend310 from Huawei \cite{ascend310}, MLU100 from Cambricon \cite{cambricon}, TPU from Google \cite{tpu} and almost all GPUs from NVIDIA \cite{cuda8}. To validate the potential of that, we utilizes the DP4A instruction (8-bit integer 4-element vector dot product) on low-end GPUs with Pascal architecture to implement efficient 8-bit kernels for calculating gradients. To the best of our knowledge, we are the first to achieve practical acceleration of INT8 training including the backward propagation. The detailed speedup results will be reported and discussed in Section \ref{section:speed}.
\begin{table}[t!]
\caption{Overhead reduced with Periodic Update (on ResNet-50).}
\label{table:period}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
period & 1 & 10 & 100 & 1000 \\
\hline
\hline
average time(s/iter)& 1.006 & 0.364 & 0.301 & 0.297 \\
\hline
\end{tabular}
\end{table}
\fi
\iffalse
In addition to ensure the stable and accurate convergence, in practice our unified INT8 training framework should also saftisfy the following three features:
\begin{itemize}
\item[F1:] Easy to plug into any convolutional neural network (CNN) architecture;
\item[F2:] Bring no much extra computational overheads on the training process;
\item[F3:] Easy to implement on off-the-shelf hardware such as GPU, TPU or any type of hardware equipped with high-performance integer arithmetic power.
\end{itemize}
For feature F1, we fortunately have the proposed general techniques that show good flexibility to integrate into any CNN architecture. Due to this reason, we can easily implement an automatic match and replacement mechanism in PyTorch that correspondingly substitutes convolutional and fully-connected layers with the 8-bit counterpart, and adopt the proposed clipping and learning rate scaling techniques in the uniform quantization. The whole workflow including both forward and backward passes is shown in Figure \ref{fig:replace}.
For feature F2, due to the heavy memory access, the time consumption of calculating clipping value is notable among the whole training process. To avoid the extra overhead, we design a Periodic Update method to optimize the clipping value periodically. As we can see in Table \ref{table:period}, the Periodic Update method dramatically reduces the computational overhead of optimizing the clipping value.
\iffalse
, which makes the computation of clipping value occurs
And we find that it brings little accuracy drop even when the period larger than 100 iterations, which will be further elaborated in Section \ref{section:update_period}.
\fi
\begin{table}[t!]
\caption{Overhead reduced with Periodic Update (on ResNet-50).}
\label{table:period}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
period & 1 & 10 & 100 & 1000 \\
\hline
\hline
average time(s/iter)& 1.006 & 0.364 & 0.301 & 0.297 \\
\hline
\end{tabular}
\end{table}
For feature F3, the practical acceleration of INT8 training relies on whether the hardware supports 8-bit efficient instruction. Since the 8-bit fixed-point operation has been widely used for inference, most existed hardware have the ability to speed up the training including backward propagation such as Ascend310 from Huawei \cite{ascend310}, MLU100 from Cambricon \cite{cambricon}, TPU from Google \cite{tpu} and almost all GPUs from NVIDIA \cite{cuda8}. To validate the potential of accelerating the backward propagation, we utilizes the DP4A instruction on low-end GPUs with Pascal architecture to implement efficient 8-bit kernels for calculating gradients.
\iffalse
To the best of our knowledge, we are the first to achieve practical acceleration with 8-bit gradients for backward propagation, and the detailed speed up results will be reported and discussed in Section \ref{section:speed}.
\fi
\fi
\section{Experiments}
We conduct extensive experiments to demonstrate that our proposed framework is unified for various network structures on popular image classification and object detection tasks with state-of-the-art accuracy, and meanwhile it can be easily deployed on the mainstream devices (NVIDIA Pascal GPU) with satisfactory speedup, compared to full-precision training.
\iffalse
\subsection{Analysis of Convergence}
In this section, we design extensive experiments to validate our theoretical derivation on convergence. Relationships among cosine distance, clipping value, learning rate and convergence are studied thoroughly.
\subsubsection{Measuring Convergence with Cosine Distance}
Our theoretical analysis shows that gradient quantization error could harm convergence seriously and we define the cosine distance $d_c$ to represent quantization error. From the curve in Figure \ref{fig:CosAcc}, we can see that as $d_c$ rises, the training accuracy drops down. Moreover, when $d_c$ increases to a certain level, the whole training crashes. This strong correlation between $d_c$ and accuracy proved that cosine distance can effectively reflect the influence of gradient quantization on the convergence.
\subsubsection{Effectiveness of Clipping}
We compared the quantization error using different clipping values. As shown in Figure \ref{fig:clip_cos}, there exists an optimal clipping value that substantially reduce the cosine distance. When clipping value is large, many small elements will be quantized to zero and when clipping value is much smaller, clipping error of large elements will increase the whole quantization error. Both of these two cases will impede the convergence of training.
\subsubsection{Influence of Learning Rate}
According to \eqref{eq:avg_regret}, decreasing learning rate can help training loss converge. However, extremely low learning rate will slow convergence and lead to a lower accuracy finally. Thus we compare our DCLRS with different learning rate configuration on CIFAR-10 dataset to explore the effects of learning rate adjustment.
Figure \ref{fig:LR} shows the accuracy of different learning rates. As can be seen, middle learning rate (0.05) could relief the crash problem that happens in high learning rate (0.1) case, but it would not converge like low learning rate. Though Low learning could converge, but it has a much slower convergence speed. Hence, it is difficult to find an appropriate learning rate for quantized training. Our Deviation Counteractive Learning Rate Scaling (DCLRS) is designed to solve this problem. DCLRS can adjust learning rate adaptively, and provide layer-wise specification. Figure \ref{fig:LR} shows our method could find a balance between crash and slow convergence.
\fi
\subsection{Ablation Study}
\noindent \textbf{Settings.} We first conduct the ablation study on CIFAR-10 dataset with MobileNetV2 \cite{MobileNet}, to validate the effectiveness of the proposed techniques. We use cosine scheduler \cite{QSGD} with initial learning rate set to 0.1 for all experiments. In the Periodic Update experiment, the $\alpha$ and $\beta$ in learning rate scaling are set to 20 and 0.1 respectively.
\noindent \textbf{Direction Sensitive Gradient Clipping.} Figure \ref{fig:cosinedistance}(a) shows the cosine distance with respect to the training steps. We can observe that conv2 (the second convolutional layer) of each block owns a much larger cosine distance than other layers of the block most of the time. This is consistent with C4 that the gradients of conv2 own sharper shape, indicating that our cosine distance can well reflect the gradient characteristics.
Moreover, as Table \ref{table:clip} lists, our proposed direction sensitive gradient clipping technique indeed prevents INT8 training from crashing, which proves the fact that optimizing a clipping value of gradients to minimize direction deviation $d_c$ can certainly ensure a stable INT8 training.
\begin{figure}[t!]
\centering
\subfigure[the cosine distance]{
\includegraphics[width=.47\linewidth]{layer_dc-eps-converted-to.pdf}
}
\subfigure[the accuracy curve]{
\includegraphics[width=.47\linewidth]{acc_lrfunc-eps-converted-to.pdf}
}
\caption{Analysis of cosine distance and learning rate scaling function.}
\vspace{-0.1in}
\label{fig:cosinedistance}
\end{figure}
\begin{table}[htbp!]
\caption{Ablation study on clipping method for INT8 training.}
\label{table:clip}
\centering
\small
\begin{tabular}{|c|c|c|}
\hline
Clipping method & No clipping & \tabincell{c}{Direction Sensitive \\Gradient Clipping} \\
\hline
\hline
Accuracy (\%) & NaN & 93.02 \\
\hline
\end{tabular}
\vspace{-0.05in}
\end{table}
\noindent \textbf{Deviation Counteractive Learning Rate Scaling.} We evaluate three forms of learning rate scaling strategies without clipping to control variable for a reasonable comparison. The results shown in Figure \ref{fig:cosinedistance}(b) reveal that linear and quadratic forms are too weak to control optimization direction within the convergence boundary and model crashes in the training process. Compared with linear and quadratic form, the scaling with exponential form is more powerful to counteract the direction deviation and prevents optimization from stepping out of the convergence boundary. We further explore its sensitivity to the selection of hyperparameter in Table \ref{table:LRParam}, and we can see that different settings of $\alpha$ and $\beta$ achieve similar accuracy, which presents the stability of our Deviation Counteractive Learning Rate Scaling.
\begin{table}[htbp!]
\vspace{-0.05in}
\centering
\caption{Comparison of different hyperparameters for learning rate scaling.}
\label{table:LRParam}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
$\alpha$ & 10 & 10 & 20 & 20 \\
\hline
$\beta$ & 0.1 & 0.2 & 0.1 & 0.2 \\
\hline
\hline
Accuracy (\%) & 92.82 & 93.28 & 93.38 & 93.27 \\
\hline
\end{tabular}
\vspace{-0.05in}
\end{table}
\noindent \textbf{Periodic Update for clipping value.} To reduce the extra computational overhead, we increase the period to update clipping value and find that it brings little hurt to the accuracy, as shown in Table \ref{table:Periodic}. This empirical conclusion brings possibilities for the practical acceleration of INT8 training. Besides, here we apply both gradient clipping and learning rate scaling, and obtain better performance (see that with period 1) than those in Table \ref{table:clip} and \ref{table:LRParam}. This further verifies the positive effects of the two general techniques.
\begin{table}[htbp!]
\vspace{-0.05in}
\caption{Ablation study on update period.}
\label{table:Periodic}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
Period & 1 & 10 & 100 & 1000 \\
\hline
\hline
Accuracy (\%) & 93.66 & 93.07 & 93.38 & 92.75 \\
\hline
\end{tabular}
\end{table}
\iffalse
\begin{table}[t!]
\caption{Comparison of different LR scaling function forms.}
\label{table:LRFunc}
\centering
\small
\begin{tabular}{|c|c|c|c|}
\hline
LR function & $d_c$ & $d_c^2$ & $\phi(d_c)$ \\
\hline
\hline
Accuracy (\%) & 81.01 & 73.37 & \textbf{93.38} \\
\hline
\end{tabular}
\end{table}
\fi
\iffalse
\begin{table}[t!]
\caption{Ablation study on combination of Gradient Clipping and Learning Rate Scaling.}
\label{table:clip}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
Gradient Clipping & $\times$ & $\checkmark$ & $\times$ & $\checkmark$ \\
\hline
Learning Rate Scaling & $\times$ & $\times$ & $\checkmark$ & $\checkmark$ \\
\hline
Accuracy (\%) & NaN & 93.02 & 93.38 & 93.66 \\
\hline
\end{tabular}
\end{table}
\fi
\iffalse
To further analyze effects of clipping method, learning rate scale and update period on quantized training, we conduct the ablation study on CIFAR-10 dataset using MobileNetV2. We specify update period as 100 except update period experiments. Default hyperparameters $\alpha$ and $\beta$ in learning rate scaling function are 20 and 0.1 respectively. Direction Sensitive Gradient Clipping are used in all experiments unless noted. We use cosine scheduler with initial learning rate set to 0.1 for all experiments.we conduct the ablation study on CIFAR-10 dataset using MobileNetV2
\subsubsection{Clipping Method}
\begin{table}[t!]
\caption{Ablation study on clipping method.}
\label{table:clip}
\centering
\begin{tabular}{|c|c|c|}
\hline
Clipping & \tabincell{c}{Bit-width \\ (W/A/G)} & Accuracy (\%) \\
\hline
\hline
\multirow{2}{*}{}
without clip & 8/8/8 & NaN \\
dynamic clip & 8/8/8 & - \\
\hline
\end{tabular}
\end{table}
In Table \ref{table:clip}, we further study the effect of clipping value on INT8 training. It is obvious that training loss of neural network could not converge without clipping method. We also compare direction sensitive gradient clipping(dynamic clip) with fixed clipping, which uses a fixed clipping ratio 0.5. Our method significantly exceeds fixed clipping method.
\subsubsection{Learning Rate Scaling Function}
In the definition of Deviation Counteractive Learning Rate Scaling, we choose exponential form as scaling function form. To validate the effectiveness of our choice, we compare exponential form with other function forms and study the sensitivity of hyperparameters in scaling function.
\textbf{Learning Rate Scaling Function Form}: Three kinds of scaling function form are studied in our experiments, including linear form, quadratic form and exponential form. These three function forms have distinct penalty intensity for quantization error of gradients. As shown in Table \ref{table:LRFunc}, exponential form function performs much better than other two function forms(12.13$\%$ higher than linear form and 19.77$\%$ higher than quadratic form).
\textbf{Learning Rate Scaling Function Hyperparameter} From Table \ref{table:LRParam} we can see that INT8 training accuracy is not very sensitive to hyperparameters $\alpha$ and $\beta$, which presents the stability of our Deviation Counteractive Learning Rate Scaling.
\begin{table}[t!]
\caption{Ablation study on LR scaling function form.}
\label{table:LRFunc}
\centering
\begin{tabular}{|c|c|c|}
\hline
LR function & \tabincell{c}{Bit-width (W/A/G)} & Accuracy (\%) \\
\hline
\hline
\multirow{3}{*}{}
$d_c$ & 8/8/8 & 81.01 \\
$d_c^2$ & 8/8/8 & 73.37 \\
$\phi(d_c)$ & 8/8/8 & \textbf{93.38} \\
\hline
\end{tabular}
\end{table}
\begin{table}[t!]
\caption{Ablation study on hyperparameters of LR scaling function.}
\label{table:LRParam}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$\alpha$ & $\beta$ & \tabincell{c}{Bit-width (W/A/G)} & Accuracy (\%) \\
\hline
\hline
\multirow{4}{*}{}
- & - & 32/32/32 & 94.55 \\
\hline
10 & 0.1 & 8/8/8 & 92.82 \\
10 & 0.2 & 8/8/8 & 93.28 \\
20 & 0.1 & 8/8/8 & 93.38 \\
20 & 0.2 & 8/8/8 & 93.27 \\
\hline
\end{tabular}
\end{table}
\subsubsection{Update Period}
\label{section:update_period}
As we mentioned above, periodic update is a necessary trick to accelerate INT8 training. So we conduct experiments to figure out how update period influence training accuracy. We start updating periodically after 3 epochs for that data range changes fiercely at early stage of training. Table \ref{table:Periodic} shows that updating periodically has slight influence on accuracy (reduce less than 1$\%$). Accuracy will not decrease much as update frequency decreases, which means our method could perform higher efficiency.
\begin{table}[t!]
\caption{Ablation study on update period.}
\label{table:Periodic}
\centering
\small
\begin{tabular}{|c|c|}
\hline
Update Period & Accuracy (\%) \\
\hline
\hline
\multirow{2}{*}{}
1 & 93.66 \\
\hline
10 & 93.07 \\
\hline
100 & 93.38 \\
\hline
1000 & 92.75 \\
\hline
\end{tabular}
\end{table}
\fi
\subsection{Image Classification}
Now we consider the popular image classification task that most prior studies choose to evaluate the quantization performance. We experiment with AlexNet \cite{AlexNet}, ResNet \cite{Resnet}, MobileNetV2 \cite{MobileNet} and InceptionV3 \cite{Inception} on CIFAR-10 \cite{CIFAR} and ImageNet (ILSVRC2012) \cite{Imagenet}. The CIFAR-10 dataset contains a training set of 50K images and a testing set of 10k images. Each image is of size 3$\times$3 with 10 classes. ImageNet (ILSVRC2012) consists of 1.2 million training images and 50K test images with 1000 classes.
\noindent \textbf{Settings.} As for the hyperparameters of ResNet, we use the same settings described in \cite{Resnet}. For other neural networks, we use cosine scheduler \cite{QSGD} with initial learning rate set to 0.1. The $\alpha$ and $\beta$ in learning rate scaling are set to 20 and 0.1 respectively. Clipping value is updated per 100 iterations for all experiments.
\noindent \textbf{CIFAR-10.} As Table~\ref{table:CIFAR-10} shows, our method achieves comparable accuracy on ResNet-20 to FP8 training, but takes much less memory and computation consumption due to the fixed-point operation. Moreover, our method performs surprisingly good on MobileNetV2 (1.01$\%$ accuracy drop) and InceptionV3 (even better than full precision model).
\noindent \textbf{ImageNet.} Table~\ref{table:IMAGENET} lists existing state-of-the-art quantized training methods including WAGE \cite{wage}, WAGEUBN \cite{wageubn} and FP8 training \cite{fp8training}. For AlexNet INT8 training, our method obtains 5.84\% improvement over DoReFa-Net \cite{zhou2016dorefa}. Free from the extra overhead like $\tanh$, our method enjoys higher efficiency than DoReFa-Net. As for the 2-bit weight and 8-bit activation/gradient case, we significantly outperform WAGE with about 3\% accuracy gain. What's more, equipped with our method, the INT8 training for ResNet architecture achieves almost no performance degradation, while none of the previous studies has done that. Compared with the FP8 training method, our method improves the accuracy by nearly 3\%. It should be noted that we can directly get a real speedup on popular off-the-shelf devices while methods like FP8 training need specially designed hardware, which means that our framework is more general for unified training acceleration.
As analyzed in \cite{li2016performance}, the convolutional layer occupies most of the training time while other layers like BatchNorm and ReLU are not computation-intensive. Therefore, we mainly focus on quantizing convolutional layers currently and do not quantize BatchNorm layer like RangeBN \cite{banner2018scalable} and WAGEUBN \cite{wageubn}. Even so, there is still a significant speedup for INT8 training. In addition, we could get comparable accuracy to full precision training, much higher than RangeBN and WAGEUBN.
\noindent \textbf{Networks using INT8 training for the first time.} To our best knowledge, we are the first to quantize gradient of MobileNetV2, which is known to be difficult in this community. Our method gets very good performance on both CIFAR-10 and ImageNet datasets using MobileNetV2, with only around 1$\%$ accuracy loss. We also try INT8 training on InceptionV3 for the first time, and achieve comparable accuracy to full precision model. Note that for InveptionV3 on CIFAR-10, our INT8 training method can even achieve better performance than the full-precision model.
\begin{table}[t!]
\caption{Results on CIFAR-10 dataset.}
\label{table:CIFAR-10}
\centering
\small
\begin{tabular}{|c|c|c|c|}
\hline
Model & Method & \tabincell{c}{Bit-width \\ (W/A/G)} & \tabincell{c}{Accuracy \\ (\%)} \\
\hline
\hline
\multirow{3}{*}{ResNet-20}
& FP & 32/32/32 & 92.32 \\
\cline{2-4}
& FP8 training \cite{fp8training} & 8/8/8 & 92.21 \\
\cline{2-4}
& Ours & 8/8/8 & 91.95 \\
\hline
\multirow{2}{*}{\tabincell{c}{MobileNetV2}}
& FP & 32/32/32 & 94.39 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{93.38} \\
\hline
\multirow{2}{*}{\tabincell{c}{InceptionV3}}
& FP & 32/32/32 & 94.89 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{95.00} \\
\hline
\end{tabular}
\vspace{-0.15in}
\end{table}
\begin{table}[t!]
\caption{Results on ImageNet dataset.}
\label{table:IMAGENET}
\centering
\small
\begin{tabular}{|c|c|c|c|}
\hline
Model & Method & \tabincell{c}{Bit-width \\ (W/A/G)} & \tabincell{c}{Accuracy \\(\%)} \\
\hline
\hline
\multirow{6}{*}{AlexNet}
& FP & 32/32/32 & 59.84 \\
\cline{2-4}
& DoReFa-Net \cite{zhou2016dorefa} & 8/8/8 & 53.00 \\
& Ours & 8/8/8 & \textbf{58.84} \\
\cline{2-4}
& WAGE \cite{wage} & 2/8/8 & 48.40 \\
& Ours & 2/8/8 & \textbf{51.28} \\
\hline
\multirow{4}{*}{ResNet-18}
& FP & 32/32/32 & 70.30 \\
\cline{2-4}
& WAGEUBN \cite{wageubn} & 8/8/8 & 66.92 \\
\cline{2-4}
& FP8 training \cite{fp8training} & 8/8/8 & 67.34 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{69.67} \\
\hline
\multirow{3}{*}{ResNet-34}
& FP & 32/32/32 & 73.68 \\
\cline{2-4}
& WAGEUBN \cite{wageubn} & 8/8/8 & 68.50 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{73.29} \\
\hline
\multirow{3}{*}{ResNet-50}
& FP & 32/32/32 & 76.60 \\
\cline{2-4}
& WAGEUBN \cite{wageubn} & 8/8/8 & 69.07 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{76.34} \\
\hline
\multirow{2}{*}{\tabincell{c}{MobileNetV2}}
& FP & 32/32/32 & 72.39 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{71.20} \\
\hline
\multirow{2}{*}{\tabincell{c}{InceptionV3}}
& FP & 32/32/32 & 72.39 \\
\cline{2-4}
& Ours & 8/8/8 & \textbf{71.20} \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
\subsection{Object Detection}
To prove the versatility of our method, we further conduct experiments with the popular object detection networks including Faster-RCNN \cite{ren2015faster}, RFCN \cite{dai2016rfcn} and RetinaNet \cite{RetinaNet} on two widely used datasets: PASCAL VOC \cite{Pascalvoc} and COCO~\cite{coco}. The PASCAL VOC dataset consists of 11k images with 20 classes. The COCO dataset contains more than 20k images and 80 object categories. Note that we are the first to successfully achieve INT8 training on the object detection task.
\noindent \textbf{Settings.} As for the hyperparameters, we follow the same rules described in \cite{Li_2019_CVPR}. And $\alpha$ and $\beta$ for learning rate scaling are the same as those used in image classification task.
\noindent \textbf{PASCAL VOC.} We test RFCN and Faster R-CNN with different backbones, and find that quantized training equipped with our method only suffers a very slight detection accuracy (mAP) drop. The result of RFCN shows that even for a deeper backbone such as ResNet-101, our INT8 training still maintains almost the same accuracy as full-precision.
\noindent \textbf{COCO.} On the large scale COCO dataset, we experiment with RetinaNet (one-stage) and Faster R-CNN (two-stage). Our method performs stably with less than 1.8$\%$ accuracy degradation on both networks. We find that RetinaNet incurs higher mAP loss than Faster R-CNN, which is inconsistent with the conclusions in the previous study \cite{Li_2019_CVPR}. This may be caused by the fact that the focal loss used in one stage detector is more sensitive to gradient quantization.
\begin{table}[t!]
\caption{Results on PASCAL VOC Dataset.}
\label{table:PASCALVOC}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
Model & Backbone & Method & \tabincell{c}{Bit-width \\ (W/A/G)} & mAP (\%) \\
\hline
\hline
\multirow{2}{*}{\tabincell{c}{Faster\\ R-CNN}}
& ResNet-50 & FP & 32/32/32 & 82.0 \\
\cline{2-4}
& ResNet-50 & Ours & 8/8/8 & \textbf{81.9} \\
\hline
\multirow{2}{*}{RFCN}
& ResNet-101 & FP & 32/32/32 & 80.8 \\
\cline{2-4}
& ResNet-101 & Ours & 8/8/8 & \textbf{79.1} \\
\hline
\end{tabular}
\vspace{-0.15in}
\end{table}
\begin{table}[t!]
\caption{Results on COCO Dataset.}
\label{table:COCO}
\centering
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
Model & Backbone & Method & \tabincell{c}{Bit-width \\ (W/A/G)} & mAP (\%) \\
\hline
\hline
\multirow{2}{*}{\tabincell{c}{Faster \\ R-CNN}}
& ResNet-50 & FP & 32/32/32 & 36.2 \\
\cline{2-4}
& ResNet-50 & Ours & 8/8/8 & \textbf{34.95} \\
\hline
\multirow{2}{*}{RetinaNet}
& ResNet-50 & FP & 32/32/32 & 36.9 \\
\cline{2-4}
& ResNet-50 & Ours & 8/8/8 & \textbf{35.1} \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{table}
\begin{figure}
\includegraphics[width=1.0\linewidth]{speed.png}
\caption{INT8 convolution speedup on GPU, where Y-axis indicates (input shape), (kernel number, kernel size) of convolution.}
\label{fig:speedup}
\vspace{-0.05in}
\end{figure}
\begin{table}[t!]
\caption{End-to-end average time for a round of INT8 training. (tested with ResNet-50 on GeForce GTX1080TI, batch size 64.)}
\label{tab:speedup}
\centering
\small
\begin{tabular}{|c|c|c|c|}
\hline
Precision & Forward (s) & Backward (s) & Iteration (s) \\ \hline \hline
FP32 (cuDNN) & 0.117 & 0.221 & 0.360 \\ \hline
INT8 (ours) & 0.101 & 0.171 & 0.293 \\ \hline
\end{tabular}
\vspace{-0.2in}
\end{table}
\subsection{Speed Result on NVIDIA GPU}
\label{section:speed}
None of the existing libraries can directly support the complete INT8 training. Thus we implement it by ourselves on NVIDIA Pascal GPU using DP4A instruction to verify the acceleration power of our method. The speedup of each convolutional layer in ResNet-50 is shown in Figure \ref{fig:speedup}. In the forward process using our solution, INT8 can bring an average \textbf{1.63$\times$} speedup, while in the backward process, it can achieve a higher \textbf{1.94$\times$} speedup.
Table \ref{tab:speedup} further reports the time consumption and speed improvement of each training round. Even if we only replace the FP32 convolutional layer with the slightly optimized INT8 one, the training time for ResNet-50 can be reduced by about 22\%.
\section{Conclusions}
In this paper, we attempt to build an INT8 training framework for common DCNNs. We found four distinctive characteristics of gradients and then gave two theoretical principles stabilizing training with the convergence bound. Based on that, we proposed Direction Sensitive Gradient Clipping and Deviation Counteractive Learning Rate Scaling. Extensive experiments prove the versatility of our method for various networks and tasks. We reduced the training time by 22\% on Pascal GPU with only trivial optimization. If each layer is sufficiently optimized, the training will achieve higher speedup and lower memory consumption. We hope our first successful attempt can help lead the community towards a fully unified INT8 training.
\nocite{caffe}
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\begin{document}
\title{A reduced order variational multiscale approach for turbulent flows}
\author{Giovanni Stabile$^1$}
\author{Francesco Ballarin$^1$}
\author{Giacomo Zuccarino$^1$}
\author{Gianluigi Rozza$^1$}
\address{$^1$ mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy}
\maketitle
\begin{abstract}
The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique.
Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection.
In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level.
The former method is denoted as \emph{consistent ROM}, while the latter is named \emph{non-consistent ROM}, in order to underline the different choices made at the two levels.
Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.
\end{abstract}
\section{Introduction}\label{sec:intro}
During the last decades, the increase of computational resources and the great improvement into numerical methods to solve partial differential equations has allowed to broaden the applicability of computational methods, originally devised in an academic environment, to industrial problems. Indeed, nowadays numerical simulation is a key ingredient in several engineering and applied sciences fields such as aeronautical, naval, civil and mechanical engineering, life sciences and medicine where computational methods are used to solve mechanics and fluid dynamics problems.
However, especially for fluid dynamics, there are still certain situations where the direct numerical simulations of the governing equations using standard discretization techniques (Finite Elements, Finite Volumes, Finite Differences, Spectral Element Methods) may become unaffordable. Such situations occurs when a large number of different system configurations are in need of being tested (shape optimization, uncertainty quantification, inverse problems) or an extremely reduced computational cost is required (real-time control).
A viable approach to solve this kind of problems, at a far lower computational cost, is given by reduced order modelling techniques \cite{bennerParSys,ChinestaEnc2017,libroRozza,quarteroniRB2016}. Reduced order models are based on the assumption that the system response is sufficiently smooth with respect to the input parameters. Given this assumption, the solution space can be restricted to a lower dimensional space given by properly chosen basis functions. This work is based on a POD-Galerkin approach which consists into the extraction of the most energetic modes representing the system dynamics and into a Galerkin projection onto the space spanned by these modes. POD-Galerkin methods have been proposed in several works starting from different full order discretization techniques and focusing the attention on different aspects. We mention \cite{deane1991,Iollo1998,ITO1998403,noack1994,peterson1989} among the first attempts to create low-dimensional models for fluid flows starting from POD basis functions.
However, it is well known that standard POD-Galerkin models are prone to possible stability issues. The type of instabilities which are analyzed an treated in literature can be classified into different classes depending on the source of instability: (i) in \cite{Akhtar2009,Bergmann2009516,taddei2017,Iollo2000,Sirisup2005218,giovannisaddam2017} the attention is focused on instability observed in transient problems during long time integration; (ii) \cite{Shafqat,Ballarin2015,Caiazzo2014598,Gerner2012,rozza2007stability,Stabile2018} are devoted to accurate pressure recovery and inf-sup instability, while (iii) in \cite{ChaconDelgadoGomezBallarinRozza2017,iliescu2013variational,Iliescu_vms} the instability due to advection dominated phenomena and transition to a turbulent regime are discussed. This work deals with the last two type of instabilities and aims to study the relationship between the stabilization techniques adopted for inf-sup pressure instabilities and the ones adopted for convection dominated problems.
Apart from the aforementioned references, ROMs based on POD-Galerkin methods have been mainly employed to model laminar flows. The main novelty of this work it thus to present a ROM specifically designed for turbulent flows, as well as discussing the role of the inf-sup pressure stabilization. It is our belief that literature on closure models for the approximation of turbulent flows using standard discretization techniques presents a large variety of options (see e.g. \cite{davidson2004,pope2001,sagaut2006}) which are waiting to be incorporated into a reduced order framework.
Among the most studied strategies we mention the Reynolds-Averaged Navier-Stokes (RANS) models \cite{pope2001}, the Large-Eddy Simulation (LES) models \cite{sagaut2006} and the Variational Multiscale (VMS) method \cite{Hughes1996,Hughes1998}.
In the development of a reduced order strategy, one must take into account that the choice of the closure model often depends on the specific application at hand. Indeed, closure models based on eddy viscosity (e.g. $k-\omega$, $k-\epsilon$, Spalart-Allmaras), mainly tailored for finite volumes discretization, seem to be particularly promising to treat developed turbulent flows with high Reynolds number in the context of industrial applications. A reduced order strategy for such applications is going to be actively pursued by the authors of this manuscript in \cite{HijaziAliStabileBallarinRozza2018,Lorenzi2016,saddam2018}. Here, however, we focus on closure model adopted for more moderate Reynolds numbers, based on a finite element discretization with VMS stabilization and it fits well moderate turbulent patterns.
The proposed methodology can still be applied to several relevant cases, such as those arising in biomedical applications. Furthermore, it builds on a stronger theoretical framework in the reduced order modelling community, which has recently seen first contributions for what concerns certification of a posteriori error estimators in turbulent flows \cite{ChaconDelgadoGomezBallarinRozza2017}.
The paper is organized as follows: the mathematical formulation of the problem is set in \autoref{sec:math_form}; the VMS finite element discretization used to numerically solve the mathematical problem is reported in \autoref{sec:FOM}. \autoref{sec:ROM} contains the formulation of the Reduced Order Model, and some consideration concerning its stabilization. The proposed methodology is tested on a numerical example in \autoref{sec:numer_example}, and finally some conclusions are drawn and perspectives for future developments are given in \autoref{sec:concl}.
\section{Mathematical Formulation of the Physical Problem}\label{sec:math_form}
The mathematical problem on which this work is focused is given by the incompressible Navier-Stokes equations. Given a space-time domain $Q=\Omega\times[0,T]\subset\mathbb{R}^2\times\mathbb{R}^+$ the initial boundary-value problem consists in solving the following equations for the velocity $\bm{u}:Q\rightarrow\mathbb{R}^2$ and the pressure $p:Q\rightarrow\mathbb{R}$ such that:
\begin{equation}\label{eq:navstokes}
\begin{cases}
\partial_t\bm{u}+ \dive (\bm{u} \otimes \bm{u})- \dive(2 \nu \nabla^s \bm{u}) + \nabla p = \bm{f} &\mbox{ in } Q,\\
\dive(\bm{u})=\bm{0} &\mbox{ in } Q,\\
\bm{u} = \bm{g} &\mbox{ on } \Gamma_{\text{in}} \times [0,T],\\
\bm{u} = \bm{0} &\mbox{ on } \Gamma_{\text{wall}} \times [0,T],\\
(2 \nu\nabla^s \bm{u} - p\bm{I})\bm{n} = \bm{0} &\mbox{ on } \Gamma_{\text{out}} \times [0,T],\\
\bm{u}=\bm{u}_0 &\mbox{ in } \Omega\times\{0\},\\
\end{cases}
\end{equation}
where $\Gamma = \Gamma_{\text{in}} \cup \Gamma_{\text{wall}} \cup \Gamma_{\text{out}}$ is the boundary of $\Omega$, composed by three non-overlapping regions $\Gamma_{\text{in}}$, $\Gamma_{\text{wall}}$ and $\Gamma_{\text{out}}$ that indicate, respectively, inlet boundary, physical walls, and outlet boundary.
Furthermore, $\nu \in \mathbb{R}^+$ is the kinematic viscosity, $\gr{f}:Q\rightarrow\mathbb{R}^2$ is the given body force (per unit volume), $\gr{g}:\Gamma_{\text{in}}\times[0,T]\rightarrow\mathbb{R}^2$ is the inlet velocity profile, and $\bm{u}_0: \Omega \to \mathbb{R}^2$ denotes the initial condition for the velocity.
The symbol $\otimes$ denotes the tensor product of two vectors in $\mathbb{R}^2$, i.e. in components $\left(\gr{u}\otimes\gr{v}\right)_{ij}=u_i v_j$, while $\nabla^s\gr{u}$ denotes the symmetric part of the velocity gradient, i.e. $\nabla^{s}\gr{u}=\frac{\nabla\gr{u}+\nabla\gr{u}^{T}}{2}$.
The weak formulation of \eqref{eq:navstokes} is obtained by multiplying the momentum equation for a test function $\gr{v}\in\gr{V} = \{\gr{v} \in \gr{H}^1(\Omega): \gr{v} = \gr{0}\text{ on }\Gamma_{\text{in}} \cup \Gamma_{\text{wall}}\}$ and the continuity equation for a test function $q\in Q = L^2(\Omega)$, and with integration by parts over the space domain $\Omega$.
The weak problem for \eqref{eq:navstokes} reads as follows:
\noindent\emph{Find $\gr{u}\in L^2([0,T],\gr{H}^1(\Omega))$ and $p\in L^2([0,T],Q)$ such that $\gr{u}=\gr{g}$ on $\Gamma_{\text{in}}\times[0,T]$ and $\gr{u}=\gr{0}$ on $\Gamma_{\text{wall}}\times[0,T]$, and the equations
\begin{equation}
\label{NS weak}
\begin{cases}
(\partial_t\gr{u},\gr{v})_{\Omega}-(\gr{u}\otimes\gr{u},\nabla \gr{v})_{\Omega}+2\nu(\nabla^s\gr{u},\nabla^s\gr{v})_{\Omega}-(p,\dive(\gr{v}))_{\Omega}=(\gr{f},\gr{v})_{\Omega},\\
(q,\dive(\gr{u}))_{\Omega}=0,\\
\gr{u}(0)=\gr{u}_0,
\end{cases}
\end{equation}
hold for all $\gr{v}\in\gr{V}$ and for all $q\in Q$.
}
\section{The Full Order Model}\label{sec:FOM}
In this section the full order discretization used through the manuscript is presented. In \autoref{subsec:FEM_GAL} the standard Galerkin finite element approximation is introduced, while \autoref{subsec:FEM_VMS} presents the Residual Based Variational Multiscale concepts used to achieve stable simulations.
\subsection{The Standard Galerkin Finite Element Approximation}\label{subsec:FEM_GAL}
Given the continuous formulation presented in \autoref{sec:math_form} here we introduce the discrete formulation and the variational multiscale finite element approximation.
We consider a finite decomposition (mesh) $\mathcal{T}_h$ of $\lbar{\Omega}=\bigcup_{K\in \mathcal{T}_h} K,\;$ composed of triangular cells $K$.
Two finite element (FE) spaces $\gr{V}_h\subset\gr{V}$ and $Q_h\subset Q$ are employed to discretize \eqref{NS weak} on $\mathcal{T}_h$.
Specifically, $\gr{V}_h$ and $Q_h$ are spaces of continuous locally polynomial functions of degree $2$ and $1$, respectively. The FE formulation on $(\gr{V}_h,Q_h)$, known as Taylor-Hood FE, satisfies the so called \emph{inf-sup} condition \cite{boffi_mixed}
\begin{equation}
\label{inf-sup}
\inf_{q_h\in Q_h\setminus\{0\}}\sup_{\gr{v}_h\in\gr{V}_h\setminus\{\gr{0}\}}\frac{(\dive(\gr{v}_h),q_h)}{\norm{\gr{v}_h}_{\gr{V}_h}\norm{q_h}_{Q_h}}\geq\beta>0.
\end{equation}
We will come back to the topic of inf-sup stability for what concerns the reduced order model. For the sake of notation in the definition of the reduction process, let us define two sets $\{\grs{\varphi}_j\}_{j=1}^{M_h}$, $\{\psi_l\}_{l=1}^{K_h}$ of Lagrangian basis functions of $\gr{V}_h$ and $Q_h$, respectively, being $M_h=\text{dim}(\gr{V}_h)$ and $K_h = \text{dim}(Q_h)$.
The semi-discrete Galerkin-FE approximation of \eqref{NS weak} reads as follows: find $\gr{u}_h\in L^2([0,T],\gr{V}_h)$ and $p_h\in L^2([0,T],Q_h)$ such that $\gr{u}_h=\gr{g}_h$ on $\Gamma_{\text{in}}\times[0,T]$ and $\gr{u}_h=\gr{0}$ on $\Gamma_{\text{wall}}\times[0,T]$, and the equations
\begin{equation}
\label{NS weak discrete}
\left\lbrace
\begin{split}
\left(\partial_t\gr{u}_h,\gr{v}_h\right)_{\Omega}&-\left(\gr{u}_h\otimes\gr{u}_h,\nabla \gr{v}_h\right)_{\Omega}+2\nu\left(\nabla^s\gr{u}_h,\nabla^s\gr{v}_h\right)_{\Omega}+\\
&-\left(p_h,\dive(\gr{v}_h)\right)_{\Omega}+\left(q_h,\dive(\gr{u}_h)\right)_{\Omega}=\left(\gr{f},\gr{v}_h\right)_{\Omega},\\
\gr{u}_h(0)&=\gr{u}_{0,h},
\end{split}
\right.
\end{equation}
hold for all $\gr{v}_h$ in $\gr{V}_h $ and for all $q_h$ in $Q_h$, being $\gr{g}_h$ and $\gr{u}_{0,h}$ suitable interpolations of $\gr{g}$ and $\gr{u}_{0}$ on the mesh $\mathcal{T}_h$.
Finally, time discretization is carried out by taking $N_T+1$ time instants $\{t_k\}_{k=0}^{N_T}$ in $[0,T]$, chosen such that $\Delta t=t_k-t_{k-1}$ is constant. For simplicity a backward Euler time stepping scheme is used in this manuscript. Since such time discretization is very standard (see e.g. \cite{quarteroni2008numerical}), and seamlessly applies at the FE, VMS and reduced order levels, in the following we will always refer to the semi-discrete formulation \eqref{NS weak discrete} omitting any further presentation of the time discretization.
\subsection{The Residual Based Variational Multiscale Formulation}\label{subsec:FEM_VMS}
In case of turbulent flows the standard Galerkin-FE approximations may fail to accurately model the physical phenomena unless a very refined mesh is employed. As such refinement is usually unaffordable for Reynolds numbers in the order of a few thousands, we resort instead to a stabilization based on a variational multiscale method.
In this work we rely on the Residual Based variational multiscale method, as presented in \cite{Bazilevs2007173}.
The Residual Based VMS was first introduced in \cite{Hughes1995}. Further theoretical development that led to the application in Computational Fluid Dynamics (CFD) were given in \cite{Hughes1998,Hughes1996}.
A residual based VMS method has been used in CFD applications for a large variety of problems (see e.g. \cite{Bazilevs2007173,codina2017variational,forti2015semi,hughes2004multiscale,masud2011heterogeneous}), and as a result such VMS method has proven to be an efficient and accurate turbulence model for applications characterized by moderately high Reynolds number flows.
The method is a two-scales method, even though other VMS variants with a larger number of scales are available in literature (e.g. the Residual Free Bubble VMS \cite{Brezzi1992}, the Projection Based VMS \cite{rebollo2015numerical} and the Algebraic VMS \cite{Gravemeier2010}).
Following the presentation in \cite{Bazilevs2007173}, we orthogonally decompose the spaces $\gr{V}$ and $Q$ as
\begin{equation}
\gr{V}=\gr{V}_h\oplus\gr{V}',\;Q=Q_h \oplus Q',
\end{equation}
using the elliptic projector $(\nabla^s\cdot, \nabla^s\cdot)_{\Omega}$ and the $L^2$ projector $(\cdot, \cdot)_{\Omega}$, respectively, as inner products.
The spaces $\gr{V}_h$ and $Q_h$ represent the coarse (resolved) scale, while $\gr{V}'$ and $Q'$ are modelling the fine (unresolved) scale.
Therefore, for any $t\in[0,T]$ we decompose the trial functions in \eqref{NS weak discrete} as
\begin{align*}
\gr{u}(t)&=\gr{u}_h(t)+\gr{u}'(t)&&\text{for }\gr{u}_h(t)\in\gr{V}_h\text{ and }\gr{u}'(t)\in\gr{V}',\\
p(t)&=p_h(t)+p'(t)&&\text{for }p_h(t)\in Q_h\text{ and }p'(t)\in Q'
\end{align*}
assuming the fine scale velocity functions to satisfy homogeneous boundary conditions on the Dirichlet boundaries,
\begin{equation*}
\gr{u}'(t)=0\text{ on }\Gamma_{\text{in}}\cup\Gamma_{\text{wall}},
\end{equation*}
so that the coarse scale velocity functions are such that
\begin{equation*}
\gr{u}_h(t)=\gr{g}_h(t)\;\text{ on }\Gamma_{\text{in}},\quad
\gr{u}_h(t)=\gr{0}\text{ on }\Gamma_{\text{wall}},
\end{equation*}
i.e. they fulfill the physical boundary conditions.
Introducing such expressions in \eqref{NS weak} with coarse scale test functions $\gr{v}_h\in\gr{V}_h$ and $q_h\in Q_h$ we obtain the so-called \emph{coarse scale equation}
\begin{equation}
\label{NS coarse}
\left\lbrace
\begin{split}
(\partial_t\gr{u}_h,\gr{v}_h)_{\Omega}&-(\gr{u}_h\otimes \gr{u}_h,\nabla \gr{v}_h)_{\Omega}+2\nu(\nabla^s \gr{u}_h,\nabla^s \gr{v}_h)_{\Omega}+\\
&-(p_h,\dive(\gr{v}_h))_{\Omega}+(q_h,\dive(\gr{u}_h))_{\Omega}
\\
&-(p',\dive(\gr{v}_h))_{\Omega}-(\nabla q_h,\gr{u}')_{\Omega}+\\
&-(\gr{u}'\otimes\gr{u}_h,\nabla \gr{v}_h)_{\Omega}-(\gr{u}_h\otimes\gr{u}',\nabla \gr{v}_h)_{\Omega}+\\
&-(\gr{u}'\otimes\gr{u}',\nabla \gr{v}_h)_{\Omega}\\
&=(\gr{f},\gr{v}_h)_{\Omega},\\
\gr{u}_h(0)&=\gr{u}_{0,h} .
\end{split}
\right.
\end{equation}
A few terms omitted in \eqref{NS coarse} drop due to the direct sum orthogonal decomposition, e.g.
\begin{equation*}
\left(\nabla^s\gr{u}',\nabla^s\gr{v}_h\right)_{\Omega}=0
\end{equation*}
while others, such as in particular
\begin{equation*}
(\partial_t\gr{u}',\gr{v}_h)_{\Omega},
\end{equation*}
are dropped as turbulence modelling assumptions \cite{Bazilevs2007173}.
Following the philosophy of the LES turbulence models the fine scale solutions are not solved explicitly, rather modeled as follows:
\begin{align*}
\gr{u}'&=-\tau_M\gr{r}_M(\gr{u}_h,p_h),\\
p'&=-\tau_Cr_C(\gr{u}_h),
\end{align*}
where such equality is to be understood as restricted to each cell in $\mathcal{T}_h$.
The structure of the stabilization terms introduced in \cite{Bazilevs2007173} and adopted in this work is the following:
\begin{equation}
\label{fine scale explicit}
\begin{cases}
\begin{split}
\gr{r}_M(\gr{u}_h,p_h)&=\partial_t\gr{u}_h+\dive(\gr{u}_h\otimes\gr{u}_h)-\dive(2\nu\nabla^{s}\gr{u}_h)+\nabla p_h-\gr{f},
\\
r_C(\gr{u}_h)&=\dive(\gr{u}_h),
\\
\tau_M&=\tau_M(\gr{u}_h)=\left(\frac{4}{{\Delta t}^2}+\gr{u}_h\cdot G \gr{u}_h+C_{\text{inv}}\nu^2 G : G \right)^{-\frac{1}{2}},
\\
\tau_C&=\tau_C(\gr{u}_h)=(\tau_M\gr{g}\cdot\gr{g})^{-1}.
\\
\end{split}
\end{cases}
\end{equation}
being $\gr{r}_M(\gr{u}_h,p_h)$ and $r_C(\gr{u}_h)$ residuals of the strong form of momentum and continuity equations, respectively, and $\tau_M$ and $\tau_C$ corresponding stabilization coefficients. Furthermore, on each cell $K$,
\begin{equation}
\left\lbrace
\begin{split}
G_{ij}&=\sum_{k=1}^2\frac{\partial\xi_k}{\partial x_i}\frac{\partial\xi_k}{\partial x_j},\quad
G : G =\sum_{i,j=1}^2 G _{ij} G _{ij},
\\
g_i&=\sum_{j=1}^2\frac{\partial\xi_j}{\partial x_i},\quad
\gr{g}\cdot\gr{g}=\sum_{i=1}^2\gr{g}_i\gr{g}_i,
\end{split}
\right.
\end{equation}
where $\gr{\xi}=(\xi_1,\xi_2)$ is the inverse of the affine mapping between the reference simplex $\hat{K}$ and the triangulation element $K$, and $C_{\text{inv}}$ is the inverse inequality constant \cite{Bazilevs2007173,quarteroni2008numerical}.
In order to proceed with an algebraic formulation of the resulting semi-discretization we expand $\gr{u}_h(t)$ and $p_h(t)$ in coordinates with respect to the Lagrangian basis on $\gr{W}_h$ and on $Q_h$. Collecting the resulting coordinates in $\mathbf{u}_h\left(t\right)=\left(u_1\left(t\right),\dots,u_{M_h}\left(t\right)\right)^T$ and $\mathbf{p}_h\left(t\right)=\left(p_1\left(t\right),\dots,p_{K_h}\left(t\right)\right)^T$, we obtain the nonlinear dynamical system
\begin{equation}
\begin{cases}
\begin{split}
M\frac{d\mathbf{u}_h\left(t\right)}{dt}&+A\mathbf{u}_h\left(t\right)+C\left(\mathbf{u}_h\left(t\right)\right)\mathbf{u}_h\left(t\right)+D\left(\mathbf{u}_h\left(t\right),\mathbf{p}_h\left(t\right)\right)
+B^T\mathbf{p}_h\left(t\right)
=\gr{F}\left(t\right),\\
B\mathbf{u}_h\left(t\right)&=E\left(\mathbf{u}_h\left(t\right),\mathbf{p}_h\left(t\right)\right),
\end{split}
\end{cases}
\end{equation}
being\footnote{With a slight abuse, in the notation we indicate for the nonlinear terms their dependence on the coefficient vectors $(\mathbf{u}_h\left(t\right), \mathbf{p}_h\left(t\right))$ on the left-hand side, while the actual expression on the right-hand side will be provided in terms of the corresponding functions $(\gr{u}_h(t), p_h(t))$.}
\begin{align*}
M_{ij}&=\left(\grs{\varphi}_j,\grs{\varphi}_i\right)_{\Omega}&&\text{ the mass matrix},\\
A_{ij}&=\left(\nabla^s\grs{\varphi}_j,\nabla^s\grs{\varphi}_i\right)_{\Omega}&&\text{ the discretized diffusion operator},\\
C\left(\mathbf{u}_h\right)_{ij}&=-\left(\grs{\varphi}_j\otimes\gr{u}_h,\nabla\grs{\varphi}_i\right)_{\Omega}&&\text{ the discretized convection operator},\\
B_{li}&=-\left(\psi_l,\dive\left(\grs{\varphi}_i\right)\right)_{\Omega}&&\text{ the discretized version of the pressure/divergence operator, and}\\
F_i\left(t\right)&=\left(\gr{f},\grs{\varphi}_i\right)_{\Omega}&&\text{ the discretized forcing term},
\end{align*}
for $i,j = 1, \hdots, M_h$ and $l = 1, \hdots, K_h$.
The remaining terms are due to the stabilization, and read:
\begin{equation*}
\begin{split}
D\left(\mathbf{u}_h,\mathbf{p}_h\right)_i&=
\sum_{K}\left(\left(\nabla\grs{\varphi}_i\right)\gr{u}_h,\tau_M\gr{r}_M\left(\gr{u}_h,p_h\right)\right)_{K}
+\sum_{K}\left(\left(\nabla\grs{\varphi}_i\right)^T\gr{u}_h,\tau_M\gr{r}_M\left(\gr{u}_h,p_h\right)\right)_{K}
\\&
-\sum_{K}\left(\grs{\varphi}_i,\tau_M\gr{r}_M\left(\gr{u}_h,p_h\right)\otimes\tau_M\gr{r}_M\left(\gr{u}_h,p_h\right)\right)_{K}
+\sum_{K}\left(\tau_Cr_C\left(\gr{u}_h\right),\dive\left(\grs{\varphi}_i\right)\right)_{K},
\\
E\left(\mathbf{u}_h,\mathbf{p}_h\right)_l&=\sum_{K}\left(\nabla{\psi}_l,\tau_M\gr{r}_M\left(\gr{u}_h,p_h\right)\right)_{K}.
\end{split}
\end{equation*}
\begin{comment}
In order to obtain the fully discretized system we discretize the time derivative with an Implicit Euler approximation scheme, thus we obtain the discretized problem that is stated as follows:
\emph{ For $k=1,\dots,N_{T}+1$ find $\gr{u}^k\in\gr{W}_h$ and $p_h\in Q_h$ such that
\begin{equation}
\label{eq:non_lin_alg_sys}
\begin{cases}
\begin{split}
M\frac{\gr{u}^k-\gr{u}^{k-1}}{\Delta t_k}&+A\gr{u}^k+C\gr{u}^k)\gr{u}^k+B^T\gr{p}^k+
\\
&+D_{1}(\tau_Cr_C(\gr{u}^k))+D_{2}(\tau_M\gr{r}_M(\gr{u}^k,\gr{p}^k))+\\
&+D_{3}(\tau_M\gr{r}_M(\gr{u}^k,\gr{p}^k))+D_{4}(\tau_M\gr{r}_M(\gr{u}^k,\gr{p}^k))+\\
&=\gr{F}(t_k),\\
B\gr{u}^k&=D_5(\tau_M\gr{r}_M(\gr{u}^k,\gr{p}^k)).
\end{split}
\end{cases}
\end{equation}}
Where
\begin{equation}
\frac{d\gr{u}(t_k)}{dt}\approx\frac{\gr{u}(t_k)-\gr{u}(t_{k-1})}{\Delta t},\;\gr{u}(t_k)=\gr{u}^k,\;\;\gr{p}(t_k)=\gr{p}^k.
\end{equation}
We remark that $\tau_M$ and $\tau_C$ depend on $\gr{u}^k$, moreover $\displaystyle\frac{d\gr{u}(t)}{dt}$ is hidden in $\displaystyle\gr{r}_M$ and it is discretized as consequence.
The system of equations \eqref{nonlinear algebraic system} is the nonlinear algebraic system that we actually solve for each time step $t_k$.
For every time step $t_k$ the nonlinear system is solved with a direct Newton method. The Newton method is implemented numerically with the usage of the MUMPS and SNES libraries \cite{MUMPS:1,MUMPS:2}.
\end{comment}
\section{The Reduced Order Models}\label{sec:ROM}
In this section we introduce the two reduced order models (ROMs) employed in this manuscript. We seek a \emph{construction-evaluation} paradigm\footnote{The reason for this terminology, rather than the more widespread \emph{offline-online} decoupling, will be clear in \autoref{subsec:Gproj}.}, as customary in the reduced order modelling community \cite{libroRozza}. A reduced basis is built by means of proper orthogonal decomposition during the construction stage, as detailed in \autoref{subsec:POD}. Afterwards, the ROM is evaluated by a Galerkin projection onto the reduced basis (\autoref{subsec:Gproj}). The two proposed ROMs will only differ during the evaluation phase.
Before laying out the details, let us underline how the proposed ROMs differ in terms of construction and evaluation when compared to previous works \cite{Bergmann2009516,wang_turb}.
The main difference is that in \cite{Bergmann2009516,wang_turb} full order snapshots employed during the construction of the reduced basis are generated using a direct numerical simulation (DNS) approach, while VMS stabilization is used as a closure model for the ROM to account for the contribution of the discarded modes. In the present case we aim to obtain snapshots for values of Reynolds numbers for which a DNS strategy becomes unfeasible. Thus, we need to resort to a VMS stabilization during the construction phase itself. Our interest is then to explore the relationship (in particular, in terms of stability) between a stabilized full order model and the consequent reduced order model built on top of it.
\subsection{Construction of the reduced basis through proper orthogonal decomposition}\label{subsec:POD}
The construction of the reduced basis is carried out by means of a Proper Orthogonal Decomposition (POD) \cite{deane1991,ITO1998403,libroRozza,peterson1989}, as follows. Velocity and pressure snapshots, denoted by $\{\gr{u}_h(t_i)\}_{i=1}^{N_T}$ and $\{p_h(t_i)\}_{i=1}^{N_T}$, are obtained solving \eqref{NS coarse}. Then, the following correlation matrices $\Sigma^{\gr{u}}$ and $\Sigma^{p}$ are assembled, as follows:
\begin{equation}
\Sigma^{\gr{u}}_{ij} = \left(\gr{u}_h(t_i) - \langle\gr{u}_h\rangle,\gr{u}_h(t_j) - \langle\gr{u}_h\rangle\right)_{1,\Omega}, \quad \Sigma^{p}_{ij} = \left({p}_h(t_i),{p}_h(t_j)\right)_{\Omega}, \quad i, j = 1, \hdots, N_T,
\end{equation}
where $(\cdot, \cdot)_\omega$ denotes the $L^2(\omega)$ inner product, while $(\cdot, \cdot)_{1, \omega} = (\nabla\cdot, \nabla\cdot)_\omega$ denotes the $\gr{H}^1(\omega)$ inner product. The corresponding induced norms will be denoted similarly as $\lVert\cdot\rVert_\omega$ and $\lVert\cdot\rVert_{1, \omega}$, being either $\omega = \Omega$ for spatial integration or (in the $L^2$ case) $\omega = [0, T]$ for time integration. Furthermore, assuming the inlet profile $\gr{g}$ to be time-independent\footnote{Indeed, non-stationary solutions can be obtained even for time-invariant input data in case of flows with large Reynolds numbers.}, the time-averaged velocity
\begin{equation}
\langle\gr{u}_h\rangle=\frac{1}{N_T}\sum_{i=1}^{N_T}\gr{u}_h(t_i) ,
\end{equation}
is subtracted from the velocity snapshots in order to for $\gr{u}_h(t_i) - \langle\gr{u}_h\rangle$, $i = 1, \hdots, N_T$, to be zero on each Dirichlet boundary. After carrying out an eigendecomposition, we denote by
\begin{equation*}
\lambda_{1}^{\gr{u}}\geq\lambda_{2}^{\gr{u}}\geq\dots\geq\lambda_{N_T}^{\gr{u}}\geq0,\qquad
\lambda_{1}^p\geq\lambda_{2}^p\geq\dots\geq\lambda_{N_T}^{p}\geq0,
\end{equation*}
the resulting eigenvalues, and by
\begin{equation*}
\lbrace\widehat{\gr{v}}_n\rbrace_{n=1}^{N_T},\qquad \lbrace\widehat{q}_n\rbrace_{n=1}^{N_T},
\end{equation*}
the corresponding eigenfunctions. The reduced basis functions are then obtained as
\begin{equation*}
\gr{\varphi}_n^{\text{rb}}=\sum_{i=1}^{N_T}\left(\gr{\hat{v}}_n\right)_i\gr{u}_h(t_i),
\qquad
\psi_n^{\text{rb}}=\sum_{i=1}^{N_T}\left(\hat{q}_n\right)_i p_h(t_i),
\qquad
n = 1, \hdots, N_T,
\end{equation*}
and possibly normalized in their respective spaces.
It is well known in the reduced basis community that spaces built from (a truncation of) $\{\gr{\varphi}_n^{\text{rb}}\}$ and $\{\psi_n^{\text{rb}}\}$ might not satisfy an inf-sup condition. It is thus customary to enrich the reduced velocity space with the so-called pressure \emph{supremizers} \cite{libroRozza,rozza2007stability,quarteroniRB2016}. Such an approach entails (for each $i = 1, \hdots, N_T$) the computation of a function $\gr{s}_h(t_i)\in\gr{V}_h$ such that
\begin{equation*}
\left(\nabla \gr{s}_h(t_i),\nabla\gr{v}_h\right)_{\Omega}=\left(\nabla p_h(t_i),\gr{v}_h\right)_{\Omega}
\end{equation*}
holds for all $\gr{v}_h\in\gr{V}_h$.
The strategy that we adopt is the \emph{approximate supremizer enrichment} as in e.g. \cite{Ballarin2015}. This approach consists in applying a POD compression to the resulting supremizer snapshots. Following the same procedure as above, we obtain a set $\lbrace\gr{\phi}_n^{\rb}\rbrace_{n=1}^{N_T}$ of supremizer basis functions.
Afterwards, the final result of the construction stage is the definition of the reduced basis spaces\footnote{Numerical results in \autoref{sec:numer_example} will compare the cases with and without supremizer enrichment. It is understood that in the latter case the reduced velocity space is defined as
\begin{equation*}
\gr{V}_{\rb} = \text{span}\{\gr{g}^{\rb}, \gr{\varphi}_1^{\text{rb}}, \hdots, \gr{\varphi}_N^{\text{rb}}\}.
\end{equation*}
We will not cover the latter case in \autoref{subsec:Gproj}, as the required changes to the formulation of the Galerkin projection are very trivial.
}
\begin{equation*}
\gr{V}_{\rb} = \text{span}\{\gr{g}^{\rb}, \gr{\varphi}_1^{\text{rb}}, \hdots, \gr{\varphi}_N^{\text{rb}}, \gr{\phi}_1^{\text{rb}}, \hdots, \gr{\phi}_N^{\text{rb}}\},
\qquad
Q_{\rb} = \text{span}\{\psi_1^{\text{rb}}, \hdots, \psi_N^{\text{rb}}\},
\end{equation*}
for some integer $N < N_T$, and $\gr{g}^{\rb} \equiv \langle\gr{u}_h\rangle$. The FE degrees of freedom of the bases in $\gr{V}_{\rb}$ and $Q_{\rb}$ are also stored as columns of the basis functions matrices $Z_{\rb}^{\gr{u}}$ and $Z_{\rb}^{p}$ for velocity and pressure, respectively.
\subsection{Evaluation of the ROM by Galerkin projection}\label{subsec:Gproj}
During the evaluation of the ROM, one seeks a solution of the form
\begin{equation*}
\gr{u}_{\rb}(t)= \gr{g}^{\rb} + \sum_{n=1}^{N}\left(u_{\rb}(t)\right)_{n}\gr{\varphi}_n^{\rb} + \sum_{n=1}^{N}\left(u_{\rb}(t)\right)_{n + N}\gr{\phi}_n^{\rb},
\qquad
p_{\rb}(t)= \sum_{n=1}^{N}\left(p_{\rb}(t)\right)_{n}\psi_n^{\rb}
\end{equation*}
through a Galerkin projection. For the sake of notation we define
$$\mathbf{u}_{\rb}\left(t\right)=\left(1, \left(u_{\rb}(t)\right)_{1},\dots,\left(u_{\rb}(t)\right)_{2N}\right)^T \in \mathbb{R}^{2N+1},$$
and
$$\mathbf{p}_{\rb}\left(t\right)=\left(\left(p_{\rb}(t)\right)_{1},\dots,\left(p_{\rb}(t)\right)_{N}\right)^T \in \mathbb{R}^N,$$
the vector of coefficients of the reduced solution.
Following \cite{Shafqat,PACCIARINI20141} we present here two approaches for the evaluation of the ROM, that we will refer here as \emph{consistent ROM} and \emph{non-consistent ROM}. In the consistent ROM, a Galerkin projection of the VMS Navier-Stokes equations \eqref{NS coarse} onto the reduced basis spaces is carried out. In contrast, in the non-consistent ROM projection of the standard Navier-Stokes equations \eqref{NS weak discrete} onto the reduced basis is sought. As snapshots were generated from \eqref{NS coarse}, the latter choice features a modelling inconsistency between the full order and reduced order levels, hence the name.
\subsubsection{Consistent ROM}
As mentioned, the aim of the \emph{consistent ROM} is to provide a reduced order model which is fully consistent with the full order model, thus including the projection of all stabilization terms of the VMS method. The resulting ROM requires therefore the solution of the following dynamical system:\footnote{In a practical implementation the row/column of the system corresponding to the first entry of $\mathbf{u}_{\rb}$ (which is a known coefficient that accounts for the non-homogeneous boundary condition) shall be condensed.}
\begin{equation*}
\begin{cases}
\begin{split}
M_{\rb}\frac{d\mathbf{u}_{\rb}(t)}{dt}&+A_{\rb}\mathbf{u}_{\rb}(t)+C_{\rb}(\mathbf{u}_{\rb}(t))\mathbf{u}_{\rb}(t)
+D_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t))
+B^T_{\rb}\mathbf{p}_{\rb}(t)
=\gr{F}_{\rb}(t),\\
B_{\rb}\mathbf{u}_{\rb}(t)&=E_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t)),
\end{split}
\end{cases}
\end{equation*}
being
\begin{align*}
&
M_{\rb} = {Z_{\rb}^{\gr{u}}}^T M Z_{\rb}^{\gr{u}},\qquad
A_{\rb} = {Z_{\rb}^{\gr{u}}}^T A Z_{\rb}^{\gr{u}},\qquad
B_{\rb} = {Z_{\rb}^{p}}^T B Z_{\rb}^{\gr{u}},\\
&
C_{\rb}\left(\mathbf{u}_{\rb}\right) = {Z_{\rb}^{\gr{u}}}^T C\left(Z_{\rb}^{\gr{u}} \mathbf{u}_{\rb}(t)\right) Z_{\rb}^{\gr{u}},\qquad
D_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t)) = {Z_{\rb}^{\gr{u}}}^T D(Z_{\rb}^{\gr{u}} \mathbf{u}_{\rb}(t),Z_{\rb}^{p}\mathbf{p}_{\rb}(t)),\\
&
E_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t)) = {Z_{\rb}^{p}}^T E(Z_{\rb}^{\gr{u}} \mathbf{u}_{\rb}(t),Z_{\rb}^{p}\mathbf{p}_{\rb}(t)),\qquad
F_{\rb} = {Z_{\rb}^{\gr{u}}}^T F.
\end{align*}
As the terms $D_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t))$ and $E_{\rb}(\mathbf{u}_{\rb}(t),\mathbf{p}_{\rb}(t))$, arising from VMS stabilization, are highly nonlinear, their evaluation should be handled with care to preserve the efficiency of the resulting ROM, enforcing the so-called offline-online decoupling (see e.g. \cite{libroRozza}). Hyper reduction techniques, such as empirical interpolation \cite{BARRAULT2004667}, GNAT \cite{Carlberg2013623} or Gappy-POD \cite{Willcox2006} methods are available to this end. Since the goal of this preliminary investigation is to compare the accuracy and feasibility of the two proposed ROMs, without aiming at the greatest amount of efficiency, we will allow an inefficient evaluation of the nonlinear terms.
\subsubsection{Non-consistent ROM}
In contrast, the goal of the \emph{non-consistent ROM} is to perform a Galerkin projection of the standard Navier-Stokes equations without any additional stabilization term. The resulting dynamical system is:
\begin{equation*}
\begin{cases}
\begin{split}
M_{\rb}\frac{d\mathbf{u}_{\rb}(t)}{dt}&+A_{\rb}\mathbf{u}_{\rb}(t)+C_{\rb}(\mathbf{u}_{\rb}(t))\mathbf{u}_{\rb}(t)
+B^T_{\rb}\mathbf{p}_{\rb}(t)
=\gr{F}_{\rb}(t),\\
B_{\rb}\mathbf{u}_{\rb}(t)&=\mathbf{0}.
\end{split}
\end{cases}
\end{equation*}
This ROM seems attractive since it only has a nonlinear contribution relative to the convective term, which is a quadratic nonlinearity.
In this case an efficient offline-online decoupling may be easily obtained with the aid of a third tensor precomputed at the end of the construction stage \cite{Ballarin2015}. This model is however not consistent with the full order model used to generate the snapshots because the VMS stabilization terms are neglected during the projection stage. Yet, such inconsistent options are being used with success in literature, as different models at the full order and reduced order levels are sought in \cite{Bergmann2009516,wang_turb}.
\section{A numerical example: flow around a circular cylinder}\label{sec:numer_example}
In this section we present some numerical results on a benchmark test case involving the flow around a circular at moderately high Reynolds numbers (Re $\approx 5000$). The resulting flow conditions are such that a standard Navier-Stokes formulation would fail (unless being provided an extremely refined mesh), and turbulence modelling is required.
The computational domain is given by a rectangular box with a circular hole. A parabolic inflow profile $\gr{g}$ is provided at the inlet, gravity is neglected $\gr{f} = \gr{0}$, and an homogeneous initial velocity is imposed. The final time is assumed to be $T = 2$ s.
The discretization of the domain is as in \autoref{fig:comp_mesh}.
The triangular tessellation counts $37572$ cells. At the full order level, the velocity and pressure fields are approximated with $P2/P1$ finite elements, resulting in $155756$ and $20153$ degrees of freedom for velocity and pressure, respectively. A backward Euler time stepping scheme is employed, with $\Delta t = 0.002$.
The full order system \eqref{NS coarse} is solved with VMS stabilization. Undersampling is performed on the flow fields, in order to contain the size of the correlation matrices $\Sigma^{\gr{u}}$ and $\Sigma^p$, still preserving all relevant features of the transient phenomena. This results in 500 snapshots, which are used for the reduced basis generation through POD. Corresponding eigenvalues decay is shown in \autoref{fig:eig_decay} for velocity, supremizers and pressure snapshots. The eigenvalue plot, respect to a lower Reynolds number case on the same physical problem, shows a slower decay of the eigenvalue and therefore a larger number of basis functions need to be considered. This fact is due to the more complex structure of the vortices that characterizes flows with higher Reynolds number.
The evaluation of the ROM is then queried. Relative errors for various quantities of interest are depicted Figures \ref{fig:error_modes}-\ref{fig:error_kinetic_enstophy}. Three possible ROMs are compared, corresponding to (i) consistent ROM with supremizers, (ii) consistent ROM without supremizers, (iii) non-consistent ROM with supremizers. The fourth option, i.e. non-consistent ROM without supremizer, is omitted because of its lack of pressure stabilization, which would result in a singular ROM. More in detail, Figure \ref{fig:error_modes} shows the velocity and pressure relative errors, integrated over time, for increasing number of modes $N$ from 10 to 100.
Furthermore, for fixed $N=100$, Figure \ref{fig:error_time} plots the velocity and pressure relative errors for each time $t \in [0, 2]$, while Figure \ref{fig:error_kinetic_enstophy} carries out a similar analysis for the kinetic energy $\mathcal{K}(\gr{u}) = \int_\Omega \lvert \gr{u} \rvert^2$ and enstrophy $\mathcal{E}(\gr{u}) = \int_\Omega \lvert \text{curl}(\gr{u})\rvert^2$.
Results in Figures \ref{fig:error_modes}-\ref{fig:error_kinetic_enstophy} show that the consistent ROM is systematically better than the non-consistent one. In particular, improvements of $1.65$\%, $4.38$\%, $0.00033$\%, $7.89$\% can be observed at the final time for the consistent ROM over the non-consistent one for velocity, pressure, kinetic energy and enstrophy, respectively. Thus, as in \cite{Shafqat_unsteady,Shafqat,PACCIARINI20141}, it is our conclusion that ensuring consistency between the full order and reduced order formulations results in the best accuracy.
Figures \ref{fig:error_modes}-\ref{fig:error_kinetic_enstophy} also highlight that the addition of supremizers is not necessary once a consistent ROM is sought. Indeed, lines corresponding to the options (i) consistent ROM with supremizer, and (ii) consistent ROM without supremizers, are always overlapping. A careful analysis of the coefficients of the reduced velocity in case (i) reveals that degrees of freedom corresponding to supremizer modes are numerically zero, thus not contributing to the solution. We claim that this is thanks to the stabilization through strong residual operators in the VMS formulation, which (through Galerkin projection) results in an inf-sup stable ROM formulation.\footnote{Nonetheless, we remark that this topic deserves further investigation. Indeed, in case of geometrical parametrization with a similar residual based SUPG stabilization we observe non-zero supremizer coefficients \cite{Shafqat}. We do mention that, in our experience, ROMs seem to be particularly sensitive to the choice of the stabilization at the full order level, especially for what concerns their consistency (or lack thereof) with the strong form. For instance, in \cite{KaBaRo18} we pursue a weaker (non-residual based, non-fully consistent) stabilization in the full order discretization, and we observe that the Galerkin projection of such stabilization terms does not result in a stable ROM, yielding a mandatory supremizer enrichment. In \cite{KaStaNouScoRo2018}, where a stabilized Shifted Boundary FEM is used as full order discretization in a Stokes setting (see also \cite{KaStaNoRoSco18_Heat} for a Poisson setting), we observe that the supremizer enrichment is not strictly needed but it leads to a better approximation of the pressure field.}\label{foot:sup}
Finally, Figures \ref{fig:plots_u} and \ref{fig:plots_p} show a comparison of the flow fields for $t=1.0$ s, $t=1.5$ s, $t=2.0$ for full order model, consistent ROM with supremizer, non-consistent ROM with supremizer; the remaining viable option (consistent ROM without supremizer) is omitted as the difference among consistent ROMs is negligible. The plots show that, especially for large times (see e.g. $t=2$ s), flow fields produced by the non-consistent ROM are qualitatively different from the full order simulation, thus resulting in larger relative errors.
\begin{figure}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{mesh_measure-crop.pdf}
\end{minipage}
\caption{The computational domain together with the geometrical dimensions. The diameter of the cylinder is equal to $D = 0.2\mbox{m}$}.
\label{fig:comp_mesh}
\end{figure}
\begin{figure}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{eigenvalue_decay.pdf}
\end{minipage}
\caption{POD eigenvalues decay for velocity, supremizer and pressure snapshots.}
\label{fig:eig_decay}
\end{figure}
\begin{figure}
\begin{minipage}{\textwidth}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{u_error_modes.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{p_error_modes.pdf}
\end{minipage}
\end{minipage}
\caption{Velocity relative error in the $L^2([0, T], \gr{H}^1(\Omega))$ norm (left) and pressure relative error in the $L^2([0, T], L^2(\Omega))$ norm (right), as a function of the number $N$ of POD modes. }
\label{fig:error_modes}
\end{figure}
\begin{figure}
\begin{minipage}{\textwidth}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{u_error_time.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{p_error_time.pdf}
\end{minipage}
\end{minipage}
\caption{Velocity relative error in the $\gr{H}^1(\Omega)$ norm (left) and pressure relative error in the $L^2(\Omega)$ norm (right), as a function of time. ROM evaluation was carried out for $N=100$. }
\label{fig:error_time}
\end{figure}
\begin{figure}
\begin{minipage}{\textwidth}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{e_ec_error_modes.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{es_ec_error_modes.pdf}
\end{minipage}
\end{minipage}
\caption{Kinetic energy (left) and enstrophy (right) relative errors as a function of the number of employed modes.}
\label{fig:error_kinetic_enstophy_modes}
\end{figure}
\begin{figure}
\begin{minipage}{\textwidth}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{e_ec_error_time.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{es_ec_error_time.pdf}
\end{minipage}
\end{minipage}
\caption{Kinetic energy (left) and enstrophy (right) relative errors as a function of time. ROM evaluation was carried out for $N=100$.}
\label{fig:error_kinetic_enstophy}
\end{figure}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{u1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_nc1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_c1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{u2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_nc2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_c2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{u3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_nc3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_c3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{u4.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_nc4.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{urb_c4.pdf}
\end{minipage}
\end{minipage}
\caption{Comparison of the flow fields in terms of velocity magnitude for $t=0.5$ s, $t=1.0$ s, $t=1.5$ s, $t=2.0$ s (from top to bottom) for full order model, non-consistent ROM with supremizer, consistent ROM with supremizer (from left to right). ROM evaluations were carried out for $N=100$.}
\label{fig:plots_u}
\end{figure}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{p1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_nc1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_c1.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{p2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_nc2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_c2.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{p3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_nc3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_c3.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{p4.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_nc4.pdf}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[width=\textwidth]{prb_c4.pdf}
\end{minipage}
\end{minipage}
\caption{Comparison of the flow fields in terms of pressure magnitude for $t=0.5$ s, $t=1.0$ s, $t=1.5$ s, $t=2.0$ s (from top to bottom) for full order model, non-consistent ROM with supremizer, consistent ROM with supremizer (from left to right). ROM evaluations were carried out for $N=100$.}
\label{fig:plots_p}
\end{figure}
\section{Conclusions and future developments}\label{sec:concl}
In the present work we introduced a novel reduced order model dealing with moderately high Reynolds numbers. The ROM has been constructed starting from VMS stabilized full order simulations and different strategies have been tested during the projection stage.
The consistent strategy, which is based on the projection of also the full order stabilization terms onto the reduced basis spaces, proved to be the best choice in terms of accuracy of the results.
The consistent strategy permits also to avoid the supremizer enrichment of the reduced velocity space. We believe that this fact is justified by the VMS full order discretization which, being inf-sup stabilized, transfers this property also to the reduced order model. However, this topic deserves still further investigation especially in the case of geometrical parametrization (see footnote 6 for further details).
As previously mentioned, as future perspective our interest is to extend the current approach also to geometrically parametrized problems and to investigate the efficiency and applicability of hyper reduction techniques.
\section*{Acknowledgements}
We acknowledge Prof. Guglielmo Scovazzi from Duke University for the fruitful discussions concerning the implementation of the VMS method. We acknowledge the support by European Union Funding for Research and Innovation - Horizon 2020 Program - in the framework of European Research Council Executive Agency: H2020 ERC Consolidator Grant 2015 AROMA-CFD project 681447 ``Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics''. We also acknowledge the INDAM-GNCS projects ``Metodi numerici avanzati combinati con tecniche di riduzione computazionale per PDEs parametrizzate e applicazioni'' and ``Numerical methods for model order reduction of PDEs''. The computations in this work have been performed with RBniCS \cite{rbnics} library, developed at SISSA mathLab, which is an implementation in FEniCS \cite{logg2012automated} of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.
\clearpage
\bibliographystyle{abbrv}
|
1,108,101,565,424 | arxiv | \section{Abstract}
\noindent
Coherent neural spiking and local field potentials are believed to be signatures of the binding and transfer of information in the brain. Coherent activity has now been measured experimentally in many regions of mammalian cortex. Synfire chains are one of the main theoretical constructs that have been appealed to to describe coherent spiking phenomena. However, for some time, it has been known that synchronous activity in feedforward networks asymptotically either approaches an attractor with fixed waveform and amplitude, or fails to propagate. This has limited their ability to explain graded neuronal responses. Recently, we have shown that pulse-gated synfire chains are capable of propagating graded information coded in mean population current or firing rate amplitudes. In particular, we showed that it is possible to use one synfire chain to provide gating pulses and a second, pulse-gated synfire chain to propagate graded information. We called these circuits synfire-gated synfire chains (SGSCs). Here, we present SGSCs in which graded information can rapidly cascade through a neural circuit, and show a correspondence between this type of transfer and a mean-field model in which gating pulses overlap in time. We show that SGSCs are robust in the presence of variability in population size, pulse timing and synaptic strength. Finally, we demonstrate the computational capabilities of SGSC-based information coding by implementing a self-contained, spike-based, modular neural circuit that is triggered by, then reads in streaming input, processes the input, then makes a decision based on the processed information and shuts itself down.
\vskip .6cm
\twocolumngrid
\section{Introduction}
Accumulating experimental evidence implicates coherent activity as an important element of cognition. Since its discovery \citep{GrayEtAl1989}, activity in the gamma band has been demonstrated to exist in numerous regions of the brain, including hippocampus \citep{BraginEtAl1995,CsicsvariEtAl2003,ColginEtAl2009},
numerous areas in cortex \citep{GrayEtAl1989,Livingstone1996,WomelsdorfEtAl2007,BroschEtAl2002,BauerEtAl2006,PesaranEtAl2002,BuschmanMiller2007,MedendorpEtAl2007,BuschmanMiller2007,GregorgiouEtAl2009,SohalEtAl2009},
amygdala and striatum \citep{PopescuEtAl2009}. Gamma band activity is associated with sharpened orientation \citep{AzouzGray2000} and contrast \citep{HenrieShapley2005} tuning in V1, and speed and direction tuning in MT \citep{LiuNewsome2006}. Attention has been shown to increase gamma synchronization between V4 and FEF \citep{GregorgiouEtAl2009}, LIP and FEF \citep{BuschmanMiller2007}, V1 and V4 \citep{BosmanEtAl2012}, and MT and LIP \citep{SaalmannEtAl2007}; In general, communication between sender and receiver neurons is improved when consistent gamma-phase relationships exist between upstream and downstream sites \citep{WomelsdorfEtAl2007}.
Theta-band oscillations are associated with spatial memory \citep{OKeefe1993,Buzsaki2002}, where neurons encoding the locations of visual objects and an animal's own position have been identified \citep{OKeefe1993,SkaggsEtAl1996}. Loss of theta results in spatial memory deficits \citep{Winson1978} and pharmacologically enhanced theta improves learning and memory \citep{MarkowskaEtAl1995}.
Classical coding mechanisms are related to neural firing rate \citep{AdrianZotterman1926}, population activity \citep{HubelWiesel1965,HubelWiesel1968,KaisslingPriesner1970}, and spike timing \citep{pmid8768391}. Firing rate \citep{AdrianZotterman1926} and population codes \citep{Knight1972,Knight2000,SirovichEtAl1999,Gerstner1995,BrunelHakim1999} are two ways to average spike number to represent graded information, with population codes capable of faster processing since they average across many fast responding neurons. Thus population and temporal codes can make use of the sometimes millisecond accuracy \citep{pmid8768391,pmid17805296,pmid23010933} of spike timing to represent dynamic averages.
New mechanisms have been proposed for short-term memory \citep{LismanIdiart1995,JensenLisman2005,Goldman2008}, information transfer via spike coincidence \citep{Abeles1982,KonigEtAl1996,Fries2005} and information gating \citep{SalinasSejnowski2001,Fries2005,RubinTerman2004,pmid24730779,pmid25503492} that rely on coherent activity in the gamma- and theta-band. For example, Fries's communication-through-coherence (CTC) model \citep{Fries2005} makes use of synchronous input that can provide windows in time during which spikes may be more readily transferred across a synapse. Additionally, synchronous firing has been used in Abeles's synfire network \citep{Abeles1982,KonigEtAl1996,pmid10591212,pmid21106815,KistlerGerstner2002} giving rise to volleys of propagating spikes.
Research to understand the network mechanisms linking coherent activity and information transfer has largely focused on synfire chains \cite{pmid12684488,pmid10591212,pmid12730700,KistlerGerstner2002,pmid24298251,pmid16641232}. The hope has been that synfire chains could be used to understand rapid, feedforward computation across multiple regions of cortex, as is seen in the response to rapid visual categorization experiments \citep{pmid11253215}. However, many studies have shown that although it is possible to transfer volleys of spikes stably from layer to layer, the spike probability waveform tends to an attractor with fixed amplitude \cite{pmid12684488,pmid10591212,KistlerGerstner2002}. Below, we refer to these chains as ``attractor synfire chains". In attractor synfire chains, although a volley of spikes can propagate, graded information, in the form of a rate amplitude, cannot.
Additional numerical studies investigating information propagation have shown that it is possible to transfer firing rates through feed-forward networks when there is sufficient background activity to keep the network near threshold \cite{pmid11880526}. This mechanism has the disadvantage that firing rate information cannot be controlled other than by increasing or decreasing background activity. Other studies have shown that additional coherent spatio-temporal structures ({\it e.g.} hubs or oscillations) can stabilize the propagation of synchronous activity and select specific pathways for signal transmission \cite{pmid24730779,pmid25503492,AkamKullmann2010,pmid24434912}.
Recently, we showed that information contained in the amplitude of a synaptic current or firing rate may be faithfully ({\it exactly} in a mean-field model) transferred from one neuronal population to another \cite{SornborgerWangTao}. In that work, coherent, non-overlapping gating pulses provided a sequence of temporal windows during which graded information was successively integrated then transferred on the synaptic time scale. We further showed that a self-contained, feed-forward network in the synfire regime can propagate graded information. This circuit, the synfire-gated synfire chain (SGSC), used an attractor synfire chain to generate gating pulses for graded information transfer. The SGSC mechanism provides a neuronal population based means of dynamically routing {\it graded} information through a neural circuit.
The SGSC mechanism naturally provides a framework in which information control and processing are separated into two complementary parts of a single neural circuit. Information processing is performed by synaptic connectivity as information propagates from layer to layer. Information control is performed by gating pulses that dynamically route information through the circuit \citep{SornborgerWangTao}.
Here, we show that, in general, overlapping gating pulses can be used to propagate temporally overlapping graded information. These solutions improve on our previous work by allowing information to cascade through a multi-layer network on a time scale that is a fraction of the synaptic time scale. We then describe a general class of time-translationally-invariant solutions to a mean-field model motivated by simulational results from the SGSC. We show a correspondence between the mean-field model and the SGSC and investigate robustness of the SGSC to finite-size effects, variability in the synaptic coupling and variability in the delay between pulses from layer to layer.
Finally, we show that by combining graded information with gating pulses, conditional decisions may be made to control information flow and the subsequent processing performed by the circuit. In order to demonstrate this information coding and decision making framework, we implement a self-contained, spike-based, modular neural circuit that is triggered by an input stream, reads in and processes the input, generates a conditional output based on the processed information, then shuts itself off.
\section{Results}
\begin{figure*}[t]
\includegraphics[width=\textwidth]{Figure1.png}
\caption{Graded information transfer in synfire-gated synfire chains. A) Circuit diagram. `g' denotes a population in the graded chain. `s' denotes a population in the gating chain. $S_{11}$, $S_{12}$ and $S_{22}$ denote synaptic couplings between and within the respective chains. The gating chain generates pulses that gate the propagation of graded information in the graded chain. B) Mean, synaptic current amplitude transferred across $12$ neural populations. N = 1000. Averaged over $50$ trials. Three amplitudes are depicted. C) Spike rasters from graded populations for one instance of graded transfer. D) Mean, synaptic current amplitude for fixed amplitude synfire chain. N = 100. Averaged over $50$ trials. E) Spike rasters from gating populations for one instance of graded transfer.}
\end{figure*}
In Fig. 1, we show how graded information may be propagated in an SGSC neural circuit (Fig. 1A). This circuit consists of two feedforward networks. One network (gating chain), set up to operate in the attractor synfire regime, generates a fixed amplitude pulse that propagates from layer to layer (Fig. 1D,E). The second network (graded chain) receives gating pulses from the gating chain and is capable of propagating graded currents and firing rates from layer to layer (Fig. 1B,C). The gating chain delivers pulses offset by time $T_0$ to the graded chain rapidly enough that there is an overlap in the integration of graded information and its transmission from one layer to the next. Graded information, in the form of synaptic currents and firing rates, is faithfully propagated across all $12$ layers in the simulation.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{Figure2.png}
\caption{Graded information transfer with overlapping pulses, exact mean-field solution. A) Graded, mean current amplitudes across $2$ populations. Two overlapping solutions are shown, one upstream (earlier in time) and one downstream (later in time). The downstream current evolution is easiest to understand: The red segment depicts the epoch when the second gating pulse has brought the downstream population to threshold. During this time, the upstream current (depicted in magenta) is integrated and the downstream current begins to rise. Once the upstream current enters the next epoch (depicted in blue), the downstream current (depicted in magenta) continues to rise. After the upstream current begins to decay exponentially (depicted in brown), the downstream current continues to rise (depicted in blue) until the gating pulse ends. At this point, the downstream current decays exponentially. So, from the point of view of the downstream population, the red segment represents the integration of the pink segment of the upstream population, the magenta segment represents the integration of the blue segment of the upstream population, and the blue segment represents the integration of the brown segment of the upstream population. $T_0/\tau = 0.6$, $T_1/\tau = 0.3$, and $T/\tau = 1.5$. $S_{exact}$ for these values is $1.582$. The coefficients of the solution polynomial are $\{ 0.733, 0.640, 0.228 \}$ (See Appendices). B) Gating pulses offset from $0$ for clarity. C) $S_{exact}$ vs. $\eta$.}
\end{figure}
The observation that spike volleys in successive layers of the SGSC overlap in time led us to consider an extension of our previous mean-field model \cite{SornborgerWangTao} in which the integration of graded information in successive populations also overlapped in time. As in our previous work, we consider the idealized case in which the gating pulses are square. In Fig. 2, we show a translationally invariant solution (Fig. 2A) and gating pulses (Fig. 2B) from such a mean-field model. Successive gating pulses of length $T$ are offset by time $T_0$. The solution is divided into segments which are the result of the integration of spikes in the corresponding segment (shifted by $T_0$) from the previous layer during the gating pulse.
For fixed $T$ and $T_0$, we find time translationally-invariant solutions for special values of the feedforward coupling strength, $S = S_{exact}$ (see Materials and Methods and Appendices). In Fig. 2C, we plot $S_{exact}$ as a function of $\eta = T/T_0$, where $\eta$ is a measure of the overlap in the integration and transmission of graded information. Note that $S_{exact}$ becomes flatter as the overlap, $\eta$, gets larger. This implies that, for large overlaps, any propagation error in the solution due to deviations from $S_{exact}$ is small. Thus, in the large $\eta$ regime, information propagation is robust to variability in both pulse timing and coupling strength. For practical purposes, we find that $\eta > 2$ or $3$ is sufficiently robust. Furthermore, for a generic feedforward network, there exists a wide range of $S$ (roughly, $S$ from $1$ to $2.7$) where we can find time translationally-invariant solutions for which graded propagation is possible.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{Figure3.png}
\caption{Fitting a square-pulse gated mean-field model of the SGSC. A) Fits of mean-field model and I \& F simulations for $3$ amplitudes for $T_0 = 0.003$. B) $T_0 = 0.003$: Blue line - $\alpha_{exact}$ as a function of $\eta$. Red triangle - $( \eta_{sim}, \alpha_{sim} )$, purple circle - $( \eta_{fit}, \alpha_{fit} )$. Inset: magnification showing location of results from fit. C) Fits of mean-field model and I \& F simulations for $3$ amplitudes for $T_0 = 0.005$. D) $T_0 = 0.005$: Blue line - $\alpha_{exact}$ as a function of $\eta$. Red triangle - $( \eta_{sim}, \alpha_{sim} )$, purple circle - $( \eta_{fit}, \alpha_{fit} )$. Inset: magnification showing location of results from fit. E) Results of fit for $T_0 = 0.003, 0.004, 0.005, 0.006$. Traces offset for clarity.}
\end{figure}
In Fig. 3, we explore whether our mean-field theory could be used to model our I\&F simulation results. First, we determined the parameters $(\eta_{fit}, \alpha_{fit})$ that gave the best-fitting mean-field solution to the simulation data, given known $T_0$. Here, we define $\alpha \equiv S (T_0/\tau) e^{-T_0/\tau}$, so that $\alpha = 1$ at $\eta = 1$. Next, using the simulational synaptic coupling, $\alpha_{sim}$, we found $\eta = \eta_{sim}$ that corresponded to the time-translationally invariant solution of the mean-field model. Closeness of these two points would give evidence that the mean-field theory, despite the simplications used to derive it ({\it e.g.} precisely timed square gating pulses, linear f-I curve, etc.), can be used to model the I\&F simulation. We show details of this fitting procedure for two different $T_0$'s (Fig. 3A--D), and summarize the results for $T_0 = 0.003, 0.004, 0.005, 0.006$ (Fig. 3E). The closeness of model fits with simulation results, for a wide range of overlaps, indicates that the mean-field theory is a good model of the SGSC simulation.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{Figure4.png}
\caption{Signal-to-Noise-Ratio as a function of the number of transfers. Red - mean current amplitude for transfer across $12$ layers. Blue - standard deviation of current amplitude. Mean and standard deviation calculated from $1000$ trials. A) $N = 1000$, B) $N = 100$, C) $S_{11}$ taken from a uniform distribution with half-width of $5\%$, $N = 100$, D) $S_{12}$ taken from a uniform distribution with half-width of $5\%$, $N = 100$, E) Pulse delay jittered by $10\%$, $N = 100$.}
\end{figure}
In Fig. 4, we investigate the robustness of pulse-gated synaptic current transfer in the SGSC to finite-size effects, variability in synaptic coupling, and inaccuracies in pulse timing. As would be expected, transfer variability decreases as $1/\sqrt{N}$ (Fig. 4A,B). Randomness in synaptic coupling either in the gating chain or the coupling between chains has little effect on the variability (compare Fig. 4C,D with Fig. 4B). As we mentioned above, this is expected due to the flatness of $S_{exact}(\eta)$ for large $\eta$. Here, $\eta = 2.5$. Similarly, jittering $T_0$ has little effect on the variability of current transfer (Fig. 4E).
Pulse-gated propagation mechanisms, such as the SGSC, naturally give rise to a probabilistic, spike-based information processing framework in which information is processed by graded chains and the flow of information is controlled by gating chains \cite{SornborgerWangTao}. Additionally, logic operations may be performed by allowing graded information to interact with the pulse generator (see Materials and Methods).
\begin{figure*}[t]
\includegraphics[width=\textwidth]{Figure5.png}
\caption{Autonomous decision making circuit. A) Neural Circuit. B) Connectivity matrix (four components) $K^{11}$ (graded to graded, upper left), $K^{12}$ (gating to graded, upper right), $K^{21}$ (graded to gating, lower left), and $K^{22}$ (gating to gating, lower right). Color bar denotes connectivities. Graded chain populations: ``Memory" ($1$ - $14$), ``Hadamard" ($15$ - $22$), ``Hadamard Copy" ($23$ - $30$), ``Input" ($31$), and ``Shutdown" ($32$). Gating chain populations are: ``Trigger" ($33$), a re-entrantly coupled population that fires until inhibited. ``Compute" ($34$ - $39$) for gating the computation of the windowed Hadamard transform in the Memory and Hadamard populations, ``Vigilance" ($40$ - $43$) a pulse loop that, along with the ``Logic - Conditional Output" ($50$) population makes a decision based on the amplitude of the output of the $8$'th Hadamard population, and ``Output Copy" ($44$ - $47$) a pulse loop that maintains a memory that the decision was made. Logic populations: ``Logic - Trigger" ($49$) a population that is conditionally excited when both Trigger and Input are excited, and ``Conditional Output" ($50$) a population that is conditionally excited when both Hadamard coefficient $8$ and a population in Output Copy are excited. C) Raster plot showing spikes from the graded chain. $T_0 = 4$ ms, $T = 7.5$ ms, $\tau = 5$ ms. Time runs from left to right. D) Raster plot showing spikes from the gating chain. E) Mean firing rates of the graded chain averaged over $50$ realizations. F) Mean firing rates for the gating chain averaged over $50$ realizations. E,F) The firing rates have been smoothed by a moving average process with width $2$ ms.}
\end{figure*}
To illustrate the capability of pulse-based information processing to perform complex computations, we show results from a neural circuit that, after being triggered by a streaming input, encodes and transforms the input then makes a decision based on the transformed input that affects subsequent processing (Fig. 5). The neural circuit consists of (see Fig. 5A) 1) a trigger, 2) a module used to keep sampled streaming input in short-term memory, 3) a $4 \times 4$ Hadamard transform (a Fourier transform using square-wave, Walsh functions as a basis), 4) a second set of Hadamard outputs (Hadamard Copy) representing output copy to a downstream circuit, 5) an Input population, 6) a Shut Down population to terminate processing, 7) a Compute gating chain to drive the computation, 8) a Vigilance gating chain that serves as a processing indicator and clock to synchronize the triggering of an output decision, 9) an Output Copy gating chain that serves as a decision indicator and is turned on based on the amplitude of the $8$'th Hadamard coefficient, and 9) Logic populations for triggering the computation and making the decision to copy the Hadamard output. Output then triggers circuit shutdown by inhibiting all gating chains.
In Fig. 5B, Memory designates Read In ($1$, $6$, $10$, $13$) and (non-cyclic) Memory populations. Hadamard designates populations holding Hadamard coefficient amplitudes. The Hadamard transform is divided into two parallel operations, one that results in positive coefficients, the other in absolute values of negative coefficients. Hadamard Copy designates populations into which the Hadamard transform may be copied. Input designates a population that linearly transduces a signal from outside the network. And Shutdown designates a population that receives summed input from the Hadamard Copy populations. Upon excitation, it shuts down the input and gating populations and terminates the computation.
Initially, the Trigger population is re-entrantly excited until the Input amplitude increases (as indicated in the top row of Fig. 5D). Input combines with Trigger to initiate firing in the Logic - Trigger population, which triggers the Compute gating chain and initiates the computation. Trigger is subsequently turned off by inhibition from the Compute gating chain. We show the computation for three successive windows, each of length $4T_0$. The gating chain binds the input into four memory chains of length $4T_0$, $3T_0$, $2T_0$ and $T_0$. Thus, four temporally sequential inputs are bound in four of the memory populations beginning at times $t = 4, 8, 12 T_0$. Hadamard transforms are performed beginning at $t = 5, 9, 13 T_0$. Each subsequent read in starts one packet length before the Hadamard transform so that the temporal windows are adjacent. At time $t = 0.06$ s, the high amplitude in Hadamard coefficient $8$ combines with gating population Conditional Output to initiate the Output Copy chain. The output is copied to Hadamard Copy populations, which then cause the shutdown of the gating chain.
This probabilistic, spike-based algorithm uses a self-exciting population coupled to a streaming input to trigger the computation (see Fig. 5B,C,D,E), then continuously gates $4$ sequential input amplitudes into $4$ read in populations and maintains the input values by gating them through working memory populations until all values are simultaneously in $4$ working memory populations. At this point, these values are gated to Hadamard populations transforming the input values into Hadamard coefficients (one set of positive coefficients and one set of absolute values of negative coefficients \cite{SornborgerWangTao}). At this point, a time-windowed Hadamard transform has been computed on the input. Gating pulses are interleaved such that this computation is performed iteratively on successive windows of length $4T$ from the streaming input.
To implement a conditional copy of the transformed data, we combine the output of the (arbitrarily chosen) eighth Hadamard coefficient in the present Hadamard output and the first population in the ``Vigilance" gating chain. This operation causes the graded pulse to activate the Output Copy chain when its amplitude is sufficiently high, conditionally causing a pulse to cascade through 4 gating populations with the last population gating the transfer from the subsequent Hadamard output to the 8 output neurons. Once the Hadamard output is copied, it activates the Shutdown population, which inhibits all populations in the gating chains, terminating the computation.
\section{Discussion}
The emerging picture from accumulating experimental evidence is that coherent activity is a fundamental contributor to cognitive function. In particular, coherence (alternatively, correlation of a signal at a given lag) is a measure of the efficacy of univariate information transfer between neuronal populations (a matrix-valued quantity is needed to measure the efficacy of multivariate information transfer). Here we have demonstrated a coherent transfer mechanism that dynamically routes graded information through a neural circuit using stereotyped gating pulses and performs computation via non-linear coupling of graded and gating pulses. As we have shown, SGSCs can be used as building blocks to implement complex information processing algorithms, including sub-circuits responsible for short-term memory, information processing and computational logic. As such, synfire-gated synfire chains should be considered as a candidate mechanism whenever coherent activity is implicated in information transfer.
Rapid visual categorization (RVC) experiments have demonstrated that objects can be recognized as early as $250 - 300$ ms after presentation. It has been conjectured that massively parallel, feedforward networks are used during RVC computations for maximum speed \citep{pmid11244543,pmid22007180,pmid25208739,pmid12684488}. At $40$ Hz, $10 - 12$ feedforward processing layers would be needed to construct such a network (Fig. 1). The signal-to-noise ratios that we demonstrated for the SGSC (Fig. 4) are good enough that it could be used for this type of information transfer at $40$ Hz. Indeed, in our examples, we show rapid propagation of graded information at $300$ Hz.
To our minds, the success of the SGSC graded information propagation mechanism rests on the structural robustness of the pulse gating mechanism. One contribution to robustness is that the synfire chain that is used for pulse generation approaches a fixed amplitude attractor with fixed temporal offset. A second contribution is that by providing overlapping temporal windows for information integration, the constraints on parametric precision to achieve graded information transfer are relaxed (Fig. 2C and related text). Having said that, the correspondence between our mean-field model and the SGSC gives weight to the idea that pulse-gating, independently of how it is implemented, is a robust mechanism for controlling information transfer in neural circuits. Thus, there is no particular reason that other pulse generators should not be entertained. For instance, experiments implicate the PVBC/OLM system of interneurons in cortical pulse generation \citep{pmid23010933}.
A conceptual framework for the manipulation of information in neural circuits arises naturally when one considers graded information transfer in the context of coherently interacting neuronal populations. In this framework, information processing and information control are conceived of as distinct components of neural circuits \citep{SornborgerWangTao}. This distinction has been used previously \cite{LismanIdiart1995,JensenLisman2005,pmid24730779,pmid25503492} in mechanisms for gating the propagation of fixed amplitude waveforms. Here, by providing a mechanism for the propagation of graded information and including computational logic by allowing graded and gating chains to interact, active linear maps (see Materials and Methods) take prominence as a key information processing structure.
It is worth mentioning that when we constructed the neural circuit example in Fig.\;5, we started at the algorithmic level, then implemented the algorithm in the mean-field firing rate model, then translated the mean-field model into the spiking, I\&F network. We feel that this is a major strength of the SGSC-based information processing framework, {\it i.e.} that it provides a practical pathway for designing computational neural circuits, either for the purpose of forming hypotheses about circuits in the brain, or for implementing algorithms on neuromorphic chips.
\section{Materials and Methods}
\subsection{The Synfire-Gated Synfire Chain Circuit}
Individual current-based, I\&F point neurons in the SGSC have membrane potentials described by
\begin{subequations}
\begin{equation}
\frac{d}{dt} v^\sigma_{i,j} = -g_{leak} \left( v^\sigma_{i,j} - V_{leak} \right) + \sum_{\sigma' = 1}^2 I^{\sigma \sigma'}_{i,j} + I^\sigma_{i,j}
\end{equation}
\begin{equation}
\tau \frac{d}{dt} I^{\sigma \sigma'}_{i,j} = -I^{\sigma \sigma'}_{i,j} + \frac{S^{\sigma \sigma'}}{p_{\sigma \sigma'}N_{\sigma'}} \sum_{i'} \sum_l \delta \left( t - t^{\sigma',l}_{i',j-1} \right)
\end{equation}
\begin{equation}
\tau \frac{d}{dt} I^{\sigma}_{i,j} = -I^{\sigma}_{i,j} + f^\sigma \sum_l \delta \left( t - s^{l}_{i,j} \right)
\end{equation}
\end{subequations}
where $\sigma, \sigma' = 1,2$ with $1$ for the graded chain and $2$ for the gating chain, $i = 1,\dots,N_\sigma$ denotes the number of neurons per population for each layer, $j = 1,\dots,M$ denotes the layer; individual spike times, $\{ t^{\sigma, l}_{i,j} \}$, with $l$ denoting spike number, are determined by the time when $v^{\sigma}_{i,j}$ reaches $V_{Thres}$. The parameters $g_{leak}$ and $V_{leak}$ denote the leak conductance and the leak potential. We have used reduced dimensional units in which time retains dimension in seconds and $V_{thresh} - V_{leak} = 1$. In these units $g_{leak} = 50$/sec. The parameter $\tau$ denotes the synaptic timescale ($\tau = 5$ ms, or approximately an AMPA synaptic timescale, in the Results above). The current $I^{\sigma \sigma'}_{i,j}$ is the synaptic current of the $\sigma$ population produced by spikes of the $\sigma'$ population. The parameter $S^{\sigma \sigma'}$ denotes the synaptic coupling strength and $p_{\sigma \sigma'}$ is the probability of coupling. $I_{i,j}^\sigma$ is a background noise current generated from Poisson spike times, $\{ s_{i,j}^l \}$, with strength $f^\sigma$ and rate $\nu_\sigma$.
\subsection{More Complex Synaptic Processing}
General SGSC circuits can incorporate a number of subcircuits, such as short-term memory and processing due to non-trivial synaptic connectivities \citep{SornborgerWangTao} such as the circuit shown in Fig. 5. In this case, more general connectivities are needed and the above equations become
\begin{subequations}
\begin{equation}
\frac{d}{dt} v^\sigma_{i,j} = -g_{leak} \left( v^\sigma_{i,j} - V_{leak} \right) + \sum_{\sigma' = 1}^2 I^{\sigma \sigma'}_{i,j} + I^\sigma_{i,j}
\end{equation}
\begin{equation}
\tau \frac{d}{dt} I^{\sigma \sigma'}_{i,j} = -I^{\sigma \sigma'}_{i,j} + \frac{S^{1}}{p_{\sigma \sigma'}N_{\sigma'}} \sum_k K^{\sigma \sigma'}_{jk} \sum_{i'} \sum_l \delta \left( t - t^{\sigma',l}_{i',k} \right)
\end{equation}
\begin{equation}
\tau \frac{d}{dt} I^{\sigma}_{i,j} = -I^{\sigma}_{i,j} + f^\sigma \sum_l \delta \left( t - s^{l}_{i,j} \right)
\end{equation}
\end{subequations}
Here, the synaptic connectivity for the graded chain is $K^{11}_{jk}$, the coupling between the chains is $K^{12}_{jk}$, and the connectivity of the gating chain is $K^{22}_{jk}$. Interaction between the graded chain and the gating chain is given by $K^{21}_{jk}$. We use $K^{21}_{jk}$ to implement conditional logic operations.
\subsection{Mean-field Solutions for Synaptic Current Propagation in the Overlapping Pulse Case}
To analyze graded propagation for the case in which the integration of graded information in successive populations overlaps in time, we assume that the gating pulse is square with amplitude sufficient to bring neuronal populations up to the firing threshold. We also assume that above threshold the activity function is linear \citep{SornborgerWangTao}.
Firing in the upstream population is integrated by the downstream population. Thus, the downstream synaptic current obeys
$$\tau \frac{d}{{dt}}{I_d} = - {I_d} + S{m_u } \; ,$$
where $S$ is a synaptic coupling strength, $I_d(t)$ is the downstream synaptic current, and $\tau$ is a synaptic timescale. In a thresholded-linear model, the upstream firing rate is
$$m_u \approx \left[ I_u(t) + I^{Exc}_0 - g_0 \right]^+ \; ,$$
where $I^{Exc}_0 = g_0 p(t)$ is an excitatory gating pulse, $p(t) = \theta(t) - \theta(t - T)$ and $\theta$ is the Heaviside step function, causing the downstream population to integrate $I_u(t)$, giving the current
$$G \left[ I_d \right] (t) \equiv S e^{-t/\tau} \left[ \int_0^t ds \; e^{s/\tau} I_u(s) + c \right] \; .$$
The graded population is pulsed for time $T$. The offset between successive gating pulses is given by $T_0$ (see Fig. 2). In \citep{SornborgerWangTao}, we studied the case where $T = T_0$. That is, the downstream pulse turned on just when the upstream pulse turned off. Here, we focus on the case where $\eta = T/T_0 > 1$, and $\eta$ need not be an integer. Let $n$ be the integer part of $\eta$. Then $T = n T_0 + T_1$, where $T_1 < T_0$.
In the Appendices, we give a general derivation of time translationally invariant solutions in this context. In brief, a translationally invariant, graded current waveform is found for a particular feedforward strength, $S = S_{exact}$, by integrating the upstream firing rate over intervals of length $T_1, T_0, \dots, T_0$, while enforcing continuity of the solution. For these solutions, $S_{exact}$ is given by the smallest root of
\begin{equation*}
\sum_{j=0}^n \frac{(-1)^j}{(n-j)!} \left[ \frac{\left((j+1) T_0 - T_1 \right)}{\tau} S e^{T_0/\tau} \right]^{n-j} = 0 \; .
\end{equation*}
\subsection{Information Processing Using Graded Transfer Mechanisms}
As we demonstrate in Results, current amplitude transfer for the SGSC may be modeled with a piecewise linear activity function, therefore synaptic connectivities between two layers each containing a vector of populations, perform a linear operation on the currents in the upstream layer \citep{SornborgerWangTao}. For instance, consider an upstream vector of neuronal populations with currents, $\mathbf{I}^{u}$, connected via a connectivity matrix $K$ to a downstream vector of neuronal populations, $\mathbf{I}^{d}$:
\begin{equation}
\mathbf{I}^{u}(t) \overset{K}{\rightarrow} \mathbf{I}^{d}(t) \; .
\end{equation}
With feedforward connectivity, $K$, the current amplitude, $\mathbf{I}^d$, obeys
\begin{equation}
\tau \frac{d}{dt} \mathbf{I}^{d} = -\mathbf{I}^{d} + S \left[ \sum_k K \mathbf{I}^u + \mathbf{p}^u(t) - g_0 \right]^+ \; ,
\end{equation}
where $\mathbf{p}^u(t)$ denotes a vector gating pulse on the upstream population. This results in the solution $\mathbf{I}^{d}(t-T) = P K \mathbf{I}^u(t)$, where $P$ is a diagonal matrix with the gating pulse vector, $\mathbf{p}$, of $0$s and $1$s on the diagonal indicating which neurons were pulsed during the transfer.
This discussion has identified three components of an information processing framework that naturally arises from mechanisms such as the SGSC:
\begin{enumerate}[itemsep = -1mm]
\item information content - graded current, $\mathbf{I}$
\item information processing - synaptic weights, $K$
\item information control - pulses, $\mathbf{p}$
\end{enumerate}
Note that the pulsing control, $\mathbf{p}$, serves as a gating mechanism for routing neural information into (or out of) a processing circuit. We, therefore, refer to amplitude packets, $\mathbf{I}$, that are guided through a neural circuit by a set of stereotyped pulses as ``bound" information. In the SGSC, information content is carried by the graded chain ({\it e.g.} Fig. 5b,d), information processing is performed by the synaptic connectivities ({\it e.g.} Fig. 5a) and information control is performed by the gating chain ({\it e.g.} Fig. 5c,e). We will refer to the combination of these control and processing structures as {\it active linear maps}.
In order to make a decision, non-linear logic circuits are introduced. A simple decision can be implemented in our framework by allowing interaction between information control and content. In our example, a graded and a gating pulse were combined to make a decision, then the output was fed as input to a gating chain. If the graded chain output a low value, the gating chain was not switched on. However, if the graded chain output was high, this initiated pulses in the gating chain, which rapidly approached an attractor. Thus, the interaction caused conditional firing in the gating chain.
\begin{acknowledgments}
{\bf Acknowledgements} L.T. thanks the UC Davis Mathematics Department for its hospitality. A.T.S. would like to thank Liping Wei and the Center for Bioinformatics at the College of Life Sciences at Peking University
for their hospitality. This work was supported by the Ministry of Science and Technology of China through the Basic Research Program (973) 2011CB809105 (W.Z. and L.T.), by the Natural Science Foundation of China grant 91232715 (W.Z. and L.T.) and by the National Institutes of Health, CRCNS program NS090645 (A.T.S. and L.T.).
\end{acknowledgments}
\bibliographystyle{apalike}
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1,108,101,565,425 | arxiv | |
1,108,101,565,426 | arxiv | \section{Introduction}\label{secIntroduction}
Credit scoring is undoubtedly one of the oldest applications of analytics where lenders and financial institutions perform statistical analysis to assess the creditworthiness of potential borrowers to help them decide whether or not to grant credit \citep{thomas2000survey}.
Fair Isaac was founded in 1956 as one of the first analytical companies offering retail credit scoring services in the US.
Its well-known FICO score (ranging between 300 and 850) has been used as a key decision instrument by financial institutions, insurers, utilities companies and even employers \citep{scheule2016credit}.
The first corporate credit scoring models date back to the late sixties with Edward Altman developing his well-known z-score model for bankruptcy prediction, which is still used to this day in Bloomberg reports as a default risk benchmark \citep{altman1968financial}.
Originally, these models were built using limited data--consisting of only a few hundred observations--and were based on simple classification techniques such as linear programming, discriminant analysis and logistic regression, which is the current industry standard given its high interpretability \citep{scheule2016credit}.
The importance of these retail and corporate credit scoring models further increased due to various regulatory compliance guidelines such as the Basel Accords and IFRS 9 which clearly stipulate the inputs and outputs of a credit scoring model together with how these models can be used to calculate provisions and capital buffers.
The emergence of more sophisticated classification techniques such as neural networks, support vector machines and random forests led to various extensive benchmarking studies aimed at improving credit scoring models in terms of their statistical performance (e.g., in terms of area under the ROC curve or classification accuracy) \citep{baesens2003benchmarking, lessmann2015benchmarking}.
Many of these studies concluded that traditional credit scoring models based on, e.g., simple logistic regression models, performed very well and newer classification techniques could only offer marginal performance gains.
In other words, research on developing high-performing credit scoring models has more or less stalled.
We believe the best investment in better credit scoring models is not to turn the attention to newer classification techniques but to leverage innovative Big Data sources instead.
While these new sources of data present the opportunity to profile potential borrowers using a wider representation of behavior, they also present an ethical challenge. Mobile phone data, e.g., in the form of call-detail records (CDR), allows constructing a very detailed social network, and using this information to profile repayment behavior can be seen as unfair to borrowers that could be punished for their mobile cell phone behavior. Recently, the use of \emph{positive information} has been put forward as a necessary source of data that should be included in scoring models \citep{worldbank2011}. Positive information is defined as all information that represents the good financial behavior, providing a clearer definition of the factors that make a good borrower. \citet{barron2003value} show, for example, that not using positive information leads to a decrease of up to 47.5\% in credit availability.
This paper introduces mobile phone data as a new Big Data source for credit scoring and shows that while it is a powerful source of information, it should be used strictly in a positive framework to increase the access to financing to borrowers who would otherwise be out of options until a much later stage. To motivate the use of this information in financial institutions, its potential is studied in both statistical and profit terms.
Big Data is typically defined in terms of its 5 Vs: Volume, Variety, Velocity, Veracity and Value.
Recent special issues of Information Systems Research \citep{agarwal2014editorial} and MIS Quarterly \citep{baesens2014transformational} indicate the explosion of interest in Big Data within the IS community.
The use of mobile phone data for credit scoring is a fitting example of this since it comes in huge volumes (Volume), has not been explored before (Variety), is generated on a continuous daily basis (Velocity) and is usually stored using a well-defined call-detail record log format (Veracity).
In this paper, its Value is also quantified by focusing both on its statistical performance (e.g., in terms of area under the ROC curve) and on its bottom line impact in terms of profit.
Additionally, an evaluation of the qualitative performance of the data in terms of positive information for enhanced financial inclusion is provided.
This study is based on a unique data set combining banking, sociodemographic and CDR data.
CDR are logs of all phone calls between the customers of a telecommunications provider (telco), see Table \ref{T:CDR}.
More specifically, the data set includes all CDR of the bank's customers, the CDR of the people they are in contact with and the banking history of these customers.
Overall, it adds up to a year and a half of banking history of over two million bank customers and the calling activity of almost 90 million unique phone numbers spanning five months.
This unique combination of data gives the opportunity to explore the potential of enriching traditional credit scoring models with social network effects reflecting calling behavior.
The three key research questions are:
\begin{itemize}
\item[Q1]
What is the added value (in terms of both AUC and profit) of including call data for credit scoring?
\item[Q2]
Can call data replace traditional data used for credit scoring?
\item[Q3]
How does default behavior propagate in the call network?
\end{itemize}
To the best of our knowledge, these questions have not been researched before.
Each of the questions will be answered from both a statistical as well as a profit perspective, which is another key contribution of this paper.
Furthermore, the implications for financial inclusion are evaluated.
The impact of this research is manifold.
A successful application of boosting the performance of credit scoring models using call data would improve credit decision-making and pricing.
The insights could also facilitate access to credit for borrowers with little or no credit history.
This is the case for young borrowers, lenders exploring new markets or in developing countries with young credit markets.
In all these cases, the borrowers are not expected to have a credit history, but they do have mobile phone records.
Knowing how default behavior propagates in a call network also has regulatory implications.
For example, the Basel Accords try to capture default correlation in order to better protect a financial institution against unexpected losses \citep{scheule2016credit}.
The research can shed new light on how default behavior is correlated.
This could lead to better provisioning and capital buffering strategies, thereby improving the resilience of the financial system against shocks and macroeconomic downturns.
Knowing how default behavior propagates in a call network also has other regulatory implications.
If CDR data is indeed useful for credit prediction, then banks and credit bureaus have a strong economic incentive to collaborate with telecommunications companies to share data in order to perform this type of analyses.
In the next section, a review on the literature on Big Data in credit scoring as well as previous research on call networks is provided.
In section \ref{sec:methodology}, the theoretical background and methodology applied in the case study is described, with the experimental setup detailed in section \ref{sec:experimentaldesign}.
The results are presented in section \ref{sec:results}, followed by a discussion on their various implications in sections \ref{sec:discuss} and \ref{sec:impact}.
The paper concludes with a summary of the contributions and discussion on possibilities for future work.
\section{Related Work}\label{sec:relatedwork}
Many analytical modeling exercises start from a flat data set, build a predictive model for a target measure of interest (e.g., churn, fraud, default) and evaluate it on an independent out-of-sample data set.
An assumption which is (oftentimes) tacitly made is that the data is independent and identically distributed.
Recent research questioned this assumption and analyzed how customers can influence each other through the different social networks that connect them \citep{sundararajan2013research}.
Various types of social behavior can be observed.
One is homophily, which states that people have a strong tendency to associate with others whom they perceive as being similar to themselves in some way.
Social influence occurs when people's behavior is affected by others with whom they interact \citep{newman2010networks, lee2016friend}.
Some of the social behavior can also be attributed to other (e.g., external) confounding factors \citep{aral2014tie}.
The idea of network learning is to embed social behavior patterns in the predictive models to successfully leverage the impact of joint customer actions \citep{macskassy2007classification}.
A key input to any social network learning exercise is the network itself, which consist of nodes and edges.
In certain settings, the definition of these networks is relatively straightforward.
As an example, consider churn prediction in telco where the network can obviously be constructed based upon data stored in the CDR.
Earlier research found significant social network effects for predicting churn in telco \citep{verbeke2014social}.
Another example is credit card fraud detection where a network can be defined by connecting credit cards to merchants.
Also in this setting, strong social network effects have been found \citep{van2015apate}.
In credit scoring, there is a firm belief amongst both researchers and practitioners that default behavior of borrowers is correlated.
To illustrate this, the Basel Accord models default correlation by means of an asset correlation term, which is set to $15\%$ for residential mortgages and $4\%$ for qualifying revolving exposures.
However, both these numbers have been set in a rather arbitrary way, or based upon some empirical but not published procedure \citep{gordy2003risk}.
This interdependency has been proven to be a significant factor amongst small and medium-sized enterprises \citep{calabrese2017birds}.
One of the key challenges in understanding network effects or default propagation in credit scoring concerns the definition of the network itself.
Preliminary attempts have been made to build networks between customers in online peer-to-peer lending.
For example, \citet{lin2013judging} illustrated that online friendships with non-defaulters increases the credit score.
These findings were also confirmed by \citet{freedman2014information}, with an additional caution that online ties on their own may not reveal true information about creditworthiness and may also be manipulated \citep{wei2015credit}.
\citet{de2018does} developed credit scoring models for microfinance using social media network information extracted from Facebook accounts.
Their results suggest that explicit networks of friends who interact are more predictive than of friends who do not, but implicit networks of people with similar behavior are better than both explicit friendship networks.
In industry, social networks are already being exploited to assess creditworthiness, by technology companies such as Lenddo, that make use of social media connections to analyse people's default risk \citep{kharif2016no}.
More recently, the interest in using call networks as a new Big Data source for credit scoring has gained traction, e.g., with \citet{wei2015credit} formulating the potential value of credit scores obtained with networks--for example, based on social media or calls--and how strategic tie-formation might affect these scores.
Although especially interesting in relation to the Chinese government's plan for a social credit system
\citep{chin2016china}, the study is only theoretical and is missing an important empirical evaluation of the proposed models \citep{wei2015credit}.
Moreover, recent press coverage on specialized smartphone applications that evaluate people's creditworthiness using the huge amount of data generated by their handsets indicates the potential of call networks as an alternative data source for credit scoring \citep{dwoskin2015lending, kharif2016no}.
Most of these studies have focused on the use of social networks in the context of social media, or have discussed the potential of CDR-induced social networks in credit scoring.
The literature on the analysis of CDR is rich \citep{naboulsi2016large}.
The idea of using CDR data for credit scoring stems from the fact that the way people use their phone is assumed to be a good proxy for their lifestyle and economic activity.
Previous research confirmed that using CDR data to build call networks by linking together individuals who are in contact with each other, results in social networks that can be used in both descriptive and predictive studies on age, gender, ethnicity, language, economic factors, geography, urbanization, and epidemics \citep{blondel2008fast,
onnela2011geographic,wesolowski2012quantifying, sarraute2014study,leo2016socioeconomic}.
For example, \citet{leo2016socioeconomic} confirm the presence of homophily in terms of economic behavior using call networks.
More specifically, they show that wealth and debt are unevenly distributed and that people are better connected with those that share their socioeconomic class.
Furthermore, \citet{haenlein2011social} investigated the distribution of customer revenue within a call network and demonstrated that high revenue customers are primarily related to other high revenue customers and the same for low revenue customers.
\section{Methodology}\label{sec:methodology}
This paper contributes to the literature by investigating the use of CDR data for credit scoring in terms of value.
Here, the proposed methodology for extracting appropriate information from the CDR data by means of social networks and influence propagation is detailed.
Furthermore, techniques for evaluating model and feature performance in terms of profit are presented.
\subsection{Call Networks: Featurization and Propagation}\label{subsec:callNetworks}
\begin{table}
\centering
\caption{An example of a CDR log. In the actual dataset the phone numbers are encrypted. \label{T:CDR}}
\scriptsize{
{\begin{tabular}{llrrr
\hline
Call Start Date&Call Start Time&Call Duration (sec)& From Number& To Number\\ \hline
01MAY2017& 14:51:14& 715 &(202) 555-0116& (701) 555-0191 \\
02MAY2017& 14:34:37& 29& (803) 555-0129& (202) 555-0116 \\
01MAY2017&20:34:14& 9 &(803) 555-0117& (406) 555-0137\\
02MAY2017& 20:03:38& 89& (701) 555-0148& (803) 555-0129 \\ \hline
\end{tabular}}}
\end{table}
A call network is a network where the nodes $\mathcal{V}=\{v_1,\dots v_n\}$ are people present in a CDR log.
These logs are kept for billing purposes and include information about every phone call made by the customers of a telecommunications operator, including the encrypted phone numbers of the customers that made and received the phone call as well as timing and length.
An example of such a log can be seen in Table \ref{T:CDR}.
Information from CDR about time and duration of phone calls (or text messages) can be used to connect the people in the network to create the edges, $e_{i,j}\in\mathcal{E}$.
The edges are either undirected, such as when two customers share a phone call but it is irrelevant which customer made the call; or directed, in which case we distinguish between outgoing and incoming edges (i.e., all phone calls made by and received by a person, respectively).
The edges are represented by an $n$ by $n$ binary matrix, called adjacency matrix $A$, where a non-zero entry denotes an existing edge between node $v_i$ and $v_j$ in an undirected network and from/to $v_i$ to/from $v_j$ in a directed network with outgoing/incoming edges.
The edges can also carry weights to indicate the intensity of the relationship between two people, for example the number or duration of phone calls they share in a given time period.
The weights are denoted by the weight matrix $W=(w_{i,j})$, where $w_{i,j}\in\mathbb{R}^{+}\cup\{0\}$.
The first order neighborhood of a node $v_i$ is the collection of nodes $v_j$ that share an edge with $v_i$, that is
\[
N^1_i=\{v_j|e_{i,j}\in\mathcal{E}, j=1,\dots,n\}.
\]
In some networks, the nodes can be labelled, or assigned to a class that is later used in a predictive analytics framework.
In this application, there are two types of labels.
The first type of label regards default, in which case the customers in the call network belong to one of two classes:
they are either defaulters, who have been in arrears for more than 90 days within a twelve month period (bad customers); or they are non-defaulters (good customers). \footnote{We use the Basel definition of default. \citep{scheule2016credit}}
When building a credit scoring model, the goal is to assign one of these two classes to each customer of interest and it is the target variable of the classification problem in this study.
In the call network there are also customers who, during the timespan of the network, have run into payment arrears for one or two months in addition to defaulters with three months of payment arrears.
For clarity, all these customers are referred to as delinquent customers and it is the second type of label.
The delinquent customers have the possibility to influence others in the network to also run into payment arrears--also referred to as default influence--and they are used when generating features as explained below.
In order to use the information that is contained in the call networks for building credit scoring models, network features are extracted for each node in the network by aggregating information about its position within the network and connectivity to other nodes.
As in similar studies, a distinction is made between direct network features, which are derived from the node's first order neighborhood, and indirect network features that take into account the whole network structure \citep{van2016gotcha}.
As stated earlier, the aim is to study how delinquent customers may influence others with whom they are connected.
Therefore, by assuming there is prior knowledge about some delinquent customers in the network (i.e., having a subset of nodes with known labels) that knowledge can be incorporated in the network features by exploiting social ties.
To this end, both direct and indirect network features are extracted as illustrated in Figure \ref{fig:propagation}.
The direct network features represent the presence and number of delinquent customers in a node's first order neighborhood.
They are easy to extract and provide a representative overview of people's social connections \citep{lu2003link}.
However, the influence of payment arrears is likely to reach further than just the first order neighborhood.
This effect is modeled using two distinct propagation methods that have been effective in previous research and are designed to simulate real-life behavior: Personalized PageRank (PR) and Spreading Activation (SPA).
The results of both methods are exposure scores which are categorized as indirect network features.
Although other propagation methods exist, such as Gibbs sampling and relaxation labelling, these were not applied here because they have been shown to be less effective for prediction in call networks \citep{oskarsdottir2017social}, are less scalable and as such did not fulfil the requirements of this study.
The features resulting from these three approaches will be used as input features when building credit scoring models, but first a more detailed explanation is provided.
\subsubsection{Link-Based Features}
\citet{lu2003link} presented a framework for inferring labels for nodes in a network based on labels of neighboring nodes.
They defined three features that can be extracted from the neighborhood of a node: count-link, mode-link and binary-link.
These represent, respectively, the frequency of classes in the neighborhood, their mode, and a binary indicator for each class.
Futhermore, using a logistic regression model, \citet{lu2003link} showed that these features are very predictive for the class of the node itself.
Extraction of link-based features is based on the presence of delinquent customers with varying number of payment arrears.
\begin{figure}
\centering
{\includegraphics[scale=0.35]{linkbasedProcess2.pdf}}
\caption{The figure demonstrates the computation of link-based measures, before and after a propagation method is applied to the network.
The figure on the left shows a network with one black node and eight white nodes.
The link-based features of the node to which the arrow points are summarized, where B means black and W means white.
The figure in the middle demonstrates the application of the propagation method with the resulting exposure scores shown for each node in the figure on the right.
After the exposure scores have been computed, a cut-off point is set at 0.5 and nodes with a score that is higher than the threshold are labelled black (B), and white (W) otherwise.
Subsequently, link-based exposure features are extracted for the node to which the arrow points.\label{fig:propagation}}
\end{figure}
\subsubsection{Personalized PageRank}
The propagation method Personalized PageRank (PR) was developed for search engines (e.g., Google) to rank webpages while also taking into account an initial source of information, such as frequently visited web pages \citep{page1999pagerank} but can also be used for different kinds of linked data \citep{van2015apate,van2016gotcha}.
For the nodes in a network with weight matrix $W$, the method iteratively computes exposure scores $\xi_{k+1}$ based on the exposure scores in the node's neighborhood $\xi_{k}$ and a random jump to other nodes in the network, determined by the information source $z$ -- also called restart vector -- using the equation
\[
\xi_{k+1}=\alpha W \xi_k +(1-\alpha)z,
\]
where $1-\alpha$, the damping factor, denotes the probability of a random jump and $k$ is the iteration step.
As a result of the initial information source, exposure scores of nodes closer to the source nodes are higher.
Here, the delinquent customers are the information source.
\subsubsection{Spreading Activation}
The propagation method Spreading Activation (SPA) originates from cognitive psychology and simulates how information, or energy, spreads through the network from a set of source nodes.
It is used to model a `word-of-mouth' scenario, where influence--in this case from delinquent customers--spreads through the network.
`Word-of-mouth' has been shown to be effective in social networks \citep{dasgupta2008social,backiel2015combining}.
Before the method begins, a set of active nodes $V^A\subset \mathcal{V}$ possesses the energy $E^0(V_A)$.
In each step $k$ of this iterative method, a part $d$ of an active node's energy $E^k(V_A)$ is spread to the nodes in its neighborhood while the rest of the energy remains.
The part that is transferred, is distributed according to the relative weights of the links to neighboring nodes, expressed by the transfer function
\[
E_{transfer}=\frac{d\cdot w_{i,j}}{\sum_{w_{i,s}\in N_i^1} w_{i,s}} E^k(V^A_i).
\]
The method stops when no more nodes are being affected and the changes in energy of the already affected nodes are smaller than a given threshold value.
The total energy always remains the same, but spreads throughout the network.
\subsubsection{Link-Based Exposure Features}
After a propagation method, such as PR or SPA, has been applied to a network, each node possesses an exposure score that can be viewed as the relative ranking of the node compared to the rest of the network.
The score can be used as a feature directly or by determining a cut-off value.
Nodes with an exposure score lower than the cut-off are defined as low-risk nodes and those with an exposure score above the cut-off as high-risk nodes \citep{van2016gotcha}.
Then, based on this re-labelling of the network, new link-based features can be extracted.
This is demonstrated in Figure \ref{fig:propagation}.
\subsection{The Expected Maximum Profit Measure}\label{subsec:EMP}
Model selection highly depends on how the performance is measured.
Traditional measures for credit scoring models include AUC, Gini coefficient and the KS statistic that either assess the discriminative ability of the models or the correctness of the categorical predictions \citep{lessmann2015benchmarking}.
The recently proposed Expected Maximum Profit (EMP) measure has an advantage over these traditional measures because it considers the expected losses and operational income generated by the loan, and is tailored towards the business goal of credit scoring \citep{verbraken2014development}.
Most importantly, when applied to credit scoring models it facilitates computing the models' value, the fifth V of Big Data.
The measure is based on the expected maximum profit measure, originally developed for customer churn prediction \citep{verbraken2013novel}, and is expressed for credit scoring by
\[
EMP=\int_{b_0}\int_{c_1} P(T(\Theta);b_0,c_1,c^{\ast})\cdot h(b_0,c_1)dc_1 db_0
\]
where
\[
P(t;b_0,c_1,c^{\ast})=(b_0-c^{\ast})\pi_0 F_0(t)-(c_1+c^{\ast})\pi_1 F_1(t)
\]
is the average classification profit per borrower given the prior probabilities of being a defaulter (non-defaulter), $\pi_0$ ($\pi_1$), and the cumulative density functions of defaulters (non-defaulters), $F_0(s)$ ($F_1(s)$).
Furthermore, $b_0$ is the benefit of correctly identifying a defaulter, $c_1$ the cost of incorrectly classifying a non-defaulter as a defaulter, $c^{\ast}$ the cost of the action, $\Theta=\frac{c_1+c^{\ast}}{b_0-c^{\ast}}$ the cost/benefit ratio and $h(b_0,c_1)$ the joint probability density function of the classification costs \citep{verbraken2014development}.
The maximum profit is achieved by optimizing the cut-off dependent average classification profit where the optimal cut-off value is
\[
T=argmax_{\forall T}P(t;b_0,c_1,c^{\ast}).
\]
As a result, the measure clearly defines an optimal fraction, expressed as
\[
\bar{\eta}_{EMP}=\int_{b_0}\int_{c_1} [\pi_0 F_0(T(\Theta))+\pi_1 F_1(T(\Theta))]\cdot h(b_0,c_1)dc_1 db_0,
\]
representing the fraction of applications that should be rejected to receive maximum profit.
\citet{verbraken2013novel} showed that the EMP corresponds to integrating over the range of the ROC curve that would be considered in a real application, discarding the segment that has a very high, unreasonable cost, and that it is an upper bound of the profit a company could achieve by applying the respective classifier.
When deriving the parameters $b_0 ,c_1$ and $c^{\ast}$ and the probability distribution $h(c_1,b_0)$, \citet{verbraken2014development} rely on the profit framework discussed in \citet{bravo2013granting}.
Thus, $b_0$ is specified as the fraction of the loan amount that is lost after default or
\begin{equation}\label{eq:lamdba}
b_0=\frac{LGD\cdot EAD}{A} =: \lambda,
\end{equation}
where $LGD$ is the loss given default, $EAD$ is the exposure at default and $A$ the loan amount.
Furthermore, $c_1$ equals the return on investment ($ROI$) of the loan and $c^{\ast}=0$ since rejecting a customer does not generate any costs.
It only remains to determine $h(b_0,c_1)$ where $ROI(c_1)$ is assumed to be constant but $\lambda(b_0)$ needs to be estimated for each dataset because it is more uncertain with a multitude of possible distributions.
\subsubsection{Model Profit}\label{subsubsec:profit}
The EMP fraction can subsequently be used to compute the profit of a given model.
First, it is translated into a cut-off value, which depends on the number of instances in the test set.
The instances are labelled as defaulters or non-defaulters depending on whether their predicted score is higher or lower than the cut-off.
Then for each customer in the test set, the confusion matrix in Table \ref{T:confusion} is used to compute the loss or gain produced by the customer.
The model profit is finally computed by aggregating the profit of all customers.
\begin{table}
\centering
\caption{Confusion matrix for computing model profit \label{T:confusion}}
\scalebox{0.85}{\begin{tabular}{llcc}
\hline
&&\multicolumn{2}{c}{Predicted class}\\
&&Non-default&Default\\ \hline
\multirow{2}{*}{Actual class }&Non-default&$ROI\cdot A$&$-ROI\cdot A$\\
&Default&$-LGD\cdot EAD$&0\\ \hline
\end{tabular}}
{}
\end{table}
\subsubsection{Feature Importance in Terms of Profit}\label{subsubsec:feature}
When a credit scoring model is built using the random forest algorithm, its properties can be used to measure the profit impact of each feature in the model.
Assuming a random forest model $RF$ was built using $N$ trees $(T_i)_{i=1}^N$ and $M$ features $(F_j)_{j=1}^M$, the feature importance in terms of profit can be computed in the following way.
\begin{enumerate}
\item
Apply the random forest model $RF$ to the test set and extract class predictions for each tree $T_i\in RF$.
\item
For each tree $T_i$ compute the profit $P(T_i)$ using the confusion matrix in Table \ref{T:confusion}.
\item
For each feature $F_j$ in the test set, compute the mean decrease in profit.
This is defined as the difference between the average profit of trees where $F_j\in T_i$ and the average profit of trees where $F_j\notin T_i$, given by the equation
\begin{equation*}
P(F_j)=\frac{\sum\limits_{i,F_j\in T_i}P(T_i)}{|\{T_i:F_j\in T_i\}|}-\frac{\sum\limits_{i,F_j\notin T_i}P(T_i)}{|\{T_i:F_j\notin T_i\}|}
\end{equation*}
where $F_j\in T_i$ means that feature $F_j$ is in tree $T_i$.
\item
Sort $P(F_j)$: the features with the highest values are those with the greatest mean decrease in profit.
\end{enumerate}
The result of this method is a ranking of the features in terms of the importance with respect to profit.
\section{Experimental Design}\label{sec:experimentaldesign}
\subsection{Data Description}\label{subsec:datadescr}
The data used in this study originates from a telecommunications operator and a commercial bank that both operate in the same country.
The datasets are anonymized, and do not contain personal information such as the name and address of customers.
The telco data contains five consecutive months of CDR data of almost 90 million unique cell phone numbers as described in Table 1.
The data from the bank includes over two million customers and it consists of three parts, namely sociodemographic information, such as age, marital status and postcode; debit account activity, including timing and amount of payments; and credit card activity.
Both sociodemographic and debit account activity span three months and conform the historic part of the dataset.
For the credit card activity, there is information about when the cards were issued, the total credit limit, monthly values of how much of the credit remains and how often the customers have failed to repay their debt until twelve months after receiving the card.
The credit card transactions serve as the key input to the credit scoring application because the data provides information about monthly payment arrears.
This is used to predict the creditworthiness of the customers.
The knowledge about the credit limit and remaining credit on the cards also allows the computation of the EMP.
\subsection{Experimental Setup}
The credit scoring models are built for customers who received a credit card within a three-month period in 2015 and they are referred to as subjects.
An overview of the experimental setup can be seen in Figure \ref{fig:experimentalSetup}.
The credit card data contains information which enables the labeling of the subjects as defaulters or non-defaulters by counting how many late payments they have in the year after signing up for the card.
As previously noted, the Basel definition is used where having three or more late payments implies default.
The label or target vector is denoted by $y_{Default}$.
\begin{figure}
\centering
{\includegraphics[scale=0.3]{creditNetBackup.pdf}}
\caption{Experimental setup for one timeframe.\label{fig:experimentalSetup}}
\end{figure}
To create the bank component of the dataset, both the sociodemographic and debit account data is used.
More precisely, sociodemographic features such as age, marital status and residency as reported at the time of the credit card application are extracted.
Furthermore, debit account activity in the month prior to receiving the credit card is considered and used to extract features representing spending behavior, as can be seen in Table \ref{T:features}.
Based on \citet{singh2015money}, two types of temporal-behavioral features that have been shown to correlate with financial well-being and consumption are included.
The first one, diversity, measures how customers spread their transactions over various bins, represented by the days of the week in this case.
For each customer $i$ and each bin $j$, the fraction of transactions $p_{ij}$ that fall within bin $j$ is computed.
The temporal diversity of customer $i$ is then defined as the normalized entropy of all transactions counted in all seven bins with $M$ being the number of non-empty bins, or
\[
D_i=\frac{-\sum_{j=1}^7 p_{ij}\log p_{ij}}{logM}.
\]
In addition, the loyalty of a customer is defined as
\[
L_i=\frac{f_i}{\sum_{j=1}^7 p_{ij}}
\]
where $f_i$ is the fraction of all transactions of customer $i$ that happen in their $k$ most frequently used bins.
In this case, loyalty characterizes the percentage of transactions that take place during a customer's three most active days.
The collection of both sociodemographic and debit account features is called `sociodemographic' features and denoted with $x_{SD}$.
\begin{figure}
\centering
{\includegraphics[scale=0.27]{creditNet.pdf}}
\caption{The figure demonstrates the various types of people that are present in the call network. \label{fig:TypesOfPeople}}
\end{figure}
The telco data is used as the key input for the social network part of the analysis.
As mentioned before, the subjects received their credit cards within a period of three months and the subjects are considered in each month separately, which results in three timeframes $t_1$, $t_2$ and $t_3$.
To build a call network for each timeframe, the CDR of three whole months prior to the card acquisition month is aggregated and people that have shared a phone call during this period are linked together after discarding any phone calls lasting less than five seconds.
Thus, there are three call networks spanning three months each.
Each network consists of all subjects that received a credit card in the month succeeding the last month in the network, everyone they shared a phone call with and all phone calls between everyone in the network.
In addition to the subjects, there are also other types of people in the network, as Figure \ref{fig:TypesOfPeople} shows.
The people-shaped entities are the subjects, whereas the diamond-shaped entities denote other bank customers (i.e., people who did not receive a credit card during the three months).
They may, however, already possess a card, and those that are known to have had payments arrears are colored black.
These are the delinquent customers in the network, as described in section \ref{subsec:callNetworks}.
Bank customers without payment arrears are colored white.
The circular entities in the network are people who are customers of the telco but not of the bank.
For all subjects in each of the three timeframes, four types of network features, both direct and indirect, are extracted.
First, features representing the calling behavior of the subjects.
Thus, the number and duration of incoming, outgoing and undirected phone calls taking place during the day and night and on different days of the week are computed.
These features are denoted by $x_{CB}$.
As described in subsection \ref{subsec:callNetworks}, information about delinquent customers in the network--the black diamonds--is used and they are labeled with respect to three distinct criteria: having one or more late payments, having two or more late payments and, having three or more late payments.
This gives the opportunity to distinguish the severity of their financial situation in relation to the influence they spread.
These three label vectors serve as the information source $z$ and active nodes $V^A$ when applying PR and SPA, respectively.
Based on these labellings the extraction of link-based features, computation of PR and SPA exposure scores together with link-based exposure features as described in subsection \ref{subsec:callNetworks} is performed.
To construct the weight matrix $W$, edges in all networks are weighted by the number of phone calls and both incoming and outgoing edges as well as undirected networks are considered.
The parameters in the propagation algorithms are set to the default values $\alpha=0.85$ (for PR) and $d=0.85$ (for SPA), based on exploration of the data which showed robust results.
For the link-based exposure features, the cut-off point is defined as the minimum exposure score of the delinquent customers with at least three late payments, since having at least three late payments defines default.
All the link-based features are viewed as one group of features denoted by $x_{LB}$.
Finally, the feature groups $x_{PR}$ and $x_{SPA}$ are respectively composed of the exposure scores of PR and SPA together with the corresponding link-based exposure features.
The result of the featurization process is a dataset of the form
\[
x=\{x_{SD},x_{CB},x_{LB},x_{PR},x_{SPA}\}, \quad y=\{y_{Default}\}
\]
Table \ref{T:features} describes some of these features.
After combining the two data sources, extracting all the features described above and cleaning up the dataset, 22,000 observations remain and over 300 features.
The fraction of defaulters is $0.0449$ or just under $5\%$ default rate.
With the datasets featurized, credit scoring models are built using binary classifiers with a $70\% / 30\%$ split into training and test set.
Before building the models, highly correlated variables are removed and undersampling of the training set conducted to reduce class imbalance, as is common when applying analytics techniques \citep{baesens2014analytics}.
Final model performance is evaluated using the test set.
The binary classifiers logistic regression, decision trees and random forests are used for the empirical analysis.
Logistic regression is the industry standard for building credit scoring models \citep{scheule2016credit}.
Decision trees are included since they are more powerful than logistic regression, while at the same time guaranteeing interpretability of the model.
They are implemented using recursive partitioning with ten-fold cross validation on the training set to tune and prune the trees.
Both the logistic regression and decision tree models are compared against random forests which are an ensemble method that constructs multiple decision trees that jointly decide upon the credit score.
Random forests are considered to be a very powerful, black-box analytical modeling technique.
As a result of parameter tuning, 500 trees were used to build each forest.
\begin{table}
\caption{Descriptions of some of the features that were extracted from the data sources. In the table IN, OUT and UD stand for networks with incoming, outgoing and undirected edges, respectively. The number in the brackets (x) indicates how delinquent customers were defnined with respect to the number of payment arrears in each case. \label{T:features}}
\scalebox{0.6}{
\begin{tabular}{m{1.5cm}lm{1.2cm}rp{18cm}}
Feature Group&Notation&Number&Feature&Description\\\hline
\multirow{8}{1.5cm}{Socio demo graphic}&\multirow{8}{*}{SD}&\multirow{8}{*}{35}&Age&Current age of the customer\\
&&&Amount Spent&Total amount spent in the month before receiving the credit card \\
&&&Mean Spent p. Day& Average amount spent per day during the month before receiving the credit card \\
&&&Diversity-NE Value&Diversity of value spent over non-empty bins during the month prior to receiving the credit card\\
&&&Diversity-ALL Number&Diversity of number of purchases over all seven bins during the month prior to receiving the credit card\\
&&&Loyalty-Number&Loyalty of number of purchases in the top three bins during the month prior to receiving a credit card\\ \hline
\multirow{3}{2cm}{Calling Behavior}&\multirow{3}{*}{CB}&\multirow{3}{*}{72}&Count IN&Total number of phone calls received during the three months of the social network \\
&&&Weekend Duration OUT&Aggregated duration of all phone calls made on weekends during the three months of the social network \\
&&&Tuesday Duration UD&Aggregated duration of all phone calls made and received on Tuesdays during the three months of the social network \\\hline
\multirow{8}{1cm}{Link-Based}&\multirow{8}{*}{LB}&\multirow{8}{*}{36}&Binary (0) IN&Binary indicator of having neighbors with no late payments, in a network with incoming edges\\
&&&Binary (1) OUT&Binary indicator of having neighbors with one late payment, in a network with outgoing edges\\
&&&Binary (2) UD&Binary indicator of having neighbors with two late payments, in a network with undirected edges\\
&&&Binary (3) UD &Binary indicator of having neighbors with three late payments, in a network with undirected edges\\
&&&Count (0) IN&Number of neighbors with no late payments, in a network with incoming edges \\
&&&Count (1) OUT&Number of neighbors with one late payment, in a network with outgoing edges\\
&&&Count (2) OUT&Number of neighbors with two late payments, in a network with outgoing edges\\
&&&Count (3) UD&Number of neighbors with three late payments, in a network with undirected edges\\\hline
\multirow{6}{1.5cm}{Persona-lized PageRank}&\multirow{6}{*}{PR}&\multirow{6}{*}{54}&Exposure (1) IN&Exposure score after applying PR on a network with incoming edges and delinquent customers with one or more late payments.\\
&&&Exposure (2) OUT&Exposure score after applying PR on a network with outgoing edges and delinquent customers with two or more late payments.\\
&&&Exposure (3) UD&Exposure score after applying PR on a network with undirected edges and delinquent customers with three or more late payments.\\
&&&Binary High Risk (1) IN&Binary indicator of having neighbors with high exposure scores after applying PR on a network with incoming edges and delinquent customers with one or more late payments. \\
&&&Binary High Risk (2) OUT&Binary indicator of having neighbors with high exposure scores after applying PR on a network with outgoing edges and delinquent customers with two or more late payments. \\
&&&Count High Risk (3) IN&Number of neighbors with high exposure scores after applying PR on a network with incoming edges and delinquent customers with three or more late payments.\\\hline
\multirow{6}{2cm}{Spreading Activation}&\multirow{6}{*}{SPA}&\multirow{6}{*}{54}&\\
&&&Exposure (1) IN&Exposure score after applying SPA on a network with incoming edges and delinquent customers with one or more late payments.\\
&&&Exposure (2) OUT&Exposure score after applying SPA on a network with outgoing edges and delinquent customers with two or more late payments.\\
&&&Exposure (3) UD&Exposure score after applying SPA on a network with undirected edges and delinquent customers with three or more late payments.\\
&&&Binary High Risk (1) IN&Binary indicator of having neighbors with high exposure scores after applying SPA on a network with incoming edges and delinquent customers with one or more late payments. \\
&&&Count High Risk (1) UD&Number of neighbors with high exposure scores after applying SPA on a network with undirected edges and delinquent customers with one or more late payments. \\
&&&Count High Risk (3) IN&Number of neighbors with high exposure scores after applying SPA on a network with incoming edges and delinquent customers with three or more late payments.\\\hline
\end{tabular}}
\end{table}
\section{Results}\label{sec:results}
The results are organized in three parts starting with empirical tests to establish the networks' relational dependency.
Subsequently, the results of the proposed methodology are detailed, first in terms of statistical performance and then in terms of economic performance.
\subsection{Homophily amongst Defaulters} \label{sec:hom}
A network is homophilic if nodes with a certain label are to a larger extent connected to other nodes with the same label.
In the default networks, homophily is present if the fraction of edges between defaulters and non-defaulters is significantly smaller than the expected fraction of such edges in the network.
A one-tailed proportion test with a normal approximation for homophily amongst defaulters resulted in a p-value of less than 0.0001, which means that there is evidence of homophily \citep{baesens2015fraud}.
Furthermore, homophily in networks can also be measured with dyadicity and heterophilicity, that is, the connectedness between nodes with the same label and of different labels, respectively, compared to what is expected in a random network \citep{baesens2015fraud}.
The networks analyzed here, have a dyadicity amongst defaulters of 0.8689 while the heterophilicity is 0.8137.
This means that the networks are not dyadic, as defaulters are not more connected amongst themselves, but they are heterophilic, i.e., there are less connections between defaulters and non-defaulters.
Based on these results, there is foundation for applying social network analytic techniques to predict default in the call networks.
\subsection{Statistical Model Performance}
\begin{table}
\centering
\caption{Statistical Model Performance (AUC). \label{T:modelperformance}}
\scalebox{0.85}{\begin{tabular}{llccc}
\hline
\multicolumn{2}{c}{Model}&\multicolumn{3}{c}{Classifier}\\
Model ID&Feature Groups&Logistic Regression&Decision Trees&Random Forest\\ \hline
A&SD&0.5869&0.7004&0.8993\\
B&CB&0.5351&0.7043 &0.8700\\
C&LB&0.5485&0.7429 &0.7697\\
D&PR&0.5163&0.7611 &0.8339\\
E&SPA&0.5281&0.7188 &0.8063\\
F&SD,CB&0.6115&0.7127&0.9227\\
G&CB,LB,PR,SPA&0.5182&0.7307&0.9154\\
H&SD,CB,LB,PR,SPA&0.6121&0.7263&0.9224\\ \hline
\end{tabular}}
\end{table}
Credit scoring models are built with the features in each feature group separately, as well as three models with a combination of feature groups, as seen in Table \ref{T:modelperformance}.
The first five models $A$, $B$, $C$, $D$ and $E$ study the main effects of each feature group.
Model $F$ combines the sociodemographic features with the calling behavior features, model $G$ includes all feature groups except the sociodemographic features and in model $H$ we consider all feature groups.
Other combinations of feature groups were tried, but they did not provide more significant results than the ones shown.
As is common practice in credit scoring, statistical model performance is measured by the area under the receiver operating curve (AUC).
The AUC summarizes the trade-off between model sensitivity and specificity in a single number between 0 and 1 with higher values meaning better performance.
From Table \ref{T:modelperformance}, it is clear that the performance with respect to the three classifiers varies substantially.
Overall, the logistic regression models perform the worst, of which models including sociodemographic features (models $A$, $F$, $H$) perform best.
Logistic regression models do not yield a better performance when using network-related features.
This hints at a non-linear behavior that cannot be properly captured by a generalized linear model.
\begin{figure}%
\centering
{ \subfloat[$95\%$ Confidence]{{\includegraphics[clip, trim=5cm 18.5cm 12.5cm 4cm, width=0.35\textwidth]{dom1} }}%
\qquad
\subfloat[$99\%$ Confidence]{{\includegraphics[clip, trim=5cm 18.5cm 12.5cm 4cm, width=0.35\textwidth]{dom2} }}}%
\caption{Domination graphs.
\label{fig:domination}}
\end{figure}
The random forests produce the best-performing models and the remaining discussion will therefore be focus on them.
First, the test of \citet{delong1988comparing} is applied to the receiver operating curves (ROC) of each pair of random forest models to compare their performance.
The results can be seen in the domination graphs in Figure \ref{fig:domination}.
The best performing models are at the top and models that perform worse are lower down.
The arrows indicate a significant improvement in statistical performance at $95\%$ and $99\%$ confidence level on the left and right, respectively.
The figure on the left demonstrates that there is not a significant difference in the performance of the three models with a combination of features ($F$, $G$, $H$), but models with only one type of features ($A$, $B$, $C$, $D$, $E$) perform significantly worse with the link-based features ($C$) performing worst overall.
\begin{figure}
\centering
{\includegraphics[scale=0.8]{ACC}}
\caption{Feature importance: Mean decrease in accuracy.
\label{Fig:varimpAcc}}
\end{figure}
Secondly, the importance of the features in model $H$ is explored to determine their ability to predict default and rank the usefulness of the features.
This is displayed in Figure \ref{Fig:varimpAcc} for the mean decrease in accuracy for the 20 most important variables.
The mean decrease in accuracy of a particular feature measures how much the accuracy of the resulting model decreases when that feature is left out of the model, and as a result, gives a score of how important it is in the model.
Figure \ref{Fig:varimpAcc} demonstrates that the calling behavior features are ranked the highest, followed by PR features and SPA features, and a single LB feature.
\subsection{Economic Model Performance}
The previous subsection showed that the statistical performance of more complex credit scoring models with a combination of feature groups is significantly better than models with only one feature group, and even better than that of models with sociodemographic features alone.
Here the economic performance of the models is evaluated and the importance of features in terms of profit by applying the EMP to the random forest models.
\begin{figure}%
\centering
{ \subfloat[EMP]{{\includegraphics[width=0.45\textwidth]{EMP2d} }}%
\qquad
\subfloat[EMP fraction]{{\includegraphics[width=0.45\textwidth]{EMPfrac2d} }}}
\caption{Sensitivity Analysis for ROI.
\label{fig:sens2D}}
\end{figure}
For the EMP (see section \ref{subsec:EMP}) various parameters need to be specified.
To compute the benefit of correctly identifying a defaulter, $\lambda$ (see Equation \ref{eq:lamdba}), the credit card limit is used as the principal ($A$) and the drawn amount on the card at the time of default as exposure at default ($EAD$).
The two remaining parameters: loss given default ($LGD$) and return on investment ($ROI$), are domain specific and not obtainable from the data directly.
Therefore, an exploration of their effect on the EMP is provided.
An analysis of the variation in EMP as a function of $LGD$ shows substantial robustness, which means that the economic performance of the models does not greatly depend on $LGD$.
Considering this, and based on expert judgement, this parameter is set to 0.8.
In contrast, EMP decreases when $ROI$ increases as is evident from Figure \ref{fig:sens2D}, which shows the EMP and its implied cutoff (EMP fraction) as a function of $ROI$ when $LGD$ is set at 0.8.
The value for $ROI$ is determined based on the `elbow' in these figures and set to 0.05.
The inflection point is the point where the $ROI$ becomes the biggest influence (thus the linear behavior) and so it is appropriate to choose a value that balances profits for the rest of the analyses.
\begin{figure}
\centering
{\includegraphics[scale=0.55]{Iris.pdf}}
\caption{Distribution of $\lambda$.\label{fig:lambda} }
\end{figure}
Subsequently, the distribution of $\lambda$ can be estimated, see Figure \ref{fig:lambda}.
As in \citet{verbraken2014development} there are two peaks in the distribution, one at each end of the unit interval and with the assumption that $\lambda$ follows a uniform distribution in between.
The peak at 0 represents credit card holders who have had payment arrears and have paid back fully, whereas the peak at 1 indicates those that never paid back their debt.
This distribution is used to determine the values for $p_0$ and $p_1$, see \citet{verbraken2014development}.
\begin{figure}%
\centering
{ \subfloat[EMP and EMP fraction]{{\includegraphics[width=0.45\textwidth]{ModelEMP} }}%
\qquad
\subfloat[Profit]{{\includegraphics[width=0.45\textwidth]{ModelProfit} }}}
\caption{Economic Model Performance. \label{fig:EMP}}
\end{figure}
\begin{comment}
\begin{table}
\caption{\label{T:EMP}}
\centering
\begin{tabular}{rrrrr}
\multicolumn{2}{c}{Model}&\multicolumn{3}{c}{Measure}\\ \cline{1-2}\cline{3-5}
ID& Variables& EMP& EMP fraction&Model Profit\\ \hline
$A$& SD& 2.51& 4.80& \usd{613628.23}\\
$B$& CB &2.35& 4.87& \usd{619327.69}\\%9319705\\
$C$& LB &1.84& 7.80& \usd{496445.40}\\%7470560\\
$D$& PR& 2.17& 3.09& \usd{613099.86}\\%9225988\\
$E$& SPA& 2.13& 4.24& \usd{573530.15}\\%8630539\\
$F$& SD,CB& 2.52& 9.34& \usd{581911.87}\\%8756668\\
$G$& SD,CB,LB,PR,SPA& 2.52& 9.47& \usd{570689.52}\\%8587793\\
$H$&CB,LB,PR,SPA& 2.52& 11.89& \usd{530940.25}\\ \hlin
\multicolumn{5}{r}{Total profit in the test set without a model equals \usd{273694.15}
\end{tabular}
\end{table}
\end{comment}
With all parameters estimated, the next step is to compute the expected maximum profit, the profit maximizing fraction of rejected loans and the model profit (as described in Section \ref{subsubsec:profit}) for the random forest models in Table \ref{T:modelperformance}.
The results can be seen in Figures \ref{fig:EMP}.
The value for EMP is expressed as a percentage of the total loan amount and measures the incremental profit relative to not building a credit scoring model.
The ranking of the values for the expected maximum profit is consistent with the ranking of the AUC values in Table \ref{T:modelperformance}, and again models $A$, $F$, $G$ and $H$ are considered best and $C$ the worst.
The EMP fraction values vary, however, and therefore so do the model profits.
The profit maximizing fraction represents the fraction of credit card applications that should be rejected in order to obtain the maximum profit.
The fact that the fraction for model $G$ is so much higher than the rest with the profit remaining the same, indicates that the model focuses on the most profitable customers.
Regarding the model profits, there is a substantial increase compared to not using a model.
Model $B$ has the highest profit, followed by models $A$ and $D$.
Models $F$ and $H$ also produce decent profits, whereas $G$ does not, at least not when compared to the rest.
Again, model $C$ performs the worst.
Of the models with only one data source, model $B$ (built with calling behavior variables) brings the higher singular results.
As no history is available for these borrowers, a possible explanation is that their socioeconomic standing can be deduced from their immediate network.
Note however that this difference is marginal, as model $A$ (sociodemographic variables) follows it.
Of the combined models, models $F$ (with both sociodemographic variables and calling behavior) and $H$, with all available variables, produce the best results in terms of profits.
\begin{figure}%
\centering
{\includegraphics[width=0.8\textwidth]{IMP} }
\caption{Feature Importance: Mean decrease in Profit.
\label{fig:varImpEMP} }%
\end{figure}
Figure \ref{fig:varImpEMP} shows the mean decrease in profit for the 20 most important features in model $H$ computed using the technique described Section \ref{subsubsec:feature}.
This profit perspective shows more variation in groups of features than the statistical one.
As for mean decrease in accuracy, more than half of the features are calling behavior features, but in contrast to Figure \ref{Fig:varimpAcc}, a quarter of the features are sociodemographic features, which in this case are features that measure consumption.
This is consistent with the result indicating that, of the combined models, model $F$ was associated to a larger profit.
To test for correlation among the ranking of features according to the two measures we computed the Spearman's $\rho$, Kendall's $\tau$ and Goodman and Kruskal's $\gamma$ correlation coefficients.
The resulting values did not indicate a correlation among the rankings.
It is interesting to see that having only network features allows us to discriminate potentially better customers. That would mean that we can look, using the network connections, beyond simple socioeconomic and sociodemographic traits, and actually profile more profitable customers.
In the models, network features help discriminate different customers, who cannot be captured by common features, and it happens that these customers bring a lot of profit.
\section{Discussion}\label{sec:discuss}
Based on these results, the three research questions in section \ref{secIntroduction} can be addressed.
The first research question Q1 assesses the value that call data adds to credit scoring models in term of AUC and profit.
For statistical performance, models including all features performed best, with the AUC value increasing by 0.023 points in the best model when compared to the sociodemographic model $A$.
The economic performance of the models in terms of EMP, EMP fraction and profit can be seen in Figure \ref{fig:EMP}.
The model with the highest profit is model $B$ and it is slightly better than the traditional model $A$.
Models with a combination of feature groups ($F$ ,$G$ and $H$) produce lower profit but their EMP values are the highest.
The reason for the lower profit is the high EMP fraction, which indicates that these models are more conservative and exclude a higher proportion of the defaulters.
These results indicate that the CDR data complements the conventional data and there is added value when including the CDR data in credit scoring models and even when used without the traditional features.
The results also provide an answer to the second research question Q2: Can call data replace traditional data used for credit scoring?
In terms of both statistical and economic performance, the results indicate that the predictive power of call data is just as good or might be even better than traditional data for these borrowers.
This is clear from the high performance of model $B$.
In addition, the importance of the calling behavior features shows that these are very predictive, much more so than the traditional features.
This result demonstrates the merit of this research. Given the high predictive power of the call data, borrowers without enough bank information can benefit from the approach by giving access to their call records to obtain credit.
Finally, the last research question Q3 about how default behavior propagates in the network, can be addressed.
The results of the homophily test in section \ref{sec:hom} showed a lower fraction of connections amongst defaulters and also between defaulters and non-defaulters.
This might partially be a consequence of the low number of defaulters in the network overall.
Furthermore, insights about the propagation of default behavior can be obtained from the importance of the features.
Firstly, for mean decrease in accuracy, see Figure \ref{Fig:varimpAcc}, a few PageRank and Spreading Activation features are important.
These are predominantly `Count Low Exposure' features which represent the number of neighbors with low exposure score.
This indicates that not having a high-risk neighbor is predictive of non-default.
The PageRank feature `Exposure (2)' is also among the 20 most important features.
It represents the PageRank exposure score when the influence comes from delinquent customers with two or more late payments and indicates a propagation effect of default behavior.
Second, for the mean decrease in profit in Figure \ref{fig:varImpEMP} there are two PageRank exposure scores, based on one and three late payments of delinquent customers.
From these observations we can say that, in terms of propagation of default influence, Personalized PageRank is more effective than Spreading Activation.
A more thorough analysis of how default propagates is needed to better understand the effect of each of these features.
\section{Impact of Research}\label{sec:impact}
The research findings presented in this paper have possible impact at various levels.
This section identifies three different levels and provides a discussion of the implications of each one.
\subsection{Regulatory Impact}
The Basel Accords model unexpected losses using a Merton single-factor model where the asset value of an obligor depends upon a systematic (e.g., the macroeconomy) and an idiosyncratic (e.g., obligor-specific) risk component \citep{scheule2016credit}.
Asset correlations are then also factored in to see how default behavior is correlated and, as such, model system risk.
A key concern relates to the exact values of these asset correlations.
For corporates, the assets can be quantified by inspecting balance sheets, and various financial models have been introduced to quantify corporate asset correlations.
For retail exposures (e.g., credit cards, mortgages, installment loans), it becomes considerably more difficult as the assets are less tangible.
Retail asset correlations have been specified in the Basel Accords using some empirical, but not published, procedure reflecting a combination of supervisory judgment and empirical evidence.
As such, they are fixed at $4\%$ for qualifying revolving exposures (e.g., credit cards) and $15\%$ for mortgages.
Given their impact on capital calculation, it would be desirable that these asset correlations are sustained by a solid theoretical framework and accompanying empirical validation.
In this research, we illustrated how default behavior on credit cards propagates in a call network.
These insights pave the way for additional research aimed at quantifying asset and default correlation for retail exposures in a more sound and solid way.
This can then lead to better regulatory asset correlation values which in turn leads to a better protection of the financial system.
\subsection{Financial Inclusion}
The results may also have a societal impact that affect borrowers in developed and developing countries in different ways.
In the former case, people who are joining the financial market for the first time, such as young people and immigrants, face troubles when applying for loans because they do not have a credit history.
Instead, they need to spend time and effort to build their credit history before financial institutions can assess whether they are creditworthy.
In developing countries where historical financial data is often nonexistent, the impact is even greater.
As reported by the World Bank, over two billion adults worldwide do not have a basic account which makes up more than $20\%$ of the adult population in some countries \citep{worldBank}.
The benefits of behavior-based microfinance in these countries are evident, as having access to small credits has a social impact on communities, helping to fight poverty and enhancing economic development \citep{copestake2007mainstreaming}.
In contrast to the lack of banking history, the high(er) availability of call data
in these countries provides an alternative for credit scoring, hereby facilitating credit access to a wider segment of the population.
According to the results, features extracted from these untraditional data sources are good predictors of credit behavior (e.g., models $B$, $G$ and $H$).
In addition, the numerous smartphone applications that are already being deployed in some developing countries are a prime example of the success of these methods.
They offer immediate small loans, that are repaid within a short period of between three weeks and six months and have lower interest rates, ranging between $6\%$ and $12\%$ as opposed to the $25\%$ interest rate in traditional microlending \citep{dwoskin2015lending}.
\subsection{Privacy and Ethical Concerns}
The results of this study are furthermore affected by privacy regulations because the implementation of some of the models depends on different parties sharing the data.
Since there are no worldwide applicable standards for data-sharing of that kind, we illustrate how this might occur by studying the reality of the US and the EU in what follows.
In the US, there is no single federal law regulating data transfer between affiliates. The transfer of financial information between a bank and a telco is protected under the Gramm-Leach-Bliley Act \citep{Cuaresma2002}. This legislation allows transfer of personally identifiable information originating from a financial service provider to a third party if the parties design a contract that disallows disclosure and use of information outside the project. In general, such a contractual framework should satisfy most other pieces of regulation that might indirectly apply to the sharing of data in the other direction (i.e., from the telco to the bank or credit bureau).
In the European Union, in contrast, there is a strong body of legislation regulating data sharing. Given that CDR data are a form of communication, and the objective of the model is to process it along with banking data in an automated way, two pieces of legislation apply: Regulation 2016/679 \citep{EU_GDPR} regarding the protection of privacy for natural persons, best known as the General Data Protection Regulation (GDPR), and the ``ePrivacy Regulation'' \citep{EU_ePrivacy} regarding the processing of personal data in the electronics communications sector.
The ePrivacy Regulation deals with if, and how, communications data at a disaggregated level can be used. Article 30 in particular mandates that a service such as a financial score, which is not only for billing or providing the mobile service, is a ``value added service'', and thus requires explicit authorization from the user. This authorization might be given in the contract, for example, or ex post to the signing of the contract via electronic authorization. In case none of these provisions can be set in place, then the sharing of CDR data cannot occur unless the data is anonymized.
The key challenge is how to make the data available to the other party, so defaulters can be correctly identified. Fortunately, there are methods that can provide privacy-preserving data linkage \citep{Clifton2004} that can be followed in order to join the data securely without compromising the individual on either side of the sharing process. Methods such as Privacy Preserving Probabilistic Record Linkage \citep[P3RL][]{Schmidlin2015}, that are in use in the medical sciences, allow secure data-sharing between partners. The secure transfer of data is also very simple to satisfy, following proper encryption and secure access protocols. The GDPR has additional provisions on data storage, forcing companies to store data only for the time necessary to provide the service, so the party receiving the linked data only for the purposes of model development must ensure proper disposal of the data after the model development. Finally, note that the model itself is considered aggregated data. Article 22 of the GDPR allows the safe use of statistical models when these are required to establish a contract with the counterparty (the financial institution), which is the case when a loan is granted.
There is as well an ethical concern in using data that depends on the social network of the borrowers to restrict funding to them. This is of course not a practice that should be recommended from the results of this model, as it would constitute unfair discrimination. However, when borrowers do not have any past behavior information that allows institutions to make a decision, or they have not accumulated enough additional information to profile them correctly, then CDR information can clearly contribute to increase financial inclusion. Thus, we propose that the use of this data be done in strict positive terms. This can be easily done when constructing a credit score: it is common practice to discretize continuous variables and give a score based on the Weight of Evidence for each of the segments \citep{scheule2016credit}. An ethical use of this information would simply assign the neutral score to those segment which would unfairly punish the borrowers, leaving the positive segments that would provide easier access to funds.
\section{Conclusion}\label{sec:conclusion}
This study presents the statistical and economic advantages of exploiting Big Data and social network analytics for credit scoring applications.
We use phone call logs are used to build call networks and social network analytics applied to enhance the performance of models that predict creditworthiness of credit cards applicants.
We do this from both a statistical and profit perspective and demonstrate how incorporating telco data has the potential of increasing the Value of credit scoring models.
Furthermore, we identify which features are most important for this predictive task, both in terms of statistical performance and profit.
According to the results, models that are built with features that represent calling behavior perform best, both when performance is measured in AUC and profit.
We also show that these features dominate other features in terms of importance.
This is an interesting result because it means that how people use their phones can be used as the sole data source when deciding whether they should be given a loan or not.
Thus we propose that the data should be used in strict positive terms, to facilitate financial inclusion for people that lack enough information for correct profiling.
The main limitation of our this is the data itself.
The scorecards that were built are for the applications of credit cards, and it is unclear how the results would generalize for other types of credits such as microloans or mortgages.
In industry, numerous applications for granting microloans via smartphones by analyzing user's behavior exist.
According to various reports, behavioral features are important in these applications as well, but that is difficult to verify without published scientific results.
Similar data could be obtained from peer-to-peer lending platforms, or through agreements between telcos and banks/credit bureaus, where there is access to both default status of users as well as behavioral features.
Behavioral data similar to the mobile phone data shown in this work could also be gathered from social media platforms such as Twitter.
The data in this study originates from a single country where a telco and a bank have a special agreement to share the data.
Therefore, an analysis of similar data from other countries or data for other types of credits would strengthen the external validity of the presented results.
In practice, lenders use credit bureau variables, such as FICO scores, when assessing creditworthiness, and unfortunately they were not available for these analyses, but would be an interesting extension of our work.
It is already clear that the mobile phone data used in this study is big in the sense of `Volume', `Velocity', `Veracity' and `Variety'.
Our analysis of the data and the resulting well-performing models show that it also has a positive effect for financial inclusion and on model profit, and as such is also important for `Value': the fifth V of Big Data!
\section*{Acknowledgments}
The authors would like to thank Ariel Berenstein for his contribution to this research.
The authors acknowledge the support of a large Belgian bank that wishes to remain anonymous.
\footnotesize{
|
1,108,101,565,427 | arxiv |
\section{Introduction}
Indirect measures of the structure of massive-star winds are possible in X-ray binaries through the analysis of the interaction between the compact companion and the stellar wind.
In this report we summarize the constraints obtained on wind clumping in HMXB using the hard X-ray variability observed by the IBIS/ISGRI instrument
on board INTEGRAL (\cite{walter:winkler03AA}). Further details can be found in \cite{walter:WalterZurita2007} and \cite{walter:Leyder2007}.
Classical wind-fed, Roche-lobe underflow, super-giant HMXB (sgHMXB) are made of a compact object orbiting within a few (1.5 to 2.5) stellar radii from a super-giant companion. Recently INTEGRAL almost tripled the number of sgHMXB systems known in the Galaxy and revealed a much more complex picture with two additional families of sources:
(1) the highly-absorbed systems which have orbital and spin periods similar to those of classical sgHMXB but much higher absorbing column densities on average (\cite{walter:walter2006}) and (2) the fast transient systems which are characterized by fast outbursts and by a very low quiescent luminosity (\cite{walter:Sguera2006,walter:Negueruela2007}).
\section{Sources and data analysis}
Several sources have now been proposed as candidate super-giant fast X-ray transient based on their hard X-ray variability characteristics, and, for a subset of them, optical counterpart spectral type. Contrasting statements have however been made on specific sources for what concerns their persistent or transient nature. In the frame of the current study we have considered all SFXT candidates together with several persistent and absorbed super-giant HMXB for comparison.
Among them, we specifically excluded known Be systems, sources detected only once by INTEGRAL,
blended INTEGRAL sources, long period systems and the sgB[e] system IGR J16318$-$4848.
We analyzed the available INTEGRAL data for 12 candidate SFXT (table \ref{walter:tab2}) that have large variability factors and compared them with the classical and absorbed sgHMXB systems that have a typical variability factor $\lesssim20$.
The sources of the sample are located along the galactic plane that has been heavily observed by INTEGRAL. All public data available until March 2007 are considered in this study.
Individual ISGRI sky images have been produced for each INTEGRAL pointing in the energy band 22--50~keV. The detection of the sources of the sample is forced in each image and the source count rate extracted.
Source flares have been detected by requiring a minimum of 25 ksec of inactivity between them. Flare duration of the order of a single INTEGRAL pointing $(2~\rm{ksec})$ have been observed in all sources (excepting IGR\,J16465$-$4507). Their typical duration is 3 ksec.
Fewer longer $(> 15~\rm{ksec})$ flares have also been detected but in most cases could be interpreted as a serie of shorter flares or a long activity period. They will not be discussed further here.
\begin{table}[H]
\vspace{-5mm}
\caption{List of SFXT candidates with quiescent flux $F_{q}$, source observing elapsed time $T_{obs}$ and flaring characteristics:
maximum count rate $F_{fl}$, number of flares $N_{fl}$ and the average flare duration $t_{fl}$.}
\label{walter:tab2}
\begin{tabular}{l|r@{.}l|c|c|l|r}
\noalign{\vspace{2mm}}
\toprule
Source &\multicolumn{2}{c|}{$F_{q}$}&\multicolumn{1}{c|}{$F_{fl}$}&$N_{fl}$&\multicolumn{1}{c|}{$t_{fl}$}&\multicolumn{1}{c}{$T_{obs}$}\\
&\multicolumn{2}{c|}{ct/s}&\multicolumn{1}{c|}{ct/s}&&\multicolumn{1}{c|}{ks}&\multicolumn{1}{c}{days}\\
\midrule
\noalign{\vspace{2mm} \bf SFXT systems\vspace{2mm}}
\tiny{IGR\,J08408$-$4503} &$<0$&1&3.9 &2 &3.6 &52.0\\
\tiny{IGR\,J17544$-$2619} & 0&06 &24&8&2.5 &127.0\\
\tiny{XTE\,J1739$-$302} & 0&08 &28&12 &4.2 &126.4\\
\tiny{SAX\,J1818.6$-$1703} & 0&18 &45&11 &2.9 &76.9\\
\tiny{IGR\,J16479$-$4514} & 0&2 &19&38 &3.6 &67.0\\
\tiny{AX\,J1841.0$-$0536} &$<0$&1&15&4 &5.8&51.9\\
\tiny{AX\,J1820.5$-$1434} &$<0$&1&5.3&4 &3.9 &59.4\\
\noalign{\vspace{2mm} \bf Intermediate systems\vspace{2mm}}
\tiny{AX\,J1845.0$-$0433} & 0&2 &6.2 &6 &4.0&55.2\\
\tiny{IGR\,J16195$-$4945} & 0&2 &4.8 &6 &2.2&71.8\\
\tiny{IGR\,J16465$-$4507} & 0&1 &6.9&3 & &66.7\\
\tiny{IGR\,J16207$-$5129} & 0&4 &9.2 &11 &4.3&73.7\\
\tiny{XTE\,J1743$-$363} & 0&5 &9.2 &19&2.5&122.9\\
\bottomrule
\end{tabular}
\end{table}
Table \ref{walter:tab2} lists the sources together with their quiescent count rate $(F_{q})$, average flare count rate $(F_{fl})$, number of flares $(N_{fl})$, range of flare durations $(t_{fl})$ and total source observing elapsed time $(T_{obs})$.
As the probability to detect a flare decreases when the source gets outside of the fully-coded field of view, the effective observing time for flare detection can be estimated as $0.6~T_{obs}$.
The sources have been separated in two categories. The SFXT include systems featuring hard X-ray variability by a factor $\gtrsim100$. ``Intermediate'' systems are candidate SFXT with smaller variability factors that could be compared with those of classical systems.
From the variability point of view, sources closer to the bottom of the table are more similar to classical sgHMXB.
\section{Discussion}
The distances to the SFXT systems has been evaluated (2--7 kpc) in a few cases.
We will assume, for the rest of the discussion, a distance of 3 kpc. The average count rate observed during flares lies between 3 and 60 ct/s which translates to hard X-ray luminosities of $(0.2-4)\times 10^{36}~\rm{erg/s}$. Such luminosities are not exceptional for sgHMXB but very significantly larger than the typical X-ray luminosity of single massive stars of $10^{30-33}~\rm{erg/s}$ at soft X-rays (\cite{walter:Cassinelli1981}).
As the sources are flaring at most once per day, their average hard X-ray luminosity is very low, reaching $(0.2-4)\times 10^{34}~\rm{erg/s} $. It is therefore very unlikely that those systems have average orbital radius lower than $10^{13}~\rm{cm}$ i.e. $\sim 10~R_*$. One expects orbital periods larger than 15 days and underflow Roche lobe systems (note that no orbital period has yet been derived in any of these systems).
\paragraph {Wind clumps}
\paragraph {}
\vspace{-0.3cm}
The interaction of a compact objet with a dense clump formed in the wind of a massive companion leads to increased accretion rate and hard X-ray emission.
The free-fall time from the accretion radius $R_a = 2\times 10^{10}~ \rm{cm}$ towards the compact object is of the order of $(2-3)\times10^2~\rm{sec}$. As the intrinsic angular momentum of the accreted gas is small (\cite{walter:Illarionov2001}) the infall is mostly radial (down to the Compton radius) and proceeds at the Bondi-Hoyle accretion rate.
With a duration of $t_{fl}=2-10$ ksec, the observed short hard X-ray flares are significantly longer than the free-fall time. The flare duration is therefore very probably linked with the thickness of the clumps which, for a clump radial velocity $V_{cl}=10^8 ~\rm{cm/s}$, is $h_{cl} = V_{cl} \times t_{fl} \sim (2-10) \times 10^{11}~\rm{cm}$.
The average hard X-ray luminosity resulting from an interaction between the compact object and the clump can be evaluated as $L_X = \epsilon~M_{acc}c^2/t_{fl}$ (where $\epsilon\sim0.1$) and the mass of a clump can then be estimated as
$ M_{cl} = ~ (R_{cl}/R_{a})^2 ~M_{acc}= (R_{cl}/R_{a})^2~L_X~t_{fl}/(\epsilon~ c^2) $
where $R_{cl}$ is the radius of the clump perpendicular to the radial distance.
In the case of a spherical clump,
$M_{cl} =
\left(\frac{L_X}{10^{36}~\rm{erg/s}}\right) \left(\frac{t_{fl}}{3~\rm{ks}}\right)^3
~7.5\times 10^{21} ~\rm{g}.$
If $\dot{N}$ is the rate of clumps emitted by the star, the observed hard X-ray flare rate is given by $T^{-1} = \dot{N}(R_{cl}^2/4R_{orb}^2).$
The rate of mass-loss in the form of wind clumps can then be estimated as
$\dot{M}_{cl} =
\left(\frac{10\rm{d}}{T}\frac{L_X}{10^{36}\rm{erg/s}}\frac{t_{fl}}{3\rm{ks}}\right)\left(\frac{R_{orb}}{10^{13}\rm{cm}}\right)^2 ~3\times 10^{-6}~\rm{M_{\odot}/y}.$
For a $\beta=1$ velocity law and spherical clumps, the number of clumps located between $1.05R_*$ and $R_{orb}$ can be evaluated as
$N=
\left(\frac{10~\rm{d}}{T}\right)\left(\frac{3~\rm{ks}}{t_{fl}}\right)^2\left( \frac{R_{orb}}{10^{13}~\rm{cm}}\right)^3~3.8\times 10^3$.
Assuming spherical clumps, the clump density at the orbital radius is $\rho_{cl}=\left(\frac{L_X}{10^{36}~\rm{erg/s}}\right) ~7\times 10^{-14} ~\rm{g~cm}^{-3}$ and the corresponding homogeneous wind density is $\rho_h=\dot{M}_{cl}/(4\pi~R_{orb}^2~V_{cl})=
\left(\frac{10~\rm{d}}{T}\frac{L_X}{10^{36}~\rm{erg/s}}\frac{t_{fl}}{3~\rm{ks}}\right)
~1.5\times 10^{-15}~\rm{g~cm}^{-3}$. The clump volume filling factor at the orbital radius is $
f_V = \frac{\rho_h}{\rho_{cl}} =
\left(\frac{10~\rm{d}}{T}\frac{t_{fl}}{3~\rm{ks}}\right)
~0.02$ and the corresponding porosity length is
$h=\frac{R_{cl}}{f_V}=
\left(\frac{T}{10~\rm{d}}\right)
~15\times 10^{12} ~\rm{cm}$.
If the density of a clump decreases with radius as $r^{-2\beta}$ and its mass remains constant, the averaged homogeneous wind density within $R_{obs}$ is $\overline{\rho_{h}}=N M_{cl}/(\frac{4}{3}\pi
R_{orb}^3
) =
\left(\frac{10~\rm{d}}{T}\frac{L_X}{10^{36}~\rm{erg/s}}\frac{t_{fl}}{3~\rm{ks}}\right)
~7\times 10^{-15} ~\rm{g~cm}^{-3}$ and the average clump volume filling factor and porosity length could be estimated as 0.1 and $3\times10^{12} ~\rm{cm}$, respectively.
The variety of $t_{fl}$, $T$ and $F_{fl}$ that are observed probably reflects a range of clump parameters and orbital radii. Several of the average clump parameters estimated above, in particular the clump density, filling factor and porosity length do not depend on the orbital radius, which is unknown, and only slowly depend on the observed quantities.
These average parameters match the macro-clumping scenario of \cite{walter:OskinovaHamannFeldmeier2007}
to reconcile clumping and mass-loss rates.
The number of clumps derived above is also comparable to evaluations by \cite{walter:Lepine1999, walter:OskinovaFeldmeierHamann2006}. The volume filling factor, porosity length and the clump mass-loss rate are also similar to those derived by \cite{walter:Bouret2005} from the study of ultraviolet and optical line profiles in two super-giant stars.
The column density through a clump can also be estimated as $N_H = \frac{M_{cl}}{R_{cl}^2m_p}=
\left(\frac{L_X}{10^{36}\rm{erg/s}}\frac{t_{fl}}{3\rm{ks}}\right)
~5\times 10^{22}\rm{cm}^{-2}$. The clumps remain optically thin in the X-rays.
\paragraph{Inter-clump medium}
\paragraph{}
\vspace{-0.3cm}
The variation of the observed X-ray flux between flares and quiescence provides in principle a direct measure of the density constrast between the wind clumps and the inter-clump medium.
Density contrasts of $>10^{2-4}$ and 15--50 have been observed in SFXT and ``Intermediate'' sources, respectively. The density contrast is larger in SFXT than in ``Intermediate'' and, of course, classical systems. Density contrasts are probably stronger when clumping is very effective.
Numerical simulations of the line driven instability (\cite{walter:Runacres2005}) predict density contrasts as large as $10^{3-5}$ in the wind up to large radii. At a distance of $10~R_*$, the simulated density can vary between $10^{-18}$ and $10^{-13}~\rm{g~cm^{-3}}$ and the separation between the density peaks are of the order of $R_*$. These characteristics are comparable to the values we have derived.
\paragraph{What about classical sgHMXB ?}
\paragraph{}
\vspace{-0.3cm}
Classical sgHMXB are characterized by small orbital radii $R_{orb}=(1.5-2.5)~R_*$, and by flux variability of a factor $\lesssim10$. Such variabilities were modelled in terms of wind inhomogeneities largely triggered by the hydrodynamic and photo-ionisation effects of the accreting object on the companion and inner stellar wind (\cite{walter:blondin91, walter:blondin94}). At small orbital radii, the companion is close to fill its Roche lobe, which triggers tidal streams. In addition the X-ray source ionizes the wind acceleration zone, prevents wind acceleration and generates slower velocities, denser winds, larger accretion radius and finally larger X-ray luminosities. Whether or not the stellar wind is intrinsically clumpy at low radius, the effect of the compact object on the wind is expected to be important.
The main difference between SFXT and classical sgHMXB could therefore be their orbital radius (\cite{walter:Leyder2007}). At very low orbital radius $(<1.5~R_*)$ tidal accretion will take place through an accretion disk and the system will soon evolve to a common envelope stage. At low orbital radius $(\sim 2~R_*)$ the wind will be perturbed in any case and efficient wind accretion will lead to copious and persistent X-ray emission $(10^{36-37}~\rm{erg/s})$. At larger orbital radius $(\sim 10~R_*)$ and if the wind is clumpy, the SFXT behavior is expected as described above. If the wind clumps do not form for any reason, the average accretion rate will remain too low and the sources will remain mostly undetected by the current hard X-ray survey instruments.
|
1,108,101,565,428 | arxiv | \section{Introduction}
Advancements in modern deep learning have resulted in substantial progress in our ability to model, reconstruct, and forecast complex time-series data, most notably through recurrent neural networks (RNNs) and neural ordinary differential equations (neural ODEs) \cite{chen2019neuralODE,rubanova2019latentODE,rangapuram2018deep,li2021learning}. Many of these techniques have been successfully
adapted to model the spatiotemporal dynamics of complex physiological systems, such as in reconstructing \cite{jiang2021label,ghimire2018generative} or forecasting the electrical activity of the cardiac system \cite{EPNET2021}.
While strong in modeling general system dynamics, however, these methods are not designed to uncover local activity with minute signal strength compared to the global dynamics. Such local activities are common and often signify important abnormal events in physiological systems, such as an extra foci that triggers an abnormal propagation of electrical waves amid normal propagation patterns (Figure \ref{fig:normalProp}).
Due to negligible optimization loss contribution given weak signal strength, reconstructing these local activities remains challenging despite deep learning advances in time-series modelling.
We present a fundamentally novel perspective of this problem: instead of general reconstruction, we stress that these local activities, while small in signal strength, are the cause of the subsequent global activities that have larger signal strength. Therefore,
we may be able to disentangle the \textit{cause} and \textit{effect} of the latent dynamics
by explicitly modeling and inferring how the latent state of a system is influenced by potential hidden internal \textit{interventions}.
Modeling the effect on system dynamics by exogenous inputs is well-studied in classic state-space systems (SSMs), and has seen recent successes in neural-network modeling of SSMs \cite{krishnan2015deep,de2019gruODEBayes,karl2016deep,gwak2020imode}.
These works, however, focus on external interventions that are either known or directly observed, whereas we consider hidden and internal causes of system's abnormal events.
As these internal causes are not directly observable, we propose to disentangle them from the observed effects on the system's dynamics. Rather than learning the complex intervened dynamics of a system from scratch as in existing works \cite{gwak2020imode}, we incorporate knowledge of the native \textit{intervention-free} dynamics of a system and focus only on learning how it is influenced by potential hidden interventions.
To this end we present a novel neural SSM with two major innovations. First, drawing inspirations from related work in intervention modeling \cite{gwak2020imode}, we introduce causal-effect modeling of the latent dynamics via a system of interacting neural ODEs that separately describe 1) the continuous-time dynamics of the internal intervention, and 2) its effect on the trajectory of the system's native state. Second, to disentangle the cause and effect from their collective observations, we 1) leverage data of intervention-free systems to pre-train a neural ODE of \textit{native} dynamics and integrate it into the intervention ODEs, and 2) infer the hidden intervention by assuming it to be responsible for differences observed between the actual and hypothetical \textit{intervention-free} dynamics at each time frame.
We demonstrated a proof-of-concept of the presented framework on reconstructing ectopic foci that disrupt normal electrical propagation using remote observations. We compared the presented method to the two most relevant time-series modeling approaches:
1) a global neural ODE to describe latent system dynamics,
and 2) a neural SSM with a latent neural ODE lacking intervention modeling, developed for reconstructing cardiac electrical propagation \cite{jiang2021label}.
Experiments were conducted on synthetic data with controlled \textit{internal interventions}, simulated in 2D settings.
Our results indicated that the presented method delivers more accurate inverse estimations in terms of localizing the triggering events.
\section{Methodology}
We formulate a neural SSM where the system dynamics is described on a lower-dimensional latent manifold in separation from its emission to the data space as illustrated in Fig.~\ref{fig:overview}.
Consider cardiac electrical propagation $\mathbf{x_t}$ and its body-surface measurement $\mathbf{Y_t}$ with a physics-based relation $\mathbf{Y_t} = \mathbf{H}\mathbf{x}_t$.
We enable causal-effect intervention modelling by sequentially learning two separate neural ODE functions, $\mathcal{F}_z$ and $\mathcal{F}_a$, that describe \textit{native} system dynamics and causal dynamics, respectively. The effect of intervention $\mathbf{a}$ on $\mathbf{z}$ is modelled through a coupled neural ODE function, $\mathcal{F}_{z, a}$.
Because intervention $\mathbf{a}$ is not directly observable,
it is not possible to separately learn $\mathcal{F}_z$, $\mathcal{F}_a$ and $\mathcal{F}_{z, a}$ all from scratch from only their collective observations.
Instead, we poise that we must leverage knowledge of native dynamics $\mathcal{F}_z$, in order to
be able to generate \textit{hypothetical observation}
$\hat{\mathbf{Y}}_t = \mathbf{H} \hat{\mathbf{x}}_t$
assuming intervention-free dynamics since last observation $\mathbf{Y}_{t-1}$. Then using the residual between the actual observation $\mathbf{Y}_t$ and hypothetical observation
$\hat{\mathbf{Y}}_t$ allows us to model and estimate intervention $\mathbf{a}_t$.
A two-stage optimization process is leveraged: First, the \textit{native} dynamics of the system are learned on an \textit{intervention-free} subset of data (termed \textit{ODE-VAE}) which is then statically integrated into a system of intervention ODEs optimized on the intervention set (termed \textit{ODE-VAE-IM}).
\begin{figure}[!tb]
\label{fig:overview}
\begin{center}
\includegraphics[width=\textwidth]{figures/overviewUpdated.png}
\caption{A) Graphical overview of the proposed network. Inputs \textbf{H} and $\mathbf{Y}_{0:T}$ are given to the intervention dynamics $\mathcal{F}_{a}$ and influence the propagation of $\mathcal{F}_{(z,a)}$. B) Schematic of a single intervention step, showing the estimation of the intervention latent variable $\mathbf{a}_k$ from data-space observations and \textit{hypothetical intervention-free observations}.
}
\label{fig:overview}
\end{center}
\end{figure}
In sections \ref{sec:naiveDynamics}-\ref{sec:dynamics} we describe the native dynamics model, the predictive model enabling causal-effect learning, the estimation of the latent intervention variable through global observations, and the model optimization loop.
\subsection{Modeling and Learning Intervention-Free Native Dynamics} \label{sec:naiveDynamics}
An ODE function $\mathcal{F}_z$ is designed to learn the \textit{native} dynamics function that handles the propagation of the system in the absence of interventions. Since we leverage this function to hypothesize the intervention-free native state of a system given at any previous system state, it must have a strong long-term forecasting ability. Therefore, inspired by \cite{yildiz2019ode2vae}, 1.) an encoding network $Enc_z$ initializes the latent ODEs vector field $\mathbf{z}_0$ using the first few frames of sequential input $\mathbf{Y}_{0:k}$ (Eq.\ref{eqn:z_ode_1}), 2.) an initial value problem is solved up to time $\mathbf{T}$ using the ODE function $\mathcal{F}_z$ (Eq.\ref{eqn:z_ode_2}), and 3.) a decoding network $Dec_z$ converts the latent trajectory $\mathbf{z}_{0:T}$ to the output space $\mathbf{X}_{0:T}$ (Eq.\ref{eqn:z_ode_3}):
\begin{align}
\mathbf{z}_0 &= Enc_z(\mathbf{Y}_{0:k})
\label{eqn:z_ode_1} \\
\mathbf{z}_i &= \mathbf{z}_{i-1} + \int_{t_{i-1}}^{t_i} \mathcal{F}_{z}(\mathbf{z}_\tau) d\tau
\label{eqn:z_ode_2} \\
\hat{\mathbf{x}}_i &= Dec_z(\mathbf{z}_i),
\label{eqn:z_ode_3}
\end{align}
where $Enc_z$ and $Dec_z$ are represented by CNNs and the sequence input length $k$ is empirically tuned.
We train this native dynamics model on datasets of \textit{intervention-free} systems.
After initializing the latent state, the full trajectory is predicted using $\mathcal{F}_{z}$, from which the reconstructed $\hat{\mathbf{x}}$
is compared with the ground truth in the loss:
\begin{align}
\mathcal{L}_{native}(\mathbf{x}_{0:T}, \hat{\mathbf{x}}_{0:T}) &=
\beta [(\mathbf{x}_0 * log(\hat{\mathbf{x}}_0)) + (1 - \hat{\mathbf{x}}_0) * log(1 - \hat{\mathbf{x}}_0)] \\
&+ \frac{1}{T} \sum_{i=1}^T (\mathbf{x}_i * log(\hat{\mathbf{x}}_i) + (1 - \hat{\mathbf{x}_i}) * log(1 - \hat{\mathbf{x}}_i),
\end{align}
where binary cross entropy (BCE) is applied to each frame and $\beta$ represents a loss weighting coefficient. We explicitly separate out the reconstruction of $\mathbf{x}_0$ to emphasize the accuracy of the initial conditions, $\mathbf{z}_0$, independent of the accuracy of the dynamic transition models. Because the network is asked to reconstruct the full sequence of $\mathbf{x}_{0:T}$ from only an estimated initial condition $\mathbf{z}_0$, it promotes learning strong, continuous dynamics models $\mathcal{F}_{z}$ that can simulate the system given any initial condition of the space.
This function, once trained, will be incorporated into the intervention-effects function $\mathcal{F}_{(z, a)}$ as a static dynamics function whose trajectory is influenced by the intervention state.
\subsection{Modelling and Learning Intervention Dynamics} \label{sec:dynamics}
The system dynamics predictive model is described by two ODE functions, $\mathcal{F}_a$ and $\mathcal{F}_{(z, a)}$, that aim to separately estimate the cause and effect of interventions in the latent space, respectively. To achieve this, we use the state-space Bayesian filtering setting where the system propagation undergoes a series of predict-from-dynamics and update-from-observations steps \cite{karl2016deep,fraccaro2017disentangled,de2019gruODEBayes}. The former comprises of the predictive model which, given a current state $\mathbf{s}_{t-1}$, predicts the next temporal state, $\hat{\mathbf{s}}_{t}$, using the transition function $\mathcal{F}$. The latter corrects the temporal prediction through an update function, $\mathcal{G}$, using an inferred state $\mathbf{s}_{enc}(t)$ from the current observation $\mathbf{Y}_i$.
\textbf{Predictive System Dynamics Model.}
$\mathcal{F}_a$ handles the prediction of the intervention state, $a_t$, while $\mathcal{F}_{(z, a)}$ handles the intervention-influenced prediction of the latent state $z_t$, denoted by the equations in Eq.~\ref{eqn:combined}, respectively.
\begin{align}
\mbox{\textbf{Prediction:} }
\begin{bmatrix}
\hat{\mathbf{a}}_i \\
\hat{\mathbf{z}}_i
\end{bmatrix}
&=
\begin{bmatrix}
\mathbf{a}_{i-1} \\
\mathbf{z}_{i-1}
\end{bmatrix}
+ \int_{t_{i-1}}^{t_i}
\begin{bmatrix}
\mathcal{F}_{a}(\mathbf{a}_\tau) \\
\mathcal{F}_{z}(\mathbf{z}_\tau) + \mathcal{F}_{a}(\mathbf{a}_\tau)
\end{bmatrix}
d\tau \label{eqn:combined}
\end{align}
\textbf{Estimating the Intervention Latent Variable.}
Different from Section 2.1 wherein our goal is to learn a native dynamic model that has strong standalone forecasting ability,
here our interest is to extract information from available observations at each time frame to uncover hidden interventions.
A dynamical function describing the temporal cause of intervention alone is not enough to produce a meaningful causal-effect model. Rather, an observational source from which the latest intervention information can be inferred is required \cite{gwak2020imode,yin2021augmentingaphyn}. We hypothesize that the latent variable associated with the interventions rises from differences observed between data space observations and \textit{hypothetical observations} continuing under \textit{intervention-free} dynamics at each timestep. For inverse image reconstruction problems, this can be formulated as the element-wise difference in the inverse solution difference of $H\mathbf{\hat{x}}^\mathbf{z}_t$ and $\mathbf{Y}_t$. This difference is then used as input to an encoder to output the estimated latent intervention state.
\begin{align}
\mbox{\textbf{Update:} } \mathbf{a}_{enc} &= Enc_a(|| \mathbf{H}\hat{\mathbf{x}_i} - \mathbf{Y}_i ||_{i:(i+k)}) \label{eqn:pienc} \\
\mathbf{a}_i &= \mathcal{G}_a(\hat{\mathbf{a}}_{i}, \mathbf{a}_{enc}) \label{eqn:z_ode}
\end{align}
We make use of the Gated Recurrent Unit (GRU) cell \cite{cho2014learningGRU}, $\mathcal{G}_a$ to learn a weighted combination of the encoded intervention variable and the prediction coming from the intervention dynamics function.
A number of methods may be applied to improve the forward intervention estimation. One such choice, and the one utilized in this work, is to predict a number of timesteps ahead and get the inverse solution difference over a sequence of frames rather than just the current predicted frame. This provides increased temporal information to the intervention encoder by exposing the longer-term differences that would arise from the current state.
\textbf{Optimizing \textit{Intervention} Models.}
The loss function is the data reconstruction loss averaged over each time frame:
\begin{align}
\mathcal{L}_{intv}(\mathbf{X}_{0:T}, \hat{\mathbf{X}}_{0:T}) &= \frac{1}{N * T} \sum_{n=0}^N \sum_{i=0}^T (\mathbf{x}_{n,i} * log(\hat{\mathbf{x}}_{n,i}) + (1 - \hat{\mathbf{x}_{n,i}}) * log(1 - \hat{\mathbf{x}}_{n,i}),
\end{align}
where BCE is applied to each time frame. As we are no longer care about the solution of an initial value problem, the emphasis on $\mathbf{x}_0$ reconstruction is lifted.
\section{Experiments}
For the task of reconstructing ectopic foci that disrupt normal electrical propagation using remote observations, we include three baseline methods: \textit{ECGI}.) a first order Tikhonov regularization for reconstruction, \textit{ODE-VAE}.) the direct application of the native dynamics network to the given intervention set, and \textit{ODE-VAE-GRU}.) a Bayesian filtering ablation where a latent update mechanism (GRU) is applied at every timestep using the next 3 observations straight as input. This represents a model optimized on native and intervention dynamics together without clear disentanglement or ability to preserve \textit{intervention-free} dynamics. We denote the proposed method as \textit{ODE-VAE-IM} throughout the experiments. The observation data $\textbf{X}_{0:T}$ is passed through a Bernoulli filter between activated and deactivated nodes following min-max normalization and all network decoders have a Sigmoid activation on the output images. The intervention variable encoder (\textit{$Enc_a$}) has non-activated outputs.
Initial learning rates were found using the cyclical learning rate estimation technique \cite{smith2017cyclical} and are decayed by $0.5$ at set points throughout training. We used AdamW optimizer \cite{loshchilov2017decoupled} with a weight decay of $1e-2$, a batch size of 16, and a latent dimension of 12 across all methods. All experiments were run on NVIDIA Tesla T4s with 16GB memory, taking $\sim$8 hours to train. Our implementation and saved models are available at \url{https://github.com/qu-gg/causal-effect-neural-ssm}, along with more examples in Supplementary Material.
\textbf{Data Generation.} Following the Fitzhugh-Nagumo model \cite{izhikevich2006fitzhugh}, we simulated the transmembrane potentials on a 100*100mm 2D grid. This synthetic dataset includes two subsets 1.) native transmembrane potential set containing 1000 voltage maps, where the initial excitation location is chosen randomly across the grid and 2.) transmembrane potentials in the presence of an extra focci. Simulating the setting where the extra Foci intervenes in the normal dynamics, we generated a dataset of 705 samples with varying initial excitation location and time and location of the extra Foci.
\begin{figure}[tb]
\centering
\includegraphics[width=\textwidth]{figures/PacingSet.png}
\caption{Synthetic TMP per-step recons in which additional excitations occur.}
\label{fig:dualProp}
\end{figure}
\subsection{Dynamics Reconstruction}
In this section, we showcase the reconstructive and temporal capabilities of the method on normal and intervention dynamics.
\textbf{Normal Dynamics.}
We first highlight the \textit{naive} dynamics model's performance on the non-intervention datasets used for pre-training. To ensure that a robust vector field is trained, our base dataset combines both single and double excitation propagations that occur within the first few frames. We posit that by introducing more propagation patterns in the base dynamics, even if they are not used in intervention, helps the ODE function and emission model to handle unstable vector state jumps. Figure~\ref{fig:normalProp} highlights two samples of per-step reconstructions, showcasing that strong dynamics are established.
\begin{figure}[tb]
\centering
\includegraphics[width=\textwidth]{figures/normalRecons.png}
\caption{Reconstruction of electrical propagation in which no interventions (foci) occur.}
\label{fig:normalProp}
\end{figure}
\textbf{Ectopic Foci.}
Figure~\ref{fig:dualProp} highlights per-timestep reconstructions of a single sample across all methods. \textit{ODE-VAE}, the base dynamics model with no means of intervention, manages to reconstruct a propagated waveform of the \textit{native} dynamics. Leveraging this fixed native dynamics function, the presented \textit{ODE-VAE-IM} is able to uncover the hidden intervention dynamics.
\begin{figure}[tb]
\centering
\includegraphics[width=\textwidth]{figures/pacingRecons.png}
\caption{Reconstruction of electrical propagation in which ectopic foci occurs.}
\label{fig:dualProp}
\end{figure}
\subsection{Localization Results}
To gauge the proposed method's numeric performance, we performed a localization test in which the time and location of interventional foci were identified from the reconstruction results across a subset of 100 samples. The mean absolute error (MAE) in time frames, the Euclidean distance in location, and the percentage of identified foci are used for quantitative metrics. Table~\ref{tab:metrics} showcases these results, in which the ability of \textit{ODE-VAE-IM} in identifying the presence of abnormal foci is highlighted.
\begin{table}[!tb]
\centering
\caption{Comparison in identifying the intervention foci's location and activation timestep. Identification percentage represents cases with clear foci reconstruction.}
\label{tab:metrics}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Model} & \textbf{\% Foci Identified} & \textbf{Timestep Activation MAE} & \textbf{Foci Location Error} \\
\hline
ODE-VAE-IM & 0.85 & 3.56 & 54.90 \\
ODE-VAE-GRU & 0.79 & 3.92 & 53.71 \\
ODE-VAE & 0.39 & 3.07 & 57.28 \\
ECGI & 0.52 & 4.10 & 52.35 \\
\hline
\end{tabular}
\end{table}
\subsection{Latent Norm Ablation}
We performed an ablation study on the latent dynamics of the cause and effect by visualizing the $L^2$-Norm of their respective latent vector states over time for two examples in Figure~\ref{fig:latentNorms}. Temporal windows are highlighted in which the intervention dynamics correctly capture the windows of starting and ending intervention activity. To facilitate comparisons between each latent space, we first perform min-max normalization over the $L^2$-Norms of each latent space to bring them within the same data range.
\begin{figure}[!tb]
\centering
\includegraphics[width=\textwidth]{figures/latentNorms.png}
\caption{Visualizations of the L2-Norm of system and intervention states over time.}
\label{fig:latentNorms}
\end{figure}
\section{Conclusion}
In this work, we propose a proof-of-concept interventional modelling framework to tackle the problem of low-strength local activity that signifies important abnormal triggering events in a dynamic system. We introduce a two-ODE system that separately models the cause and effect of system's latent state under the influence of hidden internal interventions. To this end we leverage pre-trained dynamical functions describing \textit{intervention-free} native dynamics of a system. We demonstrated the frameworks performance on reconstruction of the ectopic foci causing abnormality in cardiac electrical activity.
\textbf{Future Work.} Future work aims to extend this framework into 3-D inverse reconstruction and pacing localization using graph neural networks to enable experimentation on clinical data, following \cite{jiang2020learning,jiang2021label}. One promising direction is learning multi-interventional dynamics simultaneously via attention mechanisms.
\textbf{Limitations.} The transition between training $\mathcal{F}_{z}$ and $\mathcal{F}_{a}$ has a period of re-training up to the original performance, which identity-based initialization or loss-influence annealing $\mathcal{F}_{a}$ may alleviate. $Enc_z$ has potential generalization problems when the initial frame distributions shift between native and intervention sets and requires experimentation on its general training setup.
\bibliographystyle{splncs04}
|
1,108,101,565,429 | arxiv | \section{Parton--parton scattering amplitudes}
The parton--parton elastic scattering amplitude, at high squared energies $s$
in the center of mass and small squared transferred momentum $t$ (that is
$s \to \infty$ and $|t| \ll s$, let us say $|t| \le 1~{\rm GeV}^2$),
can be described by the expectation value of two {\it infinite lightlike}
Wilson lines, running along the classical trajectories of the colliding
particles \cite{Nachtmann91,Meggiolaro96,Meggiolaro01}.
However, this description is affected by infrared (IR) divergences, which
are typical of $3 + 1$ dimensional gauge theories.
One can regularize this IR problem by letting the Wilson lines coincide with
the classical trajectories for partons with a non--zero mass $m$
(so forming a certain {\it finite} hyperbolic angle $\chi$ in Minkowskian
space--time: of course, $\chi \to \infty$ when $s \to \infty$),
and, in addition, by considering {\it finite} Wilson lines, extending in
proper time from $-T$ to $T$ (and eventually letting $T \to +\infty$)
\cite{Verlinde,Meggiolaro02}. For example, the high--energy quark--quark
elastic scattering amplitude ${\cal M}_{fi}$ is (explicitly indicating the
colour indices $i,j$ [initial] and $i',j'$ [final] and the spin indices
$\alpha,\beta$ [initial] and $\alpha',\beta'$ [final] of the colliding quarks):
\begin{equation}
{\cal M}_{fi} \mathop{\sim}_{s \to \infty}
-i~ 2s~ \delta_{\alpha'\alpha} \delta_{\beta'\beta}~
g_M (\chi \to \infty;~T \to \infty;~t) ,
\label{scatt}
\end{equation}
\begin{equation}
g_M (\chi;~T;~t) \equiv {1 \over [Z_W(T)]^2}
\displaystyle\int d^2 \vec{z}_\perp e^{i \vec{q}_\perp \cdot \vec{z}_\perp}
\langle [ W^{(T)}_1 (\vec{z}_\perp) - {\bf 1} ]_{i'i}
[ W^{(T)}_2 (\vec{0}_\perp) - {\bf 1} ]_{j'j} \rangle ,
\label{gM}
\end{equation}
where $t = -|\vec{q}_\perp|^2$, $\vec{q}_\perp$ being the tranferred momentum,
and $\vec{z}_\perp = (z^2,z^3)$ is the distance between the two trajectories
in the {\it transverse} plane ({\it impact parameter}).
The two IR--regularized Wilson lines are defined as:
\begin{eqnarray}
W^{(T)}_1 (\vec{z}_\perp) &\equiv&
{\cal T} \exp \left[ -ig \displaystyle\int_{-T}^{+T}
A_\mu (z + {p_1 \over m} \tau) {p_1^\mu \over m} d\tau \right] ,
\nonumber \\
W^{(T)}_2 (\vec{0}_\perp) &\equiv&
{\cal T} \exp \left[ -ig \displaystyle\int_{-T}^{+T}
A_\mu ({p_2 \over m} \tau) {p_2^\mu \over m} d\tau \right] ,
\label{lines}
\end{eqnarray}
where ${\cal T}$ stands for ``{\it time ordering}'' and $A_\mu = A_\mu^a T^a$;
$z = (0,0,\vec{z}_\perp)$;
$p_1 = E(1,\beta,0,0)$ and $p_2 = E(1,-\beta,0,0)$ are the
initial four--momenta of the two quarks [$s \equiv (p_1 + p_2)^2
= 2 m^2 ( \cosh \chi + 1 )$].
Finally, $Z_W(T)$ is a sort of Wilson--line's renormalization constant:
\begin{equation}
Z_W(T) \equiv {1 \over N_c} \langle {\rm Tr} [ W^{(T)}_1 (\vec{0}_\perp) ] \rangle
= {1 \over N_c} \langle {\rm Tr} [ W^{(T)}_2 (\vec{0}_\perp) ] \rangle .
\label{ZW}
\end{equation}
The expectation values $\langle W_1 W_2 \rangle$, $\langle W_1 \rangle$,
$\langle W_2 \rangle$ are averages in the sense of the QCD functional
integrals:
\begin{equation}
\langle {\cal O}[A] \rangle =
{1 \over Z} \displaystyle\int [dA] \det(Q[A]) e^{iS_A} {\cal O}[A] ,
\end{equation}
where $Z = \displaystyle\int [dA] \det(Q[A]) e^{iS_A}$ and $Q[A]$ is the
{\it quark matrix}.
The quantity $g_M (\chi;~T;~t)$ with $\chi > 0$ can be reconstructed from the
corresponding Euclidean quantity $g_E (\theta;~T;~t)$, defined as a (properly
normalized) correlation function of two (IR--regularized) Euclidean Wilson
lines $\tilde{W}_1$ and $\tilde{W}_2$, i.e.,
\begin{eqnarray}
g_E (\theta;~T;~t) &\equiv& {1 \over [Z_{WE}(T)]^2}
\displaystyle\int d^2 \vec{z}_\perp e^{i \vec{q}_\perp \cdot \vec{z}_\perp}
\langle [ \tilde{W}^{(T)}_1 (\vec{z}_\perp) - {\bf 1} ]_{i'i}
[ \tilde{W}^{(T)}_2 (\vec{0}_\perp) - {\bf 1} ]_{j'j} \rangle_E ,\nonumber \\
Z_{WE}(T) &\equiv&
{1 \over N_c} \langle {\rm Tr} [ \tilde{W}^{(T)}_1 (\vec{0}_\perp) ] \rangle_E
= {1 \over N_c} \langle {\rm Tr} [ \tilde{W}^{(T)}_2 (\vec{0}_\perp) ] \rangle_E ,
\label{gE}
\end{eqnarray}
where:
\begin{eqnarray}
\langle {\cal O}[A^{(E)}] \rangle_E &=&
{1 \over Z^{(E)}} \displaystyle\int [dA^{(E)}] \det(Q^{(E)}[A^{(E)}])
e^{-S^{(E)}_A} {\cal O}[A^{(E)}] ,\nonumber \\
Z^{(E)} &=& \displaystyle\int [dA^{(E)}] \det(Q^{(E)}[A^{(E)}]) e^{-S^{(E)}_A} ,
\end{eqnarray}
$\theta \in ]0,\pi[$ being the angle formed by the two trajectories in
the Euclidean four--space, by an analytic continuation in the angular
variables and in the IR cutoff \cite{Meggiolaro02,Meggiolaro97,Meggiolaro98}:
\begin{eqnarray}
g_E (\theta;~T;~t) &=& g_M (\chi \to i\theta;~T \to -iT;~t) ,
\nonumber \\
g_M (\chi;~T;~t) &=& g_E (\theta \to -i\chi;~T \to iT;~t) .
\label{original}
\end{eqnarray}
This result is derived under the assumption that the function $g_M$, as a
function of the {\it complex} variable $\chi$, is {\it analytic} in a
domain ${\cal D}_M$ which includes the positive real axis $({\rm Re}\chi > 0,
{\rm Im}\chi = 0)$ and the imaginary segment $({\rm Re}\chi = 0, 0 < {\rm Im}\chi < \pi)$;
and, therefore, the function $g_E$, as a
function of the {\it complex} variable $\theta$, is {\it analytic} in a
domain ${\cal D}_E = \{ \theta \in {\bf C} ~|~ i\theta \in {\cal D}_M \}$,
which includes the real segment $(0 < {\rm Re}\theta < \pi, {\rm Im}\theta = 0)$ and the
negative imaginary axis $({\rm Re}\theta = 0, {\rm Im}\theta < 0)$.
The validity of this assumption is confirmed by explicit calculations in
perturbation theory \cite{Meggiolaro97}.
Eq. (\ref{original}) is then intended to be valid for every
$\chi \in {\cal D}_M$ (i.e., for every $\theta \in {\cal D}_E$).
The above--reported relations allow to give a nice geometrical interpretation
of the so--called {\it crossing symmetry}. Changing from a {\it quark} to an
{\it antiquark} just corresponds, in our formalism, to substitute the
corresponding Wilson line with its complex conjugate, i.e., to reverse
the orientation of the Wilson line (and the colour indices):
\begin{equation}
[W_p^*(\vec{b}_\perp)]_{lk} = [W_p^\dagger(\vec{b}_\perp)]_{kl} =
[W_{-p}(\vec{b}_\perp)]_{kl} .
\end{equation}
Changing {\it quark} nr. 2 into an {\it antiquark} corresponds, in the
Euclidean theory, to the substitution:
\begin{equation}
\theta \to \theta_2 = \pi - \theta ,
\end{equation}
and therefore, in the Minkowskian theory:
\begin{equation}
\chi \to \chi_2 = i\pi - \chi .
\end{equation}
We thus find the following {\it crossing--symmetry} relation:
\begin{equation}
g_M^{(q\bar{q})} (\chi;~T;~t)_{i'i,lk} = g_M (i\pi - \chi;~T;~t)_{i'i,kl} .
\end{equation}
[We must assume that the domain ${\cal D}_M$ also includes the half--line
$({\rm Re}\chi < 0, {\rm Im}\chi = \pi)$.]
We close this section remarking that the {\it regularized} quantities
$g_M(\chi;~T;~t)$ and $g_E(\theta;~T;~t)$, while being finite at any given
value of $T$, are divergent in the limit $T \to \infty$. In some cases this
IR divercence can be factorized out and one thus ends up with an IR--finite
(physical) quantity.
\section{Loop--loop scattering amplitudes}
Differently from the parton--parton scattering amplitudes, which are known to
be affected by infrared (IR) divergences, the elastic scattering amplitude of
two colourless states in gauge theories, e.g., two $q \bar{q}$ meson states,
is expected to be an IR--finite physical quantity.
It was shown in Refs. \cite{Nachtmann97,Dosch,Berger} that the high--energy
meson--meson elastic scattering amplitude can be approximately reconstructed
by first evaluating, in the eikonal approximation, the elastic scattering
amplitude of two $q \bar{q}$ pairs (usually called ``{\it dipoles}''), of
given transverse sizes $\vec{R}_{1\perp}$ and $\vec{R}_{2\perp}$ respectively,
and then averaging this amplitude over all possible values of
$\vec{R}_{1\perp}$ and $\vec{R}_{2\perp}$ with two proper squared
wave functions $|\psi_1 (\vec{R}_{1\perp})|^2$ and
$|\psi_2 (\vec{R}_{2\perp})|^2$, describing the two interacting mesons.
The high--energy elastic scattering amplitude of two {\it dipoles} is
governed by the (properly normalized) correlation function of two Wilson loops
${\cal W}_1$ and ${\cal W}_2$, which follow the classical straight lines for
quark (antiquark) trajectories:
\begin{equation}
{\cal M}_{(ll)} (s,t;~\vec{R}_{1\perp},\vec{R}_{2\perp}) \equiv
-i~2s \displaystyle\int d^2 \vec{z}_\perp
e^{i \vec{q}_\perp \cdot \vec{z}_\perp}
\left[ {\langle {\cal W}_1 {\cal W}_2 \rangle \over
\langle {\cal W}_1 \rangle \langle {\cal W}_2 \rangle} -1 \right] ,
\label{scatt-loop}
\end{equation}
where $s$ and $t = -|\vec{q}_\perp|^2$ ($\vec{q}_\perp$ being the tranferred
momentum) are the usual Mandelstam variables.
More explicitly the Wilson loops ${\cal W}_1$ and ${\cal W}_2$ are so defined:
\begin{eqnarray}
{\cal W}^{(T)}_1 &\equiv&
{1 \over N_c} {\rm Tr} \left\{ {\cal P} \exp
\left[ -ig \displaystyle\oint_{{\cal C}_1} A_\mu(x) dx^\mu \right] \right\} ,
\nonumber \\
{\cal W}^{(T)}_2 &\equiv&
{1 \over N_c} {\rm Tr} \left\{ {\cal P} \exp
\left[ -ig \displaystyle\oint_{{\cal C}_2} A_\mu(x) dx^\mu \right] \right\} ,
\label{QCDloops}
\end{eqnarray}
where ${\cal P}$ denotes the ``{\it path ordering}'' along the given path
${\cal C}$; ${\cal C}_1$ and ${\cal C}_2$ are two rectangular paths which
follow the classical straight lines for quark [$X_{(+)}(\tau)$, forward in
proper time $\tau$] and antiquark [$X_{(-)}(\tau)$, backward in $\tau$]
trajectories, i.e.,
\begin{eqnarray}
{\cal C}_1 &\to&
X_{(\pm 1)}^\mu(\tau) = z^\mu + {p_1^\mu \over m} \tau
\pm {R_1^\mu \over 2} , \nonumber \\
{\cal C}_2 &\to&
X_{(\pm 2)}^\mu(\tau) = {p_2^\mu \over m} \tau \pm {R_2^\mu \over 2} ,
\label{traj}
\end{eqnarray}
and are closed by straight--line paths at proper times $\tau = \pm T$, where
$T$ plays the role of an IR cutoff, which must be removed at the end
($T \to \infty$).
Here $p_1$ and $p_2$ are the four--momenta of the two quarks and of the two
antiquarks with mass $m$, moving with speed $\beta$ and $-\beta$ along, for
example, the $x^1$--direction:
\begin{eqnarray}
p_1 &=& m (\cosh {\chi \over 2},\sinh {\chi \over 2},0,0) ,
\nonumber \\
p_2 &=& m (\cosh {\chi \over 2},-\sinh {\chi \over 2},0,0) ,
\label{p1p2}
\end{eqnarray}
where $\chi = 2~{\rm arctanh} \beta > 0$ is the hyperbolic angle between the
two trajectories $(+1)$ and $(+2)$.
Moreover, $R_1 = (0,0,\vec{R}_{1\perp})$, $R_2 = (0,0,\vec{R}_{2\perp})$
and $z = (0,0,\vec{z}_\perp)$, where $\vec{z}_\perp = (z^2,z^3)$ is the
impact--parameter distance between the two loops in the transverse plane.
It is convenient to consider also
the correlation function of two Euclidean Wilson loops
$\tilde{\cal W}_1$ and $\tilde{\cal W}_2$ running along two rectangular paths
$\tilde{\cal C}_1$ and $\tilde{\cal C}_2$ which follow the following
straight--line trajectories:
\begin{eqnarray}
\tilde{\cal C}_1 &\to&
X^{(\pm 1)}_{E\mu}(\tau) = z_{E\mu} + {p_{1E\mu} \over m}
\tau \pm {R_{1E\mu} \over 2} , \nonumber \\
\tilde{\cal C}_2 &\to&
X^{(\pm 2)}_{E\mu}(\tau) = {p_{2E\mu} \over m} \tau
\pm {R_{2E\mu} \over 2} ,
\label{trajE}
\end{eqnarray}
and are closed by straight--line paths at proper times $\tau = \pm T$. Here
$R_{1E} = (0,\vec{R}_{1\perp},0)$, $R_{2E} = (0,\vec{R}_{2\perp},0)$ and
$z_E = (0,\vec{z}_\perp,0)$. Moreover, in the Euclidean theory we {\it choose}
the four--vectors $p_{1E}$ and $p_{2E}$ to be:
\begin{eqnarray}
p_{1E} &=& m (\sin{\theta \over 2}, 0, 0, \cos{\theta \over 2} ) , \nonumber \\
p_{2E} &=& m (-\sin{\theta \over 2}, 0, 0, \cos{\theta \over 2} ) ,
\label{p1p2E}
\end{eqnarray}
$\theta \in ]0,\pi[$ being the angle formed by the two trajectories
$(+1)$ and $(+2)$ in Euclidean four--space.\\
Let us introduce the following notations for the normalized correlators
$\langle {\cal W}_1 {\cal W}_2 \rangle / \langle {\cal W}_1 \rangle
\langle {\cal W}_2 \rangle$ in the Minkowskian and in the Euclidean theory,
in the presence of a {\it finite} IR cutoff $T$:
\begin{eqnarray}
{\cal G}_M(\chi;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &\equiv&
{ \langle {\cal W}^{(T)}_1 {\cal W}^{(T)}_2 \rangle \over
\langle {\cal W}^{(T)}_1 \rangle
\langle {\cal W}^{(T)}_2 \rangle } ,\nonumber \\
{\cal G}_E(\theta;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &\equiv&
{ \langle \tilde{\cal W}^{(T)}_1 \tilde{\cal W}^{(T)}_2 \rangle_E \over
\langle \tilde{\cal W}^{(T)}_1 \rangle_E
\langle \tilde{\cal W}^{(T)}_2 \rangle_E } .
\label{GM-GE}
\end{eqnarray}
As already stated in Ref. \cite{Meggiolaro02}, the two quantities in Eq.
(\ref{GM-GE}) (with $\chi > 0$ and $0 < \theta < \pi$) are expected to be
connected by the same analytic continuation in the angular variables and in
the IR cutoff which was already derived in the case of Wilson lines
\cite{Meggiolaro02,Meggiolaro97,Meggiolaro98}, i.e.:
\begin{eqnarray}
{\cal G}_E(\theta;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
{\cal G}_M(\chi \to i\theta;~T \to -iT;
~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) ,
\nonumber \\
{\cal G}_M(\chi;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
{\cal G}_E(\theta \to -i\chi;~T \to iT;
~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) .
\label{analytic}
\end{eqnarray}
Indeed it can be proved \cite{Meggiolaro05}, simply by adapting step by step
the proof derived in Ref. \cite{Meggiolaro02} from the case of Wilson lines to
the case of Wilson loops, that the analytic continuation (\ref{analytic}) is an
{\it exact} result, i.e., not restricted to some order in perturbation theory
or to some other approximation, and is valid both for the Abelian and the
non--Abelian case.
By using the analytic continuation (\ref{analytic}), one can also derive the
following {\it crossing--symmetry} relation:
\begin{eqnarray}
{\cal G}_M(i\pi-\chi;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
{\cal G}_M(\chi;~T;~\vec{z}_\perp,\vec{R}_{1\perp},-\vec{R}_{2\perp})
\nonumber \\
&=& {\cal G}_M(\chi;~T;~\vec{z}_\perp,-\vec{R}_{1\perp},\vec{R}_{2\perp}) .
\label{crossing}
\end{eqnarray}
As we have said above, the loop--loop correlation functions (\ref{GM-GE}),
both in the Minkowskian and in the Euclidean theory, are expected to be
IR--{\it finite} quantities, i.e., to have finite limits when $T \to \infty$,
differently from what happens in the case of Wilson lines.
One can then define the following loop--loop correlation functions
with the IR cutoff removed:
\begin{eqnarray}
{\cal C}_M(\chi;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &\equiv&
\displaystyle\lim_{T \to \infty} \left[
{\cal G}_M(\chi;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})
- 1 \right] , \nonumber \\
{\cal C}_E(\theta;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})
&\equiv& \displaystyle\lim_{T \to \infty} \left[
{\cal G}_E(\theta;~T;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})
- 1 \right] .
\label{C12}
\end{eqnarray}
As a pedagogic example to illustrate these considerations, we shall consider
the simple case of QED, in the so--called {\it quenched} approximation, where
vacuum polarization effects, arising from the presence of loops of dynamical
fermions, are neglected.
In this approximation, the calculation of the normalized correlators
(\ref{GM-GE}) can be performed exactly (i.e., without further approximations)
both in Minkowskian and in Euclidean theory and one finds that
\cite{Meggiolaro05} i) the two quantities ${\cal G}_M$ and ${\cal G}_E$ are
indeed connected by the analytic continuation (\ref{analytic}), and ii) the
two quantities are finite in the limit when the IR cutoff $T$ goes to
infinity:
\begin{eqnarray}
{\cal C}_M(\chi;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
\exp \left[ -i 4e^2 \coth \chi~
t(\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) \right] - 1 ,
\label{QED-M} \\
{\cal C}_E(\theta;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
\exp \left[ - 4e^2 \cot \theta~
t(\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) \right] - 1 ,
\label{QED-E}
\end{eqnarray}
where
\begin{equation}
t(\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) \equiv
{1 \over 8\pi} \ln \left(
{ |\vec{z}_\perp+{\vec{R}_{1\perp} \over 2}+{\vec{R}_{2\perp} \over 2}|
|\vec{z}_\perp-{\vec{R}_{1\perp} \over 2}-{\vec{R}_{2\perp} \over 2}| \over
|\vec{z}_\perp+{\vec{R}_{1\perp} \over 2}-{\vec{R}_{2\perp} \over 2}|
|\vec{z}_\perp-{\vec{R}_{1\perp} \over 2}+{\vec{R}_{2\perp} \over 2}| }
\right) .
\label{t-function}
\end{equation}
As shown in Ref. \cite{Meggiolaro05}, the results (\ref{QED-M}) and
(\ref{QED-E}) can be used to derive the corresponding results in the case of a
non--Abelian gauge theory with $N_c$ colours, up to the order ${\cal O}(g^4)$
in perturbation theory (see also Refs. \cite{LLCM,BB}):
\begin{eqnarray}
{\cal C}_M(\chi;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})|_{g^4} &=&
- 2g^4 \left( {N_c^2 - 1 \over N_c^2} \right) \coth^2 \chi~
[t(\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})]^2 ,
\label{QCD-pertM} \\
{\cal C}_E(\theta;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})|_{g^4} &=&
2g^4 \left( {N_c^2 - 1 \over N_c^2} \right) \cot^2 \theta~
[t(\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp})]^2 .
\label{QCD-pertE}
\end{eqnarray}
We stress the fact that both the Minkowskian quantities (\ref{QED-M}) and
(\ref{QCD-pertM}) and the Euclidean quantities (\ref{QED-E}) and
(\ref{QCD-pertE}) are IR finite, differently from the corresponding quantities
constructed with Wilson lines, which were evaluated in Ref. \cite{Meggiolaro97}
(see also Ref. \cite{Meggiolaro96}).
It is also important to notice that the two quantities (\ref{QED-M}) and
(\ref{QED-E}), as well as the two quantities (\ref{QCD-pertM}) and
(\ref{QCD-pertE}), obtained {\it after} the removal of the IR cutoff
($T \to \infty$), are still connected by the usual analytic continuation in
the angular variables only:
\begin{eqnarray}
{\cal C}_E(\theta;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
{\cal C}_M(\chi \to i\theta;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) ,
\nonumber \\
{\cal C}_M(\chi;~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) &=&
{\cal C}_E(\theta \to -i\chi;
~\vec{z}_\perp,\vec{R}_{1\perp},\vec{R}_{2\perp}) .
\label{final}
\end{eqnarray}
[Moreover, the expressions (\ref{QED-M}) and (\ref{QCD-pertM}) trivially
satisfy the crossing--symmetry relation (\ref{crossing}).]
This is a highly non--trivial result, whose general validity is discussed
in Ref. \cite{Meggiolaro05}.
(Indeed, the validity of the relation (\ref{final}) has
been also recently verified in Ref. \cite{BB} by an explicit calculation
up to the order ${\cal O}(g^6)$ in perturbation theory.)
As said in Ref. \cite{Meggiolaro05},
if ${\cal G}_M$ and ${\cal G}_E$, considered as functions of the
{\it complex} variable $T$, have in $T=\infty$ an ``eliminable {\it isolated}
singular point'' [i.e., they are analytic functions of $T$ in the {\it complex}
region $|T| > R$, for some $R \in \Re^+$, and the {\it finite} limits
(\ref{C12}) exist when letting the {\it complex} variable $T \to \infty$],
then, of course, the analytic continuation (\ref{final}) immediately derives
from Eq. (\ref{analytic}) (with $|T| > R$), when letting $T \to +\infty$.
(For example, if ${\cal G}_M$ and ${\cal G}_E$ are analytic functions of $T$
in the {\it complex} region $|T| > R$, for some $R \in \Re^+$,
and they are bounded at large $T$, i.e., $\exists B_{M,E} \in \Re^+$ such that
$|{\cal G}_{M,E}(T)| < B_{M,E}$ for $|T| > R$, then $T=\infty$
is an ``eliminable singular point'' for both of them.)
But the same result (\ref{final}) can also be derived under different
conditions. For example, let us assume that ${\cal G}_E$ is a bounded
analytic function of $T$ in the sector $0 \le \arg T \le {\pi \over 2}$,
with finite limits along the two straight lines on the border of the sector:
${\cal G}_E \to G_{E1}$, for $({\rm Re}T \to +\infty,~{\rm Im}T = 0)$, and
${\cal G}_E \to G_{E2}$, for $({\rm Re}T = 0,~{\rm Im}T \to +\infty)$.
And, similarly, let us assume that ${\cal G}_M$ is a bounded
analytic function of $T$ in the sector $-{\pi \over 2} \le \arg T \le 0$,
with finite limits along the two straight lines on the border of the sector:
${\cal G}_M \to G_{M1}$, for $({\rm Re}T \to +\infty,~{\rm Im}T = 0)$, and
${\cal G}_M \to G_{M2}$, for $({\rm Re}T = 0,~{\rm Im}T \to -\infty)$.
We can then apply the ``Phragm\'en--Lindel\"of theorem'' (see, e.g., Theorem
5.64 in Ref. \cite{PLT}) to state that $G_{E2} = G_{E1}$ and
$G_{M2} = G_{M1}$. Therefore, also in this case, the analytic continuation
(\ref{final}) immediately derives from Eq. (\ref{analytic}) when
$T \to \infty$.
The relation (\ref{final}) has been extensively used in the literature
\cite{LLCM,JP,Janik,instanton1,instanton2} in order to address, from a
non--perturbative point of view, the still unsolved problem of the asymptotic
$s$--dependence of hadron--hadron elastic scattering amplitudes and total
cross sections. (See, e.g., Ref. \cite{pomeron-book} and references therein
for a recent review of the problem. It has been also recently proved in
Ref. \cite{BB}, by an explicit perturbative calculation, that the loop--loop
scattering amplitude approaches, at sufficiently high energy, the
BFKL--{\it pomeron} behaviour \cite{BFKL}.)
An independent non--perturbative approach would be surely welcome and could be
provided by a direct lattice calculation of the loop--loop Euclidean
correlation functions.
This would surely result in a considerable progress along this line
of research.
\section*{References}
|
1,108,101,565,430 | arxiv | \section{Introduction}
\label{sec:Introduction}
The speed at which spiral galaxies rotate remains relatively constant
out to large radii \cite{RFT80,aB81}, which implies that they contain
large amounts of dark matter. The nature of this material remains
obscure, but one key diagnostic is provided by the shape of the dark
matter halo, as quantified by its shortest-to-longest axis ratio, $q =
c/a$. The roundest halos with $q \sim 0.8$ are predicted by galaxy
formation models in which hot dark matter is dominant \cite{jP93}.
Cosmological cold dark matter simulations typically result in triaxial
dark halos \cite{WQSZ92} while the inclusion of gas dynamics in the
simulations results in somewhat flattened, oblate halos
\cite{nKjeG91,sUlM94,jD94} with $q = 0.5 \pm 0.15$ \cite{jD94}.
Models for other dark matter candidates such as cold molecular gas
\cite{PCM94} and massive decaying neutrinos \cite{dS90} require halos
as flat as $q \sim 0.2$. Clearly, the determination of $q$ for real
galaxies offers a valuable test for discriminating between these
cosmological models.
\begin{figure}
\epsfxsize 1.00\hsize
\epsffile{P05_Halo_Shapes_cmp_no_MW.ps}
\caption{ \label{fig:Halo_Shapes_cmp}A summary of the existing
estimates of galaxies' dark matter halo shapes as parameterized by
their shortest-to-longest axis ratio, $q$. For each
individual galaxy we indicate the identification and technique used to
determine the halo's flattening: 1) Flaring gas layer
\protect\cite{rpO96a,BCV97}; 2) Warping gas layer
\protect\cite{HS94}; 3) X-ray isophotes, \protect\cite{BC98}; 4) Polar
ring galaxies \protect\cite{ACCHS93,SRJF94,SP95}; 5) precessing dusty
disk \protect\cite{SKD92}. The dotted line shows the predicted
distribution for the shapes of halos in a cold dark matter cosmology
($q=0.5 \pm 0.15$; Dubinski 1994).
}
\end{figure}
In Fig.~\ref{fig:Halo_Shapes_cmp}, we summarize the existing estimates
of halo shape, derived using a variety of techniques. It is evident
from this figure that there is only a very limited amount of data
available for measuring the distribution of $q$ in galaxies. Rather
more worrying, though, is the fact that different techniques seem to
yield systematically different answers. The warping gas layer method,
in which the shape of the halo is inferred by treating any warp in a
galaxy's gas layer as a bending mode in a flattened potential
\cite{HS94}, seems to imply that dark halos are close to spherical,
with $q \ga 0.8$. Conversely, the flaring gas layer technique, which
determines the halo shape by assuming that the gas layer is in
hydrostatic equilibrium in the galaxy's gravitational potential and
uses the thickness of the layer as a diagnostic for the distribution
of mass in the galaxy \cite{rpO96a}, yields much flatter halo shape
estimates with $q \la 0.4$. Although the numbers involved are rather
small, there do seem to be real differences between the results
obtained by the different methods. Thus, either these techniques are
being applied to systematically different classes of galaxy, or at
least some of the methods are returning erroneous results.
In order to determine which techniques are credible, we really need to
apply several methods to a single galaxy, to see which produce
consistent answers. As a first step towards such cross-calibration,
this paper compares the shape of our own galaxy's dark matter halo as
inferred by two distinct techniques:
\begin{enumerate}
\item {\it Stellar kinematics}. Our position within the Milky Way
means that we have access to information for this galaxy that cannot
be obtained from other systems. In particular, it is possible to
measure the total column density of material near the Sun using
stellar kinematics. After subtracting off the other components, we
can infer the local column density of dark matter. By comparing this
density close to the plane of the Galaxy to the over-all mass as
derived from its rate of rotation, we can obtain a direct measure of
the shape of the halo.
\item {\it The flaring gas layer method}. As outlined above, this
technique assumes that the H{\sc i} emission in the Milky Way comes from
gas in hydrostatic equilibrium in the Galactic potential, from which the
shape of the dark halo is inferred. Since the results of previous
applications of this method give results somewhat out of line with the
other techniques, it is important to assess the method's credibility.
\end{enumerate}
As well as providing a check on the validity of the flaring gas layer
method, these analyses will also provide another useful datum on the
rather sparsely populated Fig.~\ref{fig:Halo_Shapes_cmp}.
The remainder of the paper is laid out as follows. Both of the above
methods rely on knowledge of the Milky Way's rotation velocity as a
function of Galactic radius -- its ``rotation curve'' -- so
Section~\ref{sec:rot_curve} summarizes the data available for
estimating this quantity, and the dependence of the inferred rotation
curve on the assumed distance to the Galactic centre and local
rotation velocity. The analysis of the shape of the dark halo
requires that we decompose the Milky Way into its visible and dark
matter components, so Section~\ref{sec:mass_mod} discusses the
construction of a set of models consistent with both the photometric
properties of the Galaxy, and its mass properties as inferred from the
rotation curve. In Section~\ref{sec:q_from_stars} we show how these mass
models can be combined with the local stellar kinematic measurements
to determine the shape of the dark halo. Section~\ref{sec:q_from_gas}
presents the application of the gas layer flaring technique to the
mass models. Section~\ref{sec:q_from_both} combines the results
derived by the two techniques and assesses their consistency. The
broader conclusions of this work are drawn in
Section~\ref{sec:conclusions}.
\section{The observed rotation curve}
\label{sec:rot_curve}
Our position within the Milky Way complicates the geometry when
studying its structure and kinematics. It is therefore significantly
harder to determine our own galaxy's rotation curve, $\Theta(R)$, than
it is to derive those for external systems. More directly accessible
to observation than $\Theta(R)$ is the related quantity
\begin{equation}
W(R) = R_0\left[{\Theta(R) \over R} - {\Theta_0 \over R_0}\right],
\label{eq:WofR}
\end{equation}
where $R_0$ and $\Theta_0$ are the distance to the Galactic centre and
the local circular speed. If one assumes that material in the galaxy
is in purely circular motion, then some simple geometry shows that, for
an object at Galactic coordinates $\{l,b\}$ with a line-of-sight
velocity $v_{\rm los}$,
\begin{equation}
W = {v_{\rm los} \over \sin l \cos b}
\label{eq:Wdef}
\end{equation}
\noindent (Binney \& Merrifield 1998, \S9.2.3). Thus, if one measures
the line-of-sight velocities for a series of objects at some radius
$R$ in the Galaxy, one has an estimate for $W(R)$. By adopting values
for $R_0$ and $\Theta_0$, one can then use equation~(\ref{eq:WofR}) to
determine the rotation speed at that radius.
In practice, the difficulty lies in knowing the Galactic radii of the
objects one is looking at. One solution is to look at standard
candles, whose distances can be estimated, and hence whose radii in
the Galaxy can be geometrically derived. Alternatively, one can
select the subset of some tracer -- usually H{\sc i} or H$_2$ gas -- whose
line-of-sight velocities and Galactic coordinates places it at the
same value of $W$, and hence at the same radius. One can then use the
properties of this cylindrical slice through the Galaxy to infer its
radius. For example, in the inner Galaxy, all the material in a ring
of radius $R$ will lie at Galactic longitudes of less than $l_{\rm
max} = \sin^{-1} (R/R_0)$, so one can use the extent of the
emission on the sky of each $W$-slice to infer its radius -- an
approach traditionally termed the ``tangent point method'' [see, for
example, Malhotra (1994,1995)]. At radii greater than $R_0$ this
method is no longer applicable as the emission will be visible at all
values of $l$. However, by assuming that the thickness of the gas
layer does not vary with azimuth, one can use the observed variation
in the angular thickness of the layer with Galactic longitude to
estimate the radii of such $W$ slices (Merrifield 1992).
For the remainder of this paper, we use Malhotra's (1994, 1995)
tangent point analysis to estimate $W(R)$ in the inner Galaxy. For
the outer Galaxy, we have combined Merrifield's (1992) data with Brand
\& Blitz's (1993) analysis of the distances to H{\sc II} regions, from which
standard candle analysis the rotation curve can be derived.
In order to convert $W(R)$ into $\Theta(R)$ using
equation~(\ref{eq:WofR}), we must adopt values for the Galactic
constants, $R_0$ and $\Theta_0$. Unfortunately, there are still
significant uncertainties in these basic parameters. In the case of
$R_0$, for example, the extensive review by Reid (1993) discussed
measurements varying between $R_0 = 6.9 \pm 0.6\,{\rm kpc}$ and $R_0 =
8.4 \pm 0.4\,{\rm kpc}$. Even more recently there have been few signs
of convergence: Layden et al.\ (1996) used RR Lyrae stars as standard
candles to derive $R_0 = 7.2 \pm 0.7\,{\rm kpc}$, while Feast \&
Whitelock (1997) used a Cepheid calibration to obtains $R_0 = 8.5 \pm 0.5\,{\rm
kpc}$. The constraints on $\Theta_0$ are similarly weak: a recent
review by Sackett (1997) concluded that a value somewhere in the range
$\Theta_0 = 210 \pm 25\,{\rm km}\,{\rm s}^{-1}$ provided the best
current estimate. It should also be borne in mind that the best
estimates for $R_0$ and $\Theta_0$ are not independent. Analysis of the
local stellar kinematics via the Oort constants gives quite a
well-constrained value for the ratio $\Theta_0/R_0 = 26.4 \pm 1.9\,{\rm
km}\,{\rm s}^{-1}\,{\rm kpc}^{-1}$ (Kerr \& Lynden-Bell 1986), so a
lower-than-average value of $R_0$ is likely to be accompanied by a
lower-than-average value for $\Theta_0$. Currently, the best available
measure of the local angular velocity of the Milky Way is based on VLBI
proper motion measurements of SgrA$^*$. Assuming that SgrA$^*$ is at
rest with respect to the Galactic centre, Reid et al.\ (1999) find
$\Theta_0/R_0 = 27.25 \pm 2.5 {\rm km}\,{\rm s}^{-1}\,{\rm kpc}^{-1}$,
consistent with the value proposed by Kerr \& Lynden-Bell.
\begin{figure}
\epsffile{P05_fMW_Rotation_Curves.ps}
\caption{
\label{fig:MW_Rotation_Curves} The rotation curve for the Milky Way
for values of $R_0 = 7.1\,{\rm kpc}$, $\Theta_0 = 185\,{\rm km}\,{\rm
s}^{-1}$, and $R_0 = 8.5\,{\rm kpc}$, $\Theta_0 = 220\,{\rm km}\,{\rm
s}^{-1}$. The figure also shows one of the ways in which the rotation
curve can be decomposed into the contributions from different mass
components: the bulge (dotted line); the stellar disk (filled
circles); the H{\sc i} layer (crosses, where negative values mean that
the force is directed outwards); the $H_2$ layer (circles); and the
dark halo (dashed line). The best fit model, which is obtained by
summing the individual components in quadrature, is shown as a full
line.
}
\end{figure}
To illustrate the effect of the adopted values of $\Theta_0$ and $R_0$
on the derived rotation curve, Fig.~\ref{fig:MW_Rotation_Curves} shows
$\Theta(R)$ for two of the more extreme plausible sets of Galactic
parameters. Clearly, the choice of values for these quantities
affects such basic issues as whether the rotation curve is rising or
falling in the outer Galaxy. Given the current uncertainties, it
makes little sense to pick fixed values for the Galactic constants,
so in the following analysis we consider models across a broad
range of values, $5.5\,{\rm kpc} < R_0 < 9\,{\rm kpc}$, $165\,{\rm
km}\,{\rm s}^{-1} < \Theta_0 < 235\,{\rm km}\,{\rm s}^{-1}$.
\section{Mass models}
\label{sec:mass_mod}
In order to relate the rotation curve to the shape of the dark halo, we
must break the gravitational potential responsible for the observed
$\Theta(R)$ into the contributions from the different mass components.
As is usually done in this decomposition
\cite{egvAS86,smK87,kBeg89,LF89,ahB92,rpO95,rpO96b,wDjB98}, we adopt a
model consisting of a set of basic components:
\begin{enumerate}
\item {\it A stellar bulge}. Following Kent's (1992) analysis of the
Galaxy's K-band light distribution, we model the Milky Way's bulge by a
``boxy'' density distribution,
\begin{equation}
\rho_b(R,z) \propto K_0(s/h_b),\ \hbox{where}\ s^4 = R^4 + (z/q_b)^4.
\end{equation}
\noindent This modified Bessel function produces a bulge that appears
exponential in projection. The observed flattening of the K-band light
yields $q_b = 0.61$, and its characteristic scalelength is $h_b =
670\,{\rm pc}$ (Kent 1992). The constant of proportionality depends on
the bulge mass-to-light ratio, $\Upsilon_b$, which we leave as a free
parameter.
\item {\it A stellar disk}. The disk is modelled by a density
distribution,
\begin{equation}
\rho_d(R,z) \propto \exp(-R/h_d) {\rm sech}(z/2z_d).
\end{equation}
\noindent The first term gives the customary radially-exponential
disk. The appropriate value for the scalelength, $h_d$, is still
somewhat uncertain -- Kent, Dame \& Fazio (1991) estimated $h_d = 3
\pm 0.5\,{\rm kpc}$, while Freudenreich (1998) found a value of
$2.5\,{\rm kpc}$. We therefore leave this parameter free to vary
within the range $2\,{\rm kpc} \le h_d \le 3\,{\rm kpc}$. The
$z$-dependence adopts van der Kruit's (1988) compromise between a
${\rm sech}^2$ isothermal sheet and a pure exponential. For
simplicity, we fix the scaleheight at $z_d = 300\,{\rm pc}$. However,
the exact $z$-dependence of $\rho_d$ was found to have no impact on
any of the following analysis. Once again, the constant of
proportionality depends on the mass-to-light ratio of the disk,
$\Upsilon_d$, which we leave as a free parameter.
\item {\it A gas disk}. From the H{\sc i} data given by Burton (1988)
and Malhotra (1995) and the H$_2$ column densities from Bronfman et
al.\ (1988) and Wouterloot et al.\ (1990), we have inferred the
density of gas as a function of radius in the Galaxy. This analysis
treats the gas as an axisymmetric distribution, and neglects the
contribution from ionized phases of the interstellar medium, but does
include a 24\% contribution by mass from helium (Olive \& Steigman
1995).
\item {\it A dark matter halo}. We model the dark halo as a flattened
non-singular isothermal sphere, which has a density distribution
\begin{equation}
\rho_h(R,z) = \rho_h {R_h^2 \over R_h^2 + R^2 + (z/q)^2},
\end{equation}
\noindent where $\rho_h$ is the central density, $R_h$ is the halo core
radius, and $q$ is the halo flattening, which is the key parameter in
this paper. \end{enumerate}
The procedure for calculating a mass model from these components is
quite straightforward. For each pair of Galactic constants, $R_0$ and
$\Theta_0$, we vary the unknown parameters $\{\Upsilon_b, h_d,
\Upsilon_d, \rho_h, R_h, q\}$ to produce the mass model that has a
gravitational potential, $\Phi(R, z)$, such that \begin{equation} v(R) =
\left(R {\partial\Phi\over\partial R}\biggr|_{z=0}\right)^{1/2}
\end{equation} provides the best fit (in a minimum $\chi^2$ sense) to
the observed rotation curve, $\Theta(R)$. Examples of two such best-fit
models are shown in Fig.~\ref{fig:MW_Rotation_Curves}.
As is well known
\cite{egvAS86,smK87,kBeg89,LF89,ahB92,rpO95,rpO96b,wDjB98}, such mass
decompositions are by no means unique: there is near degeneracy between
the various unknown parameters, so many different combinations of the
individual components can reproduce the observed rotation curve with
almost equal quality of fit. We have therefore searched the entire
$\{\Upsilon_b, h_d, \Upsilon_d, \rho_h, R_h, q\}$ parameter space to
find the complete subset of values that produce fits in which the
derived value of $\chi^2$ exceeded the minimum value by less than unity.
\section{Halo flattening from local stellar kinematics}
\label{sec:q_from_stars}
Although the analysis of the previous section tells us something about
the possible range of mass models for the Milky Way, it does not place
any useful constraint on the shape of the dark halo: for any of the
adopted values of $R_0$ and $\Theta_0$, one can find acceptable mass
models with highly-flattened dark halos ($q \sim 0$), round dark halos
($q \sim 1$), and even prolate dark halos ($q > 1$). We therefore
need some further factor to discriminate between these models.
One such constraint is provided by stellar kinematics in the solar
neighbourhood. Studies of the motions of stars in the Galactic disk
near the Sun imply that the total amount of mass within 1.1 kpc of the
Galactic plane is $\Sigma_{1.1} = (71 \pm 6) M_{\odot}\,{\rm pc}^{-2}$
(Kuijken \& Gilmore 1991). Clearly, the value of $\Sigma_{1.1}$
provides an important clue to the shape of the Milky Way's dark halo:
in general, a model with a highly-flattened dark halo will place a lot
of mass near the Galactic plane leading to a high predicted value for
$\Sigma_{1.1}$, while a round halo will distribute more of the dark
matter further from the plane, depressing $\Sigma_{1.1}$.
The dark halo is not the only contributor to $\Sigma_{1.1}$. In
particular, the stellar disk has a surface density in the solar
neighbourhood, $\Sigma_*$, which may contribute a significant fraction
of $\Sigma_{1.1}$. There must therefore be a fairly simple relation
between the adopted value of $\Sigma_*$ and the inferred halo
flattening, $q$. Specifically, as one considers larger possible
values of $\Sigma_*$, the amount of dark matter near the plane must
decrease so as to preserve the observed value of $\Sigma_{1.1}$. Such
a decrease can be achieved by increasing $q$, thus making the dark
halo rounder.
\begin{figure}
\epsffile{P05_fq_Sigma_shade.ps} \caption{ \label{fig:q_Sigma_star}
The flattening of the model Galactic halo, $q$, as a function of the
adopted model's column of stellar disk mass in the Solar
neighbourhood, $\Sigma_*$. We show the best fits to the stellar
kinematic constraints (dotted line) and the flaring of the H{\sc i}
layer (full line). The shaded region shows values of $\Sigma_*$ that
lie more than 1-$\sigma$ from the observed value. This particular set
of calculations has been made assuming $R_0 = 7.1\,{\rm kpc}$ and
$\Theta_0=185{\rm\,km \,s}^{-1}$.}
\end{figure}
This inter-relation is illustrated in Fig.~\ref{fig:q_Sigma_star}.
Here, for one of the sets of possible Galactic constants, we have
considered all the mass models that produce an acceptable fit to the
rotation curve and reproduce the meaured value of $\Sigma_{1.1}$. For
each acceptable model, we have extracted the value of the halo
flattening $q$ and the mass of the stellar disk in the solar
neighbourhood, $\Sigma_*$. For the reasons described above, these
quantities are tightly correlated, and Fig.~\ref{fig:q_Sigma_star}
shows the linear regression between the two.
Clearly, we have not yet derived a unique value for the halo
flattening: by selecting models with different values of $\Sigma_*$,
we can still tune $q$ to essentially any value we want. However,
$\Sigma_*$ is not an entirely free parameter. From star-count
analysis, it is possible to perform a stellar mass census in the solar
neighbourhood, and thus determine the local stellar column density.
Until relatively recently, this analysis has been subject to
significant uncertainties, with estimates as high as $\Sigma_* =
145M_{\odot}\,{\rm pc}^{-2}$ \cite{jnB84b}. However, there is now
reasonable agreement between the various analyses, with more recent
published values of $35 \pm 5M_{\odot}\,{\rm pc}^{-2}$ (Kuijken \&
Gilmore 1989) and $26 \pm 4M_{\odot}\,{\rm pc}^{-2}$ (Gould, Bahcall
\& Flynn 1997). If we adopt Kuijken \& Gilmore's slightly more
conservative error bounds on $\Sigma_*$, the shaded regions in
Fig.~\ref{fig:q_Sigma_star} no longer represent acceptable models as
they predict the wrong local disk mass density, so we end up with a
moderately well-constrained estimate for the halo flattening of $q =
0.7 \pm 0.1$.
Note, though, that although this analysis returns a good estimate for
$q$, Fig.~\ref{fig:q_Sigma_star} only shows the value obtained for a
particular set of Galactic constants. As discussed above, changing
the adopted values for $R_0$ and $\Theta_0$ alters the range of
acceptable mass models, which, in turn, will alter the derived
correlation between $\Sigma_*$ and $q$. The uncertainties in the
Galactic constants are still sufficiently large that the absolute
constraints on $q$ remain weak. Nevertheless, it is to be hoped that
the measurements of $R_0$ and $\Theta_0$ will continue to improve over
time, leading to a unique determination of $q$ by this method.
Further, as we shall see below, it may be possible to attack the
problem from the other direction by using other estimates of $q$ to
help determine the values of the Galactic constants.
\section{Halo flattening from gas layer flaring}
\label{sec:q_from_gas}
We now turn to the technique developed by Olling (1995) for measuring
the shape of a dark halo from the observed thickness of a galaxy's gas
layer. In essence the approach is similar to the stellar-kinematic
method described above: the over-all mass distribution of the Milky
Way is inferred from its rotation curve, while the degree to which
this mass distribution is flattened is derived using the properties of
a tracer population close to the Galactic plane. In this case, the
tracer is provided by the H{\sc i} emission from the Galactic gas
layer. The thickness of the Milky Way's gas layer is dictated by the
hydrostatic balance between the pull of gravity toward the Galactic
plane and the pressure forces acting on the gas. As the density of
material in the Galaxy decreases with radius, the gravitational force
toward the plane becomes weaker, so the equilibrium thickness of the
layer becomes larger, giving the gas distribution a characteristic
``flared'' appearance. The exact form of this flaring depends on the
amount of mass close to the plane of the Galaxy. Thus, by comparing
the observed flaring to the predictions of the hydrostatic equilibrium
arrangement of gas in the mass models developed in
Section~\ref{sec:mass_mod}, we can see what degree of halo flattening
is consistent with the observations.
\begin{figure*}
\epsffile{P05_fObserved_Flaring.ps}
\caption{ \label{fig:Observed_Flaring} The thickness of the
H{\sc i} layer of the Milky Way. The widths in the inner Galaxy were
taken from Malhotra (1995). For the outer Galaxy, we plot the widths
from Diplas \& Savage (1991; open triangles), Wouterloot et al.\
(1992; crosses), Merrifield (1992; open circles), and the average (thick
full line \& filled squares). The top panel represents the ``raw''
measurements, in bottom panel we present the beam-size corrected widths.
}
\end{figure*}
\subsection{The observed flaring of the gas layer} Before we can apply
this technique, we need to summarize the observational data available
on the flaring of the Galactic H{\sc i} layer. Merrifield (1992)
calculated the thickness of the gas layer across a wide range of
Galactic azimuths in his determination of the outer Galaxy rotation curve.
As a check on the validity of that analysis, we have also drawn on the
work of Diplas \& Savage (1991) and Wouterloot et al.\ (1992), which
derived the gas layer thicknesses across a more limited range of
azimuths. For completeness, we have also included the data for the
inner Galaxy as derived by Malhotra (1995). The resulting values for
the full-width at half maximum (FWHM) of the density of gas
perpendicular to the plane are given in the upper panel in
Fig.~\ref{fig:Observed_Flaring}. Note that, once again, the results
depend on the adopted values of the Galactic constants -- since the
radii in the Galaxy of the various gas elements were derived from
their line-of-sight velocities via equations~(\ref{eq:WofR}) and
(\ref{eq:Wdef}), the values of $R$ in Fig.~\ref{fig:Observed_Flaring}
depend on $R_0$ and $\Theta_0$\footnote{Merrifield's method for
determining the thickness of the gas layer also exposes some of the
shortcomings of more traditional methods. If the wrong values for
$R_0, \Theta_0$ and $\Theta(R)$ are chosen, the inferred thickness of
the gas layer (at a given $R$) will show a systematic variation with
Galactocentric azimuth. It is possible to correct published data for
this effect, but only if the assumed rotation curve, Galactic
constants, as well as the thickness of the gas layer as a function of
azimuth are specified.}.
It would appear from this figure that there are some discrepancies
between the various measurements in the outer Galaxy. After some
investigation, we established that these differences can be attributed
to the effects of the beam sizes of the radio telescopes with which
the observations were made. Such resolution effects will mean that
the FWHM of the gas will tend to be overestimated. In the lower
panel, we show what happens when the appropriate beam correction is
made to the data from Diplas \& Savage (1991) and Wouterloot et al.\
(1992). No similar correction is required for the Merrifield (1992)
analysis, as in that work the derived value of the FWHM was dominated
by gas towards the Galactic anti-centre, which lies at relatively small
distances from the Sun, and so the beam correction is small. Clearly,
this correction brings the various data sets into much closer agreement.
We therefore adopt the mean curve shown in this panel for the
following analysis; the error bars shown represent the standard error
obtained on averaging the various determinations.
\subsection{Sources of pressure support}
In order to compare the observed gas layer flaring to the predictions
for a gas layer in hydrostatic equilibrium, we must address the source
of the pressure term in the hydrostatic equilibrium equation. The
most obvious candidate for supporting the H{\sc i} layer comes from
its turbulent motions. In the inner Galaxy, the H{\sc i} is
observed to have a velocity dispersion of $\sigma_g = 9.2{\rm\,km
\,s}^{-1}$ independent of radius \cite{sM95}, and we assume that this
value characterizes the turbulent motions of the gas throughout the
Galaxy, providing a kinetic pressure term.
Potentially, there may be other forces helping to support the Galactic
H{\sc i} layer: non-thermal pressure gradients associated with magnetic
fields and cosmic rays may also provide a net force to resist the pull
of gravity on the Galactic H{\sc i}. However, the analysis we are doing
depends most on the properties of the H{\sc i} layer at large radii in the
Milky Way, where star formation is almost non-existent, so energy
input into cosmic rays and magnetic fields from stellar processes is
likely to be unimportant.
Our concentration on the properties of gas at large radii also
eliminates another potential complexity. In the inner Galaxy, the
interstellar medium comprises a complicated multi-phase mixture of
molecular, atomic and ionized material. A full treatment of the
hydrostatic equilibrium of such a medium is complicated, as gas can
transform from one component to another, so all components would have
to be considered when calculating hydrostatic equilibrium. For the
purpose of this paper, it is fortunate that at the low pressures
characteristic of the outer Galaxy it is not possible to maintain both
the cold molecular phase and the warm atomic phase
\cite{pM93,WHKTB95}. Braun \& Walterbos (1992) and Braun (1997,1998)
have shown that the cold phase disappears when the B-band surface
brightness of a galaxy falls below the 25th magnitude per square
arcsecond level, which occurs at $\sim 1.5 R_0$ in the Milky Way
(Binney \& Merrifield 1998, \S10.1). Further, the ionized fraction of
the ISM is expected to decrease with distance as it is closely
associated with sites of star formation \cite{FWGH96,WHL97}.
Ultimately, the ionizing effects of the extragalactic background
radiation field become significant, but only when the H{\sc i} column
density falls below about $1 M_{\odot}\,{\rm pc}^{-2}$
\cite{pM93,DS94,rpO96b}, which lies well beyond the radii we are
considering here.
For the current analysis, it therefore seems reasonable to treat the
Milky Way's gas layer as a single isothermal component supported
purely by its turbulent motions. Moreover, this assumption has been
made in the previous implementations of the gas flaring method
\cite{rpO96a,BCV97}. A principal objective of this paper is to test
the validity of those analyses by comparing results obtained by the
flaring technique to those obtained by other methods. It is therefore
important that we make the same assumption of a single isothermal
component in the present study.
\subsection{Fitting to model gas layer flaring}
We are now in a position to compare observation and theory. The
technique for calculating the gas layer thickness in different mass
models has been described in detail by Olling (1995). In brief, for
each model, at each radius $R$, one integrates the hydrostatic
equilibrium equation,
\begin{equation}
{\partial\Phi \over \partial z}
= -{1 \over \rho_g} {\partial \rho_g \sigma_g^2 \over \partial z},
\end{equation}
to obtain the gas density distribution, $\rho_g(R, z)$. The FWHM of
this model gas distribution can then be compared directly with the
observations.
\begin{figure}
\epsffile{P05_fObs_Mod_Flaring.ps}
\caption{\label{fig:Obs_Mod_Flaring} The observed flaring of the
Galaxy's gas layer (triangles) compared with the flaring predicted for
a number of mass models that reproduce the observed rotation curve and
value of $\Sigma_*$. The models in the top panel have been calculated
assuming $R_0 = 8.5\,{\rm kpc}$ and a disk mass-to-light ratio
$\Upsilon_{\rm d, K-band} = 0.60$; the bottom panel shows models with
$R_0 = 7.1\,{\rm kpc}$ and $\Upsilon_{\rm d,K-band} = 0.41$. The
models shown here have a disk scale-length of $h_d = 2.5\,{\rm kpc}$.
The halo flattening of the model and the reduced $\chi^2$ value of the fit
is shown for each case.}
\end{figure}
The results of such calculations are illustrated by the examples shown
in Fig.~\ref{fig:Obs_Mod_Flaring}. The basic trends in this analysis are
clearly demonstrated by these examples. The decrease in total density
with radius leads to a dramatic flaring in the model-predicted gas layer
thickness, just as is seen in the observations. For a flatter model
dark halo, the mass is more concentrated toward the plane of the Galaxy,
squeezing the H{\sc i} layer into a thinner distribution.
Once again, the results depend quite sensitively on the choice of
Galactic constants, since these values affect both the gas
distribution as inferred from observations and the acceptable mass
models as inferred from the rotation curve. As is apparent from
Fig.~\ref{fig:Obs_Mod_Flaring}, none of the $R_0 = 8.5\,{\rm kpc}$
models fits the observations. In fact, to match the observed layer
width one would require a substantially prolate dark matter halo with
$q \sim 1.5$, which none of the current dark matter scenarios predict.
For $R_0 = 7.1\,{\rm kpc}$, on the other hand, a very good fit is
obtained for models with a halo flatness of $q \sim 0.7$. Such models
even reproduce the observed inflection in the variation of the gas
layer width with radius at $R \sim 10\,{\rm kpc}$.
For each plausible set of Galactic constants, we can carry out a
similar analysis to that in Section~\ref{sec:q_from_stars} to see what
range of values of $q$ are consistent with the observed gas layer
flaring. Because this technique relies on data from large radii in
the Galaxy, where the dark matter halo is the dominant source of mass,
the flaring predicted by the models depends very little on the
properties of the disk and bulge. Unlike the stellar kinematic
analysis, therefore, one cannot trade off the mass in the disk against
the mass near the plane from a more-flattened dark halo. This
difference is illustrated in Fig.~\ref{fig:q_Sigma_star}, which shows
the way that the value of $q$ inferred from the gas layer flaring
depends on the properties of the stellar disk (as parameterized by the
model's column density of stars in the Solar neighbourhood). As for
the stellar-kinematic analysis, there is a well-defined correlation
between $q$ and $\Sigma_*$, but, for the reasons described above, the
trend for the current method is very much weaker, and, within the
observationally-allowed range for $\Sigma_*$, $q$ is tightly
constrained to $0.73 \pm 0.03$.
Although this constraint is remarkably good, it should be borne in
mind that it is still dependent on the adopted values for the Galactic
constants. As we have seen above, larger values for $R_0$ lead to
rounder, or even prolate estimates for halo shape, so we will not
obtain an unequivocal measure for $q$ from this method until the
Galactic constants are measured more accurately
\section{Combining the techniques}
\label{sec:q_from_both}
The different slopes of the two relations in
Fig.~\ref{fig:q_Sigma_star} raises an interesting possibility.
Clearly, for a consistent picture, one must use a single mass model to
reproduce both the stellar-kinematic constraint on the mass in the
solar neighborhood and the observed flaring of the gas layer. Thus,
although there are whole linear loci in this figure of models with
different values of $q$ and $\Sigma_*$ that satisfy each of these
constraints individually, there is only the single point of
intersection between these two lines where the model fits both the
stellar-kinematic constraint and the observed flaring of the gas
layer. Hence, for given values of $R_0$ and $\Theta_0$, one predicts
unique values for $q$ and $\Sigma_*$.
We have therefore repeated the analysis summarized in
Fig.~\ref{fig:q_Sigma_star} spanning the full range of plausible values
for $R_0$ and $\Theta_0$, and calculated the mutually-consistent
estimates for $\Sigma_*$ and $q$ for each case. In order to reduce the
computational complexity of this large set of calculations to manageable
proportions, we made use of Olling's (1995) fitting formula,
which showed that, if self-gravity is negligible, one can approximate
the model-predicted thickness of the gas layer by the relation
\begin{equation}
{\rm FWHM}(R) \approx \sqrt{13.5q \over 1.4 + q} {\sigma_g \over v_{h, \infty}}
\sqrt{R_h^2 + R^2},
\label{eq:FWHMapprox}
\end{equation}
\noindent where $v_{h, \infty}$ is the circular rotation speed of the
dark halo component at large radii. Allowing for the self-gravity of
the gas layer, one obtains a similar formula. Comparing the values
derived from this formula to sample results obtained by the full
integration process described above, we found that the approximation
based on equation~(\ref{eq:FWHMapprox}) matches the detailed
integration for $R > 1.75R_0$. We therefore only used the data from
beyond this radius in the following analysis, in which the model gas
layer thickness was estimated using the approximate formula.
\begin{figure*}
\epsffile{P05_Sstr_q_R0_T0_xf.ps} \caption{ \label{fig:Sstr_q_R0_T0}
Contours of the stellar column density in the Solar neighbourhood
($\Sigma_*$; long dashed lines) and halo flattening ($q$; dotted
lines) as a function of the adopted values for the Galactic constants,
$R_0$ and $\Theta_0$. The heavy full line and the heavy dashed line
corresponds to Kuijken \& Gilmore's (1989) determination of
$\Sigma_*$, and the $\pm1-\sigma$ values. The heavily-shaded region
corresponds to parts of parameter space consistent with these values
of $\Sigma_*$ that produce an oblate halo. The lightly-shaded area
gives the corresponding region if we adopt Gould, Bahcall and Flynn's
(1997) values for $\Sigma_*$. The cross shows the determination of
the Galactic constants based on an analysis of the Oort constants
(Olling \& Merrifield 1998). }
\end{figure*}
The values obtained for $q$ and $\Sigma_*$ for each possible pair of
Galactic constants are presented in Fig.~\ref{fig:Sstr_q_R0_T0}.
Thus, for example, for values of $R_0 = 7.1\,{\rm kpc}$ and $\Theta_0
= 185\,{\rm km}\,{\rm s}^{-1}$, one finds $\Sigma_* = 35
M_{\odot}\,{\rm pc}^{-2}$ and $q = 0.7$, corresponding to the
intercept that we previously calculated in
Fig.~\ref{fig:q_Sigma_star}.
Figure~\ref{fig:Sstr_q_R0_T0} places some interesting limits on the
properties of the Milky Way. For example, if we maintain our
prejudice that the halo should be oblate ($q < 1$), then, unless we
adopt a particularly extreme value for $R_0$, we find that $\Theta_0$
must be less than $\sim 190\,{\rm km}\,{\rm s}^{-1}$. If we also
adopt Kuijken \& Gilmore's (1989) measurement of $\Sigma_* = 35 \pm
5M_{\odot}\,{\rm pc}^{-2}$, we find that only models within the
heavily-shaded region of Fig.~\ref{fig:Sstr_q_R0_T0} are acceptable,
placing an upper limit on $R_0$ of $\sim 7.6\,{\rm kpc}$.
Conversely, if one forces the Galactic constants to the IAU standard
values of $R_0 = 8.5\,{\rm kpc}$ and $\Theta_0 = 220\,{\rm km}\,{\rm
s}^{-1}$ (Kerr \& Lynden-Bell 1987), one finds barely-credible values
of $\Sigma_* \sim 60 M_{\odot}\,{\rm pc}^{-2}$ and $q \sim 1.5$.
\section{Conclusions}
\label{sec:conclusions}
The prime objective of this paper has been to check the validity of
techniques for measuring halo flattening by asking whether two
different techniques return consistent values when applied to the
Milky Way. As we have seen in the last section, the answer is a
qualified ``yes.'' The qualification is that consistency with the
measured stellar column density requires values of the Galactic
constants that differ from those conventionally adopted. However, as
was discussed in Section~\ref{sec:rot_curve}, the true values of these
constants remain elusive, with estimates spanning the ranges $ 7\,{\rm
kpc} < R_0 < 8.5\,{\rm kpc}$ and $185\,{\rm km}\,{\rm s}^{-1} <
\Theta_0 < 235\,{\rm km}\,{\rm s}^{-1}$. With such gross
uncertainties, it quite straightforward to pick values that produce an
entirely self-consistent picture. To underline this point, we have
included on Fig.~\ref{fig:Sstr_q_R0_T0} the results of Olling \&
Merrifield's (1997) estimates for $R_0$ and $\Theta_0$ derived from an
analysis of the Oort constants. If that analysis is valid, then we
have a consistent model for the Milky Way in which the dark halo has a
flattening of $q \sim 0.8$.
Ultimately, this analysis will allow us to come to one of two
conclusions:
\begin{enumerate}
\item If future studies confirm low values for $R_0$ and $\Theta_0$
similar to those derived by Olling \& Merrifield (1997), then the
consistency of the two analyses for calculating $q$ imply that the gas
layer flaring technique is valid, adding confidence to the previous
determinations by this method.
\item Conversely, if we learn in future that $R_0$ and $\Theta_0$ are
closer to the more conventional larger values, then the implied values
of $\Sigma_*$ are so far from the observed estimates that one has to
suspect that at least one of the techniques for measuring $q$ is
compromised. In this case, one would have to look more closely at
some of the assumptions that went into the analysis. For example,
perhaps the non-thermal pressure forces from cosmic rays and magnetic
fields have a role to play even at large radii in galactic disks.
Alternatively, perhaps the H{\sc i} layer is not close enough to
equilibrium for the hydrostatic analysis to be valid. Finally, our
assumption of azimuthal symmetry may be invalid. Strong departures
from axisymmetry could mean that our determination of the thickness
and column density of the gas is compromised, and that the locally
determined values of $\Sigma_{1.1}$ and $\Sigma_*$ may not be
representative for the Galactocentric radius of the Sun.
\end{enumerate}
Assuming for the moment that the analysis is valid, we have another
datum to add to Fig.~\ref{fig:Halo_Shapes_cmp}. Since the two
previous flaring analyses returned systematically rather small values
of $q \sim 0.3$, it is reassuring that the Milky Way seems to indicate
a larger value of $\sim 0.8$ -- it appears that the low values
are simply a coincidence arising from the very small number
statistics. This larger value is inconsistent with the very flat
halos that are predicted by models in which the dark matter consists
of either decaying neutrinos (Sciama 1990) or cold molecular hydrogen
(Pfenniger, Combes \& Martinet 1994).
With the addition of the Milky Way to the data presented in
Fig.~\ref{fig:Halo_Shapes_cmp}, the only technique that stands out as
giving systematically different values for $q$ is the bending mode
analysis of warped gas layers. In this regard, it is notable that
simulations cast some doubt on the validity of such analyses. The
method assumes that warps are manifestations of persistent bending
modes that occur when the flattening of a disk and the surrounding
dark halo are misaligned. However, the simulations show that
gravitational interactions between a misaligned disk and halo rapidly
bring the two back into alignment, effectively suppressing this
mechanism (e.g.\ Dubinski \& Kuijken 1995, Nelson \& Tremaine 1995,
Binney, Jiang \& Dutta 1998). The number of measurements is still
rather small, but it is at least interesting that if one neglects the
warped gas layer results, the remaining data appear entirely
consistent with the dotted line in Fig.~\ref{fig:Halo_Shapes_cmp},
which shows Dubinski's (1994) prediction for the distribution of halo
shapes that will be produced in a cold dark matter cosmology.
\section*{acknowledgments}
We would like to thank Andy Newsam, Irini Sakelliou, Konrad Kuijken,
Marc Kamionkowski and Jacqueline van Gorkom for useful discussions. We
are also very grateful to the referee, James Binney, for his major
contribution to the clarity of this paper.
|
1,108,101,565,431 | arxiv |
\section{The CMS Collaboration \label{app:collab}}\begin{sloppypar}\hyphenpenalty=5000\widowpenalty=500\clubpenalty=5000\input{EXO-10-010-authorlist.tex}\end{sloppypar}
\end{document}
|
1,108,101,565,432 | arxiv | \section{Introduction}
Discrete symmetries play an important role in
model building of particle physics.
For example, abelian and non-abelian discrete flavor symmetries
are useful to derive realistic quark/lepton masses and their
mixing \cite{Altarelli:2007cd}.
Discrete non-abelian flavor symmetries can also be used to
suppress flavor changing neutral current processes in
supersymmetric models \cite{dbkaplan,babu}.
Furthermore, discrete symmetries can be introduced to forbid unfavorable couplings
such as those leading to fast proton decay \cite{murayama,kakizaki}.
Superstring theory is a promising candidate for
unified theory including gravity and
may provide with an origin of
such discrete symmetries \cite{Dine:1988kq}.
It is widely assumed that
superstring theory leads to anomaly-free effective theories.
In fact the anomalous $U(1)$ symmetries
are restored by the Green-Schwarz (GS)
mechanism \cite{Green:1984sg,Witten:1984dg,Ibanez:1998qp}.
For this mechanism to work,
the mixed anomalies between the anomalous $U(1)$ and other continuous
gauge symmetries have to satisfy a certain set of conditions,
the GS conditions,
at the field theory level.
In particular, in heterotic string theory the mixed anomalies between
the anomalous $U(1)$ symmetries and other continuous gauge
symmetries must be universal for different gauge
groups up to their Kac-Moody
levels \cite{Schellekens:1986xh,Kobayashi:1996pb}.
A well-known discrete symmetry in heterotic string theory is
T-duality symmetry, and
its effective theory has T-duality anomalies \cite{Derendinger:1991hq}.
It has been shown that
the mixed anomalies between T-duality symmetry and
continuous gauge symmetries are universal except
for the sector containing an $N=2$ subsector
and are exactly canceled by the GS mechanism \cite{Ibanez:1992hc}.
That has phenomenologically interesting
consequences which have been studied in early 90's
\cite{Ibanez:1992hc,Ibanez:1991zv,Kawabe:1993pz}.
Heterotic orbifold construction is one of interesting
4D string models \cite{Dixon,IMNQ}.
(See also for resent works Ref.~\cite{Kobayashi:2004ud,Forste:2004ie}
and for review \cite{Choi-Km}.)
Geometrical structures of their compact spaces are simple compared with
other types of 4D string model constructions.
Phenomenological aspects in effective theory are related
with geometrical aspects of orbifolds.
Discrete symmetries which may be used as non-abelian flavor symmetries
and also certain discrete $R$-symmetries originate from the geometrical
structure of
orbifolds \cite{Dine:1988kq,Ibanez:1992uh,Kobayashi:2004ud,Kobayashi:2006wq}.
In this paper
we consider discrete R-symmetries.
Stringy-originated discrete symmetries are strongly constrained
due to stringy consistency, and
it is phenomenologically and theoretically
important to study anomalies of discrete symmetries,
as it is pointed out in \cite{Ibanez:1991hv}
and the example of T-duality shows.
We shall investigate the mixed anomalies between the discrete R-symmetries
and the continuous gauge symmetries in concrete orbifold models.
We will also study relations between the discrete $R$-anomalies,
one-loop beta-function coefficients (scale anomalies) and
T-duality anomalies.
This paper is organized as follows.
In section 2, we give a brief review on heterotic
orbifold models to fix our notation.
In section 3, we define discrete $R$-charges, which is
one of our main interests.
In section 4, we calculate the mixed anomalies between
the discrete $R$-symmetries and the continuous gauge symmetries
in concrete models.
We also study the relations of R-anomalies with
one-loop beta-function coefficients and T-duality anomalies.
In section 5, we discuss phenomenological implications of
our results.
Section 6 is devoted to conclusion and discussion.
\section{Heterotic orbifold models}
Here we review briefly heterotic orbifold models.
First we give a review on ${\bf Z}_N$ orbifold models, and next
explain ${\bf Z}_N \times {\bf Z}_M$ orbifold models.
Heterotic string theory consists of
10D right-moving superstrings and 26D left-moving
bosonic strings.
Their common 10 dimensions correspond to our 4D space-time
and 6D compact space.
The other 16D left-moving bosonic strings correspond to
a gauge part.
Here, we consider the $E_8 \times E_8$ heterotic string theory,
where momenta of 16D left-moving bosonic strings span
$E_8 \times E_8$ root lattice.
The following discussions are also applicable to
$SO(32)$ heterotic string theory.
In orbifold models, the 6D compact space is chosen to be
6D orbifold.
A 6D orbifold is a division of 6D torus $T^6$ by a twist $\theta$,
while the torus $T^6$ is obtained as $R^6/\Lambda^6$,
where $\Lambda^6$ is 6D lattice.
Eigenvalues of the twist $\theta$ are denoted as
$e^{2\pi i v_1}, e^{2\pi i v_2}$
and $e^{2\pi i v_3}$ in the complex basis $Z_i$ ($i=1,2,3$).
To preserve 4D N=1 supersymmetry (SUSY), they must satisfy
the following condition,
\begin{equation}
v_1+v_2+v_3= {~~\rm integer}.
\end{equation}
When one of $v_i$ is integer, N=2 SUSY is preserved.
In the case with $v_i\neq {\rm integer}$, only N=1 SUSY
is preserved.
Such ${\bf Z}_N$ orbifolds are classified into
${\bf Z}_3$, ${\bf Z}_4$, ${\bf Z}_6$-I, ${\bf Z}_6$-II, ${\bf Z}_7$, ${\bf Z}_8$-I, ${\bf Z}_8$-II,
${\bf Z}_{12}$-I and ${\bf Z}_{12}$-II, and their twists are explicitly shown
in Table 1 and Table 2.
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& ${\bf Z}_3$ & ${\bf Z}_4$ & ${\bf Z}_6$-I & ${\bf Z}_6$-II & ${\bf Z}_7$ \\
$v_i$ & $(1,1,-2)/3$ & $(1,1,-2)/4$ & $(1,1,-2)/6$ & $(1,2,-3)/6$ &
$(1,2,-3)/7$ \\ \hline \hline
$T_1$ & $(1,1,1)/3$ & $(1,1,2)/4$ & $(1,1,4)/6$ & $(1,2,3)/6$ &
$(1,2,4)/7$ \\
$T_2$ & --- & $(2,2,0)/4$ & $(2,2,2)/6$ & $(2,4,0)/6$ & $(2,4,1)/7$ \\
$T_3$ & --- & --- & $(3,3,0)/6$ & $(3,0,3)/6$ & --- \\
$T_4$ & --- & --- & --- & $(4,2,0)/6$ & $(4,1,2)/7$ \\ \hline
\end{tabular}
\end{center}
\caption{$H$-momenta for ${\bf Z}_3$, ${\bf Z}_4$, ${\bf Z}_6$-I, ${\bf Z}_6$-II and
${\bf Z}_7$ orbifolds} \label{tab:H-momenta-1}
\end{table}
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{|c|c|c|c|c|}
\hline
& ${\bf Z}_8$-I & ${\bf Z}_8$-II & ${\bf Z}_{12}$-I & ${\bf Z}_{12}$-II \\
$v_i$ & $(1,2,-3)/8$ & $(1,3,-4)/8$ & $(1,4,-5)/12 $ &
$(1,5,-6)/12$ \\ \hline \hline
$T_1$ & $(1,2,5)/8$ & $(1,3,4)/8$ & $(1,4,7)/12$ & $(1,5,6)/12$ \\
$T_2$ & $(2,4,2)/8$ & $(2,6,0)/8$ & $(2,8,2)/12$ & $(2,10,0)/12$ \\
$T_3$ & --- & $(3,1,4)/8$ & $(3,0,9)/12$ & $(3,3,6)/12$ \\
$T_4$ & $(4,0,4)/8$ & $(4,4,0)/8$ & $(4,4,4)/12$ & $(4,8,0)/12$ \\
$T_5$ & $(5,2,1)/8$ & --- & --- & $(5,1,6)/12$ \\
$T_6$ & --- & --- & $(6,0,6)/12$ & $(6,6,0)/12$ \\
$T_7$ & --- & --- & $(7,4,1)/12$ & --- \\
$T_8$ & --- & --- & --- & --- \\
$T_9$ & --- & --- & $(9,0,3)/12$ & --- \\
$T_{10}$ & --- & --- & --- & $(10,2,0)/12$ \\
\hline
\end{tabular}
\end{center}
\caption{$H$-momenta for ${\bf Z}_8$-I, ${\bf Z}_8$-II, ${\bf Z}_{12}$-I and
${\bf Z}_{12}$-II orbifolds} \label{tab:H-momenta-2}
\end{table}
On the orbifold, closed string satisfies the following boundary
condition,
\begin{equation}
X(\sigma = \pi) = \theta^k X (\sigma = 0) +V,
\end{equation}
where $V$ is a shift vector on the 6D lattice $\Lambda^6$.
The complex basis of $X$ corresponds to $Z_i$.
The $\theta^k$-twisted sector is denoted by $T_k$, while
the sector with $k=0$ is the so-called untwisted sector.
It is convenient to bosonize right-moving fermionic strings.
Here we write such bosonized fields by $H^t$ ($t=1,\cdots,5$).
Their momenta $p_t$ are quantized and span the SO(10) weight lattice.
Space-time bosons correspond to SO(10) vector momenta,
and space-time fermions correspond to SO(10) spinor momenta.
The 6D compact part, i.e. the SO(6) part, $p_i$ ($i=1,2,3$)
is relevant to our study.
All of ${\bf Z}_N$ orbifold models have three untwisted
sectors, $U_1$, $U_2$ and $U_3$, and their massless bosonic modes
have the following SO(6) momenta,
\begin{equation}
U_1:(1,0,0), \qquad U_2:(0,1,0), \qquad U_3:(0,0,1).
\label{H-momenta-U}
\end{equation}
On the other hand, the twisted sector $T_k$ has
shifted $SO(6)$ momenta, $r_i=p_i+kv_i$.
Table 1 and Table 2 show explicitly $H$-momenta $r_i$ of
massless bosonic states.
That implies their $SO(6)$ $H$-momenta are obtained as
\begin{equation}
r_i = |kv_i|-{\rm Int}[|kv_i|],
\label{H-momentum:Zn}
\end{equation}
where ${\rm Int}[a]$ denotes an integer part of fractional number $a$.
This relation is not available for the untwisted sectors,
and $r_i$ is obtained as Eq.~(\ref{H-momenta-U}).
The gauge sector can also be broken and
gauge groups smaller than $E_8 \times E_8$ are obtained.
Matter fields have some representations under such
unbroken gauge symmetries.
Massless modes for 4D space-time bosons correspond to
the following vertex operator \cite{Friedan:1985ge,Dixon:1986qv},
\begin{equation}
V_{-1} = e^{-\phi}\prod_{i=1}^3(\partial Z_i)^{{\cal N}_i} (\partial \bar
Z_i)^{\bar {\cal N}_i}e^{ir_tH^t}e^{iP^IX^I}e^{ikX}
\sigma_k,
\end{equation}
in the $(-1)$-picture, where $\phi$ is the bosonized ghost,
$kX$ corresponds to the 4D part and $P^IX^I$ corresponds
to the gauge part.
Oscillators of the left-mover are denoted by
$\partial Z_i$ and $\partial \bar Z_i$, and
${\cal N}_i$ and $\bar {\cal N}_i$ are oscillator numbers, which are included
in these massless modes.
In addition, $\sigma_k$ denotes the twist field for
the $T_k$ sector.
Similarly, we can write the vertex operator for 4D space-time
massless fermions as
\begin{equation}
V_{-\frac12} = e^{-\frac12 \phi}\prod_{i=1}^3(\partial Z_i)^{N_i}
(\partial \bar Z_i)^{\bar N_i}e^{ir_t^{(f)}H_t}e^{iP^IX^I}e^{ikX}
\sigma_{k},
\end{equation}
in the $(-1/2)$-picture.
The $H$-momenta for space-time fermion and boson, $r_i^{(f)}$ and
$r_i$ in the same supersymmetric multiplet are
related each other as
\begin{equation}
r_i = r_i^{(f)} + (1,1,1)/2.
\end{equation}
We need vertex operators $V_0$ with the 0-picture when we compute
generic n-point couplings.
We can obtain such vertex operators $V_0$ by operating
the picture changing operator, $Q$, on $V_{-1}$,
\cite{Friedan:1985ge},
\begin{equation}
Q=e^\phi (e^{-2 \pi i r^v_i H_i}\bar \partial Z_i + e^{2 \pi i r^v_i
H_i}\bar \partial \bar Z_i), \label{p-change}
\end{equation}
where $r^v_1=(1,0,0)$, $r^v_2=(0,1,0)$ and $r^v_3=(0,0,1)$.
Next we briefly review on ${\bf Z}_N \times {\bf Z}_M$ orbifold
models \cite{Font:1988mk}.
In ${\bf Z}_N \times {\bf Z}_M$ orbifold models, we introduce two independent
twists $\theta$ and $\omega$, whose twists are represented by
$e^{2\pi i v^1_i}$ and $e^{2\pi i v^2_i}$, respectively
in the complex basis.
Two twists are chosen such that
each of them breaks 4D N=4 SUSY to 4D N=2 SUSY and
their combination preserves only N=1 SUSY.
Thus, eigenvalues $v^1_i$ and $v^2_i$ are chosen as
\begin{equation}
v^1_i=(v^1,-v^1,0), \qquad v^2_i=(0,v^2,-v^2),
\end{equation}
where $v^1,v^2 \neq {\rm integer}$.
In general, ${\bf Z}_N \times {\bf Z}_M$ orbifold models
have three untwisted sectors, $U_1$, $U_2$ and $U_3$,
and their massless bosonic modes have the same $SO(6)$ $H$-momenta $r_i$ as
Eq.~(\ref{H-momenta-U}).
In addition, there are $\theta^k \omega^\ell$-twisted sectors,
and their $SO(6)$ $H$-momenta are obtained as
\begin{equation}
r_i = |kv^1_i|+|\ell v^2_i| - {\rm Int}[|kv^1_i|+|\ell v^2_i|].
\label{H-momentum:ZnZm}
\end{equation}
Vertex operators are also constructed in a similar way.
Recently, non-factorizable ${\bf Z}_N \times {\bf Z}_M$ orbifold
models have been studied \cite{Faraggi:2006bs}.
The above aspects are the same for such non-factorizable models.
\section{Discrete R-symmetries}
Here we define R-charges.
We consider n-point couplings including two fermions.
Such couplings are computed by the following
n-point correlation function of vertex operators,
\begin{equation}
\langle V_{-1}V_{-1/2}V_{-1/2}V_0\cdots V_0 \rangle .
\end{equation}
They must have the total ghost charge $-2$, because the background
has the ghost number 2.
When this n-point correlation function does not vanish,
its corresponding n-point coupling in effective theory is
allowed.
That is, selection rules for allowed n-point correlation functions
in string theory correspond to symmetries in effective theory.
The vertex operator consists of several parts, the
4D part $e^{kX}$, the gauge part $e^{iPX}$,
the 6D twist field $\sigma_k$, the 6D left-moving oscillators
$\partial Z_i$ and the bosonized fermion $e^{irH}$.
Each part has its own selection rule for allowed couplings.
For the 4D part and the gauge part,
the total 4D momentum $\sum k$ and the total momentum
of the gauge part $\sum P$ should be conserved.
The latter is nothing but the requirement of gauge invariance.
The selection rule for 6D twist fields $\sigma_k$ is controlled by
the space group selection rule
\cite{Dixon:1986qv,Kobayashi:1991rp}.
Similarly, the total $H$-momenta can be conserved
\begin{equation}
\sum r_i =1.
\end{equation}
Here we take a summation over the $H$-momenta for
scalar components, using the fact that
the $H$-momentum of fermion component differs by $-1/2$.
Another important symmetry is the twist symmetry of
oscillators.
We consider the following twist of oscillators,
\begin{eqnarray}
& & \partial Z_i \rightarrow e^{2 \pi i v_i}\partial Z_i, \qquad
\partial \bar Z_i \rightarrow e^{-2 \pi i v_i}\partial \bar Z_i,
\nonumber \\
& & \bar \partial Z_i \rightarrow e^{2 \pi i v_i}\bar \partial Z_i,
\qquad \bar \partial \bar Z_i \rightarrow e^{-2 \pi i v_i}\bar
\partial \bar Z_i.
\end{eqnarray}
Allowed couplings may be invariant under the above $Z_N$ twist.
Indeed, for 3-point couplings corresponding to
$\langle V_{-1}V_{-1/2}V_{-1/2}\rangle$, we can require
$H$-momentum conservation and $Z_N$ twist invariance of oscillators
independently.
However, we have to compute generic n-point couplings
through picture changing, and the picture changing operator $Q$
includes non-vanishing $H$-momenta and right-moving oscillators
$\bar \partial
Z_i$ and
$\bar \partial \bar Z_i$.
Consequently, the definition of the H-momentum
of each vertex operator depends on the choice of
the picture and so its physical meaning remains somewhat obscure.
We therefore use a picture independent quantity as
follows,
\begin{equation}
R_i \equiv r_i + {\cal N}_i - \bar {\cal N}_i,
\end{equation}
which can be interpreted as an R-charge \cite{Kobayashi:2004ud}.
This R-symmetry is a discrete
surviving symmetry of the continuous $SU(3)~(\subset SU(4))$ R-symmetry
under orbifolding.
Here we do not
distinguish oscillator numbers for the left-movers and right-movers,
because they have the same phase under $Z_N$ twist. Indeed,
physical states with $-1$ picture have vanishing oscillator number
for the right-movers, while the oscillator number for the
left-movers can be non-vanishing. Thus, hereafter ${\cal N}_i$ and
$\bar {\cal N}_i$ denote the oscillator number for the left-movers,
because we study the physical states with $-1$ picture from now. For
simplicity, we use the notation $\Delta {\cal N}_i = {\cal N}_i -
\bar {\cal N}_i$.
Now, we can write the selection rule due to $R$-symmetry as
\begin{equation}
\sum R_i = 1 \quad {\rm mod} \quad N_i,
\end{equation}
where $N_i$ is the minimum integer satisfying $N_i = 1/\hat v_i$,
where $\hat v_i= v_i + m$ with any integer $m$.
For example, for $Z_6$-II orbifold, we have $v_i=(1,2,-3)/6$, and
$N_i=(6,3,2)$.
Thus, these are discrete symmetries.
Note that the above summation is taken over scalar components.
Discrete R symmetry itself is defined as the following
transformation,
\begin{equation}
\vert R_i \rangle \rightarrow e^{2\pi i v_i R_i} \vert R_i \rangle,
\label{eq:R-trans}
\end{equation}
for states with discrete $R$-charges, which are defined
mod $N_i$.
For later convenience, we show discrete $R$-charges for
fermions in Table~\ref{tab:R}.
As shown there, gaugino fields always have
$R$-charge $(1/2,1/2,1/2)$.
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{|c|c|} \hline
& $R_i$ \\ \hline
gaugino & $(1/2,1/2,1/2)$ \\
$U_1$ & $(1/2,-1/2,-1/2)$ \\
$U_2$ & $(-1/2,1/2,-1/2)$ \\
$U_3$ & $(-1/2,-1/2,1/2)$ \\
$T_k$ & $kv_i - {\rm Int}[kv_i]-1/2+\Delta {\cal N}_i$ \\ \hline
\end{tabular}
\end{center}
\caption{Discrete $R$-charges of fermions in
${\bf Z}_N$ orbifold models} \label{tab:R}
\end{table}
\section{Anomalies of R-symmetry}
\subsection{Discrete R anomalies}
Let us study anomalies of discrete R-symmetry.
Under the R-transformation like Eq.~(\ref{eq:R-trans}), the
path integral measure of fermion fields is not
invariant, but changes as
\begin{equation}
{\cal D}\psi {\cal D}\psi^\dagger \rightarrow
{\cal D}\psi {\cal D}\psi^\dagger exp \left[-2\pi i v_i \sum_{G_a} A^{R_i}_{G_a}
\int d^4x \frac{1}{16\pi^2}F^{(G_a)}_{\mu \nu}\tilde F^{{(G_a)} \mu \nu}\right],
\end{equation}
where $\tilde F^{{(G_a)} \mu \nu}=\frac{1}{2}\varepsilon^{\mu \nu \rho
\sigma} F^{(G_a)}_{\rho \sigma}$.
The anomaly coefficients $A^{R_i}_{G_a}$ are obtained as
\begin{equation}
A_{G_a}^{R_i}=\sum R_i T({\bf R}_{G_a}),
\end{equation}
where $T({\bf R}_{G_a})$ is the Dynkin index for ${\bf R}_{G_a}$
representation under $G_a$.
The winding number of the gauge field configuration,
i.e., the Pontryagin index,
\begin{equation}
\nu \equiv \frac{T({\bf R}^{(f)}_{G_a})}{16\pi^2}\int d^4x F^{(G_a)}_{\mu \nu}\tilde F^{{(G_a)} \mu \nu},
\end{equation}
is integer, where $T({\bf R}^{(f)}_{G_a})$ denotes
the Dynkin index of a fundamental representation of $G_a$.
Thus, the anomaly coefficients $A^{R_i}_{G_a}$
are defined modulo $N_iT({\bf R}^{(f)}_{G_a})$.
By use of our discrete $R$ charge, the anomaly coefficients are written as
\begin{equation}
A_{G_a}^{R_i}= \frac{1}{2} C_2(G_a) +
\sum_{\rm matter} (r^{}_i-\frac{1}{2} +\Delta {\cal N}_i) T({\bf R}_{G_a}),
\end{equation}
where $C_2(G_a)$ is quadratic Casimir.
Note that $r_i$ denotes the SO(6) shifted momentum for
bosonic states.
The first term in the right hand side is a contribution from
gaugino fields and the other is the contribution from matter fields.
If these anomalies are canceled by the
Green-Schwarz mechanism, these mixed anomalies must
satisfy the following condition,
\begin{equation}
\frac{A_{G_a}^{R_{i}}}{k_a}= \frac{A_{G_b}^{R_i}}{k_b},
\label{GS-R}
\end{equation}
for different gauge groups, $G_a$ and $G_b$, where
$k_a$ and $k_b$ are Kac-Moody levels.
In the simple orbifold construction, we have the Kac-Moody level
$k_a=1$ for non-abelian gauge groups.
Note again that anomalies are defined modulo
$N_iT({\bf R}^{(f)}_{G_a})$.
The above GS condition has its meaning mod
$N_iT({\bf R}^{(f)}_{G_a})/k_a$.
As illustrating examples, let us study explicitly one $Z_3$ model
and one $Z_4$ model.
Their gauge groups and massless spectra are shown
in Table~\ref{tab:Z3} and Table~\ref{tab:Z4}.\footnote{
See for explicit massless spectra Ref.~\cite{Katsuki:1989cs},
where a typographical error is included in the $U_3$ sector
of the $Z_4$ orbifold model.
It is corrected in Table~\ref{tab:Z4}.}
First, we study R-anomalies in the $Z_3$ orbifold model.
Since $v_i=(1,1,-2)/3$, we have $N_i=3$.
For both $E_6$, mixed R-anomalies are computed as
\begin{equation}
A^{R_{i}}_{E_6}= \frac{3}{2}+9n^i_{E_6},
\end{equation}
where $n^i_{E_6}$ is integer.
The second term in the right hand side appears because
anomalies are defined modulo $N_iT(27)$ with
$N_i=3$ and $T(27)=3$ for $E_6$.
Similarly, mixed R-anomalies for $SU(3)$ are computed as
\begin{equation}
A^{R_i}_{SU(3)}=-12 +\frac{3}{2}n^i_{SU(3)},
\end{equation}
where $n^i_{SU(3)}$ is integer.
The second term in the right hand side appears through
$N_iT(3)$ with $N_i=3$ and $T(3)=1/2$ for $SU(3)$.
Thus, in this model, mixed R-anomalies satisfy
\begin{equation}
A^{R_i}_{E_6}=A^{R_i}_{SU(3)} \qquad ({\rm mod}~~3/2).
\end{equation}
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{|c|c|} \hline
gauge group & $E_6 \times SU(3) \times E_6 \times SU(3)$ \\ \hline \hline
sector & massless spectrum \\ \hline
$U_1$ & (27,3;1,1)+ (1,1;27,3)\\
$U_2$ & (27,3;1,1)+ (1,1;27,3) \\
$U_3$ & (27,3;1,1)+ (1,1;27,3) \\
$T_1$ & $27(1,\bar 3;1,\bar3)$ \\ \hline
\end{tabular}
\end{center}
\caption{Massless spectrum in a
${\bf Z}_3$ orbifold model} \label{tab:Z3}
\end{table}
\begin{table}[t]
\begin{center}
\small
\begin{tabular}{|c|c|} \hline
gauge group & $SO(10) \times SU(4) \times SO(12) \times SU(2)
\times U(1)$ \\ \hline \hline
sector & massless spectrum \\ \hline
$U_1$ & $(16_c,4;1,1)+ (1,1;32_c,1)+(1,1;12_v,2)$ \\
$U_2$ & $(16_c,4;1,1)+ (1,1;32_c,1)+(1,1;12_v,2)$ \\
$U_3$ & $(10_v,6;1,1)+ (1,1;32_c,2) +2(1,1,;1,1)$ \\
$T_1$ & $16(1,4;1,2)$ \\
$T_2$ & $16(10_v,1;1,1)+16(1,6;1,1)$ \\ \hline
\end{tabular}
\end{center}
\caption{Massless spectrum in a
${\bf Z}_4$ orbifold model} \label{tab:Z4}
\end{table}
Next, we study R-anomalies in the $Z_4$ orbifold model
with the gauge group
$SO(10)\times SU(4) \times SO(12) \times SU(2) \times U(1)$.
Since the $Z_4$ orbifold has $v_i=(1,1,-2)/4$,
we have $N_i=(4,4,2)$.
Mixed anomalies between $R_{1,2}$ and $SO(10)$ are
computed as
\begin{equation}
A^{R_{1,2}}_{SO(10)} = 1 + 4 n^{1,2}_{SO(10)},
\end{equation}
with integer $n^{1,2}_{SO(10)}$,
where the second term appears through $N_iT({\bf R}_a)$
with $N_i=4$ and $T(10)=1$ for $SO(10)$.
Similarly, mixed anomalies between $R_3$ and $SO(10)$
is computed as
\begin{equation}
A^{R_{3}}_{SO(10)} = -9 + 2 n^{3}_{SO(10)},
\end{equation}
with integer $n^{3}_{SO(10)}$.
Furthermore, mixed R-anomalies for other non-abelian groups
are obtained as
\begin{eqnarray}
& & A^{R_{1,2}}_{SU(4)} = -7 + 2 n^{1,2}_{SU(4)}, \qquad
A^{R_{3}}_{SU(4)} = -9 + n^{3}_{SU(4)}, \nonumber \\
& & A^{R_{1,2}}_{SO(12)} = 1 + 4 n^{1,2}_{SO(12)}, \qquad
A^{R_{3}}_{SO(12)} = 3 + 2n^{3}_{SO(12)}, \\
& & A^{R_{1,2}}_{SU(2)} = -15 + 2 n^{1,2}_{SU(2)}, \qquad
A^{R_{3}}_{SU(2)} = 3 + n^{3}_{SU(2)},
\nonumber
\end{eqnarray}
with integer $n^{i}_{G_a}$, where the second terms
appear through $N_iT({\bf R}_a)$ with $N_i=(4,4,2)$, and
$T(12)=1$ for $SO(12)$, $T(4)=1/2$ for $SU(4)$ and
$T(2)=1/2$ for $SU(2)$.
These anomalies satisfy the GS condition,
\begin{eqnarray}
& & A^{R_{1,2}}_{SO(10)}=A^{R_{1,2}}_{SU(4)} = A^{R_{1,2}}_{SO(12)} =
A^{R_{1,2}}_{SU(2)} \qquad ({\rm mod}~~2), \nonumber \\
& & A^{R_{3}}_{SO(10)}=A^{R_{3}}_{SU(4)} = A^{R_{3}}_{SO(12)} =
A^{R_{3}}_{SU(2)} \qquad ({\rm mod}~~1).
\end{eqnarray}
\subsection{Relation with beta-function}
Here we study the relation between discrete R anomalies and
one-loop beta-functions.
We find
\begin{equation}
\sum_{i=1,2,3}r_i=1,
\end{equation}
{}from Eqs.~(\ref{H-momentum:Zn}) and (\ref{H-momentum:ZnZm})
as well as Table~\ref{tab:H-momenta-1} and
Table~\ref{tab:H-momenta-2}.
By using this, we can write the sum of R-anomalies as
\begin{eqnarray}
A^R_{G_a} &=& \sum_{i=1,2,3}A^{R_i}_{G_a} \nonumber \\
&=& \frac{3}{2}C_2(G_a) + \sum_{\rm matter} T({\bf R}_{G_a})(-\frac{1}{2}+
\sum_i\Delta {\cal N}_i).
\end{eqnarray}
Thus, when $\sum_i\Delta {\cal N}_i=0$, the total anomaly
$A^R_{G_a}$ is proportional to the one-loop beta-function
coefficient, i.e. the scale anomaly, $b_{G_a}$,
\begin{equation}
b_{G_a} = 3 C_2(G_a) - \sum_{\rm matter} T({\bf R}_{G_a}).
\end{equation}
When we use the definition of R charge $\tilde R_i = 2 R_i$,
we would have $A^{\tilde R}_{G_a} = b_{G_a}$.
It is not accidental that
$A^R_{G_a}$ is proportional to $b_{G_a}$
\cite{jones,piguet1}.
The sum of the R-charges $\sum_{i=1,2,3}R_i$
of a supermultiplet is
nothing but the R-charge (up to an overall normalization)
associated with the R-current
which is a bosonic component of the supercurrent \cite{ferrara},
when the R-charge is universal for all of matter fields,
i.e. $\sum_i\Delta {\cal N}_i=0$.
Using the supertrace identity \cite{piguet2} it is
in fact possible to show \cite{piguet1} that
$A^R_{G_a}$ is proportional to $b_{G_a}$ to all orders in perturbation
theory.
In explicit models, non-abelian groups except $SU(2)$
have few massless matter fields with non-vanishing oscillator
numbers, while massless matter fields with oscillators
can appear as singlets as well as $SU(2)$ doublets.
Thus, in explicit models the total R-anomaly $A^R_{G_a}$
is related with the one-loop beta-function coefficient $b_{G_a}$,
\begin{equation}
2A^R_{G_a} = b_{G_a},
\label{anomR-b}
\end{equation}
modulo $N_iT({\bf R}_a)$ for most of non-abelian groups.
Since the total R-anomalies satisfy the GS condition,
$A^R_{G_a}=A^R_{G_b}$, the above relation between
$A^R_{G_a}$ and $b_{G_a}$ leads to
\begin{equation}
b_{G_a} = b_{G_b},
\label{GS-b}
\end{equation}
modulo $2N_iT({\bf R}_a)$.
For example, the explicit $Z_3$ orbifold model
and $Z_4$ orbifold model in Table~\ref{tab:Z3} and Table~\ref{tab:Z4}
have only non-oscillated massless modes except singlets.
The $Z_3$ orbifold model has the following total R-anomalies and
one-loop beta-function coefficient,
\begin{eqnarray}
& & A^R_{E_6}=\frac{9}{2}+9n_{E_6}, \qquad
b_{E_6} = 9, \nonumber \\
& & A^R_{SU(3)}=-36 + \frac{3}{2}n_{SU(3)}, \qquad
b_{SU(3)}=-72.
\end{eqnarray}
Hence, this model satisfy $2A^R_{G_a}=b_{G_a}$ and its
one-loop beta-function coefficients satisfy
\begin{equation}
b_{E_6}=b_{SU(3)} \qquad ({\rm mod}~~3).
\end{equation}
Similarly, the $Z_4$ orbifold model in Table~\ref{tab:Z4}
has the total R-anomalies and
one-loop beta-function coefficients as,
\begin{eqnarray}
& & A^R_{SO(10)}= -7 + 2 n_{SO(10)}, \qquad b_{SO(10)}= -14
\nonumber \\
& & A^R_{SU(4)}= -23 + n_{SU(4)}, \qquad b_{SU(4)} = -46
\nonumber \\
& & A^R_{SO(12)}=5 + 2n_{SO(10)}, \qquad b_{SO(12)}= 10 \\
& & A^R_{SU(2)}=-27 + n_{SU(2)}, \qquad b_{SU(2)}= -54.
\nonumber
\end{eqnarray}
Thus, this model also satisfies $2A^R_{G_a}=b_{G_a}$
and its one-loop beta-function coefficients satisfy
\begin{equation}
b_{SO(10)}=b_{SU(4)}=b_{SO(12)}=b_{SU(2)} \qquad ({\rm mod}~~2).
\end{equation}
\subsection{Relation with T-duality anomaly}
Here we study the relation between R-anomalies and T-duality anomalies.
The relation between R-symmetries and T-duality has also been
studied in Ref.~\cite{Ibanez:1992uh}.
The T-duality anomalies are obtained
as \cite{Derendinger:1991hq,Ibanez:1992hc}
\begin{equation}
A^{T_i}_{G_a} = -C_2({G_a}) +\sum_{\rm matter} T({\bf R}_{G_a})
(1+2n_i),
\end{equation}
where $n_i$ is the modular weight of matter fields for
the $i$-th torus.
The modular weight is related with $r_i$ as
\begin{eqnarray}
n_i &=& -1 {\rm~~for~~} r_i=1,\nonumber \\
&=& 0 {\rm~~for~~} r_i=0,\\
&=& r_i-1 - \Delta {\cal N}_i {\rm~~for~~} r_i \neq 0,1.
\nonumber
\end{eqnarray}
Note that $n_i = -r_i$ for $r_i=0,1/2,1$.
Thus, in the model, which includes only matter fields with
$r_i=0,1/2,1$, the T-duality anomalies and R-anomalies are
proportional to each other,
\begin{equation}
A^{T_i}_{G_a} = -2A^{R_i}_{G_a}.
\label{relation:T-R}
\end{equation}
In generic model, such relation is violated, but
T-duality anomalies and R-anomalies are still related with each other
as
\begin{equation}
A^{T_i}_{G_a} = -2A^{R_i}_{G_a} -2 \sum_{r_i \neq 0,1/2,1}
(2 r_i -1).
\end{equation}
T-duality should also satisfy the GS condition,
\begin{equation}
\frac{A^{T_i}_{G_a}}{k_a} =\frac{A^{T_i}_{G_b}}{k_b},
\end{equation}
for the $i$-th torus, which does not include the N=2 subsector.
Thus, the requirement that T-duality anomalies and R-anomalies
should satisfy the GS condition, leads to a similar condition for
\begin{equation}
\Delta_a^i= 2 \sum_{r^b_i \neq 0,1/2,1}
(2 r^b_i -1).
\end{equation}
For the $i$-th torus, which includes
N=2 subsector, T-duality anomalies can be canceled by
the GS mechanism and T-dependent threshold correction \cite{Dixon:1990pc}.
Thus, for such torus, the T-duality anomalies has no
constrain from the GS condition.
However, even for such torus, R-anomaly should satisfy
the GS condition.
For example, the $Z_4$ orbifold model in Table~\ref{tab:Z4}
has the following T-duality anomalies,
\begin{eqnarray}
& & A^{T_{1,2}}_{SO(10)}=-2, \qquad A^{T_3}_{SO(10)}=18, \nonumber \\
& & A^{T_{1,2}}_{SU(4)}=-2, \qquad A^{T_3}_{SU(4)}=18, \nonumber \\
& & A^{T_{1,2}}_{SO(12)}=-2, \qquad A^{T_3}_{SO(12)}=-6, \\
& & A^{T_{1,2}}_{SU(2)}=-2, \qquad A^{T_3}_{SU(2)}=-6.
\nonumber
\end{eqnarray}
They satisfy the GS condition,
\begin{equation}
A^{T_{1,2}}_{SO(10)}=A^{T_{1,2}}_{SU(4)}=A^{T_{1,2}}_{SO(12)}=
A^{T_{1,2}}_{SU(2)}.
\end{equation}
On the other hand, for the third torus, T-duality anomalies $A^{T_3}_{G_a}$
do not satisfy the GS condition, that is, anomalies $A^{T_3}_{G_a}$
are not universal, because there is the N=2 subsector
and one-loop gauge kinetic functions depend on the $T_3$ moduli
with non-universal coefficients \cite{Dixon:1990pc}.
However, they satisfy
\begin{eqnarray}
& & A^{T_3}_{SO(10)}=-2A^{R_3}_{SO(10)}, \qquad
A^{T_3}_{SU(4)}=-2A^{R_3}_{SU(4)}, \nonumber \\
& & A^{T_3}_{SO(12)}=-2A^{R_3}_{SO(12)}, \qquad
A^{T_3}_{SU(2)}=-2A^{R_3}_{SU(2)},
\end{eqnarray}
because this model has only massless modes with $r_3=0,1/2,1$.
Indeed, all of $Z_4$ orbifold models include only
massless modes with $r_3=0,1/2,1$.
Furthermore, all of $Z_N$ orbifold models with $v_i=1/2$
have only massless modes with $r_i=0,1/2,1$.
Thus, the above relation (\ref{relation:T-R})
holds true in such $Z_N$ orbifold models.
That is also true for $R_1$-anomalies in $Z_2 \times Z_M$
orbifold models with $v_1=(1/2,-1/2,0)$ and $v_2=(0,v_2,-v_2)$.
Such relation between T-duality anomalies and
R-anomalies (\ref{relation:T-R}) would be important,
because the GS condition on R-anomalies leads to a certain
condition on the T-duality anomalies even including the
N=2 subsector.
For example, in the above $Z_4$ orbifold model,
the following condition is required
\begin{equation}
A^{T_3}_{SO(10)}=A^{T_3}_{SU(4)}=
A^{T_3}_{SO(12)}=A^{T_3}_{SU(2)} \qquad ({\rm mod}~~2).
\end{equation}
\section{Phenomenological implications}
\subsection{Symmetry breaking of
the discrete R-symmetries}
\subsubsection{Nonperturbative breaking}
If the discrete R-symmetries are anomalous, they
are broken by nonperturbative effects at low energy.
This is because, for
the GS mechanism to take place,
the axionic part of the dilaton $S$ should transform
non-linearly under the anomalous symmetry.
This means that a term like $e^{-aS}$
with a constant $a$ has a
definite charge $R_i^S$ under the anomalous symmetry.
Nonperturbative effects can therefore induce
terms like $e^{-aS}\Phi^1 \cdots \Phi^n$
with matter fields $\Phi^a$, where the total
charge satisfies the condition for allowed couplings,
i.e. $R^S_i+\sum_a R^a_{i}=1$ (mod $N_i$).
This implies that below the scale of the
vacuum expectation value (VEV) of $S$,
such non-invariant terms can
appear in a low-energy effective Lagrangian.
The canonical dimension of the non-invariant operator
$e^{-aS}\Phi^1 \cdots \Phi^n$ that can be generated by the
nonperturbative effects depends of course on the R charge $R^S$.
If the smallest dimension is lager than four, they will be
suppressed
by certain powers of the string scale.
However, the operator can produce
non-invariant mass terms like $m \Phi \Phi'$,
because some of the chiral superfields may acquire VEVs.
One should worry about such cases.
Needless to say that small higher dimensional terms would be useful
in phenomenological applications such as explaining
fermion masses.
In the case that
the smallest dimension is smaller than three,
the anomalous discrete R symmetry
has less power to constrain the low-energy theory.
\subsubsection{Spontaneous breaking}
In the discussion above, we have considered
R-symmetry breaking by nonperturbative effects when
R-symmetries are anomalous.
Here we comment on another type of symmetry breaking;
they can be broken spontaneously by the VEVs of scalar fields in the form
$U(1)\times R \rightarrow R'$.
That is, we consider a spontaneous symmetry breaking, where
some scalar fields with non-vanishing $U(1)$ and $R$ charges
develop their VEVs and they break
$U(1)$ and $R$ symmetries in such a way that an unbroken $R'$ symmetry
remains intact.
(Its order is denoted by $N'$ below.)
Even in such symmetry breaking, we can obtain
the GS condition for the unbroken $R'$ from the GS condition for the $U(1)$
and R-anomalies.
Suppose that we have the GS condition for the $U(1)$ symmetry as
\begin{equation}
Tr Q T({\bf R}_{G_a})/k_a= Tr Q T({\bf R}_{G_b})/k_b,
\end{equation}
where $Q$ is the $U(1)$ charge.
Since the unbroken $R'$ charge is a linear combination of $R_i$ and $Q$,
the mixed anomalies for $R'$ should also satisfy the
GS condition,
\begin{equation}
Tr R' T({\bf R}_{G_a})/k_a= Tr R' T({\bf R}_{G_b})/k_b.
\end{equation}
Here the anomaly coefficients $Tr R' T({\bf R}_{G_a})$ are
defined modulo $N'T({\bf R}^{(f)}_{G_a})$.
Through the symmetry breaking $U(1)\times R \rightarrow R'$,
some matter fields may gain mass terms like
\begin{equation}
W\sim m \Phi \bar \Phi.
\end{equation}
Such a pair of the matter fields $\Phi$ and $\bar \Phi$ should form
a vector-like representation of $G_{a}$ and have
opposite $R'$ charges of the unbroken $R'$ symmetry.
The heavy modes of this type have therefore no contribution to the
mixed anomalies between the gauge symmetry $G_a$ and
the unbroken $R'$ symmetry.
This implies that the above GS condition for the unbroken $R'$ remains
unchanged even after the spontaneous symmetry breaking.
The symmetry breaking
$U(1)\times R \rightarrow R'$ also allows
Majorana mass terms like
\begin{equation}
W\sim m \Phi \Phi.
\end{equation}
This type of Majorana mass terms can appear for an even
order $N'$ of the $R'$ symmetry if the $R'$
charge of $\Phi$ is $N'/2$ and $\Phi$ is
in a real representation ${\bf R}_{G_a}$
of the unbroken gauge group $G_a$.
The field $\Phi$ contributes to the anomaly coefficient as
$\frac{N'}{2}T({\bf R}_{G_a})$.
That however may change only the modulo-structure of the anomaly
coefficients.
For $SU(N)$ gauge group, this contribution is
obtained as $\frac{N'}{2}\times
({\rm integer})$.
Thus, the modulo-structure does not change, that is,
the anomaly coefficients $Tr R' T({\bf R}_{G_a})$ are defined
modulo $N'/2$.
However, for other gauge groups, the modulo-structure
of the anomaly coefficients may change.
\subsection{
Gravity-induced supersymmetry breaking and
Gauge symmetry breaking}
The most important difference of the discrete R-symmetries
compared with T-duality in phenomenological applications
comes from the fact that (for the heterotic orbifold
string models) the moduli and dilaton superfields have vanishing
R-charges.
The VEVs of their bosonic components do not therefore violate
the discrete R-symmetries in the perturbation theory.
(We have discussed above the nonperturbative effects due to the VEV of
the dilaton, which may be small in a wide class of models.)
However,
the F-components of the moduli and dilaton superfields
have non-zero R-charges. Therefore, since the VEVs of these
F-components generate soft-supersymmetry breaking (SSB) terms
at low energy, the SSB terms do not have to respect the discrete
R-symmetries. \footnote{
Whether
the nonperturbative effects due to the VEV of the dilaton
do play an important roll in the SSB sector depends on the
R charge of the dilaton, and one has to check it explicitly
for a given model.}
Fortunately, in the visible sector,
the scale of the R-symmetry breaking must be of the same
order as that of supersymmetry breaking.
If the order of the discrete R-symmetry is even,
the VEVs of these F-components break the discrete R-symmetry
down to its subgroup $Z_2$, an R-parity.
That is an interesting observation because it may be an origin
of the R-parity of the minimal supersymmetric standard model (MSSM).
Gauge symmetry breaking can be achieved by VEVs of chiral
supermultiplets in a non-trivial representation of the gauge group
or by non-trivial Wilson lines. Clearly, if the chiral supermultiplets
have vanishing R-charges and only their scalar
components acquire VEVs, the discrete R-symmetries remain
unbroken. Similarly, the Wilson lines do not break the discrete R-symmetries
because gauge fields have no R charge.
As a consequence, the discrete R-symmetries have a good chance
to be intact at low energy if the nonperturbative effects are small.
\subsection{Constraints on low-energy beta-functions}
Only anomaly-free discrete R-symmetries
remain as intact symmetries in a low-energy effective theory.
Obviously, the model with anomaly-free
discrete R-symmetries corresponds to $A^{R_i}_{G_a}=0$
(mod $N_iT({\bf R}^{(f)}_{G_a}))$.
Consider for instance $SU(N)$ gauge groups for which
$T({\bf R}^{(f)}_{G_a})=1/2$ is usually satisfied.
Then in models, which have
no oscillator mode in a non-trivial representations of $SU(N)$,
the relation between R-anomalies
and beta-function coefficients lead to
\begin{equation}
b_a = 2 A^{}_{G_a}=0,
\end{equation}
mod $N_i$ for any gauge group $G_a$.
For example, the $Z_3$ orbifold model
with anomaly-free R-symmetries leads to
$b_a=3n_a$ with integer $n_a$, while
the $Z_4$ orbifold model with anomaly-free R-symmetries
leads to $b_a=2n_a$.
Similarly,
$b_a=1$ would be possible in $Z_6$-II orbifold models
because $N_i=(6,3,2)$ as one can see from Table 1.
Even for anomalous discrete R-symmetries,
the GS condition for R-anomalies and the relation between
beta-function coefficients (\ref{GS-R}), (\ref{anomR-b}),
(\ref{GS-b}) would have phenomenological
implications. As discussed at the beginning in this section,
the non-perturbative effects can generate
operators like $e^{-aS}\Phi^1 \cdots \Phi^n$.
If its canonical dimension is larger than four, its contribution
to low-energy beta-functions may be assumed to be small.
\footnote{If the operator produces
non-invariant mass terms like $M \Phi \Phi'$ with $M$
larger than the low-energy scale,
the low-energy spectrum may change. Then
the power of the discrete R-symmetries decreases.}
As for the MSSM
we find $b_3=-3$ and $b_2=1$ for $SU(3)$ and $SU(2)$, respectively.
That is, we have $b_2 - b_3=4$, implying
the MSSM can not be realized, e.g. in $Z_3$ orbifold models,
because $Z_3$ orbifold models require
$b_a - b_b=0$ mod $3$ if the effects of the symmetry breaking
of the discrete R-symmetries can be neglected.
Similarly, the model with $b_2 - b_3=4$ can not be
obtained in the $Z_6$-I, $Z_7$ or $Z_{12}$-I orbifold models.
\section{Conclusion}
We have studied anomalies of the discrete R-symmetries
in heterotic orbifold models.
They are remnants of $SU(4)_R$ symmetry which,
along with extended $N=4$ supersymmetry,
is explicitly broken by orbifolding.
We have found that
the mixed anomalies for different gauge groups
satisfy the universal GS condition.
Therefore,
these anomalies
can be canceled by the GS mechanism,
which remains to be proven at the string theory level.
As a byproduct,
we have found a relation between
the anomaly coefficients
of the discrete R-symmetries
and one-loop beta-function coefficients.
In particular, in the case that
the contribution coming from the oscillator modes for
the chiral matter fields in
non-trivial representations of a gauge group
vanishes, the anomaly coefficient corresponding to
the sum of the discrete R-symmetry anomaly
is exactly proportional to the one-loop beta-function
coefficient of the corresponding gauge coupling.
In a wide class of models, the discrete R-symmetries
may be unbroken at low energy.
The main reason for this is that the moduli
superfields have vanishing R-charges.
This should be contrasted to the case of T-duality,
where the moduli fields transform non-trivially
under the T-duality transformation.
We have studied the relation between anomalies of
the discrete R-symmetries and T-duality.
We have argued that the discrete R-symmetries
have a good chance to be unbroken down to
the supersymmetry breaking scale.
Even below this scale a $Z_2$ subgroup
is unbroken, which may be an origin of
the R-parity of the MSSM.
In fact, the R-parity of the MSSM is completely anomaly-free,
indicating that it has a stringy origin.
Our investigation
on the discrete R-symmetries in heterotic orbifold models
could be extended to other types of heterotic models, e.g.
free fermionic construction \cite{Antoniadis:1989zy} and
Gepner models \cite{Gepner:1987vz} as well as
Calabi-Yau models.
Furthermore, our studies
can be extended to type IIA and IIB string theories
with D-branes, e.g. intersecting/magnetized D-brane models.
This however would be beyond the scope of the present paper,
and we will leave it to our future study.
At last we emphasize that
string models have other discrete symmetries.
For example, heterotic orbifold models have non-abelian discrete flavor
symmetries \cite{Kobayashi:2006wq}.
They may be identified with the non-abelian discrete flavor
symmetries which have been recently introduced
in constructing low-energy flavor models \cite{Altarelli:2007cd}.
Further investigations in this direction are certainly
necessary to link the non-abelian discrete flavor symmetries
from the top and the bottom with each other.
\subsection*{Acknowledgement}
K.~S.~C. \/ is supported in part by the European Union 6th framework program
MRTN-CT-2004-503069 "Quest for unification",
MRTN-CT-2004-005104 "ForcesUniverse", MRTN-CT-2006-035863
"UniverseNet and SFB-Transregio 33 "The Dark Univeres"
by Deutsche Forschungsgemeinschaft (DFG).
T.~K.\/ and J.~K.\/ are supported in part by the Grand-in-Aid for
Scientific Research \#1754025,\#18540257 and \#19034003, respectively.
T.~K.\/ is also supported in part by the Grant-in-Aid for the 21st Century
COE ``The Center for Diversity and Universality in Physics'' from the
Ministry of Education, Culture, Sports, Science and Technology of Japan.
|
1,108,101,565,433 | arxiv | \section{Introduction}
\label{s:int}
Suppose we want to transmit or store a block of $l$ qubits (i.e. two-state
quantum systems) in a noisy environment. Here `noisy' means that each qubit
may become entangled with the environment. Thus due to spurious interactions
with the environment the actual state of the $l$ qubits, described by a
density operator $\rho(t)$, will differ from the original state $\ket{\Psi}$.
This deviation can be quantified by the fidelity
\begin{equation}
F(t)=\bra{\Psi}\rho(t)\ket{\Psi} =1-\epsilon(t).
\end{equation}
In order to maximize this fidelity we may try all sorts of tricks ranging
from the most obvious one i.e. isolating the qubits from the environment to
more sophisticated methods such as ``symmetrisation" \cite{Deutsch1993,Barenco
etal}, ``purification" \cite{Bennett etal,QPA}, and ``quantum error
correction" \cite{Shor}.
The last method seems to be the most popular one at
the moment and relies on encoding the state of $l$ qubits into a set of $n$
qubits and trying to disentangle a certain number of qubits from the
environment after some period of time. In the following we describe, very
briefly, how some of these techniques work.
We will assume that in the block of $l$ qubits each qubit is coupled to a different environment.
This is a perfectly reasonable assumption, which is valid if the coherence length of the
environment/reservoir is less than the spatial separation between the qubits~\cite{palma}, and
introduces a great deal of simplifications to the calculations. Basically it allows us
to view any dissipation of $l$ qubits as a set of independent dissipations of $l$
single qubits (i.e. we ignore collective phenomena such as superradiance etc.).
The qubit--environment interaction leads to the qubit--environment
entanglement, which in its most general form is given by
\begin{eqnarray}
\ket{0}\ket{R} & \longrightarrow & \ket{0}\ket{R_{00}(t)} +
\ket{1}\ket{R_{01}(t)},\\ \ket{1}\ket{R} & \longrightarrow &
\ket{0}\ket{R_{10}(t)} + \ket{1}\ket{R_{11}(t)},
\label{dissipation}
\end{eqnarray}
where states of the environment $\ket{R}$ and $\ket{R_{ij}}$ are
neither normalised nor orthogonal to each other (thus we have to take
additional care at the end of our calculations and normalise the final
states). The r.h.s. of the formulae above can also be written in a
matrix form as
\begin{equation}
\left( \begin{array}{cc}
\ket{R_{00}} & \ket{R_{01}} \\
\ket{R_{10}} & \ket{R_{11}}
\end{array}
\right)
\left( \begin{array}{c}
\ket{0}\\ \ket{1}\end{array}\right ),
\end{equation}
and the 2 $\times$ 2 matrix can can be subsequently decomposed into
some basis matrices e.g. into the unity and the Pauli matrices
\begin{equation}
\ket{R_0}1+\ket{R_1}\sigma_x + i\ket{R_2}\sigma_y + \ket{R_3}\sigma_z,
\end{equation}
where $\ket{R_0}= (\ket{R_{00}}+\ket{R_{11}})/2$, $\ket{R_3}=
(\ket{R_{00}}-\ket{R_{11}})/2$, $\ket{R_1} = (\ket{R_{01}}+\ket{R_{10}})/2$,
and $\ket{R_2} =(\ket{R_{01}}-\ket{R_{10}})/2$. Thus the qubit initially
in state $\ket{\Psi}$ will evolve as
\begin{equation}
\ket{\Psi}\ket{R} \longrightarrow \sum_{i=0}^3\sigma_i\ket{\Psi}\ket{R_i}
\label{action}
\end{equation}
becoming entangled with the environment (we have relabelled the unity
operator and the Pauli matrices $\{1, \sigma_x, \sigma_y, \sigma_z\}$
respectively as $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$). Its
fidelity with respect to the initial state $\ket{\Psi}$ evolves as
\begin{equation}
F(t) = \sum_{i,j} \bra{\Psi}\sigma_i\ket{\Psi}
\bra{\Psi}\sigma_j\ket{\Psi}\bra{R_j(t)}R_i(t)\rangle .
\label{fidelity}
\end{equation}
The formula (\ref{action}) describes how the environment affects any
quantum state of a qubit and shows that a general qubit--environment interaction
can be expressed as a superposition of unity and Pauli operators acting
on the qubit. As we will see in the following, in the language of error
correcting codes this means that the qubit state is evolved into a superposition
of an error-free component and three erroneous components, with errors
of the $\sigma_x$, $\sigma_y$ and $\sigma_z$ type.
We can carry on this description even if the qubit itself is not in a pure
state $\ket{\Psi}$ but is
entangled with some other qubits. For example, if in a three qubit
register initially in state $\ket{\tilde\Psi} =
\ket{0}\ket{0}\ket{0}-\ket{1}\ket{1}\ket{1}$ the second qubit
interacted with its environment then the state of the register at some
time $t$ is given by
\begin{equation}
\sum_{i=0}^3\sigma_i^{(2)} \ket{\tilde\Psi}\ket{R_i (t)} =
\sum_{i=0}^3(\ket{0}(\sigma_i\ket{0})\ket{0} -
(\ket{1}(\sigma_i\ket{1})\ket{1})\ket{R_i (t)},
\label{int}
\end{equation}
where the superscript $(2)$ reminds us that the Pauli operators act
only on the second qubit. We can then say that the second qubit was affected
by quantum errors which are represented by the Pauli operators
$\sigma_i$. Errors affecting classical bits can only change their
binary values ($0\leftrightarrow 1$), in contrast
quantum errors operators $\sigma_i$ acting on qubits can change their
binary values ($\sigma_x$), their phases ($\sigma_z$) or both
($\sigma_y$).
In general, a batch of $n$ qubits initially in some state
$\ket{\tilde\Psi}$, each of them interacting with different
environments, will evolve as
\begin{equation}
\prod_{k=1}^n\sum_{i=0}^3\sigma_i^{(k)} \ket{\tilde\Psi}\ket{R_i^{(k)} (t)}\;,
\end{equation}
namely multiple errors of the form
$\sigma_i\otimes\sigma_j\cdot\cdot\cdot\otimes\sigma_k$ may occur, affecting
several qubits at the same time.
So much about unwelcome dissipation, what about remedies?
\section{Stabilization via symmetrisation}
The first proposed remedy was based on a symmetrisation procedure
\cite{Deutsch1993}. The basic idea is as follows. Suppose you have a quantum
system, you prepare it in some initial state $\ket{\Psi_i}$ and you
want to implement a prescribed unitary evolution $\ket{\Psi (t)}$ or
simply you want to preserve $\ket{\Psi_i}$ for some period of time
$t$. Now, suppose that instead of a single system you can prepare $R$
copies of $\ket{\Psi_i}$ and subsequently you can project the state of
the combined system on the symmetric subspace i.e. the subspace
containing all states which are invariant under any permutation of the
sub-systems. The claim is that frequent projections on the symmetric
subspace will reduce errors induced by the environment. The intuition
behind this concept is based on the observation that a prescribed
error-free storage or evolution of the $R$ independent copies starts
in the symmetric sub-space and should remain in that
sub-space. Therefore, since the error-free component of any state
always lies in the symmetric subspace, upon successful projection it
will be unchanged and part of the error will have been removed. Note
however that the projected state is generally not error--free since
the symmetric subspace contains states which are not of the simple
product form $\ket{\psi}\ket{\psi}\ldots\ket{\psi}$. Nevertheless it
has been shown that the error probability will be suppressed by a
factor of $1/R$ \cite{Barenco etal}.
We illustrate here this effect in the simplest case of two qubits.
The projection into the symmetric subspace is performed in this case
by introducing the symmetrisation operator:
\begin{equation}
\label{symop2}
S= \frac{1}{2} (P_{12}+P_{21})\;,
\end{equation}
where $P_{12}$ represents the identity and $P_{21}$ the permutation
operator which exchanges the states of the two qubits. The
symmetric--projection of a pure state $\ket{\Psi} $ of two qubits is
just $S\ket{\Psi}$, which is then renormalised to unity. It follows
that the induced map on mixed states of two qubits (including
renormalisation) is:
\begin{equation}
\label{symproj2}
\rho_1 \otimes \rho_2 \longrightarrow
\frac{S(\rho_1 \otimes\rho_2 )
S^{\dagger}}{\mbox{Tr}{S(\rho_1 \otimes\rho_2 )
S^{\dagger}}}
\end{equation}
The state of either qubit separately is then obtained by partial trace
over the other qubit.
Consider for example the symmetric projection of $\rho\otimes \rho$
followed by renormalisation and partial trace (over either qubit) to
obtain the final state ${\rho_s}$ of one qubit, given that the
symmetric-projection was successful. A direct calculation based on
(\ref{symproj2}) yields:
\begin{equation}
\label{twid}
\rho \mapsto {\rho_s}= \frac{\rho +\rho^2}{\mbox{Tr}{(\rho +\rho^2 ) }}
\end{equation}
For any mixed state $\xi$ of a qubit the expression $\mbox{Tr}{\xi ^2
}$ provides a measure of the purity of the state, ranging from
$\frac{1}{4}$ for the completely mixed state $I/2$ (where $I$ is the
unit operator) to 1 for any pure state. From (\ref{twid}) we get
\begin{equation}
\mbox{Tr}{{\rho_s}^2} > \mbox{Tr}{\rho ^2}
\end{equation}
so that ${\rho_s}$ is {\em purer} than $\rho$. This illustrates that
successful projection of a mixed state into the symmetric subspace
tends to enhance the purity of the individual systems.
To be more specific, let us assume now that the two copies are
initially prepared in pure state $\rho_0 = \proj{\Psi}$ and that they
interact with independent environments. After some short period of
time $\delta t$ the state of the two copies $\rho^{(2)}$ will have
undergone an evolution
\begin{equation}
\label{dec}
\rho^{(2)}(0) = \rho_0 \otimes \rho_0 \hspace{5mm}
\longrightarrow
\hspace{5mm} \rho^{(2)}(\delta t) = \rho_1\otimes\rho_2
\end{equation}
where $\rho_i = \rho_0 + \varrho_i$ for some Hermitian traceless
$\varrho_i$.
We will retain only terms of first order in the perturbations
$\varrho_i$ so that the overall state at time $\delta t$ is
\begin{eqnarray}
\rho^{(2)} = \rho_0 \otimes\rho_0
+ \varrho_1 \otimes \rho_0 +\rho_0 \otimes \varrho_2
+ O(\varrho_1 \varrho_2 )\;.
\label{decst}
\end{eqnarray}
We can calculate the average purity of the two copies before symmetrisation by calculating the
average trace of the squared states:
\begin{equation} \label{this}
\frac{1}{2} \sum_{i=1}^2 \mbox{Tr}( (\rho_0+\varrho_i)^2) = 1 + 2
\mbox{Tr}(\rho_0\tilde\varrho ),
\end{equation}
where $\tilde\varrho=\frac{1}{2}(\varrho_1+\varrho_2)$.
Note that $\mbox{Tr}({\rho_0 \tilde\varrho})$ is negative, so that the
expression above does not exceed 1.
After symmetrisation each qubit is in state
\begin{equation}
\rho_s=[1- \mbox{Tr}(\rho_0\tilde\varrho )] \rho_0 +\frac{1}{2}\tilde\varrho
+\frac{1}{2}(\rho_0\tilde\varrho+ \tilde\varrho\rho_0)
\end{equation}
and has purity
\begin{equation}
\mbox{Tr}(\rho_s^2)=1 + \mbox{Tr}(\rho_0\tilde\varrho).
\end{equation}
Since $\mbox{Tr}{\rho_s^2 }$ is closer to 1 than (\ref{this}),
the resulting symmetrised
system $\rho_s$ is left in a purer state.
Let us now see how the fidelity changes by applying the symmetrisation
procedure. The average fidelity before symmetrisation is
\begin{equation} \label{fid1}
F_{bs}=\frac{1}{2} \sum_i \bra{\Psi}\rho_0 +\varrho_i\ket{\Psi} =
1+ \bra{\Psi} \tilde\varrho \ket{\Psi}\;,
\end{equation}
while after successful symmetrisation it takes the form
\begin{equation}
F_{as}=\bra{\Psi}\rho_s \ket{\Psi} =
1+ \frac{1}{2} \bra{\Psi}\tilde\varrho \ket{\Psi}\;.
\end{equation}
The state after symmetrisation is therefore closer to the initial state
$\rho_0$.
For the generic case of $R$ copies the purity of each qubit after
symmetrisation is given by \cite{Barenco etal}
\begin{equation}
\mbox{Tr}(\rho_s^2)=1 + 2\frac{1}{R} \mbox{Tr}(\rho_0\tilde\varrho)\;,
\label{purR}
\end{equation}
where now $\tilde\varrho=\frac{1}{R}\sum_{i=1}^R\varrho_i$,
and the fidelity takes the form
\begin{equation} \label{fid2}
\bra{\Psi}\rho_s \ket{\Psi} = 1+ \frac{1}{R} \mbox{Tr}
({\rho_0 \tilde\varrho })\;.
\end{equation}
Formulae (\ref{purR}) and (\ref{fid2}) must be compared with the
corresponding ones before symmetrisation, i.e. (\ref{this}) and (\ref{fid1}).
As we can see, $\rho_s$ approaches the unperturbed state
$\rho_0$ as $R$ tends to infinity.
Thus by choosing $R$ sufficiently large and the rate of symmetric projection
sufficiently high, the residual error at the end of a computation can,
in principle, be controlled to lie within any desired small tolerance.
The efficiency of the symmetrisation procedure depends critically on
the probability that the state of the $R$ qubits is successfully
projected into the symmetric subspace. It has been shown that if the
projections are done frequently enough, then the cumulative
probability that they all succeed can be made as close as desired to
unity. This is a consequence of the fact that the fidelity of the
state of the $R$ computers with respect to the corresponding error
free state for small times $\delta t$ has a parabolic behaviour (see
section \ref{s:dyn}). Therefore the probability of successful projection, which
is unity at the initial time, begins to change only to second order in
time. If we project $n$ times per unit time interval, i.e. we choose
the time interval between two subsequent projections $\delta t = 1/n$,
then the cumulative probability that all projections in one unit time
interval succeed is given by
\begin{eqnarray}
[1-k(\delta t)^2 ]^n = (1-\frac{k}{n^2})^n \rightarrow 1
\mbox{ as } n\rightarrow \infty\;.
\end{eqnarray}
Here $k$ is a constant depending on the rate of rotation of the state
out of the symmetric subspace. This effect is known as the ``quantum
watch-dog effect'' or the ``quantum Zeno effect".
\section{Quantum encoding and decoding}
\label{s:enc-dec}
The idea of protecting information via encoding and decoding lies at
the foundations of the classical information theory. It is based on a
clever use of redundancy during the data storage or transmission. For
example, if the probability of error (bit flip) during a single bit
transmission via a noisy channel is $p$ and each time we want to send
bit value 0 or 1 we can encode it by a triple repetition i.e. by
sending 000 or 111. At the receiving end each triplet is decoded as
either zero or one following the majority rule - more zeros means 0,
more ones means 1. This is the simplest error correcting protocol
which allows to correct up to one error.
In the triple repetition code the signalled bit value is recovered
correctly both when there was no error during the transmission of the
three bits, which happens with probability $(1-p)^3$, and when there
was one error at any of the three locations, which happens with
probability $3p(1-p)^2$. Thus the probability of the correct
transmission (up to the second order in $p$) is $1-3p^2$ i.e. the
probability of error is now $3p^2$, which is much smaller when
compared with the probablity of error without encoding and decoding
$p$ ($p\ll 1$).
This way we can trade the probability of error in the
signalled message for a number of transmissions via the channel.
In our example the reduction of the error rate
from $p$ to $3p^2$ required to send three times more bits.
If sending each bit via the channel costs us money we
have to decide what we treasure more, our bank account or our
infallibility. The triple repetition code encodes one bit into three
bits and protects against one error, in general we can construct codes
that encode $l$ bits into $n$ bits and protect against $t$ errors. The
best codes, of course, are those which for a fixed value $l$ minimize
$n$ and maximize $t$.
Quantum error correction which protects quantum states is a little bit
more sophisticated simply because the bit flip is not the only
''quantum error" which may occur, as we have seen in the previous
sections. Moreover, the decoding via the majority rule does not
usually work because it may involve measurements which destroy quantum
superpositions. Still, the triple repetition code is a good starting
point to investigate quantum codes and even to construct the simplest
ones.
The simplest interesting case of the most general qubit--environment
evolution (\ref{dissipation}) is the case of decoherence \cite{Zurek1991}
where the environment effectively acts as a measuring apparatus
\begin{eqnarray}
\ket{0}\ket{R} & \longrightarrow & \ket{0}\ket{R_{00}(t)},\\
\ket{1}\ket{R} & \longrightarrow & \ket{1}\ket{R_{11}(t)}.
\label{decoherence}
\end{eqnarray}
Following our discussion in Section \ref{s:int} we can see that this model
leads only to dephasing errors of the $\sigma_z$ type. It turns out
that a phase flip can be handled almost in the same way as a classical
bit flip. Again, consider the following scenario: we want to store, in
a computer memory, one qubit in an {\em unknown} quantum state of the
form $\alpha\ket{0}+\beta\ket{1}$ and we know that any single qubit
which is stored in a register may, with a small probability $p$,
undergo a decoherence type entanglement with an environment
(Eq. \ref{decoherence}); in particular
\begin{equation}
(\alpha\ket{0}+\beta\ket{1})\ket{R}\longrightarrow
\alpha\ket{0}\ket{R_{00}}+\beta\ket{1}\ket{R_{11}}.
\end{equation}
Let us now show how to reduce the probability of decoherence to be of
the order $p^2$.
Before we place the qubit in the memory register we {\em encode} it:
we can add two qubits, initially both in state $\ket{0}$, to the
original qubit and then perform an encoding unitary transformation
\begin{eqnarray}
\ket{000}&\longrightarrow &\ket{C_0}
=(\ket{0}+\ket{1})(\ket{0}+\ket{1})(\ket{0}+\ket{1}),\\
\ket{100}&\longrightarrow &\ket{C_1}
=(\ket{0}-\ket{1})(\ket{0}-\ket{1})(\ket{0}-\ket{1}),
\end{eqnarray}
generating state $\alpha\ket{C_0}+\beta\ket{C_1}$.
Now, suppose that only the second stored
qubit was affected by decoherence and became entangled with the environment:
\begin{eqnarray}
\alpha (\ket{0}+\ket{1})(\ket{0}\ket{R_{00}}
+\ket{1}\ket{R_{11}})(\ket{0}+\ket{1}) +\nonumber\\
\beta (\ket{0}-\ket{1})(\ket{0}\ket{R_{00}}
-\ket{1}\ket{R_{11}})(\ket{0}-\ket{1}),
\end{eqnarray}
which, following Eq. (\ref{int}), can be written as
\begin{equation}
(\alpha\ket{C_0} + \beta\ket{C_1})\ket{R_0}
+ \sigma_z^{(2)}(\alpha\ket{C_0} + \beta\ket{C_1})\ket{R_3}.
\end{equation}
If vectors $\ket{C_0}$, $\ket{C_1}$, $\sigma_z^{(k)}\ket{C_0}$, and
$\sigma_z^{(k)}\ket{C_1}$ are orthogonal to each other we can try to
perform a measurement on the qubits and project their state either on
the state $\alpha\ket{C_0} + \beta\ket{C_1}$ or on the orthogonal one
$\sigma_z^{(2)}(\alpha\ket{C_0} + \beta\ket{C_1})$. The first case yields
the proper state right away, the second one requires one application
of $\sigma_z$ to compensate for the error. In this simple case one can
even find a direct unitary operation which can fix all one qubit phase
flips regardless their location. For example the transformation
\begin{eqnarray}
\ket{000}\to \ket{000} & & \ket{100} \to \ket{011}\nonumber \\
\ket{001}\to \ket{001} & & \ket{101} \to \ket{110}\nonumber \\
\ket{010}\to \ket{010} & & \ket{110} \to \ket{101}\nonumber \\
\ket{011}\to \ket{111} & & \ket{111} \to \ket{100}
\end{eqnarray}
corrects any single bit flip $0\leftrightarrow 1$ and when applied in
the conjugate basis ($\ket{0'}=\ket{0}+\ket{1}$,
$\ket{1'}=\ket{0}-\ket{1}$) it corrects any single phase flip (the bit
flips become phase flips in the new basis). The snag is that using the
scheme above we can correct up to one phase error $\sigma_z$ or we can
go to a conjugate basis and the same scheme will correct up to one
amplitude error $\sigma_x$ but it cannot correct up to one general
error, be it amplitude or phase.
To fix this problem Peter Shor in 1995 combined the phase and the
amplitude correction schemes into one constructing the following nine
qubit code \cite{Shor}:
\begin{eqnarray}
&&\ket{0}\to \frac{1}{2\sqrt{2}}(\ket{000}+\ket{111})(\ket{000}+\ket{111})
(\ket{000}+\ket{111})
\label{shor1}\\
&&\ket{1}\to \frac{1}{2\sqrt{2}}(\ket{000}-\ket{111})(\ket{000}-\ket{111})
(\ket{000}-\ket{111})\;.
\label{shor2}
\end{eqnarray}
This code involves double encoding, first in base $\ket{0}$ and
$\ket{1}$ and then in base $\ket{0'}$ and $\ket{1'}$, and it
allows to correct up to one either bit or phase flip. It turns out
that the ability to correct both amplitude and phase errors suffices
to correct any error due to entanglement with the environment. In
other words the action of the environment on qubits can be viewed in
terms of bit and phase flips.
\section{Quantum error-correcting codes}
The original nine qubit code of Shor can be further simplified. It has
been shown that a five qubit code suffices to correct a single error
of any type. Let us now specify the conditions for the existence of
quantum error-correcting codes.
We say we can correct a single error $\sigma_i^{(k)}$ (where $i=0\ldots 3$
refers to the type of error)
if we can find a transformation such that it maps all states
with a single error $\sigma_i^{(k)}\ket{\tilde\Psi}$ into the proper error
free state $\ket{\tilde\Psi}$:
\begin{equation}
\sigma^{(k)}_i \ket{\tilde\Psi} \longrightarrow \ket{\tilde\Psi}
\end{equation}
To make it unitary we may need an ancilla
\begin{equation}
\sigma_i^{(k)} \ket{\tilde\Psi}\ket{0} \longrightarrow \ket{\tilde\Psi}
\ket{a_i^k}\;.
\end{equation}
For encoded basis states of a single qubit
$\ket{C_0}$ and $\ket{C_1}$ this implies \cite{bdws}
\begin{eqnarray}
A_k \ket{C_0 }\ket{0} & \longrightarrow & \ket{C_0}\ket{a_k}\\
A_k \ket{C_1 }\ket{0} & \longrightarrow & \ket{C_1}\ket{a_k},
\end{eqnarray}
where $A_k$ denotes all the possible types of independent errors
affecting at most one of the qubits.The above requirement leads to the
following unitarity conditions
\begin{eqnarray}
\bra{C_0} A^\dagger_k A_l\ket{C_0}&=&\bra{C_1}
A^\dagger_k A_l\ket{C_1} = \bra{a_k} a_l\rangle\;,
\label{condec}\\
\bra{C_0}A^\dagger_k A_l\ket{C_1}&=& 0 \;.
\end{eqnarray}
The above conditions are straightforwardly generalised to an arbitrary
$t$ error correcting code, which corrects any kind of transformations affecting
up to $t$ qubits in the encoded state. In this case the operators $A_k$ are
all the possible independent errors affecting up to $t$ qubits, namely
operators of the form $\Pi_{i=1}^t\sigma_i$ acting on $t$ different qubits.
In the case of the so-called ``nondegenerate codes''
Eq. (\ref{condec}) takes the simple form \cite{nostro}
\begin{equation}
\bra{C_0} A^\dagger_k A_l\ket{C_0} =\bra{C_1} A^\dagger_k A_l\ket{C_1}=0\;.
\label{orth}
\end{equation}
This condition requires
that all states which are obtained by affecting up to $t$ qubits in the
encoded states are all orthogonal to each other, and therefore distinguishable.
This ensures that by performing suitable projections of the encoded
state we are able to detect the kind of error which occurred and ``undo'' it
to recover the desired error free state.
Condition (\ref{orth}), even if more restrictive than (\ref{condec}), is
particularly useful because it allows to establish bounds on the resources
needed in order to have efficient nondegenerate codes.
Let us assume that the initial state of $l$ qubits
is encoded in a redundant Hilbert space of $n$ qubits.
If we want to encode $2^l$ input basis states and correct up to $t$ errors
we must choose the dimension of the encoding Hilbert space $2^n$ such that
all the necessary orthogonal states can be accomodated.
According to Eq. (\ref{orth}), the total number of orthogonal states that we
need in order to be able to correct $i$ errors of the three types $\sigma_x$,
$\sigma_y$ and $\sigma_z$ in an $n$-qubit state is
$3^i\left(\begin{array}{c} n \\ i \end{array}\right)$ (this is the number of
different ways in which the errors can occur).
The argument based on counting orthogonal states then leads to the following
condition
\begin{eqnarray}
2^l\sum_{i=0}^t 3^i\left(\begin{array}{c} n \\ i
\end{array}\right)\leq 2^n.\label{hamming}
\end{eqnarray}
Eq. (\ref{hamming}) is the quantum version of the Hamming bound for
classical error-correcting codes~\cite{macw}; given $l$ and $t$ it
provides a lower bound on the dimension of the encoding Hilbert space
for nondegenerate codes. Let us mention that an explicit construction for
quantum codes for some values $(l,n,t)$ which saturate the quantum Hamming
bound has been provided \cite{gottesman}.
It is interesting that this bound has not been beaten so far by degenerate
codes \cite{barolo}.
The quantum version of the classical
Gilbert-Varshamov bound~\cite{macw} can be also obtained, which gives
an upper bound on the dimension of the encoding Hilbert space for optimal
non degenerate codes:
\begin{eqnarray}
2^l\sum_{i=0}^{2t} 3^i\left(\begin{array}{c} n \\ i
\end{array}\right)\geq 2^n.\label{gvbound}
\end{eqnarray}
This expression can be proved from the observation that in the $2^n$
dimensional Hilbert space with a maximum number of encoded basis vectors
(or code-vectors) ${\left | \, C^k \right \rangle}$
any vector which is orthogonal to ${\left |
\, C^k \right \rangle}$ (for any $k$) can be reached by applying up to
$2t$ error operations of $\sigma_x$, $\sigma_y$, and $\sigma_z$ type to
any of the $2^l$ encoded basis vectors. Clearly all vectors which cannot be
reached in the $2t$ operations can be added to the encoded basis states
${\left | \, C^k \right \rangle}$ as all the vectors into which they can be
transformed by applying up to $t$ amplitude and/or phase transformations
are orthogonal to all the others. This situation cannot happen because
we have assumed that the number of code-vectors is maximal. Thus the
number of orthogonal vectors that can be obtained by performing up to
$2t$ transformations on the code-vectors must be at least equal to the
dimension of the encoding Hilbert space.
It follows from Eq. (\ref{hamming}) that a one-bit quantum error
correcting code to protect a single qubit ($l=1$, $t=1$) requires at
least $5$ encoding qubits and, according to Eq. (\ref{gvbound}), this
can be achieved with less than $10$ qubits. Indeed, Shor's nine qubit
code can be simplified to the seven qubit code \cite{steane}, and
ultimately to the quantum Hamming bound \cite{bdws,lafl}. We will consider
explicitly one form of the five qubit code in Section \ref{s:beyond}.
The asymptotic form of the quantum Hamming bound (\ref{hamming}) in the
limit of large $n$ is given by
\begin{eqnarray}
\frac{l}{n}\leq 1-\frac{t}{n}\log_2 3 -H(\frac{t}{n}),\label{hamasym}
\end{eqnarray}
where $H$ is the entropy function $H(x)=-x \log_2 x-(1-x)\log_2(1-x)$.
The corresponding asymptotic form for the quantum Gilbert-Varshamov bound
(\ref{gvbound}) is
\begin{equation}
\frac{l}{n}\geq 1-\frac{2t}{n}\log_2 3 -H(\frac{2t}{n}).\label{gvasym}
\end{equation}
As we can see from eq. (\ref{hamasym}), in quantum error correction there is
an upper bound on the error rate $t/n$ which a code can tolerate.
In fact, differently from the classical case, where any arbitrary error rate
can be corrected by a suitable code, in the quantum world the ratio $t/n$
cannot be larger than 0.18929 for nondegenerate codes.
\section{System-environment dynamics}
\label{s:dyn}
In order to provide a tangible illustration of some abstract ideas
discussed in the text we have picked up the most popular
quantum-optical model of dissipation commonly used to describe
spontaneous emission. A two level atom, with two energy eigenstates
$\ket{0}$ and $\ket{1}$ separated by $\hbar\omega_0$, interacting with
an environment modelled as a set of quantised harmonic oscillators,
e.g. a set of quantised modes of radiation with frequencies
$\omega_m$. The Hamiltonian of the combined system $H= H_0 +V$
includes both the free evolution of the qubit and the environment. The
free evolution Hamiltonian is given by
\begin{equation}
H_0 = \hbar\omega_0 \proj{1} +\sum_m \hbar\omega_m a^\dagger_m a_m\;,
\end{equation}
where $a_m$ and $a^\dagger_m$ represent the annihilation and
creation operators of the radiation mode of frequency $\omega_m$.
The interaction (in the rotating wave approximation) is described by
\begin{equation}
V = \sum_m \lambda_m \ket{0}\bra{1} a^\dagger_m + \lambda^\star_m
\ket{1}\bra{0} a_m\;,
\end{equation}
where $\lambda_m$ is
the coupling constant between the qubit and the mode of frequency $\omega_m$.
In order to find the time evolution of the relative states of the
environment $\ket{R_i(t)}$ we need some knowledge about the
qubit-environment interaction. Let us then have a closer look at a
dissipative dynamics in our model of a qubit coupled to a continuum
of field modes or harmonic oscillators. If all the oscillators in the
environment are in their ground states and the qubit is initially
prepared in state $\ket{\Psi}=\alpha\ket{0}+\beta\ket{1}$ then the
dynamics described by the Hamiltonian $H=H_0+V$ does not affect state
$\ket{0}$. It is state $\ket{1}$ which undergoes a decay. Let us then
consider a case when the initial state of the combined system
(qubit+environment) is
\begin{equation}
\ket{\phi_i}=\ket{1}(\ket{0}_1\ket{0}_2\ldots\ket{0}_f\ldots\ket{0}_{max}),
\end{equation}
meaning the qubit is in state $\ket{1}$ and all the harmonic
oscillators in their ground states $\ket{0}$ (we will denote the state where
all harmonic oscillators are in the ground state as the vacuum
$\ket{\bf 0}$). Possible final states of the combined system are
\begin{equation}
\ket{\phi_f}=\ket{0}(\ket{0}_1\ket{0}_2\ldots\ket{1}_f\ldots\ket{0}_{max}),
\end{equation}
where the qubit decayed to state $\ket{0}$ and one of the harmonic
oscillators got excited. Let us note that
\begin{equation}
H_0 \ket{\phi_i} = \hbar\omega_0 \ket{\phi_i} ,\quad H_0 \ket{\phi_f}
= \hbar\omega_f \ket{\phi_f},\quad \bra{\phi_f} H_0 \ket{\phi_i} = 0 ,
\quad \bra{\phi_f} V \ket{\phi_i} = \lambda_f .
\label{Ham}
\end{equation}
Let us write $\ket{\phi (t)}$ as
\begin{equation}
\ket{\phi (t)} = c_i(t)e^{-i\omega_0 t}\ket{\phi_i} + \sum_f c_f (t)
e^{-i\omega_f t}\ket{\phi_f}
\end{equation}
which, using our notation from the previous section, implies
$\ket{R_{00}}=\ket{\bf{0}}$, $\ket{R_{01}}=0$, $\ket{R_{10}(t)}=\sum_f
c_f(t)e^{-i\omega_ft}\ket{1_f}$ and $\ket{R_{11}(t)}=c_i(t)e^{-i\omega_0
t}\ket{\bf{0}}$.
In order to find the relevant time dependance we have to solve the
Schr\"odinger equation
\begin{eqnarray}
i\hbar \dot{c}_i (t)
& =& \sum_f \lambda_f^\star e^{-i(\omega_f - \omega_0) t} c_f (t)
\label{Schr1}\\
i\hbar \dot{c}_f (t)
& =& \lambda_f e^{i(\omega_f - \omega_0) t} c_i (t).
\label{Schr2}
\end{eqnarray}
The second equation can be solved formally for $c_f (t)$
\begin{equation}
c_f(t) = - \frac{i}{\hbar} \int_0^t dt' \lambda_f e^{i(\omega_f -
\omega_0) t'} c_i (t')
\label{eqcf}
\end{equation}
and after substituting this expression for $c_f(t)$ in
Eq.(\ref{Schr1}) we obtain
\begin{equation}
\dot {c}_i(t) = - \int_0^t dt' K(t-t') c_i(t'), \qquad K(\tau) =
\frac{1}{\hbar^2}\sum_f|\lambda_f|^2 e^{-i(\omega_f - \omega_0) \tau}.
\end{equation}
It is the function $\lambda_f = \lambda (\omega_f)$ which determines
the character of the evolution.
\begin{itemize}
\item Parabolic Decay.
At short times, the exponential in $e^{-i(\omega_f - \omega_0)
(t-t')}$ in $K(t-t')$ can be replaced by $1$. This is justified when
$t\ll \frac{1}{\Delta}$, where $\Delta$ is a typical width of the
$\lambda (\omega_f)$ curve. Usually, for a bell-shaped $\lambda
(\omega_f)$ curve the order of $\Delta$ is pretty well approximated by
$\omega_0$. For example if we analyse spontaneous emission in the
optical domain then $\omega_0 = \Delta = 10^{15}\mbox{Hz}$ thus the
short time means here much less than $10^{-15}$ s. The integration in
Eq. (\ref{eqcf}) together with the initial condition $c_i(t=0) = 1$
gives
\begin{equation}
|c_i(t)|^2 = |\bra{\phi_i}\phi(t)\rangle|^2 =
1 - 2\frac{t^2}{\hbar^2} \sum_f \lambda^2_f.
\end{equation}
The same result can be otained obtained directly by writing
\begin{equation}
\ket{\phi(t)} = e^{-iHt/\hbar}\ket{\phi_i} = (1 - \frac{i}{\hbar} Ht -
\frac{1}{\hbar^2} H^2 t^2 + \ldots)\ket{\phi_i}
\end{equation}
which, together with Eq.(\ref{Ham}) gives
\begin{equation}
|\bra{\phi_i}\phi(t)\rangle|^2 = 1- 2\frac{t^2}{\hbar^2} (\langle H^2
\rangle - \langle H \rangle^2) \ldots = 1-2\frac{t^2}{\hbar^2}\sum_f
\lambda^2_f + \ldots
\end{equation}
Thus for short times the decay is always parabolic. Let us mention in
passing that from a purely mathematical point of view we have assumed
here that expression $(\langle H^2\rangle -\langle H\rangle^2) =
\sum_f \lambda^2_f $, i.e. the variance of the energy in the initial
state $\ket{\phi_i}$, is finite. Needless to say in reality it is
always finite but there are mathematical models in which, due to
various approximations, this may not be the case (e.g. the Lorentzian
distribution which has no finite moments).
\item Exponential Decay.
Expression $|\lambda_f|^2 e^{-i(\omega - \omega_0) \tau}$ viewed as a
function of $\omega_f -\omega_0$ oscillates with frequency $1/\tau$
whereas $\lambda_f = \lambda (\omega_f)$ varies smoothly in the
frequency domain. Again taking $\Delta$ as the typical width of the
$\lambda (\omega_f)$ curve for $\tau >> 1/ \Delta$ the sum in
$K(\tau)$ averages out to zero. This allows to substitute $c_i(t)$ for
$c_i(t')$ in Eq.(\ref{Schr1}) which gives
\begin{equation}
\dot {c}_i(t) \approx -c_i(t)\int_0^t d\tau K(\tau) \approx
-c_i(t)\int_0^\infty d\tau K(\tau).
\end{equation}
Now we can calculate $\int_0^\infty d\tau K(\tau)$ using the identity
\begin{equation}
\int_0^\infty d \tau e^{i\omega\tau} = \lim_{\epsilon \rightarrow 0^+}
\int_0^\infty d \tau e^{i(\omega + i\epsilon)\tau} = \lim_{\epsilon
\rightarrow 0^+}\frac{i}{\omega+i\epsilon} = i\mbox{\cal
P}\frac{1}{\omega} + \pi \delta (\omega).
\end{equation}
It gives
\begin{equation}
\int_0^\infty d\tau K(\tau) = \frac{\gamma}{2} + i \delta,
\quad \frac{\gamma}{2} =
\frac{\pi}{\hbar^2} |\lambda (\omega_f=\omega_0)|^2, \quad \delta =\mbox{\cal
P}\sum_f\frac{|\lambda_f|^2}{\omega_0-\omega_f}.
\end{equation}
Incorporating the energy shift $\hbar \delta$ into the modified energy
separation $\hbar (\omega_0+\delta)$ we finally obtain
\begin{equation}
\dot{c}_i(t) = -\frac{\gamma}{2} c_i(t)\qquad \mbox{that is} \qquad
c_i(t) = e^{-\frac{\gamma t}{2}}
\end{equation}
and consequently
\begin{equation}
c_f(t)=\frac{\lambda_f}{\hbar} \frac{1-e^{i(\omega_f-\omega'_0 +
i\gamma/2)t}}{\omega_f-\omega'_0 + i\gamma/2}
\end{equation}
\end{itemize}
Let us now go back to the language introduced in section \ref{s:int}.
The states of the environment $\ket{R_0(t)},\\ \ket{R_1(t)}, \ket{R_2(t)}$
and $\ket{R_3(t)}$ in the present context take the explicit form
\begin{eqnarray}
\ket{R_0(t)}&=&\frac{1}{2}[1+c_i(t)e^{-i\omega_0 t}]\ket{\bf 0}\;,
\label{R0}\\
\ket{R_1(t)}&=&\frac{1}{2}\sum_f c_f(t) e^{-i\omega_f t}\ket{1}_f\;,
\label{R1}\\
\ket{R_2(t)}&=&-\frac{1}{2}\sum_f c_f(t) e^{-i\omega_f t}\ket{1}_f\;,
\label{R2}\\
\ket{R_3(t)}&=&\frac{1}{2}[1-c_i(t)e^{-i\omega_0 t}]\ket{\bf 0}\;.
\label{R3}
\end{eqnarray}
By formula (\ref{fidelity}), the fidelity of this process is given by
\begin{eqnarray}
F(t)&=&\bra{R_0(t)} R_0(t)\rangle + \bra{R_3(t)} R_3(t) \rangle
-2\mbox{Re}\bra{R_0(t)}R_3(t)\rangle\nonumber\\
&=&|c_i(t)|^2\;.
\end{eqnarray}
Therefore, the fidelity in the case of a parabolic decay takes the form
\begin{eqnarray}
F_{par}(t)=
1 - 2\frac{t^2}{\hbar^2} \sum_f \lambda^2_f\;,
\end{eqnarray}
while in the case of an exponential decay it has the exponential form
\begin{eqnarray}
F_{exp}(t)= e^{-\gamma t}\;.
\end{eqnarray}
\section{Benefits of quantum error correction}
\label{s:beyond}
In order to get an idea about the efficiency of quantum error correction, we will now discuss
an explicit construction of the single error-correcting five qubit code.
The initial state of the qubit $\alpha\ket{0}+\beta\ket{1}$ is encoded in state
$\alpha\ket{C_0}+\beta\ket{C_1}$, where \cite{lafl}
\begin{eqnarray}
\ket{C_0}&=&\ket{00010}+\ket{00101}-\ket{01011}+\ket{01100}
+\ket{10001}-\ket{10110}-\ket{11000}-\ket{11111}\\
\ket{C_1}&=&\ket{00000}-\ket{00111}+\ket{01001}+\ket{01110}
+\ket{10011}-\ket{10100}+\ket{11010}-\ket{11101}.
\end{eqnarray}
(To see the benefits of quantum error correction we do not need to use
the explicit form of the code, we wrote it down here for those
curious readers who may want to play with quantum error correcting codes.)
These encoded
states are chosen in such a way that conditions (\ref{orth}) are
satisfied. Since this code can correct any type of error affecting one
qubit, it is suitable for protecting quantum states against
spontaneous emission. We notice
that the spontaneous emission process described in Sect. \ref{s:dyn},
unlike decoherence, involves both phase and amplitude errors and
therefore it cannot be successfully defeated with less than five
bit codes.
The probability that the state undergoes exponential decay in the presence
of spontaneous emission is approximately given by
\begin{eqnarray}
P_{dec}(t)=1-F_{exp}(t)=1-e^{-\gamma t}\;.
\end{eqnarray}
If we assume that the five qubits decay independently from each other, the
probability that none of them decays is given by
\begin{eqnarray}
P_{no\;dec}(t)=e^{-5\gamma t}\;
\end{eqnarray}
while the probability that only one of them decays is
\begin{eqnarray}
P_{1\;dec}(t)=e^{-4\gamma t}(1-e^{-\gamma t})\;.
\end{eqnarray}
Since by construction the above error correction scheme corrects perfectly
the encoded state when only one of the qubits is affected, the fidelity
of reconstruction of the state after the error correction is at least as
high as the probability of having at most one qubit decay during the process,
that is
\begin{eqnarray}
F_{ec}(t)\ge P_{no\;dec}(t)+ 5P_{1\;dec}(t)=e^{-4\gamma t}(5-4e^{-\gamma t}).
\end{eqnarray}
In order to have a successful error correction the such fidelity must be greater than the fidelity
$F_{exp}(t)$ corresponding to a single qubit in the absence of error correction. This is true when
the decay probability
$P_{dec}(t)$ is much smaller than one, namely when the correction procedure is applied at times
$t\ll 1/\gamma$. Actually, for $t\ll 1/\gamma$ the fidelity of reconstruction after error correction
is bounded by
\begin{eqnarray}
F_{ec}(t)\ge 1-10\gamma^2 t^2 + O(t^3)\;,
\end{eqnarray}
namely it has parabolic form, while the single qubit decay probability is
\begin{eqnarray}
P_{dec}(t)\simeq 1-\gamma t\;.
\end{eqnarray}
\section{Concluding remarks}
Research in quantum error correction in its all possible variations has become
vigorously active and any comprehensive review of the field must be obsolete as soon
as it is written. Here we have decided to provide only some very basic knowledge,
hoping that this will serve as a good starting point to enter the field. The reader
should be warned that we have barely scratched the surface of the current activities
in quantum error correction neglecting topics such as group theoretical ways of
constructing good quantum codes \cite{GF4}, concatenated codes
\cite{knill}, quantum fault tolerant computation \cite{divi-shor} and many others.
Many interesting papers in these and many related areas can be found at the Los
Alamos National Laboratory e-print archive (http://xxx.lanl.gov/archive/quant-ph).
This work was supported in part by the European TMR Research Network
ERP-4061PL95-1412, the TMR Marie Curie Fellowship Programme,
Hewlett-Packard, The Royal Society London and Elsag-Bailey,
a Finmeccanica Company.
|
1,108,101,565,434 | arxiv | \section{Introduction}
The fundamental nature of Einstein equations as well as beautiful
discovery of the existence of a large class of two-dimensional
completely integrable systems made very natural various
expectations and conjectures of the integrability of the Einstein
equations, at least for the space-times with an
Abelian two-dimensional isometry group, when the reduced dynamical
equations are effectively two-dimensional. First of all, this
concerned the Einstein equations for gravitational fields in vacuum,
which integrability was conjectured and even motivated partially
long ago (see the papers of Geroch, Maison)~\footnote{The lack of
space urges the author to avoid a detail citation and to refer the
reader to the references in a few papers cited below, but mainly --
to a large and very useful F.J.Ernst's collection of
related references and abstracts, accessible throw {\sf
http://pages.slic.com/gravity}, as well as to gr-qc and hep-th data
bases.}. However, the actual discovery of very rich internal
structure of these equations and development of effective methods for
the construction of infinitely large classes of their solutions
actually have been started more then twenty years ago. It is
necessary to mention here a variety of more or less general and well
known now methods and results, such as Belinskii and Zakharov
formulation of the inverse scattering method and their construction
of vacuum $N$-soliton solutions, the constructions of B\"acklund
transformations of Harrison and of Neugebauer, the infinite
dimensional algebra of internal symmetries of stationary axisymmetric
electrovacuum Einstein - Maxwell equations, found by Kinnersley and
Chitre and "exponentiation" of some of these symmetries made by
Hoenselaers, Kinnersley and Xanthopoulos. Later it was shown, that
besides the vacuum case, the integrability properties are possessed
by two-dimensional space - time symmetry reductions of Einstein
equations in the presence of the massless matter fields -- the
electromagnetic fields (Kinnersley and Chitre, Hauser and Ernst,
GA), or/and Weyl massless two-component spinor (neutrino) field
(GA), or/and minimally coupled scalar field, or/and stiff fluid with
$p=\varepsilon$ (Belinskii), or electromagnetic field with dilaton
(Belinski and Ruffini), as well as of some string theory induced
gravity models with axion, dilaton and electromagnetic fields (e.g.,
Bakas, Sen, Gal'tsov and Kechkin).
In this communication we present a sketch of general approach to the
analysis and solution of all mentioned above integrable space - time
symmetry reductions of Einstein equations (of both, hyperbolical and
elliptical types). This approach, called a "monodromy transform
approach", is based on and it develops the results of the author's
papers~\cite{GA:1980a}${}^-$\cite{GA:1988}. It leads a) to a
definition in the most general context of a convenient set of
functional parameters -- "monodromy data", which analytical
properties on the spectral plane are closely related to various
physical and geometrical properties of solutions, and b) to a
construction of pure linear integral equations (of Cauchy and then,
of Fredholm types) equivalent to the original reduced field equations
and admitting a construction of their general local solutions in
terms of homogeneously converging functional
series~\footnote{In~\cite{GA:1983,GA:1985,GA:1988}
many intermediate statements were argued very briefly and for the
analytical case only. However, all these considerations are valid
also for a larger class of solutions with very low order of
differentiability (namely, $C^3$ for the metric components). The
corresponding rigorous proof can be found in the
recently published paper of Hauser and Ernst (I.~Hauser, F.J.~Ernst,
gr-qc/9903104), where many closely related statements were proved in
fullest detail, but in a different context of the analysis of the
group structure of the solution space of the Ernst equations and
characteristic initial value problem.}. Various applications of this
approach to a classification of solutions, exact linearization of
various boundary value problems and to explicit construction of new
classes of exact solutions are expected to be considered elsewhere.
\section{Generalized Ernst equations}
The space - time symmetry ansatz of existence of an Abelian two -
dimensional space - time isometry group, provided all field
components and potentials also possess this symmetry, provides
a reduction of all mentioned above cases or eventually, of Einstein
- Maxwell - Weyl equations to generalized form of the Ernst
equations, except for axion - dilaton gravity, which leads
(as it is already known) to a matrix analog of these equations.
In the differential form notation the reduced Einstein
- Maxwell - Weyl equations can be written as~\footnote{These
equations follow immediately from generalized
Kinnersley equations derived in~\cite{GA:1983}.}
\begin{equation}\label{ErnstEquations}
\left\{\begin{array}{l} d\,{}^{\ast}d{\cal
E}+\displaystyle{d(\alpha+i\delta)\over\alpha}\,{}^{\ast}d{\cal
E} -\displaystyle{(d {\cal E}+2\overline{\Phi} d\Phi) \over
\mbox{Re\,}{\cal E}+\Phi \overline{\Phi}}\,{}^{\ast}d{\cal E}
=0\\[1em]
d\,{}^{\ast}d\Phi+\displaystyle{d(\alpha+i\delta)\over\alpha}\,
{}^{\ast}d\Phi -\displaystyle{(d {\cal E}+2\overline{\Phi} d\Phi)
\over \mbox{Re\,}{\cal E}+\Phi
\overline{\Phi}}\,{}^{\ast}d\Phi=0\\[1.2em]
d\,{}^{\star}d\alpha=0,\quad\qquad d\beta \equiv-\epsilon\,
{}^{\star}d\alpha\\[1em] d\,{}^{\star}d\gamma=0,\quad\qquad
d\delta \equiv {}^{\star}d\gamma.
\end{array}\right.
\end{equation}
where ${\cal E}(x^1,x^2)$ and $\Phi(x^1,x^2)$ are complex scalar
Ernst potentials; "${}^{\ast}$" is a Hodge star operator,
such that $d\,{}^{\ast}d$ is the two-dimensional d'Alambert or
Laplace operator in the hyperbolical ($\epsilon=1$) or elliptical
($\epsilon=-1$) case respectively, defined on the orbit space
$(x^1,x^2)$. The real functions $\alpha(x^1,x^2)$ -- a measure of
area on the orbits and $\gamma(x^1,x^2)$ -- a potential for neutrino
current vector, are arbitrary "harmonical" functions, provided
$d\alpha\wedge {}^{\ast}d\alpha\ne 0$. These functions determine two
other auxiliary real functions -- their "harmonical" conjugates
$\beta(x^1,x^2)$ and $\delta(x^1,x^2)$.
\section{Equivalent "spectral" $N\times N$ - matrix
problem}
For each of the integrable reductions of Einstein equations
considered above we use similar associated complex $N\times N$ -
matrix problems ($N=2$ for vacuum fields, $N=3$ for the models with
electromagnetic and Weyl spinor fields and $N=4$ for string theory
induced gravity models with axion, dilaton and electromagnetic
fields) for the four unknown matrix functions
\begin{equation}\label{Matrices}
{\bf U}(\xi,\eta),\,\, {\bf V}(\xi,\eta),\,\,{\bf
\Psi}(\xi,\eta,w),\,\, {\bf W}(\xi,\eta,w)
\end{equation}
which should satisfy two groups of conditions. The first
one is a deformation problem for a linear system
with given (case dependent) structures of canonical forms
of coefficients and normalization at some reference point
$(\xi_0,\eta_0)$:
\begin{equation}\label{Linsys} \begin{array}{lccl}
\left\{\begin{array}{l}
2 i (w-\xi)\partial_\xi {\bf \Psi}={\bf U}(\xi,\eta) {\bf
\Psi}\\[1ex]
2 i (w-\eta)\partial_\eta {\bf \Psi}={\bf V} (\xi,\eta)
{\bf \Psi}\end{array}\right.
&\left.\vphantom{\begin{array}{l}
2 i (w-\xi)\partial_\xi {\bf \Psi}={\bf U}(\xi,\eta) {\bf
\Psi}\\[1ex]
2 i (w-\eta)\partial_\eta {\bf \Psi}={\bf V} (\xi,\eta)
{\bf \Psi}\end{array}}\hskip1ex\right\Vert\hskip1ex&
\left.\begin{array}{l}
({\bf U})_{can}={\bf U}_{(0)}\\[1ex]
({\bf V})_{can}={\bf V}_{(0)}\end{array}
\hskip1ex\right\Vert\hskip1ex& {\bf
\Psi}(\xi_0,\eta_0,w)={\bf I} \end{array}
\end{equation}
The second group of conditions implies the existence for
(\ref{Linsys}) of a Hermitian integral of certain structure with case
dependent constant matrix ${\bf\Omega}$:
\begin{equation}\label{Wequations} \begin{array}{rcl}
\left\{\begin{array}{l}
{\bf \Psi}^\dagger {\bf W} {\bf \Psi} = {\bf W}_0(w)\\[1ex]
{\bf W}_0^\dagger (w)={\bf W}_0 (w)\end{array}\right.&\left.
\vphantom{\begin{array}{l}
{\bf \Psi}^\dagger\cdot {\bf W}\cdot{\bf \Psi} = {\bf W}_0(w)\\[1ex]
{\bf W}_0^\dagger (w)={\bf W}_0 (w)\end{array}}
\hskip3ex\right\Vert\hskip3ex &
\displaystyle{\partial {\bf W}\over\partial w} = 4 i {\bf \Omega}
\end{array}
\end{equation}
where $w$ is complex ("spectral") parameter and $\xi$, $\eta$ are
geometrically defined space-time coordinates: $\xi=\beta+ j\alpha$,
$\eta=\beta-j\alpha$ with $j=1$ for $\epsilon=1$ and $j=i$ for
$\epsilon=-1$. Thus, for the hyperbolic case ($\epsilon=1$) the
coordinates $(\xi,\eta)$ are two real light cone coordinates, while
for the elliptical case ($\epsilon=-1$) these coordinates are complex
conjugated to each other. The canonical forms of ${\bf U}$ and ${\bf
V}$ matrices (up to a permutation of diagonal elements) are
$$\left.\begin{array}{l}
{\bf U}_{(0)}^{N=2}=\mbox{diag}\,(i,0)\\[1ex]
{\bf V}_{(0)}^{N=2}=\mbox{diag}\,(i,0)
\end{array}\hskip0.2ex\right\Vert\hskip0.2ex
\left.\begin{array}{l}
{\bf U}_{(0)}^{N=3}=\mbox{diag}\,(i+a(\xi),0,0)\\[1ex]
{\bf V}_{(0)}^{N=3}=\mbox{diag}\,(i+b(\eta),0,0)\end{array}
\hskip0.2ex\right\Vert\hskip0.2ex
\begin{array}{l}
{\bf U}_{(0)}^{N=4}=\mbox{diag}\,(i,i,0,0)\\
{\bf V}_{(0)}^{N=4}=\mbox{diag}\,(i,i,0,0) \end{array}
$$
where $a(\xi)=2\partial_\xi\gamma$, $b(\eta)=2\partial_\eta\gamma$ and
a spinor field potential $\gamma$ is an arbitrary
real solution of
$\partial_\xi\partial_\eta\gamma=0$, provided, for $\epsilon=-1$,
$\vert\mbox{Im}\, a(\xi)\vert<1$
and $\vert\mbox{Im}\,b(\eta)\vert<1$ at least for $\xi$, $\eta$
close enough to $\xi_0$, $\eta_0$.
The matrices ${\bf \Omega}$ are constant:
$${\bf \Omega}^{N=2}=\left(\hskip-1.5ex\begin{array}{rr}
0&1\\-1&0
\end{array}\right)
\hskip1ex\left\Vert\hskip1ex
{\bf \Omega}^{N=3}=\left(\hskip-1.5ex\begin{array}{rrr}
0&1&0\\-1&0&0\\0&0&0 \end{array}\right)
\hskip1ex\right\Vert\hskip1ex
{\bf\Omega}^{N=4}=\left(\hskip-1.5ex\begin{array}{rrrr}
0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{array}\right)$$
For any solution of (\ref{ErnstEquations}) the matrices
(\ref{Matrices}) can be calculated explicitly and for any solution
(\ref{Matrices}) of (\ref{Linsys}), (\ref{Wequations}) the solution
of the (generalized) Ernst equations can be easily calculated
using the identifications
$\partial_\xi{\cal E}=-{\bf U}_1{}^2$, $\partial_\eta{\cal E}=-{\bf
V}_1{}^2$ and $\partial_\xi\Phi={\bf U}_1{}^3$,
$\partial_\eta\Phi={\bf V}_1{}^3$. Besides that,
(\ref{Matrices}) -- (\ref{Wequations}) imply ${\bf W}=4
i(w-\beta){\bf \Omega}+{\bf G}(\xi,\eta)$, where the components of
${\bf G}(\xi,\eta)$ are algebraically related to the
components of metric and electromagnetic
potential~\cite{GA:1980a}${}^-$\cite{GA:1983}.
\section{Direct problem of the monodromy
transform and definition of the monodromy data}
Let us consider at first the linear system (\ref{Linsys}).
It can be shown~\cite{GA:1988}, that any its solution ${\bf
\Psi}(\xi,\eta,w)$ is holomorphic on the spectral plane outside a cut
$L=L_+\cup L_-$, which structure is shown on the
Figure \ref{fig:wplane}. Four endpoints of this cut
are the branchpoints of ${\bf \Psi}$. The local behaviour of ${\bf
\Psi}$ at the cuts $L_+$ and $L_-$ is characterized by
monodromy matrices $T_+(w)$ and $T_-(w)$, which describe the
transformations of ${\bf \Psi}$ along the paths, surrounding
the branchpoints on $L_+$ and $L_-$:
\begin{equation}\label{Tmatrices}
{\bf \Psi}\stackrel {{\bf
T}_\pm}\longrightarrow \widetilde{\bf\Psi}={\bf\Psi}\cdot {\bf
T}_\pm(w),\qquad {\bf T}_\pm(w)={\bf I}-(1+e^{-2 i
[\sigma]_\pm})\displaystyle{{\bf l}_\pm(w)\otimes{\bf k}_\pm(w)\over
({\bf l}_\pm(w)\cdot{\bf k}_\pm(w))}
\end{equation}
where $2 i [\sigma]_+=\pi a(w)$ and
$2 i [\sigma]_-=\pi b(w)$. If a spinor field vanishes, i.e. for
$a(w)=b(w)=0$, the branchpoints at the ends of $L_+$ and $L_-$ are
algebraic branchpoints of the orders $\frac12$ or $-\frac12$ and
therefore, we have ${\bf T}_\pm^2(w)\equiv{\bf I}$.
The structure (\ref{Tmatrices}) of monodromy matrices $T_\pm(w)$
allows to associate with any fundamental solution ${\bf \Psi}$ four
complex vector functions ${\bf k}_\pm(w)$, ${\bf l}_\pm(w)$, defined
(due to a homogeneity of these expressions) in a projective sense and
depending upon the spectral parameter only:
\begin{equation}\label{Mdata}
{\bf k}_\pm(w)=(1,{\bf u}_\pm(w),{\bf v}_\pm(w)),\qquad
{\bf l}_\pm(w)=(1,{\bf p}_\pm(w),{\bf q}_\pm(w)),\qquad
\end{equation}
Following~\cite{GA:1985,GA:1988} we can find that
(\ref{Wequations}) are equivalent to some constraint on the
monodromy data (\ref{Mdata}), which unambiguously relates the
components of the vectors ${\bf l}_\pm$ and ${\bf k}_\pm$ and hence,
the functions ${\bf u}_\pm(w)$ and ${\bf v}_\pm(w)$ can represent a
complete set of the monodromy data for the entire problem
(\ref{Linsys}) - (\ref{Wequations}).
In the case of axion - dilaton gravity instead of vectors
(\ref{Mdata}) we have $2\times 4$ - matrices of the
structure ${\bf k}_\pm(w)=({\bf I},{\bf u}_\pm(w))$, where ${\bf I}$
is a $2\times 2$ unit matrix and ${\bf u}_\pm(w)$ are arbitrary
$2\times 2$ - matrix functions. These monodromy
data can be associated with any local solution of the reduced field
equations and therefore, this construction solves the direct problem
of our monodromy transform.
\begin{figure}[h]
\begin{center} \epsfxsize=.97\textwidth \epsfbox{fig_1_it.eps}
\caption{The structures of the cut $L=L_+\cup L_-$ on the spectral
plane $w$ and
domains of holomorphicity of the monodromy data functions ${\bf
u}_\pm(w)$ and ${\bf v}_\pm (w)$ in the hyperbolic ($\epsilon=1$) and
the elliptic ($\epsilon=-1$) cases. \label{fig:wplane}}
\end{center}
\end{figure}
\section{Inverse problem of the monodromy
transform and equivalent singular integral
equations}
Simple arguments, similar to once used
in~\cite{GA:1985,GA:1988}, show that
certain components of local algebraic structure of
${\bf \Psi}$ on $L_\pm$ -- the components of two complex
vectors $\mbox{\grb\char'047}\,_\pm(\xi,\eta,w)$ should satisfy the similar sets of
linear singular integral equations. (In the case of gravity with
axion, dilaton and electromagnetic fields $\mbox{\grb\char'047}\,_\pm(\xi,\eta,w)$ are
$2\times 4$ - matrices.) Omitting farther the suffices $\pm$ and
keeping in mind that, for example, ${\bf k}(\tau)\equiv{\bf
k}_+(\tau)$ for $\tau\in L_+$ and ${\bf k}(\tau)\equiv{\bf
k}_-(\tau)$ for $\tau\in L_-$, we can write these integral equations
in the form \begin{equation}\label{Line}
\nu(\xi,\eta,\tau)\mbox{\grb\char'047}\,(\xi,\eta,\tau)+\displaystyle{{1\over\pi
i}\mathop{\int\hskip-0.8pc\mbox{\it /}}\limits_L}\,{[\,\lambda e^{i\sigma}\,]_\zeta
\over\zeta-\tau}\,{\cal H}(\tau,\zeta)\,
{\mbox{\grb\char'047}\,}(\xi,\eta,\zeta)\,d\,\zeta =-{\bf k}(\tau) \end{equation}
where a Cauchy principal value integral is used, and the
coefficients are
$$\nu(\xi,\eta,\tau)=-\{\lambda e^{i\sigma}\}_\tau{\cal
H}(\tau,\tau),\qquad
{\cal H}(\tau,\zeta)=({\bf k}(\tau)\cdot{\bf l}(\zeta);$$
$[\ldots]_\zeta$ and $\{\ldots\}_\zeta$ are a "jump"
and a "continuous part" of functions at the point $\zeta\in L$.
The functions $\lambda(\xi,\eta,w)=\sqrt{(w-\xi)(w-\eta)
/(w-\xi_0)(w-\eta_0)}$ and
$2\sigma(\xi,\eta,w)=\int_{L_+}a(\zeta)/(w-\zeta)d\zeta+
\int_{L_-} b(\zeta)/(w-\zeta)d\zeta$.
Thus, the integral equation (\ref{Line}) is determined completely in
terms of functions (\ref{Mdata}). Besides that, the Ernst potentials
and all of the field components can be calculated as path integrals,
which are also determined in terms of monodromy data and the
corresponding solution of (\ref{Line})~\cite{GA:1985,GA:1988}.
Therefore, the solution of (\ref{Line}) solves the inverse problem
of our monodromy transform.
\section{Equivalent Fredholm equation: the existence and uniqueness
of local solutions for arbitrary chosen monodromy
data}
In accordance with the well known theory of linear singular integral
equations, the equation (\ref{Line}) within the class of solutions
regular on the cut, possesses an important property, that
the index of its characteristic part is equal to zero for arbitrary
chosen monodromy data functions. This means that this equation
admits various equivalent regularizations. We present two
equivalent forms of the corresponding (quasi-) Fredholm equations
which are left and right regularizations of (\ref{Line})
(the dependence upon $\xi$, $\eta$ is not shown here explicitly):
\begin{equation}\label{Flines}
\mbox{\grb\char'036}\,(\tau)+\int\limits_L{\cal
F}(\zeta,\tau)\mbox{\grb\char'036}\,(\zeta)\,d\,\zeta={\bf h}(\tau),\qquad
\mbox{\grb\char'041}\,(\tau)+\int\limits_L{\cal
G}(\zeta,\tau)\mbox{\grb\char'041}\,(\zeta)\,d\,\zeta={\bf k}(\tau) \end{equation}
where $\mbox{\grb\char'036}\,(\tau)=-{\cal H}(\tau,\tau)\mbox{\grb\char'047}\,(\tau)$ and the following
relations take place
$$\mbox{\grb\char'036}\,(\tau)=-\displaystyle{1\over
B(\tau)Z(\tau)} {\cal R}_\tau \left[B(\tau)\mbox{\grb\char'041}\,(\tau)\right],\qquad
{\bf h}(\tau)=-\displaystyle{1\over
B(\tau)Z(\tau)} {\cal R}_\tau \left[B(\tau){\bf k}(\tau)\right]$$
The kernals ${\cal F}(\zeta,\tau)$ and ${\cal G}(\zeta,\tau)$
are determined by the expressions
$$\left.\begin{array}{l}
{\cal F}(\zeta,\tau)=\displaystyle{B(\zeta)Z(\zeta)\over
B(\tau)Z(\tau)}{\cal R}_\tau \left[B(\tau){\cal
S}(\tau,\zeta)\right]\\[2ex] {\cal
G}(\zeta,\tau)=B(\zeta)\widetilde{\cal R}_\zeta \left[{\cal
S}(\tau,\zeta)\right]
\end{array}\hskip2ex\right\Vert\hskip2ex{\cal
S}(\tau,\zeta)=\displaystyle{{\cal H}(\tau,\zeta)-{\cal
H}(\zeta,\zeta)\over i\pi {\cal H}(\zeta,\zeta)(\zeta-\tau)}$$
where the operators ${\cal R}_\tau$, $\widetilde{\cal R}_\tau$ and auxiliary
functions possess the expressions
$$\left.\begin{array}{l}
{\cal R}_\tau
\left[f(\tau)\right]=A(\tau)
f(\tau)-B(\tau)Z(\tau)\displaystyle{{1\over
i\pi}\mathop{\int\hskip-0.8pc\mbox{\it /}}\limits_L {f(\zeta)\,d\,\zeta\over
Z(\zeta)(\zeta-\tau)}}\\[1ex]
\widetilde{\cal R}_\tau
\left[f(\tau)\right]=A(\tau) f(\tau)+\displaystyle{{1\over
Z(\tau)}{1\over i\pi}\mathop{\int\hskip-0.8pc\mbox{\it /}}\limits_L\displaystyle{B(\zeta)
Z(\zeta)\over (\zeta-\tau)}f(\zeta)\,d\,\zeta}
\end{array}\hskip2ex\right\Vert\hskip1ex
\begin{array}{l}
A(\tau)=\sin [\sigma]_\tau\\[1ex]
B(\tau)=i \cos [\sigma]_\tau\\[1ex]
Z(\tau)=i[\,\lambda\,]_\tau e^{i\{\sigma\}_\tau}
\end{array}$$
(We note here, that for electrovacuum ($\sigma\equiv0)$ we have
$A(\tau)=0$, $B(\tau)=i$.)
The local solution of each of the equations (\ref{Flines}) for any
given set of monodromy data can be constructed by the known
iterative method. In particular,
\begin{equation}\label{Series}
\begin{array}{lcl}
\mbox{\grb\char'036}\,(\tau)=\mbox{\grb\char'036}\,_0(\tau)+\sum\limits_{n=1}^\infty
\left(\mbox{\grb\char'036}\,_n(\tau)-\mbox{\grb\char'036}\,_{n-1}(\tau)\right),\\[2ex]
\mbox{\grb\char'036}\,_0(\tau)={\bf h}(\tau),\qquad
\mbox{\grb\char'036}\,_n(\tau)={\bf h}(\tau)-\displaystyle{\int\limits_L}{\cal
F}(\tau,\zeta)\mbox{\grb\char'036}\,_{n-1}(\zeta)\,d\,\zeta \end{array}
\end{equation}
For local solutions, when the coordinates $\xi$ and $\eta$
are close enough to their initial values $(\xi_0,\eta_0)$, it is
easy to prove a homogeneous convergence of the series (\ref{Series})
and therefore, the existence as well as the uniqueness of the
solution.
\section*{Acknowledgments}
This work was supported in part by the Russian Foundation for Basic
Research Grants 99-01-01150, 99-02-18415.
|
1,108,101,565,435 | arxiv | \section{Introduction}
In \cite{lip} O.~Lipovan proved some retarded versions of the
inequalities of Ou-Iang \cite{ou} and Pachpatte \cite{Pach}. From
Lipovan's retarded inequalities, some criteria are obtained
ensuring the global existence of solutions to the generalized
Li{\'e}nard equation with time delay, and to a retarded Rayleigh
type equation \cite{lip}. More recently, Y.G.~Sun generalized the
Lipovan's retarded inequalities \cite{yuan}. In this note we prove two theorems that give as corollaries all the results obtained in
the above cited papers. Our results can be used to further study
the global existence of solutions to differential equations with a time delay. Several generalizations in different directions than
that followed here are found in the literature (see \textrm{e.g.}
\cite{agar,Jiang,lip06,Ma,Ma07,Ma08,Xu,Zhao}).
\section{Main results}
\label{sec:mainResults}
Throughout we use the notation $\mathbb{R}_0^+=[0,\infty)$,
$\mathbb{R}^+=(0,\infty)$, and $Dom(f)$ and Im$(f)$ to denote, respectively, the domain and the image of a function $f$.
\begin{thm}
\label{teor1} Let $f(t,s)$ and $g(t,s)$ $\in
C(\mathbb{R}_0^+\times\mathbb{R}_0^+,\mathbb{R}_0^+)$ be
nondecreasing in $t$ for every $s$ fixed. Moreover, let $\phi\in
C(\mathbb{R}_0^+,\mathbb{R}_0^+)$ be a strictly increasing
function such that $\lim_{x\rightarrow\infty}\phi(x)=\infty$ and
suppose that $c\in C(\mathbb{R}_0^+,\mathbb{R}^+)$ is a
nondecreasing function. Further, let $\eta, w\in
C(\mathbb{R}_0^+,\mathbb{R}_0^+)$ be nondecreasing with
$\{\eta,w\}(x)>0$ for $x\in(0,\infty)$ and
$\int_{x_0}^\infty\frac{ds}{\eta(\phi^{-1}(s))}=\infty$, with
$x_0$ defined as below. Finally, assume that $\alpha\in
C^1(\mathbb{R}_0^+,\mathbb{R}_0^+)$ is nondecreasing with
$\alpha(t)\leq t$. If $u\in C(\mathbb{R}_0^+,\mathbb{R}_0^+)$
satisfies
\begin{equation}
\label{in1} \phi(u(t))\leq
c(t)+\int_0^{\alpha(t)}\left[f(t,s)\eta(u(s))w(u(s))+g(t,s)\eta(u(s))\right]ds,\
t\in\mathbb{R}_0^+,
\end{equation}
then there exists $\tau\in\mathbb{R}^+$ such that, for all
$t\in[0,\tau]$, we have
$$\Psi(p(t))+\int_0^{\alpha(t)}f(t,s)ds\in Dom(\Psi^{-1}),$$
and
\begin{equation}
\label{in2} u(t)\leq
\phi^{-1}\left\{G^{-1}\left(\Psi^{-1}\left[\Psi(p(t))
+\int_0^{\alpha(t)}f(t,s)ds\right]\right)\right\},
\end{equation}
where
$$G(x)=\int_{x_0}^x\frac{ds}{\eta(\phi^{-1}(s))},$$
with $x\geq c(0)>x_0>0$ if
$\int_{0}^x\frac{ds}{\eta(\phi^{-1}(s))}=\infty$ and $x\geq
c(0)>x_0\geq 0$ if
$\int_{0}^x\frac{ds}{\eta(\phi^{-1}(s))}<\infty$,
\begin{align*}
p(t)&=G(c(t)) +\int_0^{\alpha(t)}g(t,s)ds,\\
\Psi(x)&=\int_{x_1}^x\frac{ds}{w(\phi^{-1}(G^{-1}(s)))},\ x>0,\
x_1>0.
\end{align*}
Here, $G^{-1}$ and $\Psi^{-1}$ are the inverse functions of $G$
and $\Psi$, respectively.
\end{thm}
\begin{rem}
We note that $\Psi$ is a strictly increasing function. Hence, if
$\Psi$ is unbounded we obviously have that
$$\Psi(p(t))+\int_0^{\alpha(t)}f(t,s)ds\in Dom(\Psi^{-1})$$ for all
$t\in\mathbb{R}^+_0$. Consider the case that Im$(\Psi)=(m,M)$,
where $m<\Psi(p(0))$ and $M=\sup\{\Psi(x):x\in(0,\infty)\}$. Let
us fix a number $\delta>0$ and consider $\tau\in(0,\delta)$ such
that
$$0\leq\int_0^{\alpha(t)}f(t,s)ds<M-\Psi(p(\delta)),\ t\in[0,\tau].$$
This number $\tau$ certainly exists since
$\int_0^{\alpha(t)}f(t,s)ds$ is a continuous function and
$\int_0^{\alpha(0)}f(0,s)ds=0$. From the above inequality we can
write
$$\Psi(p(t))+\int_0^{\alpha(t)}f(t,s)ds\leq\Psi(p(\delta))+\int_0^{\alpha(t)}f(t,s)ds<M,\ t\in[0,\tau],$$
that is,
$$\Psi(p(t))+\int_0^{\alpha(t)}f(t,s)ds\in Dom(\Psi^{-1}),\ t\in[0,\tau].$$
\end{rem}
\begin{proof}
Letting $t=0$ in (\ref{in1}), we observe that inequality
(\ref{in2}) holds trivially for $t=0$. Fixing an arbitrary number
$t_0\in(0,\tau]$, we define on $[0,t_0]$ a positive and
nondecreasing function $z(t)$ by
\begin{equation*}
z(t)=c(t_0)+\int_0^{\alpha(t)}\left[f(t_0,s)\eta(u(s))w(u(s))+g(t_0,s)\eta(u(s))\right]ds.
\end{equation*}
Then, $z(0)=c(t_0)$,
\begin{equation}
\label{in4} u(t)\leq\phi^{-1}(z(t)),\ t\in[0,t_0],
\end{equation}
and
\begin{align*}
z'(t)&=[f(t_0,\alpha(t))\eta(u(\alpha(t)))w(u(\alpha(t)))
+g(t_0,\alpha(t))\eta(u(\alpha(t)))]\alpha'(t)\\
&\leq\eta(\phi^{-1}(z(\alpha(t))))[f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))
+g(t_0,\alpha(t))]\alpha'(t)\, .
\end{align*}
Since $\alpha(t)\leq t$, we deduce that
\begin{equation*}
\frac{z'(t)}{\eta(\phi^{-1}(z(t)))}\leq[f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))
+g(t_0,\alpha(t))]\alpha'(t).
\end{equation*}
Integrating the above relation on $[0,t]$ yields
\begin{equation*}
G(z(t))\leq G(c(t_0)) +\int_0^{\alpha(t_0)}g(t_0,s)ds +
\int_0^{\alpha(t)}f(t_0,s)w(\phi^{-1}(z(s)))ds,
\end{equation*}
which implies that
\begin{equation}
\label{in3} z(t)\leq G^{-1}\left[p(t_0) +
\int_0^{\alpha(t)}f(t_0,s)w(\phi^{-1}(z(s)))ds\right] \, .
\end{equation}
Defining $v(t)$ on $[0,t_0]$ by
$$v(t)=p(t_0) +
\int_0^{\alpha(t)}f(t_0,s)w(\phi^{-1}(z(s)))ds,$$ we have that
$v(0)=p(t_0)$ and
\begin{align*}
v'(t)&=[f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))]\alpha'(t)\\
&\leq[f(t_0,\alpha(t))w(\phi^{-1}(G^{-1}(v(\alpha(t)))))]\alpha'(t),
\end{align*}
\textrm{i.e.},
\begin{equation*}
\frac{v'(t)}{w(\phi^{-1}(G^{-1}(v(t))))}\leq
f(t_0,\alpha(t))\alpha'(t) \, .
\end{equation*}
Integrating this last inequality from $0$ to $t$, we obtain
$$\Psi(v(t))\leq\Psi(v(0))+\int_0^{\alpha(t)}f(t_0,s)ds,$$
and from this we get
\begin{equation}
\label{in5} v(t_0)\leq
\Psi^{-1}\left[\Psi(p(t_0))+\int_0^{\alpha(t_0)}f(t_0,s)ds\right].
\end{equation}
From (\ref{in4}), (\ref{in3}) and (\ref{in5}) we deduce that
$$u(t_0)\leq \phi^{-1}\left\{G^{-1}\left(\Psi^{-1}\left[\Psi(p(t_0))
+\int_0^{\alpha(t_0)}f(t_0,s)ds\right]\right)\right\}.$$ Since
$t_0\leq\tau$ is arbitrary we are done with the proof.
\end{proof}
\begin{thm}
\label{teor2} Let functions $f$, $g$, $\phi$, $c$, $\eta$, $w$,
$\alpha$, $u$, $G$ and $\Psi$ be as in Theorem~\ref{teor1}. If for
all $t\in\mathbb{R}_0^+$ the inequality
\begin{equation}
\label{in6} \phi(u(t))\leq
c(t)+\int_0^{\alpha(t)}f(t,s)\eta(u(s))w(u(s))ds+\int_0^t
g(t,s)\eta(u(s))w(u(s))ds
\end{equation}
holds, then
\begin{equation}
\label{in7} u(t)\leq
\phi^{-1}\left\{G^{-1}\left(\Psi^{-1}\left[\Psi(G(c(t)))
+\int_0^{\alpha(t)}f(t,s)+\int_0^{t}g(t,s)ds\right]\right)\right\}
\end{equation}
for all $t\in[0,\tau]$, where $\tau>0$ is chosen in such a way that
$$
\Psi(G(c(t)))
+\int_0^{\alpha(t)}f(t,s)+\int_0^{t}g(t,s)ds\in Dom(\Psi^{-1}) \, .
$$
\end{thm}
\begin{proof}
Letting $t=0$ in (\ref{in6}), we observe that inequality
(\ref{in7}) holds trivially for $t=0$. Fixing an arbitrary number
$t_0\in(0,\tau]$, we define on $[0,t_0]$ a positive and
nondecreasing function $z(t)$ by
\begin{equation*}
z(t)=c(t_0)+\int_0^{\alpha(t)}f(t_0,s)\eta(u(s))w(u(s))ds+\int_0^t
g(t_0,s)\eta(u(s))w(u(s))ds\, .
\end{equation*}
Then, $z(0)=c(t_0)$,
\begin{equation}
\label{in8} u(t)\leq\phi^{-1}(z(t)), \quad t\in[0,t_0] \, ,
\end{equation}
and, since $\alpha(t)\leq t$,
\begin{align*}
z'(t)&=f(t_0,\alpha(t))\eta(u(\alpha(t)))w(u(\alpha(t)))\alpha'(t)
+g(t_0,t)\eta(u(t))w(u(t))\\
&\leq\eta(\phi^{-1}(z(t)))[f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))\alpha'(t)
+g(t_0,\alpha(t))w(\phi^{-1}(z(t))]\, ,
\end{align*}
\textrm{i.e.},
\begin{equation*}
\frac{z'(t)}{\eta(\phi^{-1}(z(t)))}\leq
f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))\alpha'(t)+g(t_0,t)w(\phi^{-1}(z(t))
\, .
\end{equation*}
Integrating the above relation on $[0,t]$ yields
\begin{equation*}
G(z(t))\leq
G(c(t_0))+\int_0^{\alpha(t)}f(t_0,s)w(\phi^{-1}(z(s)))ds+\int_0^{t}g(t_0,s)w(\phi^{-1}(z(s))ds
,
\end{equation*}
which implies that
\begin{equation}
\label{in9} z(t)\leq G^{-1}(v(t)), \quad t\in[0,t_0]
\end{equation}
where $v(t)$ is defined by
$$v(t)=G(c(t_0))+\int_0^{\alpha(t)}f(t_0,s)w(\phi^{-1}(z(s)))ds
+ \int_0^{t}g(t_0,s)w(\phi^{-1}(z(s)))ds.$$ Note that
$v(0)=G(c(t_0))$ and
\begin{align*}
v'(t)&=[f(t_0,\alpha(t))w(\phi^{-1}(z(\alpha(t))))]\alpha'(t)
+g(t_0,t)w(\phi^{-1}(z(t)))\\
&\leq
w(\phi^{-1}(G^{-1}(v(t))))[f(t_0,\alpha(t))\alpha'(t)+g(t_0,t)],
\end{align*}
\textrm{i.e.},
\begin{equation*}
\frac{v'(t)}{w(\phi^{-1}(G^{-1}(v(t))))}\leq
f(t_0,\alpha(t))\alpha'(t)+g(t_0,t).
\end{equation*}
Integrating this last inequality from $0$ to $t$, we obtain
$$\Psi(v(t))\leq\Psi(v(0))+\int_0^{\alpha(t)}f(t_0,s)ds+\int_0^{t}g(t_0,s)ds,$$
from which we get
\begin{equation}
\label{in10} v(t_0)\leq
\Psi^{-1}\left[\Psi(G(c(t_0)))+\int_0^{\alpha(t_0)}f(t_0,s)ds+\int_0^{t_0}g(t_0,s)ds\right].
\end{equation}
From (\ref{in8}), (\ref{in9}) and (\ref{in10}), we deduce that
$$u(t_0)\leq \phi^{-1}\left\{G^{-1}\left(\Psi^{-1}\left[\Psi(G(c(t_0)))
+\int_0^{\alpha(t_0)}f(t_0,s)ds+\int_0^{t_0}g(t_0,s)ds\right]\right)\right\}.$$
Since $t_0$ is arbitrary, inequality (\ref{in7}) is true.
\end{proof}
\section{Corollaries}
Theorem~\ref{teor1} and Theorem~\ref{teor2} generalize previous
results in the literature \cite{lip,yuan}.
Let $m>n>0$ be some constants. Define $\phi(x)=x^m$,
$c(t)=c^{m/(m-n)}$, $c>0$, and $\eta(x)=m/(m-n)x^n$ for
$x\in\mathbb{R}_0^+$. Then,
$$G(x)=\int_0^x\frac{ds}{\eta(\phi^{-1}(s))}
=\frac{m-n}{m}\int_0^x\frac{ds}{s^{n/m}}=x^{(m-n)/m}.$$ We have
$\lim_{x\rightarrow\infty}G(x)=\infty$. Assume that $f(t,s),\
g(t,s)$ do not depend on the variable $t$. Finally, let $x_1=1$.
Then we have the following from Theorem~\ref{teor1}:
\begin{cor}\cite[Theorem~2.1]{yuan}
\label{cor:y} If
$$u^m(t)\leq
c^{m/(m-n)}+\frac{m}{m-n}\int_0^{\alpha(t)}\left[f(s)u^n(s)w(u(s))+g(s)u^n(s)\right]ds,\
t\in\mathbb{R}_0^+,$$ then
$$u(t)\leq
\left\{\Psi^{-1}\left[\Psi\left(c+\int_0^{\alpha(t)}g(s)ds\right)
+\int_0^{\alpha(t)}f(s)ds\right]\right\}^{1/(m-n)},\
t\in[0,\tau],$$ where
$$\Psi(x)=\int_{1}^x\frac{ds}{w(s^{1/(m-n)})},\ x>0,$$
and $\tau\in\mathbb{R}^+$ is chosen such that
$$\Psi\left(c+\int_0^{\alpha(t)}g(s)ds\right)+\int_0^{\alpha(t)}f(s)ds\in Dom(\Psi^{-1}),\ t\in[0,\tau].$$
\end{cor}
\begin{rem}
Setting $m=2$ and $n=1$ in Corollary~\ref{cor:y}, we obtain
Lipovan's Theorem~1 of \cite{lip}.
\end{rem}
Theorem~\ref{teor1} permits also to enunciate, for example, the
following corollary:
\begin{cor}
Let $\phi(x)=x^n$ and $\eta(x)=(x^n+1)\ln(x^n+1)$ with $n>0$.
Further, let $c(t)=c>0$ and assume that the functions $f$, $w$ and
$\alpha$ are as in Theorem~\ref{teor1}, with $g\equiv 0$. If
$$u^n(t)\leq
c+\int_0^{\alpha(t)}\left[f(t,s)(u^n(s)+1)\ln(u^n(s)+1)w(u(s))\right]ds,\
t\in\mathbb{R}_0^+,$$ then there exists a number
$\tau\in\mathbb{R}^+$ such that
$$\Psi(G(c))+\int_0^{\alpha(t)}f(t,s)ds\in Dom(\Psi^{-1}),\ t\in[0,\tau],$$
and
$$u(t)\leq
\left\{G^{-1}\left(\Psi^{-1}\left[\Psi(G(c))
+\int_0^{\alpha(t)}f(t,s)ds\right]\right)\right\}^{1/n},\
t\in[0,\tau],$$ where
$$G(x)=\int_{x_0}^x\frac{ds}{(s+1)\ln(s+1)}, \quad x\geq c>x_0>0,$$
and
$$\Psi(x)=\int_{1}^x\frac{ds}{w(e^{e^s\ln(x_0+1)}-1)^{1/n})},\quad x>0,$$
since
$$G^{-1}(x)=e^{e^x\ln(x_0+1)}-1,\quad x>0.$$
\end{cor}
Interesting corollaries can also be obtained from
Theorem~\ref{teor2}. For instance, if we define $\phi$, $c$,
$\eta$, $f$ and $g$ as in the beginning of this section, we are
lead to a result proved in \cite{yuan}:
\begin{cor}\cite[Theorem~2.2]{yuan}
If $$u^m(t)\leq
c^{m/(m-n)}+\frac{m}{m-n}\left(\int_0^{\alpha(t)}f(s)u^n(s)w(u(s))ds+\int_0^t
g(s)u^n(s)w(u(s))ds\right)$$ for all $t\in\mathbb{R}_0^+$, then
\begin{equation}\label{in11}u(t)\leq \left\{\Psi^{-1}\left[\Psi(c)
+\int_0^{\alpha(t)}f(t,s)+\int_0^{t}g(t,s)ds\right]\right\}^{1/(m-n)}
\end{equation}
for all $t\in[0,\tau]$, where $\tau>0$ is chosen such that
$$\Psi(c)
+\int_0^{\alpha(t)}f(t,s)+\int_0^{t}g(t,s)ds\in Dom(\Psi^{-1}).$$
Here $$\Psi(x)=\int_{1}^x\frac{ds}{w(s^{1/(m-n)})},\ x>0.$$
\end{cor}
\begin{rem}
In \cite[Theorem~2.2]{yuan} the author assumed that
$\lim_{x\rightarrow\infty}\Psi(x)=\infty$. That means that
inequality (\ref{in11}) is valid for all $t\in\mathbb{R}_0^+$.
\end{rem}
Following some techniques introduced in \cite{GronTS},
it is our belief that the results of Theorems~\ref{teor1}
and \ref{teor2} admit a generalization for time scales.
This is under study and will be addressed
in a forthcoming paper.
|
1,108,101,565,436 | arxiv | \section{$G$-stable modules for abstract groups}
\label{s2}
In this chapter we study ${\mathbb A} G$-modules where $G$ is a group, ${\mathbb A}$
is an associative ring.
\subsection{Automorphisms of indecomposable modules}
\label{s2.1}
Let ${\mathbb B}$ be a finite dimensional algebra over a field ${\mathbb K}$
(of any characteristic), $M$ a finite-dimensional ${\mathbb B}$-module, ${\mathbb E} ={{\mbox{\rm End}}}(M)$ its endomorphism ring, $J=J({\mathbb E})$ its Jacobson radical, and $H={{\mbox{\rm Aut}}} (M)$
its automorphism group. We start with the following useful observation:
\begin{prop}
\label{aut_mod}
\begin{enumerate}
\item The quotient algebra ${\mathbb E}/J$ is a division algebra if and only if $M$ is indecomposable.
\item
If $M$ is indecomposable and ${\mathbb E} /J$ is separable,
then $H\cong {{\mbox{\rm GL}}}_1({\mathbb D})\ltimes U$ where
${\mathbb D}={\mathbb E} /J$ is a division algebra and
$U=1+J$ is a connected unipotent group.
\item Further in the conditions of (2),
if ${\mathbb D}={\mathbb K}$, then $H= {{\mbox{\rm GL}}}_1 ({\mathbb K}) \times U$.
\end{enumerate}
\end{prop}
\begin{proof} (1) It is a standard fact that a finite length module
is indecomposable if and only if its endomorphism ring is local.
Since ${\mathbb E}$ is finite-dimensional, this is equivalent to ${\mathbb E}/J$ being
a division ring.
(2) By (1), ${\mathbb D}={\mathbb E}/J$. Since ${\mathbb D}$ is separable, we can use the Malcev-Wedderburn Theorem to split off the radical, i.e., to realize ${\mathbb D}$ as a subalgebra
of ${\mathbb E}$ such that ${\mathbb E}= {\mathbb D} \oplus J$.
Clearly, $H={{\mbox{\rm GL}}}_1 ({\mathbb E})$. Consider an element $x=d+j$, $d\in{\mathbb D}$, $j\in J$.
Since $x^n = d^n + j^\prime$ for some $j^\prime\in J$, $x$ is nilpotent if and only if $d=0$. By the Fitting Lemma, $x\in H$ if and only if $d\neq 0$. The key isomorphism
is given by the multiplication map:
$$ {{\mbox{\rm GL}}}_1({\mathbb D})\ltimes U \xrightarrow{\cong} H = {{\mbox{\rm GL}}}_1 ({\mathbb E}), \ \
(d, 1+ j) \mapsto d+dj \; , $$
$$ H = {{\mbox{\rm GL}}}_1 ({\mathbb E}) \xrightarrow{\cong} {{\mbox{\rm GL}}}_1({\mathbb D})\ltimes U, \ \
d + j \mapsto (d, 1 +d^{-1}j) \; . $$
It remains to observe that $U=1+J$ is a connected unipotent algebraic group.
It is connected because it is isomorphic to $J$ as a variety.
It is unipotent because each of its elements is unipotent in ${{\mbox{\rm GL}}} (M)$.
(3) The Malcev-Wedderburn decomposition turns $J$ into a
${\mathbb D}$-${\mathbb D}$-bimodule. Our condition forces
${\mathbb D}\otimes_{\mathbb K} {\mathbb D}^{op} = {\mathbb K}\otimes_{\mathbb K} {\mathbb K}^{op} = {\mathbb K}$
so that the bimodule structure is just the ${\mathbb K}$-vector space structure.
Hence, ${{\mbox{\rm GL}}}_1({\mathbb D})={{\mbox{\rm GL}}}_1 ({\mathbb K})$ and $U$ commute.
\end{proof}
\subsection{$(L,H)$-Morphs}
\label{s2.2}
Let $G\geq L$, $K\geq H$ be two group-subgroup pairs.
Let $N=N_K(H)$ and $C_K(H)$ be the normaliser and the centraliser
of $H$ in $K$.
By {\em an $(L,H)$-morph from $G$ to $K$}
we understand
a function $f:G\rightarrow K$ satisfying the following four conditions:
\begin{enumerate}
\item $f\mid_L$ is a group homomorphism.
\item $f(G)\subset N_K(H)$.
\item $f(x)f(y)\in f(xy)H$ for all $x,y \in G$.
\item $f(L)\subset C_K(H)$.
\end{enumerate}
By a {\em weak $(L,H)$-morph from $G$ to $K$}
we understand
a function $f:G\rightarrow K$ satisfying only the first three conditions.
One can observe that a weak $(L,H)$-morph is just
a homomorphism $G\rightarrow N/H$ with a choice of lifting to $N$
satisfying an additional condition.
For instance, weak $(G,1)$-morphs are the same as homomorphisms $G\rightarrow K$
and weak $(1,K)$-morphs are just functions $G\rightarrow K$ which preserves the identity. Furthermore, the same statements also hold if we replace weak morphs with morphs in the previous sentence.
Commonly $(L,H)$-morphs originate from
$K$-$G$-sets $X=\,_KX_G$, i.e.,
$G$ acts on the right, $K$ on the left and the actions commute.
Let $\theta\in X$ such that its $G$-orbit is inside its $K$-orbit.
Let $H$ be the stabiliser of $\theta$ in $K$. Choose a section
$K/H\rightarrow K$ which sends the coset $H$ to $1_K$. The composition of
the section with the $G$-orbit map of $\theta$ is a function
$$
f:G\rightarrow K \ \ \mbox{ characterised by } \ \
\,^{f(x)}\theta = \theta^{x}
\ \ \mbox{ for all } x\in G.$$
\begin{lemma}
\label{1Hhom}
The map $f$ defined above is a $(1,H)$-morph.
\end{lemma}
\begin{proof}
By definition, $\,^{f(xy)}\theta = \theta^{xy}$.
On the other hand,
$\theta^{xy}= (\theta^x)^y = (\,^{f(x)}\theta)^y =
\,^{f(x)f(y)}\theta$.
Hence, $\theta = \,^{f(xy)^{-1}f(xy)}\theta
= \,^{f(xy)^{-1}f(x)f(y)}\theta$
and $f(xy)^{-1}f(x)f(y)\in H$.
Now pick $h\in H$. Then
$\,^{f(x)^{-1}h f(x)}\theta =
\,^{f(x)^{-1}h}\theta^x =
\,^{f(x)^{-1}}\theta^x =
\,^{f(x)^{-1}f(x)}\theta= \theta$
so that
${f(x)^{-1}h f(x)}\in H$.
\end{proof}
We would like to identify weak $(L,H)$-morphs that
define the same homomorphisms $G\rightarrow N/H$.
More precisely, we say that two weak $(L,H)$-morphs $f$ and $f^\prime$ are equivalent
if $f^\prime (x)\in f(x)H$ for all $x \in G$.
We denote the set of equivalence classes of weak $(L,H)$-morphs by
$[LH]{\mbox{\rm mo}}(G,K)$.
Furthermore, given a fixed homomorphism $\theta:L\rightarrow K$ we denote by
$\LH(G,K)$ the set of equivalence classes of those
weak $(L,H)$-morphs that restrict to $\theta$ on $L$.
Let $A$ be an additive abelian group with a $G$-action (a ${\mathbb Z} G$-module).
We consider a subcomplex $(\widetilde{C}^\bullet (G,L;A),d)$
of the standard complex $(C^\bullet (G;A),d)$
that consists of such cochains $\mu_n$
that are trivial on $L^n$, i.e., $\mu_n\mid_{L\times \ldots \times L} \equiv 0_A$.
We observe that this cochain complex fits into an exact sequence of cochain complexes
$$0\rightarrow \widetilde{C}^\bullet(G,L;A)\rightarrow C^\bullet (G;A)\rightarrow C^\bullet (L;A)\rightarrow 0 \; .$$
This then allows us to form a long exact sequence of cohomology
$$\ldots\rightarrow H^{n-1}(G;A)\rightarrow H^{n-1}(L;A)\rightarrow \widetilde{H}^n(G,L;A)\rightarrow
H^n(G;A)\rightarrow H^n(L;A)\rightarrow\ldots$$
For our purposes, we have to modify this subcomplex slightly. We consider a subcomplex $(C^\bullet (G,L;A),d)$
of the standard complex $(C^\bullet (G;A),d)$ which is obtained from $(\widetilde{C}^\bullet (G,L;A),d)$ in the following way:
for $n>0$, $C^n(G,L;A)=\widetilde{C}^n (G,L;A)$,
whilst $C^0(G,L;A)=A^L$. We can furthermore replace the complex $C^\bullet(L;A)$ with the
complex $\widetilde{C}^\bullet(L;A)$, defined by $\widetilde{C}^n(L;A)={{\mbox{\rm Coker}}}(C^n(G,L;A)\to C^n(G;A))$ for all $n\geq 0$. In particular, we observe that
$\widetilde{C}^n(L;A)=C^n(L;A)$ for all $n\geq 1$. This then recovers
an exact sequence of cochain complexes:
$$0\rightarrow C^\bullet(G,L;A)\rightarrow C^\bullet (G;A)\rightarrow \widetilde{C}^\bullet (L;A)\rightarrow 0 \; .$$
In particular, observing that for the cochain complex $\widetilde{C}^\bullet(L;A)$ we have $\widetilde{H}^0(L;A)=0$ and $\widetilde{H}^n(L;A)=H^n(L;A)$ for $n\geq 1$,
we can form the long exact sequence of cohomology
$$0\rightarrow H^1(G,L;A)\rightarrow\ldots\rightarrow H^{n-1}(L;A)\rightarrow H^n(G,L;A)\rightarrow
H^n(G;A)\rightarrow H^n(L;A)\rightarrow\ldots$$
What can we say about the natural map $f_n :H^n (G,L;A) \rightarrow H^n(G;A)$?
From this long exact sequence, the following proposition is clear.
\begin{prop}
\label{cohom_compare}
\begin{enumerate}
\item For $n>0$, $H^{n} (L;A)=0$ if and only if $f_n$ is surjective and $f_{n+1}$ is injective.
\item For $n>1$, $f_n$ is injective if and only if the restriction map $Z^{n-1} (G;A)\rightarrow Z^{n-1} (L;A)$ is surjective.
\end{enumerate}
\end{prop}
\begin{proof}
{(1)} This follows from the exact sequence.
{(2)} Suppose $Z^{n-1} (G;A)\rightarrow Z^{n-1} (L;A)$ is surjective.
Pick $\mu \in Z^n(G,L;A)$ such that $[\mu]\in\ker (f_n)$.
Then $\mu\in B^n(G;A)$ and $\mu = d \eta$ for some $\eta\in C^{n-1}(G;A)$.
Moreover, $d (\eta|_{L})=\mu|_{L}\equiv 0$ so that $\eta|_{L}\in Z^{n-1}(L;A)$.
Our assumption gives $\zeta\in Z^{n-1}(G;A)$ such that $\zeta|_L=\eta|_L$.
Hence, $\eta-\zeta \in C^{n-1}(G,L;A)$ and $\mu = d(\eta-\zeta) \in B^{n}(G,L;A)$.
Now suppose $f_n$ is injective. Pick $\mu\in Z^{n-1}(L;A)$, and extend it to $\chi\in C^{n-1}(G;A)$. Hence $d\chi\in Z^n(G,L;A)$ and $[d\chi]\in \ker(f_n)$. So $d\chi=d\zeta$ for some $\zeta\in C^{n-1}(G,L;A)$. Now $\chi-\zeta\in Z^{n-1}(G;A)$ and $(\chi-\zeta)|_L=\mu$.
\end{proof}
\begin{cor}
For $n>1$, $H^n(G,L;A)=0$ if and only if $H^{n-1}(G;A)\rightarrow H^{n-1}(L;A)$ is surjective and $H^{n}(G;A)\rightarrow H^{n}(L;A)$ is injective. Furthermore, $H^1(G,L;A)=0$ if and only if $H^{1}(G;A)\rightarrow H^{1}(L;A)$ is injective.
\end{cor}
The next theorem clarifies the origin of this new complex.
Let us fix a homomorphism
$\theta=f|_L : L \rightarrow N$
and choose a subgroup
$\widetilde{H} \leq H$, normal in $N=N_K(H)$ such that $A\coloneqq H/\widetilde{H}$ is abelian.
Notice that the conjugation
$\,^{g H}h \widetilde{H} \coloneqq ghg^{-1} \widetilde{H}$
defines a structure of an $N/H$-module (and a $G$-module via any weak $(L,H)$-morph) on $A$.
Informally, we should think of the next theorem as ``an exact sequence''
\begin{equation}
\label{exact_seq0}
H^1(G,L;A)
\dashrightarrow
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)
\longrightarrow
\LH(G,N)
\longrightarrow
H^2(G,L;A)
\end{equation}
keeping in mind that the second and the third terms are sets (not even pointed sets)
and the first arrow is an ``action'' rather than a map.
Let us make it more precise: a weak $(L,H)$-morph defines a $G$-module structure $\rho$ on $A$.
For each particular $\rho$ (not just its isomorphism class) we define
$$
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho \subseteq
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N), \ \ \
\LH(G,N)_\rho
\subseteq
\LH(G,N)
$$
as subsets of those weak $(L,H)$-morphs that define this particular $G$-action $\rho$.
These subsets could be empty, in which case we consider the following theorem true for trivial reasons. The reader should consider this theorem and its proof as a generalisation of the results in sections 1 and 2 in \cite{Thev} to the situation of weak $(L,H)$-morphs.
\begin{theorem}
\label{exact_seq}
We are in the notations preceding this theorem. For each $G$-action $\rho$ on $A$
the following statements hold:
\begin{enumerate}
\item There is a restriction map
$$
{{\mbox{\rm Res}}}:
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho
\longrightarrow
\LH(G,N)_\rho, \ \ \
{{\mbox{\rm Res}}} (\langle f\rangle ) = [f]
$$
where
$\langle f\rangle$ and $[f]$
are the equivalence classes
in
$[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
and
$\LH(G,N)_\rho$.
\item The abelian group $Z^1(G,L;(A,\rho))$
acts freely on the set $[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
by
$$
\gamma \cdot \langle f \rangle \coloneqq \langle \dot{\gamma} f \rangle
\ \mbox{ where } \
\dot{\gamma}f (x) = \dot{\gamma} (x) f (x)
\ \mbox{ for all } \ x\in G
$$
and $\dot{\gamma}: G \xrightarrow{\gamma} A \rightarrow H$
is a lift of $\gamma$ to a map $G\rightarrow H$ with
$\dot{\gamma} (1)=1$.
\item The corestricted restriction map
$
{{\mbox{\rm Res}}}:
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho
\longrightarrow
\mbox{\rm Im}({{\mbox{\rm Res}}})
$
is a quotient map by the $Z^1(G,L;(A,\rho))$-action.
\item Two classes $\langle f\rangle, \langle g\rangle\in[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
lie in the same $B^1(G,L;(A,\rho))$-orbit if and only if
there exist $h\in H$, $f^\prime\in \langle f\rangle$, $g^\prime\in \langle g\rangle$
such that $[f(L),h]\subset\widetilde{H}$ and $f^\prime (x) = h g^\prime (x) h^{-1}$
for all $x \in G$.
\item
There is an obstruction map
$${{\mbox{\rm Obs}}}:
\LH(G,N)_\rho
\longrightarrow
H^2(G,L;(A,\rho)), \ \ \
{{\mbox{\rm Obs}}} ([f]) = [f^\sharp]
$$
where the cocycle $f^\sharp$ is defined by
$
f^\sharp(x,y)= f(x)f(y)f(xy)^{-1}\widetilde{H}
$.
\item The sequence (\ref{exact_seq0}) is exact, i.e., the image of ${{\mbox{\rm Res}}}$ is equal to ${{\mbox{\rm Obs}}}^{-1}([0])$.
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose $\langle f\rangle = \langle g\rangle$. This gives a function
$\alpha :G \rightarrow \widetilde{H}$ such that $\alpha |_L \equiv 1$ and
$f(x) = \alpha (x) g(x)$ for all $x \in G$. Since $H\supseteq \widetilde{H}$,
we conclude that $[f]=[g]$ and the map ${{\mbox{\rm Res}}}$ is well-defined. This proves (1).
Suppose ${{\mbox{\rm Res}}} (\langle f\rangle) = {{\mbox{\rm Res}}} (\langle g\rangle)$.
Then
$[f] = [g]$ gives a function
$\alpha :G \rightarrow H$ such that $\alpha |_L \equiv 1$ and
$f(x) = \alpha (x) g(x)$ for all $x \in G$.
We can also obtain such a function from a cochain
$\gamma \in C^1(G,L;(A,\rho))$ by lifting
$\alpha = \dot{\gamma}$.
Let us compute in the group $N/\widetilde{H}$
denoting
$a\widetilde{H}$ by $\overline{a}$.
The weak $(L,H)$-morph condition for $f$ is
equivalent to
the following equality:
$$
\overline{\alpha (xy)} \; \overline{g(xy)}
=
\overline{f (xy)}
=
\overline{f (x)} \; \overline{f(y)}
=
\overline{\alpha (x) g(x)} \;
\overline{\alpha (y) g(y)}
=
\overline{
\alpha (x) g(x)
\alpha (y) g(x)^{-1}} \; \overline{g(x) g(y)}.
$$
Now notice that
$$
\overline{g(xy)}
=
\overline{g(x) g(y)}
=
\overline{g(x)} \; \overline{g(y)}
$$
is the weak $(L,H)$-morph condition for $g$, while
$$
\overline{\alpha (xy)}
=
\overline{\alpha (x) g(x)
\alpha (y) g(x)^{-1}}
=
\overline{\alpha (x)}\;
\overline{g(x)\alpha (y) g(x)^{-1}}
=
\overline{\alpha (x)}\;
[\rho (x) (\overline{\alpha})] (y)
$$
is the cocycle condition for $\overline{\alpha}=\alpha \widetilde{H}$.
Any two of these three conditions imply the third one,
which proves both (2) and (3), except the action freeness.
Suppose $\langle f\rangle=\gamma\cdot\langle f\rangle=\langle \dot{\gamma}f\rangle$.
This gives a function
$\alpha :G \rightarrow \widetilde{H}$ such that $\alpha |_L \equiv 1$ and
$\dot{\gamma}(x)f(x) = \alpha (x) f(x)$ for all $x \in G$.
Hence, $\dot{\gamma}=\alpha$ and
$\gamma=\overline{\alpha}\equiv 1$.
Thus, the action is free.
Let us examine
$da\cdot \langle f\rangle=
\langle \dot{da}f\rangle$ for some $a\in A^L$.
Since $d a (x) = -a +\rho (x)(a)$
and $\rho (x)$ can be computed by conjugating with $f(x)$,
we immediately conclude that
$$
[\dot{da}f] (x) =
\dot{a}^{-1} f(x)\dot{a}f(x)^{-1}f(x) =
\dot{a}^{-1} f(x)\dot{a}.
$$
It is easy to see that $[f(L),\dot{a}]\subset \widetilde{H}$.
The argument we have just given is reversible, i.e., if
$f (x) = h g (x) h^{-1}$ then $\langle g\rangle = d\overline{h} \cdot \langle f \rangle$ and $\overline{h}\in A^L$.
This proves (4).
Suppose
$[f] = [g]$. This gives a function
$\alpha :G \rightarrow H$ such that $\alpha |_L \equiv 1$ and
$f(x) = \alpha (x) g(x)$ for all $x \in G$.
Let us compute the cocycles in $N/\widetilde{H}$,
keeping in mind that $H/\widetilde{H}$ is abelian:
\begin{align*}
f^\sharp (x, y) = \overline{f(x)}\overline{f(y)}\overline{f(xy)^{-1}}
= & \
\overline{\alpha(x)}\; \overline{g(x)}\; \overline{\alpha(y)} \; \overline{g(y)} \;
\overline{g(xy)}^{-1} \overline{\alpha(xy)}^{-1}
= \\
(\overline{\alpha(xy)}^{-1} \overline{\alpha(x)}\; \overline{g(x)}
\overline{\alpha(y)} \overline{g(x)^{-1}})
\overline{g(x)}\overline{g(y)} \overline{g(xy)^{-1}}
& =
d\,\overline{\alpha} (x,y) + g^\sharp(x, y).
\end{align*}
Thus $[f^\sharp] = [g^\sharp]$, proving (5).
It is clear that $f^\sharp \equiv 1$ for
$f\in [L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$.
Hence, ${{\mbox{\rm Obs}}} ({{\mbox{\rm Res}}} (\langle f \rangle))=[0]$.
Suppose now that ${{\mbox{\rm Obs}}} ( [f])=[0]$.
This gives a function
$\alpha :G \rightarrow H$ such that $\alpha |_L \equiv 1$ and
$d \overline{\alpha} = f^\sharp$
Consider $g:G\rightarrow N$ defined by $g(x) = \alpha (x)^{-1}f(x)$ for all $x \in G$.
Then $[g]=[f]$ and we can verify that $g\in [L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
by checking $g^\sharp \equiv 1$ in $N/\widetilde{H}$:
\begin{align*}
g^\sharp (x,y)=
\overline{\alpha(x)}^{-1}
\overline{f(x)}\;
\overline{\alpha(y)}^{-1}
\overline{f(y)}\;
\overline{f(xy)}^{-1}
\overline{\alpha(xy)}\;
&
\sim
\overline{\alpha(xy)}\;
\overline{\alpha(x)}^{-1}
(\overline{f(x)}\;
\overline{\alpha(y)}
\; \overline{f(x)}^{-1} )^{-1}
f^\sharp (x,y)
\\
=
(d\,\overline{\alpha}(x,y))^{-1}
f^\sharp (x,y)
&
\equiv 1.
\end{align*}
This proves (6).
\end{proof}
Let us quickly re-examine how the last section works
for $(L,H)$-morphs.
All of its results including
Theorem~\ref{exact_seq} clearly work,
although the objects that appear have additional properties.
Most crucially,
since $f(L)\subseteq C_K(H)$,
the $L$-action
on the abelian group $A$ is trivial.
If $L$ is normal in $G$,
this just means that $A$ is a ${\mathbb Z} G/L$-module.
An important feature is that $Z^1(L;A)$ consists of homomorphisms $L\rightarrow A$ in this case.
This means that Proposition~\ref{cohom_compare}
yields the following corollary:
\begin{cor}
If the group $L$ is perfect, then
$f_1 :H^1 (G,L;A) \rightarrow H^1(G;A)$ is surjective
and
$f_2 :H^2 (G,L;A) \rightarrow H^2(G;A)$ is injective.
\end{cor}
\subsection{Module extensions}
\label{s2.4}
We now assume that $L$ is a normal subgroup of $G$.
Let ${\mathbb A}$ be an associative ring, $(V,\theta)$ an
${\mathbb A} L$-module, $K={{\mbox{\rm Aut}}}_{{\mathbb A}}V$ and $H={{\mbox{\rm Aut}}}_{{\mathbb A} L}V$
its automorphism groups. We can think of $\theta$
as an element of the set of ${\mathbb A} L$-structures
$X= \hom (L,K)$. Then $H$ is the centraliser in $K$ of $\theta (L)$.
By $N$, as before, we denote the normaliser of $H$ in $K$.
Naturally, $X$ is a $K$-$G$-set: $G$ acts by conjugation on $L$
twisting the ${\mathbb A} L$-module structure. $K$ acts by conjugations
on the target, while $H={{\mbox{\rm Stab}}}_K (\theta)$.
The module $V$ is called {\em $G$-stable}
if $(V,\theta)\cong (V,\theta^g)$ for all $g\in G$.
This is equivalent to the orbit inclusion $\theta^G\subseteq \,^K\theta$.
By Lemma~\ref{1Hhom} this gives a $(1,H)$-morph
$f : G \rightarrow K$.
If $g \in L$, the isomorphism $f(g) : (V,\theta) \rightarrow (V,\theta^g)$
can be chosen to be $\theta (g)$. Indeed,
$$
\theta (g) (\theta(h) v)=\theta (gh) (v) =
\theta (ghg^{-1}) (\theta(g)(v)) =
\theta^{g}(h)(\theta(g)(v))
$$
for all $g,h\in L$.
Then, without loss of generality $f\vert_L=\theta$,
and $f$ is an $(L,H)$-morph in $\LH(G,N)$.
Suppose that the group $H={{\mbox{\rm Aut}}}_{{\mathbb A} L}V$ is soluble.
We can always find its subnormal series
$H=H_0 \rhd H_1 \rhd \ldots \rhd H_k = \{1\}$
with abelian quotients $A_j=H_{j-1}/H_j$ such that
each $H_j$ is normal in $N$. For instance, we can
use the commutator series $H_j=H^{(j)}$.
In this case, every abelian group $A_j$ becomes an $N$-module.
If ${\mathbb A}$ is finite-dimensional over the field ${\mathbb K}$
and $V$ is a finite-dimensional indecomposable
${\mathbb A} L$-module, we can use Proposition~\ref{aut_mod}
to derive useful information about its automorphisms.
In particular, if ${\mathbb D}={{\mbox{\rm End}}}_{{\mathbb A} L}(V)/J$ is a separable field extension of ${\mathbb K}$,
then $H= {{\mbox{\rm GL}}}_1 ({\mathbb D}) \ltimes (1+J)$ is soluble.
It admits another standard
$N$-stable subnormal series:
$$
H_m = 1+J^m, \ m\geq 1, \ \ \
A_m = (1+J^m)/(1+J^{m+1}).
$$
As groups, we have $A_m=((1+J^m)/(1+J^{m+1}),\cdot)\cong (J^m/J^{m+1},+)$.
The following theorem is the
direct application of Theorem~\ref{exact_seq}.
It determines the uniqueness and existence of a $G$-module structure on a $G$-stable $L$-module.
The proof is obvious.
\begin{theorem}\label{branching}
Let $V=(V,\theta)$ be a $G$-stable ${\mathbb A} L$-module with a soluble automorphism
group $H$, where ${\mathbb A}$ is an associative ring. Let $H=H_0 \rhd H_1 \rhd \ldots \rhd H_k = \{1\}$
be a subnormal $N$-stable series with abelian factors $A_j=H_{j-1}/H_j$.
Any ${\mathbb A} G$-module structure $\Theta$ on $(V,\theta)$ compatible
with its ${\mathbb A} L$-structure (i.e., $\Theta |_{{\mathbb A} L} = \theta$)
can be discovered by the following recursive process in $k$ steps.
One initialises the process with an $(L,H_0)$-morph $f_0=f$
coming from the $G$-stability. The step $m$ is the following.
\begin{enumerate}
\item The $(L,H_{m-1})$-morph $f_{m-1}:G\rightarrow N$ such that
$f_{m-1}|_L=\theta$ determines a $G$-module structure $\rho_m$ on $A_m$.
\item If ${{\mbox{\rm Obs}}} ([f_{m-1}])\neq 0\in H^2 (G, L;(A_m,\rho_m))$, then
this branch of the process terminates.
\item If ${{\mbox{\rm Obs}}} ([f_{m-1}])=0\in H^2 (G, L;(A_m,\rho_m))$, then
we choose an $(L,H_{m})$-morph $f_{m}:G\rightarrow N$
such that ${{\mbox{\rm Res}}} ([f_m])=[f_{m-1}]$.
\item For each element of $H^1 (G, L;(A_m,\rho_m))$ we choose
a different $f_m$ branching the process. (The choices
different by an element of $B^1 (G, L;(A_m,\rho_m))$
are equivalent, not requiring the branching.)
\item We change $m$ to $m+1$ and go to step (1).
\end{enumerate}
An ${\mathbb A} G$-module structure $\Theta$ on $(V,\theta)$ compatible
with its ${\mathbb A} L$-structure
is equivalent to $f_k$ for one of the non-terminated branches.
Distinct non-terminated branches produce (as $f_k$)
non-equivalent compatible ${\mathbb A} G$-module structures.
\end{theorem}
This process is subtle as $\rho_m$ is revealed only
when $f_{m-1}$ is computed.
It would be useful to
have stability, i.e., the fact the $G$-modules
$(A_m,\rho_m)$ are the same (isomorphic) for different
branches. The actions $\rho_m$ on $A_m=H_{m-1}/H_m$
on different branches
differ by conjugation via a function $G\rightarrow H_{m-2}$.
Thus, one needs all two-step quotients $H_{m-1}/H_{m+1}$
to be abelian to ensure stability.
Having said that, we can still have some easy criteria
for existence, uniqueness and non-uniqueness.
\begin{cor} {\rm (Existence Test)}
Suppose $H^2 (G, L;(A_m,\rho_m))=0$ for all $m$ for one of the branches.
Then this branch does not terminate and an ${\mathbb A} G$-module structure exists.
\end{cor}
\begin{cor}\label{Uniq} {\rm (Uniqueness Test)}
Suppose $H^1 (G, L;(A_m,\rho_m))=0$ for all $m$
for one of the non-terminating branches.
Then this branch is the only branch. Moreover, if
an ${\mathbb A} G$-module structure exists, it is unique up to an isomorphism.
\end{cor}
\begin{cor} {\rm (Non-Uniqueness Test)}
Suppose $H^1 (G, L;(A_k,\rho_k))\neq0$
for one of the non-terminating branches.
Then there exist non-equivalent ${\mathbb A} G$-module structures.
\end{cor}
\subsection{Extension from not necessarily normal subgroups}
In Section \ref{s2.4} we restrict our attention to the case of $L$ being a normal subgroup of $G$. Let us take a moment to examine how Section \ref{s2.4} works if $L$ is not normal.
Set $P\coloneqq\bigcap_{g\in G}L^g$, where $L^g\coloneqq g^{-1}Lg$.
Let ${\mathbb A}$ be an associative ring, $(V,\theta)$ an
${\mathbb A} L$-module. Note that $(V,\theta)$ is also an ${\mathbb A} P$-module under restriction, so we can view $\theta$
as an element of the set
$X= \hom (P,K)$. Let $K={{\mbox{\rm Aut}}}_{{\mathbb A}}V$ and $H={{\mbox{\rm Aut}}}_{{\mathbb A} P}V$ be
its automorphism groups, so $H$ is the centraliser in $K$ of $\theta (P)$.
By $N$, as before, we denote the normaliser of $H$ in $K$.
As in Section \ref{s2.4}, $X$ is a $K-G$-set.
The ${\mathbb A} L$-module $V$ is called {\em $G$-stable-by-conjugation}
if $(V,\theta)\cong (V,\theta^g)$ as ${\mathbb A}[L\cap L^g]$-modules for all $g\in G$. Note that this condition guarantees that $V$ is $G$-stable as an ${\mathbb A} P$-module.
This is equivalent to the orbit inclusion $\theta^G\subseteq \,^K\theta$.
By Lemma~\ref{1Hhom} this gives a $(1,H)$-morph
$f : G \rightarrow K$.
If $g \in L$, the ${\mathbb A}[L\cap L^g]$-isomorphism $f(g) : (V,\theta) \rightarrow (V,\theta^g)$
can be chosen to be $\theta (g)$. Indeed,
$\theta (g) (\theta(h) v)=\theta (gh) (v) =
\theta (ghg^{-1}) (\theta(g)(v)) =
\theta^{g}(h)(\theta(g)(v))$ for $g\in L$, $h\in L\cap L^g$.
Then, without loss of generality $f\vert_L=\theta$,
and $f$ is an $(L,H)$-morph in $\LH(G,N)$.
This then allows us to proceed with the inductive process of Theorem \ref{branching} as before, when $H={{\mbox{\rm Aut}}}_{{\mathbb A} P}V$ is soluble.
\subsection{Comparison with $C^\bullet(G/L;A)$}
\label{s2.5}
When studying the question of extending representations from a normal subgroup, Dade and Th\'{e}venaz use the cohomology of the cochain complex $(C^\bullet(G/L;A),d)$ to control existence and uniqueness of such extensions.
In this paper, however, we use the cohomology complex
$(C^\bullet(G,L;A),d)$ instead. It is worth taking a moment
to compare the cohomology of these two complexes, and see where the difference in approaches arises. We use the notation of Section~\ref{s2.2},
assuming that cochains are normalised
since this does not affect the cohomology groups.
In order for the action of $G/L$ on $A$ to make sense,
we need to make the assumption that $L$ acts on $A$ trivially.
The reader can observe that this assumption holds in the case considered in Section~\ref{s2.4}, and, in fact, holds whenever one obtains the $G$-action on $A$
from an $(L,H)$-morph as opposed to a weak $(L,H)$-morph. With this assumption, we have the following proposition.
\begin{prop}\label{H1Map}
Under the aforementioned conditions
we have isomorphisms of groups
\newline
$H^0(G,L;A) \cong H^0 (G/L;A)$
and
$H^1 (G,L;A)\cong H^1 (G/L;A)$.
\end{prop}
\begin{proof}
It is easy to see that
$H^0(G,L;A) = A^G = H^0 (G/L;A)$.
The natural map from the group of normalised cochains
$$
\inf : \widehat{C}^1(G/L;A) \rightarrow C^1 (G,L;A), \ \ \
\inf (\mu) (g) = \mu (g L).
$$
defines a map
${{\mbox{\rm Inf}}}\coloneqq[\inf] : H^1(G/L;A) \rightarrow H^1 (G,L;A)$
of cohomology groups.
It is injective because ${{\mbox{\rm Inf}}}([\mu])=0$ means that $\inf (\mu) = da$ for some $a\in A$.
Then $\mu = da$ and $[\mu]=0$.
It is surjective because for $\eta \in Z^1 (G,L;A)$ we have $d\eta =0$ that translates as
$$
\eta (gh) = \,^g(\eta (h)) + \eta (g) \ \
\mbox{ for all } \ \
g,h\in G.
$$
If one chooses $h\in L$, then it tells us that $\eta (gh) = \eta (g)$,
i.e., that $\eta$ is constant on $L$-cosets.
Thus, the cocycle
$$
\mu \in \widehat{Z}^1(G/L;A) , \ \ \
\mu (g L) \coloneqq \eta (g)
$$
is well-defined. By definition $\inf (\mu) = \eta$.
\end{proof}
Considering the second cohomology of these complexes,
it is still possible to construct the inflation map ${{\mbox{\rm Inf}}}:H^2(G/L;A)\to H^2(G,L;A)$ in the natural way, but this map is no longer an isomorphism in general. We can still view $H^2(G/L;A)$ as a subgroup of $H^2(G,L;A)$:
\begin{prop}\label{H2inj}
The map ${{\mbox{\rm Inf}}}:H^2(G/L;A)\to H^2(G,L;A)$ is injective.
\end{prop}
\begin{proof}
If ${{\mbox{\rm Inf}}}([\eta])=0\in H^2(G,L;A)$ then there exists $\mu\in C^1(G,L;A)$ such that $d\mu=\inf(\eta)$. Note that $\inf(\eta)$ is constant on $L\times L$-cosets by construction. In particular, for $g\in G$ and $h\in L$, we have
$$\mu (g) - \mu (gh) = \,^g(\mu (h)) + \mu (g) - \mu (gh) = \inf(\eta)(g,h)=\inf(\eta)(g,1)=\inf(\eta)(1,1)=0\, ,$$
using the cocycle condition in the penultimate equality.
Hence, $\mu$ is constant on cosets of $L$ in $G$. In particular,
if we define $\widetilde{\mu}\in \widehat{C}^1(G/L;A)$ by $\widetilde{\mu}(gL)=\mu(g)$ then we obtain that $\eta=d\widetilde{\mu}$ and so $[\eta]=0\in H^2(G/L;A)$.
\end{proof}
In the context of Theorem~\ref{exact_seq}, we can see that $H^2(G/L;A)$ and $H^2(G,L;A)$ can be made to play the same role in certain key cases.
To that end, we say that an $(L,H)$-morph $f$ is {\em normalised} if $f(gh)=f(g)f(h)$ whenever $g\in G$ and $h\in L$. Note that this definition is independent of the subgroup $H$.
\begin{lemma}\label{Norm}
In the context of Theorem~\ref{branching}, the $(L,H_i)$-morphs $f_i$ can be assumed to be normalised for each $i$. Furthermore, with this assumption, the cocycles $f_i^\sharp \in Z^2(G,L;A_{i+1})$ are constant on cosets of $L\times L$ in $G\times G$.
\end{lemma}
\begin{proof}
These results follow easily from Lemma 9.2 and Lemma 9.4(i) in Karpilovsky \cite{Kar}.
\end{proof}
For the remainder of this section
we assume that all morphs are normalised. The second statement of
Lemma~\ref{Norm}
immediately yields that, given an $(L,H)$-morph $f$, ${{\mbox{\rm Obs}}}([f])$ lies in the image of the natural homomorphism ${{\mbox{\rm Inf}}}:H^2(G/L;A)\to H^2(G,L;A)$.
The discussion in this section yields the following result.
\begin{cor}\label{H2Map}
Let $f$ be a normalised $(L,H)$-morph. Then there exists $\eta\in Z^2(G/L;A)$
with ${{\mbox{\rm Inf}}}([\eta])={{\mbox{\rm Obs}}}([f])$. Furthermore, ${{\mbox{\rm Obs}}}([f])=0\in H^2(G,L;A)$ if and only if $[\eta]=0\in H^2(G/L;A)$.
\end{cor}
Combining Proposition~\ref{H1Map} and Corollary~\ref{H2Map}, we observe that Sections~\ref{s2.2} and \ref{s2.4} could be interpreted using the cochain complex $C^\bullet(G/L;A)$ at all points instead of the complex $C^\bullet(G,L;A)$ (although doing so would force us to work exclusively with normalised morphs instead of not-necessarily-normalised weak morphs). Indeed, this is the approach taken by Dade and Th\'{e}venaz in the contexts they consider. Our reasons for not taking this approach are threefold. Firstly, our new complex fits nicely into an exact sequence as described in Section~\ref{s2.2}.
Secondly, this complex is more natural to work with --
Dade and Th\'{e}venaz essentially move from the complex $C^\bullet(G/L;A)$ to the complex $C^\bullet(G,L;A)$ as described in this section, and then proceed as we do.
Finally, our main motivation in studying the case for abstract groups is to gain insight into the question for algebraic groups, where the procedures described in this section do not work smoothly (cf. Section~\ref{s3.5}).
In particular, the reader should note that if $H$ is abelian then the corollaries at the end of Section \ref{s2.4} give precisely Corollary 1.8 and Proposition 2.1 in \cite{Thev}.
\section{$G$-stable modules for algebraic groups}
\label{s3}
In this chapter we consider algebraic groups over an algebraically closed field ${\mathbb K}$ of positive characteristic $p$. Algebraic groups are affine and reduced,
groups schemes are affine and not necessarily reduced.
\subsection{Rational and algebraic $G$-modules}
\label{s3.1}
We distinguish algebraic and rational maps of algebraic varieties.
In particular, we can talk about algebraic and rational homomorphisms
of algebraic groups $f:G\rightarrow H$.
The latter are defined on an open dense subset $U={{\mbox{\rm dom}}}(f)$ of $G$ containing $1$ and satisfy
$f(x)f(y)=f(xy)$ whenever $x,y,xy\in U$.
A rational automorphic $G$-action on a commutative algebraic group $H$ is a rational map $G\times H\rightarrow H$, defined on an open set $U\times H$ containing $1\times H$, with the usual action conditions and also such that for each $g\in U$ the map $x\mapsto \,^gx$ is a group automorphism of $H$. An algebraic $G$-action on $H$ is the same, but where the map $G\times H\rightarrow H$ is algebraic.
In an important case, the distinction between rational and algebraic maps can be essentially forgotten, as observed by Rosenlicht \cite{Ros2}.
\begin{lemma}\cite[Theorem 3]{Ros2}
\label{rat_hom}
Let $G$ and $H$ be algebraic groups with $G$ connected.
Suppose $f:G\rightarrow H$
is a rational homomorphism.
Then $f$ extends uniquely to an algebraic group homomorphism $G\rightarrow H$.
\end{lemma}
When $H$ is commutative, this lemma is a special case of the next lemma. Indeed, if one takes the $G$-action on $H$ to be trivial, then the condition in the following lemma is precisely the condition for a map to be a homomorphism.
\begin{lemma}\label{rat_cocycle}
Suppose that $G$ is a connected algebraic group and $(H,+)$ is a commutative algebraic group with an algebraic automorphic $G$-action $\rho$. Let $f:G\rightarrow H$ be a rational map such that $f(xy)=f(x)+ \,^x{f(y})$ for all $x,y,xy\in {{\mbox{\rm dom}}}(f)$ (where $\,^x f(y)\coloneqq\rho(x)(f(y))$). Then $f$ extends to an algebraic map satisfying $f(xy)=f(x)+ \,^x{f(y})$ for all $x,y\in G$.
\end{lemma}
\begin{proof}
Since $f$ is rational and $G$ is connected, ${{\mbox{\rm dom}}}(f)=U\subset G$ is a dense open subset. Set $V=U\cap U^{-1}$.
Fix $x\in V$. Consider the rational map
$$f_x:G\rightarrow H, \quad \quad \quad f_x(y)\coloneqq f(yx) + \,^{yx}f(x^{-1}).$$
This map is rational since it is defined on the dense open set $Vx^{-1}$.
Observe that on $V\cap Vx^{-1}$ we have that $f_x=f$ by the assumption on $f$.
Now, let $x,z\in V$ and define the rational map
$$f_{x,z}:G\rightarrow H, \quad \quad \quad f_{x,z}(y)\coloneqq f_x(y)-f_z(y).$$
Then $f_{x,z}$ is defined on $Vx^{-1} \cap Vz^{-1}$. If the set $f_{x,z}^{-1}(H\setminus\{0\})$ is non-empty, it is open dense. Hence, it has non-empty intersection with $V\cap Vx^{-1}\cap Vz^{-1}$. However, since on $V\cap Vx^{-1}\cap Vz^{-1}$ we have $f=f_x=f_z$, this is impossible. Thus, we must have $f_{x,z}\equiv 0$ on $Vx^{-1}\cap Vz^{-1}$. In particular, if $y\in Vx^{-1}\cap Vz^{-1}$ then $f_x(y)=f_z(y)$.
Therefore, the following map is a well-defined locally-algebraic, and hence algebraic, map
$$\widehat{f}:G\rightarrow H, \quad \quad \quad \widehat{f}(y)\coloneqq f_w(y) \ \ \mbox{where} \ \ w\in y^{-1}V.$$
This map clearly restricts to $f$ on $V$. Furthermore, it satisfies the condition from the lemma:
Let $a,b\in G$. Choose $w\in b^{-1}a^{-1}V\cap b^{-1}V$ -- this exists since both these sets are open dense in $G$. We then have $abw\in V$ and $bw\in V$. The condition on $f$ tells us that
$0=f(1)=f(bw)+ \,^{bw}f(w^{-1}b^{-1})$.
Hence, we have the equations
$$\widehat{f}(ab)=f_w(ab)=f(abw)+\,^{abw}f(w^{-1}),$$
$$\widehat{f}(a)=f_{bw}(a)=f(abw)+\,^{abw}f(w^{-1}b^{-1}),$$
$$\,^a\widehat{f}(b)=\,^a f_w(b)=\,^{a}f(bw)+ \,^{abw}f(w^{-1}).$$
This then gives us that $\widehat{f}(ab)=\widehat{f}(a)+ \,^{a}\widehat{f}(b)$, as required.
\end{proof}
Recall that a rational\footnote{It is a standard terminology,
which slightly disagrees with our usage of the adjective {\em rational}.}
representation of an algebraic group $G$ is a vector space $V$, equipped
with an algebraic homomorphism $\theta: G\rightarrow {{\mbox{\rm GL}}}(V)$.
An immediate consequence of Lemma \ref{rat_hom} is that
if $G$ is connected, then $\theta$ is uniquely determined by
any of its restrictions to an open subset
and
any rational homomorphism of algebraic groups $G\to GL(V)$
determines a representation.
Similar to the case of abstract groups, we have the following proposition:
\begin{prop}\cite[Section 4.3]{Xan}(cf. Proposition~\ref{aut_mod}.)\label{alg_aut}
Suppose that $V$ is a finite-dimensional indecomposable ${\mathfrak g}$-module, where ${\mathfrak g}$ is the Lie algebra of the algebraic group $G$ over ${\mathbb K}$.
Then as algebraic groups we have $${{\mbox{\rm Aut}}}_{\mathfrak g}(V)={\mathbb K}^\times\times (1+J)$$ where $J$ is the Jacobson radical of ${{\mbox{\rm End}}}_{\mathfrak g}(V)$. Furthermore, $1+J$ is a connected unipotent algebraic subgroup of ${{\mbox{\rm Aut}}}_{\mathfrak g}(V)$.
\end{prop}
\subsection{Rational and algebraic cohomologies}
\label{s3.2}
Let $H$ be an affine group scheme acting on
an additive algebraic group $(A,+)$
algebraically by automorphisms. The following easy lemma
shall be useful in what follows.
\begin{lemma}\label{prim}
Let $H$ be an irreducible affine group scheme. Then $H$ is primary, i.e., every zero-divisor
in ${\mathbb K}[H]$ lies inside the nilradical.
\end{lemma}
\begin{proof}
The affinity of $H$ tells us that ${\mathbb K}[H]={\mathbb K}[y_1,\ldots,y_n]/I$ for some $n\geq 1$ and some Hopf ideal $I$. In particular,
$I$ has a primary decomposition $I=Q_0\cap\ldots\cap Q_r$ (which we assume to be normal) with associated primes $P_0=\sqrt{I},P_1,\ldots,P_r$. From the perspective of group schemes, this uniquely endows $H$ with a finite collection $p_0,p_1,\ldots,p_r$ of embedded
points of $H$,
where $p_i$ is a generic point of
the irreducible closed subscheme given by $Q_i$.
Furthermore, for $i>0$ each $p_i$ is of codimension at least one.
If $x$ is a closed point in $H$, then the set $xp_0,xp_1,\ldots,xp_r$ corresponds to the associated primes of another primary decomposition of $I$.
Hence, by uniqueness, $x$ acts on the set $p_0,p_1,\ldots,p_r$ by permutation. Thus, $H_{red}=\bigcup_{i=1}^{r}(\bigcup_{x {\tiny \mbox{ closed point}}}xp_i)_{red}=\bigcup_{i=1}^{r}(p_i)_{red}$. However, over an algebraically closed field, $H_{red}$ cannot be a finite union proper subvarieties. Hence, $r=0$ and the result follows.
\end{proof}
Define the cochain complex $(C^n_{Rat}(H;A),d)$ to consist of the rational maps
$H^n\rightarrow A$ defined at $(1,1,\ldots, 1)$
with the standard differentials of group cohomology.
A rational function $f$ on $H^n$ is defined
on an open dense subset $U\subseteq H^n$,
thus, $U$ has a non-empty intersection
$U_\alpha = U \cap H^n_\alpha$
with each irreducible component
$H^n_\alpha$ of $H^n$.
Since $H^n$ is a group scheme, its irreducible components are connected components
that yields the direct sum decomposition of functions:
$$
{\mathbb K} [H^n] = \oplus_\alpha {\mathbb K} [H^n_\alpha].
$$
Note that each $H_{\alpha}$ is isomorphic to an irreducible affine group scheme, so we can apply Lemma~\ref{prim}.
Thus, $U_\alpha$ is of the form $U(s_\alpha)$ for
a non-zero-divisor $s_\alpha \in {\mathbb K} [H^n_\alpha]$
and $f= hs^{-1}$ for some $h \in {\mathbb K} [H^n]$
and a non-zero-divisor $s \coloneqq (s_\alpha) \in {\mathbb K} [H^n]$.
Thus, $f\in {\mathbb K}[H^n]_S$,
the localised ring of functions
obtained by inverting the set $S$ of all non-zero-divisors.
Writing functions on the algebraic group $A$ as
${\mathbb K}[A]={\mathbb K}[x_1, \ldots x_m]/I$, a rational $n$-cochain $\mu$
is uniquely determined by an $m$-tuple of rational functions
$(\mu_i)\in {\mathbb K}[H^n]_S^m$ satisfying the relations of $I$.
In particular, if each component of $H$ is infinitesimal,
$$
{\mathbb K}[H^n]_S= {\mathbb K}[H^n]
\ \ \mbox{ and } \ \
C^n_{Rat}(H;A)=C^n_{Alg}(H;A) \; ,
$$
where, in general,
$(C^n_{Alg}(H;A),d)$ is the cochain subcomplex if $(C^n_{Rat}(H;A),d)$
that consists of those rational maps $H^n\rightarrow A$
which are, in fact, algebraic.
Let us now concentrate on a connected algebraic group $G$ and
its connected subgroup scheme $L$.
There is another subcomplex of $(C^n_{Rat}(G;A),d)$ which we are interested in:
we define $(\widetilde{C}^\bullet_{Rat}(G,L;A),d)$ to consist of rational maps $G^n\rightarrow A$
that are trivial on $L^n$ (i.e., everywhere $0$ on $L^n$). As in the case of abstract groups, we define $(C^\bullet_{Rat}(G,L;A),d)$ by
$$C^n_{Rat}(G,L;A)=
\begin{cases}
\widetilde{C}^n_{Rat}(G,L;A), & \text{if}\ n>0, \\
A^L, & \text{if}\ n=0 .
\end{cases}
$$
There is a natural inclusion of cochain complexes $C^\bullet_{Rat}(G,L;A)\to C^\bullet_{Rat}(G;A)$. We can hence define the cochain complex $\widetilde{C}^\bullet_{Rat}(L;A)$ such that $\widetilde{C}^n_{Rat}(L;A)\coloneqq {{\mbox{\rm Coker}}}(C^n_{Rat}(G,L;A)\to C^n_{Rat}(G;A))$ for all $n\geq 0$.
In particular, this gives us the short exact sequence of cochain complexes
$$0\to C^\bullet_{Rat}(G,L;A)\to C^\bullet_{Rat}(G;A)\to \widetilde{C}^\bullet_{Rat}(L;A)\to 0.$$
We define the algebraic complexes $C^\bullet_{Alg}(G,L;A)$ and $\widetilde{C}^\bullet_{Alg}(L;A)$ in the expected way, and once again get a short exact sequence of cochain complexes.
In either case, this allows us to construct the long exact sequence in cohomology (suppressing the `Rat' and `Alg'):
\begin{equation}
\label{Seq_star}
0\to H^1(G,L;A)\to\ldots\rightarrow \widetilde{H}^{n-1}(L;A)\rightarrow H^n(G,L;A)\rightarrow
H^n(G;A)\rightarrow \widetilde{H}^n(L;A)\to\ldots
\end{equation}
Note that $\widetilde{H}^0_{Rat}(L;A)=\widetilde{H}^0_{Alg}(L;A)=0$, hence,
this exact sequence starts in degree one.
These long exact sequences can be connected, using the maps induced by the inclusions $C^n_{Alg}(G,L;A)\hookrightarrow C^n_{Rat}(G,L;A)$ and $C^n_{Alg}(G;A)\hookrightarrow C^n_{Rat}(G;A)$:
$$
\begin{CD}
\ldots @>>> H^n_{Alg}(G,L;A) @>>> H^n_{Alg}(G;A) @>>> \widetilde{H}^n_{Alg}(L;A) @>>> {H}^{n+1}_{Alg}(G,L;A) @>>> \ldots\\
@. @VVV @VVV @VVV @VVV @.\\
\ldots @>>>H^n_{Rat}(G,L;A) @>>> H^n_{Rat}(G;A) @>>> \widetilde{H}^n_{Rat}(L;A) @>>> {H}^{n+1}_{Rat}(G,L;A) @>>> \ldots.
\end{CD}
$$
Since we identify $C^0_{Alg}(G;A)$ with algebraic maps from the trivial algebraic group to $A$ (and similarly in the other complexes), there is no distinction between rational and algebraic maps. Hence,
$$H^0_{Rat}(G;A)=H^0_{Alg}(G;A)=H^0_{Rat}(G,L;A)=H^0_{Alg}(G,L;A)=A^G.$$
The cocycle condition on $f\in C^1_{Rat}(G;A)$ is precisely the condition considered in Lemma~\ref{rat_cocycle} for a rational map $f:G\rightarrow A$.
Since $G$ is connected, Lemma~\ref{rat_cocycle} tells us the map extends to an algebraic map. Hence, in this case
$$H^1_{Rat}(G;A)=H^1_{Alg}(G;A)\
\mbox{ and } \
H^1_{Rat}(G,L;A)=H^1_{Alg}(G,L;A).$$
This leads to the following proposition. The first part of it follows from the exact sequence. The second part has a similar proof as Proposition~\ref{cohom_compare}.
\begin{prop}(cf. Proposition~\ref{cohom_compare})
\label{alg_cohom_compare}
\begin{enumerate}
\item If $\widetilde{H}_{Rat}^{1} (L;A)=0$, then
$H^1_{Rat}(G,L;A)=H^1_{Rat}(G;A)$.
\item For $n>0$, if the natural map $Z^{n-1}_{Rat} (G;A)\rightarrow \widetilde{Z}^{n-1}_{Rat} (L;A)$ is surjective, then the natural map
$H^n_{Rat}(G,L;A)\rightarrow H^n_{Rat}(G;A)$ is injective.
\end{enumerate}
\end{prop}
The appropriate long exact sequence yields the following.
\begin{cor}
$H^2_{Rat}(G,L;A)=0$ if and only if $H^{1}_{Rat}(G;A)\rightarrow \widetilde{H}^{1}_{Rat}(L;A)$ is surjective and $H^{2}_{Rat}(G;A)\rightarrow \widetilde{H}^{2}_{Rat}(L;A)$ is injective.
\end{cor}
When the action is trivial, we can learn more about what these cohomology groups are.
\begin{lemma}\label{zero}
If $G$ acts trivially on $A$ and ${{\mbox{\rm Hom}}}(L,A)=0$, then $\widetilde{Z}^{1}_{Rat} (L;A)=0$.
\end{lemma}
\begin{proof}
Let $\mu + C^1_{Rat}(G,L;A)\in \widetilde{Z}^{1}_{Rat} (L;A)$, so $d\mu\in C^2_{Rat}(G,L;A)$. In particular, $d\mu\vert_{L^2}=0$. However, since the action is trivial, $d\mu\vert_{L^2}=0$ if and only if $\mu\vert_{L}$ is a rational homomorphism $L\to A$ if and only if $\mu\vert_L$ is a homomorphism $L\to A$ (since $L$ is connected, by assumption). Since ${{\mbox{\rm Hom}}}(L,A)=0$, we conclude that $\mu + C^1_{Rat}(G,L;A)=0+C^1_{Rat}(G,L;A)$. Hence, $\widetilde{Z}^{1}_{Rat} (L;A)=0$.
\end{proof}
\begin{lemma}\label{H1}
Let $G$ be a connected algebraic group which acts trivially on a commutative algebraic group $A$. Let $L\leq G$ be a closed connected subgroup scheme. Then $H^1_{Rat}(G;A)={{\mbox{\rm Hom}}}(G,A)$ and
$H^1_{Rat}(G,L;A)=\{\mu\in {{\mbox{\rm Hom}}}(G,A)\,\vert\,\mu\vert_{L}\equiv 0\}$.
\end{lemma}
\begin{proof}
Since the $G$-action on $A$ is trivial, the coboundary map $C^0_{Rat}(G;A)\rightarrow C^1_{Rat}(G;A)$ is just the trivial map. Hence, we get that $H^1_{Rat}(G;A)=Z_{Rat}^1(G;A)$, the rational 1-cocycles of $G$. However, as the action is trivial, rational 1-cocycles of $G$ on $A$ are the same as homomorphisms of algebraic groups $G\rightarrow A$.
Hence, $H^1_{Rat}(G;A)={{\mbox{\rm Hom}}}(G,A)$.
Essentially the same argument gives $H^1_{Rat}(G,L;A)=\{\mu\in {{\mbox{\rm Hom}}}(G,A)\,\vert\,\mu\vert_{L}\equiv 0\}$.
\end{proof}
Combining Lemma~\ref{H1} with Lemma~\ref{zero} and Proposition~\ref{alg_cohom_compare}(2), we get the following corollary.
\begin{cor}\label{H1+H2}
Let $G$ be a connected algebraic group acting algebraically (not necessarily trivially) by automorphisms on a commutative algebraic group $A$. Let $L\leq G$ be a connected closed subgroup scheme of $G$ such that the action of $L$ on $A$ is trivial, and ${{\mbox{\rm Hom}}}(L,A)=0$. Then $H^1_{Rat}(G,L;A)= H^1_{Alg}(G;A)$ and $H^2_{Rat}(G,L;A)\rightarrow H^2_{Rat}(G;A)$ is injective.
\end{cor}
The following lemma by van der Kallen \cite[Prop. 2.2]{vdKa}
is useful in what follows.
\begin{lemma}\label{perfect}
Let $G$ be a semisimple, simply-connected algebraic group.
Suppose further that, if $p=2$, the Lie algebra ${\mathfrak g}$ of $G$ does not contain $A_1, B_2$ or $C_l$ ($l\geq3$) as a direct summand. Then ${\mathfrak g}$ is perfect, i.e., ${\mathfrak g}=[{\mathfrak g},{\mathfrak g}]$.
\end{lemma}
\begin{proof}
It is enough to prove this result for $G$ simple and simply-connected, with irreducible root system $\Phi$. It is well known that ${\mathfrak g}$ is simple and non-abelian (and so ${\mathfrak g}=[{\mathfrak g},{\mathfrak g}]$) in the following cases: $p\nmid l+1$ in type $A_l$, $p\neq 2$ in types $B_l,C_l,D_l$, $p\neq 2,3$ in types $E_6,E_7,F_4,G_2$, and $p\neq 2,3,5$ in type $E_8$. It is further known \cite{CKR} that ${\mathfrak g}$ is simple and non-abelian in the following cases: $p=2$ in types $E_6,G_2$, $p=3$ in types $E_7,F_4$, and $p= 2,3,5$ in type $E_8$.
Furthermore, it is known from Table 1 in \cite{Hog} that ${\mathfrak g}=[{\mathfrak g},{\mathfrak g}]$ in all the remaining cases except for $p=2$ in types $A_1,B_2,C_l$ ($l\geq3$).
\end{proof}
\begin{lemma}\label{H2}
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field ${\mathbb K}$ of characteristic $p$
which acts trivially on a commutative algebraic group $A$. Suppose further that, if $p=2$, the Lie algebra ${\mathfrak g}$ of $G$ does not contain $A_1, B_2$ or $C_l$ ($l\geq3$) as a direct summand.
Let $G\1$ be the first Frobenius kernel of $G$. Then $H^2_{Rat}(G,G\1;A)=0$.
\end{lemma}
\begin{proof}
Let us first show that $H^2_{Rat}(G;A)=0$.
Let $\mu:G\times G\to A$ be a rational cocycle defined on the open set $U\times U$ with $U^{-1}=U$. We can define a local group structure on the set $A\times G$ by setting
$$(a,g)(b,h)=(a+b+\mu(g,h),gh)
\ \mbox{ and } \
(a,g)^{-1}=(-a-\mu(g,g^{-1}),g^{-1}).$$
In the language of Weil \cite{Wei}, $A\times U$ is a group-chunk in the pre-group $A\times G$.
By Weil's theorem \cite{Wei}, there exists an algebraic group $H$ birationally equivalent to $A\times U$ with $\Phi:A\times U\to \Phi(A\times U)$
an isomorphism of algebraic group-chunks
and $\Phi(A\times U)$ a dense open set in $H$.
Since $H$ is connected it is generated by $\Phi(A\times U)$. Let $f:A\to H$ be the natural algebraic group homomorphism coming from $A\to A\times U$. This is clearly injective and, since $A$ commutes with each element of $A\times U$, $f(A)\subset Z(H)$. Furthermore, the natural projection $A\times U\to G$ extends to a rational (and so algebraic) homomorphism $\pi:H\to G$, which is surjective as $U$ generates $G$ (since $G$ connected). Finally, it is clear that $f(A)=\ker\pi\cap \Phi(A\times U)$. Hence, $\pi$ descends to a homomorphism $\bar{\pi}:H/f(A)\to G$, whose kernel is discrete (since $\Phi(A\times U)$ is dense in $H$) and, hence, central (as $G$ connected).
In other words, we have a central extension $1\to A\to H\to G\to 1$ of algebraic groups, which corresponds to an algebraic cocycle $\widetilde{\mu}:G\times G\to A$. It is straightforward to see that $\widetilde{\mu}\vert_{U\times U}=\mu\vert_{U\times U}$, and hence $[\mu]$ lies in the image of the natural map $H^2_{Alg}(G;A) \rightarrow H^2_{Rat}(G;A)$. Therefore, the map $H^2_{Alg}(G;A) \rightarrow H^2_{Rat}(G;A)$ is surjective.
It suffices to prove that $H^2_{Alg}(G;A)=0$ when $A$ is ${\mathbb G}_a$ or ${\mathbb G}_m$ or a finite group:
the long exact sequence in cohomology reduces the case of arbitrary $A$ to one of these cases.
It is known that $H^2_{Alg}(G;{\mathbb G}_a)=H^2(G;{\mathbb K}_{triv})=0$ \cite[II.4.11]{Jan}.
Consider a non-trivial cohomology class in $H^2_{Alg}(G;A)$ when $A$ is ${\mathbb G}_m$ or a non-trivial finite group.
It yields a non-split central extension $1\rightarrow A \rightarrow \widetilde{G} \rightarrow G \rightarrow 1$.
Pick a non-trivial character $\chi: A \rightarrow {\mathbb G}_m$. There exists an irreducible representation of $\widetilde{G}$
with a central character $\chi$. It is an irreducible projective representation of $G$.
By the original version of
Steinberg's tensor product theorem \cite{Ste}
it is linear. Hence, $\chi$ is trivial. This contradiction proves that $H^2_{Alg}(G;A)=0$ for these two particular $A$.
We have finished the proof that $H^2_{Rat}(G;A)=0$ for an arbitrary $A$.
Since $G\1$ is a height 1 group scheme, rational homomorphisms of schemes $G\1\rightarrow A$ are fully controlled by the corresponding restricted homomorphisms
of Lie algebras ${\mathfrak g}\rightarrow {{\mbox{\rm Lie}}}(A)$.
By Lemma \ref{perfect},
${\mathfrak g}=[{\mathfrak g},{\mathfrak g}]$ and thus all such homomorphism of Lie algebras are trivial. Hence,
we can apply Corollary~\ref{H1+H2} to get that
$H^2_{Rat}(G,G\1;A)\rightarrow H^2_{Rat}(G;A)$
is injective, and so $H^2_{Rat}(G,G\1;A)=0$.
%
\end{proof}
\subsection{$G$-Stable bricks}
\label{s3.3}
In Chapter~\ref{s2}, we have introduced the notions of weak
$(L,H)$-morphs and $(L,H)$-morphs for abstract groups. In this section, we discuss how these notions apply to algebraic groups and see how they can be used to shed some light on the lifting of ${\mathfrak g}$-modules to $G$-modules.
Suppose that $G,K$ are algebraic groups over ${\mathbb K}$, where $G$ is connected, and that $L,H$ are closed subgroup schemes of $G,K$ respectively. We say that a rational map $f:G\rightarrow K$ is {\em a (weak) $(L,H)$-morph of algebraic groups}
if it satisfies the conditions for a (weak) $(L,H)$-morph of abstract groups, where condition (3) is interpreted for only those $x,y,xy\in {{\mbox{\rm dom}}}(f)$.
In analogy with the case of abstract groups, a weak $(L,H)$-morph of algebraic groups is a homomorphism $G\rightarrow N/H$ with a rational lifting $N/H\rightarrow N$ which satisfies an additional condition. It is clear that if $H$ is normal in $K$ then condition (2) is trivially satisfied. We again have that weak $(L,1)$-morphs are just homomorphisms $G\rightarrow K$, and that weak $(1,K)$-morphs are rational maps $G\rightarrow K$ which preserve the identity.
We say that two weak $(L,H)$-morphs of algebraic groups, $f$ and $g$, are equivalent if $f(x)g(x)^{-1}\in H$ for all $x\in {{\mbox{\rm dom}}}(f)\cap{{\mbox{\rm dom}}}(g)$. Given a homomorphism of algebraic groups $\theta:L\rightarrow K$, we denote by $\LH(G,K)$ the quotient by this equivalence relation of the set of weak $(L,H)$-morphs of algebraic groups from $G$ to $K$ which restrict to $\theta$ on $L$.
Suppose that $X$ is a separated algebraic scheme on which $G$ acts rationally on the right (i.e. the action $X\times G\rightarrow X$ is a rational map), $K$ acts algebraically on the left, and the actions commute. Suppose further that $\theta\in X({\mathbb K})$ is such that $\theta^G\subset \,^{K}\theta$, and that there exists a rational section $K/H\rightarrow K$ where $H={{\mbox{\rm Stab}}}_K(\theta)$ is the scheme-theoretic stabiliser of $\theta$.
As in the case for abstract groups, this gives us a rational map
$$
f:G\rightarrow K \ \ \mbox{ characterised by } \ \
\,^{f(x)}\theta = \theta^{x}
\ \ \mbox{ for all } x\in U\overset{open}{\subset} G.$$
\begin{lemma}\label{alg_1Hhom}
The map $f$ defined above is a $(1,H)$-morph of algebraic groups.
\end{lemma}
\begin{proof}
We can think of $f$ as the composition of the following rational maps
$$G\hookrightarrow \{\theta\}\times G\rightarrow\,^K\theta\rightarrow K/H\rightarrow K.$$
Note that $^K\theta\rightarrow K/H$ is an algebraic map by Demazure-Gabriel \cite[Proposition 3.2.1]{DG}. We then have that the composition is rational since each domain of definition intersects the previous map's image.
The proof that $f(x)f(y)\in f(xy)H$ for $x,y\in G$ with $f(x),f(y)$ and $f(xy)$ defined is exactly the same as in the abstract case, as is the proof that $f(G)\subset N_K(H)$.
\end{proof}
Now we fix algebraic (group, subgroup scheme) pairs $(G,L)$ and $(K,H)$ with $H$ soluble and $G$ connected. Denote by $m_G, m_K$ the corresponding multiplication maps, $\Delta_G, \Delta_K$ the diagonal embeddings, and $inv_G, inv_K$ the inverse maps. Let $\theta:L\rightarrow K$ be a homomorphism of algebraic group schemes. Furthermore, choose $\widetilde{H}$ to be an algebraic subgroup of $H$, characteristic in $N=N_K(H)$ such that $A\coloneqq H/\widetilde{H}$ is commutative. We denote the quotient map $H\rightarrow A$ by $\pi$.
We can define an $N$-action on $H$ by conjugation. Note that since $\widetilde{H}$ is characteristic in $N$, so preserved by conjugation, this passes to an algebraic $N$-action on $A$. Hence, we have an algebraic action of $N$ on $A$ which is trivial on $H$ (since $A$ is commutative). This gives us an algebraic $N/H$-action on $A$. For an element $f\in \LH(G,K)$, we get a rational homomorphism $G\rightarrow N/H$ which is, in fact, algebraic by Lemma \ref{rat_hom}. Thus, every element of $\LH(G,K)$ induces an algebraic $G$-action on $A$. This $G$-action respects the multiplication operation of $A$, i.e. it is an algebraic automorphic $G$-action.
As in the case for abstract groups, we can form something resembling an exact sequence. Let $\rho$ be a rational $G$-action on $A$, and define
$$[L\widetilde{H}]^\theta {\mbox{\rm mo}}(G,N)_\rho\subset [L\widetilde{H}]^\theta {\mbox{\rm mo}}(G,N),\,\,\quad [LH]^\theta {\mbox{\rm mo}}(G,N)_\rho\subset [LH]^\theta {\mbox{\rm mo}}(G,N)$$
as the subsets of weak morphs which induce the action $\rho$.
We get the following theorem.
\begin{theorem}(cf. Theorem \ref{exact_seq})
\label{alg_exact_seq}
For a rational $G$-action $\rho$ on $A$
the following statements hold:
\begin{enumerate}
\item There is a restriction map
$$
{{\mbox{\rm Res}}}:
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho
\longrightarrow
[LH]^\theta {\mbox{\rm mo}}(G,N)_\rho, \ \ \
{{\mbox{\rm Res}}} (\langle f\rangle ) = [f]
$$
where
$\langle f\rangle$ and $[f]$
are the equivalence classes
in
$[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
and
$[LH]^\theta {\mbox{\rm mo}}(G,N)_\rho$.
\item The abelian group $Z_{Rat}^1(G,L;(A,\rho))$
acts freely on the set $[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
by
$$
\gamma \cdot \langle f \rangle \coloneqq \langle \dot{\gamma} f \rangle
\ \mbox{ where } \
\dot{\gamma}f = m_K\circ (\dot{\gamma}\times f)\circ\Delta_G$$
and $\dot{\gamma}: G \xrightarrow{\gamma} A \rightarrow H$
comes from a rational Rosenlicht section $A\rightarrow H$ (cf. \cite[Theorem 10]{Ros2}) with
$\dot{\gamma} (1)=1$.
\item The corestricted restriction map
$
{{\mbox{\rm Res}}}:
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho
\longrightarrow
\mbox{\rm Im}({{\mbox{\rm Res}}})
$
is a quotient map by the $Z_{Rat}^1(G,L;(A,\rho))$-action.
\item If $H$, $\widetilde{H}$ and $A$ are reduced, two classes $\langle f\rangle, \langle g\rangle\in[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho$
lie in the same $B_{Rat}^1(G,L;(A,\rho))$-orbit if and only if
there exist
$h\in H$, $f^\prime\in \langle f\rangle$, $g^\prime\in \langle g\rangle$
such that $[f(L),h]\subset\widetilde{H}$ and $f^\prime (x) = h g^\prime (x) h^{-1}$
for all $x \in G$.
\item
There is an obstruction map
$${{\mbox{\rm Obs}}}:
[LH]^\theta\mbox{\rm mo}(G,N)_\rho
\longrightarrow
H_{Rat}^2(G,L;(A,\rho)), \ \ \
{{\mbox{\rm Obs}}} ([f]) = [f^\sharp]
$$
where the cocycle $f^\sharp$ is defined by $$G\times G\xrightarrow{(p_1,p_2,m_K)} G\times G\times G \xrightarrow{(f,f,inv_K f)} K\times K\times K\xrightarrow{m_K} H\xrightarrow{\pi} A$$
Here, $p_1$ and $p_2$ denote projection to the first and second coordinate respectively.
\item The sequence (cf. Sequence~(\ref{exact_seq0}))
$$
[L\widetilde{H}]^\theta{\mbox{\rm mo}} (G,N)_\rho
\longrightarrow
\LH(G,N)_\rho
\longrightarrow
H^2_{Rat}(G,L;(A,\rho))
$$
is exact, i.e., the image of ${{\mbox{\rm Res}}}$ is equal to ${{\mbox{\rm Obs}}}^{-1}([0])$.
\end{enumerate}
\end{theorem}
\begin{proof}
If $\langle f\rangle =\langle g \rangle$ then the map $$\alpha: G\xrightarrow{(f, inv_K g)} K\times K\xrightarrow{m} K$$ has image in $\widetilde{H}$ and is trivial on $L$. It is rational as it is a composition of rational maps, and the identity is in the domain of definition and image of each map.
We also observe that given an analogous $\alpha:G\rightarrow H$ (i.e. corresponding to $[f]=[g]$) we get $\pi\alpha:G\rightarrow A$. Denoting the Rosenlicht section \cite[Theorem 10]{Ros2} $A\rightarrow H$ by $\tau$, we see that $\tau\pi\alpha=\alpha$ and thus $\dot{(\pi\alpha)}=\alpha$. Note that we may assume that the Rosenlicht section is defined at $0_A$ by composing with a translation if necessary. All the maps here are rational. In particular, $\pi\alpha\in C^1_{Rat}(G,L;(A,\rho))$.
With these observations in mind, the remainder of the proof follows in the same way as in the proof of Theorem \ref{exact_seq} does for abstract groups, doing everything diagrammatically.
\end{proof}
Before going any further, let's consider the following case where we can use this exact sequence directly.
A restricted ${\mathfrak g}$-module $(V,\theta)$ satisfying the condition that ${{\mbox{\rm Aut}}}_{\mathfrak g}(V)={\mathbb K}^\times$ is called a {\em brick}. A brick is necessarily an indecomposable ${\mathfrak g}$-module.
\begin{theorem}
\label{brick}
Suppose $G$ is a semisimple, simply-connected algebraic group over an algebraically closed field ${\mathbb K}$ of characteristic $p>0$, with Lie algebra ${\mathfrak g}$. Suppose further that, if $p=2$, ${\mathfrak g}$ does not contain $A_1, B_2$ or $C_l$ ($l\geq3$) as a direct summand. Let $(V,\theta)$ be a finite-dimensional $G$-stable brick. Then there exists a unique $G$-module structure $\Theta$ on $V$ with $\Theta\vert_{G_{1}}=\theta$.
\end{theorem}
\begin{proof}
We use Theorem \ref{alg_exact_seq} in the following situation:
\begin{itemize}
\item $L=G_{1}$, the first Frobenius kernel of $G$,
\item $K={{\mbox{\rm GL}}}(V)$,
\item $H={{\mbox{\rm Aut}}}_{\mathfrak g}(V)={\mathbb K}^\times$,
\item $N=N_K(H)$,
\item $X={{\mbox{\rm Hom}}}_{{\mathbb K}}({\mathfrak g},{\mathfrak g}{\mathfrak l}(V))$, a separated algebraic scheme with $\theta\in X({\mathbb K})$.
\end{itemize}
Observe that $G$ acts on $X$ on the right via the adjoint map on the domain and ${{\mbox{\rm GL}}}(V)$ acts on $X$ on the left via conjugation on the image. Furthermore, the actions commute, and the $G$-stability of $V$ gives us that $\theta^G\subset \,^{\mbox{\tiny GL}(V)}\theta$.
Hence, Lemma \ref{alg_1Hhom} gives us a $(1,H)$-morph of algebraic groups, say $f:G\rightarrow {{\mbox{\rm GL}}}(V)$. In particular, it gives a homomorphism of algebraic groups $f:G\to\PGL(V)$, together with a rational lifting $\eta:\PGL(V)\to {{\mbox{\rm GL}}}(V)$. This rational lifting can be defined as follows: fix a basis of $V$ and let $U$ be the open subset of $\PGL(V)$ consisting of all cosets which can be represented by a (unique) matrix $A=(a_{ij})\in{{\mbox{\rm GL}}}(V)$ with $a_{11}=1$. Then define the map $\eta:U\to {{\mbox{\rm GL}}}(V)$ by assigning to each coset this representative.
Currently $f$ and $\theta$ give the same maps from $G_{1}$ to $N/H$ -- since
$$
\,^{\theta(x)}\theta(a)(v)=\theta(x)\theta(a)\theta(x^{-1})(v)=\theta(xax^{-1})(v)=\theta^{x}(a)(v)
$$
for $x,a\in G_{1}({\mathbb S})$, $v\in V({\mathbb S})$ for any commutative
${\mathbb K}$-algebra ${\mathbb S}$. Note, however, that the maps $G\1\rightarrow K$ do not necessarily agree.
To fix this potential disagreement, we define a rational map $R:G_{1}\rightarrow H={\mathbb K}^\times$ by $R(g)=f(g)^{-1}\theta(g)$ for $g\in G_{1}({\mathbb S})$.
There exists a rational map $\widetilde{R}:G\rightarrow H={\mathbb K}^\times$ which restricts to $R$ on $G_{1}$. Indeed, we have $R\in{\mathbb K}[G_1]$ (as $G_1$ is infinitesimal), so we can lift it to $\widetilde{R}\in{\mathbb K}[G]$ (since ${\mathbb K}[L]$ is a quotient of ${\mathbb K}[G]$). Let $U=G\setminus\widetilde{f}^{-1}(0)$. This is open in $G$, and on $U$ we have that the image of $\widetilde{R}$ lies inside ${\mathbb K}^\times$, so $\widetilde{R}$ is a rational map $G\to{\mathbb K}^\times$. If now we define $\widetilde{f}:G\rightarrow {{\mbox{\rm GL}}}(V)$ by $\widetilde{f}(g)=f(g)\widetilde{R}(g)$, we get that $\widetilde{f}$ is a $(G_{1},H)$-morph which restricts to $\theta$ on $G_{1}$, fixing the disagreement.
Observe that with $\widetilde{H}\coloneqq 1$, we get (in the notation of the Theorem~\ref{alg_exact_seq}) $A=H$ and $G$ acting on $A$ trivially. Hence, the ``exact sequence'' from Theorem \ref{alg_exact_seq} is
$$H^1_{Rat}(G,G_{1};{\mathbb K}^\times)\dashrightarrow [G_{1} 1]^\theta\mbox{\rm mo}(G,N)_1\rightarrow [G_{1}H]^\theta\mbox{\rm mo}(G,N)_1\rightarrow H^2_{Rat}(G,G_{1};{\mathbb K}^\times)$$
By Lemma \ref{H2}, $H^2_{Rat}(G,G_{1};{\mathbb K}^\times)=0$. Hence $[\widetilde{f}]\in [G_{1}H]^\theta\mbox{\rm mo}(G,N)_1$ can be lifted to $\widehat{f}\in[G_{1} 1]^\theta\mbox{\rm mo}(G,N)_1$. This means that $\Theta\coloneqq\widehat{f}:G\rightarrow {{\mbox{\rm GL}}}(V)$ is a homomorphism of algebraic groups which restricts to $\theta$ on $G_{1}$. Furthermore, this representation is unique (up to equivalence) if $H^1_{Rat}(G,G_{1};{\mathbb K}^\times)=0$.
By Lemma \ref{H1}, $H^1_{Rat}(G,G_{1};{\mathbb K}^\times)=\{\mu\in {{\mbox{\rm Hom}}}(G;{\mathbb K}^\times)\,\vert\,\mu\vert_{G_{1}}\equiv 1\}$. Since $G$ is perfect, $H^1_{Rat}(G,G_{1};{\mathbb K}^\times)=0$ and the extension is unique.
\end{proof}
\subsection{$G$-Stable modules with soluble automorphisms}
\label{s3.4}
We return to the general situation, where $(G,L), (K,H)$ are algebraic (group, subgroup scheme) pairs with $H$ soluble, $G$ connected, and $H$ reduced. However, from now on we suppose that $L$ is a normal subgroup scheme of $G$. We also fix a homomorphism of algebraic groups $\theta:L\rightarrow K$, where the image commutes with $H$, so we are now dealing with $(L,H)$-morphs.
Everything in the previous section can be reformulated in terms of $(L,H)$-morphs without difficulty - the key difference is that the $G$-action on $A$ is now trivial on $L$. Since $H$ is soluble, we can find a subnormal series $H=H_0\rhd H_1\rhd\ldots\rhd H_k=\{1\}$ with commutative quotients $A_j=H_{j-1}/H_j$ and each $H_j$ characteristic in $N=N_K(H)$ and reduced.
Suppose that $f$ is an $(L,H)$-morph of algebraic groups such that $f\vert_L=\theta$. As in the case of abstract groups, we get the following theorem -- it generalises the procedure which we have used for bricks in the previous section.
\begin{theorem}(cf. Theorem \ref{branching})
\label{alg_branching}
Given an $(L,H)$-morph of algebraic groups $f=f_0$ with $f\vert_L=\theta$, we obtain any $(L,1)$-morph extending $\theta$ by applying the following procedure. Step $m$ is the following:
\begin{enumerate}
\item The $(L,H_{m-1})$-morph $f_{m-1}:G\rightarrow N$ such that
$f_{m-1}|_L=\theta$ determines a rational $G$-action $\rho_m$ on $A_m$.
\item If ${{\mbox{\rm Obs}}} ([f_{m-1}])\neq 0\in H^2_{Rat} (G, L;(A_m,\rho_m))$, then
this branch of the process terminates.
\item If ${{\mbox{\rm Obs}}} ([f_{m-1}])=0\in H^2_{Rat} (G, L;(A_m,\rho_m))$, then
we choose an $(L,H_{m})$-morph $f_{m}:G\rightarrow N$
such that ${{\mbox{\rm Res}}} ([f_m])=[f_{m-1}]$.
\item For each element of $H_{Rat}^1 (G, L;(A_m,\rho_m))$ we choose
a different $f_m$ branching the process. (The choices
different by an element of $B_{Rat}^1 (G, L;(A_m,\rho_m))$
are conjugate by an element of $H$.)
\item We change $m$ to $m+1$ and go to step (1).
\end{enumerate}
An $(L,1)$-morph which restricts to $\theta$ on $L$ is equivalent to $f_k$ for one of the non-terminated branches. Two $(L,1)$-morphs $f, g$ come from different branches if and only if
there is no $h\in H$ such that $f(x)=h g(x)h^{-1}$ for all $x\in G$.
\end{theorem}
We get the following corollaries, similarly to Section~\ref{s2.4}:
\begin{cor}
Suppose $H_{Rat}^2(G,L;(A_m,\rho_m))=0$ for all $m$ for one of the branches. Then this branch does not terminate and there is a homomorphism $f:G\rightarrow K$ which restricts to $\theta$ on $L$.
\end{cor}
\begin{cor}
Suppose $H_{Rat}^1(G,L;(A_m,\rho_m))=0$ for all $m$ for one of the non-terminating branches. Then this branch is the only branch. Moreover, if a homomorphism of algebraic groups $f:G\rightarrow K$ restricting to $\theta$ exists, then it is unique up to conjugation by an element of $H$.
\end{cor}
\begin{cor}
Suppose $H_{Rat}^1(G,L;(A_k, \rho_k))\neq 0$ for one of the non-terminating branches. Then there exist algebraic homomorphisms $G\rightarrow K$ which are not conjugate by an element of $H$.
\end{cor}
We apply this theorem (and these corollaries) in the following case - a generalisation of the case from the previous section:
\begin{itemize}
\item $G$ -- connected algebraic group over ${\mathbb K}$ with Lie algebra ${\mathfrak g}$,
\item $L=G_{1}$,
\item $K={{\mbox{\rm GL}}}(V)$, where $(V,\theta)$ is a finite-dimensional $G$-stable indecomposable ${\mathfrak g}$-module,
\item $H={{\mbox{\rm Aut}}}_{\mathfrak g}(V)$,
\item $X={{\mbox{\rm Hom}}}_{{\mathbb K}}({\mathfrak g},{\mathfrak g}{\mathfrak l}(V))$, a separated algebraic scheme with $\theta\in X({\mathbb K})$.
\end{itemize}
Applying exactly the same argument as in Theorem~\ref{brick}, we only start to encounter problems when trying to extend the rational map $R:G_{1}\rightarrow H$ to a rational map on the whole of $G$. This can be fixed without much difficulty.
As a variety, we have that $H={\mathbb K}^\times\times {\mathbb K}^n\subset {\mathbb K}^{n+1}$ for some $n$ [Proposition \ref{alg_aut}]. Hence, we get $R=(R_0,R_1,\ldots,R_n)$ where $R_i\in {\mathbb K}[G_{1}]$ for $i=0,1,\ldots,n$. We can then lift each of these to elements of ${\mathbb K}[G]$, so we obtain $\widetilde{R}=(\widetilde{R_0},\widetilde{R_1}\ldots,\widetilde{R_n}):G\rightarrow {\mathbb K}^{n+1}$. We would like the image to lie in $H$.
Thus, we define $U=G\setminus R_0^{-1}(0)$. This is an open set in $G$, so we can view $\widetilde{R}$ as a rational map from $G$ to ${\mathbb K}^\times\times {\mathbb K}^n=H$ which is defined on $U$, and restricts to $R$ on $G_{1}$.
Now we can define $\widetilde{f}:G\rightarrow {{\mbox{\rm GL}}}(V)$ as $\widetilde{f}(g)=f(g)\widetilde{R}(g)$.
This is a $(G_{1},H)$-morph of algebraic groups, which restricts to $\theta$ on $G_{1}$. Hence, we are in the situation of Theorem \ref{alg_branching}. Observe that $\theta:G_{1}\rightarrow {{\mbox{\rm GL}}}(V)$ extends to a homomorphism of algebraic groups $\Theta:G\rightarrow {{\mbox{\rm GL}}}(V)$ if and only if there exists a $(G_{1},1)$-morph of algebraic groups extending $\theta$. In particular, the corollaries to Theorem \ref{alg_branching} can be used to determine the existence and uniqueness of a $G$-module structure on $V$.
\begin{cor}{\rm (Existence Test)}
Suppose that $G$ is a connected algebraic group over ${\mathbb K}$ with Lie algebra ${\mathfrak g}$, and that $V$ is an indecomposable $G$-stable finite-dimensional ${\mathfrak g}$-module. Then there exists a $G$-action on $V$, which respects the ${\mathfrak g}$-module structure, if and only if there is a branch (in the terminology of Theorem \ref{alg_branching}) which does not terminate; for instance, a branch such that $H^2_{Rat}(G,G_{1};(A_m,\rho_m))=0$ for all $(A_m,\rho_m)$ on that branch.
\end{cor}
\begin{cor}\label{alg_Uniq} {\rm (Uniqueness Test)}
Suppose that $G$ is a connected algebraic group over ${\mathbb K}$ with Lie algebra ${\mathfrak g}$, and that $V$ is an indecomposable $G$-stable finite-dimensional ${\mathfrak g}$-module. Suppose further that there exists a $G$-action on $V$
which extends the ${\mathfrak g}$-module structure. This $G$-action is unique (up to isomorphism) if and only if there is a branch (in the terminology of Theorem \ref{alg_branching}) such that $H^1_{Rat}(G,G_{1};(A_m,\rho_m))=0$ for all $(A_m,\rho_m)$ on that branch.
\end{cor}
Observe that combining Corollary~\ref{alg_Uniq} with Corollary \ref{H1+H2} for the $N$-stable subnormal series $H_m=1+J^m$, $m\geq 1$, we get a similar result to Proposition 4.3.1 in \cite{Xan}.
\subsection{Comparison with $C^\bullet_{Rat}(G/L;A)$}
\label{s3.5}
Let us now mimic the approach we took in Section \ref{s2.5} and examine how our cochain complex $(C^\bullet_{Rat}(G,L;A),d)$ compares with the complex $(C^\bullet_{Rat}(G/L;A),d)$ on the level of cohomology. We use the notation of Section~\ref{s3.3}. As with our discussion in Section~\ref{s2.5} we have to assume that $L$ acts trivially on $A$ for this discussion to be meaningful -- a condition which holds in the examples considered.
Similar to the case for abstract groups, we have the following proposition.
\begin{prop}\label{H1Map2}
Under the aforementioned conditions
we have isomorphisms of groups
\newline
$H^0_{Alg}(G,L;A) \cong H^0_{Alg} (G/L;A)$
and
$H^1_{Alg} (G,L;A)\cong H^1_{Alg} (G/L;A)$.
\end{prop}
\begin{proof}
Making use of the universal property of the quotient for algebraic groups, the proof follows word-for-word as in Proposition~\ref{H1Map}.
\end{proof}
Recalling the observation that there is no distinction between $H^i_{Alg}$ and $H^i_{Rat}$ for $i=0,1$ this tells us that $H^0_{Rat}(G,L;A) \cong H^0_{Alg} (G/L;A)$
and
$H^1_{Rat} (G,L;A)\cong H^1_{Alg} (G/L;A)$ in these circumstances.
The universal property of the quotient for algebraic groups further yields an analogue of Proposition \ref{H2inj}.
\begin{prop}
The maps ${{\mbox{\rm Inf}}}_{Alg}:H^2_{Alg}(G/L;A)\to H^2_{Alg}(G,L;A)$ and \newline
${{\mbox{\rm Inf}}}_{Rat}:H^2_{Rat}(G/L;A)\to H^2_{Rat}(G,L;A)$ are injective.
\end{prop}
\begin{proof}
The proof follows as in Proposition \ref{H2inj}.
\end{proof}
In the case of abstract groups, Section~\ref{s2.5} shows that by making careful choices of $(L,H)$-morphs in Theorem~\ref{branching} the image of the obstruction maps ${{\mbox{\rm Obs}}}:
\LH(G,N)_{\rho_i}
\longrightarrow
H^2(G,L;(A_i,\rho_i))$
always lies inside $H^2(G/L;(A_i,\rho_i))\hookrightarrow H^2(G,L;(A_i,\rho_i))$. As such, it is possible to reinterpret Theorem~\ref{branching} using the complex $(C^\bullet(G/L;A),d)$ instead of $(C^\bullet(G,L;A),d)$ at all points. This conclusion for abstract groups, however,
relies on the observation that it is always possible to assume that the $(L,H)$-morphs being considered are normalised. When translating the results to the case of algebraic groups it is far from clear that the analogues of Lemma~\ref{Norm} and Corollary~\ref{H2Map} hold.
{\bf Question:} Can the $(L,H)$-morphs considered in Sections~\ref{s3.3} and \ref{s3.4} be chosen to be normalised?
|
1,108,101,565,437 | arxiv | \section{Clock Postulate and Global Reference Frame}
One of the widely known incompatibilities between special relativity and general relativity is how each theory sees red-shift. In special relativity, the moving observers notice clocks rate change by a Lorentz factor $\gamma$. But stationary observers share the same clocks rate even if they are at different spatial positions as the electromagnetic wave frequency is only a source-dependent property. Therefore, if those stationary observers notice any shift in the wavelength, then it is justified only if the observer's clock ticks with different rates at different spacetime points, which is not compatible of special relativity. In fact the general theory of relativity justifies the change in clock rate as a response to the change in the gravitational potential from point to another, i.e., gravitational red-shift is a property of general covariance, which is global principle of general relativity. That is why A. Einstein realized that special theory of relativity and the equivalence principle hold locally, not globally \cite{Einstein:1911}. This is clearly manifested in accelerated Rindler spacetime with metric $ds^2=-r^2dt^2+dr^2$ \cite{Rindler:1966zz} such that the observer's clock rate is determined by $R(r)=\big(1-2V(r)\big)^{1/2}$, where $V(r)$ is the gravitational potential. Therefore, when $r=0$, the observer clock rate is also zero; $R(r)=0$, which matches with the expected observation of the clock rate at the horizon: time freezes. As long as the gravitational field is uniform and weak, we can approximately comparing $1/\gamma=\sqrt{1-v^2/c^2}$ of special relativity to $R(r)\sim1-\big(V(r)/c^2\big)$ in general relativity, where $g_{tt}=-R^2(r)$. By a uniform field we mean it has one direction and constant value like the electric field between two infinitely long parallel plates. And by weak we mean $V(r)/c^2<<1$. This means locally we can explain the redshift assuming special relativity and the equivalence principle. But when the full theory of general relativity is assumed, things change. In special relativity you should detect redshift while the source is moving with constant speed from spatial point to another. This is Doppler shift and it's constant because it's characterized by the speed only. However in the presence of gravity and without source movement, we still can detect a gravitational redshift when the time-varying gravitational potential itself changes at the same point through time. The redshift is characterized by the change in the gravitational potential. So if we want to study the change in the gravitational potential, we need to consider some differential equation. If you want to solve this differential equation from SR viewpoint, you don't get correct answer. Rather you get Abraham theory or Nordstrom theory. And if you do the differential equation from GR viewpoint, i.e., general covariance differential equation, you get the answer that matches with experiments. But that answer is not compatible with SR as SR is based on inertial frames only. More details on this issue is reported here \cite{GRdisbute}.
In order to correlate how special relativity and general relativity see redshift, we should ask how acceleration/gravitation affect the clock rate. According to Don Koks \cite{Koks:1998}, we need to take into consideration what he calls is \emph{clock postulate}. The clock postulate generalizes the special relativistic comparison between the different clocks rates according to the Lorentz factor $\gamma$. This generalization is based on that even if the moving clock accelerates, the ratio of the rate of stationary clock compared to its rate is still scaled or compared by $\gamma$ as in relativistic Doppler effect. Consequently, ratio depends only on $v$ not the derivatives of $v$. Therefore, an accelerating clock counts out its time in a way that fore every one moment, its rate will slow by a factor $\gamma$ which is determined by instantaneous velocity at that moment; its acceleration has no effect at all. It is important to emphasize on that the clock postulate does not say that the counting out rate of a moving clock is unaffected by its acceleration. It says that the quantitative measurement of the relative rate does not depend on acceleration. Equivalently, the rate of such accelerated clock is indistinguishable from that of a clock in a \emph{momentarily comoving inertial frame}. For that, we can imagine an observer, in normal frame, is holding an inertial clock that for a brief moment slows to a stop alongside another accelerated observer with a clock, so that their relative velocity is momentarily zero. This is crucial as we shall see when we consider The Arnowitt-Deser-Misner (ADM) formalism. At that moment they are ticking at the same rate. If that all is still confusing, you may consider the example of riding a bicycle on an icy morning that is mentioned in Ref. \cite{Koks:1998}. Contemplation about the clock postulate is not that obvious as the postulate can not be proved. Rather, simply it describes how we observe the physical world. It might be thought that the clock postulate mandates abandoning the Equivalence Principle as the Equivalence Principle equates between gravitational fields and acceleration. However, this is not true for the reasons explained in Ref. \cite{Koks:1998} by the example of a rocket empty of fuel and the two astronauts.
\section{One preferred frame}
The absolute space and time means by definition a preferred frame, in which physics laws appear to be recognizably different from they appear in other frames. This Newtonian concept is contradicted with the assumed principle of relativity in inertial frames and the principle of covariance in general theory of relativity. However, all inertial frames are preferred over non-inertial frames in special relativity as they observe only cause-and-effect relations between events in closed intervals, a property that general relativity lacks as it can observe a cause-and-effect relations between internal and external events. In this paper, we use the Rindler observer and the ADM formalism to restore the compatibility between the inertial privileged frames and relative space of general relativity, at least slice by slice. For more details on this issue, see \cite{Wheeler,GRdisbute}. We consider spacetime foliation in ADM formalism \cite{Arnowitt:1962hi}, at every given moment according to canonical observers, each with its speed at that moment. These observers are related to each other by ``synchronicity'' \cite{LachiezeRey:2001gn}.
This approach comes with \textit{a global reference frame} that stays Minkowskian along the worldline of such observer. Despite that the absolute synchronicity is not attainable either in special relativity or in general relativity, a local synchronicity is extended beyond local neighborhoods as in Ref. \cite{LachiezeRey:2001gn}. This extension shows how the observed redshift can be useful in constructing a black hole universal clock with the help of relative gravitational redshift, as how we show in this letter, and the idea that the black hole is fully entangled with the space around it from Hawking radiation. The entanglement ``monogamy'' is saved by the cool horizon approach \cite{Maldacena:2013xja}. Like any clock, time will be measured in terms of lengths. This clock will form a triangle that gives all measurements this clock can yield. The Rindler observer \cite{Rindler:1966zz} would analyze the ADM formalism as if the shift function equal zero $N_i=0$. Therefore, the normal vector $\hat{n}$ to the spatial slice is proportional to the time basis $\partial_t$ of Rindler frame field with proportionality factor that is equal to the lapse function $\alpha$. In this case the coordinate observer is moving with velocity $(g_{tt})^{-1/2}e_t$ while the normal observer moves with velocity $\hat{n}$. The factor similar to $\gamma$ in ADM formalism is in form of $(1-N_iN^i/\alpha^2)^{-1/2}$. See Fig \ref{ADM}. Rindler observer, the previous factor is equal to 1. However this comparison between observers is NOT a physical relativist Lorentz boost, and the frame of reference can be dragged with any speed the observer chooses. It turns out that the normal observer and the Rindler observer are \emph{the same} or \emph{One Observer}. They both see each other as if they are both belong to the same inertial Minkowski frame. This matches with the global reference frame described in Ref. \cite{LachiezeRey:2001gn}. The one observer idea may match with Wheeler idea of One-electron universe that he described to Feynman and on which Feynman depend to assume that "positrons could simply be represented as electrons going from the future to the past in a back section of their world lines" as Feynman stated in his Nobel speech \cite{Wheeler-Feynman}.
\begin{figure}
\centering
\includegraphics[width=80mm]{Picture1.jpg}
\caption{ADM slices of spacetime}
\label{ADM}
\end{figure}
\section{The clock measurements}
We assume an existence of Schwarzschild black hole with an event horizon. We investigate the gravitational red-shift which is a property of general covariance. The Rindler's Observer measures the gravitational red-shift which is a property of general covariance and its relative values between any two points or slices of spacetime. Therefore, we consider two points ($A$ and $R$) in the gravitational field of black hole as shown in the following Fig. (\ref{blackholetriangle}). Notice here these two points form a triangle that follow a geodesics geometry of the considered black hole when connecting the two points with black hole center. If $R$ and $A$ are far enough from $K$, the triangle become approximately Euclidean triangle.
\begin{figure}[h!]
\centering
\includegraphics[width=80mm]{Everything.pdf}
\caption{Black hole universal clock }
\label{blackholetriangle}
\end{figure}
Between these two points $A$ and $R$, there are two possible clock measurements as follows:
\begin{enumerate}
\item Relative gravitational red-shift which is represented by the ratio at two different points
\begin{equation}
\frac{z_A}{z_R}=\frac{(1-\frac{r_s}{r_A})^{-1/2}-1}
{(1-\frac{r_s}{r_R})^{-1/2}-1}
\label{Relative}
\end{equation}
\item The difference in gravitational red-shift at two different points.
\begin{equation}
\Delta z= z_A-z_R= (1-\frac{r_s}{r_A})^{-1/2}-(1-\frac{r_s}{r_R})^{-1/2}\label{Difference}
\end{equation}
\end{enumerate}
\section{Weak Gravitational Approximation}
\subsection{Relative gravitational red-shift}
We consider the weak gravitational approximations, $r_s<<r_K$ and $r_s<<r_R$. The gravitational red-shift for both $A$ and $R$ can be approximated as follows
\begin{eqnarray}
z_A&=&(1-\frac{r_s}{r_A})^{-1/2}-1\approx \frac{r_s}{2r_A} \\
z_R&=&(1-\frac{r_s}{r_R})^{-1/2}-1\approx \frac{r_s}{2r_R}.\label{shiftsAR}
\end{eqnarray}
We compute the relative gravitational red-shift using Eq(\ref{Relative}). We express it in terms of all lengths measured at $R$ including the distance between $A$ and $R$ ($r_{AR}$).
\begin{equation}
\frac{z_R}{z_A}=\frac{1}{\sqrt{1-\frac{r_{AR}^2}{r_A^2}+2 \frac{r_R r_{AR}}{r_A^2} \cos{\alpha}}}=\delta
\label{resolutionrelativity1}
\end{equation}
Notice the value of $\alpha$ can be $0 \leq \alpha \leq \pi/2$. This equation represents the relative gravitational-red-shift between two points $A$ and $R$ in a weak gravitational field. For the case $\alpha=\pi/2$. The relative gravitational red-shift will be given by
\begin{equation}
\frac{z_R}{z_A}=\frac{1}{\sqrt{1-(\frac{r_{AR}}{r_A})^2}}=\delta \label{gammafactor}
\end{equation}
On the other side, the measurement in local inertial frames are determined in terms of relative time dilation as follows
\begin{equation}
\frac{t_R}{t_A}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\end{equation}
where $v$ is the relative speed between the two points $A$ and $R$ in the local inertial frames and $c$ is the speed of light.
We notice the relative gravitational red-shift or relative gravitational time dilation matches with the definition of time dilation in special relativity if the ratio $r_{AR}^2/{r_A}^2$ can be replaced by ratio $v^2/c^2$. The match is legitimate and mathematically consistent since the relative gravitational red-shift introduces a local measurement which is the time dilation in local inertial frames. Therefore, for mathematical consistency of general relativity, its local measurement should be equivalent to measurement special relativity that hold only in local inertial frames. This means when $\alpha=\pi/2$, the gamma factor of special relativity emerges as a ratio between the gravitational red-shift at $A$ and $R$. The clock measurements depends only on \textbf{``one variable''}; the distance from the gravitational source, which is the reason for velocity ratios turned to be lengths ratios in this delta factor in Eq. (\ref{gammafactor}). The ratio $r_{AR}^2/{r_A}^2$ can be considered as a geometric or gravitational interpretation of the ratio $v^2/c^2$. This comparison can be written as
\begin{equation}
\frac{r_{AR}}{r_A}=\frac{r_{AR}/t}{r_A/t}=\frac{v}{c} \label{wholeandart}
\end{equation}
This would support the approach of time varying speed of light as a solution of cosmological puzzles that was suggested in \cite{Albrecht:1998ir}. It may support also the experimental findings of changing physical constants such as fine structure constant in gravitational field as shown recently in \cite{Wilczynska:2020rxx}. In our case, the ratio $v/c$ varies depending on the distance from the gravitational source.
We note that time can be inserted easily in the previous equation as a ``redundant variable'' which suggest a possible timeless state which is consistent mathematically. The timeless state has been proposed in many contexts such as shape dynamics which introduce a gravitational origin of arrow of time
\cite{Barbour:2011dn,Barbour:2013jya}. The timeless also emerged in Thermal time hypothesis
which assume that time only flow in thermodynamics or statistical patterns \cite{Connes:1994hv}. It has been mathematically intuited as well that timeless universe is possible \cite{timelessmath}. In other words, space may be a frozen time \cite{ibnarabi}.
To realize the effect of other values of angle $\alpha$ in weak gravitational field, we consider an approximation which is $r_{AR}<<r_A$, $r_{AR}<<r_R$. In that case, the delta factor in Eq. (\ref{resolutionrelativity1}) is approximated as following
\begin{eqnarray}
\delta \approx 1-\frac{r_R r_{AR}}{r_A^2}\cos{\alpha} \label{Kepler}
\end{eqnarray}
It is found that this equation matches with the derivative of Kepler equation.
\begin{equation}
\frac{dm}{dE}= 1-e \cos{E}
\end{equation}
where $m$ is the mean anomaly, $E$ is the eccentric anomaly, and $e$ is the eccentricity. In our approximation, the eccentricity $e$ is approximately equal to $r_R r_{AR}/r_A^2$, and $E$ refers to the angle $\alpha$. This gives a geometric interpretation of Kepler equation from the relative gravitational red-shift.
\subsection{Difference in Gravitational red-shift}
In this section, we compute the clock measurement as difference in gravitational red-shift. For weak gravitational approximation, we get
\begin{equation}
\Delta z= z_R-z_A= \frac{r_s}{2r_R} -\frac{r_s}{2r_A} \label{differnce}
\end{equation}
Let us make an approximation as following $r_A=r_R+x$, where $x<<r_A$ and $x<<r_R$. In that case, Eq. (\ref{differnce}) will be rewritten as follows. We use the value of Schwarzschild radius $r_s=2 GM/c^2$
\begin{equation}
\Delta z=G M \frac{x}{c^2 r_R^2} \label{mass1}
\end{equation}
where $G$, is the gravitational constant, $M$, is the black hole mass and $c$ is the speed of light. From Eq.(\ref{wholeandart}), $c$ can be set to equal to $r_A$ if we take $t$ to be unity since we agree that t is a redundant factor through matching local gravity measurement with local inertial frames. We find that Eq. (\ref{mass1}) can be arranged to take the following form
\begin{equation}
\Delta z~M=\Delta M=GM^2 \frac{x}{r_A^2 r_R^2} \label{emc3}
\end{equation}
where $\Delta M= \Delta z~M$. $\Delta M$ represents a possible mass between any two different points in the gravitational field. This would give a geometric representation for mass concept in terms of difference between different points to the black hole. We want to understand the physical meaning of the factor $G M^2$ in r.h.s of Eq. (\ref{emc3}). When we look at Bekenstein-Hawking entropy equation \cite{Bekenstein:1973ur,Hawking:1974sw}.
\begin{equation}
S_{BH}= \frac{c^3 A}{4 G \hbar}=\frac{4 \pi}{c \hbar} G M^2
\end{equation}
where $A= 16 \pi (G M/c^2)^2$ stands for surface area of a black hole. We found that the factor $GM^2$ in r.h.s of Eq. (\ref{emc3}) between any two different points can be expressed in terms of black hole entropy as follows
\begin{equation}
\Delta z~M= \Delta M=\frac{\hbar}{4 \pi} \frac{x}{r_R^2 r_A} S_{BH} \label{mass}
\end{equation}
We assumed that time is a unity. let us consider this unit as the Planck time. This means that the Planck constant in previous equation can be replaced through the following process
\begin{equation}
t_p=\sqrt{\frac{\hbar G}{c^5}}=1
\end{equation}
Since the Planck time is our unity, then c can set to be $r_A$. Therefore, the Planck constant in this geometric picture will be given by
\begin{equation}
\hbar G= r_A^5
\end{equation}
This equation gives a geometric or gravitational interpretation of Planck constant. The relative mass between any two different points is therefore given by
\begin{equation}
\Delta z M= \Delta M= \frac{1}{4 \pi G}\frac{x~ r_A^4}{r_R^2} S_{BH}=\frac{1}{16 \pi G}\frac{x ~~r_A^2}{r_R^2}~~ A \label{InformationMatter}
\end{equation}
The previous equation gives purely a geometric expression for the relative mass in terms of the gravitational source area of its full entropy. It is experimentally proved that the difference in gravitational potential has an effect on the apparent weight of the 14.4-keV ray of Iron (Fe) \cite{Pound:1960zz,Pound:1964zz}. This may be an experimental support for the derived relation that connect the difference in gravitational red-shift and emergence of mass in this section.
\section{Gravity and Uncertainty}
In previous sections, we have shown that the concept of velocity is replaced with the relative distance between any any two different points in the space time in the black hole universal clock. This would generate a timeless state in a mathematically consistent way. In that state, the gravitational measurements happens in terms of only one variable which is the distance from the gravitational source. Time variable appear to be a redundant variable. Since time, and therefore velocity dissolves, therefore there is no meaning to define uncertainty in this timeless state. We propose that the distance from the gravitational source may form the \emph{hidden variable} of every observation process in quantum mechanics. This may complete the connection between quantum mechanics and gravity in one unified theory, which is the timeless state. This may complete the picture that was introduced in EPR \cite{Einstein:1935rr}.
This implies that the uncertainty amount would decrease as the measurement happens closer to the gravitational source. The uncertainty \emph{emerges} due to the difference in information between point $A$ and point $R$ without knowing the distance to the source. This difference is encoded in Eq. \ref{InformationMatter}. The difference in information (uncertainty) would be represented by the difference in gravitational red-shift as follows:
\begin{equation}
\frac{4 \pi \Delta M r_R^2 r_A}{x S_{BH}}= \hbar
\end{equation}
Notice that the difference in information between Point $A$ and point $R$ depends on the distances $r_A$ and $r_R$. If we do not know these values, this difference will be hidden in our local measurements, and therefore uncertainty emerges. Notice that the variables on the left hand side are greater than or equal to the Planck constant. This relation represent the hidden variables which is reason for emergence of uncertainty principle inequality in local measurements.
\section{Strong Gravity Case}
In strong gravity case, we can use Eqs. {\ref{Relative} and \ref{Difference}} without any approximation. These relations can be computed for any two points, and it gives a wide spectrum of measurements of relative gravitational red-shifts and masses in strong gravity field. In strong gravity field, the triangle will not be perfectly Euclidean but can be computed for every kind of measurement by knowing the length of this triangle.
\section{Why Clock Postulate/Timeless State of Gravity may be important?}
Time in principle is the change happening for any physical system. To approach a unified picture of physics, we can investigate states in which no change happening, a frozen moment of the physical system. In that state, everything will be reduced to geometry. In this geometric picture, we can find a geometric picture of mass, speed of light, etc. Timeless state in that sense could introduce a unified picture of different concepts in physics. We hope to evolve this study in the future.
\section{Conclusion}
We investigate how Rindler observer in ADM formalism make measurements in the slices of space-time. We found that the Rindler observer in ADM formalism restores the compatibility between the inertial privileged frames and relative space of general relativity, at least slice by slice. We determined the relative or local measurements that Rindler observer could make in each slice of the space-time. We got a timeless state of the universe in which we found a geometric interpretations of speed of light and mass. The timeless state may form a correlation between relative gravitational red-shift and internal symmetries that are independent of time. We hope that this equivalence may open the door for a gravitational technology. It is worth mentioning that timeless state of gravity can shed a light on the nature of problem of time in Wheeler-De-Witt equation. We hope to report on these in the future.
\vspace{10 mm}
{\textbf Acknowledgment}
This work is supported by
the quantum gravity research grant, Los Angeles, California.
|
1,108,101,565,438 | arxiv | \section{Introduction}
Many hard diffractive and high-energy processes are under intense study in the last two decades. One of the main subjects in high-energy physics is the improvement of our knowledge about quantum chromodynamics (QCD). Additionally, the Higgs mechanism is one of most important subjects to be investigated at the LHC, being a cornerstone in the electroweak sector of the standard model (SM). The Higgs boson is expected to be produced by the gluon fusion process ($gg \to H$), making the data analysis of this process an important topic in the project for the LHC experiments, like ATLAS and CMS.
Recent analyses presented an updated estimation of the mass range where it is expected to observe the Higgs boson, which, combining the data coming from CDF and D0 experiments at the Tevatron, have excluded the range $158 < M_{H} < 175$ GeV with $95\%$ of confidence level \cite{:2010ar}. Furthermore, very recent simulations of the ATLAS experimental group have shown that a wider mass range can be excluded with the future LHC data. For instance, with an integrated luminosity of 2 fb$^{-1}$ and a 8 TeV beam energy the range 114 $< M_{H}<$ 500 GeV can be excluded with 95\% confidence level \cite{Collaboration:2010dk}.
The diffractive processes are well described by the Regge theory, where it is considered that a family of resonances is exchanged by the colliding protons \cite{Collins:1977jy}. The leading pole that accounts for this interaction will drive the high-energy behavior of the total cross section, being particularly labeled Pomeron, that has the vacuum quantum numbers \cite{Foldy:1963zz}. However, the nature of the Pomeron is not completely known, as well as its reaction mechanisms, but it is a successful formalism to describe hard diffraction data \cite{Donnachie:1992ny}. Moreover, based on the parton model, it was proposed that the Pomeron could have a partonic content, i.e., quarks and gluons as its constituents, by the Ingelman-Schlein (IS) formalism \cite{Ingelman:1984ns}. Then, systematical observations of diffractive deep inelastic scattering at HERA have increased the knowledge about the Pomeron, providing a diffractive distribution of singlet quarks and gluons into the Pomeron as well as the diffractive structure functions \cite{Aktas:2006hy}.
In this work we are interested in the single diffractive (SD) processes, characterized by the emission of a Pomeron from one of the colliding hadrons that scatters off the other hadron. The cross sections for the SD process are computed at next-to-leading order (NLO) accuracy with QCD and electroweak (EW) corrections, and we use the gap survival probability (GSP) from two different models that accounts for the survival factor for the diffractive Higgs boson production. The cross sections and the diffractive ratios are estimated for the process $p + p(\bar{p}) \to p + H + [LRG] + p(\bar{p})$ for the kinematical regime of the Tevatron ($\sqrt{s}$ = 1.96 TeV) and for those expected to be reached in the LHC ($\sqrt{s}$ = 7, 8 and 14 TeV). In this approach the hard processes will occur by the interaction of the content of one hadron and the content of the Pomeron. In other words, the diffractive cross section is the convolution of the diffractive parton distribution functions (DPDF) and the corresponding partonic cross section, in a similar way as the inclusive case. In addition, diffractive events with a large momentum transfer are also characterized by the absence of hadronic energy in a certain angular regions of the final state, the so-called rapidity gaps. So, the SD processes will present in the final state a large rapidity gap between one proton and the Higgs boson as its main signature.
For the Tevatron kinematical regime, it is known that the data are not correctly predicted with the use of the IS formalism \cite{GayDucati:2007ps,*Kopeliovich:2005ym}, however there are important contributions from unitarity effects to the single-Pomeron exchange cross section that can be considered. These absorptive (unitarity) corrections take into account the fraction of large rapidity gap processes, except elastic scattering, being quite important for the reliability of the predictions for hard diffractive processes. The multi-Pomeron contributions depend on the particular hard process, and one is able to compute the GSP \cite{Chehime:1992bp,*Bjorken:1991xr,*Bjorken:1992er} for a specific production process, which accounts for the fraction of events where the rapidity gaps will be present in the final state after the rescattering events. In this way, the application of a survival factor in the diffractive cross section can correctly describe the high-energy data. For instance, some predictions for $W^{\pm}$, $Z^{0}$, heavy quarks, $\Upsilon$ and $J/\psi$ were presented in Refs.\cite{GayDucati:2007ps,GayDucati:2010vu,*GayDucati:2009rr} for the LHC energies, and it was possible to see that this approach describes very well the Tevatron data.
This paper is organized as follows: in Sec. \ref{inc}, we present the main equations for the inclusive production of the Higgs boson at NLO accuracy. Next, in Sec. \ref{sd-dpe}, we rewrite the parton luminosity in order to introduce the Pomeron exchange from the colliding proton, taking into account the $gg \to H$ production. Further, in Sec. \ref{gsp}, we present the models for the GSP applied in this work, showing the probabilities for each energy regime. Then, in Sec. \ref{results}, we present the estimations for the inclusive and diffractive cross sections as a function of the Higgs boson mass for different collider energies, and also the rapidity distributions of the Higgs boson. Finally, in Sec. \ref{concl}, we summarize our conclusions.
\section{Inclusive production}
\label{inc}
Let us present the main formulas for the inclusive cross sections for the production of Higgs boson in proton-proton collisions. The production process considered in this work is the gluon fusion $pp \to gg \to H$, since it is the leading production mechanism of the Higgs bosons in the high-energy regime \cite{Carena:2002es,*Hahn:2006my,*Duperrin:2008in}. The gluon coupling to the SM Higgs boson is mediated by a triangular loop of quarks, with the leading contribution of the quark top. The production cross section at lowest order is given by \cite{Spira:1995rr}
\begin{eqnarray}
\sigma_{LO}(pp\to H + X)=\sigma_{0}\tau_{H}\frac{\dif{\cal{L}}^{gg}}{\dif\tau_{H}},
\label{equacao34}
\end{eqnarray}
with the Drell-Yan variable defined as $\tau_{H} = M^{2}_{H}/s$, where $s$ is the invariant $pp$ collider energy squared. The gluon-gluon luminosity has the form
\begin{eqnarray}
\frac{\dif{\cal{L}}^{gg}}{\dif\tau}=\int^{1}_{\tau}\frac{\dif x}{x}g(x,M^{2})g(\tau/x,M^{2}),
\label{luminosity}
\end{eqnarray}
with $g(x,M^{2})$ being the gluon distribution function into the proton, where we apply the MSTW2008 parametrization at NLO accuracy for such distribution \cite{Martin:2009bu,*Martin:2009iq}, with $M$ as the factorization scale. In Eq.(\ref{equacao34}), the function $\sigma_{0}$ reads
\begin{eqnarray}
\sigma_{0} = \frac{G_{F} \alpha^{2}_{s}(\mu^{2})}{288\sqrt{2}\pi}\left |\frac{3}{4}\sum_{q}A_{Q}(\tau_{Q}) \right |^{2},
\label{SIGMAzero}
\end{eqnarray}
where $A_{Q}(\tau_{Q}) = 2[\tau_{Q} + (\tau_{Q} - 1) f(\tau_{Q})]/\tau_{Q}^{2}$, and $\tau_{Q} = M^{2}_{H}/4m^{2}_{q}$. In this work it is considered only the leading contribution of the top quark ($m_{q}$ $\equiv$ $m_{t}$ = 172.5 GeV), called heavy-quark limit in Ref.\cite{Spira:1995rr}, and then we are taking the approximation $\tau_{Q} \leq 1$, which means the use of $f(\tau_{Q}) = \arcsin^{2}{\sqrt{\tau_{Q}}}$.
The NLO QCD corrections for the fusion process $gg \to H$ correspond to the processes $gg \to H(g)$, $gq \to Hq$ and $q\bar{q} \to Hg$ \cite{Dawson:1990zj,Spira:1995rr}, introducing virtual and real corrections to the scattering amplitude. The production cross section for the Higgs boson at NLO accuracy in $pp$ collisions is written as \cite{Spira:1995rr}
\begin{eqnarray}
\sigma_{NLO}(pp \to H + X) = \sigma_{0} \left[ 1 + {\cal{C}}\frac{\alpha_s(\mu^{2})}{\pi} \right ]\tau_{H}\frac{d{\cal{L}}^{gg}}{d\tau_{H}} + \Delta\sigma_{gg} + \Delta\sigma_{gq} + \Delta\sigma_{q\bar{q}},
\label{equation38}
\end{eqnarray}
with the renormalization scale in the strong coupling constant $\alpha_{s}$ and the factorization scale in the parton densities to be fixed properly. Particularly, in Eq.(\ref{SIGMAzero}) the strong coupling constant is applied at lowest order accuracy; however, for the NLO contributions the $\alpha_{s}$ is applied at NLO accuracy through the exact numerical solution \cite{Gluck:1998xa}
\begin{eqnarray}
\frac{\dif \alpha_{s}(\mu^{2})}{\dif \ln \mu^{2}} = - \frac{\beta_{0}}{4\pi} \alpha_{s}^{2}(\mu^{2}) - \frac{\beta_{1}}{16\pi^{2}}\alpha_{s}^{3}(\mu^{2}),
\label{nlo_alphas}
\end{eqnarray}
where $\beta_{0} = (11N_{c} - 2N_{F})/3$ and $\beta_{1} = (102N_{c} - 38N_{F})/3$, with $N_{c} = 3$. The $\Lambda$ scale is fixed by the threshold of the quark masses during the $\mu^{2}$ evolution, and fixing the value of $N_{F}$ properly.
The coefficient ${\cal{C}}(\tau_{Q})$ denotes the contributions from two-loop virtual corrections, regularized by the infrared singular part of the cross section for real gluon emission, and is expressed by \cite{Spira:1995rr}
\begin{eqnarray}
{\cal{C}}(\tau_{Q}) = \pi^{2} + c(\tau_{Q}) + \left( \frac{11 N_{c} - 2N_{F}}{6} \right) \log\frac{\mu^{2}}{M^{2}_{H}} ,
\label{equacao39}
\end{eqnarray}
where $\pi^{2}$ refers to the infrared part, and $c(\tau_{Q})$ is a finite function, which, solved analytically, results in $c(\tau_{Q}) = 11/2$ for $\tau_{Q} = M^{2}_{H}/4m^{2}_{q} \ll 1$ \cite{Graudenz:1992pv}.
The $\Delta\sigma_{ij}$ are the hard contributions from gluon radiation in the $gg$ scattering and the $q\bar{q}$ annihilation, and they depend on the renormalization scale $\mu$ and the factorization scale $M$ in the parton densities. These contributions can be expressed by \cite{Spira:1995rr}
\begin{subequations}
\begin{eqnarray}\nonumber
\Delta\sigma_{gg} &=& \int^{1}_{\tau_{H}}\dif\tau\frac{\dif{\cal{L}}^{gg}}{\dif\tau}\frac{\alpha_{s}}{\pi}\sigma_{0} \left \{ -\hat{\tau}P_{gg}(\hat{\tau}){\text{log}}\frac{M^{2}}{s} + d_{gg}(\hat{\tau},\tau_{Q}) \right. \\ &+& \left. 12\left [ \left(\frac{{\text{log}}(1-\hat{\tau})}{1-\hat{\tau}}\right )_{+} - \hat{\tau}[2-\hat{\tau}(1-\hat{\tau})]{\text{log}}(1-\hat{\tau}) \right ] \right\}, \\
\Delta\sigma_{gq} &=& \int^{1}_{\tau_H}\dif\tau\sum_{q,\bar{q}}\frac{\dif{\cal{L}}^{gq}}{\dif\tau}\frac{\alpha_{s}}{\pi}\sigma_{0}\left \{ d_{gq}(\hat{\tau},\tau_{Q}) + \hat{\tau}P_{gq}(\hat{\tau})\left [ -\frac{1}{2}{\text{log}}\frac{M^{2}}{\hat{s}}+{\text{log}}(1-\hat{\tau})\right]\right\}, \\
\Delta\sigma_{q\bar{q}} &=& \int^{1}_{\tau_{H}}\dif\tau \sum_{q}\frac{\dif{\cal{L}}^{q\bar{q}}}{\dif\tau}\frac{\alpha_{s}}{\pi}\sigma_{0}d_{q\bar{q}}(\hat{\tau},\tau_{Q}),
\label{equacao40}
\end{eqnarray}
\end{subequations}
where $\hat{\tau} = \tau_{H}/\tau$, and $P_{gg}(\hat{\tau})$ and $P_{gq}(\hat{\tau})$ are the standard Altarelli-Parisi functions \cite{Altarelli:1977zs}
\begin{subequations}
\begin{eqnarray}
P_{gg}(\hat{\tau}) &=& 6 \left\{ \left( \frac{1}{1-\hat{\tau}} \right)_{+} + \frac{1}{\hat{\tau}} - 2 + \hat{\tau}(1 - \hat{\tau}) \right\} + \frac{11N_{c}-2N_{F}}{6} \delta(1-\hat{\tau}), \\
P_{qg}(\hat{\tau}) &=& \frac{4}{3} \frac{1 + (1 - \hat{\tau})^{2}}{\hat{\tau}}.
\label{ap-eq}
\end{eqnarray}
\end{subequations}
The $F_{+}$ denotes the usual $+$ distribution, such that $F(\hat{\tau})_{+} = F(\hat{\tau}) - \delta(1 - \hat{\tau}) \int^{1}_{0}d\hat{\tau}^\prime F(\hat{\tau}^\prime)$. As we are considering the heavy-quark limit, the $d_{ij}$ functions can be solved analytically, resulting in a simpler set of expressions \cite{Spira:1995rr}
\begin{subequations}
\begin{eqnarray}
d_{gg}(\hat{\tau},\tau_{Q}) & = & -\frac{11}{2}(1-\hat{\tau})^{3}, \\
d_{gq}(\hat{\tau},\tau_{Q}) & = & -1+2\hat{\tau}-\frac{\hat{\tau}^{2}}{3}, \\
d_{q\bar{q}}(\hat{\tau},\tau_{Q}) & = & \frac{32}{27}(1-\hat{\tau})^{3}.
\label{dfunctions}
\end{eqnarray}
\end{subequations}
Finally, also included are the electroweak two-loop corrections \cite{Actis:2008ts,*Actis:2008ug,*Actis:2008uh}, which enhance the total cross section by 5\% in comparison to the NNLO QCD cross section. In this way, the total cross section is computed with the addition of the EW corrections by
\begin{eqnarray}
\sigma_{\textrm{NLO}} \equiv \sigma_{\textrm{QCD+EW}} = \sigma_{\textrm{QCD}}(1 + \delta_{\textrm{EW}}).
\label{ew-corr}
\end{eqnarray}
The total cross sections for the inclusive process are shown by the solid curves in the Figs. \ref{fig_lhc}-\ref{fig_tev} for different collider energies. The gray bands around these curves express the variation of the renormalization and the factorization scales in the range 0.5$M_{H} < (\mu=M) < 4.0M_{H}$. Looking particularly to the results for the LHC, our results reproduce the values obtained in Ref.\cite{Spira:1995rr,*Dittmaier:2011ti}, although it is not the case for Ref.\cite{Erhan:2003za}\footnote{Comparing the results obtained in the Ref.\cite{Spira:1995rr,*Dittmaier:2011ti} and the curve presented in the Fig.6 in Ref.\cite{Erhan:2003za} for the total cross section in inclusive process, one can see that there is a disagreement between the results at $\sqrt{s}$ = 14 TeV, since the NNLO cross section for $M_{H}$ = 200 GeV in Ref.\cite{Erhan:2003za} is clearly smaller than that predicted in Ref.\cite{Spira:1995rr,Dittmaier:2011ti} at NLO.}.
\section{Diffractive production}
\label{sd-dpe}
For the diffractive process, the calculations are based on the IS formalism for diffractive hard scattering \cite{Ingelman:1984ns}. In this case, the Pomeron structure is taken into account by its quark and gluon content through the parametrization of the DPDF. The SD cross section is assumed to factorize into the Pomeron-hadron cross section and the Pomeron flux factor. In other words, it consists of three steps: first, a hard Pomeron is emitted from one of the protons in a small momentum transfer $|t|$, being this hadron detected in the final state; then, the second hadron scatters off the emitted Pomeron; during the Pomeron-hadron interaction, partons from the Pomeron interact with partons of the hadron, producing the Higgs boson. Accordingly, we will take into account absorptive effects, multiplying the diffractive cross section by a specific survival factor for each collider energy. The luminosity for the SD process reads
\begin{eqnarray}\nonumber
\frac{\dif{\cal{L}}_{\textrm{SD}}^{gi}}{\dif\tau} &=& \int^{1}_{\tau} \frac{\dif x}{x} \int \frac{\dif x_{{\tt I \! P}}}{x_{{\tt I \! P}}} F_{i/{\tt I \! P}/p}\left(x_{{\tt I \! P}},\frac{x}{x_{{\tt I \! P}}},M^{2}\right) g(\tau/x,M^{2}) \\
&+& \int^{1}_{\tau} \frac{\dif x}{x} \int \frac{\dif x_{{\tt I \! P}}}{x_{{\tt I \! P}}} g(x,M^{2}) F_{i/{\tt I \! P}/p}\left(x_{{\tt I \! P}},\frac{\tau}{xx_{{\tt I \! P}}},M^{2}\right),
\label{sdexp}
\end{eqnarray}
The Pomeron structure function $F_{i/{\tt I \! P}/p}$ is expressed by
\begin{eqnarray}
F_{i/{\tt I \! P}/ p} = f_{{\tt I \! P}/ p}(x_{{\tt I \! P}})f_{i/{\tt I \! P}}\left (\frac{x}{x_{{\tt I \! P}}},M^{2} \right ),
\label{func_pom}
\end{eqnarray}
with $f_{{\tt I \! P}/ p}(x_{{\tt I \! P}})$ being the Pomeron flux, and $f_{i/{\tt I \! P}} (\beta,\mu^{2})$ the parton distribution into the Pomeron, where $i$ stands for $g$, $q$, and $\bar{q}$.
In the estimates for the cross sections in Eq.(\ref{sdexp}), we consider a standard Pomeron flux from Regge phenomenology, which is constrained from the experimental analysis of the diffractive structure function. In this case, we apply the flux obtained with the H1 parametrization \cite{Aktas:2006hy}. The Pomeron structure function has been modeled in terms of a light flavor singlet distribution $\Sigma(x)$, i.e., the $u$, $d$ and $s$ quarks with their respective antiquarks. Also, it has a gluon distribution $g(z)$, with $z$ being the longitudinal momentum fraction of the parton in the hard subprocess. The gluon density is a constant at the starting evolution scale $Q^{2}_{0} = 2.5$ GeV$^{2}$. In our numerical calculations, we apply the cut $x < x_{{\tt I \! P}} \leq 0.05$ in agreement with the H1 parametrization. The Pomeron trajectory is assumed to be linear, $\alpha_{{\tt I \! P}}(t) = \alpha_{{\tt I \! P}}(0) + \alpha^{\prime}_{{\tt I \! P}}t$, with $\alpha^{\prime}_{{\tt I \! P}}$ and their uncertainties obtained from fits to H1 forward proton spectrometer (FPS) data \cite{Aktas:2006hx}. We choose $x_{{\tt I \! P}} \int^{t_{min}}_{t_{cut}} f_{{\tt I \! P}/p}\dif t = 1$ at $x_{{\tt I \! P}} = 0.003$, where $|t_{min}| \approx m^{2}_{p}x^{2}_{{\tt I \! P}}/(1 - x_{{\tt I \! P}})$ is the minimum kinematically accessible value of $|t|$, $m_{p}$ is the proton mass, and $|t_{cut}| = 1.0$ GeV$^{2}$ is the limit of the measurement. The H1 parametrization provides two different inputs for the fit of the partonic structure functions. As our curves show very close results using both fits, we chose the fit A to perform our predictions in this work.
\section{Gap Survival Probability}
\label{gsp}
In the diffractive cross sections [Eq.(\ref{sdexp})], we are further including the GSP $<\!|S|^{2}\!>$, being described in terms of absorptive corrections \cite{Bjorken:1991xr,Bjorken:1992er}. It can be estimated using the equation
\begin{eqnarray}
<\!|S|^2\!> = \frac{\int|{\cal{A}}\,(s,b)|^2\,e^{-\Omega (s,b)}\,\dif^{\,2}\!\boldsymbol{b}}{\int|{\cal{A}}\,(s,b)|^2\,\dif^{\,2}\!\boldsymbol{b}} ,
\end{eqnarray}
where $\cal{A}$ is the amplitude of the particular process of interest at the center-of-mass energy squared $s$ described in the impact parameter space $b$. The quantity $\Omega$ is the opacity (or optical density) of the interaction of the incoming hadrons. This suppression factor of a hard process accompanied by a rapidity gap does not depend only on the probability of the initial state survival, but it is also sensitive to the spatial distribution of partons inside the incoming hadrons, i.e., on the dynamics of the whole diffractive part of the scattering matrix.
There are distinct approaches in the literature to compute the value of the $<\!|S|^2\!>$, predicting different probabilities for the diffractive Higgs boson production. Applying a survival factor to diffractive processes brings an uncertainty to the predictions for the production cross sections \cite{GayDucati:2007ps}, since there is no accurate prediction for the GSP, resulting in an imprecise predictions. Hence, we compare two different models for the GSP, being the most applied in other works, in order to investigate the available calculations of the survival factor to drive our predictions, and certainly the ones that will be studied to describe the future data. The first one is that of Refs. \cite{Khoze:2000vr,*Kaidalov:2001iz} (labeled KKMR), which considers a two-channel eikonal model that embodies pion-loop insertions in the Pomeron trajectory, diffractive dissociation and rescattering effects. Then, the survival probability is computed for single, central and double diffractive processes at different collider energies, assuming that the spatial distribution in impact parameter space is driven by the slope $B$ of the Pomeron-proton vertex. We will consider the value $<\!|S|^2\!>_{\mathrm{KKMR}}^{\mathrm{SD}}$ = 6\% (10\%) for the SD process in the LHC (Tevatron).
The second estimation for the survival factor is the model presented in Ref. \cite{Gotsman:1999xq,*Gotsman:2005rt} (labeled GLM), with a calculation for an eikonal single-channel approach. We take the case where the soft input is obtained directly from the measured values of $\sigma_{tot}$, $\sigma_{el}$ and hard radius $R_{H}$. The F1C approach was chosen to perform our predictions, resulting in a probability of $<\!|S|^2\!>^{\mathrm{SD}}_{\mathrm{GLM}}$ = 8.1\% (12.6\%) for the LHC (Tevatron) energy. We quote Ref. \cite{Gotsman:2005rt} for a detailed comparison between this approach and the Kaidalov-Khoze-Martin-Ryskin (KKMR) one, including further discussions on model dependence of inputs and consideration of multichannel calculations.
Unfortunately, these models only account for the GSP in the kinematical regime of the Tevatron or the LHC energies, i.e., for $\sqrt{s} = 1.8$ TeV and $14$ TeV. In order to make precise estimations with reliable values for the GSP, we chose to adopt a similar way to estimate the survival factor, in \% for the desired energy, following the approach of Ref.\cite{Machado:2007fr}
\begin{eqnarray}
<\!|S|^{2}\!>(\%) = \frac{a}{b + \ln\sqrt{s}},
\label{func-form}
\end{eqnarray}
with the parameters $a$ = 46.52 (30.77) and $b$ = -3.80 (-4.41) for the GLM (KKMR) model. Then, the Table \ref{tab-gsp} summarizes all the survival factors applied in the predictions for the SD process.
These particular models were chosen in order to indicate the uncertainty (model dependence) of the soft interaction effects. It is worth to mention that some implementations of GLM model include the results of a two- or three-channel calculation for $<\!|S|^2\!>$, which are considerably smaller than the one-channel approach \cite{Gotsman:2005rt}.
\section{Results and comments}
\label{results}
In this work we are mainly interested in the analysis of the cross sections for the SD Higgs boson production for different collider energies, bringing a rapidity gap in the final state as its main signature. Furthermore, the diffractive production can be an alternative way to detect the Higgs boson in hadron colliders, since it provides a higher signal-to-background ratio \cite{Khoze:2006uj}. Moreover, the SD production cross sections are presented in Figs. \ref{fig_lhc}-\ref{fig_tev}, being the results presented with no survival factor by the dashed curves, and including the GLM (dot-dashed) and KKMR models (double-dot-dashed). Additionally, some values of the production cross section are presented in Table \ref{tab1} for selected Higgs masses, showing specifically the values for the cross section with the adopted survival probabilities. As one can see, the production cross section in the kinematical regime of the Tevatron is very small, as expected. However, for higher energies the cross section reaches values of the order of 100 fb, showing that it may be possible to detect the Higgs boson in the LHC through the SD process. Besides, there are some detectors to be set up at the LHC experiments, and they will make it possible to tag the outgoing proton \cite{Albrow:2008pn,*Bonnet:2007pw,*Roland:2010ch} or to detect forward showers \cite{Albrow:2008az,*Lamsa:2009ej}. Then, the SD events can be an effective way to look for the Higgs boson at the LHC.
Still, to have a clear analysis of these results and to estimate the fraction of diffractive events, we compute the diffractive ratio for the Higgs boson mass of $M_{H}$ = 120 GeV, being presented in Table \ref{tab2}. As expected, the diffractive ratios are small and growing with the collider energy. Specifically, the GLM model shows a ratio nearly constant from the Tevatron energy until 7 TeV, and then growing for higher energies. However, the KKMR model shows a different behavior, presenting a decrease in the same region, and growing slowly at higher energies. This effect is observed due to our assumption for the survival factor for the collider energy of 7 and 8 TeV. However, a proper calculation of the survival factor for these energies may show a higher GSP than those presented in the Table \ref{tab-gsp}, which will increase the ratio, reaching a similar behavior as the result for the Tevatron.
In fact, these values show that the SD events will have a very small rate in the LHC kinematical regime for a luminosity of a few fb$^{-1}$. It encourages the implementation of specific detectors in order to detect the rapidity gaps or the forward protons, since the Higgs boson discovery from its decay products is going to be more difficult in the inclusive production due to the huge background signal \cite{Rainwater:2002hm}.
Finally, in Figs. 5 and 6 we present the rapidity distribution of the Higgs boson for different collider energies. For higher energies, the distributions are clearly central, which shows that the contributions from the parton distribution function and the DPDF have larger values in central rapidity. Particularly, as the momentum fractions of the parton into the hadron A increases, the one of the parton into the hadron B (or the Pomeron) decreases uniformly, achieving a higher combined contribution for $y_{H} = 0$. However, this same behavior does not occur in the Tevatron kinematical regime, showing two distinct peaks in the distribution where the parton distribution function and DPDF have its higher combined contribution. In the results for mid energies (7 and 8 TeV), one can see that the distributions are still central, however showing very small peaks around $|y_{H}|$ = 2.
\section{Conclusions}
\label{concl}
In summary, we have evaluated estimations for the SD Higgs boson production in $pp$ collisions at the Tevatron and the LHC, considering the IS formalism with the introduction of rescattering corrections. We are using the Regge factorization to calculate the SD cross sections at NLO accuracy (QCD+EW). In particular, we take a parametrization from H1 Collaboration for the Pomeron structure function, extracted from their measurements of $F^{D(3)}_{2}$, with the results directly dependent on the quark singlet and gluon content of the Pomeron. For the available fits in this parametrization, we chose the fit A to perform our predictions. For instance, the cross sections are of about $\sigma_{\mathrm{SD}}$ = 0.4 (0.1) pb for $M_{H}$ = 120 GeV for $\sqrt{s}$ = 14 TeV (7 TeV). These cross sections are higher than that obtained from the $\gamma\gamma$ production mechanism, predicting a production cross section of 0.12 -- 0.18 fb \cite{Khoze:2001xm,*d'Enterria:2009er,*Miller:2007pc}, and even for the exclusive Higgs boson production \cite{Khoze:1997dr,*GayDucati:2010xi}. Moreover, comparing our estimations with no survival factor, we predict a cross section for $\sqrt{s}$ = 14 TeV higher than the previous results for the SD process \cite{Erhan:2003za}. In addition, the two models considered for the GSP have computed a survival factor that has a variation of 25\%. This difference is significative in order to perform reliable predictions for the Higgs boson production; however, it is expected that they are going to be tuned with the future data from the LHC experiments. In any case, our predictions with different survival factors may give a good estimation for the production cross section for the presented collider energies.
The SD production of the Higgs boson may not provide significative advantage in comparison to the inclusive production, since the background can not be suppressed using the same statements as in the double Pomeron exchange case. Nevertheless, the hadronic activity in the final state will be reduced in the SD events, increasing the possibility of observing the Higgs boson. Furthermore, the rapidity gaps may be observed in the LHC with the use of specific detectors, and then it can bring new data to be compared to the SD estimations. These results are the first NLO predictions for the single diffractive Higgs boson production, applying the GSP in the diffractive factorization. Besides, we have feasible values for the diffractive cross sections, and diffractive ratios as well, but the difference in the predictions is a bit high, which reveals that a study of the GSP for the multiple-Pomeron interactions in SD events is highly necessary. Therefore, we have presented updated estimations for the diffractive Higgs boson production, allowing the possibility to compare them to the future LHC data.
\begin{acknowledgments}
This work was supported by CNPq and FAPERGS, Brazil. We want to thanks M. V. T. Machado for useful comments. GGS would like to thank the Center for Particle Physics and Phenomenology (CP3) at Universit\'e catholique de Louvain for the hospitality.
\end{acknowledgments}
|
1,108,101,565,439 | arxiv | \section{Introduction}
\label{sec:Intro}
The number, reliability and coverage of evolutionary trees are growing rapidly \cite{Maddison07,TreeFam}. However, knowing organisms' evolutionary relationships through phylogenetics is only one step in understanding the evolution of their characteristics \cite{Yang12}. Three issues are particularly challenging. The first is limited information: empirical information is typically only available for extant taxa, represented by tips of a phylogenetic tree, whereas evolutionary questions frequently concern unobserved ancestors deeper in the tree. The second is dependence: the available information for different organisms in a phylogeny is not independent since a phylogeny describes a complex pattern of non-independence; observed variation is a mixture of this inherited variation and specific variation \cite{Cheverud85}. The third is high dimensionality: the emerging literature on function-valued traits \cite{Kirkp89,TFPG12,Stinchcombe12} recognises that many characteristics of living organisms are best represented as a continuous function rather than a single factor or a small number of correlated factors. Such characteristics include growth or mortality curves \cite{Pletcher99}, reaction-norms \cite{Kingsolver01} and distributions \cite{Zhang11}, where the increasing ease of genome sequencing has greatly expanded the range of species in which distributions of gene \cite{Moss11} or predicted protein \cite{Knight04} properties are available. Therefore, a function-valued trait is defined as a phenotypic trait that can be represented by a continuous mathematical function \cite{Kingsolver01}.
Previous work \cite{JM12} proposed an evolutionary model for function-valued data $d$ related by a phylogeny $\bf{T}$. The data are regarded as observations of a phylogenetic Gaussian Process (PGP) at the tips of $\bf{T}$. That work shows that a PGP can be expressed as a stochastic linear operator $X$ on a fixed set $\phi$ of basis functions (independent components of variation), so that
\begin{equation} d = X^T \phi
\label{lip}
\end{equation}
However, the study does not address the linear inverse problem of obtaining estimates $\hat{\phi}$ and $\hat{X}$ of $\phi$ and $X$: our first contribution in this paper is to provide an approach to this problem in section \ref{sec:DimensionRed} via independent principal components analysis (IPCA \cite{Yao12}).
We refer to $X$ as the {\em mixing matrix}, and to the $(i,j)$th entry of $X$ as the {\em mixing coefficient} of the $i$th basis function at the $j$th taxon. It is these mixing coefficients that we model as evolving. For each fixed value of $i$, the $X_{ij}$ are correlated (due to phylogeny) as $j$ varies over the taxa; the basis functions themselves do not evolve in our model.
In section \ref{sec:PhyloGP} we address the problem of estimating the statistical structure of the mixing coefficients by performing phylogenetic Gaussian process regression (PGPR) on each of the rows of $\hat{X}$ separately. This corresponds to assuming independence between the rows (i.e. that the coefficients of the different basis functions evolve independently). It is commonly argued in the quantitative genetics literature \cite{Butler04} that evolutionary processes can be modelled as
Ornstein-Uhlenbeck (OU) processes. Under these assumptions the estimation of the forward operator reduces to the estimation of a small vector $\gamma$ of parameters \cite{JM12}. In section \ref{sec:Simulation} we clarify the interpretation of these parameters in evolutionary contexts. The explicit PGPR posterior likelihood function is then used to obtain maximum
likelihood (MLE) estimates for $\gamma$. The estimation of $\gamma$ is known to be a challenging statistical problem \cite{Beaulieu12}. We suggest an approach based on the principal of \emph{bagging} \cite{Breiman96} in section \ref{sec:HypEst}.
Our final contribution (section \ref{sec:AncRec}) addresses the problem of estimating the function-valued traits of ancestral taxa. The PGPR step above also returns a posterior distribution for the mixing coefficient of each basis function at each ancestral taxon in the phylogeny. At any particular ancestor the estimated basis functions may be combined statistically, using the posterior distributions of their respective mixing coefficients, to provide a function-valued posterior distribution. Since the univariate posterior distributions are Gaussian, and the mixing is linear, the posterior for the function-valued trait has a closed form representation as a Gaussian process (Eq. \ref{ancestorGP}) which provides a major analytical and computational advantage for the approach.
We can verify the methods proposed by using a PGP as a stochastic generative model. This simulates correlated function-valued traits across the taxa of $\bf{T}$.
Given only the phylogeny and the function-valued traits of taxa at its tips, our estimates for $\hat{\phi}$ and the ancestral functions are then compared to the simulation.
Overall, our three methods (in \ref{sec:DimensionRed}, \ref{sec:HypEst}, \ref{sec:AncRec}) appropriately combine developments in functional data analysis with the evolutionary dynamics of quantitative phenotypic traits, allowing nonparametric Bayesian inference from phylogenetically correlated function-valued traits. An outline of the framework presented in the current work can be found in Fig. \ref{FIG0}.
\begin{figure}[!ht]
\includegraphics[width=.5\textwidth]{Figure_0_5.pdf}
\caption{The three methods presented in this paper (ovals) and their interrelationships.}
\label{FIG0}
\end{figure}
\section{Methods \& Implementation}
\label{sec:Methods}
\subsection{Artificial evolution of function-valued traits}
\label{sec:Simulation}
We begin by generating a random phylogenetic tree $\bf{T}$ with 128 tips, shown in Fig. \ref{FIG_tree}. This fixes the experimental design for our simulation and inference, but further simulations given in the Supplementary Material confirm that the statistical performance of our methods is consistent across a range of choices for $bf{T}$. Branch length distributions are surprisingly consistent across organisms \cite{Venditti10}; branch lengths were drawn from the empirical branch length distribution\footnote{See Supplementary Material section 1} extracted from TreeFam 8.0 \cite{TreeFam}.
\begin{figure}[!t]
\includegraphics[width=.5\textwidth]{FinalFigureTree-crop.pdf}
\caption{The random phylogenetic tree used \& examples of the function-valued traits shown at the tips (extant taxa) and the internal nodes (ancestral taxa). A subset of these is used in Fig. \ref{AncRecFig}.} \label{FIG_tree}
\end{figure}
Secondly we chose a basis $\phi$ in Eq. \ref{lip}. We have no reason {\em a priori} to suppose that this basis is orthogonal and, in general, there is no reason for our inference procedure to be sensitive to the particular shape of the basis functions. The three simple non-orthogonal, unimodal functions shown in Fig. \ref{FIG_IPCA} were therefore chosen as examples. For computational purposes each basis function was stored numerically as a vector of length 1024, so that the basis matrix $\phi$ was of size $3 \times 1024$ and its $i$th row stored the $i$th basis function.
\begin{figure}[ht]
\includegraphics[width=.5\textwidth]{Figure_IPCA.pdf}
\caption{From Top Left: Original Basis signals, $\phi$; Mixed Sample at the tips, $d$ (four individual function-valued traits are shown; red line and grey band show respectively the mean and two standard deviations for all 128 function-valued data at the tips); IPCA Basis, $\hat{\phi}$; PCA Basis.} \label{FIG_IPCA}
\end{figure}
Thirdly, different mixing coefficients were generated by a phylogenetic OU process for each basis function and stored in the respective row of $X$. Our modelling assumption is that the mixing coefficients for distinct basis functions $\phi_1,\phi_2,\phi_3$ are statistically independent of each other: in Eq. \ref{lip} this means that the rows of $X$ are independent. It is therefore sufficient to describe the stochastic process generating $X_i$, the $i$th row of $X$ with $i \in \{1,2,3\}$. We calculated the mixing matrix at the 128 tip taxa so $X$ is of size $3\times 128$. The ``true'' ancestral values were established by generating phylogenetic Ornstein-Uhlenbeck (OU) processes over the whole phylogeny.
The values of this process at tip taxa were stored in a row vector $\overline{X}_i$ ($\overline{X}_i$ is a simulation of the tip taxa mixing coefficients $X_i$ excluding the non-phylogenetic variation)
and its values at internal taxa were stored in a row vector $W_i$ for performance analyses in section \ref{sec:AncRec}. To simulate the additional effect of non-phylogenetic variation (due, for example, to measurement error or environmental effects), independent (that is, non-phylogenetic) variation was added to each entry of $\overline{X}_i$:
\[X_i = \overline{X}_i + {\bf \epsilon}_i\]
where $\epsilon_i$ is a $1 \times 128$ vector of independent Gaussian errors with mean 0 and variance $\sigma_n^i$ and finally the matrix multiplication in Eq. \ref{lip} was performed to obtain the simulated data $d$. The `extant' function-valued trait at tip taxon $j$ is thus $\sum_{i=1}^3 X_{ij}\phi_i$ (a vector of length 1024), while the ancestral function-valued trait at internal taxon $g$ is $\sum_{i=1}^3 W_{ig}\phi_i$. The ancestral function-valued traits therefore exhibit only the phylogenetic part of simulated variation, while the extant function-valued traits exhibit both phylogenetic and non-phylogenetic variation. Of course, it is not possible to reconstruct non-phylogenetic variation using phylogenetic methods: we simulate non-phylogenetic variation only to demonstrate that it does not prevent the reconstruction of the phylogenetic part of variation for ancestral taxa in sections \ref{sec:DimensionRed} to \ref{sec:AncRec}.
We now comment on the specific parameters chosen for the phylogenetic OU processes above. As in \cite{Hansen97} we refer to the {\em strength of selection parameter} $\alpha$ and the {\em random genetic drift} $\sigma$: we add superscripts to these parameters to distinguish between the three different OU processes. With this notation, the mixing coefficients for the row $X_i$ have the following covariance function :
\begin{align}
K_{\bf T}^i({\bf t}_1,{\bf t}_2)=& E[X_{ij} X_{ig}] \label{oucov}
\\ \quad =& (\sigma_f^{i})^2 \exp\left( -2\alpha^i D_T({\bf t}_j,{\bf t}_g)\right) + (\sigma_n^i)^{2} \delta_{{\bf t}_j,{\bf t}_g}^e \nonumber \end{align}
where $\sigma^i_f=\sqrt{\frac{(\sigma^i)^2}{2\alpha^i}}$
, $D_T({\bf t}_j,{\bf t}_g)$ denotes the phylogenetic or patristic distance (that is, the distance in ${\bf T}$) between the $j$th and $g$th tip taxa, $\sigma_n$ is defined as above, and
\begin{eqnarray*}
\delta_{t_j,t_g}^e = \left\{ \begin{array}{ll}1 & \text{iff } t_j = t_g \text{ and } t_j \text{ is a tip taxon}, \\
0 & \text{ otherwise}
\end{array}\right.
\end{eqnarray*}
adds non-phylogenetic variation to extant taxa as discussed above, ie. $\delta^e$ evaluates to 1 only for extant taxa, thus $\sigma_n$ quantifies within-species genetic or environmental effects and measurement error in the ith mixing coefficient.
We see from Eq. \ref{oucov} that the proportion of variation in the row $X_i$ attributable to the phylogeny is $\frac{(\sigma_f^i)^2}{(\sigma_f^i)^2 + (\sigma_n^i)^2}$.
In the Gaussian process regression literature in Machine Learning, $\frac 1 {2\alpha}$ is equivalent to $\ell$, the characteristic length-scale \cite{Rasmussen06} of decay in the correlation function and in the following we work with the latter.
For all of the OU processes we used characteristic length scales relative to 8.22, the maximum patristic distance ($\ell_{max}$) between two extant taxa for our simulated tree (Fig. \ref{FIG_tree}).
The values we used are given in Table \ref{Tab1}. In particular, $\sigma_f^i=0$ when $i=2$ and it follows that the characteristic length scale $\ell$ plays no role for this OU process, and equally we do not define the strength-of-selection parameter $\alpha^i$ when $i=2$.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{cccc}
$i$ & $\sigma_{f}^i$ & $\ell^i$ & $\sigma_{n}^i$ \\ \hline
1 & 2.5 & 6.17 & .5 \\
2 & 0 & NA & 1 \\
3 & 1.5 & 2.06 & .5 \\
\end{tabular}
\end{center}
\caption{The fixed values used for the parameters in Eq. \ref{oucov} to generate the mixing coefficients $X_{ij}$. Each row constitutes a value of $\gamma^i$. 6.17 \& 2.06 correspond to .75 and .25 of the tree's $\ell_{max}$ respectively. When $i=2$, $\ell^{i}$ is not applicable since there is no phylogenetic variation in the sample.
}
\label{Tab1}
\end{table}
\subsection{Dimensionality reduction and source separation for function-valued traits}
\label{sec:DimensionRed}
Given a dataset $d$ of function-valued traits, we would like to find appropriate estimates $\hat{X}$ and $\hat{\phi}$ of the mixing matrix $X$ and the basis set $\phi$ respectively. The first task is to identify a good linear subspace $S$ of the space of all continuous functions by choosing basis functions appropriately. The purpose is to work, not with the function-valued data directly, but with their projections in $S$. We may say that the chosen subspace $S$ is good if the projected data approximate the original data well while the number of basis functions is not unnecessarily large, so that $S$ has the `effective' dimension of the data.
We then face a linear inverse problem: given the dataset $d$ of function-valued traits, the task is to generate estimates $\hat{X}$ and $\hat{\phi}$ (Eq. \ref{lip}). This task is also known as {\em source separation} \cite{Hyvarinen00}, which has a variety of implementations making different assumptions about the basis $\phi$ and mixing coefficients $X$. One widely used approach is PCA \cite{Bishop06},
which returns orthogonal sets of basis functions to explain the greatest possible variation. PCA has been extended to take account of phylogenetic relationships \cite{Revell09PCA}, however, if a sample of functions is generated by mixing non-orthogonal basis functions, the principal components of the sample (whether or not they account for phylogeny) will not equal the basis curves, due to the assumption of orthogonality: see Fig. \ref{FIG_IPCA}.
In independent components analysis (ICA), the alternative assumption is made that the rows $X_i$ of $X$ are statistically independent. This assumption fits more naturally with our modelling assumptions, since we assume that the rows $X_i$ are mutually independent \cite{Hyvarinen00}. ICA has proved fruitful in other biological applications \cite{Scholz04} as has passing the results of PCA to ICA, which has been termed IPCA \cite{Yao12}.
PCA \textit{is} an appropriate tool for identifying the effective dimension of a high-dimensional dataset \cite{Minka00}. So, to achieve both dimension reduction and source separation, we first applied PCA to the dataset $d$ (the 128 function-valued traits at the tips of ${\bf T}$) to determine the appropriate number of basis functions. The principal components were then passed to the \emph{CubICA} implementation of ICA \cite{Blaschke04}. CubICA returned a new set of basis functions (Fig. \ref{FIG_IPCA}, lower-right panel) which were taken as the estimated basis $\hat{\phi}$.
\subsection{Phylogenetic Gaussian process regression}
\label{sec:PhyloGP}
ICA also returns the estimated mixing coefficients at tip taxa, $\hat{X}$. Our next step was to perform PGPR \cite{JM12} separately on each row $\hat{X}_i$, assuming knowledge of the phylogeny $\bf{T}$, in order to obtain posterior distributions for all mixing coefficients
throughout the tree $\bf{T}$.
Gaussian process regression (GPR) \cite{Rasmussen06} is a flexible Bayesian technique in which prior distributions are placed on continuous functions. Its range of priors includes the Brownian motion and Ornstein-Uhlenbeck (OU) processes, which are by far the most commonly used models of character evolution \cite{Hansen96,Butler04}.
Its implementation is particularly straightforward since the posterior distributions are also Gaussian processes and have closed forms. We now give a brief exposition of GPR, using notation standard in the Machine Learning literature (see, for example, \cite{Rasmussen06}).
A Gaussian process may be specified by its mean surface and its covariance function $K(\gamma)$, where $\gamma$ is a vector of parameters. Since the components of $\gamma$ parameterise the prior distribution, they are referred to as {\em hyper}parameters. The Gaussian process prior distribution is denoted
\[f\sim\mathcal{N}(0,K(\gamma))\]
If $x^*$ is a set of unobserved coordinates and $x$ is a set of observed coordinates, the posterior distribution of the vector $f(x^*)$ given the observations $f(x)$ is
\begin{align}\label{dangermouse}
f(x^*)|f(x) \sim \mathcal N (A,B)
\end{align}
where
\begin{align}
A =& K(x^*,x,\gamma)K(x,x,\gamma)^{-1}f(x), \label{boselecta} \\
B =& K(x^*,x^*,\gamma)\notag\\ -& K(x^*,x,\gamma)K(x,x,\gamma)^{-1}K(x^*,x,\gamma)^{T} \label{chas and dave}
\end{align}
and $K(x^*,x,\gamma)$ denotes the $|x^*| \times |x|$ matrix of the covariance function $K$ evaluated at all pairs $x^*_i \in X^*, x_j \in X$. Equations \ref{boselecta} and \ref{chas and dave} convey that the posterior mean estimate will be a linear combination of the given data and that the posterior variance will be equal to the prior variance minus the amount that can be explained by the data. Additionally, the log-likelihood of the sample $f(x)$ is
\begin{align}\label{log1}
\log p(f(x)|\gamma) = &-\frac{1}{2}f(x)^T K(x,x,\gamma)^{-1}f(x) \notag\\
- &\frac{1}{2}\log(det(K(x,x,\gamma)))- \frac{|x|}{2} \log 2\pi.
\end{align}
It can be seen from Eq. \ref{log1} that the maximum likelihood estimate is subject both to the fit it delivers (the first term) and the model complexity (the second term).
Thus, Gaussian process regression is non-parametric in the sense that no assumption is made about the structure of the model: the more data gathered, the longer the vector $f(x)$, and the more intricate the posterior model for $f(x^*)$.
PGPR extends the applicability of GPR to evolved function-valued traits.
A \emph{phylogenetic} Gaussian process is a Gaussian process indexed by a phylogeny ${\bf T}$, where the function-valued traits at each pair of taxa are conditionally independent given the function-valued traits of their common ancestors. When the evolutionary process has the same covariance function along any branch of $\bf{T}$ beginning at its root (called the {\em marginal covariance function}), these assumptions are sufficient to uniquely specify the covariance function of the PGP, $K_{\bf T}$. As we assume that $\bf{T}$ is known in our inverse problem, the only remaining modelling choice is therefore the marginal covariance function. As can be seen from Eq. \ref{oucov}, $K$ is a function of patristic distances on the tree rather than Euclidean distances as standard in spatial GPR.
In comparative studies, where one has observations at the tips of ${\bf T}$, the covariance function $K_{\bf T}$ may be used to construct a Gaussian process prior for the function-valued traits, allowing functional regression. In the model that we use
this is equivalent to specifying a Gaussian prior distribution for the mixing coefficients $Y_{ij}$ and $X_{ij}$. This may be done by regarding the row vectors $Y_i$ and $X_i$ as observations of a univariate PGP. As noted in \cite{JM12}, if we assume that the evolutionary process is Markovian and stationary then the modelling choice vanishes and the marginal covariance function is specified uniquely: it is the stationary OU covariance function. If we also add explicit modelling of non-phylogenetically related variation at the tip taxa, the univariate prior covariance function has the unique functional form presented in Eq. \ref{oucov}. We do not assume knowledge of the parameters of Eq. \ref{oucov} however: their estimation is the subject of the next section.
\subsection{Hyperparameter estimation}
\label{sec:HypEst}
Since the posterior distributions returned by PGPR depend on the hyperparameter vector $\gamma$, we must estimate $\gamma$ in order to reconstruct ancestral function-valued traits, and the estimation procedure should correct for the dependence due to phylogeny. Maximum likelihood estimation (MLE) of the phylogenetic variation, non-phylogenetic variation and characteristic-length-scale hyperparameters $\sigma_f^i$, $\sigma_n^i$ and $\ell^i$ respectively may be attempted numerically using the explicit prior likelihood function (Eq. \ref{log1}). Because estimating $\sigma_f^i$ and $\ell^i$ alone is challenging \cite{Beaulieu12} (although the estimation improves significantly with increased sample size), and we have further increased the challenge by introducing non-phylogenetic variation, we propose an improved estimation procedure using the machine learning technique {\em bagging} \cite{Breiman96}, which a member of the \emph{boosting} framework \cite{Bishop06}.
We show that these estimates may be further improved if one knows the value of the ratio $\frac{(\sigma_f)^2}{(\sigma_n)^2}$, which is closely related to Pagel's $\lambda$ \cite{Pagel97}.
Bagging (bootstrap aggregating) seeks to reduce the variance of an estimator by generating multiple estimates and averaging. It is simple to implement given an existing estimation procedure: one adds a loop front end that selects a bootstrap sample and sends it to the estimation procedure and a back end that aggregates the resulting estimates \cite{Breiman96}. We generated 100 (sub)trees of 100 taxa by sampling without replacement our original 128 taxa tree, obtained the MLE for $\gamma$ on each subtree, and averaged these estimates to obtain the aggregated estimate $\hat{\gamma}$. Our results are shown in Table \ref{Tab2}: for $i=1$ and $i=3$, given our moderate sample size (128 taxa), the accuracy of these results is at least in line with the state of the art \cite{Beaulieu12} despite the additional challenge posed by non-phylogenetic variation. For $i=2$, where phylogenetic variation is absent from the generative model ($\sigma_f^i$=0), our estimation procedure indicates its absence by returning estimates for $\ell^i$ whose magnitude is unrealistically small for the examined tree (less than the 1st percentile of the tree's patristic distances).
Commenting further on this matter, exceptionally \emph{small} characteristic length-scales relative to the tree patristic distances, as seen here, practically suggest taxa-specific phylogenetic variation, ie. non-phylogenetic variation. This holds also in its reverse: exceptionally \emph{large} characteristic length-scales suggest a stable, non-decaying variation across the examined taxa that is indifferent to their patristic distances, again suggesting the absence of phylogenetic variance among the nodes.
To assess the robustness of this hyperparameter estimation method we performed 1024 simulations, randomly regenerating the tree and parameter vector $\gamma$ each time\footnote{See Supplementary Material section 2}. The accuracy of these estimates is shown in Fig. \ref{Fig_BaggingResults}. Improved results when the ratio $\frac{(\sigma_f)^2}{(\sigma_n)^2}$ is known {\em a priori} (for example, through knowledge of Pagel's $\lambda$) are also given in the supplementary material (Sect. 2 \& 3). Our ultimate aim is ancestor reconstruction rather than hyperparameter estimation {\em per se}, and this is the subject of the next section.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{cccc}
$i$ & $\hat{\sigma}_{f}^i$ & $\hat{\ell}^i$ & $\hat{\sigma}_{n}^i$ \\ \hline
1 & 3.41 (.62) & 2.83 (.47)& 0.78 (.47) \\
2 & 0.55 (.33)& 0.05 (.02) & 0.84 (.34) \\
3 & 2.83 (.33) & 2.06 (.50) & 0.73 (.29) \\
\end{tabular}
\end{center}
\caption{The bagging estimates for the hyperparameters in Eq. \ref{oucov} (standard deviations of bagging estimates in parentheses). Each row corresponds to a given estimate of the vector $\hat{\gamma}^i$. These estimates provide the maximum likelihood value for Eq. \ref{log1} and are comparable with the original ones from Table \ref{Tab1}.}\label{Tab2}
\end{table}
\begin{figure*}[!htb]
\includegraphics[width=.95\textwidth]{Bagging_MLE_Results.pdf}
\caption{Kernel Density estimates of the relative errors in 1024 runs of the $\gamma$ estimation procedure, each time for a different tree, a different set of mixing coefficients and a different set of parameters in $\gamma$; no components of $\gamma$ are assumed to be known beforehand. Estimation results are commented on in the Discussion.
The median values shown by the dotted line are (-0.073, -0.131 and 0.001) respectively.
} \label{Fig_BaggingResults}
\end{figure*}
\subsection{Ancestor reconstruction}
\label{sec:AncRec}
Having generated function-valued data (Sect. \ref{sec:Simulation}), extracted mixing coefficients $\hat{X}$ (Sect. \ref{sec:DimensionRed}) and performed hyperparameter estimation (Sect. \ref{sec:HypEst}), we may now perform PGPR (Sect. \ref{sec:PhyloGP}) on each row $\hat{X}_i$, to obtain the univariate Gaussian posterior distribution for the mixing coefficient $W_{i{\bf t}^*}$ at any internal taxon ${\bf t}^*$. As discussed in Sect. \ref{sec:PhyloGP}, the Gaussian process prior distribution has covariance function (Eq. \ref{oucov}). We have assessed the accuracy of our bagging estimate $\hat{\gamma}$ in Sect. \ref{sec:HypEst} and we now
substitute $\hat{\gamma}^i$ into Eq. \ref{oucov}. Taking a simple and direct approach, our estimate $\hat{\phi}$ obtained in Sect. \ref{sec:DimensionRed} may then be substituted into Eq. \ref{lip} to obtain the function-valued posterior distribution $f_{{\bf t}^*}$ for the function-valued trait at taxon ${\bf t}^*$. Since our estimated basis functions are stored numerically as vectors of length 1024, this gives the same discretision for the ancestral traits.
Conditioning
on our estimated mixing coefficients $\hat{X}_i$ for the tip taxa, the posterior distribution of $W_{i {\bf t}^*}$ is \[W_{i {\bf t}^{*}}\sim\mathcal{N}(\hat{A_i},\hat{B_i})\]
where the vector $\hat{A_i}$ and matrix $\hat{B_i}$ are obtained from Eq.'s \ref{boselecta} and \ref{chas and dave}, taking $x=\hat{X}_i$, $x^*=W_{i{\bf t}^*}$ and $\hat{\gamma^i}$ respectively for our observation coordinates, estimation coordinates and hyperparameter vector. Since our prior assumption is that the rows of $X$ are statistically independent of each other, it follows from Eq. \ref{lip} that
\begin{equation}\label{ancestorGP}
f_{{\bf t}^*}\sim\mathcal{N}(\Sigma_{i=1}^k \hat{A_i} \hat{\phi_i}, \Sigma_{i=1}^k \hat{\phi}_i^T \hat{B_i} \hat{\phi_i})
\end{equation}
The marginal distributions of this representation (mean and standard deviation) are shown in Fig. \ref{AncRecFig}.
Fig. \ref{AncRecFig} compares the function-valued estimates $\hat{f_{{\bf t}^*}}$ to the simulated function-valued traits at the root (left panel), an internal node (centre), and at a tip (right panel). In the centre and left panels the simulated function-valued data is shown in black, and can be seen typically to lie within two posterior standard deviations. In the right panel, the black line is the observed function-valued trait at that tip: the red line and dark grey band represent the posterior distribution of its phylogenetic component, and the light grey band represents the estimated magnitude of the additional non-phylogenetic variation.
Uncertainty over the phylogenetic part of variation (dark grey band) decreases from root to tip, as all observations are at the extant tip taxa. We note that the posterior distributions, even at the root, put clear statistical constraints on the phylogenetic part of ancestral function-valued data: in this (admittedly simulated and highly controlled) setting we can reason effectively about ancestral function-valued traits.
\begin{figure*}[!ht]
\includegraphics[width=.95\textwidth]{Fig4_151_13Dec_EstimatedThetas.pdf}
\caption{Posterior distributions at three points in the phylogeny using the estimated $\hat{\phi}$ and $\hat{\gamma}$.
The prediction made by the regression analysis is shown via the posterior mean (red line), the component of posterior variance due to phylogenetic variation (two standard deviations, dark grey band) and non-phylogenetic variation (two standard deviation, light grey band). The black line shows the simulated data enabling visual validation of the ancestral predictions. In the right panel, the black line is the training data at a tip taxon the red line and dark grey band represent the posterior distribution of its phylogenetic component, while the light grey band represents the estimated magnitude of non-phylogenetic variation. The root and internal taxon here are the same as those indicated in Fig.\ref{FIG_tree} \& \ref{FIG_IPCA}, and the tip is the second from bottom on the same figure.} \label{AncRecFig}
\end{figure*}
\section{Discussion}
In Sec. \ref{sec:Simulation} we have appealed to Eq. \ref{lip} in the setting of mathematical inverse problems where, given data $d$, the challenge is to infer a forward operator $G$ and model $\phi$ such that:
\begin{equation} d = G(\phi)
\end{equation}
and such problems are typically under-determined and require additional modelling assumptions \cite{Jaynes84}.
Given a phylogeny $\bf{T}$ and function-valued data $d$ at its tips, we wish to infer the forward operator $G_{\bf{T}}$ and model $\phi$ such that
\begin{equation} d = G_{\bf{T}}(\phi)
\end{equation}
When the data $d$ are a small number of correlated factors per tip taxon, a variety of statistical approaches are available (e.g. see \cite{Salamin10}, \cite{Hadfield10}). When the data are functions, the phylogenetic Gaussian processes (PGP) \cite{JM12,Kerr12} has been proposed as the forward operator and this is the approach we have taken in this work.
Our dimensionality reduction methodology in Sect. \ref{sec:DimensionRed} can be easily varied or extended. For example, any suitable implementation of PCA may be used to perform the initial dimension reduction step: in particular, if the data have an irregular design (as happens frequently with function-valued data), the method of Yao et al. \cite{Yao05} may be applied to account for this; the ICA step then proceeds unchanged. We also note that while we find the {\em CubICA} implementation of ICA to be the most successful in our signal separation task, other implementations like {\em FastICA} \cite{Hyvarinen00} or {\em JADE} \cite{Cardoso99} can also be employed.
In general, ICA gives rows $\hat{X}_i$ of the estimated mixing matrix that are maximally independent under a particular measure of independence involving, for example, higher sample moments or mutual information,
in order to approximate the solution of the inverse problem in Eq. \ref{lip} under our assumption of independence between the rows of $X$.
PCA and ICA have different purposes (respectively, orthogonal decomposition of variation and separation of independently mixed signals) and we employ them sequentially in IPCA. IPCA is nonparametric and, in particular, both distributionally and phylogenetically agnostic. This means that unlike PCA, IPCA is robust to non-Gaussianity in the data and, unlike phylogenetically corrected PCA, IPCA is robust to mis-specification of the phylogeny and to mixed phylogenetic and non-phylogenetic variation in the data: any of these can be features of biological data.
It can be seen in Fig. \ref{Fig_BaggingResults} that the estimation of $\ell$ is more challenging than the estimation of $\sigma_n$ or $\sigma_f$, having greater bias and variance. This corresponds to the documented difficulty of estimating the parameter $\alpha$ in the Ornstein-Uhlenbeck model, particularly for smaller sample sizes.
Our work on hyperparameter estimation in Sec. \ref{sec:HypEst} mitigates these difficulties due to small sample size \cite{Beaulieu12,Collar09} by employing bagging in order to bootstrap our sample. Somewhat unintuively, bagging ``works'' exactly because the subsample $\hat{\gamma}$ estimates are variable and thus we avoid overfitted final estimates\footnote{See Supplementary Material Section 2}.
Conceptually our work on hyperparameter estimation, when taken together with Sec. \ref{sec:DimensionRed}, relates to the character process models of \cite{Pletcher1999} and orthogonal polynomial methods of \cite{Kirkpatrick89}, which give estimates for the autocovariance of function-valued traits. Writing out Eq. \ref{lip} for a single function-valued trait (at the $j$th tip taxon, say), our model may be viewed as
\begin{equation}
f(x) = \sum_{i=1}^3 g_{ij} \phi_i(x) + \sum_{i=1}^3 e_{ij} \phi_i(x)
\end{equation}
where the mixing coefficient $X_{ij}$ has been expressed as the sum of $g_{ij}$, the genetic (i.e. phylogenetic) part of variation, plus $e_{ij}$, the non-phylogenetic (eg. environmental) part of variation, just as in these references.
Then the autocorrelation of the function-valued trait is
\begin{equation}\label{buzz}
E[f(x_1)f(x_2)] = \sum_{i=1}^3 \left((\sigma^f_{i})^2 + (\sigma^n_i)^2\right) \phi_i(x_1)\phi_i(x_2)
\end{equation}
The estimates of $\sigma^f_{i}$ and $\sigma^n_{i}$ obtained in section \ref{sec:HypEst} may be substituted into Eq. \ref{buzz} to obtain an estimate of the autocovariance of the function-valued traits under study. This estimate has the attractions both of being positive definite (by construction) and of taking phylogeny into account.
Various frameworks exist which could be used to generalise the method presented in Sec. \ref{sec:HypEst}, to model heterogeneity of evolutionary rates along the branches of a phylogeny \cite{Revell09} or for multiple fixed \cite{Butler04} or randomly evolving \cite{Hansen08, Beaulieu12} local optima of the mixing coefficients. For the stationary OU process the optimum trait value appears only in the mean, and not in the covariance function, and so does not play a role as a parameter in GPR (see \cite{Rasmussen06}). We have not implemented such extensions here, effectively assuming that a single fixed optimum is adequate for each mixing coefficient. Nonetheless our framework is readily extensible to include such effects, either implicitly through branch-length transformations \cite{Pagel99}, or explicitly by replacing the OU model with the more general Hansen model \cite{Hansen08}.
R Code for the IPCA, ancestral reconstruction and hyperparameter estimation is available from \\ \url{https://github.com/fpgpr/}
|
1,108,101,565,440 | arxiv | \section*{ACKNOWLEDGMENTS}
The author wishes to acknowledge useful conversations
and correspondence with S.Jain, J.Morehead, and M.Srednicki.
|
1,108,101,565,441 | arxiv | \section{Introduction}
The concept of synchronistion is based on the adjustment of rhythms of
oscillating systems due to their interaction \cite{pikovsky01}.
Synchronisation phenomenon was recognised by Huygens in the 17th century, time
when he performed experiments to understand this phenomenon \cite{bennett02}.
To date, several kinds of synchronisation among coupled systems were reported,
such as complete \cite{li16}, phase \cite{pereira07,batista10}, lag
\cite{huang14}, and collective almost synchronisation \cite{baptista12}.
Neuronal synchronous rhythms have been observed in a wide range of researches
about cognitive functions \cite{wang10,hutcheon00}. Electroencephalography and
magnetoencephalography studies have been suggested that neuronal
synchronization in the gamma frequency plays a functional role for memories in
humans \cite{axmacher06,fell11}. Steinmetz et al. \cite{steinmetz00}
investigated the synchronous behaviour of pairs of neurons in the secondary
somatosensory cortex of monkey. They found that attention modulates
oscillatory neuronal synchronisation in the somatosensory cortex. Moreover,
in the literature it has been proposed that there is a relationship between
conscious perception and synchronisation of neuronal activity \cite{hipp11}.
We study spiking and bursting synchronisation betwe\-en neuron in a neuronal
network model. A spike refers to the action potential generated by a neuron
that rapidly rises and falls \cite{lange08}, while bursting refers to a
sequence of spikes that are followed by a quiescent time \cite{wu12}. It was
demonstrated that spiking synchronisation is relevant to olfactory bulb
\cite{davison01} and is involved in motor cortical functions \cite{riehle97}.
The characteristics and mechanisms of bursting synchronisation were studied in
cultured cortical neurons by means of planar electrode array \cite{maeda95}.
Jefferys $\&$ Haas discovered synchronised bursting of CA1 hippocampal
pyramidal cells \cite{jefferys82}.
There is a wide range of mathematical models used to describe neuronal activity,
such as the cellular automaton \cite{viana14}, the Rulkov map
\cite{rulkov01}, and differential equations \cite{hodgkin52,hindmarsh84}.
One of the simplest mathematical models and that is widely used to depict
neuronal behaviour is the integrate-and-fire \cite{lapicque07}, which is
governed by a linear differential equation. A more realistic version of it is
the adaptive exponential integrate-and-fire (aEIF) model which we consider in
this work as the local neuronal activity of neurons in the network. The aEIF is
a two-dimensional integrate-and-fire model introduced by Brette $\&$ Gerstner
\cite{brette05}. This model has an exponential spike mechanism with an
adaptation current. Touboul $\&$ Brette \cite{touboul08} studied the
bifurcation diagram of the aEIF. They showed the existence of the Andronov-Hopf
bifurcation and saddle-node bifurcations. The aEIF model can generate multiple
firing patterns depending on the parameter and which fit experimental data from
cortical neurons under current stimulation \cite{naud08}.
In this work, we focus on the synchronisation phenomenon in a randomly
connected network. This kind of network, also called Erd\"os-R\'enyi network
\cite{erdos59}, has nodes where each pair is connected according to a
probability. The random neuronal network was utilised to study oscillations in
cortico-thalamic circuits \cite{gelenbe98} and dynamics of network with
synaptic depression \cite{senn96}. We built a random neuronal network with
unidirectional connections that represent chemical synapses.
We show that there are clearly separated ranges of parameters that lead to
spiking or bursting synchronisation. In addition, we analyse the robustness to
external perturbation of the synchronisation. We verify that bursting
synchronisation is more robustness than spiking synchronisation. However,
bursting synchronisation requires larger chemical synaptic strengths, and
larger voltage potential relaxation reset to appear than those required for
spiking synchronisation.
This paper is organised as follows: in Section II we present the adaptive
exponential integrate-and-fire model. In Section III, we introduce the neuronal
network with random features. In Section IV, we analyse the behaviour of
spiking and bursting synchronisation. In the last Section, we draw our
conclusions.
\section{Adaptive exponential integrate-and-fire}
As a local dynamics of the neuronal network, we consider the adaptive
exponential integrate-and-fire (aEIF) model that consists of a system of two
differential equations \cite{brette05} given by
\begin{eqnarray}\label{eqIF}
C \frac{d V}{d t} & = & - g_L (V - E_L) + {\Delta}_T
\exp \left(\frac{V - V_T}{{\Delta}_T} \right) \nonumber \\
& & +I-w , \nonumber \\
\tau_w \frac{d w}{d t} & = & a (V - E_L) - w,
\end{eqnarray}
where $V(t)$ is the membrane potential when a current $I(t)$ is injected, $C$
is the membrane capacitance, $g_L$ is the leak conductance, $E_L$ is the
resting potential, $\Delta_T$ is the slope factor, $V_T$ is the threshold
potential, $w$ is an adaptation variable, $\tau_w$ is the time constant, and
$a$ is the level of subthreshold adaptation. If $V(t)$ reaches the threshold
$V_{\rm{peak}}$, a reset condition is applied: $V\rightarrow V_r$ and
$w\rightarrow w_r=w+b$. In our simulations, we consider $C=200.0$pF,
$g_L=12.0$nS, $E_L=-70.0$mV, ${\Delta}_T=2.0$mV, $V_T=-50.0$mV, $I=509.7$pA,
$\tau_w=300.0$ms, $a=2.0$nS, and $V_{\rm{peak}}=20.0$mV \cite{naud08}.
The firing pattern depends on the reset parameters $V_r$ and $b$. Table
\ref{table1} exhibits some values that generate five different firing patterns
(Fig. \ref{fig1}). In Fig. \ref{fig1} we represent each firing pattern with a
different colour in the parameter space $b\times V_r$: adaptation in red, tonic
spiking in blue, initial bursting in green, regular bursting in yellow, and
irregular in black. In Figs. \ref{fig1}a, \ref{fig1}b, and \ref{fig1}c we
observe adaptation, tonic spiking, and initial burst pattern, respectively, due
to a step current stimulation. Adaptation pattern has increasing inter-spike
interval during a sustained stimulus, tonic spiking pattern is the simplest
regular discharge of the action potential, and the initial bursting pattern
starts with a group of spikes presenting a frequency larger than the steady
state frequency. The membrane potential evolution with regular bursting is
showed in Fig. \ref{fig1}d, while Fig. \ref{fig1}e displays irregular pattern.
\begin{table}[htbp]
\caption{Reset parameters.}
\centering
\begin{tabular}{c c c c c}
\hline
Firing patterns & Fig. & b (pA) & $V_r$ (mV) & Layout \\ \hline
adaptation &\ref{fig1}(a) & 60.0 & -68.0 & red \\
tonic spiking & \ref{fig1}(b) & 5.0 & -65.0 & blue\\
initial burst & \ref{fig1}(c) & 35.0 & -48.8 & green \\
regular bursting & \ref{fig1}(d) & 40.0 & -45.0 & yellow\\
irregular & \ref{fig1}(e) & 41.2 & -47.4 & black \\ \hline
\end{tabular}
\label{table1}
\end{table}
\begin{figure}[hbt]
\centering
\includegraphics[height=7cm,width=10cm]{fig1.eps}
\caption{(Colour online) Parameter space for the firing patterns as a function
of the reset parameters $V_r$ and $b$. (a) Adaptation in red, (b) tonic spiking
in blue, (c) initial bursting in green, (d) regular bursting in yellow, and (e)
irregular in black.}
\label{fig1}
\end{figure}
As we have interest in spiking and bursting synchronisation, we separate the
parameter space into a region with spike and another with bursting patterns
(Fig. \ref{fig2}). To identify these two regions of interest, we use the
coefficient of variation (CV) of the neuronal inter-spike interval (ISI), that
is given by
\begin{eqnarray}\label{CV}
{\rm CV}=\frac{{\sigma}_{\rm{ISI}}}{\rm{\overline{ISI}}},
\end{eqnarray}
where ${\sigma}_{\rm{ISI}}$ is the standard deviation of the ISI normalised by
the mean $\bar{\rm ISI}$ \cite{gabbiani98}. Spiking patterns produce
$\rm{CV}<0.5$. Parameter regions that represent the neurons firing with spiking
pattern are denoted by gray colour in Fig. \ref{fig2}. Whereas, the black
region represents the bursting patterns, which results in $\rm{CV} \geq 0.5$.
\begin{figure}[hbt]
\centering
\includegraphics[height=7cm,width=9cm]{fig2.eps}
\caption{Parameter space for the firing patterns as a function of the reset
parameters $V_r$ and $b$. Spike pattern in region I ($\rm{CV}<0.5$) and
bursting pattern in region II ($\rm{CV}\geq 0.5$) are separated by white
circles.}
\label{fig2}
\end{figure}
\section{Spiking or bursting synchronisation}
In this work, we constructed a network where the neurons are randomly connected
\cite{erdos59}. Our network is given by
\begin{eqnarray}\label{eqIFrede}
C \frac{d V_i}{d t} & = & - g_L (V_i - E_L) + {\Delta}_T \; \rm{exp}
\left(\frac{V_i - V_T}{{\Delta}_T} \right) \nonumber \\
& + & I_i - w_i + g_{\rm{ex}} (V_{\rm{ex}} - V_i) \sum_{j=1}^N A_{ij} s_j + \Gamma_i,
\nonumber \\
\tau_w \frac{d w_i}{d t} & = & a_i (V_i - E_L) - w_i, \nonumber \\
\tau_{\rm{ex}} \frac{d s_i}{d t} & = & - s_i.
\end{eqnarray}
where $V_i$ is the membrane potential of the neuron $i$, $g_{\rm{ex}}$ is the
synaptic conductance, $V_{\rm{ex}}$ is the synaptic reversal potential,
$\tau_{\rm{ex}}$ is the synaptic time constant, $s_i$ is the synaptic weight,
$A_{ij}$ is the adjacency matrix, $\Gamma_i$ is the external perturbation, and
$a_i$ is randomly distributed in the interval $[1.9,2.1]$.
The schematic representation of the neuronal network that we have considered
is illustrated in Fig \ref{fig3}. Each neuron is randomly linked to other
neurons with a probability $p$ by means of directed connections. When $p$ is
equal to 1, the neuronal network becames an all-to-all network. A network with
this topology was used by Borges et al. \cite{borges16} to study the effects
of the spike timing-dependent plasticity on the synchronisation in a
Hodgkin-Huxley neuronal network.
\begin{figure}[hbt]
\centering
\includegraphics[height=6cm,width=9cm]{fig3.eps}
\caption{Schematic representation of the neuronal network where the neurons
are connected according to a probability $p$.}
\label{fig3}
\end{figure}
A useful diagnostic tool to determine synchronous behaviour is the complex
phase order parameter defined as \cite{kuramoto03}
\begin{equation}
z(t)=R(t)\exp({\rm i}\Phi(t))\equiv\frac{1}{N}\sum_{j=1}^{N}\exp({\rm i}\psi_{j}),
\end{equation}
where $R$ and $\Phi$ are the amplitude and angle of a centroid phase vector,
respectively, and the phase is given by
\begin{equation}
\psi_{j}(t)=2\pi m+2\pi\frac{t-t_{j,m}}{t_{j,m+1}-t_{j,m}},
\end{equation}
where $t_{j,m}$ corresponds to the time when a spike $m$ ($m=0,1,2,\dots$) of a
neuron $j$ happens ($t_{j,m}< t < t_{j,m+1}$). We have considered the beginning
of the spike when $V_j>-20$mV. The value of the order parameter magnitude goes
to 1 in a totally synchronised state. To study the neuronal synchronisation of
the network, we have calculated the time-average order-parameter, that is given
by
\begin{equation}
\overline{R}=\frac{1}{t_{\rm fin}-{t_{\rm ini}}}\sum_{t_{\rm ini}}^{t_{\rm fin}}R(t),
\end{equation}
where $t_{\rm fin}-t_{\rm ini}$ is the time window for calculating $\bar{R}$.
Figs. \ref{fig4}a, \ref{fig4}b, and \ref{fig4}c show the raster plots for
$g_{\rm ex}=0.02$nS, $g_{\rm ex}=0.19$nS, and $g_{\rm ex}=0.45$nS, respectively,
considering $V_r=-58$mV, $p=0.5$, and $b=70$pA, where the dots correspond to
the spiking activities generated by neurons. For $g_{\rm ex}=0.02$nS (Fig.
\ref{fig4}a) the network displays a desynchonised state, and as a result,
the order parameter values are very small (black line in Fig. \ref{fig4}d).
Increasing the synaptic conductance for $g_{\rm ex}=0.19$nS, the neuronal network
exhibits spike synchronisation (Fig. \ref{fig4}b) and the order parameter values
are near unity (red line in Fig. \ref{fig4}d). When the network presents
bursting synchronisation (Fig. \ref{fig4}c), the order parameter values vary
between $R\approx 1$ and $R\ll 1$ (blue line in Fig. \ref{fig4}d). $R\ll 1$ to
the time when the neuron are firing.
\begin{figure}[hbt]
\centering
\includegraphics[height=11cm,width=10cm]{fig4.eps}
\caption{(Colour online) Raster plot for (a) $g_{\rm ex}=0.02$nS, (b)
$g_{\rm ex}=0.19$nS, and (c) $g_{\rm ex}=0.45$nS, considering $V_r = -58$mV,
$p=0.5$, and $b=70$pA. In (d) the order parameter is computed for
$g_{\rm ex}=0.02$nS (black line), $g_{\rm ex}=0.19$nS (red line), and
$g_{\rm ex}=0.19$nS (blue line).}
\label{fig4}
\end{figure}
In Fig. \ref{fig5}a we show ${\bar R}$ as a function of $g_{\rm ex}$ for
$p=0.5$, $b=50$pA (black line), $b=60$pA (red line), and $b=70$pA (blue line).
The three results exhibit strong synchronous behaviour (${\bar R}>0.9$) for
many values of $g_{\rm ex}$ when $g_{\rm ex}\gtrsim 0.4$nS . However, for
$g_{\rm ex}\lesssim 0.4$nS, it is possible to see synchronous behaviour only for
$b=70$pA in the range $0.15{\rm nS}<g_{\rm ex}<0.25{\rm nS}$. In addition, we
calculate the coefficient of variation (CV) to determine the range in
$g_{\rm ex}$ where the neurons of the network have spiking or bursting behaviour
(Fig. \ref{fig5}b). We consider that for CV$<0.5$ (black dashed line) the
neurons exhibit spiking behaviour, while for CV$\geq 0.5$ the neurons present
bursting behaviour. We observe that in the range
$0.15{\rm nS}<g_{\rm ex}<0.25{\rm nS}$ for $b=70$pA there is spiking
sychronisation, and bursting synchronisation for $g_{\rm ex}\gtrsim 0.4$nS.
\begin{figure}[hbt]
\centering
\includegraphics[height=7cm,width=9cm]{fig5.eps}
\caption{(Colour online) (a) Time-average order parameter and (b) CV for
$V_r=-58$mV, $p=0.5$, $b=50$pA (black line), $b=60$pA (red line), and $b=70$pA
(blue line).}
\label{fig5}
\end{figure}
\section{Parameter space of synchronisation}
The synchronous behaviour depends on the synaptic conductance and the
probability of connections. Fig. \ref{fig6} exhibits the time-averaged order
parameter in colour scale as a function of $g_{\rm ex}$ and $p$. We verify a
large parameter region where spiking and bursting synchronisation is strong,
characterised by ${\bar R}>0.9$. The regions I and II correspond to spiking and
bursting patterns, respectively, and these regions are separated by a white
line with circles. We obtain the regions by means of the coefficient of
variation (CV). There is a transition between region I and region II, where
neurons initially synchronous in the spike, loose spiking synchronicity to give
place to a neuronal network with a regime of bursting synchronisation.
\begin{figure}[hbt]
\centering
\includegraphics[height=6cm,width=9cm]{fig6.eps}
\caption{(Colour online) $g_{\rm ex} \times p$ for $V_r=-58$mV and $b=70$pA,
where the colour bar represents the time-average order parameter. The regions I
(spike patterns) and II (bursting patterns) are separated by the white line
with circles.}
\label{fig6}
\end{figure}
We investigate the dependence of spiking and bursting synchronisation on the
control parameters $b$ and $V_r$. To do that, we use the time average order
parameter and the coefficient of variation. Figure \ref{fig7} shows that the
spike patterns region (region I) decreases when $g_{\rm ex}$ increases. This
way, the region I for $b<100$pA and $V_r=-49$mV of parameters leading to no
synchronous behaviour (Fig. \ref{fig7}a), becomes a region of parameters that
promote synchronised bursting (Fig. \ref{fig7}b and \ref{fig7}c). However, a
large region of desynchronised bursting appears for $g_{\rm ex}=0.25$nS about
$V_r=-45$mV and $b>100$pA in the region II (Fig. \ref{fig7}b). For
$g_{\rm ex}=0.5$nS, we see, in Fig. \ref{fig7}c, three regions of desynchronous
behaviour, one in the region I for $b<100$pA, other in region II for $b<200$pA,
and another one is located around the border (white line with circles) between
regions I and II for $b>200$pA.
\begin{figure}[hbt]
\centering
\includegraphics[height=12cm,width=7cm]{fig7.eps}
\caption{(Colour online) Parameter space $b \times V_r$ for $p=0.5$, $\gamma=0$
(a) $g_{\rm ex}=0.05$nS, (b) $g_{\rm ex}=0.25$nS, and (c) $g_{\rm ex}=0.5$nS, where
the colour bar represents the time-average order parameter. The regions I
(spike patterns) and II (bursting patterns) are separated by white circles.}
\label{fig7}
\end{figure}
It has been found that external perturbations on neuronal networks not only can
induce synchronous behaviour \cite{baptista06,zhang15}, but also can suppress
synchronisation \cite{lameu16}. Aiming to study the robustness to perturbations
of the synchronous behaviour, we consider an external perturbation $\Gamma_i$
(\ref{eqIFrede}). It is applied on each neuron $i$ with an average time
interval of about $10$ms and with a constant intensity $\gamma$ during $1$ms.
Figure \ref{fig8} shows the plots $g_{\rm ex} \times p$ for $\gamma>0$, where
the regions I and II correspond to spiking and bursting patterns, respectively,
separated by white line with circles, and the colour bar indicates the
time-average order parameter values. In this Figure, we consider $V_r=-58$mV,
$b=70$pA, (a) $\gamma=250$pA, (b) $\gamma=500$pA, and (c) $\gamma=1000$pA. For
$\gamma=250$pA (Fig. \ref{fig8}a) the perturbation does not suppress spike
synchronisation, whereas for $\gamma=500$pA the synchronisation is
completely suppressed in region I (Fig. \ref{fig8}b). In Fig. \ref{fig8}c, we
see that increasing further the constant intensity for $\gamma=1000$pA, the
external perturbation suppresses also bursting synchronisation in region II.
Therefore,the synchronous behavior in region II is more robustness to
perturbations than in the region I, due to the fact that the region II is in a
range with high $g_{\rm ex}$ and $p$ values, namely strong coupling and high
connectivity.
\begin{figure}[hbt]
\centering
\includegraphics[height=12cm,width=7cm]{fig8.eps}
\caption{(Colour online) $g_{\rm ex} \times p$ for $V_r=-58$mV, $b=70$pA,
(a) $\gamma=250$pA, (b) $\gamma=500$pA, and (c) $\gamma=1000$pA.}
\label{fig8}
\end{figure}
In order to understand the perturbation effect on the spike and bursting
patterns, we consider the same values of $g_{\rm ex}$ and $p$ as Fig.
\ref{fig7}a. Figure \ref{fig9} exhibits the space parameter $b\times V_r$,
where $\gamma$ is equal to $500$pA. The external perturbation suppresses
synchronisation in the region I, whereas we observe synchronisation in
region II. The synchronous behaviour in region II can be suppressed if the
constant intensity $\gamma$ is increased. Therefore, bursting synchronisation
is more robustness to perturbations than spike synchronisation.
\begin{figure}[hbt]
\centering
\includegraphics[height=5cm,width=7cm]{fig9.eps}
\caption{(Colour online) $b \times V_r$ for $g_{\rm ex}=0.05$nS, $p=0.5$, and
$\gamma=500$pA, where the colour bar represents the time-average order
parameter. The regions I (spike patterns) and II (bursting patterns) are
separated by white line with circles.}
\label{fig9}
\end{figure}
\section{Conclusion}
In this paper, we studied the spiking and bursting synchronous behaviour in a
random neuronal network where the local dynamics of the neurons is given by the
adaptive exponential integrate-and-fire (aEIF) model. The aEIF model can exhibit
different firing patterns, such as adaptation, tonic spiking, initial burst,
regular bursting, and irregular bursting.
In our network, the neurons are randomly connected according to a probability.
The larger the probability of connection, and the strength of the synaptic
connection, the more likely is to find bursting synchronisation.
It is possible to suppress synchronous behaviour by means of an external
perturbation. However, synchronous behaviour with higher values of
$g_{\rm ex}$ and $p$, which typically promotes bursting synchronisation, are more
robust to perturbations, then spike synchronous behaviour appearing for
smaller values of these parameters. We concluded that bursting synchronisation
provides a good environment to transmit information when neurons are stron\-gly
perturbed (large $\Gamma$).
\section*{Acknowledgements}
This study was possible by partial financial support from the following
Brazilian government agencies: CNPq, CAPES, and FAPESP (2011/19296-1 and
2015/07311-7). We also wish thank Newton Fund and COFAP.
|
1,108,101,565,442 | arxiv | \section{Introduction}
\label{sec1}
Semiclassical gravity describes the interaction of the
gravitational field as a classical field with quantum matter
fields. For a free quantum field this theory is robust in the
sense that it is consistent and fairly well understood
\cite{birrell82,wald94}. The gravitational field is described by
the semiclassical Einstein equation which has as a source the
expectation value in some quantum state of the matter stress tensor
operator. The semiclassical theory is in some
sense unique as a theory where the gravitational field is
classical. In fact, a classical gravitational field interacts with
other fields through their stress tensors, and the only reasonable
c-number stress tensor that one may construct
\cite{Wal77,Wal78a,Wal78b} with the stress tensor operator of a
quantum field is its expectation value in some quantum state.
However, the scope and limits of the theory are not so well
understood because we still lack a fully well understood quantum
theory of gravity. It is assumed that the semiclassical theory
should break down at Planck scales, which is when simple order of
magnitude estimates suggest that the quantum effects of gravity
cannot be ignored: the gravitational energy of a quantum
fluctuation of energy in a Planck size region, determined by the
Heisenberg uncertainty principle, is of the same order of
magnitude as the energy of the fluctuation itself.
{}From the semiclassical Einstein equations it seems also clear that
the semiclassical theory should break down when the quantum
fluctuations of the stress tensor are large. Ford \cite{ford82}
was among the first to have emphasized the importance of these
quantum fluctuations. It is less clear, however, how to quantify
the size of these fluctuations. Thus, Kuo and Ford \cite{kuo93}
used the variance of the fluctuations of the stress tensor
operator compared to the mean value as a measure of the validity
of semiclassical gravity. As pointed out by Hu and Phillips
\cite{hu00,phillips00} such a criterion should be refined by
considering the back reaction of those fluctuations on the metric.
Ford and collaborators also noticed that the metric fluctuations
associated to the matter fluctuations can be meaningfully
classified as ``active'' \cite{ford97,yu99,yu00} and ``passive''
\cite{ford82,kuo93,ford99,ford03,borgman03}.
A different approach to the validity of semiclassical gravity was
taken by Horowitz \cite{horowitz80,horowitz81} who studied the
stability of a semiclassical solution with respect to linear
metric perturbations. In the case of a free quantum matter field
in its Minkowski vacuum state, flat spacetime is a solution of
semiclassical gravity. The equations describing those metric
perturbations involve higher order derivatives, and Horowitz found
unstable ``runaway'' solutions that grow exponentially with
characteristic timescales comparable to the Planck time; see also
the analysis by Jordan \cite{jordan87a}. Later, Simon
\cite{simon90,simon91}, argued that those unstable solutions lie
beyond the expected domain of validity of the theory and
emphasized that only those solutions which resulted from
truncating perturbative expansions in terms of the square of the
Planck length are physically acceptable \cite{simon90,simon91}.
Further discussion was provided by Flanagan and Wald
\cite{flanagan96}, who advocated the use of an ``order reduction''
prescription first introduced by Parker and Simon \cite{parker93}.
More recently Anderson, Molina-Par\'\i s and
Mottola have taken up the issue of the validity of semiclassical
gravity \cite{anderson03} again. Their starting point is the fact
that the semiclassical Einstein equation will fail to provide a
valid description of the dynamics of the mean spacetime geometry
whenever the higher order radiative corrections to the effective
action, involving loops of gravitons or internal graviton
propagators, become important. Next, they argue qualitatively that such
higher order radiative corrections cannot be neglected if the
metric fluctuations grow without bound. Finally, they propose a
criterion to characterize the growth of the metric fluctuations,
and hence the validity of semiclassical gravity, based on the
stability of the solutions of the linearized semiclassical
equation. Following these approaches the Minkowski
metric is shown to be a
stable solution of semiclassical gravity with respect to small
metric perturbations.
As emphasized in Ref. \cite{anderson03} the above criteria may be
understood as criteria within semiclassical gravity itself. It is
certainly true that stability is a necessary condition for the
validity of a semiclassical solution, but one may also look for
criteria within extensions of semiclassical gravity. In the
absence of a quantum theory of gravity such criteria may be found
in more modest extensions. Thus, Ford \cite{ford82} considered
graviton production in linearized quantum gravity and compared the
results with the production of gravitational waves in
semiclassical gravity. Ashtekar \cite{Ash96} and Beetle
\cite{Bee98} found large quantum gravity effects in
three-dimensional quantum gravity models. In a recent paper
\cite{HuRouVer04a} (see also Ref. \cite{HuRouVer04b}) we advocate
for a criteria within the stochastic gravity approach. Stochastic
semiclassical gravity extends semiclassical gravity by
incorporating the quantum stress tensor fluctuations of the matter
fields; see Refs. \cite{hu03a,hu04a} for reviews.
It turns out that this validity criteria is equivalent to the
validity criteria that one might advocate within the large $N$
expansion, that is the theory describing the interaction of the
gravitational field with $N$ identical matter fields. In the
leading order, namely the limit in which $N$ goes to infinity and
the gravitational constant is appropriately rescaled, the theory
reproduces semiclassical gravity. Thus, a natural extension of
semiclassical gravity is provided by the next to leading order. It
turns out that the symmetrized two-point quantum correlations of
the metric perturbations in the large $N$ expansion are equivalent
to the two-point stochastic metric fluctuations predicted by
stochastic gravity. Our validity criterion can then be
summarized as follows: a solution of semiclassical gravity is
valid when it is stable with respect to quantum metric
perturbations. This criterion implies to consider the quantum
correlation functions of the metric perturbations.
It is important to emphasize that the above validity criterion
incorporates in a unified and self-consistent way the two main
ingredients of the criteria exposed above. Namely, the criteria
based on the quantum stress tensor fluctuations of
the matter fields, and the criteria based on the stability of
semiclassical solutions against classical metric perturbations. In
the following discussion we will argue that the former is
incorporated through the so called \textit{induced} fluctuations
and the later though the so called \textit{intrinsic}
fluctuations. These correspond to Ford's ``passive'' and
``active'' fluctuations, respectively. We will see that
symmetrized quantum two-point metric fluctuations can always be
decomposed as a sum of induced and intrinsic fluctuations.
The paper is organized as follows. In section \ref{2} we briefly
review the main ingredients of semiclassical gravity. In section
\ref{3} we introduce stochastic gravity as a theory that goes
beyond semiclassical theory by incorporating the fluctuations of
the quantum stress tensor operator. In section \ref{sec6} our
validity criterion is applied to the study of flat spacetime as a
solution of semiclassical gravity. The problem of the runaway
solutions and methods to deal them is discussed. Throughout the
paper in order to emphasize the qualitative aspects
we use a simplified notation without
tensorial indices and for a few points we also use qualitative arguments
and order of magnitude estimates. We refer the reader to the
papers \cite{hu03a,hu04a,HuRouVer04a,HuRouVer04b} were the
technical details, as well as many subtleties that cannot be
summarized here, are provided. Our metric and curvature
conventions are those of Ref. \cite{misner73}, and we use
$\hbar=c=1$.
\section{Semiclassical gravity}
\label{2}
At present semiclassical gravity cannot be rigorously derived,
but, it can be formally justified in several ways. One of them is
the leading order in the large $N$ expansion
\cite{hartle81}, where $N$ is the number of independent free
quantum fields which interact with gravity only. In this limit,
after path integration one arrives at a theory in which formally
the gravitational field can be treated as a c-number
and the quantum fields are fully quantized.
Semiclassical gravity can be summarized as follows. Let $g$ be the
metric tensor and $\hat\phi$ a scalar field operator. The
semiclassical Einstein equation as the dynamical equation that
describes the back-reaction of quantum matter on the metric
$g$ can be written as
\begin{equation}
G_g=\kappa \langle \hat T^R\rangle_g,
\label{1.1}
\end{equation}
where $\hat T=T[\hat\phi^2]$ is the matter stress tensor in a simplified
notation, which is quadratic in the field operator $\hat \phi$,
and $\kappa=8\pi G $, where $G$ is Newton's constant. This
operator, being the product of distribution valued operators, is
ill defined and needs to be regularized and renormalized, the $R$
in $\hat T^R$ means that the operator has been renormalized. The
angle brackets on the right hand side mean that the expectation
value of the stress tensor operator is computed in some quantum
state, say $|\psi\rangle$, compatible with the geometry described
by the metric $g$. On the left hand side $G_g$ stands for the
Einstein tensor of the metric $g$ together with the cosmological
constant term and other terms quadratic in the curvature which are
generally needed to renormalize the matter stress tensor operator.
The quantum field operator $\hat\phi$ propagates in the background
defined by the metric $g$, it thus satisfies a Klein-Gordon
equation,
\begin{equation}
(\Box_g -m^2)\hat\phi=0,
\label{1.2}
\end{equation}
where $\Box_g$ stands for the D'Alambert operator in the
background of $g$ and $m$ is the mass of the scalar field. A
solution of semiclassical gravity consists of the set $(g,\hat
\phi,|\psi\rangle)$ where $g$ is a solution of Eq.~(\ref{1.1}),
$\hat \phi$ is a solution of Eq.~(\ref{1.2}) and $|\psi\rangle$ is
the quantum state in which the expectation value of the stress
tensor in Eq.~(\ref{1.1}) is computed.
As we recalled in the introduction this theory is in some sense
unique as a theory that describes the interaction of a classical
gravitational field with quantum matter. As an effective theory it
should break down at Planck scales. Also, from the right hand side
of the semiclassical Einstein equation it seems clear that the
theory should also break down when the fluctuations of the quantum
stress tensor are large. This has been emphasized by Ford and
collaborators, and may be illustrated by the example of Ref.
\cite{ford82} as follows.
Let us assume a quantum state formed by an isolated system which
consists of a superposition with equal amplitude of one
configuration with mass $M_1$ and another with mass $M_2$.
Semiclassical theory as described in Eq. (\ref{1.1}) predicts that
the gravitational field of this system is produced by the average
mass $(M_1+M_2)/2$, that is a test particle will move on the
background spacetime produced by such a source. However one would
expect that if we send a succession of test particles to probe the
gravitational field of the above system half of the time they
would react to the field of a mass $M_1$ and the other half to the
field of a mass $M_2$. If the two masses differ substantially the
two predictions are clearly different, note that the fluctuations
in mass of the quantum state is of the order of $(M_1-M_2)^2$.
Although the previous example is suggestive a word of caution
should be said in order not to take it too literary. In fact, if
the previous masses are macroscopic the quantum system decoheres
very quickly \cite{Zur91} and instead of a pure quantum state it
is described by a density matrix which diagonalizes in a certain
pointer basis. Thus for observables associated to this pointer
basis the matrix density description is equivalent to that
provided by a statistical ensemble. In any case, however, from the
point of view of the test particles the predictions differ from
that of the semiclassical theory.
\section{Stochastic gravity}
\label{3}
The purpose of stochastic (semiclassical) gravity is to be able to
deal with the situation of the previous example when the
predictions of the semiclassical theory may be inaccurate.
Consequently, our first point is to characterize the quantum
fluctuations of the stress tensor.
The physical observable that measures these fluctuations is
$\langle \hat T^2\rangle-\langle \hat T\rangle^2$. To make this
more precise let us introduce the tensor operator $\hat t\equiv\hat T- \langle
\hat T\rangle\hat I$, where $\hat I$ is the identity operator, then we
introduce the {\it{noise kernel}} as the four-index bi-tensor defined as
the expectation value of the anticommutator of the operator $\hat t$:
\begin{equation}
N(x,y)=\frac{1}{2}\langle\{ \hat t(x),\hat t(y)\} \rangle_g.
\label{1.3}
\end{equation}
Thus, the noise kernel is the symmetrized connected part of the
two-point quantum correlation function of the stress tensor
operator with respect to the state of the matter fields. The
subindex $g$ here means that this expectation value in taken in a
background metric $g$. An important property of the symmetric
bi-tensor $N(x,y)$ is that it is finite because the tensor
operator $\hat t$ is finite since the ultraviolet divergences of
$\hat T$ are cancelled by the substraction of $\langle \hat
T\rangle$. Since the operator $\hat T$ is selfadjoint $N(x,y)$,
which is the expectation value of an anticommutator, is real and
positive semi-definite \cite{hu03a}. Thus, when considering the
inverse kernel $N^{-1}(x,y)$, one must work in the subspace
obtained from the eigenvectors which have strictly positive
eigenvalues when the noise kernel is diagonalized. The last
property allows for the introduction of a classical Gaussian
stochastic tensor $\xi$ defined by
\begin{equation}
\langle\xi(x)\rangle_s=0,\ \ \ \langle\xi(x)\xi(y)\rangle_s=N(x,y).
\label{1.4}
\end{equation}
This stochastic tensor is symmetric and divergenceless,
$\nabla\cdot \xi=0$, as a consequence of the fact that the stress
tensor operator is divergenceless. The subindex $s$ means that the
expectation value is just a classical stochastic average. Note
that we assume that $\xi$ is Gaussian just for simplicity in order
to include the main effect of the quantum fluctuations.
The idea now is simple we want to modify the semiclassical
Einstein equation (\ref{1.1}) by introducing a linear correction
to the metric tensor $g$, such as $g+h$, which accounts
consistently for the fluctuations of the stress tensor. The
simplest equation is,
\begin{equation}
G_{g+h}=\kappa (\langle \hat T^R\rangle_{g+h}+\xi),
\label{1.5}
\end{equation}
where $g$ is assumed to be a solution of equation (\ref{1.1}).
This stochastic equation must be thought of as a linear equation
for the metric perturbation $h$ which will behave, consequently,
as a stochastic field tensor. Note that the tensor $\xi$ is not a
dynamical source, since it has been defined in the background
metric $g$ which is a solution of the semiclassical equation. Note
also that this source is divergenceless with respect to the
metric, and it is thus consistent to write it on the right hand
side of the Einstein equation. This equation is gauge invariant
with respect to diffeomorphisms defined by any field on the
background spacetime \cite{martin99b}. If we take the statistical
average of equation~(\ref{1.5}) it becomes just the semiclassical
equation for the perturbed metric $g+h$ where now the expectation
value of $\hat T$ is taken in the perturbed spacetime.
The stochastic equation (\ref{1.5}) is known as the
Einstein-Langevin equation. To linear order in $h$ we have
\cite{martin99b},
\begin{equation}
\langle \hat T^R\rangle_{g+h}(x)=-2\int H(x,x^\prime)\cdot
h(x^\prime), \label{1.5a}
\end{equation}
where the kernel $H(x,x^\prime)$ has three terms, one of them is
proportional to the imaginary part of the expectation value of the
time ordered two-point stress tensor, $\mathrm{Im}\langle T(\hat
T(x)\hat T(x^\prime))\rangle$, the second term is proportional to
the expectation value of the stress tensor commutator, $\langle
[\hat T(x),\hat T(x^\prime)]\rangle$, and the third is
proportional to the functional derivative of $\langle \hat
T\rangle $ with respect to the metric (excluding the implicit
dependence on the metric of the field $\hat\phi$).
Of course, this kernel is
also the main ingredient of the linearized semiclassical Einstein
equation around a given background metric $g$. The other key
ingredient in the Einstein-Langevin equation is the noise kernel
$N(x,y)$ which defines the stochastic inhomogeneous source of the
equation. This kernel should be thought of as a distribution
function, the limit of coincidence points has meaning only in the
sense of distributions. Explicit expressions of this kernel in
terms of the two point Wightman functions are given in Ref.
\cite{martin99b} on a general background. Detailed expressions for
this kernel in the Minkowski background are given in Ref.
\cite{martin00}, and expression based on point-splitting methods
have also been given in Refs. \cite{RouVer00,phillips00} in other
backgrounds.
The Einstein-Langevin equation has been previously derived making
use of a formal analogy with open quantum systems and employing
the influence functional formalism \cite{feynman63,feynman65}. The
basis for this approach is a functional formalism known as closed
time path, first introduced by Schwinger
\cite{schwinger61,keldysh65,chou85}, which is an effective action
method suitable to derive dynamical equations for expectation
values of quantum operators; rather than transition elements as in
the standard effective action method. The closed time path
formalism was later applied to the problem of back-reaction of
quantum fields on the spacetime metric
\cite{jordan86,calzetta87,campos94}, in order to derive
semiclassical Einstein equations. The formalism was then applied
along the lines of the influence functional formalism to derive
Einstein-Langevin equations in several contexts
\cite{calzetta94,hu95a,hu95b,campos96,calzetta97c,martin99b,martin99c}.
In Ref.~\cite{martin99a} the Einstein-Langevin equation was
derived by an axiomatic approach by arguing that it is the only
consistent generalization of the semiclassical Einstein equation
which takes into account the back-reaction of the
matter stress tensor fluctuations to lowest order.
We have summarized the axiomatic approach in this section.
The solution of the Einstein-Langevin equation (\ref{1.5}), taking
into account Eq. (\ref{1.5a}), may be expressed as,
\begin{equation}
h(x)=h^0(x)+\kappa\int G_{R}(x,x^\prime)\cdot \xi(x^\prime),
\label{1.5b}
\end{equation}
where $h^0$ is a solution of the homogeneous part of equation
(\ref{1.5}) which contains all the information on the initial
conditions, and $G_{R}(x,x^\prime)$ is the retarded propagator
with vanishing initial conditions associated with the equation
(\ref{1.5}). The two-point correlation function for the metric
perturbation which is the physically most relevant observable can
then be written as:
\begin{eqnarray}
&&\langle h(x)h(y)\rangle_s =\langle h^0(x)h^0(y)\rangle_s +
\nonumber\\
&&\quad\quad\quad\quad\quad\kappa^2 \int G_R(x,x^\prime)\cdot
N(x^\prime,y^\prime)\cdot G_R(y,y^\prime), \label{1.5c}
\end{eqnarray}
where the first average, $\langle h^0(x)h^0(y)\rangle_s$, is
taken with respect to the initial conditions.
It turns out that going to leading order in $1/N$, in the large
$N$ expansion, one can show that the stochastic correlation
functions for the metric perturbations obtained from the
Einstein-Langevin equation coincide with the symmetrized two-point
quantum correlation functions of the metric perturbations. The
details of the derivation will be given in Ref.~\cite{roura03b}
and are summarized in Ref.~\cite{HuRouVer04a} for the particular
case of a Minkowski background, to which we will restrict in
section \ref{sec6}. In this case $\kappa$ in Eq. (\ref{1.5b}) has to be
replaced by the rescaled gravitational coupling constant
$\bar{\kappa} = N \kappa$ and the noise kernel for a single field
$N(x,y)$ must be replaced by $(1/N)N(x,y)$. Thus, we have that the
symmetrized two-point quantum correlation function for the metric
perturbation is
\begin{equation}
\frac{1}{2}\langle\{\hat h(x),\hat h(y)\}\rangle=\langle h(x)h(y)\rangle_s.
\label{1.5d}
\end{equation}
where the Lorentz gauge condition $\nabla\cdot (h - (1/2) \eta
\mathrm{Tr}h) = 0$ ($\eta$ is the Minkowski metric) as well as
some initial condition to fix completely the remaining gauge
freedom of the initial state should be implicitly understood.
It should be emphasized that there are two different contributions to
the symmetrized quantum correlation function, which are clearly
distinguished in Eq.~(\ref{1.5c}). The first contribution is
related to the quantum fluctuations of the initial state of the
metric perturbations and corresponds to the so called
\emph{intrinsic} fluctuations; here the stochastic average must be taken
with respect to the Wigner distribution function
that describes the initial quantum state. The second contribution is
proportional to the noise kernel, it accounts for the fluctuations of
the stress tensor of the matter fields and corresponds to the so
called \emph{induced} fluctuations. These two contributions to the
two-point correlation functions is also seen in the description
of some quantum Brownian motion models which are typically used as
paradigms of open quantum systems
\cite{calzetta03a,CalRouVer01,CalRouVer02}. Both, the
intrinsic and induced fluctuations, play a role in our stability
criterion for the solutions of semiclassical gravity.
The full two-point quantum correlation function
for the metric $\langle\hat h(x)\hat h(y)\rangle$ can, in fact, be obtained
{}from the Einstein-Langevin equation. Since this correlation can be
given in terms of the antisymmetrized and the symmetrized quantum
correlation function we only need the commutator that to leading
order in $1/N$ is independent of the initial state of the metric
perturbation and is given by
\begin{equation}
\frac{1}{2}\langle[\hat h(x),\hat h(y)]\rangle=i\kappa
[G_R(y,x)-G_R(x,y)]. \label{1.5e}
\end{equation}
Note that the information on the retarded propagator is already in
the linearized semiclassical Einstein equation. That is, Eq.
(\ref{1.5}) without the stochastic source.
\subsection{A toy model}
To justify Eq. (\ref{1.5d}) which plays an essential role in our
criteria for the validity of semiclassical gravity it is useful to
introduce a simple toy model for gravity which minimizes the
technical complications. The model is also useful to clarify the
role of the noise kernel and illustrate the relationship between
the semiclassical, stochastic and quantum descriptions. Let us
assume that the gravitational equations are described by a
massless scalar field $h$ whose source is another massless scalar
field $\phi$ which satisfies the Klein-Gordon equation in flat
spacetime $\Box \phi=0$. The field stress tensor is quadratic in
the field, and independent of $h$. The classical gravitational
field equations will be given by
\begin{equation} \Box h=\kappa T,
\label{2.12a}
\end{equation}
where $T$ is now the (scalar) trace of the stress tensor. Note that
this is not a self-consistent theory since $\phi$ does not react
to the gravitational field $h$. This model obviously differs from the
standard linearized theory of gravity discussed previously, where
$T$ is also linear in $h$, but it captures some of its key
features.
In the Heisenberg representation the quantum scalar field $\hat h$ satisfies
\begin{equation}
\Box \hat h=\kappa\hat T.
\label{2.12}
\end{equation}
Since $\hat T$ is quadratic in the field operator $\hat\phi $ some
regularization procedure has to be assumed in order for Eq.
(\ref{2.12}) to make sense. Since we work in flat spacetime we may
simply use a normal ordering prescription to regularize the
operator $\hat T$. The solutions of this equation, i.e. the field
operator at the point $x$, which we call $\hat h_x$ in this
subsection to avoid confusion with
the more standard notation, $\hat h(x)$,
used in the rest of the paper, may be written in terms of the
retarded propagator $G_{xx^\prime}$ of the D'Alambertian as,
\begin{equation} \hat h_x=\hat
h^0_x+\kappa\int G_{xx^\prime}\hat T_{x^\prime}, \label{2.13}
\end{equation}
where $\hat h^0_x$ is the free field which carries information on
the initial conditions and the state of the field. {}From this
solution we may compute, for instance, the symmetric two-point
quantum correlation function (the anticommutator)
\begin{equation}
\langle \{\hat h_x,\hat h_y\}\rangle = \langle \{\hat h^0_x,\hat
h^0_y\}\rangle + \kappa^2 \int G_{xx^\prime}G_{yy^\prime}
\langle\{\hat T_{x^\prime},\hat T_{y^\prime}\}\rangle,
\label{2.14}
\end{equation}
where the expectation value is taken with respect to the quantum
state in which both fields $\phi$ and $h$ are quantized.
We have assumed $\langle \hat h^0\rangle=0$ for the free
field.
We can now consider the semiclassical theory for this problem. If
we assume that $h$ is classical and the matter field is quantum
the semiclassical limit may just be described by substituting
into the classical equation (\ref{2.12a}) the stress trace by the
expectation value of the scalar stress operator $\langle \hat
T\rangle$, in some quantum state of the field $\hat \phi$. We may simply
renormalize the expectation value of $\hat T$ using normal ordering, then for
the vacuum state of the field $\hat\phi$, we would simply have
$\langle\hat T\rangle_0=0 $. The semiclassical theory thus reduces
to
\begin{equation}
\Box h=\kappa \langle \hat T\rangle.
\label{2.15a}
\end{equation}
The two point function $h_xh_y$ that one may derive from this
equation depends on the two point function $\langle \hat
T_x\rangle \langle \hat T_y\rangle $ and clearly cannot reproduce
the quantum result of Eq.~(\ref{2.14}) which depends on the
expectation value of two-point operator $\langle\{\hat T_x,\hat
T_y\}\rangle$. That is, the semiclassical theory entirely misses
the fluctuations of the scalar stress operator $\hat T$.
To extend this semiclassical theory in order to account for such
fluctuations, we introduce the noise kernel as we did in the
previous section. Thus, we define
\begin{equation}
N_{xy}= \frac{1}{2}\langle\{\hat
t_x,\hat t_y\}\rangle
\label{2.14a}
\end{equation}
where $\hat t\equiv\hat T-\langle\hat T\rangle$, and we have used again
the sub-index notation to avoid confusion with the noise kernel of
the previous section. The bi-scalar $N_{xy}$ is real and
positive-semidefinite, as a consequence of $\hat t$ being
self-adjoint \cite{hu03a}. Consequently we can introduce a
Gaussian stochastic field as:
\begin{equation}
\langle\xi\rangle_s=0,\quad \langle\xi_x\xi_y\rangle_s=N_{xy}.
\label{2.14b}
\end{equation}
where the subscript $s$ means a statistical average.
The extension of the semiclassical equation may be simply
performed by adding to the right-hand side of the semiclassical
equation (\ref{2.15a}) the stochastic source $\xi$, which accounts
for the fluctuations of $\hat T$ as follows,
\begin{equation}
\Box h=\kappa\left( \langle \hat T\rangle+\xi\right).
\label{2.15}
\end{equation}
This equation is in the form of a Langevin equation: the field $h$
is classical but stochastic and the observables we may obtain from
it are correlation functions for $h$. In fact, the solution of
this equation may be written in terms of the retarded propagator
as, \begin{equation} h_x=h^0_x+\kappa\int
G_{xx^\prime}\left(\langle\hat T_{x^\prime}\rangle
+\xi_{x^\prime}\right) , \label{2.16}
\end{equation}
{}from where the two point correlation function for the classical field
$h$, after using the definition of $\xi$ and that $\langle
h^0\rangle_s=0$, is given by
\begin{equation}
\langle h_x h_y\rangle_s = \langle h^0_x h^0_y\rangle_s
+\frac{\kappa^2}{2}\int G_{xx^\prime}G_{yy^\prime}\langle\{\hat
T_{x^\prime},\hat T_{y^\prime}\}\rangle. \label{2.17}
\end{equation}
Note that in writing $\left<\dots\right>_s$ here we are assuming a
double stochastic average, one is related to the stochastic
process $\xi$ and the other is related to the free field $h^0$
which is assumed also to be stochastic with an initial
distribution function to be specified.
Comparing Eqs.~(\ref{2.14}) and (\ref{2.17}) we see that the
respective second term on the right-hand side are identical
(except for a factor of $2$ due to the symmetrization) provided
the expectation values are computed in the same quantum state for
the field $\hat \phi$. The fact that the field $h$ is also quantized
in (\ref{2.14}) does not change the previous statement; recall that
$T$ does not depend on $h$. The nature
of the first term on the right-hand sides of equations
(\ref{2.14}) and (\ref{2.17}) is different: in the first case it
is the two-point quantum expectation value of the free quantum
field $\hat h^0$ whereas in the second case it is the stochastic
average of the two point classical homogeneous field $h^0$, which
depends on the initial conditions. Now we can still make these
terms equal to each other (with the factor of $2$) if we assume
for the homogeneous field $h^0$ a Gaussian distribution of initial
conditions such that
\begin{equation}
\langle h^0_x
h^0_y\rangle_s= \frac{1}{2}\langle\{\hat h^0_x,\hat
h^0_y\}\rangle.
\label{2.17a}
\end{equation}
This Gaussian stochastic field $h^0$ can always be defined due to
the semi-positivity of the anti-commutator. Thus, under this
assumption on the initial conditions for the field $h$ the two
point correlation function of Eq.~(\ref{2.17}) equals the quantum
expectation value of Eq.~(\ref{2.14}) exactly. Thus, we have
\begin{equation}
\frac{1}{2}\langle \{\hat h_x,\hat h_y\}\rangle=\langle h_x
h_y\rangle_s,
\label{2.17b}
\end{equation}
which may be compared to Eq. (\ref{1.5d}).
Comparing with the linearized theory of gravity described in
the previous section we see that $\langle T\rangle$ depends also
on $h$, both explicitly and also implicitly through the coupling
of $\phi$ with $h$. The retarded propagator here $G_{xx^\prime}$
is then replaced by the propagator $G_R(x,x^\prime)$ of the
previous section and the functions $h^0$, which are here the free
metric perturbations are replaced by the homogeneous solutions of
the previous section.
\section{Stability of flat spacetime}
\label{sec6}
Let us now apply our validity criterion to flat spacetime. One
particularly simple and interesting solution of semiclassical
gravity is the Minkowski metric. In fact, when the quantum fields
are in the Minkowski vacuum state one may take the renormalized
expectation value of the stress tensor $\langle T^R\rangle=0$
(this is equivalent to assuming that the cosmological constant is
zero) and the Minkowski metric $\eta$ is a solution of the
semiclassical Einstein equation (\ref{1.1}). Thus, we can look for
the stability of flat spacetime against quantum matter fields.
According to the criteria we have established we have to look for
the behavior of the two-point quantum correlations for the metric
perturbations $h$ over the Minkowski background which are given by
Eqs.~(\ref{1.5c}) and (\ref{1.5d}). As we have emphasized several
times these fluctuations separate in two parts: the first term on
the right hand side of Eq.~(\ref{1.5c}) corresponds to the
\textit{intrinsic} fluctuations, and the second term corresponds
to the \textit{induced} fluctuations.
\subsection{Intrinsic fluctuations}
Let us first consider the intrinsic fluctuations,
\begin{equation}
\langle h^0(x) h^0(y)\rangle_s, \label{6.1}
\end{equation}
where $h^0$ are the homogeneous solutions of the
Einstein-Langevin equation (\ref{1.5}), or equivalently the
linearly perturbed semiclassical equation, and the statistical
average is taken with respect to the Wigner distribution that
describes the initial quantum state of
the metric perturbations. Since these solutions are
described by the linearized semiclassical equation around flat
spacetime we can make use of the results derived in
Refs.~\cite{horowitz80,flanagan96,anderson03}. The solutions for
the case of a massless scalar field were first discussed in
Ref.~\cite{horowitz80} and an exhaustive description can be found
in Appendix~A of Ref.~\cite{flanagan96}. Decomposing the metric
perturbation into scalar, vectorial and tensorial parts and
computing the linearized Einstein tensor, one gets a vanishing
result for the vectorial part of the metric perturbation; the
scalar and tensorial components of the metric perturbation give
rise, respectively, to the scalar and tensorial components of the
linearized Einstein tensor.
The vectorial part is found to vanish whereas the scalar
and tensorial contributions for a massless and conformally coupled
scalar field (see Ref.~\cite{flanagan96} for the massless case
with arbitrary coupling and Refs.~\cite{martin00,anderson03} for
the general massive case) satisfy the following equations:
\begin{equation}
\left( 1 + 12 \kappa \bar{\beta} p^2 \right) \tilde{G}^{
\mathrm{(S)}}(p) = 0 \label{scalar},
\end{equation}
\begin{equation}
\lim\limits_{\;\epsilon \rightarrow 0^+} \left( 1 + \frac{\kappa
p^2}{960 \pi^2} \ln \frac{p^2 }{\mu^2} \right) \tilde{G}^{
\mathrm{(T)} }(p) = 0 \label{tensorial},
\end{equation}
where in the last equation the prescription that the time
component of $p$ has a small imaginary part, $p^0+i \epsilon$, is
taken. Here $\tilde G(p)$ stands for the Fourier transform of the
linearized Einstein tensor, the upper indices $S$ and $T$ stand
for scalar and tensorial respectively, $\bar\beta$ is a
dimensionless renormalized parameter that multiplies some of
the quadratic terms in the curvature in the effective action for the
gravitational field, and $\mu$ is a renormalization mass scale.
See Ref.~\cite{HuRouVer04a} for a more complete description.
For the scalar component when $\bar{\beta} = 0$ the only solution
is $\tilde{G}^{ \mathrm{(S)} }(p) = 0$. When $\bar{\beta}
> 0$ the solutions for the scalar component exhibit an oscillatory
behavior in spacetime coordinates which corresponds to a massive
scalar field with $m^2 = (12 \kappa |\bar{\beta}|)^{-1}$; for
$\bar{\beta} < 0$ the solutions correspond to a tachyonic field
with $m^2 = - (12 \kappa |\bar{\beta}|)^{-1}$: in spacetime
coordinates they exhibit an exponential behavior in time, growing
or decreasing, for wavelengths larger than $4 \pi (3 \kappa
|\bar{\beta}|)^{1/2}$, and an oscillatory behavior for wavelengths
smaller than $4 \pi (3 \kappa |\bar{\beta}|)^{1/2}$. On the other
hand, the solution $\tilde{G}^{ \mathrm{(S)} }(p) = 0$ is
completely trivial since any scalar metric perturbation
$\tilde{h}(p)$ giving rise to a vanishing linearized Einstein
tensor can be eliminated by a gauge transformation as explained in
Ref. \cite{HuRouVer04a}.
For the tensorial component, when $\mu \le \mu_\mathrm{crit} =
l_p^{-1} (120\pi)^{1/2} e^{\gamma}$, where $l_p$ is the Planck length
($l_p^2\equiv \kappa/8\pi$) the first factor in
Eq.~(\ref{tensorial}) vanishes for four complex values of $p^0$ of
the form $\pm \omega$ and $\pm \omega^*$, where $\omega$ is some
complex value. We will consider here the case in which $\mu <
\mu_\mathrm{crit}$; a detailed description of the situation for
$\mu \ge \mu_\mathrm{crit}$ can be found in Appendix~A of
Ref.~\cite{flanagan96}. The two zeros on the upper half of the
complex plane correspond to solutions in spacetime coordinates
exponentially growing in time, whereas the two on the lower half
correspond to solutions exponentially decreasing in time. Strictly
speaking, these solutions only exist in spacetime coordinates,
since their Fourier transform is not well defined. They are
commonly referred to as runaway solutions and for $\mu \sim
l_p^{-1}$ they grow exponentially in time scales comparable to the
Planck time.
In order to deal with those unstable solutions, one possibility is
to employ the \textit{order reduction} prescription
\cite{parker93}, which we will briefly summarize in the last
subsection. With such a prescription we are left only with the
solutions which satisfy $\tilde{G}(p)=0$. The solutions for
$\tilde{h}(p)$ simply correspond to free linear gravitational
waves propagating in Minkowski spacetime expressed in the
transverse and traceless (TT) gauge. When substituting back into
Eq.~(\ref{6.1}) and averaging over the initial conditions we
simply get the symmetrized quantum correlation function for free
gravitons in the TT gauge for the state given by the
Wigner distribution. As far as the intrinsic fluctuations are
concerned, it seems that the order reduction prescription is too
drastic, at least in the case of Minkowski spacetime, since no
effects due to the interaction with the quantum matter fields are
left.
A second possibility, proposed by Hawking \emph{et al.}
\cite{hawking01,hawking02}, is to impose boundary conditions which
discard the runaway solutions that grow unbounded in time and
correspond to a special prescription for the integration contour
when Fourier transforming back to spacetime coordinates. Following
that procedure we get, for example, that for a massless
conformally coupled scalar field with $\bar{\beta} > 0$ the
intrinsic contribution to the symmetrized quantum correlation
function coincides with that of free gravitons plus an extra
contribution for the scalar part of the metric perturbations which
renders Minkowski spacetime stable but plays a crucial role in
providing a graceful exit for inflationary models driven by the
vacuum polarization of a large number of conformal fields. Such a
massive scalar field would not be in conflict with present
observations because, for the range of parameters
considered, the mass would be far too large to have observational
consequences \cite{hawking01}.
\subsection{Induced fluctuations}
The induced fluctuations are described by the second term in
Eq.~(\ref{1.5c}). They are induced for the noise kernel that describes
the stress tensor fluctuations of the matter fields,
\begin{equation}
\frac{\bar\kappa^2}{N}\int G_R(x,x^\prime)\cdot N(x^\prime,y^\prime)\cdot
G_R(y,y^\prime) , \label{6.2}
\end{equation}
where we write the expression in the large $N$ limit. The
contribution corresponding to the induced quantum fluctuations is
equivalent to the stochastic correlation function obtained by
considering just the inhomogeneous part of the solution to the
Einstein-Langevin equation: the second term on the right-hand side
of Eq.~(\ref{1.5c}). Taking all that into account, it is clear
that we can make use of the results for the metric correlations
obtained in Ref.~\cite{martin00} by solving the Einstein-Langevin
equation. In fact, one should simply take $N=1$ to transform our
expressions to those of Ref.~\cite{martin00} and, similarly,
multiply the noise kernel in the expressions of that reference by
$N$ so that they can be used here, which follows
{}from the fact that we have $N$ independent matter fields.
The same kind of exponential instabilities in the runaway
solutions of the homogeneous part of the Einstein-Langevin
equation also arise when computing the retarded propagator
$G_\mathrm{R}$. In order to deal with those instabilities, similar
to the case of the intrinsic fluctuations, one possibility is to
make use of the order reduction prescription. The
Einstein-Langevin equation becomes then $\tilde G (p)=
\bar\kappa\tilde \xi (p)$. The second possibility, following the
proposal of Hawking \emph{et al.}, is to impose boundary
conditions which discard the exponentially growing solutions and
translate into a special choice of the integration contour when
Fourier transforming back to spacetime coordinates the expression
for the propagator. In fact, it turns out that the propagator
which results from adopting that prescription coincides with the
propagator that was employed in Ref.~\cite{martin00}. Note,
however, that this propagator is no longer a strictly retarded
propagator since it exhibits causality violations at Planck
scales. A more detailed discussion on all these points can be
found in Appendix E of Ref.~\cite{HuRouVer04a}.
Following Ref.~\cite{martin00}, the Einstein-Langevin equation can
be entirely written in terms of the linearized Einstein tensor.
The equation involves second derivatives of that tensor, and in
terms of its Fourier components $\tilde{G}(p)$ takes the form
\begin{equation}
(1+ F(p))\cdot \tilde{G}(p) = \bar{\kappa} \tilde{\xi} (p) ,
\label{6.3}
\end{equation}
where $F$ is a four-index tensor which depends on $p^2\ln p^2$
when the field is massless and conformally coupled. This reflects
the fact that we have second derivatives of the Einstein tensor
and the nonlocality of the Einstein-Langevin equation (or also of
the perturbed semiclassical equation). {}From equation (\ref{6.3})
one may obtain the correlation functions for $\tilde{G}(p)$,
$\langle \tilde G (p)\tilde G (q)\rangle_s$, which are invariant
under gauge transformations of the metric perturbations. Writing
the linearized Einstein tensor in terms of the metric
perturbation, which takes a particularly simple form in the
Lorentz gauge, one may derive the correlation functions for
$\tilde{h}(p)$: $\langle \tilde h (p)\tilde h (q)\rangle_s$.
Finally, the correlation functions in spacetime coordinates can be
easily obtained by Fourier transforming these correlations. For
massless and conformally coupled matter fields explicit results
are given in Ref.~\cite{martin00}, they have the general
expression:
\begin{equation}
\langle h(x)h(y)\rangle_s=\frac{\bar\kappa^2}{720\pi N}\int
\frac{e^{ip\cdot
(x-y)}P\theta(-p^2)}{|1+(\bar\kappa/960\pi^2)p^2\ln(p^2/\mu^2)|^2}
\label{6.4}
\end{equation}
where $P$ is a four-index projection tensor. This correlation
function for the metric perturbations is in agreement with the
real part of the propagator obtained by Tomboulis in
Ref.~\cite{tomboulis77} using a large $N$ expansion.
To estimate this integral let us consider spacelike separated
points $x-y=(0,\mathbf{r})$ and introduce the Planck length
$l_p$. It is not difficult to see \cite{hu04a}, that for space
separations $|\mathbf{r}|\gg l_p$ we have
\begin{equation}
\langle h(x)h(y)\rangle_s\sim \frac{l_p^4}{|\mathbf{r}|^4},
\label{6.5}
\end{equation}
and for $|\mathbf{r}|\sim N l_p$ we have
\begin{equation}
\langle h(x)h(y)\rangle_s\sim
e^{-|\mathbf{r}|/l_p}\frac{l_p}{|\mathbf{r}|}.
\label{6.6}
\end{equation}
Since these fluctuations are induced by the matter stress fluctuations we
infer that the effect of the matter fields is to suppress metric
fluctuations at small scales. On the other hand, at large scales
the induced metric fluctuations are small compared to the free
graviton propagator which goes like $l_p^2/|\mathbf{r}|^2$.
We thus conclude that, once the instabilities giving rise to the
unphysical runaway solutions have been discarded, the fluctuations
of the metric perturbations around the Minkowski spacetime induced
by the interaction with quantum scalar fields are indeed stable.
Instabilities would lead to a divergent result when Fourier
transforming back to spacetime coordinates. Note that when the
order reduction prescription is used the $p^2\ln p^2$ terms are
absent in the corresponding Eq. (\ref{6.4}). Thus, in contrast to
the intrinsic fluctuations, there is still a nontrivial
contribution to the induced fluctuations due to the quantum matter
fields in this case.
\subsection{Order reduction prescription and large $N$}
Runaway solutions are a typical feature of equations describing
back-reaction effects, such is in classical electrodynamics, and
are due to higher than two time derivatives in the dynamical
equations. In a very schematic way the semiclassical Einstein
equations have the form
\begin{equation}
G_h+l_p^2\ddot G_h=0,
\label{6.7}
\end{equation}
where $G_h$ stands for the linearized Einstein tensor
over the Minkowski background, say,
and we have simplified the equation as much as
possible. The second term of the equation is due to the vacuum
polarization of matter fields and contains four time derivatives
of the metric perturbation. Some specific examples of such an equation
are, in momentum space, Eqs.~(\ref{scalar}) and (\ref{tensorial}).
The order reduction
procedure is based on treating perturbatively the terms involving
higher order derivatives, differentiating the equation under
consideration and substituting back the higher derivative terms in
the original equation keeping only terms up to the required order
in the perturbative parameter. In the case of the semiclassical
Einstein equation, the perturbative parameter is $l_p^2$. If we
differentiate twice Eq.~(\ref{6.7}) with respect to time it is
clear that the second order derivatives of the Einstein tensor are
of order $l_p^2$. Substituting back into the original equation, we
get the following equation up to order $l_p^4$: $ G_h=0+ O(l_p^4). $
Now, there are certainly no runaway solutions but also no effect
due to the vacuum polarization of matter fields. Note that the
result is not so trivial when there is an inhomogeneous term on
the right hand side of Eq.~(\ref{6.7}), this is what happens with the
induced fluctuations predicted by the Einstein-Langevin equation.
Semiclassical gravity is expected to provide reliable results as
long as the characteristic length scales under consideration, say
$L$, satisfy that $L\gg l_p$ \cite{flanagan96}. This can be
qualitatively argued by estimating the magnitude of the different
contributions to the effective action for the gravitational field,
considering the relevant Feynman diagrams and using dimensional
arguments. Let us write the effective gravitational action, again
in a very schematic way, as
\begin{equation}
S_{\mathrm{eff}}=\int \sqrt{-g}\left( \frac{1}{l_p^2}R
+\alpha R^2+l_p^2 R^3+\dots \right),
\label{6.8}
\end{equation}
where $R$ is the Ricci scalar. The first term is the usual
classical Einstein-Hilbert term, the second stands for terms
quadratic in the curvature (square of Ricci and Weyl tensors) this
terms appear as radiative corrections due to vacuum polarization
of matter fields, here $\alpha$ is an dimensionless parameter
presumably of order 1, the $R^3$ terms are higher order
corrections which appear for instance when one considers internal
graviton propagators inside matter loops. Let us assume that
$R\sim L^{-2}$ then the different terms in the action are of the
order of $R^2\sim L^{-4}$ and $l_p^2R^3\sim l_p^2L^{-6}$.
Consequently when $L\gg l_p^2$, the term due to matter loops is a
small correction to the Einstein-Hilbert term $(1/l_p^2)R\gg R^2$,
and this term can be treated as a perturbation. The justification
of the order reduction prescription is actually based on this
fact. Therefore, significant effects from the vacuum polarization
of the matter fields are only expected when their small
corrections accumulate in time, as would be the case, for
instance, for an evaporating macroscopic black hole all the way
before reaching Planckian scales.
However if we have a large number $N$ of matter fields the
estimates for the different terms change in a remarkable way. This
is interesting because the large $N$ expansion seems the best
justification for semiclassical gravity. In fact, now the vacuum
polarization terms involving loops of matter are of order
$NR^2\sim NL^{-4}$. For this reason the contribution of the
graviton loops, which is of order $R^2$, can be neglected in front
of the matter loops; this justifies the semiclassical
limit. Similarly higher order corrections are of order
$Nl_p^2R^3\sim Nl_p^2L^{-6}$. Now there is a regime, when $L\sim
\sqrt{N}l_p$, where the Einstein-Hilbert term is comparable to the
vacuum polarization of matter fields, $(1/l_p^2)R\sim NR^2$, and
yet the higher correction terms can be neglected because we still
have $L\gg l_p$, provided $N\gg 1$. This is the kind of situation
considered in trace anomaly driven inflationary models
\cite{hawking01}, such as that originally proposed by Starobinsky
\cite{starobinsky80}, where the exponential inflation is driven by
a large number of massless conformal fields. The order reduction
prescription would completely discard the effect from the vacuum
polarization of the matter fields even though it is comparable to
the Einstein-Hilbert term. In contrast, the procedure proposed by
Hawking \emph{et al.} keeps the contribution from the matter
fields. Note that here the actual physical Planck length $l_p$ is
considered, not the rescaled one, $\bar{l}_p^2 = \bar\kappa/8\pi$,
which is related
to $l_p$ by $l_p^2 = \kappa/8\pi= \bar{l}_p^2/ N$.
\section{Conclusions}
An analysis of the stability of any solution of semiclassical
gravity with respect to small quantum corrections should consider
not only the evolution of the expectation value of the metric
perturbations around that solution, but also their fluctuations,
encoded in the quantum correlation functions. Making use of the
equivalence (to leading order in $1/N$, where $N$ is the number of
matter fields) between the stochastic correlation functions
obtained in stochastic semiclassical gravity and the quantum
correlation functions for metric perturbations around a solution
of semiclassical gravity, the symmetrized two-point quantum
correlation function for the metric perturbations can be
decomposed into two different parts: the intrinsic fluctuations
due to the fluctuations of the initial state of the metric
perturbations itself, and the fluctuations induced by their
interaction with the matter fields. If one considers the
linearized perturbations of the semiclassical Einstein equation,
only information on the intrinsic fluctuations can be retrieved.
On the other hand, the information on the induced fluctuations
naturally follows from the solutions of the Einstein-Langevin
equation.
As a specific example, we have analyzed the symmetrized two-point
quantum correlation function for the metric perturbations around
the Minkowski spacetime interacting with $N$ scalar fields
initially in the Minkowski vacuum state. Once the ultraviolet
instabilities which are ubiquitous in semiclassical gravity and
are commonly regarded as unphysical, have been properly dealt with
by using the order reduction prescription or the procedure
proposed by Hawking \emph{et al.} \cite{hawking01,hawking02}, both
the intrinsic and the induced contributions to the quantum
correlation function for the metric perturbations are found to be
stable \cite{HuRouVer04a}. Thus, we conclude that Minkowski
spacetime is a valid solution of semiclassical gravity.
\section{Acknowledgments}
I am very grateful to Hans-Thomas Elze and co-organizers for
giving me the opportunity to participate at the Second International Workshop
DICE2004 on {\it
{}From Decoherence and Emergent Classicality to Emergent Quantum Mechanics},
and for their kind and generous hospitality. The research reported
here is mainly based in work made in collaboration with Bei-Lok Hu
and Albert Roura I am grateful to them for their invaluable
contribution to this topic and for having enjoyed many fruitful discussions.
I thank Daniel Arteaga, Lajos Diosi, Jonathan Halliwell,
Jim Hartle, and Renaud Parentani
for discussions on this topic.
I also thank Albert Roura for a critical reading of the
manuscript and many fruitful discussions. This work has been partially
supported by the MEC Research Projects No. FPA2001-3598 and FPA2004-04582.
|
1,108,101,565,443 | arxiv | \section{Introduction}
A core-collapse supernova (SN) explosion marks the disruption of a massive star by an energetic shock wave followed by the
formation of a neutron star or a black hole. In the first hundreds of milliseconds (ms) of a supernova, temperatures in the range
of tens of MeV, densities beyond nuclear matter saturation density $n_0 \sim 0.145\:$fm$^{-3}$ and proton fractions $Y_p \leq 0.3$
are reached. With such properties, supernovae (SNe) are astrophysical laboratories for dense nuclear matter which,
in the phase diagram of strongly interacting matter, possess overlap regions with heavy-ion experiments, such as in the future
FAIR facility at GSI, Darmstadt (Germany) and the NICA facility at the JINR in Dubna (Russia). The modeling of core-collapse
SNe represents a great computational challenge. It requires as input amongst others a nuclear matter equation of state (EoS),
which provides information about the thermodynamic properties and compositions of matter for a large range of baryon
number densities $n_b$, temperatures $T$, and isospin states, characterized by $Y_p$. Being discussed to populate neutron
star interiors, hyperons and quark matter can also be included in SN equations of state (EoSs). At present, both components are
tested on their impacts in SN simulations. Hereby, the applied quark and hyperon EoSs should be compatible with
observed neutron star properties, e.g. pulsar masses. As we will argue in the next section, the latter indicate a stiff nuclear matter
EoS at high density.
\newline
In the following we will give a short summary on recent pulsar mass measurements and their implications for quark matter
in neutron star interiors. We will proceed with an overview of quark matter studies in core-collapse SNe. In the
last section, we will focus on the impact of low density quark-hadron phase transitions on the gravitational collapse of light
and intermediate mass progenitor stars.
\section{Strange quark matter in massive neutron stars}
Neutron star masses can be deduced for pulsars in binary systems, if effects from general relativity such as the advance of
periastron $\dot{\omega}$ and Shapiro delay are observable in the pulsar signal \cite{Lorimer08}. For a long time the Hulse-Taylor
pulsar PSR B1913+16 had the highest precisely known mass of $M = 1.4414 \pm 0.0002\:$M$_\odot$, with most pulsars
clustering around this value \cite{Thorsett99}. However, the observation of $\dot{\omega}$ and Shapiro delay in the low mass
X-ray binary J1903-0327 \cite{Champion08} allowed the determination of the mass for the corresponding millisecond pulsar to
$M = 1.667 \pm 0.021\:$M$_\odot$. In 2010, an even higher pulsar mass was obtained for the millisecond pulsar PSR J1614-2230.
A massive binary partner and the large inclination angle of the system allow the measurement of the Shapiro delay to a high
accuracy and reveal a pulsar mass of $M = 1.97 \pm 0.04$M$_\odot$ \cite{Demorest10}. At present, this value represents the
highest robust mass ever measured for a compact star and poses tight constraints on the nuclear matter EoS restricting it to
stiff models. Hybrid and strange stars are only compatible with such a high mass if stiffening effects from the strong interaction
between the quarks, e.g. in form of the strong interaction coupling constant $\alpha_s$ and/or color superconductivity -
have a large impact on the quark matter EoS (see \cite{Weissenborn11} and references therein).
\section{Strange quark matter in core-collapse supernovae}
The dynamical effects of phase transitions on neutron stars and core-collapse SNe were first discussed by Migdal \cite{Migdal79}
in 1979, and later studied by Takahara and Sato \cite{Takahara86} as well as Brown \cite{Brown88}. In the early 90s,
Gentile et al. \cite{Gentile93} performed general relativistic hydrodynamic core-collapse SN simulations including a phase
transition to strange quark matter for critical densities of $\sim (2-3)\: n_0$. The authors tested different setups of the
quark-hadron mixed phase, finding the formation of two shock waves. The first shock wave is caused by the usual stiffening
of the hadronic EoS for $n_b > n_0$, which halts and reverts the infall of matter in the center of the collapsing iron core
(core-bounce). The second wave is formed due to the softening of the EoS in the quark-hadron mixed phase together with the
subsequent stiffening in pure quark matter. As neutrino transport was not included in the calculations, the dynamics of the
shock waves could only be followed for a few ms. Simulations including neutrino transport were performed by e.g. Nakazato
et al. \cite{Nakazato08, Nakazato10}. The authors applied the hadronic SN Shen EoS \cite{Shen98} extending it by the inclusion
of quark matter at higher densities as well as thermal pions. For the quark matter EoS, Nakazato et al. chose the simple bag
model \cite{Chodos74b} with a bag constant of $B^{1/4} \sim 209\:$MeV. The quark-hadron phase transition was modeled
by a Gibbs construction \cite{Glendenning92}. The studied progenitor stars had masses of $\geq 40\:$M$_\odot$, whereas
the gravitational collapse of such massive stars usually ends with the production of a black hole \cite{Heger03}. Due to the
softer hybrid EoS, the onset of quark matter accelerates the black hole formations by tens to hundreds of ms in comparison
to simulations with the original Shen EoS. While the earlier collapse results in a shortening of the neutrino emission, the softening
of the EoS influences the neutrino spectra. Both effects can be used as indicators for a quark matter phase transition.
However, as hyperons and different nuclear interactions \cite{Sumiyoshi10, Hempel11} can have similar imprints on the neutrino
signal, further studies are required.
\newline
In this work we focus on quark matter phase transitions with low critical densities around $n_0$ for SN conditions.
It was shown by Drago et al. \cite{Drago99} and later argued in Sagert et al. \cite{Sagert09} and Fischer et al. \cite{Fischer11}
that quark models with such low $n_{crit}$ for SN conditions can be compatible with higher transition densities in heavy ion
collisions, caused by the larger $Y_p$ in the latter and different strangeness production mechanisms.
\section{Early quark-hadron transition in core-collapse supernova}
In our study we implement a quark-hadron phase transition into the Shen EoS via the Gibbs construction. For the quark
matter EoS, we apply a quark bag model which is extended by first order corrections in $\alpha_s$. The thermodynamic grand potential is
given by \cite{Fischer11}:
\begin{equation}
\Omega_{QM}= \sum_{i=u,d,s} \left[\Omega_i + \frac{\alpha_s}{\pi} \left( T^2 \mu_i^2 + \frac{\mu_i^2}{2 \pi^2} \right) + \frac{35 \pi}{126} T^4 \alpha_s \right] + B .
\label{bag_eos}
\end{equation}
Hereby, $\Omega_i$ are the fermi contributions for the up, down, and strange quarks. First order corrections from $\alpha_s$
are given by the second and third terms, while $B$ is the bag constant which represents all non-perturbative effects of the strong
interaction. The quark masses are chosen to $m_s=100\:$MeV for strange quarks and $m_u=m_d=0\:$MeV for up and down quarks.
In the very simple bag model, first order corrections from $\alpha_s$ are not present. As a consequence, the corresponding quark
EoS is very soft. The maximum masses of the resulting hybrid stars are low and do not fulfill the mass constraint of PSR
J1614-2230 \cite{Weissenborn11}. The inclusion of $\alpha_s$ correction terms stiffen the quark EoS and can increase the hybrid
star maximum masses up to $\geq 2\:$M$_\odot$ \cite{Weissenborn11}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, angle=270]{plot1.eps}
\caption{Hybrid star mass radius relations for the quark bag model in eq.(\ref{bag_eos}) and the hadronic EoS TM1.
Solid lines are stars with a pure quark matter core, while dashed lines indicate mixed phase cores. The horizontal
lines show pulsar mass measurements discussed in the text.}
\label{mr_plot1}
\end{center}
\end{figure}
For low critical densities, the resulting hybrid stars have an extended quark-hadron mixed phase in their interior and only for
sufficiently stiff hadronic EoSs possess a pure quark matter core \cite{Weissenborn11}. Figure \ref{mr_plot1} shows the mass-radius
relations for different parameter sets in eq.(\ref{bag_eos}). Hadronic matter is described by the relativistic mean field TM1 model
\cite{Sugahara94} which is also the basis for the Shen EoS. The three parameter sets with $\alpha_s=0$ and $\alpha_s=0.3$, shown
in Fig.~\ref{mr_plot1}, have been tested on their impact in one dimensional SN simulations based on general relativistic radiation
hydrodynamics and three flavor Boltzmann neutrino transport \cite{Liebendoerfer04}. The simulations show that due to the low
critical densities, the quark-hadron mixed phase sets in already at core-bounce, when matter in the center of the collapsing iron
core reaches densities around $n_0$. However, as the quark fraction is small and $Y_p \sim 0.3$, the hybrid EoS is very similar
to the one of hadronic matter \cite{Fischer11}, and the dynamics proceed like in a normal core-collapse SN for the first $(200-400)\:$ms.
The core-bounce launches a hydrodynamic shock wave which starts to propagates outwards. On its way, it loses energy due to
the disintegration of infalling heavy nuclei and production of neutrinos. As the shock wave moves across the neutrinospheres,
the neutrinos are emitted in a burst dominated by electron neutrinos $\nu_e$. This energy loss turns the expanding shock into a
standing accretion shock (SAS). As shock-heated matter continues to be accreted through the SAS on the surface of the proto neutron
star (PNS), the density and temperature in its interior rise and a growing volume of the PNS enters the mixed phase. The quark fraction in
the mixed phase increases which leads to a softening of the EoS. This, together with the growing gravitational mass of the PNS eventually
triggers its collapse to a more compact hybrid star configuration. Similar to the study of Gentile et al., a second shock wave forms and
propagates outwards. Shock heating of the infalling neutron rich matter leads to the production of $\bar{\nu}_e$, as well as $\nu_{\mu / \tau }$,
and $\bar{\nu}_{\mu / \tau }$, which are released in a second burst as the second shock wave propagates across the neutrinospheres.
\begin{figure}
\begin{center}
\includegraphics[width=0.42\textwidth, angle=270]{plot2.eps}
\caption{Phase diagram for transitions from hadronic matter (TM1) to strange quark matter at $Y_p=0.1,0.3$, and $0.5$.
The quark matter parameters are $B^{1/4}=145\:$MeV and $\alpha_s=0.7$. Thin lines mark the onset of the quark-hadron mixed phase,
thick lines give the transition to pure quark matter.}
\label{phase_diag145_as07}
\end{center}
\end{figure}
For all three parameter sets the second shock wave eventually leads to a SN explosion, whereas the explosion energy and the time
difference between the first and second neutrino bursts depend on $n_{crit}$ and the model of the progenitor star \cite{Sagert09,Fischer11}.
To test the effects of a stiffer quark EoS on the SN dynamics we chose $B^{1/4}=145\:$MeV and $\alpha_s=0.7$. As can be seen from
Fig.~\ref{mr_plot1}, the corresponding hybrid star maximum mass is $\sim 1.97\:$M$_\odot$ and thereby compatible PSR J1614-2230.
Fig.~\ref{phase_diag145_as07} shows the phase diagram for different values of $Y_p$ (for details of phase diagram calculations
see \cite{Fischer11}). The critical densities for the onset of the mixed phase for $Y_p = 0.3$ are around $n_{crit} < 1.5\:n_0$, however,
due to the similar stiffness of the quark and hadron EoSs, the mixed phase extends up to $\geq 10\: n_0$ for
low T. As the temperature rises above $60\:$MeV, the mixed phase experiences a significant reduction and the critical densities
become low. However, for such high values of $T$, the inclusion of pions and hadron resonances becomes important and would most
likely change the shape of the phase diagram.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, angle=270]{plot3.eps}
\caption{Onset of the mixed phase for different quark EoS parameter sets and $Y_p=0.3$. The thin solid line
show the temperatures and densities which are reached in the center of the PNS shortly before and during the first $500\:$ms after
core-bounce.}
\label{phase_diag_as07_a}
\end{center}
\end{figure}
Similar to the soft bag EoS parameter sets, we tested the new hybrid EoS in one dimensional SN simulations
for the collapse of a $15\:$M$_\odot$ and a $30\:$M$_\odot$ progenitor. In both cases we find that quark matter appears very
late - around $1\:$s after core bounce and does not influence the SN dynamics \cite{Sagert12}. The reason for this outcome
can be seen in Fig.~\ref{phase_diag_as07_a} which shows the onset of the mixed phases for the discussed quark EoSs.
Though the critical densities for $B^{1/4}=145\:$MeV and $\alpha_s=0.7$ are low, it can be seen that they
are higher in comparison to the soft EoSs. In addition we plot the densities and temperatures in the center of the PNS
shortly before and during the first $500\:$ms after the core-bounce. While it can be seen that for the soft models, matter in the PNS
enters the quark-hadron mixed phase, the critical densities for the new parameter set are too high.
However, the reduction of the bag constant to $B^{1/4}=139\:$MeV leads to sufficiently low values of $n_{crit}$ as is illustrated in
Fig.~\ref{phase_diag_as07_a} and also fulfills the maximum mass constraint of PSR J1614-2230 (see Fig.~\ref{mr_plot1}).
This parameter set is currently tested in one dimensional SN simulations and will give further insights in the role and detectability
of a quark matter phase transition in core-collapse SNe.
\section{Summary}
The onset of strangeness in core-collapse supernovae (SNe) can be studied by implementing hyperons and strange quark matte in SN
equations of state (EoSs). Hereby, the chosen parameter sets must fulfill restrictions from the recent finding of a two solar mass pulsar
PSR J1614-2230, which is only compatible with a stiff high density nuclear matter EoS. Within currently applied quark EoS models,
high critical densities were shown to shorten the time to black hole formation in the gravitational collapse of massive progenitor stars.
For the onset of quark matter with a soft EoS at critical densities $n_{crit}$ around nuclear matter saturation density $n_0\sim 0.145$fm$^{-3}$,
the proto neutron star collapses to a more compact hybrid star configuration within hundreds of ms after core-bounce. This launches a
shock wave which leads to the SN explosion and releases a neutrino burst, dominated by $\bar{\nu}_e$. The properties of the neutrino
burst are dependent on $n_{crit}$ as well as the model of the progenitor star. A first study with a stiff quark EoS shows no
impacts on the SN dynamics as the critical densities are too high to be reached in the early post-bounce phase. However, studies are
on the way, in which we apply a stiff quark EoS parameter set where SN matter enters the quark-hadron mixed phase at $n_{crit} \sim n_0$.\\
\newline
\textit{Acknowledgements}\\
The project was funded by the Swiss National Science Foundation (SNF) under project numbers PP00P2 - 124879/1, 200020 - 122287.
T.F is support by HIC for FAIR and by the SNF under project~no.~PBBSP2-133378. The work of G.P. is supported by
the DFG under Grant No. PA 1780/2-1 and J.S.-B. is supported by the DFG through the Heidelberg Graduate School of Fundamental
Physics. M. H. is supported by the SNF under project number no. 200020-132816/1. M. H. is also grateful for
participating in the EuroGENESIS collaborative research program of the
European Science Foundation (ESF) and the ENSAR/THEXO project. I.S. is supported by the AvH foundation via a Feodor Lynen fellowship
and wishes to acknowledge the HPCC of MSU and the iCER. The authors are additionally supported by CompStar, a research networking
program of the ESF, and the scopes project funded by the SNF grant. no. IB7320-110996/1.
\bibliographystyle{nar}
|
1,108,101,565,444 | arxiv | \section{Supplementary Data}
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\end{document}
\section{Keywords:} Network Analysis, STEM Education, Assessment, Enrolment, Entropy}
\end{abstract}
\section{Introduction}
There is an increasing demand to understand the choices that students make when it comes to selecting courses in secondary school and further education. Obtaining a clear picture of the skills that students leave school with is an important goal for governments across the world, and this is especially true regarding Science, Technology, Engineering and Mathematics (STEM). For example, the New Zealand Qualifications Authority (NZQA) \citep[p.8]{NZQA2016} specifically stated that:
\begin{quote}
To meet the demand for essential skills for the twenty first century, New Zealand needs to grow the number and diversity of skilled workers in Science, Technology, Engineering and Maths.
\end{quote}
Governments are pushing to not only increase the number of students participating in STEM education, but also to increase the representation of students who have been historically underrepresented in STEM. While trends may differ across countries, disparities in STEM participation tend to be found at the intersection of gender, ethnicity and social class \citep{Archer2015b, PISA_NZ_2017}. Globally, women are typically underrepresented in subjects such as physics and computer science, while there tends to be gender parity in subjects such as biology and medicine. In the case of Aotearoa New Zealand, similar disparities in STEM participation are found \citep{NZQA2016,EducationCounts_2016a,EducationCounts_2016b}. In addition, students from M\={a}ori and Pacific Island backgrounds have also been underrepresented in post-compulsory STEM education \citep{MoH2014, NZQA2016}.
Student attrition from STEM education is often viewed in terms of a leaky pipeline, with students from groups who are under represented in STEM being more likely to drop out of STEM education with each advance from one educational stage to the next. However, participation in STEM education is complex. Not only is it important to consider the socio-cultural context in which students are placed when they make their subject choices, it is also important to consider the structural context of the education system. We are increasingly able to draw upon rich, complex, education-related administrative data to achieve this, but we must consider how we can analyse these data in a manner that preserves complex structures and provides new and useful insights. By meeting this goal, we can increase our understand of what participation in STEM looks like.
As detailed by \citet{hipkins2005staying}, there are many ways in which STEM participation can be reported on. At a broad level, we can summarise the number of students enrolled in each subject (e.g., how many students of each demographic group study biology?). We can also explore patterns at finer-grained levels by summarising participation per high school (e.g., which high schools have higher proportions of students studying science?), or by reporting participation at the level of assessment (e.g., how many students took a specific biology exam?). While it is relatively easy to summarise and interpret participation at broad levels, untangling and understanding patterns of subject participation at fine-grained levels can be a difficult task. This task is especially difficult in the context of Aotearoa New Zealand, which operates a particularly complex high school assessment system.
The goal of the current study is to develop and employ a novel method of reporting on student participation in STEM by looking specifically at students' co-enrolments at the level of assessment. We begin by summarising the insights that can be gained by exploring STEM participation at a broad level. We then provide a brief summary of the National Certificate of Educational Achievement (NCEA), Aotearoa New Zealand's internationally unique high school qualification. We then move on to demonstrate how the quantitative technique of network analysis can be employed to reveal structures in NCEA participation. Finally, we discuss the novel insights provided by network analysis of STEM co-enrolments in NCEA assessments spanning the previous decade. \\
\subsection{Broad Understandings of Student Participation in STEM}
Student participation in STEM is often reported at a broad level, with information detailing the counts of students who are enrolled in each subject, and how this differs across demographic groups. In Aotearoa New Zealand, data is readily available by sex (male or female) and Socio-economic status (SES) from 2004 to 2018 \citep{EducationCounts_2018}. Exploring these data can provide a surface level description of what the field of STEM education looks like in Aotearoa New Zealand.
As shown in Figure \ref{fig:STEM_Participation_Sex}, in Year 13 (final year of high school), female students in Aotearoa New Zealand are less likely to take physics, with this under-representation being steady across years. The same figure shows that female students are more likely to take biology, and more recently chemistry, with this over-representation becoming increasingly more pronounced over time.
Figure \ref{fig:STEM_Participation_Sex_Maths} shows that female students continue to be underrepresented in mathematics subjects, such as accounting and calculus. However, female students have higher levels of representation in statistics than male students in recent years \citep{EducationCounts_2018}. The same data shows that, in technology subjects, the computer and engineering subjects have continually been male-dominated, with this becoming more pronounced over time\citep{EducationCounts_2018}. Food technology and textiles are the only female dominated technology domains.
Data from \cite{EducationCounts_2018} also allows us to see trends in STEM participation by school decile, a proxy measure of SES. In Aotearoa New Zealand, school decile refers to the affluence of the neighborhood in which a school is located. High decile schools are located in more affluent areas, whilst low decile schools are located in less affluent areas. As shown in Figure \ref{fig:STEM_Participation_Decile}, students who attended higher decile schools had greater rates of participation in science subjects, and this pattern was also evident for calculus and statistics. The relationship between student enrolment in technology learning domains and decile has no discernible pattern.
Broad level data, such as those discussed above, allow us to interpret trends in subject enrolments over time. However, they provide only a surface level understanding of STEM participation. Beneath the aggregation of counts per subject label hides important information that is useful for policy makers and researchers. Each subject consists of many different assessments, each covering unique content and following different assessment criteria. The following section will provide a brief introduction to Aotearoa New Zealand's main high school assessment system, the National Certificate of Educational Achievement (NCEA).\\
\subsection{A Brief Introduction to the National Certificate of Educational Achievement}
The National Certificate of Educational Achievement (NCEA) is the main form of secondary school assessment in Aotearoa New Zealand. First introduced to students in 2002, NCEA was designed to replace norm-referenced assessment. In norm-referenced assessments student achievement is judged against the average achievement of the student population \citep{Mahoney2005}. Instead, achievement in NCEA is based on the competencies of individual students \citep{hipkins2016ncea}, meaning that achievement is an indicator of what a student \textit{knows}, and not just how they rank amongst their peers. Therefore, it is possible for all students to pass if they all meet the assessment criteria. Assessment operates at the level of specific skills, or \textit{standards}, that comprise a subject discipline. For example, instead of just receiving an overall grade for biology, students take several standards in the subject discipline of biology that demonstrate their competence in particular areas (e.g., ``\textit{Demonstrate understanding of biological ideas relating to micro-organisms}''). By successfully completing standards, students accumulate credits, the value of which is linked to the amount of work needed to fulfil a standard. The three levels of the NCEA typically correspond to the final three years of high school. NCEA Level 1 is typically taken in Year 11 (age $\sim$ 15), NCEA Level 2 in Year 12, and NCEA Level 3 in Year 13.
What makes the NCEA a unique assessment system is its flexibility. Compared to the systems it replaced (School Certificate, Sixth Form Certificate, and Bursary), there is more variety in the assessments/standards that students may be enrolled in \citep{Mahoney2005}. In providing increased choice to students and their educators, and more flexible pathways through high school, it was hoped that the NCEA would benefit students from a range of backgrounds. As stated in the New Zealand curriculum \citep[p.41]{NZCurriculum2007}: \begin{quote}
Schools recognise and provide for the diverse abilities and aspirations of their senior students in ways that enable them to appreciate and keep open a range of options for future study and work. Students can specialise within learning areas or take courses across or outside learning areas, depending on the choices that their schools are able to offer.
\end{quote}
The NCEA meets these goals by providing students with more learning pathways through high school, which aims to serve both students who wish to progress to tertiary study, and those who want to enter into vocational careers. These two pathways are reflected in the two main types of assessment offered: unit and achievement standards.\\
\subsubsection{Unit and Achievement Standards}
Unit standards tend to assess more vocational subjects (e.g., plumbing, hairdressing, agriculture). Unit standards have strict criteria that need to be achieved in order to pass \citep{hipkins2016ncea}, and are thus suited to assessing skills that follow a procedure. If a student meets the criteria they pass; if they fail a step, they fail the standard. All unit standards are assessed internally by the institution where the student is placed, offering the opportunity to teach and learn in a manner that caters more to students' contexts. Internal assessments are moderated by the NZQA, according to the New Zealand Qualification Framework\citep{NZQF}, to ensure the assessment is consistent and rigorous. That being said, schools often provide the opportunity for students to retake failed internal assessments at a later time.
Achievement standards assess more traditional subjects that are tied to the New Zealand curriculum, such as science, mathematics, and English. While many achievement standards are assessed internally, a number of them are taken under standardised conditions and assessed by an external body (i.e., the NZQA). Unlike unit standards, where students can only be judged to have passed or failed, achievement standards often have assessment criteria that can be interpreted more subjectively and require a different grading structure \citep{hipkins2016ncea}. Instead of pass or fail, achievement standards have four outcomes: not achieved, achieved, merit, and excellence. This grading structure seeks to reward students who demonstrate knowledge at a higher level than simply showing competence. The introduction of different grading levels in achievement standards provides increased opportunity to rank students by performance \cite{shulruf2010new}, a process that NCEA was not initially designed to accommodate \citep{hipkins2016ncea}.
The relevance of achievement and unit standards can be tied to students' future aspirations in the context of STEM. \citet[p.20]{wong2016science} differentiates these aspirations as being tied to either careers \textit{in} science, or careers \textit{from} science. Careers \textit{in} science may be defined as: ``...occupations that are involved with the research or discovery of science as their primary purpose'' \citep[p.20]{wong2016science}. Achievement standards may be more closely linked to these types of careers as they provide the means to assess theoretical work, and provide the pathway to university. Careers \textit{from} science may be defined as ``careers that are related to science'' but prioritise other aspects of STEM \cite[p.20]{wong2016science}. This includes careers in technology, and also careers in horticulture and farming that are even more applied. The vocational slant of unit standards may prepare students better for these types of careers \textit{from} science. With that being said, students can take a combination of unit and achievement standards.
While there are many potential pathways through the NCEA, the eventual goal for students is to accumulate enough credits to achieve NCEA Level 3. Students who wish to attend university must meet a separate goal over and above the requirements for NCEA Level 3. To be eligible to enrol at a university, students must attain University Entrance (UE), which is the equivalent of achieving NCEA Level 3 with a specified number of credits coming from three subjects on an approved subjects list (these include subjects such as biology, physics, mathematics, and English) with specific achievement standards \citep{NZQA_Approved}, and a higher standard of literacy than regular NCEA Level 3 \citep{hipkins2016ncea}. Specific university programs may also have their own requirements for enrolment. For example, to transition from NCEA to engineering at the University of Auckland, students must attain specific externally assessed achievement standards in Level 3 calculus and physics \citep{UoA_Engineering}. Alternate pathways to university STEM study are possible, such as completion of university foundation courses, but these take additional time. The decisions that students make regarding the selection of STEM standards in NCEA Level 3 can thus have long-lasting implications. It is therefore especially important to understand how NCEA Level 3 is structured, and how this relates to student outcomes.
Given the complexity of the NCEA, exploring participation in STEM at the level of individual assessments can provide additional insights that complement our broad level understandings of STEM participation discussed previously. Doing so allows us to explore factors related to individual standards (such as the type of standard assessment and whether it was assessed internally or externally) as well as co-enrolment patterns and pathways through assessments. To build on the broad level understandings outlined above, we now adopt the following research questions:
\begin{itemize}
\item Can we identify patterns in the NCEA Level 3 standards taken by students in STEM?
\item If so, how do the patterns of NCEA Level 3 standard enrolments differ across demographic characteristics, SES, and time?
\end{itemize}
Given that the NCEA can be considered ``one of the most complicated education system in the world'' \citep{hipkins2016ncea}, unpacking details at a more fine-grained level can be a daunting task. To explore this complicated system and answer our research questions, we employ quantitative techniques based in the field of network analysis. We explain how network analysis can be used as a tool to understand patterns of assessment, especially in contexts where the system is complex (as with the NCEA). The following sections will discuss how network analysis can help us explore what participation looks like for students studying STEM. \\
\section{A Network Methodology}
\subsection{Data}
We make use of Statistics NZ's Integrated Data Infrastructure (IDI) to access administrative data pertaining to students' high school and census information \citep{IDI}. The IDI is a collection of government data sets, containing micro-data on student enrolment and demographics, linked at the level of individuals for the population of Aotearoa New Zealand. We focus on students taking NCEA Level 3 from 2010 to 2016, as this is the most up to date data available at the time of writing. We focus on NCEA Level 3 as this level is the most highly specialised, and precedes entrance to university and employment. Years prior to 2010 are available, but were omitted due to processing constraints. The years spanning 2010 to 2016 were also of specific interest, due to education policy reforms introduced around 2012 and 2013 \citep{hipkins2016ncea}.
We apply several rules when selecting student cohorts to be included, in order to minimise the risk of adding statistical noise to our analysis. In order to focus our analysis on students who have had the majority of their education in Aotearoa New Zealand, we only select individuals who are identified as having tax, birth or visa records present in the IDI. We also only include students who had NCEA records when they were 15 or 16 and during NCEA Level 1. These filters help focus our sample on the resident population of Aotearoa New Zealand, and minimise the chances of including visitors or foreign exchange students. We also limit our sample to students who attended state schools in Aotearoa New Zealand. This is because private schools in Aotearoa New Zealand are more likely to offer a combination of the NCEA and other formal qualifications (such as Cambridge or International Baccalaureate), introducing additional layers of complexity. For the purposes of our analysis we also assign each student a single cohort year based on the most frequent year in which they took standards. This is because students are able to take NCEA Level 3 standards over multiple years. For example, if a student took two NCEA Level 3 standards during 2015, and ten NCEA Level 3 standards during 2016, we would assign the student to the 2016 cohort. We choose not to exclude Level 3 standards taken in a different year from the overall cohort year, as these standards would still contribute to the student's qualification.
We include the following variables in our analysis:
\begin{itemize}
\item Students' sex (male or female). Due to limitations in the administrative data used, we are not able to include gender (and non-binary classifications of gender) in our analysis.
\item Students' ethnicity. Each student is able to identify with multiple ethnic groups, following the classification set out by Stats NZ \citep{StatsEthnicity}. The main ethnic groups include European; M\={a}ori; Pacific Island; Asian; Middle Eastern, Latin American, or African (MELAA); and Other. For the purposes of this study, we do not report results for MELAA and Other populations as they include a broad cross section of individuals, but typically involve relatively small numbers.
\item High school decile. This is a rating out of 10 for the affluence of the area where the school is located. For the purposes of the following analysis, we categorise high school decile into 3 groups. Deciles 1-3 are low decile, deciles 4-7 are medium decile, and deciles 8-10 are high decile.
\item NCEA Level 3 standards taken. For each student, we have records of all of the standards taken at NCEA Level 3. We only include standards from the New Zealand curriculum learning areas of Science, Technology and Mathematics \citep{NZCurriculum2007}. For each standard, we have information on its subject area (e.g., physics, biology, mathematics etc.), whether it was a unit or achievement standard, and whether it was assessed internally or externally.\\
\end{itemize}
\subsection{Network Analysis}
We employ network analysis to understand STEM enrolment at NCEA Level 3 at a fine-grained level. At its fundamental level, a network is a collection of nodes and edges. Nodes can represent an agent (e.g., a student) or an object (e.g., a standard); edges link two nodes together to indicate some form of relationship. Networks can be used to represent anything from human relationships and transport networks, to biological and computer systems \citep{barabasi2003linked}. In education research, network analysis has tended to focus on the relationships shared between students in the classroom \citep{tranmer2014multiple}, or communication between staff at educational institutions \citep{daly2010social}. There are few examples of education research that use network analysis to investigate non-social relationships. We seek to expand this area of research by applying network analysis to high school assessment enrolment data. As we will outline in the following section, network analysis can help us identify patterns in NCEA standard co-enrolments.
In our analysis, nodes take the form of students and standards. Edges in our network represent any recorded instance where a student was enrolled in a NCEA Level 3 STEM standard during high school. This creates a bipartite network (also commonly referred to as a two-mode network). A bipartite network is any network where there are two types of node, and nodes can only connect to a node of a different type. In our case, a standard cannot be connected directly to another standard, and a student cannot be connected to another student. For example, in A in Figure \ref{fig:BipariteNetwork}, standards may be represented by nodes in set \textit{U}, and students may be represented by nodes in set \textit{V}.
We create a network of students and the standards they were enrolled in for the whole of our student population. We structure this network so that it is multidimensional. Each student node belongs to a specific year, region, and decile, while standards can exist across multiple years, regions, and school deciles. In order to analyse the properties of our network, we are required to `project' onto one set of nodes. This means that we take the node set belonging to a single node type, and generate edges between these nodes when they are linked to a common node of the other node set. For example, B and C in Figure \ref{fig:BipariteNetwork} shows the projection of the network in Figure \ref{fig:BipariteNetwork}. In the projections, standards represented in set \textit{U} are now connected to one another (B), and students in set \textit{V} are also now connected (C).
As we are interested in the patterns of standards that students took, we project onto the standard nodes (Figure \ref{fig:BipariteNetwork}:B). This results in a network of standards that are connected by edges indicating that students took those two standards together within their NCEA Level qualification. The edges of the projected standard network can also take on a weighting that corresponds to the frequency that two standards were taken together by students.\\
\subsection{Normalization and Community Detection}
Our goal is to use the co-enrolment network to understand the standards that tend to be taken together, and by which students. To do this, we employ community detection. Community detection is a process in which we identify sets of nodes that are clustered together by the edges in the network. Previous research by \citet{ferral2005clustering} has employed similar clustering techniques to investigate communities of subjects that tend to be taken together in the NCEA, but this was limited by the number of high schools sampled, the response rate of schools, and the availability of demographic and standard information. Usually, community detection methods identify communities by maximising the \textit{modularity} score within communities. Modularity refers to the tendency of nodes to connect to other nodes within the same community relative to nodes that are outside the community. While there are many different community detection algorithms, the current study makes use of the infomap algorithm \citep{rosvall2009map}.
In order for our communities to more truly represent the standards that tend to be taken together, we need to normalize our edges so that weights do not refer to the raw counts of students' co-enrolments. The raw weighting does not consider the fact that standards have different populations of students. As a result, community detection may group two standards together simply because one standard has a large number of students. To explain more clearly, we can use the hypothetical case of English Standard A, Physics Standard A and Physics Standard B. If English Standard A has a population of 1000 students, and 10\% of those students take Physics Standard B, the raw weight is 100 students. If 100 students took Physics Standard A, and 90\% of those students took Physics Standard B, the raw weight is 90 students. While we would expect the two physics standards to be grouped together, using raw counts of students as edge weights may not result in grouping that meet these expectations. Instead, we make use of a normalization technique called Revealed Comparative Preference (RCP) to provide more consistent communities. RCP measures the fraction of students from standard $j$ who also took a second standard $i$, relative to the overall fraction of students taking standard $i$, across all other standards. More specifically:
$$RCP(i,j) = \frac{x_{ij}/\sum\limits_{j}x_{ij}}{\sum\limits_{i}x_{ij}/\sum\limits_{ij}x_{ij}}$$
where $x_{i,j}$ is the number of students taking both standard $i$ and $j$, $x_j$ ( or $x_i$) is the total number of students taking standard $j$ (respectively, standard $i$), and $x$ is the total number of unique students enrolled in any standard. This RCP metric is based on the measure Revealed Comparative Advantage, used in economics \citep{Balassa1965}, and was calculated using the EconGeog package \citep{balland2017economic} in R \citep{team2013r}. The RCP calculation returns a value where anything greater than 1 indicates a `preference' for two standards being taken together. A value below 1 indicates that, given the number of students in either standard, there was no preference for the two standards being taken together.
We remove any edge in the network where the RCP value is below 1, and subsequently any node that no longer has any edges (isolated nodes with a degree of 0). This results in a network that consists of standards connected by edges with a weighting relative to the preference for each standard being taken together with its neighbors in NCEA Level 3. We then identify communities of standards that are grouped together in our network using the infomap community detection algorithm \citep{rosvall2009map}. In simple terms, the infomap algorithm partitions the network in a way that maximizes the number of edges within a community, relative to the edges between communities. \\
\subsection{Exploring Participation}
To compare student participation across the educational fields detected, we can consider the relative proportion of students from particular years, and across school deciles and social groups. One of our goals is to establish an idea of how each network is structured. Are the enrolments for a specified demographic group spread more evenly across a network, or are they focused in particular areas? To answer this question, we employ the metric of entropy. Entropy is a concept originating from the field of thermodynamics, and provides an indication of how organised or disorganised a system is. In the case of the current study, we use entropy to assess how participation is spread across the network. Using the measures of entropy as signals of disparities, we then explore the rates of participation across communities and standards in finer detail. The following section will outline our measure of entropy.\\
\subsubsection{Entropy}
Entropy provides an aggregated metric of how ordered a system is. Systems that are highly ordered have a lower level of entropy, while disordered systems have higher entropy. To use an analogy, a crystalline solid with atoms focused together on a regular grid has low entropy, while a gas with atoms randomly spread across a grid has higher entropy. Following this analogy, we may explain low entropy as an indication that a pattern of standard enrolments is more focused or specialised in specific areas. In contrast, high entropy in the network of standards indicates that a patterns of enrolment is more diverse. By partitioning our network into different social groups (e.g., across sex, ethnicity, and school decile) we can explore similarities and differences in network structures.
We calculate entropy in two steps. Firstly, we work out the probability of a sub-population enrolling in a specific standard given the total number of enrolments in the network for that sub-population. This probability is given by:
$$p^q_i = \frac{\sum\limits_j x_{ij}^q}{{\sum\limits_{ij} x_{ij}^q}}$$
where $x^q_i$ is the number of students in a sub-population~$q$ enrolled in standard~$~i$, and $x^q$ is the total number of enrolments for that sub-population. Using this measure of probability, we calculate entropy using the following formula:
$$S^q = -\sum_{i}^{N}{
\frac{p^q_i\log{p^q_i}}{\log{x^q}}
}$$
Where $p^q_i$ is the probability of a student from sub-population~$q$ enrolling in standard~$i$, given the overall total number of enrolments for that sub-population($x^q$). The resulting score $S^q$ provides an single positive value that indicates the entropy in the network, where a lower value indicates lower entropy (i.e., ordered patterns of enrolments), and a higher value indicates higher entropy ( i.e., more disordered enrolments). While we choose to normalise by the total number of enrolments for a sub-population, alternative methods of normalisation are also possible. For example we could normalise by the number of standards in the network. This would be equivalent to assuming that the probability of a student enrolling in a specific standard is independent of the standard. This assumption does not hold for two reasons. Firstly, standards have very different numbers of student enrolments. Secondly, different student groups are differently represented in different standards. Normalising by $x^q$ accounts for the number of students from a sub-population enrolled in a specific standard. However, it does not let us distinguish between effects due to the size/popularity of a standard and those due to differing preferences of specific populations for specific standards. A downfall of this approach is that it does not account for standards that have no enrolments for students in a specific sub-population.
We ascertain a level of confidence by using a bootstrapping method, where we vary the count $x^q_i$ in each standard $i$ by a uniform random amount of up to $\pm20\%$, and recalculate entropy. We repeat this process 1000 times for each entropy measure. \\
\subsubsection{Trends}
Following the entropy measure, we investigate how participation differs across demographic groups per standard by comparing raw counts, proportions, and probabilities. The communities identified provide a good indication of the standards that tend to be taken together, which allows us to explore rates of participation across groups of standards as well as individual standards. We are able to explore a range of attributes, such as such as the probabilities of sub-populations enrolling in a standard, with respect to sex, ethnicity, school decile, and type of standard (achievement/unit standards, internally/eternally assessed).
Following the identification of different communities of standards, our goal is to explore the student enrolment patterns in these communities. Based on trends outlined previously by \citep{hipkins2016ncea} and based on data from \citet{EducationCounts_2018}, we make the following hypotheses:
\begin{itemize}
\item Female students will be more likely to have enrolled in standards in communities related to biology.
\item Male students will be more likely to have enrolled in standards in communities related to physics, calculus and computer science.
\item Students who attended high decile schools will be more likely to have enrolled in externally assessed standards.
\end{itemize}
Less research has investigated the relationship between assessment type (achievement or unit) and STEM enrolment, but we may expect that students groups who historically succeed in traditional forms of education (high SES, European and Asian students) to be more likely to have enrolled in externally assessed achievement standards. Student groups who have historically been under-served by traditional assessment may be more likely to have enrolled in unit standards. \\
\section{Results and Discussion}
The complete co-enrolment network across all years, regions, and deciles is shown in Figure \ref{fig:NetworkAll}).
Across all years the infomap algorithm identified 42 communities of Level 3 STEM standards. As NCEA Level 3 is the most specialised stage of high school education, we would expect our network to be strongly partitioned into different community structures. This is reflected in a high modularity score of 0.83. The modularity score indicates that the nodes tend to share more edges with nodes within the same community than with nodes in different communities. The structure of the network changed over the period of time considered in the analysis, with a significant change taking place between 2012 and 2013. During this time, a change in education policy resulted in a reform in assessment. Science and mathematics linked unit standards were phased out, and a new set of achievement standards were introduced. Post the education reform in 2013, the overall number of standards diminished, and the network is mainly dominated by one community of mathematics and science standards (see Figure \ref{fig:NetworkYear}). This policy change is also reflected in changing levels of entropy in the network over time. As shown in Figure \ref{fig:Entropy_Gender}, the overall entropy of the network of assessments (taking all students into account) decreased over time. This gives an indication that, following the reforms to standards in 2013, student enrolments were more standardised and focused, and less flexible.
Through the use of network analysis we are able to delineate the main fields of study that comprise NCEA Level 3 STEM. Our method of using RCP and community detection separates out standards according to their propensity for being taken together, rather than simply classifying by subject label. The resulting network is partitioned according to two main pathways, communities of standards reflecting progression to university study (i.e., mainly achievement standards), and communities of standards orientated towards vocations (unit standards, and internally assessed standards). The detected communities thus provide a clearer picture of NCEA enrolment than broader subject labels. To provide an example, the chemistry standard ``\textit{Evaluate the interaction of a chemical process with society and/or the environment}'' may not assess the same content knowledge as another chemistry standard ``\textit{Demonstrate understanding of the properties of organic compounds}''. Despite both standards belonging to the chemistry domain, the community detection algorithm assigned them to different communities in the network. While standards assessing applications of science to other vocations or to societal issues may help in the pathway to careers \textit{from} science, standards assessing scientific theory are more representative of the pathway to university and careers \textit{in} science.
On the whole, the communities in the network tended to be comprised of standards from the same domain of study. For example, biology standards tend to be taken in conjunction with other biology standards, physics with physics, and so on. However, the community detection algorithm mainly grouped science and mathematics subjects in the two large communities. These two communities, which occurred at different time periods (one before 2013, and one after) can be viewed as the pathway to university study. They consist mainly of achievement standards (many of them externally assessed, especially after 2013), and include physics, biology, chemistry, and calculus.
The following sections will outline some patterns that can be observed from 2010 to 2016 by sex, ethnicity and school decile. While there are a vast number of patterns to be explored and discussed further, we focus our discussion on the main patterns. We provide the reader with full access to an interactive web application that can be used to explore the network in depth (\url{https://stur600.shinyapps.io/ExploreNCEA_L3_STEM/}). This application allows the user to filter the network by subject disciplines, types of standard, as well as school decile and demographic criteria. Through the patterns that we highlight, we seek to demonstrate the additional insights that can be gained through investigating the NCEA at a finer-grained level, and how they can further inform our understanding of what STEM participation looks like in Aotearoa New Zealand. We begin our discussion by focusing on the patterns that were seen based on students' sex, and then move on to discuss patterns by students' ethnicity and school decile.\\
\subsection*{Patterns by Sex}
Overall, there were small differences in the entropy in the network by sex, with entropy being slightly higher for the male sub-population (see Figure \ref{fig:Entropy_Gender}). This finding suggests that the male sub-population of the network had more enrolments spread across the network, while the female sub-population were more focused in specific areas. Further investigation of communities in the network showed clear examples of disparities in subject enrolments by sex which may explain the difference in entropy. Male students tended to dominate communities defined by standards in the agriculture, engineering, and practical technology (welding, furniture making etc.) domains, while female students had greater rates of enrolments in standards related to life sciences and textiles.
Corroborating the broad level trends outlined previously in the current study, and the trends detailed across other international contexts \citep{Else_Quest_2013, Sheldrake_2015, NSF, InstituteofPhysics_2013}, female students were more likely to enrol in biology standards, and less likely to enrol in physics and calculus standards. The majority of biology standards had around 60-70\% female students across years, while female students were also more likely to have enrolled in standards in the \textit{Core Science} domain. This domain includes standards such as \textit{Research a current scientific controversy} (61.5\% female) and \textit{Describe genetic processes} (67.3\% female). Female students were less likely to be represented in calculus and physics standards than male students. Investigating these disparities at the standard-level provides additional insights (see Table \ref{table:StdattainmentGender}).
\begin{table}[htbp]
\begin{tabular}{|c|c|c|c|}
\hline
Standard & Assessment Type & Domain & Female (\%) \\ \hline
\parbox[c]{50mm}{\textit{Differentiate functions and use derivatives to solve problems}} & EX & Calculus & 38.2\\
& & & \\
\parbox[c]{50mm}{\textit{Integrate functions and use integrals to solve problems}} & EX & Calculus & 38.3\\
& & & \\
\parbox[c]{50mm}{\textit{Differentiate functions and use differentiation to solve problems}} & IN (Unit) & Calculus & 42.3 \\
& & & \\
\parbox[c]{50mm}{\textit{Integrate functions and use integration to solve problems}} & IN (Unit) & Calculus & 46.3\\
& & & \\
\parbox[c]{50mm}{\textit{Demonstrate understanding of wave systems}} & EX & Physics & 35.2\\
& & & \\
\parbox[c]{50mm}{\textit{Demonstrate understanding of electrical systems}} & EX & Physics & 34.7 \\
& & & \\
\parbox[c]{50mm}{\textit{Demonstrate understanding of mechanical systems}} & EX & Physics & 36.2
\\ \hline
\end{tabular} \caption{\textbf{Calculus and Physics Standard Enrolments by Sex.} Female students were underrepresented in calculus and physics standards in general, but especially in the external achievement standards that were part of the pathway to university science. The representation of female students in calculus unit standards was closer to even.} \label{table:StdattainmentGender}
\end{table}
The rates of enrolment for female students in the physics standards were low, with the proportion of female students in externally assessed physics standards being around 35\% overall. The participation of female students in the standards related to calculus were also low compared to male students, with the proportion of female students being around 35-38\% in the main externally assessed standards. Interestingly, the internally assessed unit standard equivalents of the calculus standards, which were available to students prior to 2013, had an increased proportion of female students (around 42-46\%). Much research has been dedicated to understanding why disparities persist in physics and calculus by sex, with research often suggesting that female students tend to be less confident in mathematics and calculus compared to male students \citep{Hofer_2016, Heilbronner_2012, Simon_2015}. It may be that the calculus unit standards, which are assessed internally, in a familiar space with the opportunity to resit, offers a safer assessment environment where female students are more comfortable (see \citet{cheryan2017some} for a comprehensive review of the issues impacting on gender differences in STEM choice).\\
\subsection*{Patterns by Ethnicity and School Decile}
We report the results for ethnicity and school decile together, given that they are inextricably linked; M\={a}ori and Pacific students are over-represented in low decile schools. With the exception of students attending low decile schools, the M\={a}ori and Pacific sub-populations had higher entropy, while Asian and European students had lower entropy (see Figure \ref{fig:Entropy_Decile}).
As can be seen in Figure \ref{fig:NetworkEthnicity} the higher entropy for the M\={a}ori and Pacific sub-populations can be attributed to the fewer enrolments for M\={a}ori and Pacific students in science and mathematics (especially in the communities reflecting the pathways to university science), and the relatively increased enrolments in vocationally orientated standards. Asian and European students had more enrolments focused in science and mathematics, and were also over-represented in externally assessed standards, which are fewer in number than internally assessed standards.
Although enrolments for the European and Asian sub-populations were focused in the communities of science and mathematics standards needed for university, the Asian sub-population had slightly higher entropy across years. The Asian sub-population had more enrolments in accounting standards, as well as mathematics and science standards overall, while the European sub-population had a narrower range of standards and relatively fewer enrolments in internally assessed science and mathematics standards. The higher entropy for Asian sub-population may relate to the categorical grouping of ``Asian'', which contains an extremely diverse population of students. This categorisation ranges from Pakistan and Bangladesh to China, and also some Pacific Island nations (e.g., Fijian Indians). The diversity of the population, including the cultures and social backgrounds, may have been reflected in an increased diversity of enrolments.
Figure \ref{fig:Entropy_Decile} shows that entropy is more similar across ethnic groups for low decile schools, and that overall, the baseline entropy (indicated by the black line) for low decile schools is greater than the baseline entropy for higher decile schools. The higher entropy for low decile schools may be a consequence of increased enrolments in internally assessed standards, and fewer enrolments in standardised, externally assessed standards. Previous research has commented on this pattern \citep{hipkins2016ncea}. \citet{wilson2017subject} observed that lower decile schools were less likely to have students enrolled in Subject Literacy Achievement Standards, which are achievement standards that can be used as indicators of subject-specific literacy. After exploring the rates of enrolment, we also confirm that lower decile schools are less likely to have students enrolled in key externally assessed science and mathematics standards.
While each ethnic group sub-population tends to have a similar entropy in low decile schools, a wider gap is present for middle and higher decile schools. For these schools, entropy is lower for Asian and European sub-populations, and higher for M\={a}ori and Pacific sub-populations. This suggests that the enrolment in STEM standards for M\={a}ori and Pacific students in middle and high deciles schools tends to be more diverse and less standardised. The lower entropy for Asian and European students is likely related to the focused participation in science and mathematics achievement standards required for university entrance, while enrolment for M\={a}ori and Pacific sub-populations is less focused in these standards and more balanced across other domains (including communities of unit standards). Table \ref{table:StdattainmentEthnicityDecile} shows the rates of enrolment in some key externally assessed science and calculus standards for each ethnic group sub-population, split by school decile and comparing 2010 to 2013.
\begin{landscape}
\begin{table}[htbp]
\begin{tabular}{lll|l|l|l|l|l|l|l|l|}
\cline{4-11}
&&& \multicolumn{2}{l|}{\% of Asian} & \multicolumn{2}{l|}{\% of Euro} & \multicolumn{2}{l|}{\% of M\={a}ori} & \multicolumn{2}{l|}{\% of Pacific} \\ \hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \centering Standard}} & \multicolumn{1}{l|}{\centering Domain} & \centering Year & Low & High & Low & High & \multicolumn{1}{l|}{Low} & High & Low & High \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Differentiate functions and use\\ derivatives to solve problems}}} & \multicolumn{1}{l|}{Calculus} & 2010 & 36.2 & 52.2 & 17.7 & 23.1 & \multicolumn{1}{|l|}{11.3} & 15.9 & 13.4 & 15.2 \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Apply differentiation methods in solving\\ problems}}} & \multicolumn{1}{l|}{Calculus} & 2016 & 35.4 & 57.3 & 19.0 & 25.7 & \multicolumn{1}{l|}{9.9} & 16.3 & 13.6 & 18.3 \\ \hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Describe processes and patterns of\\ evolution}}} & \multicolumn{1}{l|}{Biology} & 2010 & 19.6 & 32.8 & 18.8 & 29.0 & \multicolumn{1}{l|}{10.9} & 23.3 & 7.4 & 17.0 \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Demonstrate understanding of evolutionary processes leading to speciation}}} & \multicolumn{1}{l|}{Biology} & 2016 & 27.2 & 33.7 & 24.0 & 31.8 & \multicolumn{1}{l|}{14.6} & 27.3 & 13.6 & 27.6 \\ \hline
\multicolumn{1}{|l|}{\parbox[c]{70mm}{\strut \textit{Describe aspects of organic chemistry}}} & \multicolumn{1}{l|}{Chemistry} & 2010 & 25.0 & 40.0 & 14.8 & 24.7 & \multicolumn{1}{l|}{7.4} &16.0 & 10.0 & 12.1 \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Demonstrate understanding of the\\ properties of organic compounds}}} & \multicolumn{1}{l|}{Chemistry} & 2016 & 32.0 &42.6 &19.9 &27.9 & \multicolumn{1}{l|}{13.0} & 19.6 &14.2 & 19.6 \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Demonstrate understanding of mechanical systems}}} & \multicolumn{1}{l|}{Physics} & 2010 & 25.4&38.0 &14.7 &23.9 & \multicolumn{1}{l|}{7.0} &16.7 & 8.4&11.0 \\
\hline
\multicolumn{1}{|l|}{\parbox[c]{75mm}{\strut \textit{Demonstrate understanding of mechanical systems}}} & \multicolumn{1}{l|}{Physics} & 2016 & 30.3&46.4 & 17.4& 27.3& \multicolumn{1}{l|}{8.5} &15.8 &9.6 &17.0 \\ \hline
\end{tabular}
\caption{\textbf{Key Standard Enrolments by Ethnicity and Decile}. The percentages of students enrolled in key externally assessed achievement standards in STEM by ethnic group and school decile (low/high). The percentage indicates the number of students from that ethnic group in a particular year who enrolled in the standard, as a fraction of the total number of students from that ethnic group in a particular year who took a STEM standard. For example, of the Asian students attending a low decile school in 2010 who took a STEM standard, 36.2\% took the calculus standard \textit{Differentiate functions and use derivatives to solve problems}. These percentages show that rates of enrolment differed across ethnic groups, with varied differences within these groups by school decile, and also comparing standards offered in 2010 and 2016.} \label{table:StdattainmentEthnicityDecile}
\end{table}
\end{landscape}
While Table \ref{table:StdattainmentEthnicityDecile} shows that Asian and European had higher rates of participation in key externally assessed science and calculus standards, the differences between low and high school deciles appears to be considerable for the Asian sub-population compared to other groups. For example, the difference in participation for Asian students by decile in the calculus standard on differentiation offered in 2016 is 22\%, compared to 6.7\% for European, 6.4\% for M\={a}ori, and 4.7\% for Pacific Islands. This may once again point to the diversity of the categorisation of ``Asian'', but importantly highlights the importance of considering ethnic group categorisations in tandem with SES.
The fact that low decile and M\={a}ori and Pacific sub-populations had fewer enrolments in key science and mathematics standards provides evidence that the pathway to university science is dominated by higher decile schools, and especially Asian and European students at these schools. In contrast, students from lower decile schools, and also M\={a}ori and Pacific students in higher decile schools, had relatively more enrolments in a larger and more disparate pool of internally assessed unit standards. Unit standards provide a valuable type of assessment that prepares students for vocational careers, and it may be that a higher proportion of students from less affluent areas seek vocational careers after high school. However, this does not necessarily explain why M\={a}ori and Pacific sub-populations attending higher decile schools are less likely to be channelled into science and mathematics standards.
The differing patterns of enrolment for M\={a}ori and Pacific Island sub-populations and Asian and Pak\={e}ha is complex, but may be explained in several ways. Firstly, higher decile schools primarily serving M\={a}ori and Pacific students may choose to offer more internal assessments that provide increased opportunity to assess in culturally appropriate way (e.g., less competition, more formative feedback). Secondly, and less optimistically, it may be that teachers hold lower expectations for M\={a}ori and Pacific students \citep{turner2015teacher}, and are less likely to place them in the pathway towards university science. This idea was reflected by a participant in a study by \citep{graham2010Maori}: \blockquote{The teachers decide where the class is at in terms of choosing which standards [Unit versus Achievement]. It's a disadvantage on you because
it depends on what the teacher thinks you can do and what the kids in your class can do.}
Our analysis also highlights the impact of a pivotal reform in the NCEA, where curriculum-linked unit standards were phased out, and the system became more standardised and less flexible. Our results suggest that this change did not result in a decrease in participation in science and mathematics for M\={a}ori and Pacific Island sub-populations who were over-represented in the curriculum-linked unit standard in earlier years. Instead, as can be seen in Table \ref{table:StdattainmentEthnicityDecile} enrolment often increased at a greater rate than other ethnic groups, especially in higher decile schools. For example, in high decile schools the Pacific Island sub-population saw an larger increase of around 10.6\% in external biology standard relating to evolution, compared to 0.9\% for Asian, 2.8\% for European, and 4\% for M\={a}ori. Although we cannot comment on how this educational reform impacted on students' outcomes in science and mathematics, the reduced flexibility may actually help students by making NCEA less complex. Previous research has found that the complexity of NCEA can be confusing for students and parents to navigate \citep{graham2010Maori,jensen2010ncea}.\\
\subsection{Implications and Future Directions}
The current study fills a gap in the previous literature by investigating patterns of co-enrolments in NCEA Level 3 STEM standards by students' sex, ethnicity, and a proxy measure of SES. We believe that this study is the first of its kind to use bipartite networks to represent high school assessment data. Through our methodological approach, we are able to take into account a wealth of information related to students and the standards that they enrolled in. This includes demographic information (such as sex and ethnicity) and specific NCEA Level 3 standard information, such as the manner in which standards were assessed (externally or internally), and whether the standard was an achievement standard (traditional curriculum based subjects, such as English or science) or a unit standard (more vocational subjects, such as farming or practical technology).
The NCEA is very complex, but our method of analysis allows us to consider the different pathways that students follow based on the assessments they enrolled in. The communities of standards highlighted through our analysis reflect two main pathways, either towards vocations and careers \textit{from} science, or the pathway towards university and careers \textit{in} science. Despite growing discussion regarding the outcomes of different types of standards in the Aotearoa New Zealand context \citep{hipkins2016ncea,Lipson2017}, there has been a lack of research into how this information relates to student background. The methodology and results outlined in the current study enables us to represent the NCEA as a complex education system, and this can provide detailed insights into what science participation looks like.
A limitation of our analysis is the fact that we do not have access to students' level of achievement in the standards they enrolled in Level 3, or in previous years. As detailed by \citet{jensen2010ncea}, achievement outcomes in standards would be highly influential in shaping the pathways that open up or close off for students as they go through NCEA. Furthermore, the disparities seen in participation in key science standards may be tied to the development of academic identity \citep{bolstad2008seeing} which we are also unable to quantify. \citep[p.216]{Archer2014} argue that \blockquote{`cleverness' [can be viewed] as a racialized, gendered, and classed discourse, such that the identity of the `ideal' or `clever' student is not equally open to all students as a viable and authentic identity.} This notion of `cleverness' may explain the disparities found in the current study. More specifically, it may be that the `clever' pathway through NCEA is not open to all students. As described by \citet{hipkins2016ncea}, NCEA informally developed into a two-tiered system, with curriculum-linked unit standards commonly being viewed an easy pathway, and achievement standards, and especially externally assessed achievement standards, being viewed as a tougher pathway. Students who identify as less academic may purposefully seek easy pathways through NCEA, without fully understanding that doing so can reduce educational opportunities later on \citep{jensen2010ncea}.
Students with a family background of success in education may be more likely to view the academic pathway as normal or even expected. This idea is described in a related study of high school science pathways in the United Kingdom, where \citet{archer2017stratifying} found that students from more affluent backgrounds were more likely to see the science-orientated pathway as an `obvious' choice. Students from less affluent backgrounds may also be more motivated to seek full time employment, rather than pursue a pathway towards university study and the debt it may entail. However, the question remains as to the extent to which student from less affluent backgrounds knowingly choose vocational pathways and are not channeled down this pathway by simply attended a school in a low SES area.\\
\section{Conclusion}
The current study uses network science methods to explore disparities in science participation in Aotearoa New Zealand. It summarises the broad rates of participation by sex and school decile, and also participation at a finer-grained level through a network analysis of STEM standard co-enrolments for the final year of high school. The initial summary of science participation showed that male students have been more likely to take `physical' subjects (e.g., physics, calculus, practical technology), while female students have been more likely to take life science subjects (e.g., biology, health). A network analysis of NCEA enrolment data corroborated these findings, and added additional insights that showed that participation by sex were more equal in calculus unit standards. Our use of network analysis also allowed us to characterise the structure of co-enrolments for different sub-populations. Through the combination of Revealed Comparative Preference (RCP) and community detection, we were to explore the specific pathways that students participate in during high school STEM education, while a metric of entropy provided a description of how ordered or disordered co-enrolments were. This use of entropy to characterise co-enrolment provides a novel approach to understanding student pathways through education, and revealed valuable insights. We found that Asian and European sub-populations were had patterns of enrolments focused in science and mathematics standards reflecting the pathway to university study. In contrast, the M\={a}ori and Pacific Island sub-populations, and lower decile school sub-population in general, had more disorganised patterns of enrolments with fewer enrolments in externally assessed science and mathematics standards. Our findings suggest that while policy changes have impacted on the structure of NCEA enrolments over time, disparities by sex, ethnicity, and school decile continued to be evident. While it is difficult to explain how much of standard enrolment is due to student choice, and how much of it is due to structural inequities present in the school system, our findings reveal disparities in STEM at a fine-grained level. Our findings suggest that the pathway to university science has been dominated by higher decile schools, and especially Asian and European students at these schools. These results provide a detailed picture of what STEM participation looks like in Aotearoa New Zealand.
\section{Disclaimer}
The results in this paper are not official statistics. They have been created for research purposes from the Integrated Data Infrastructure (IDI), managed by Statistics New Zealand. The opinions, findings, recommendations, and conclusions expressed in this paper are those of the author(s), not Statistics NZ. Access to the anonymised data used in this study was provided by Statistics NZ under the security and confidentiality provisions of the Statistics Act 1975. Only people authorised by the Statistics Act 1975 are allowed to see data about a particular person, household, business, or organisation, and the results in this paper have been confidentialised to protect these groups from identification and to keep their data safe. Careful consideration has been given to the privacy, security, and confidentiality issues associated with using administrative and survey data in the IDI. Further detail can be found in the Privacy impact assessment for the Integrated Data Infrastructure available from www.stats.govt.nz.
\section*{Author Contributions}
SMT contributed to the conception, formulation, analysis, and writing of the manuscript. DO'N contributed to the formulation of the manuscript, and provided feedback.
\section*{Funding}
ST was supported by a University of Auckland Doctoral Scholarship (https://www.auckland.ac.nz/en.html). DO'N received funding from Te P\={u}naha Matatini (https://www.tepunahamatatini.ac.nz/) grant number UOA 9167-3705716. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
\section*{Acknowledgments}
We acknowledge the contribution of Adrian Ortiz-Cervantes who aided in the conceptualisation of this research.
\section*{Data Availability Statement}
The datasets generated for this study can be found at \url{https://stur600.shinyapps.io/ExploreNCEA_L3_STEM/} and R code is available on request.
\bibliographystyle{frontiersinSCNS_ENG_HUMS}
|
1,108,101,565,445 | arxiv | \section{Introduction}
Microtubules (MTs) are fascinating biological macromolecules that are
essential for intracellular trafficking, cell division and maintaining the
cell shape. They display unique elastic and dynamic behavior, inherited by
their complex self-assembling nanotube structure. Surprisingly, despite an
enormous accumulation of knowledge about their structure, the MTs static and
dynamic properties have challenged all attempts of a fully coherent
interpretation. In this paper we develop the line of thoughts leading towards
a new theory that provides a more comprehensive understanding of these MT's
properties. The fundamental result is that stabilized MTs spontaneously form
large scale superhelices of micron size pitches and diameters. The MT's
super-helical structure turns out to be a consequence of a cooperative
interaction between its individual subunits that can sustain several stable
curved conformations. Cooperativity of fluctuating internal degrees of freedom
in combination with the cylindrical MT symmetry lead to a helical state with
very unique characteristics in the world of macromolecules: MTs are helices
that are permanently but coherently reshaping -i.e. changing their reference
ground state configuration- by thermal fluctuations. In particular when
clamped by one end MTs undergo an unexpected zero energy motion. As we will
see, this could be the key for a consistent interpretation of certain
challenging experimental results not captured by the conventional scenarios
and models. It is worth remarking, that the large majority of experiments
probing the above mentioned properties are in-vitro experiments on MTs with
stabilizing drugs. We stress this point, as the experimentally prevalent
presence of stabilizing agents is not innocent and could modify MT's
properties. Nevertheless there are reasons to believe that the theory
developed here might be valid also for non treated MTs.
Before embarking on the road to a "polymorphic MT theory", we first review the
conventional understanding of the common MT properties as well as some key
experiments which will be our necessary guides towards the model we propose.
This paper which extends and deepens a previous short presentation of the idea
of polymorphic MTs \cite{PAPER1} is written in two parts. In the first part,
we provide conceptual and graphical explanations of the ideas behind the model
and investigate consequences of the here developed polymorphic MT model. We
hope that this part is self consistent and didactic enough, so that a general
reader can grasp the basic underlying ideas. In the second more technical
part, the mathematical model is developed and quantitative results are
presented. The details and derivations are left to an extensive appendix.
\section{Short review of what we understand about microtubules}
Microtubules are cytoskeletal protein filaments of eukaryotic cells fulfilling
different structural and mechanical functions in the cell: MTs act as
"cellular bones" strongly influencing the cell shape, constitute the main
routes for molecular motor mediated intracellular cargo transport \cite{Genref
MTs, GeneralAmos} and perform other important tasks like stirring the
cytoplasm \cite{MTStirringRod}. Besides, MTs play a central role in the
assembly of the mitotic spindle during cell division and are at the heart of
the functioning of cilia and flagella \cite{Alberts}. This versatility of MTs
in a variety of biological functions mainly relies on their unique high
stiffness and on their dynamics of assembly and disassembly. The high rigidity
of MTs (similar to hard plastic) is due to their structure that is known in
exquisite detail from $3$D electron microscopy reconstructions
\cite{NogalesMTubule,NogalesMT2} : MTs are hollow tubes whose walls are formed
by assembly of a variable number $N$ of parallel protofilaments (PFs). The PFs
themselves are built by head-to-tail self association of the $\alpha\beta
$-tubulin heterodimer protein subunit (yielding a polarity to MTs) whose
structure has been resolved by electron crystallography
\cite{NogalesTubulin,NogalesTubulin2}.
In vivo, MTs most commonly appear with $13$ PFs \cite{Bouchet} (although there
are exceptions depending of the cell type), whereas in vitro a variety of
structures with PF number ranging from $N=9$ to $16$ were observed
\cite{Wade,Chretien}. Transitions of different lattice types (mostly with the
gain or loss of one PF) within a single MT are also frequent
\cite{Ray,ChretienMetoz,ChretienFuller}. The MT lattice can accommodate for
these different structures by twisting the PFs around the central MT axis
although this process is energetically costly. The typical lattice twist
repeat lengths (pitches) are $:+3.4,+25,-6\mu m$ for $N=12$, $13$ and $14$ PF
MTs respectively, with +/- denoting right/lefthanded twist \cite{Wade,
Chretien, Ray, ChretienMetoz, ChretienFuller}. As we will see the lattice
twist is an essential property of the model that we will propose later on. Any
deviation from the most frequent, energetically favorable and thus less
twisted configuration $N=13$ implies an internal prestrain in the MT lattice.
The latter stress can locally deform the end portions of the lattice or even
destabilize it \cite{HunyadiMechanical}.
Another internal prestrain in the MT lattice is believed to be caused by the
tubulin subunit which upon incorporation into the lattice hydrolyzes a bound
GTP\ molecule converting it quickly into a GDP-tubulin form which has the
tendency to form a curved state - curling radially away from the axis
\cite{Mandelkow,NogalesRings}. The constraint imposed by the MT lattice
however maintains the GDP-tubulin dimers in a straight unfavorable state - in
turn trapping mechanical prestrain. In this conventional view MTs are seen as
internally prestrained but intrinsically straight Euler beams. The GDP-tubulin
prestrain is also believed to trigger rapid depolymerization called the
polymerization "catastrophe" \cite{Mitchison}. The stability of MT is
regulated either by the presence of a thin layer of yet unhydrolyzed GTP
tubulin dimers at the growing MT end (the so called GTP-cap-model,
\cite{EricksonCap,JanosiCap}) or by the binding of MT-associated proteins
(MAP) or of drugs such as taxol.
\section{What we don't understand about microtubules}
\subsection{Mechanical properties of stabilized MTs}
To investigate the mechanical properties of the MT lattice the presence of the
polymerization dynamics is often an obstacle. As previously mentioned it can
be switched off by stabilizing agents like taxol or MAPs. In the bulk of the
available in vitro studies taxol stabilization has been the experimental
method of choice for investigating the elastic properties of MT. It is
believed that taxol maintains tubulin dimers in an approximately straight
conformation and thus prevents depolymerization \cite{ArnalTaxol, AmosTaxol,
Xiao}. However, direct EM investigations of single taxol-GDP-PFs
\cite{Multistable Tub EM} reveals a more complex and interesting picture. A
taxol stabilized single PF can in fact coexist in several conformational
states with comparable free energies: a straight state $\kappa_{PF
^{st}\approx0$ and a weakly curved state with intrinsic curvature $\kappa
_{PF}^{wc}\approx1/250nm$ (see Fig. 1a). A third highly curved state with
$\kappa_{PF}^{hc}\approx1/20nm$ additionally appears after longer observation
periods. Notably Elie-Caille et al. \cite{Multistable Tub EM} pointed out a
cooperative nature of the straight to curved transition within single PFs.
These important findings -several conformational tubulin dimer states and
cooperative interaction between them- will be one central ingredient of our
model later on.
Going from a single protofilament to the whole tube, a central mechanical
property of interest often measured for stabilized MTs, is its persistence
length defined as $l_{p}=B/k_{B}T.$ Here $B\propto Y$ stands for the flexural
rigidity which is for an isotropic beam proportional to its elastic\ Young
modulus $Y$. The persistence length is the length scale characterizing the
filament's resistance to thermally induced bending moments. Several
experimental approaches have been developed to measure bending stiffness of
taxol stabilized MTs. One method consists in measuring MT's thermal shape
fluctuations via dark-field microscopy \cite{GITTES,Venier} or fluorescence
light microscopy \cite{Mickey}. Alternatively in several other experiments,
$l_{p}$ has been determined by applying controlled bending forces via
hydrodynamics \cite{Venier}, optical tweezers
\cite{Felgner,Kukimoto,ActtiveMTBending, TAKASONEbuckling} and atomic force
microscope tips \cite{Kis}. Interestingly the authors in \cite{GITTES}
observed that stabilized MTs are not perfectly straight, but contain some form
of quenched curvature disorder which needs to be subtracted from the
measurements. In the same vein, the observations from dark-field images of the
thermal fluctuations of the free end of axoneme-bound MTs, show that taxol
stabilized MTs adopt a three-dimensional helicoid structure \cite{Venier} (we
will return to this point later on).
The persistence length obtained from these different experiments is set in the
range of $1$ to $8$ $mm$ with some of newer studies going down to $0.1mm$ (cf
below). For a standard semiflexible polymer (like e.g. actin or DNA) we expect
$l_{p}$ to be a material constant, in particular independent of the filament's
length. However, for MTs, the experimental $l_{p}$ data are not only highly
scattered but extremely confusing on this point. That the persistence length
could indeed depend on the MT length was first mentioned and observed in
references \cite{ActtiveMTBending, TAKASONEbuckling} and confirmed by other
experimental measurements - probing either the thermal movement
\cite{GlidingAssay} or the active bending deformations by electrical fields of
individual taxol stabilized MTs gliding on a kinesin-coated surface
\cite{GlidingAssay2,GlidingAssay3}. In particular these techniques gave for
short MT segments with submicron length a persistence length between $0.08$
and $0.24$ $mm$.
This intriguing "length dependent stiffness" was also investigated by
Pampaloni et al who measured the lateral fluctuations of MTs grafted to a
substrate \cite{Pampaloni}. These authors found a $l_{p}$ falling within a
range of $0.11$ to $5.04$ $mm$ for MT lengths varying from $2.6$ to $47.5$
$\mu m$ - with a strong linear length-$l_{p}$ correlation. A similar
experiment done in \cite{Taute} measured the longest relaxation time
$\tau_{\max}\left( L\right) $ of MTs for various MT lengths $L$. It was
found that MTs exhibit unusually slow thermal dynamics compatible with
$\tau_{\max}\propto L^{3}$ (cf. Fig 9) - in sharp contrast to standard
semiflexible filaments with $\tau_{\max}\propto L^{4}$. In \cite{Brangwyne},
the relaxation time for long MTs ($L>10\mu m$) extracted from a 2-D shape
analysis of taxol stabilized fluorescent MTs shows an anomalous dynamics on
short scales as well. Furthermore it has also been experimentally found that
the rigidity of MTs depends on their growth velocity \cite{Janson}. All these
experiments measuring the bending stiffness and the bending dynamics brought
the community to the conclusion that the "beam of Life" cannot be seen as a
simple Euler beam. Its complex internal structure should determine its elastic
and dynamical properties. But which internal mode contributes to the now
obvious elastic complexity, is the important question - a convincing answer of
which is still missing.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
width=3.5 in ]{Fig1.eps}
\end{center}
\caption{The empirical evidence for tubulin bistability: (a) A single taxol
stabilized protofilament can coexist in a straight $\kappa_{PF}\approx0$ and a
slightly curved $\kappa_{PF}\approx1/250nm$ state (reproduction from
\cite{Multistable Tub EM}) (b) Taxol stabilized microtubuls in gliding assay
experiments can switch to a stable circular state and move on circular tracks
(from \cite{Amos}\cite{ValeCoppin}). Microtubules are occasionally observed to
switch back and forth between the circular and straight states. (c) Taxol
stabilized microtubules form a three-dimensional helicoid structure with a
$15\mu m$ pitch (from \cite{Venier}).
\end{figure}
\subsection{Helices and Rings}
Even more intriguing than the issue of MT elasticity is perhaps the question
:\ What is the ground state of a MT? While the naive answer - a straight rod -
would be the most accepted view, a number of experiments put this mundane
simplistic picture in doubt. For instance, wavy sinusoidal and circular shapes
are frequently observed when MTs are adsorbed to glass surfaces or confined
between them. In this confined case the Fourier mode analysis of MT
deformations systematically reveals that a few discrete modes have a larger
amplitude than the fluctuations around them \cite{GITTES,Brangwyne,Janson},
cf. also \cite{VanHeuvel} -supplementary material. This is a strong hint
towards the presence of some type of "frozen in" curvature - dynamically
quenched on experimental timescales. Likewise in motility assays, it is often
seen that when MTs glide over a surface coated with molecular motors they
follow wavy sinusoidal and often circular tracks \cite{Amos,ValeCoppin}, cf.
Figs. 1b. In this context, particularly interesting is the observation by Amos
\& Amos \cite{Amos} of the formation of permanently circling MTs (see Figs
1b). These unusual stable circular gliding structures persisted for many
cycles and occasionally straightened again. As written by the authors
\cite{Amos}, this suggests that "an intact tubular polymer is capable of
holding more than one conformational state without the help of an external
force". From EM images it was inferred that the underlying mechanism stems
from the balance of individual taxol-GDP-tubulin dimers between two or more
different stable conformations \cite{Amos EM}.
An even more clear hint towards the real nature of the "frozen in "\ curvature
was -as already mentioned- discovered by Venier et al \cite{Venier} (Fig.1c)
who described stabilized MTs as \textit{wavy periodic shapes} with a half
period of about $7-8\mu m$. This observation combined with the fact that MTs
often went out of focus, lead the authors to "suggest that taxol-treated
microtubules may adopt a three-dimensional helicoid structure of $15\mu m$ pitch".
Therefore it seems that the MT curvature is a persistent attribute attached to
its lattice. Any serious attempt to fully understand the complexity of
MT\ elasticity can not avoid the question of its ground state. Indeed
understanding fluctuations around a particular state will be futile\ as long
as the origin of the state itself remains in the dark.
\subsection{The Soft Shear Model}
A first theoretical attempt to cope with some aspects of the described MT
mechanical complexity was the "soft shear model" (also called "Timoshenko beam
model", or "anisotropic composite material model")
\cite{Kis,Pampaloni,Frey,KulicMTshear}. In this model the MT\ is considered as
an anisotropic fiber-reinforced material \cite{Kis,Pampaloni} with the tubulin
protofilaments acting as strong fibers weakly linked with easily shearable
inter-protofilament bonds. Some specific equilibrium statistical and
mechanical properties of that model were investigated in
\cite{Frey,KulicMTshear}. An interesting peculiarity and inherent consequence
of this model is that any local lattice deformation gives rise to a long
distance curvature relaxation \cite{KulicMTshear} and can lead to a long range
interaction along the MT contour. This aspect of the "soft shear model" (SSM)
is in phenomenological agreement with cooperative deformations induced by
enzymes like katanin. Furthermore this model predicts a length-dependent
persistence length which approximately resembles the measured behavior
\cite{Pampaloni,Taute}. However in detail it suffers a number of difficulties
and inconsistencies in particular :
-The ground state of SSM is straight - in conflict with the helical ground
state observation \cite{Venier}.
- The SSM\ does not allow for lattice multistability as observed by Amos\&
Amos \cite{Amos}.
- The predicted value of the shear modulus is extremely small
\cite{Pampaloni,Taute,KulicMTshear} ($10^{5}-10^{6}$ times smaller than the
Young modulus). This would imply very strong shearing in bent microtubule
structures. This however is unsupported by other experimental evidence. Indeed
observations of straight and highly bent MTs shows that bending does not
significantly modify the relative position of the inter-protofilament bonds
\cite{Chretienlateralbonds}.
-The dynamics of clamped MTs \cite{Pampaloni,Taute} does not come naturally
out from the SSM. To reach agreement and fit the experimental dynamics the
shear model needs to introduce an ad hoc internal dissipation of unclear origin.
- For short MTs ($<4\mu m$) that model is very far off and disagrees with the
"plateau" region of the bending stiffness vs length relations ( cf. Fig 3 in
\cite{Taute}).
A careful reanalysis of clamped MT experiments, cf. Figs. 2,3 in
\cite{Pampaloni,Taute}, reveals two features not captured by the SSM : the
persistence length scales for large $L\ $approximately as $\sim L$ (without
signs of saturation) while the relaxation times scale as $\sim L^{3}$. This
exotic behavior naively suggests the presence of a limited angular hinge at
the MT\ clamping point. On the other hand artifacts that could trivially lead
to a \textquotedblright hinged behavior\textquotedblright\ (like loose
MT\ attachment and punctual MT\ damage) were specifically excluded in
experiments \cite{Pampaloni,Taute}.
Facing all these obstacles it becomes increasingly clear that the solution to
all puzzles requires a cut and a radically different hypothesis.
\section{Idea of Polymorphic Microtubule Dynamics}
The new scenario proposed here is based on the hypothesis of
\textit{cooperative} \textit{internal} MT lattice dynamics. The two central
assumptions of our model are as follows:
(I) The taxol-GDP-tubulin dimer is a conformationally multistable entity and
fluctuates between at least 2 states on experimental time scales :\ a straight
and an outwards curved state (Figs. 1a and 2a).
(II) There is a nearest-neighbor cooperative interaction of tubulin states
along the PF axis.
Note that assumption I is very different from the conventional picture where
GDP-tubulin has only \textit{one} energetically favorable (curved) state. We
will show that a model based on I and II straightforwardly leads us to the
very origin of MT (super) helicity and provides a coherent explanation for
static and dynamic measurements in thermal fluctuation experiments.
In contrast to the soft shear model, the present model is elastically
isotropic but the monomer curvature is bistable. As we will see, the ground
state in this new model is a highly degenerate 3 dimensional helix fluctuating
between many equivalent configurations.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=3.979in,
width=3.1081in
]{Fig2.eps}
\end{center}
\caption{Elements of the "polymorphic tube model". (a)\ The GDP-tubulin
protofilaments can fluctuate cooperatively between two discrete states. The
curved $\sigma=1$ state is energetically preferred over the straight state
$\sigma=0$ with an energy gain $E=-\Delta G$. The junction between straight
and curved states along the same protofilament are penalized by a coupling
constant $E=+J.$ b) Competition between tubulin switching energy and elastic
lattice strain energy leads to spontaneous symmetry breaking: MT bends to a
randomly chosen direction and assumes a non-zero polymorphic order parameter
$P$. The energy becomes invariant with respect to an arbitrary rotation of the
polymorphic phase angle $\phi$.
\end{figure}
\subsection{Conformational Symmetry Breaking and Helix Formation}
In this chapter we will give a simple pictorial panorama over the consequences
of assumptions I and II. What happens when tubulin dimers obeying assumption
I\ are trapped in the circularly symmetric MT lattice? Starting from
assumption I we imagine that the straight and curved GDP-tubulin state have a
certain energy difference $\Delta G>0,$ with the curved state being slightly
more favorable. The strict preference for the curved state is however only
true if the tubulin dimers are free i.e. not confined to the lattice. The
situation becomes more interesting once they are incorporated into
the\ lattice. Obviously the outwards bending tendency of the curved state is
in conflict with the geometry of the lattice. Switching a dimer on one
MT\ side will frustrate the dimers on the opposite side of the tube and
prevent them from switching at the same time. On the other hand the direct
neighbors of the curved dimer (on the same MT side) will profit and switch
easier to the curved state, as the lattice is already slightly "pre-bent" in
the correct direction. This peculiar interplay of negative and positive
interaction gives rise to a clustering of curved dimer states into a single
block on one side of the tube cf. Fig. 2b.
This immediately raises the question about the orientation of this curved
dimer block. Which MT\ side will be selected, and in which direction will the
MT\ overall bend, can only be decided by the process of \textit{spontaneous
symmetry breaking}. That is, if the ground state is a curved dimer block, it
will be a highly degenerate state (see Fig. 3). In turn it can be expected to
thermally move through the lattice at next to no energy cost (apart from some
friction)! This is the most essential feature of the present model.
So far we have considered only a single MT\ cross-section. If the assumption
II (cooperativity) would not hold the curved state blocks would pick their
sides at each section completely independently. The tube would locally curve
in random uncorrelated directions and the effect of tubulin multistability
would stay essentially invisible on the larger scale. However according to
assumption II\ the blocks become correlated in orientation and prefer to stack
on the top of each other. This then leads to macroscopically a bent - in fact
a circular MT - if the PF\ were not twisted around the central MT\ axis (see
Fig. 2b).\ This provision brings us to the last interesting point. As already
mentioned MT lattices are generically twisted i.e. their PFs are not strictly
parallel. With a cooperative interaction along the PFs - which are now
twisting around the tube axis - the final product of assumption I\ and II will
be a long pitched helix! The pitch of the helix should coincide with the
lattice twist repeat length $:+3.4,+25,-6\mu m$ for $N=12$, $13$ and $14$ PF
MTs respectively.\ The created "polymorphic helix", however will not be unique
and will be able to reshape between its $N\ $indistinguishable orientation states.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=3.6331in,
width=3.1081in
]{Fig3.eps}
\end{center}
\caption{The straight state of the microtubule becomes unstable and forms a
spontaneous bend with fixed curvature pointing towards a randomly chosen
direction. The microtubule can assume one of the N degenerate ground states
and switches between them at no energy cost - the effective potential has a
shape reminiscent of a Mexican hat.
\end{figure}
When we graft one end of such a polymorphic helix onto a surface (as e.g.
performed in \cite{Pampaloni,Taute}) the helix will still be able to switch
between the equivalent orientations and perform a motion that we call
"wobbling" (see Fig. 4). It is exactly this type of motion that can give rise
to the static and dynamic effects measured in \cite{Pampaloni,Taute}.
To see this we can approximate the movement of a clamped polymorphic helix
that is switching between its equivalent ground states with a "rigid conical
rotor" (see Fig. 4). For such a rotor the transverse displacement $\rho$ of
the MT end grows linearly with its length $L.$ Using the naive definition of
persistence length $l_{p}\approx L^{3}/3\left\langle \rho^{2}\right\rangle $
(where $\left\langle {}\right\rangle $ is the ensemble average) and the fact
that $\left\langle \rho^{2}\right\rangle \propto L^{2}$ this apparent
persistence length becomes proportional to the length $l_{p}\propto L.$ This
scaling (cf. Fig. 11) is in agreement with the experimental results in
\cite{Pampaloni,Taute} giving us the first hint that the model is on the right
track. Appendix E comments on the robustness of this conical hinge-like motion
against a limited local hindrance of the wobbling mode due to the adsorbed
part of the grafted MT.
A further encouragement comes from the study of the dynamics of the model.
Making again the approximation of a rigid conical rotation (induced by
wobbling) the observed unusual scaling of the longest relaxation time
$\tau_{\max}\propto L^{3}$ can also be easily understood. In fact a slender
object of length $L$ rotated along a conical surface has a friction constant
$\xi_{rot}\propto L^{3}$. In turn the longest relaxation time is given by
diffusional equilibration time on the cone, i.e. $\tau_{\max}\propto\xi
_{rot}/k_{B}T\propto L^{3}$ (cf. Fig. 12). It turns out that the model
correctly predicts both the exponent and the prefactor of the experimental
relaxation time. \begin{figure}[ptbh]
\begin{center}
\includegraphics[
width=3.0in ]{Fig4.eps}
\end{center}
\caption{Clamped polymorphic microtubule with intrinsic lattice twist attached
at its end to a substrate \cite{Pampaloni}\cite{Taute} performs a peculiar
movement. It forms a polymorphic helix with N degenerate ground states and
switches between them at no energy cost. The approximately conical motion with
opening angle $\alpha$ leads to anomalous lateral fluctuations $\left\langle
\rho^{2}\right\rangle \sim\alpha^{2}L^{2}$ - radically different from all
other semiflexible filaments.
\end{figure}
In addition it is further reassuring that the helical ground state(s) of the
model provides a rationale for the observation of MT\ helices by Venier et al.
\cite{Venier}.
\subsection{In Vivo Implications of Microtubule Polymorphism}
The bulk of observations cited here is made in vitro on taxol stabilized MTs.
It is known that taxol inhibits MT disassembly by maintaining the tubulin
dimer in a state that strongly favors polymerization. But from the experiments
reported above it is also clear that taxol does not suppress all tubulin
conformational changes : taxol-GDP-tubulin dimer has multiple stable
conformations \cite{Multistable Tub EM}. How do these in vitro findings relate
to MTs in vivo?
It is a common empirical observation in many in vivo systems that despite
their high bending stiffness MTs are often seen curved or highly wavy on
micron scales
\cite{Brangwyne,BicekInvivo,BrangwynneWavyBucking,Borisy,Kaech,SamsonovTau}.
For instance in \cite{BicekInvivo}, highly sinusoidal MTs on the periphery of
living LLC-PK1 cells were observed. These shapes are usually explained as a
consequence of motor induced buckling opposed by a gel-like environment that
leads to finite wavelength- buckling \cite{BrangwynneWavyBucking}. While the
"buckling in a gel" interpretation is physically appealing a closer look at
the data in \cite{BrangwynneWavyBucking} (in particular the accompanying movie
material) reveals an absence of correlation in buckling \textit{directions} of
neighboring MTs. This observation puts a question mark on the participation of
a background continuum gel - as in this case the strains in the gel would
necessarily propagate to neighboring microtubules and lead to spatially
correlated buckling events - in sharp contrast with observations. \ Therefore
we are left with a robust phenomenon of sinusoid, constant wavelength MTs -
however without a definite interpretation so far.
The phenomenology of stable rings \cite{Amos} and wavy sinusoid MTs forming in
gliding assays \cite{ValeCoppin} is strikingly similar and visually
indistinguishable from the pure in vivo observations
\cite{Brangwyne,BicekInvivo,BrangwynneWavyBucking,Borisy,Kaech,SamsonovTau}.
Indeed, in both situations highly curved lattice states of very similar
magnitudes $\kappa_{MT}\approx1-2\mu m^{-1}$ are readily observed. This
analogy between in vivo and vitro cases suggests that tubulin dimers - both in
vivo and taxol stabilized (in vitro) posses an identical highly curved state
($\kappa_{PF}^{h.c.}\approx1/20nm$). This highly curved state appears to be
activated within the lattice only under compressive loads and seems to be a
universal property of GDP tubulin itself - independent of taxol stabilization.
On the other hand the weakly curved state of taxol-GDP-dimer ($\kappa
_{PF}^{w.c.}\approx1/300nm$)) is soft enough to be activated by thermal
fluctuations. The empirical evidence for the weakly curved state in vivo is
far less obvious than for the highly curved state. The slight deformations
induced by the former would be more difficult to distinguish from other MT
deforming effects in living cells like motors, polymerization forces, presence
of lattice defects, bundling and microtubule associated protein action.
However the observed phenomenology of length dependent persistence length of
MTs growing from centrosomes in egg extracts \cite{Keller} (no taxol present)
resembles qualitatively and quantitatively the in vitro observations
\cite{Pampaloni,Taute}. This is one more indication that the dynamic
MT\ polymorphism could persist also in vivo.
At least two possible biological implications of the helicoidal polymorphic MT
nature come straight to mind. First, a curved or helical beam under
compressive load responds like a mechanical spring and is therefore much
softer (in tension, compression and bending) than a straight beam. Therefore a
network of helical MTs might be important for the overall mechanical
compliance of the cell. A perfectly straight MT buckled by an extracellular
load would be much more susceptible to mechanical failure and depolymerization
than a soft compliant helix. Second, helical shapes are geometrically
(topologically) prevented from side-by-side aggregation, can thus evade the
formation of bundles and instead form lose networks with much fewer contacts.
It appears that a tuned helicity (that can be switched on or off) could be a
good mechanical control parameter for the formation of different cytoskeletal
structures. While the bulk of the cytoplasmic MTs is preferably in the lose
network state (favored by helicity) in occasional situations like the neuronal
axons straight aligned MTs are required. In such a situation the bundling
could be triggered by switching the lattice to the straight state. Remarkably
in the process of axonal retraction the straight axonal bundle is destabilized
(and eventually contracted towards the cell soma)\ by an apparent transition
of the MTs to a wavy coiled state \cite{Baas} very reminiscent in shape to
single wavy MTs from cell soma
\cite{Brangwyne,BicekInvivo,BrangwynneWavyBucking,Borisy,Kaech,SamsonovTau}.
In general MT's polymorphism might have other, less obvious biological
implications that still have to be identified. In particular a more
speculative possibility is that the tubulin's allosteric multistability might
also be a piece in the puzzle of MT "catastrophes". A cooperative curvature
switch might trigger a transition from growth to depolymerization. Maybe the
most fascinating aspect could lie in the possibility of signal transmission
along single MTs via a long range conformational switch. If our model is
correct this is the most inherent and distinct consequence of the underlying mechanism.
We conclude here the qualitative description of essential ideas behind the
polymorphic tube model. In the following we switch gears and present in more
quantitative detail the mathematical model. The mathematically less inclined
reader is invited to fly over some figures, comparisons with experiments,
maybe pick up additional concepts (such as polymorphic defects and their
dynamics) and jump to the perspectives section which will underline the
essential biological consequences.
\section{The Polymorphic Tube Model}
Until now the discussion was qualitative and in the following we build the
mathematical model of the polymorphic MT. We will provide quantitative
arguments and model experimental data, justifying thus a posteriori the
previous discussion. In this section, we will focus on the thermally induced
weakly curved state in taxol-stabilized MTs and leave the case of mechanically
induced highly curved MTs for further works.
We model the GDP-tubulin dimer state by a two state variable $\sigma
_{n}\left( s\right) =0,1$ which corresponds to the straight and$\ $curved
state at each lattice site. The $n=1,...N$ \ is the circumferential PF index
and $s\in\left[ 0,L\right] $ is the longitudinal position variable along the
MT centerline. We recall that our model is based on the following assumptions:
(I) The taxol-GDP-tubulin dimer fluctuates between 2 states - straight and
curved - (Fig. 2a) with an energy difference $\Delta G>0$ favoring the curved
state. The energy density resulting from the switching of tubulin dimers (at a
given MT section) is then given by
\begin{equation}
e_{trans}\left( s\right) =-\frac{\Delta G}{b}\sum\nolimits_{n=1}^{N
\sigma_{n}(s) \label{e_trans
\end{equation}
with$\ b\approx8nm$ the dimer length.
(II) There is an Ising type nearest-neighbor cooperative interaction of
tubulin states \textit{along} the PF axis with an interaction energy $J>0$
favoring nearest neighbor dimers on the same PF to be in the same state. This
leads to the interaction energy density
\begin{equation}
e_{inter}\left( s\right) =-\frac{J}{b}\sum\nolimits_{n=1}^{N}\left(
2\sigma_{n}\left( s\right) -1\right) \left( 2\sigma_{n}\left( s+b\right)
-1\right) \label{eJ
\end{equation}
The last term missing in our description is the elastic energy density of the
MT lattice. For a usual isotropic Euler beam the material deformations
$\varepsilon$ are related to the centerline curvature vector $\vec{\kappa}$
via $\varepsilon=-\vec{\kappa}\cdot\vec{r}$ with $\vec{r}$ the radial material
vector in the cross-section. For a polymorphic MT, modelled as a continuum
material made of $N$ PFs ($N=11-16$), its actual deformation will depend on
the polymorphism-induced prestrain $\varepsilon_{pol}.$ In this case the
elastic energy density of the MT can be written as
\begin{equation}
e_{el}\left( s\right) =\frac{Y}{2}\int_{R_{i}}^{R_{o}}\int_{0}^{2\pi}\left(
\varepsilon-\varepsilon_{pol}\right) ^{2}rdrd\alpha\label{e_el
\end{equation}
where the integration in $e_{el}$ goes over the annular MT cross-section with
$R_{i}\approx7.5nm,$ $R_{o}\approx11.5nm$ the inner and outer MT radii
respectively. The prestrain $\varepsilon_{pol}$ is a function of the
polymorphic state of the tubulin dimers. Its definition requires a
decomposition of the tubulin dimer in an inner part (facing the tube axis) and
an outer part (facing from the tube axis outwards), cf. Fig 5. We assume that
each curved dimer state generates a positive prestrain $+\varepsilon_{PF}$ on
its inner part and an equal but negative prestrain $-\varepsilon_{PF}$ on its
outer part. We can then write $\varepsilon_{pol}\left( s,r,\alpha\right)
=\varepsilon_{PF}\sigma_{n}\left( s\right) $ $\left[ I_{[R_{i},R_{o
-d_{PF}/2]}\left( r\right) -I_{[R_{o}-d_{PF}/2,R_{o}]}\left( r\right)
\right] $ $\cdot I_{[\frac{2\pi}{N}n+q_{0}s,\frac{2\pi}{N}\left( n+1\right)
+q_{0}s]}\left( \alpha\right) $ where $I_{[.]}\left( x\right) =1$ if
$x\in\lbrack.]$ and $0$ otherwise ( Heaviside function ) and $d_{PF}$ is the
PF diameter. The parameter $q_{0}$ appearing in the angular part of
$\varepsilon_{pol}$ is the natural lattice twist that leads to the proper
geometric rotation of a PF around the tube axis. This parameter is lattice
type dependent and takes discrete values $2\pi/q_{0}=+3.4\mu m,+25\mu m,-6\mu
m$ for $N=12$, $13$ and $14$ PF MTs respectively \cite{Wade,Chretien
\cite{Ray}\cite{ChretienFuller}. We can estimate the prestrain $\varepsilon
_{PF}$ from the experimental value of the single switched PF's curvature
$\kappa_{PF}\approx\left( 250nm\right) ^{-1}$ -measured in
Ref.\cite{Multistable Tub EM} on single taxol-stabilized PFs - to be
$\varepsilon_{PF}=d_{PF}\kappa_{PF}/2\approx10^{-2}.$ Collecting all energy
contributions together the total elastic +\ polymorphic energy of the MT is
then given b
\begin{equation}
E_{MT}=\int\nolimits_{0}^{L}\left( e_{el}+e_{trans}+e_{inter}\right) ds.
\label{E_MTTotal
\end{equation}
The ground state within this model is determined by the interplay of the first
two terms $e_{el}$ and $e_{trans}.$ The last term $e_{inter}$ rules over
cooperativity and is responsible for the suppression of defects in the ideal
polymorphic order (cf. Fig. 7). A large value of the cooperativity constant
with $J>>k_{b}TL/b$ would imply a defect free lattice. However the presence of
the latter defects (and their motion) are a necessary ingredient for the
overall rearrangement of the helix as discussed later on.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=1.4512in, width=3.0735in ]{Fig5.eps}
\end{center}
\caption{Strains and deformations in the polymorphic tube model. Each tubulin
dimer can fluctuate between a straight state $\sigma=0$ and a curved state
$\sigma=1$ of intrinsic curvature $\kappa_{PF}$. The curved tubulin dimer
generates a positive prestrain $+\varepsilon_{PF}$ on its inner part and an
equal but negative prestrain $-\varepsilon_{PF}$ on its outer part with
strains related to observed dimer curvature via $\varepsilon_{PF}=(R_{o
-R_{i})\kappa_{PF}/2$.
\end{figure}
To understand the basic behavior of the ideal helical ground state without
defects - i.e. PFs are individually in a uniform state (either curved or
straight)- we initially restrict ourselves to the simplified energy density
$e=e_{el}+e_{trans}$. To investigate the MT geometry we first introduce two
reference frames (Fig. 2b). One is the material frame with base vectors
$(\vec{u}_{1},\vec{u}_{2},\vec{u}_{3})$ attached to the MT cross-section. The
other is an external fixed laboratory frame with base vectors $(\vec{u
_{x},\vec{u}_{y},\vec{u}_{z}).$ Putting the MT along the $\vec{u}_{z}$ axis
direction and considering small MT angular deflections we have $\vec{u
_{z}\approx\vec{u}_{3}$. In this case the two frames are simply related to
each other by a rotation transformation $R(s)$ given by the internal MT
lattice twist $q_{0}$, such that $(\vec{u}_{x},\vec{u}_{y})=R(s)(\vec{u
_{1},\vec{u}_{2})$ with
\begin{equation}
R(s)=\left(
\begin{array}
[c]{cc
\cos q_{0}s & -\sin q_{0}s\\
\sin q_{0}s & \cos q_{0}s
\end{array}
\right) \label{R(s)
\end{equation}
To rewrite $e$ in a more illuminating fashion, we define two important order
parameters at each MT cross-section. The first one of them is the
\textit{vectorial polymorphic order parameter }
\[
\vec{P}\left( s\right) =\sum_{n=1}^{N}\left( \vec{u}_{1}\cos\dfrac{2\pi
n}{N}+\vec{u}_{2}\sin\frac{2\pi n}{N}\right) \sigma_{n}\left( s\right)
\]
Physically, $\vec{P}$ - a 2D vector at each local material frame section
attached to the MT (cf Fig. 2b)- describes the asymmetry of distribution - a
kind of "polarization" - of the dimer states. It acquires a non zero value
only in the case when the curved and non-curved states are azimuthally
separated on opposite MT\ sides. For instance the "all-straight" or
"all-curved" PF state correspond both to the same value $\vec{P}=0$. Besides
the vector $\vec{P}$ we need to define a second (scalar) quantity
\[
M\left( s\right) =\sum_{n=1}^{N}\sigma_{n}\left( s\right)
\]
which counts the total number of dimers in the curved state at cross-section
$s$ - or in the Ising model terminology : the "magnetization". After
integration of Eq. \ref{e_el} over the cross-section and some algebra the
energy density $e=e_{el}+e_{trans}$ can be written in a more appealing form:
\begin{equation}
e=\frac{B}{2}\left[ \left( \vec{\kappa}-\vec{\kappa}_{pol}\right)
^{2}+\kappa_{1}^{2}\left( \frac{\pi}{N}\gamma M-\sin^{2}(\pi/N)\vec{P
^{2}\right) \right] \label{EnergyPandM
\end{equation}
with the elastic bending modulus $B=\frac{Y\pi}{4}\left( R_{o}^{4}-R_{i
^{4}\right) $, with $\kappa_{1}=\frac{\left( R_{0}-R_{1}\right) ^{2}
{\pi\left( R_{o}^{2}+R_{i}^{2}\right) }\kappa_{FP}$ and a dimensionless
parameter
\begin{equation}
\gamma=\frac{\kappa_{PF}}{\kappa_{1}}-\frac{2N\Delta G}{bB\kappa_{1}^{2}}
\label{gamma
\end{equation}
For small deviations of the tube axis from the $\vec{u}_{z}$ direction, the
polymorphic curvature vector $\vec{\kappa}_{pol}$ in the external coordinate
frame $\left( \vec{u}_{x},\vec{u}_{y}\right) $ is\ related to $\vec{P}$ (in
the internal frame $\left( \vec{u}_{1},\vec{u}_{2}\right) $) via the
transformation in Eq. \ref{R(s)}
\begin{equation}
\vec{\kappa}_{pol}=cR(s)\vec{P} \label{Rot
\end{equation}
with $c=\sin\left( \pi/N\right) \kappa_{1}$ a geometric proportionality constant.
\subsection{Phase Diagram}
In the absence of defects the energy expression Eq. \ref{EnergyPandM} allows
us to determine the conformational ground state of a polymorphic MT. To this
end, we first resort to one further small simplification and make the "single
block ansatz", i.e. at each cross-section we assume only a single continuous
block of $p$ switched PFs. This ansatz was previously used by Calladine and
has been proven very useful in modelling bacterial flagellin polymorphic
states \cite{Asakura,Calladine}. In the ground-state configuration the
curvature is given by $\vec{\kappa}=\vec{\kappa}_{pol}(p)$ whose absolute
value obtained from Eq. \ref{Rot} is $\kappa_{pol}\left( p\right)
=\kappa_{1}\sin\left( \pi p/N\right) $. The optimal switched block size
$p=p^{\ast}$ can be determined by minimizing the second term in Eq.
\ref{EnergyPandM} which within this ansatz become
\begin{equation}
e=\frac{B\kappa_{1}^{2}}{2}\left( \gamma\frac{\pi}{N}p-\sin^{2}\left(
\frac{\pi}{N}p\right) \right) \label{EnergyL
\end{equation}
This gives rise to an interesting MT phase behavior (cf. Fig. 6). The latter
only depends on the polymorphic-elastic competition parameter $\gamma$ from
Eq. \ref{gamma}\ -that measures the ratio between polymorphic energy of
tubulin switching and the purely elastic cost of this transition.
For $\gamma<-1$ the chemical switching potential $\Delta G$ strongly dominates
the elastic energy cost $bB\kappa_{1}^{2}$. Therefore switching is highly
favorable and all the PFs will be found in the state $\sigma=1$. This gives
rise to a straight but highly prestrained configuration.
Analogously for $\gamma>1$ the bending energy contribution is too costly for
PFs to switch at all. Therefore in this regime the PFs are all in the straight
state with $\sigma=0$ and the MT is consequently straight as well.
For $-1<\gamma<1$ the situation is more interesting. In this interval we have
a coexistence of two locally (meta) stable MT conformations : the\ straight
tube (prestressed or not - depending on the sign of $\gamma$)\ and a curved
lattice state with $p^{\ast}$ switched protofilaments. For $-\overline{\gamma
}<\gamma<\overline{\gamma}$ with $\overline{\gamma}\approx0.72$ the curved
lattice state is the absolute energy minimum and the straight state is only
metastable. Therefore in this regime and in the absence of twist $\left(
q_{0}=0\right) $ the ground state configuration of the whole tube would be a
simple circular arc section (cf. Fig. 2b). On the other hand, the ground state
of a microtubule bearing natural lattice twist $q_{0}\neq0$ will be helical
(cf. Fig. 4).
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=2.3981in, width=3.237in ]{Fig6.eps}
\end{center}
\caption{The phase diagram of the polymorphic microtubule model as function of
the polymorphic-elastic interaction parameter $\gamma$ from Eq. \ref{gamma}.
Depending of the magnitude and sign of $\gamma$ the microtubule can be either
in an "all PF switched state" (black), "no PF switched state" (grey) or in a
mixed curved-helical state. Only the latter "mixed state" will display net
curvature and lead to an observable helical appearance.
\end{figure}
It is easy to see that a stable helical state as observed in \cite{Venier} is
only possible for a switching ratio $p^{\ast}/N\in\lbrack1/4,3/4]$. This
together with $\kappa_{PF}=1/250nm$ from \cite{Multistable Tub EM} and
$\kappa_{pol}\left( p^{\ast}\right) =\kappa_{1}\left\vert \sin\left( \pi
p^{\ast}/N\right) \right\vert $ provides us with a direct estimate of the
radius of curvature $\kappa_{pol}^{-1}\approx9-14\mu m$. Very strikingly the
latter range reproduces closely the observed MT helical curvatures as
estimated from Venier at al. work \cite{Venier} $\kappa_{measured}^{-1
\approx11\mu m$ - lending strong support to the model. Furthermore, taking the
helical state stability as an empirical fact (implying that $\left\vert
\gamma\right\vert <0.72$) and assuming a typical protein Young modulus
$Y\approx1-10GPa,$ allows us a simple estimate of the transition energy per
monomer $\Delta G\approx+1.1$ to $+11kT$ $-$ a reasonable range for a soft
biological object.
In general, the full energy expression -including the cooperativity energy
term- Eq. \ref{E_MTTotal}\ gives rise to a very complex behavior. Here we will
focus on some basic new phenomena. It turns out that the most remarkable
deviation from standard wormlike chain (WLC) behavior arises from the
fluctuation dynamics of the polymorphic order parameter's angular phase that
we consider in the following.
\subsection{Polymorphic phase fluctuations}
Here we introduce a sightly different phenomenological model that simplifies
the study of Eq. \ref{E_MTTotal} while still reflecting important aspects and
physical properties of it. In this section we assume as before that the
helical state is the ground state and consider now the effect of the
fluctuations around it. To this end we decompose the polymorphic order
parameter
\[
\overrightarrow{P}\left( s\right) =P\left( s\right) \left[ \cos\left(
\phi\left( s\right) \right) \vec{u}_{1}+\sin\left( \phi\left( s\right)
\right) \vec{u}_{2}\right]
\]
with $P\left( s\right) $ being the \textquotedblright polymorphic
amplitude\textquotedblright\ and $\phi\left( s\right) $ the
\textquotedblright polymorphic phase variable\textquotedblright. The latter
one determines the orientation of the switched block tubulin dimers at each
cross section with respect to the MT material frame. From Eq. \ref{Rot}, the
centerline curvature with respect to a fixed external $\left( \vec{u
_{x},\vec{u}_{y}\right) $ frame is then
\begin{equation}
\overrightarrow{\kappa}_{pol}(s)=\kappa_{0}\left[ \cos\left( q_{0
s+\phi\left( s\right) \right) \vec{u}_{x}+\sin\left( q_{0}s+\phi\left(
s\right) \right) \vec{u}_{y}\right] \label{kappaPol
\end{equation}
with $\kappa_{0}=cP(s)$.
In general both the polymorphic phase $\phi$ and amplitude $P$ can fluctuate
along the MT contour and give contributions to the polymorphic energy. The
phase fluctuations are induced by creation and motion of polymorphic "double
defects", cf. Fig. 7. The double defect - a kind of "polymorphic dislocation"
- that can be either left or right handed - maintains the number of switched
protofilaments constant while reorienting the direction of curvature. At zero
temperature the lattice would be defect-free, $\phi$ would be constant and the
polymorphic order parameter $\vec{P}$ would strictly follow the lattice twist.
The phase change $\phi^{\prime}\equiv d\phi/ds$ will deviate from zero if on
relevant length scales there are enough polymorphic double defects to allow
for a reorientation of the polymorphic order parameter away from optimum. The
double defects carry only a limited local energy cost $\Delta E=2J$ per defect
and can be easily thermally excited if $J\lesssim k_{B}T$. In the
approximation of a large number of PFs, assuming that $\phi$ can change
continuously we can write the phase contribution to the energy as
\begin{equation}
E_{pol}(\phi)=\frac{C_{\phi}}{2}\int_{0}^{L}ds\phi^{\prime2} \label{Ephi
\end{equation}
with the polymorphic phase stiffness $C_{\phi}\approx k_{B}T\frac{N^{2}b
{8\pi^{2}}\left( 2+e^{2J/k_{B}T}\right) $ which can be related to the
density of double defects with energy $2J\;($cf. Fig. 7 and Appendix A),
giving rise to a new length scale - the \textit{polymorphic phase coherence
length} $l_{\phi}=C_{\phi}/k_{B}T.$ For MTs shorter than $l_{\phi}$ we will
observe coherent helices while on longer length scales the helix softens
significantly and looses eventually its helical appearance.
In contrast to the just discussed polymorphic dislocations which can be easily
thermally excited, the variation of the polymorphic amplitude $P$, i.e, change
of the number of switched PFs is more energetically costly. Any deviation of
$P$ from its optimum state $P^{\ast}$ (given by the phase diagram) is
associated with an energy cost $E\propto(\left\vert P\right\vert -\left\vert
P^{\ast}\right\vert ))^{2}\cdot l$ proportional to the length $l$ of the
region in the unfavorable state, cf. Fig. 7 (see Appendix B). Therefore we
conclude that on large enough scales the polymorphic phase fluctuations will
be the dominant effect. Based on this and on the observation of stable helical
states \cite{Venier} we will in the following assume $P=const.$
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=1.8204in, width=2.9525in ]{Fig7.eps}
\end{center}
\caption{Defects in ideal polymorphic order soften the helical states and give
rise to "polymorphic dynamics". Single defects carry a cost proportional to
their length, double defects only a local energy contribution. The coexistence
of left- and right handed defects (LH and RH) along the length leads to a
"random walk" of the polymorphic curvature direction and in turn to an
effective torsional deformation.
\end{figure}
For small deflections around the $z$-axis, the unit vector tangent to the MT's
centerline is approximately given by $\overrightarrow{t}\approx(\theta
_{x},\theta_{y},1)$\ in the laboratory frame $\left( \vec{u}_{x},\vec{u
_{y},\vec{u}_{z}\right) $\ where $\overrightarrow{\theta}=\left( \theta
_{x},\theta_{y}\right) $\ are the centerline deflection angles in x/y
direction. The global centerline curvature $\overrightarrow{\kappa
}=d\overrightarrow{t}/ds$\ can then be approximated as $\overrightarrow
{\kappa}\approx\left( \theta_{x}^{\prime},\theta_{y}^{\prime},0\right) $ and
the total MT energy can be written as follows:\
\begin{equation}
E_{tot}=E_{pol}\left( \phi\right) +E_{el}\left( \theta,\phi\right) .
\label{ET
\end{equation}
The first energy term is the polymorphic phase contribution Eq. \ref{Ephi}.
The second term is the elastic bending energ
\begin{equation}
E_{el}\left( \theta,\phi\right) =\frac{B}{2}\int_{0}^{L}(\overrightarrow
{\theta^{\prime}}-\overrightarrow{\kappa}_{pol})^{2}ds. \label{Eel
\end{equation}
From Eqs. \ref{Ephi}-\ref{Eel}, we see that the zero-temperature ground state
corresponds to $\phi=const$ and $\overrightarrow{\theta^{\prime
}=\overrightarrow{\kappa}_{pol}$ - that is to a defect-free helix with a pitch
given by the natural lattice twist $q_{0}.$ At finite temperature, both
elastic and polymorphic fluctuations will be excited so that the curvature can
be decomposed as $\overrightarrow{\theta^{\prime}}=\overrightarrow{\kappa
}_{pol}+\overrightarrow{\theta^{\prime}}_{el}$ with $\overrightarrow
{\theta^{\prime}}_{el}$ the purely elastic contribution. This gives rise to a
helical MT shape described by the curvature $\overrightarrow{\kappa
_{pol}+\overrightarrow{\theta^{\prime}}_{el}$ and torsion $\tau\sim
\phi^{\prime}+q_{0}.$ The MT lateral displacements away from\ the $z$ axis can
be written as $\overrightarrow{\rho}(s)=(x(s),y(s))=\int_{0}^{s}\left(
\theta_{x}(s^{\prime})),\theta_{y}(s^{\prime}))\right) ds^{\prime}$ that for
small deflections decouples into elastic and polymorphic displacements
$\overrightarrow{\rho}(s)=\overrightarrow{\rho}_{pol}+\overrightarrow{\rho
}_{el}$, where $\overrightarrow{\rho}_{el}\approx\int_{0}^{s}\overrightarrow
{\theta}_{el}(s^{\prime})ds^{\prime}$ and $\overrightarrow{\rho}_{pol
\approx\int_{0}^{s}\overrightarrow{\theta}_{pol}(s^{\prime})ds^{\prime}.$ The
latter can also be written from Eq. \ref{kappaPol} as
\begin{align}
\overrightarrow{\rho}_{pol}(s) & =\kappa_{0}\int_{0}^{s}ds^{\prime}\int
_{0}^{s^{\prime}}d\widetilde{s}\left( \cos\left( q_{0}\widetilde{s
+\phi\left( \widetilde{s}\right) \right) \overrightarrow{e}_{x}\right.
\nonumber\\
& \left. +\sin\left( q_{0}\widetilde{s}+\phi\left( \widetilde{s}\right)
\right) \overrightarrow{e}_{y}\right) \label{Polydeplacement
\end{align}
In Fig. 8, snapshots of configurations of clamped MT obtained from Monte Carlo
simulations are plotted for different concentrations of double defects (i.e.
different values of $l_{\phi}$) with twist and no twist. It is interesting to
remark at this point that based on the symmetry in the problem any MT
configuration can be rotated around the $z-$axis with no energy cost. This
seemingly trivial feature - the energetic degeneracy- is in fact the most
distinctive and unusual property of a polymorphic chain. We consider the
consequences in the following.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=1.8213in, width=3.2387in ]{Fig8.eps}
\end{center}
\caption{The conical "wobbling" motion due to the rotational energy degeneracy
of the polymorphic MT model : Snapshots of Monte-Carlo simulated lattice
states. (a) Twisted MTs free of defects $L/l_{\phi}\ll1$ and with numerous
defects $L/l_{\phi}>1$. For larger number of defects, the helix looses its
coherent look. (b) For non-twisted MTs the movement has a typical parabolic
"trumpet-like" shape.
\end{figure}
\subsection{The Wobbling Mode}
By construction a polymorphic chain as we describe it here has a $N$ fold
symmetry. Therefore, there are $N$ different helical states of different
orientations with the same energy i.e. $N$ grounds states. This energy
degeneracy is also reflected in the continuum model (where the $N$ fold
symmetry is approximated as continuous) by the rotational invariance of
$E_{pol}\left( \phi\right) $ (which depends only on $\phi^{\prime}$). The
broken cylindrical to helical symmetry of the straight state is then restored
by the presence of a \textquotedblright Goldstone mode\textquotedblrigh
\ $\phi\rightarrow\phi+\phi_{0}$ consisting of a rotation of $P$ by an
arbitrary angle $\phi_{0}$ in the material frame (cf. Figs. 2 and 3). This
mode comes energetically for free and leads to dramatic effects on chain's
fluctuations. For instance for a MT clamped at one end, this symmetry implies
that the MT will randomly rotate much like a rigid rotor as shown in Fig. 4.
Note however that this rotation will still be associated with a certain
dissipation as the system has to go over energy barriers between two helical
states. This barrier due to the flipping of lattice states can be overcome at
nonzero temperatures by the creation of double defects which diffusively
propagate along the MT and eventually angularly reorient the polymorphic order
parameter $\vec{P}$. These dynamic phenomena and the dissipation mechanisms
will be discussed in a later section.
In summary, the zero-energy mode that we will also call the \textquotedblright
wobbling mode\textquotedblright\ is an inherent feature of a helically
polymorphic filament and as we will see now could be the culprit causing
anomalous fluctuations of clamped MTs.
\subsection{Persistence Length Anomalies}
Among several definitions of the persistence length, we consider here - for
direct comparison with clamped MTs experiments \cite{Pampaloni,Taute} - the
"lateral fluctuation persistence length" defined as
\begin{equation}
l_{p}^{\ast}(s)=\left( 2/3\right) s^{3}/V(s) \label{LpStar
\end{equation}
with $V(s)=\left\langle \rho\left( s\right) ^{2}\right\rangle -\left\langle
\rho\left( s\right) \right\rangle ^{2}$ the variance of $\rho^{2
=x^{2}+y^{2}$ - the transverse displacement at position $s$ and $\left\langle
..\right\rangle $ the ensemble average. We assume as in experiments
\cite{Pampaloni,Taute} a rigid clamping point at the position $s=0$ preventing
the microtubule from translating and rotating at that point.
For an ideal semiflexible wormlike chain we expect that the persistence length
$l_{p}^{\ast}=l_{B}$ is a position independent and definition invariant
quantity equal to the bending persistence length $l_{B}=B/k_{B}T$. (For
another more classical definition of the persistence length - coming from the
tangent tangent correlation function, see also Appendix C). However for a
polymorphic chain the strict equivalence of $l_{p}^{\ast}$ and $l_{B}$ is not
correct. To see that, we can decompose the polymorphic and elastic
fluctuations $\overrightarrow{\rho}(s)=\overrightarrow{\rho}_{pol
+\overrightarrow{\rho}_{el}$. Inserting this in Eq. \ref{LpStar} and taking
into account that for small deflections the two components decouple
$\left\langle \overrightarrow{\rho}_{pol}\overrightarrow{\rho}_{el
\right\rangle =0$ leads us to the following relation for the persistence
length
\begin{equation}
l_{p}^{\ast}=\left( l_{pol}^{-1}+l_{B}^{-1}\right) ^{-1} \label{lptotal
\end{equation}
with the polymorphic persistence length given by $l_{pol}=\left( 2/3\right)
s^{3}/V_{pol}$ and $V_{pol}=\left\langle \rho_{pol}{}^{2}\right\rangle
-\left\langle \rho_{pol}\right\rangle ^{2}$. The average $\left\langle
..\right\rangle $ is now performed over the phase $\phi$ governed by the
energy Eq. $\ref{Ephi}$. More precisely, the average of any arbitrary
functional $A\left[ \phi\right] $ of the polymorphic phase can be performed
by first selecting one of the equivalent ground states denoted $\phi_{0}$
characterized by $\phi(0)=\phi_{0}$ and performing the average over the
polymorphic angle distribution Eq. $\ref{Ephi}$ around the chosen ground
state, i.e.,
\begin{equation}
\left\langle A\left[ \phi\right] \right\rangle |_{\phi_{0}}=\frac{1}{Z
\int\emph{D}\tilde{\phi}A\left[ \tilde{\phi}+\phi_{0}\right] \exp
(-\frac{l_{\phi}}{2}\int_{0}^{L}ds\tilde{\phi}^{\prime2}) \label{Aphi
\end{equation}
where $\tilde{\phi}=\phi-\phi_{0}$ (and thus $\tilde{\phi}(0)=0$) and
$Z=\int\emph{D}\tilde{\phi}\exp(-\frac{l_{\phi}}{2}\int_{0}^{L}ds\tilde{\phi
}^{\prime2})$ is the partition function. In a second step - for a freely
rotating polymorphic phase- we integrate over the rotational zero mode
$\phi_{0}$ : $\left\langle A\left[ \phi\right] \right\rangle =\frac{1}{2\pi
}\int\left\langle A\left[ \phi\right] \right\rangle |_{\phi_{0}}d\phi_{0}$.
This operation correctly takes into account the phase fluctuations over (and
around) all equivalent ground states related by the transform $\phi
\rightarrow\tilde{\phi}+\phi_{0}$. The rotational symmetry around the z-axis
(integration on $\phi_{0}$) readily implies $\left\langle \rho_{pol}\left(
s\right) \right\rangle =0$ and $\left\langle x_{pol}^{2}\right\rangle
=\left\langle y_{pol}^{2}\right\rangle .$ Therefore the polymorphic
persistence length can be written as
\begin{equation}
l_{pol}(s)=\left( 1/3\right) s^{3}/\left\langle y_{pol}^{2}(s)\right\rangle
. \label{lpol
\end{equation}
with
\begin{equation}
\left\langle y_{pol}^{2}(s)\right\rangle =\int_{0}^{2\pi}\frac{d\phi_{0}
{2\pi}\int_{0}^{s}\int_{0}^{s}\left\langle \theta_{y,pol}(s_{1})\theta
_{y,pol}(s_{2})\right\rangle |_{\phi_{0}}ds_{1}ds_{2} \label{ypolsquare
\end{equation}
whose computation (for details cf. Appendix C) leads to the following mean
square displacement
\begin{align}
\left\langle y_{pol}(s)^{2}\right\rangle & =\frac{2\kappa_{0}^{2}l_{\phi
}{3(1+4l_{\phi}^{2}q_{0}^{2})^{4}}\left\{ P_{1}\left( s\right)
-e^{-\frac{s}{2l_{\phi}}}P_{2}\left( s\right) \cos\left( q_{0}s\right)
\right. \nonumber\\
& \left. -e^{-\frac{s}{2l_{\phi}}}P_{3}\left( s\right) \sin(q_{0
s)\right\} \label{y2pol
\end{align}
where $P_{i}(s)$ are polynomial functions given in Appendix C.\
A typical curve of $l_{p}^{\ast}$ vs $s$ is provided in Fig. 9. In general it
shows three different characteristic regimes denoted I, II and III in the
figure :
I.\ At short distances to the attachment point $s<s_{\min}\approx\pi/q_{0}$
(half the polymorphic wavelength) the total persistence length can be
approximately given b
\begin{equation}
l_{p}^{\ast}\approx l_{B}-\frac{3\kappa_{0}^{2}l_{B}^{2}s}{8}
\label{lp short L
\end{equation}
In the limit of very short distances $s<<l_{B}^{-1}\kappa_{0}^{-2},$ the
polymorphic fluctuations become negligible and are completely dominated by
purely "classical" semiflexible chain fluctuations. Not surprisingly the
persistence length coincides then with the classical bending persistence
length $l_{p}^{\ast}\left( 0\right) =l_{B}$. Starting from $l_{B}$,
polymorphic fluctuations begin to contribute reducing $l_{p}^{\ast}$ that
attains a global minimum at $s_{\min}\approx\pi/q_{0}$.
II. For intermediate length values $s_{\min}<s<$ $l_{\phi}$, the total
persistence length displays a non-monotonic oscillatory behavior around a
nearly linearly growing averag
\begin{equation}
l_{p}^{\ast}\left( s\right) \approx\frac{2}{3}\frac{q_{0}^{2}}{\kappa
_{0}^{2}}s+\frac{4}{3}\frac{q_{0}}{\kappa_{0}^{2}}\sin\left( q_{0}s\right)
\label{lp longue L
\end{equation}
This result is worth deeper understanding. A moment of thinking reveals that
the oscillatory part with wavelength $2\pi/q_{0}$ is related to the helicity
of the ground state. At the same time the linear growth $l_{p}^{\ast}\left(
s\right) \propto\alpha^{2}s$ can be associated with the roughly conical
rotation of the clamped chain (wobbling mode) which acts as an effective
\textquotedblright rotational hinge\textquotedblright\ at the attachment
point, cf. Fig. 4. The sinusoidally modulated rotation cone which builds an
approximate envelope for the chains' motion has an opening angle $\alpha$
which is related to the geometric features of the helix $\alpha=2\kappa
_{0}q_{0}^{-1}$.
III. Finally for very large distances from the attachment point $s\gg l_{\phi
}$ we expect to recover classical results of a semiflexible chain again.
Indeed in this asymptotic regime the effective persistence length reaches
saturation with a renormalized constant value $l_{p}^{\ast}\left(
\infty\right) =1/\left( l_{pol}^{-1}+l_{B}^{-1}\right) $\ where
\begin{equation}
l_{pol}=2l_{\phi}q_{0}^{2}\kappa_{0}^{-2}+\frac{1}{2}\kappa_{0}^{-2}l_{\phi
}^{-1} \label{lpinfinity
\end{equation}
Intuitively the helix looses then its "coherent nature" - due to strong
variations of $\phi^{\prime}$ and elastic fluctuations $\theta_{el}$ - and the
collective rigid rotational (\textquotedblright conical\textquotedblright)
motion is finally replaced by an uncorrelated segment movement. Not
surprisingly the persistence length becomes then length independent again.
Curves for different values of $l_{\phi}$ are provided in Fig. 10.
\subsubsection{Untwisted MTs}
While there are no completely twist-free MTs and every lattice will have
generically a small twist, one can still formally study the interesting
limiting case $q_{0}=0$. Note that the large estimated pitch of $13$PF MTs is
finite and in the range of $\gtrsim25\mu m$
\cite{Wade,Chretien,Ray,ChretienFuller} while often assumed to be
approximately infinite. In such an ideal case the theory still applies,
however the overall behavior of $l_{p}^{\ast}\left( L\right) $ will
substantially change and become much less consistent with the linear scaling
found in experiments. While for $q_{0}L>1$ the chain in leading order moves on
a linear cone (with fixed opening angle $\alpha$) for $q_{0}L<<1$ while still
$L<<l_{\phi}$ the \textquotedblleft wobbling\textquotedblright\ motion is
happening on a quadratic cone (a \textquotedblleft trumpet
shaped\textquotedblright\ cone, cf. Fig. 8b). More precisely for the
exceptional case of untwisted MTs the polymorphic part of the lateral
fluctuations Eq. \ref{y2pol} behaves as $\left\langle y_{pol}(s)^{2
\right\rangle =2/3\kappa_{0}^{2}l_{\phi}\left( P_{1}-e^{-\frac{s}{2l_{\phi}
}P_{2}\right) $ with $P_{1}(s)=24l_{\phi}^{3}-3l_{\phi}s^{2}+s^{3}$,
$P_{2}(s)=24l_{\phi}^{3}+12l_{\phi}^{2}s{\small .}$ Therefore for short MTs
$L<<l_{\phi},$ the lateral fluctuations grow with the length in fourth power
$\left\langle y_{pol}(L)^{2}\right\rangle \approx\kappa_{0}^{2}{\small L
^{4}/8$, whereas for long ones $L>>l_{\phi},$ the deviation grows cubically,
$\left\langle y_{pol}(L)^{2}\right\rangle ={\small 2/3}\kappa_{0}^{2}l_{\phi
}{\small L}^{3}.$ From this, the persistence length has consequently two
typical regimes. For $L<<l_{\phi}$ we deduce from Eqs. \ref{lptotal} and
\ref{lpol} that $l_{p}^{\ast}\left( L\right) \approx\left( l_{B
^{-1}+3/8\kappa_{0}^{2}L\right) ^{-1}$. The latter expression implies that
long, untwisted MTs appear increasingly softer with growing length, and the
effective persistence length decays inversely $l_{p}^{\ast}\propto1/L$ for
$L>>1/(3/8\kappa_{0}^{2}l_{B})$ and reaches a limiting value $l_{p}^{\ast
}=l_{B}/\left( 1+2l_{B}l_{\phi}\kappa_{0}^{2}\right) $ for $L>>l_{\phi}.$
For a MT of $L\sim10-20
\mu m$ we would expect $l_{p}^{\ast}\sim10-20\mu m$ - a value $2$ orders of
magnitude smaller than observed in \cite{Pampaloni,Taute}. This decreasing
behavior is in contrast with observations $l_{p}^{\ast}\propto L$ - leading us
to the conclusion that the $0$-twist MTs does not constitute a significant
portion of the experimental data \cite{Pampaloni,Taute} and twist is
necessarily required for the growth of $l_{p}^{\ast}$ with $L.$
Having developed some static consequences of the polymorphic MT we now turn to
its dynamical aspects.
\begin{figure}[ptb]
\begin{center}
\includegraphics[
height=2.2061in ]{Fig9.eps}
\end{center}
\caption{A typical shape of the effective persistence length $l_{p}^{\ast
}\left( L\right) $ for a clamped microtubule as obtained from Eqs.
\ref{lptotal}, \ref{lpol} and \ref{y2pol}. Most generically the curve displays
three different regimes: (I) An initial rapidly decreasing regime where
polymorphic effects become more effective with growing $L$ (softening the
chain), (II) a linearly-growing oscillatory regime - corresponding to the
coherent wobbling movement of the clamped microtubule, (III) an asymptotic
plateau regime, where the helix progressively loses its coherence with growing
$L.$ In this regime, the behavior tends to that of a classical semiflexible
chain yet with a renormalized effective persistence length given by cf. Eq.
\ref{lpinfinity}.
\end{figure}
\begin{figure}[ptb]
\begin{center}
\includegraphics[
height=2.0773in, width=3.3157in ]{Fig10.eps}
\end{center}
\caption{Comparison of theoretical persistence lengths for different values of
the polymorphic phase coherence length $l_{\phi},$ ($l_{B}=25mm,$ $\kappa
_{0}=0.03\mu m^{-1}$ and $q_{0}=0.8\mu m^{-1}$). For $l_{\phi}=\infty,$ the MT
is a (defect free) coherent helix performing the "wobbling motion"(as in Figs.
4 and 8a left panel) . The plateau regime - where elastic fluctuations become
dominant over polymorphic- is reached for very long MT only (not seen in the
Fig.). Finite $l_{\phi}$ reduces the coherent wobbling motion - shortens
region (II) of Fig. 9 - and the plateau regime is reached earlier with
decreasing $l_{\phi}$.
\end{figure}
\subsection{Polymorphic Phase Dynamics}
To describe the MT fluctuation dynamics we consider the total dissipation
functional $P_{diss}=P_{ext}+P_{int}$ which is composed of an internal
dissipation $P_{int}=\frac{1}{2}\xi_{int}\int\dot{\phi}^{2}ds$ (with
$\dot{\phi}\equiv d\phi/dt$) coming from the flipping of lattice states and an
external hydrodynamic\ dissipation $P_{ext}=\frac{1}{2}\xi_{\perp}\int
\overset{\cdot}{\rho}^{2}ds$ associated with the time variation of the MT
deflection $\overrightarrow{\rho}\left( s,t\right) =(x(s,t),y(s,t))$. We
assume that the friction constant (per unit length) $\xi_{\perp}$ of the
helical MT is approximately the friction constant $\xi_{\perp}=4\pi
\eta/\left( \ln\left( 2L/R\right) -1/2\right) $ of a long slender body of
length $L,$ (small)\ radius $R<<L$ moving in a fluid with viscosity $\eta$ at
low Reynolds numbers. The time evolution equation of the phase variable
$\phi\left( s,t\right) $ and the lateral displacement $y\left( s,t\right)
$ (and $x\left( s,t\right) $) are given by the coupled Langevin equations
\begin{equation}
\frac{\delta E_{tot}}{\delta\phi}=-\frac{\delta P_{diss}}{\delta\dot{\phi
}+\Gamma_{\phi} \label{LangevinPhi
\end{equation}
and
\begin{equation}
\frac{\delta E_{tot}}{\delta y}=-\frac{\delta P_{diss}}{\delta\dot{y}
+\Gamma_{\rho
\end{equation}
with $\Gamma_{\phi/\rho}$ the thermal noise terms. In general the lateral
displacement $y\left( s,t\right) $ has contributions from both polymorphic
$y_{pol}(s,t)\approx\kappa_{0}\int_{0}^{s}ds^{\prime}\int_{0}^{s^{\prime}
\sin\left( \phi(\tilde{s},t)+q_{0}s\right) d\widetilde{s}$ and elastic
fluctuations $y_{el}(s,t)\approx\int_{0}^{s}\theta_{el}(s^{\prime
},t)ds^{\prime}$ and the dynamics is highly non-linear. In the regime
$L>>l_{\phi}$ where the helix looses its coherence one expects to retrieve the
dynamics of the usual semiflexible filament with $\tau\sim L^{4}$. However in
the opposite and physically more interesting regime $L<<l_{\phi}$ a new and
different dynamic behavior can be expected. As we learnt from the study of the
static case the effects of polymorphism become more pronounced at shorter
lengths. As we have seen in this regime, the dominant motion is the wobbling
rotation of a coherent helix on a cone where elastic fluctuations become
negligible compared to polymorphic ones i.e. $y(s,t)\approx y_{pol}(s,t)$. In
this regime few polymorphic defects $L<<l_{\phi}$ are present and the phase
can be approximated as $\phi\left( s,t\right) \approx\phi_{0}(t)+\delta
\phi\left( s,t\right) .$Using this decomposition with $\delta\phi\left(
s,t\right) <<1$ we can expand $P_{ext}$ to leading order
\begin{equation}
P_{diss}\approx\frac{1}{2}L\left( \xi_{int}+\xi_{ext}\right) \dot{\phi
_{0}^{2}+O\left( \delta\dot{\phi}^{2}\right)
\end{equation}
with a external friction constant $\xi_{ext}$ given by
\begin{equation}
\xi_{ext}=\frac{\xi_{\perp}\kappa_{0}^{2}}{q_{0}^{4}}(2\left( 1+\cos
Lq_{0}\right) -4\frac{\sin Lq_{0}}{Lq_{0}}+\frac{q_{0}^{2}L^{2}}{3})
\label{Xiexternal
\end{equation}
The evolution of the zero mode $\phi_{0}\left( t\right) $ reduces from the
Langevin equation, Eq. \ref{LangevinPhi}, to $0=-\frac{\delta P_{diss}
{\delta\dot{\phi}_{0}}+\Gamma_{\phi}$ which leads to the equation of motio
\begin{equation}
\frac{d}{dt}\phi_{0}\left( t\right) =\frac{1}{\xi_{tot}}L^{-1}\int_{0
^{L}\Gamma_{\phi}\left( s,t\right) ds \label{phiotime
\end{equation}
with a friction constant $\xi_{tot}=\xi_{int}+\xi_{ext}.$ Therefore $\phi
_{0}\left( t\right) $ satisfies the simple Langevin equation Eq.
\ref{phiotime} corresponding to a simple potential free Brownian motion with
its mean square displacement given by (see Appendix D for a more detailed
explanation)
\begin{equation}
\left\langle \left( \phi_{0}\left( t\right) -\phi_{0}\left( 0\right)
\right) ^{2}\right\rangle =\frac{2k_{B}T}{L\xi_{tot}}t. \label{MSDphio
\end{equation}
In this limit (wobbling mode dominant) we have a roughly rigid helix moving
randomly along a cone and all the physics is contained in the effective
friction coefficient $\xi_{tot}\left( \xi_{int},\kappa_{0},q_{0},L\right) $
and its dependence on the internal dissipation $\xi_{int},$ the helix
parameters $\kappa_{0},$ $q_{0}$ and the length $L$.
For later comparison with experiments we compute the longest relaxation time
$\tau$\ given by the auto-correlation function $\left\langle y_{pol
(s,0)y_{pol}(s,t)\right\rangle \propto e^{-t/\tau}$. Using Eq. \ref{MSDphio},
a short computation (Appendix D) leads t
\begin{equation}
\left\langle y_{pol}(L,0)y_{pol}(L,t)\right\rangle =\left\langle y_{pol
^{2}(L)\right\rangle e^{-t/\tau(L)
\end{equation}
with $\left\langle y_{pol}^{2}(L)\right\rangle =\frac{\kappa_{0}^{2}
{q_{0}^{2}}\left( \frac{L^{2}}{2}+\frac{1-\cos(q_{0}L)}{q_{0}^{2}
-\frac{L\sin(q_{0}L)}{q_{0}}\right) $\ the static mean square displacement
Eq. \ref{y2pol} in the limit $L<<l_{\phi}$\ and with $\tau\left( L\right)
=L\xi_{tot}/k_{B}T$ the longest relaxation time - proportional to the total
friction constant $\xi_{tot}$ - inheriting its length dependence in two
different length regimes. For very short lengths $L\ll l_{c}=\left(
3\xi_{int}\xi_{\perp}^{-1}\right) ^{1/2}q_{0}\kappa_{0}^{-1}$ when the
hydrodynamic dissipation is entirely dominated by internal dissipation we have
a linear scaling
\begin{equation}
\tau\left( L\right) \approx L\xi_{int}/k_{B}T. \label{tau1
\end{equation}
For larger $L>l_{c}$ the wobbling movement through the fluid is the dominant
source of dissipation an
\begin{equation}
\tau\left( L\right) \approx L\xi_{ext}(L)/k_{B}T. \label{Tau2
\end{equation}
with $\xi_{ext}(L)$ given by Eq. \ref{Xiexternal}. Note that the $L$
dependence of $\xi_{ext}$ relies also on the $L$ dependence of $\xi_{\perp
}(L)$ which for simplicity has been modelled as the friction of an ideal
slender tube moving in a liquid. A more precise (but difficult) determination
of $\xi_{\perp}$ could slightly change its variation with $L,$ although not
the general trends of $\xi_{ext}(L).$ In particular if we assume that
$\xi_{\perp}$ is $L$ independent and that $Lq_{0}\gg1$ we have in this regime
the scaling
\begin{equation}
\tau\left( L\right) \approx\frac{\xi_{\perp}}{3k_{B}T}\left( \kappa
_{0}/q_{0}\right) ^{2}L^{3}. \label{Tau3
\end{equation}
So in summary for $\tau$ we expect a cross-over from a linear, internal
dissipation dominated $L$ dependence at short lengths to a cubic length
dependence given by hydrodynamic friction alone.
\section{Comparison with Clamped MT Experiments}
The comparison with experiments \cite{Pampaloni,Taute} which measure lateral
fluctuations of clamped MTs reveals several interesting characteristics that
are in agreement with predictions, cf. Fig. 11. First, the predicted mean
linear growth of $l_{p}^{\ast}\left( L\right) $ agrees with experiments as a
single exponent fit $l_{p}^{\ast}\sim L^{\delta}$ of the data provides
$\delta=1.05$. Besides the linear growth the experimental data reveal a large
spread of $l_{p}^{\ast}$ data points which seems to grow approximately in
proportion to the length. This linearly growing experimental spread is likely
linked to the intrinsic spread of $q_{0}$ values of different MT lattice
populations \cite{Wade,Chretien,Ray,ChretienFuller}. MTs with different number
of protofilaments will display different lattice twists ranging from
$q_{0}\approx2\pi/3\mu m$ \ (for 12 PF\ MTs) to $q_{0}\lesssim2\pi/25\mu m$
\ (for 13 PF\ MTs). Keeping in mind the scaling $l_{p}^{\ast}\left( s\right)
\propto q_{0}^{2}s$ (cf. Eq. \ref{lp longue L}) we would expect more than an
order of magnitude variation of measured $l_{p}^{\ast}$ while the slope
$l_{p}^{\ast}\left( s\right) /s$ should display a constant spread.
Second, the data of Taute et al. \cite{Taute} (Fig. 11, circle) indicate a
non-monotonic dependence with systematic trends over several consecutive data
points. This seems phenomenologically well captured by the oscillatory
behavior $l_{p}^{\ast}\left( L\right) $ from Eq. \ref{lp longue L}. On the
other hand, the data of Pampaloni et al. \cite{Pampaloni} are definitely much
more spread and don't allow such a clear conclusion. Therefore the
non-monotonicity of $l_{p}^{\ast}\left( L\right) $ is at present
experimentally difficult to infer from the two existing experimental data sets
taken together, however it is consistent with data within the error bars. As
mentioned the presence of different lattice populations (even within single
MTs) could give rise to a large spread of experimental data points and an
effective "washing out" of the non-monotonic behavior for different lattices
within the same statistics.
Remarkably the experimental data reveal that the large length plateau $s\gg
l_{\phi}$ where $l_{p}^{\ast}$ would become length independent is not reached
even for longest MTs ($\sim50\mu m$). This is in phenomenological agreement
with the theory - as based on the observed long coherent helices by Venier et
al \cite{Venier} it would imply a very long $l_{\phi}$. The absence of the
plateau in Pampaloni and Taute's data allows a lower estimate of the coherence
length: $l_{\phi}>55\mu m$ which in turn would imply a large coupling constant
$J>4k_{B}T$. The best comparison between theory and experiments (cf. Fig. 11)
gives $l_{B}=25mm$ corresponding to a rather high Young modulus $Y\approx9GPa$
higher than typically reported before
\cite{GITTES,Venier,Mickey,Felgner,Kukimoto,ActtiveMTBending,TAKASONEbuckling,Brangwyne,Janson
. However the present value is well within the range for proteins and protein
tubes with $Y\ $up to $19GPa$ are reported in literature \cite{15 GPa Tubes}.
The higher value of the bare Young modulus extracted form the present theory
should also not be a surprise as in previous models all MT conformational
fluctuations were interpreted as originating from bending deformations alone.
In our model - both elastic and polymorphic fluctuations contribute with the
latter being much softer and giving therefore dominant contribution.
The best fitting helix wave length $\lambda\approx7.5\mu m$ is close to the
expected $6\mu m$ corresponding to the twist
\cite{Wade,Chretien,Ray,ChretienFuller} of the $14$ PF MT population. This is
in agreement with the fact that, - in contrast to the in vivo situation- the
large pitch ($\lambda\approx25\mu m$) $13$PF MTs are likely underrepresented
in the data \cite{Pampaloni,Taute}. Indeed in vitro studies of
taxol-copolymerized MTs display a MT population consisting of a majority of
$14$ PFs ($61\%$) while $13$ PFs ($32\%$) are less represented
\cite{Wade,Chretien,Ray,ChretienFuller}. The in vitro conditions therefore
strongly shift the PF population away from the preferred low twist $13$ PF MT
towards the highly twisted $14$PF MT.
\begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=2.8003in, ]{Fig11.eps}
\end{center}
\caption{Effective persistence length $l_{p}^{\ast}$ as a function of the
position from the attachment point along the clamped MT contour. The
experimental data (stars and circles) \cite{Pampaloni,Taute} and the
theoretical prediction with $l_{B}=25mm,$ $\lambda=7.5\mu m,$ $\kappa_{0
^{-1}=18\mu m,$ $q_{0}l_{\phi}\gg1$.
\label{persistence length
\end{figure}
The estimated $l_{B}$ is larger than in previous studies ($l_{B}\sim1-6mm)$
where however polymorphic fluctuations were neglected. This result leads us to
the conclusion that if polymorphism is partially suppressed one would measure
much larger effective $l_{p}$ - as in fact observed. For example in studies
where 2D slab geometry is used (MTs between 2 close glass slides) an effective
suppression of the 3 dimensional polymorphic helices or their reduced mobility
is expected. A typical observation in such cases is an extensive
\textquotedblleft intrinsic curvature\textquotedblright\ (of previously
unknown origin). Within our theory one could interpret this curvature as
pinned / quenched polymorphic helices prevented from free fluctuations by the
confinement. These effects could in general explain the dramatic variations of
measured $l_{p}$ values based on the presence/ absence of polymorphic
\textquotedblleft softening\textquotedblright\ in different experimental
setups and geometries.
Now, let us consider the clamped MT dynamics as investigated in \cite{Taute}.
A careful analysis of Taute et al's data reveals a peculiar scaling of the
longest relaxation time with the length. Indeed an independent single exponent
fit of the Taute et al data \cite{Taute} gives $\tau\propto L^{\alpha}$ with
$\alpha=2.9$ in the experimental range considered $2.2\mu m<L<28\mu m$. This
peculiar scaling can be understood as originating from hydrodynamic relaxation
of a "wobbling" polymorphic chain. Using only the previously best fitting
parameters of the static data (Fig. 11) $\left( \kappa_{0}/q_{0}\right)
^{2}\approx4.\,8\times10^{-3}$ and $\eta=10^{-3}Pa\cdot s$ (water viscosity)
we find a remarkable correspondence between the theoretical prediction Eq.
\ref{Tau2}, i.e., $\tau_{th}\left( L\right) \approx L\xi_{ext}(L)/k_{B}T$
and data of \cite{Taute}, as shown in Fig.\ 12.\ This 0-parameter prediction
matches well the data for larger lengths. For scaling comparison, it is
interesting to compare the data with the approximate theoretical relaxation
time Eq. \ref{Tau3}, i.e., $\tau\left( L\right) \approx\frac{\xi_{\perp
}{3k_{B}T}\left( \kappa_{0}/q_{0}\right) ^{2}L^{3}$ (expression strictly
valid for $L\gg0.8$ $\mu m$) which has a scaling law in agreement with the
single exponent fit of the data. Considering that $\xi_{\perp}\approx
1.6\eta-2.3\eta$ is roughly length independent in the experimental $L$ range
one can assume $\xi_{\perp}\approx2\eta$ and obtains the prefactor $\tau
_{th}/L^{3}=7.9\times10^{14}s/m^{3}.$ Keeping in mind the simplicity of the
interpretation (and the lack of free parameters therein), this compares very
favorably with the best fit of experimental data slope $\tau_{fit
/L^{3}=6.\,25\times10^{14}s/m^{3}$ (cf. Fig. 12). Note that the approximate
value of $\tau_{th}$ seems to correspond a little better to the data but this
is likely of no physical significance at this level of approximation. Indeed
the neglected elastic modes other than the wobbling as well as the
approximation of the hydrodynamic friction should change the details of the
$L$ dependence of $\xi_{ext}$ and thus of $\tau_{th}$ (but not the general
trends). Without a more precise computation of the dynamics we can be
satisfied with the rather astonishing agreement between theory and experiment
for long MTs. This leads us again to the conclusion that in these experiments
long MTs behave as almost rigid helical polymorphic rotors whose motion is
dominated by the zero energy \textquotedblright wobbling\textquotedblrigh
\ mode and its hydrodynamic dissipation\cite{NOTE}.
For very short MTs we should expect deviation from this simple interpretation.
In this regime the linearly scaling internal dissipation,coming from the
migration of polymorphic defects, should start to dominate over pure
hydrodynamic friction and for sufficiently short MT lengths $L\rightarrow0$ we
could measure $\xi_{int}$ from the limit value of $\tau_{th}/L$. It appears
that for the presently available data $L>2$ $\mu m$ \cite{Taute} this
plateau-regime is not yet fully developed, enabling us to provide only an
upper numerical estimate for the inner dissipation $\xi_{int}\lesssim
4\times10^{-17}Ns$. \begin{figure}[ptbh]
\begin{center}
\includegraphics[
height=3.0684in, width=3.2292in ]{Fig12.eps}
\end{center}
\caption{The experimental microtubule relaxation times \cite{Taute} and the
no-adjustable-parameter theoretical prediction (full line) as obtained from
static data in Fig.11 (with $l_{B}=25mm,$ $\lambda=7.5\mu m,$ $\kappa_{0
^{-1}=18\mu m,$ $q_{0}l_{\phi}\gg1$ ). The dashed line illustrates the long
length approximation Eq. \ref{Tau3} displaying the characteristic cubic
scaling with length.
\end{figure}
\subsection{Very Short MTs}
Comparison with experiments for even shorter chains ($L<\pi/q_{0}$) is more
difficult due to the lack of data in clamped MT experiments ($L>2\mu m$). We
can nevertheless try a comparison with the results by the Dekker group for the
kinesin motor gliding assay of short MTs \cite{GlidingAssay,GlidingAssay2}.
Besides some similarities with the clamped fluctuating MTs, there are a number
of differences between the modelled situation of free 3D MTs and the 2D
gliding assay. In particular the 2D geometry will strongly perturb the
preferentially 3-dimensional helical ground state. The active contribution of
strong motor forces on the trajectory of short MTs is an additional potential
perturbation. Effects of MT buckling and axial MT rotation by kinesins become
likely important. This said and ignoring the differences we can still compare
(in order of magnitude) our and Dekker groups' results
\cite{GlidingAssay,GlidingAssay2}. For micron sized MTs we obtain $l_{p
\sim0.8\mu m$ in approximate agreement with $\sim0.2\mu m$ obtained in
\cite{GlidingAssay}. One should mention that the two cited studies
\cite{GlidingAssay,GlidingAssay2} give strongly different results depending on
whether free gliding or gliding assay with additional electric field were
considered. This difference comes from the larger deformations induced by the
electric field. Therefore the comparison of our theory with the passive
gliding assay appears more appropriate and gives a closer agreement.
\section{Summary}
In summary, we have suggested a new model that connects some of the most
persistent and confusing experimental findings concerning microtubules.
Starting from a rather broad spectrum of (apparently) disconnected
observations we have progressively built the case for a new hypothesis : the
existence of an internal switching of the GDP tubulin dimer within the
microtubule lattice. Why do microtubules become helically wavy, why do they
switch to permanently bent circular states, why do they fluctuate anomalously
when clamped? These three dangling questions became the central pillars for
the present model. Surprisingly, the simple assumption of a bistable
GDP-tubulin seems to explain these otherwise disparate phenomena in a unified
manner. As we know from recent experiments - the bistability hypothesis of
taxol stabilized protofilaments is an empirical fact indeed \cite{Multistable
Tub EM}. We have shown here, that the incorporation of such a bistable tubulin
into a closed elastic lattice changes its free behavior - it introduces strong
conformational competition among the tubulin dimers. Tubulin units on opposite
sides of the tube now start to compete for who is going to switch to the
curved state. The lattice induced frustration does not allow all the tubulin
dimers to minimize their energy individually and to switch to their preferred
states at the same time. The symmetry breaking induced by this frustration
mechanism leads to a global microtubule lattice curving. Remarkably the
curving direction is chosen randomly - and this has profound consequences. The
microtubule can chose between many energetic ground states (as many states as
protofilaments in the lattice). When we graft one end of the microtubule onto
a substrate while still allowing it to chose its bending direction freely the
strange energy degeneracy generates a very unusual thermal motion. In this
case the microtubule's motion follows -roughly speaking- a cone and it rotates
- or "wobbles"- at no energy cost around its attachment tangent. This mode of
motion - which is not to be confused with material frame rotation (which is
strictly prohibited by grafting) - is probably among the most striking
outcomes of the two state GDP-tubulin model. It is exactly this behavior that
allows to consistently explain the measurements of unusual lateral
fluctuations of grafted microtubules \cite{Pampaloni,Taute}.
\section{Perspectives}
We have focussed here on modelling taxol stabilized microtubules and the
question naturally arises if the model proposed here affects the 'real' in
vivo microtubules. On the one hand the 'weakly curved' state that is involved
in the soft polymorphic dynamics as described here seems to be (so far) a
specific signature of taxol stabilized GDP tubulin state. On the other hand we
have argued that the naturally occurring 'high curvature' GDP-tubulin state
could coexist with the straight state in the lattice under in vivo conditions
where MTs are stabilized with MAPs. The involvement of this 'high curvature'
state switching seems to manifest itself in motor driven straight to wavy
transitions of MTs in many living cell systems
\cite{Brangwyne,BicekInvivo,BrangwynneWavyBucking,Borisy,Kaech,SamsonovTau}. A
particularly impressive instance of such a polymorphic switching event in vivo
could be found in the process of axonal retraction where the whole MT
cytoskeleton of the axon undergoes a straight to helical transition and in
turn retracts towards the soma \cite{Baas}. To understand these dramatic
transitions in vivo the present theory has to be advanced and modified in two
manners. First, the effect of large active motor forces (rather than thermal
ones) has to be taken into account. In particular, one expects that under
strong buckling forces even thermodynamically unfavorable states can become
activated and constitute the ground state upon large loads.
Second, in virtually all in vivo experiments MTs are essentially confined in
2D as the containing cells adsorb to the glass substrate and assume a vary
flat 'fried egg' configuration. Consequently, the measured properties will not
necessarily reflect the three-dimensional properties of the molecule. This is
particularly important for a MT transformed to a polymorphic helix state where
the confinement entails naturally a strong deformation (of the initially 3
dimensional ground state). Under confinement the helical bending and torsional
modes become strongly coupled and bring about new physical effects. In
particular a torsionally very soft helix will have a tendency to unwind and
form in extreme cases circular arcs - reminiscent of the rings observed in
gliding assay experiments \cite{Amos}.
Finally, the local action of molecular motors could trigger a switching to the
highly curved state. While for classical motors like kinesin 1 direct evidence
for such a mode of action is still missing its relative kinesin 13
\cite{MCAK,Multistable Tub EM} has a well documented ability to actively
trigger radial bending of protofilaments. Other molecules like katanin
\cite{Katanin,Katanin2} have also been suggested to perturb the lattice and
trigger longer range transitions \cite{KulicMTshear}. This opens the
intriguing question : could classical motors (kinesin 1 and dynein) trigger
cooperative state transitions and even transmit conformational signals along
the tube? Considering the present model for stabilized MTs (where high
cooperativity is inherent to the data interpretation) this idea might not be
far fetched. In fact some evidence towards long range cooperativity of kinesin
binding along the MT was presented by Muto et al.
\cite{KinesinCooperativeBinding} - however these results still await robust reproduction.
This brings us to the question of what experiments should be performed in
order to nail down the polymorphic mechanism or any other mechanism for MT
dynamics. With microtubules being such delicate, subtle and possibly long
range correlated objects (as suggested here) a general rule of thumb for
experiments should be: Treat them more gently (do not confine) and observe
more carefully (look for correlated motion). A simple yet important experiment
would be a systematic direct observation of one-side grafted but otherwise
\emph{completely unconfined} MTs fluorescently labelled \emph{along their full
contour length}. As mentioned the presence of a quasi 2D confinement in thin
chambers as used in most MT experiments so far would perturb the native
helical MT state and should be therefore explicitly avoided. The freely
suspended gold-EM nanogrid attachment geometry as used by Pampaloni et al.
seems particularly suited for that task. Going beyond Pampaloni et al. who
labelled and traced the MT end only (via a bead), the microtubules should be
visualized along the full contour in this geometry. Tracing of several or all
points along the contour should reveal the predicted sinusoid- helical nature
of MT states. The present model predicts a peculiar cooperatively rearranging
helix state with characteristic tell tale curvature correlations between
different lattice positions which are entirely absent for usual semiflexible
filaments. Directly observing such collective motions -like the suggested
"wobbling mode" - while prohibiting trivial spacial rotations that could mask
the effects (by MT grafting) would constitute smoking gun evidence for a
polymorphism related mechanism.
In conclusion, we have proposed a novel model for internal MT lattice
dynamics. We have shown that it accounts for the otherwise mysterious MT
helicity \cite{Venier}, the anomalous length dependent lateral fluctuation
static \cite{Pampaloni,Taute} and dynamic scaling\ \cite{Taute}. The latter
two phenomena appear as mere consequences of the peculiar \textquotedblright
wobbling motion\textquotedblright\ of the polymorphic cooperatively switching
MT lattice. Although most of the observations discussed here are made in vitro
on taxol-stabilized MTs, we provided arguments in favor the existence of
polymorphic MT states in vivo. We speculate that the implied conformational
bistability of tubulin and the allosteric interaction are more than just
nature's way to modulate the elastic properties of its most important
cytoskeletal mechano-element. It could also be a missing piece in the puzzle
of polymerization "catastrophes". Even more intriguingly the predicted
structural cooperativity could allow for long range conformational signalling
along single MTs and turn the latter into an efficient "confotronic" wire
transmitting regulatory signals across the cell.
\section{Acknowledgements}
We acknowledge fruitful discussions with Francesco Pampaloni, Denis
Chr\'{e}tien, Thomas Surrey, Francois N\'{e}d\'{e}lec, Jean-Francois Joanny,
Sergey Obukhov, Linda Amos, Andr\'{e} E.X. Brown and thank Falko Ziebert for
discussion and useful comments on the manuscript.
\section{Appendix}
\subsection{A. Polymorphic phase coherence length}
In this section, we derive the formula $l_{\phi}=\frac{N^{2}b}{8\pi^{2
}\left( 2+e^{2J/k_{B}T}\right) $ for the polymorphic phase coherence length.
To this end we want to calculate the distribution of double junctions that
leads to angular orientation change $\Delta\Phi$ on a scales $l$ much larger
than the tubulin dimer $b$, yet still much smaller than the total length:
$b\ll l\ll L.$ In this domain, at each cross section we have 3 possibilities:
1) State $j=0$ with no double defect. The rotation angle $\Delta\Phi$ is
attached to the internal lattice rotation,\ $\frac{\Delta\Phi}{b}-q_{0}=0$
2)\ State $j=-1$ for a left handed double defect, $\Delta\Phi$ deviates from
the internal twist :\ $\frac{\Delta\Phi}{b}-q_{0}=-\frac{1}{b}\frac{2\pi}{N}$
3)\ State $j=+1$ for a right handed double defect with $\frac{\Delta\Phi
{b}-q_{0}=+\frac{1}{b}\frac{2\pi}{N}.$
On a length $l$ we are throwing a 3 sided dice $l/b$ times and the total
rotation of $\Delta\Phi$ away from optimal twist is $\Delta\Phi-q_{0
l=\frac{2\pi}{N}\sum_{n=1}^{l/b}j_{n}.\ $The variation of the polymorphic
phase with respect to the internal twist is the
\[
\Delta\phi=\frac{\Delta\Phi}{l}-q_{0}=\frac{1}{l}\frac{2\pi}{N}\sum
_{n=1}^{l/b}j_{n
\]
For $l/a>>1$ the law of large numbers implies that the random variable
$\Delta\phi=\frac{1}{l}\frac{2\pi}{N}\sum_{n=1}^{l/b}j_{n}$ becomes Gaussian
distributed
\[
p\left( \Delta\phi\right) \propto e^{-\frac{\Delta\phi^{2}}{2\left\langle
\Delta\phi^{2}\right\rangle }
\]
with mean $\left\langle \Delta\phi\right\rangle =0$ and $\left\langle
\Delta\phi^{2}\right\rangle =\left( \frac{1}{l}\frac{2\pi}{N}\right)
^{2}\left( \frac{l}{b}\right) \left\langle j^{2}\right\rangle $ (as
$\left\langle j_{n}j_{m}\right\rangle =\delta_{nm}\left\langle j^{2
\right\rangle $). The average $\left\langle j^{2}\right\rangle $ is given from
the Boltzmann factors of the three different states $p_{0}=\frac
{1}{1+2e^{-2\beta J}}$ and $p_{\pm1}=\frac{e^{-2\beta J}}{1+2e^{-2\beta J}}$
so that $\left\langle j^{2}\right\rangle =\frac{2e^{-2\beta J}}{1+2e^{-2\beta
J}}$. We can now interpret the quantity $1/\left( 2\left\langle \Delta
\phi^{2}\right\rangle \right) $ as coming from an effective elastic energy
over the interval $l$ by writing $\frac{\Delta\phi^{2}}{2\left\langle
\Delta\phi^{2}\right\rangle }=\frac{1}{2}\beta C_{\phi}l\left( \Delta
\phi\right) ^{2}$ which allows us to identify the effective stiffness
\[
C_{\phi}=kT\frac{N^{2}}{8\pi^{2}}\left( 2+e^{2J\beta}\right) b.
\]
Note that this expression is valid for large enough $J\ $suppressing higher
order defects i.e. in the limit when multiple double defects sitting on a
single lattice site (i.e. $\left\vert j\right\vert >1$)\ can be ignored.
\subsection{B. The variation of the polymorphic modulus}
In this appendix we compute the energy variation due to a deviation of the
polymorphic modulus $\left\vert P\right\vert $ away from its optimal value
$\left\vert P^{\ast}\right\vert $ minimizing the energy; i.e., the change of
the number of switched PFs. We start with the energy density of a MT cross sectio
\begin{equation}
e=\frac{B}{2}\left( \left( \kappa-\kappa_{pol}(p)\right) ^{2}+\kappa
_{1}^{2}\left( \gamma\frac{\pi}{N}p-\sin^{2}\left( \frac{\pi}{N}p\right)
\right) \right)
\end{equation}
whose minimum energy is reached for $p^{\ast}=\frac{N}{2}-\frac{N}{2\pi
}\arcsin\gamma$. Assuming a continuous number of PFs, the energy of a state
with $p=p^{\ast}+\Delta p$ switched PFs reads to quadratic order
\begin{align*}
e(p^{\ast}+\Delta p) & \approx e(p^{\ast})-\frac{\pi^{2}B}{N^{2}}\kappa
_{1}^{2}\cos\left( \frac{2\pi}{N}p^{\ast}\right) \Delta p^{2}\\
& =e(p^{\ast})+\frac{\pi^{2}B}{N^{2}}\kappa_{1}^{2}\sqrt{1-\gamma^{2}}\Delta
p^{2
\end{align*}
where we used $\cos(\pi-\arcsin\gamma)=-\sqrt{1-\gamma^{2}}$. Therefore the
energy variation of a segment of length $l$ reads
\[
\Delta E\approx\frac{\pi^{2}B}{N^{2}}\kappa_{1}^{2}\sqrt{1-\gamma^{2}
{\textstyle\int_{0}^{l}}
ds(p(s)-p^{\ast}(s))^{2
\]
Now using $\left\vert P\left( s\right) \right\vert =\left\vert \sin\left(
\frac{\pi}{N}p\right) \right\vert /\sin(\pi/N)$ we can write the energy
variation to the same (quadratic) order as
\[
\Delta E\approx B\kappa_{1}^{2}\sin^{2}(\pi/N)\sqrt{1-\gamma^{2}
{\textstyle\int_{0}^{l}}
ds((\left\vert P\left( s\right) \right\vert -\left\vert P^{\ast}\right\vert
))^{2
\]
Therefore any deviation of $P$ from its optimum state $P^{\ast}$ is associated
with an energy cost proportional to the length $l$ of the region in the
unfavorable state.
\subsection{C. Persistence length(s)}
A definition of the persistence length, often used in single molecule
experiments, is expressed in terms of the lateral deviation $\overrightarrow
{\rho}=(x(s),y\left( s\right) )$ of a MT clamped at $s=0$ from its
attachment axis : $l_{p}^{\ast}\left( s\right) =2/3s^{3}/\left\langle
\left( \overrightarrow{\rho}\left( s\right) -\left\langle \overrightarrow
{\rho}\left( s\right) \right\rangle \right) ^{2}\right\rangle $ and
$\left\langle ..\right\rangle $ is the statistical average. The equivalence of
the $x$ and $y$ directions implies that $l_{p}^{\ast}\left( s\right)
=1/3s^{3}/\left\langle \left( y\left( s\right) -\left\langle y\left(
s\right) \right\rangle \right) ^{2}\right\rangle .$ The second often used
alternative but more standard definition of the persistence length - the
tangent persistence length- is related to the angular correlation
$l_{p}\left( s-s^{\prime}\right) =\left\vert s-s^{\prime}\right\vert
/V(s-s^{\prime})$ with the variance $V=\left\langle \left( \theta_{y}\left(
s\right) -\theta_{y}\left( s^{\prime}\right) \right) ^{2}\right\rangle $
(by symmetry we have the same expression with $\theta_{x}$). Whereas for an
ideal WLC $l_{p}^{\ast}=l_{p}=l_{B}$ is position and definition independent
this is not the case for a polymorphic chain (see Fig. 13). For small angular
deformations the decoupling of chain's fluctuations into polymorphic and
purely elastic contributions allows to decompose the persistence length as
$l_{p}^{-1}=l_{pol}^{-1}+l_{B}^{-1}$ - this result being valid for both
definitions of the persistence length.
We first focus on the first definition - the clamped persistence length. In
this case the polymorphic persistence length $l_{pol}^{\ast}(s)$ is given b
\begin{equation}
l_{pol}^{\ast}(s)=1/3s^{3}/\left\langle \left( y_{pol}\left( s\right)
-\left\langle y_{pol}\left( s\right) \right\rangle \right) ^{2
\right\rangle \label{lpol1
\end{equation}
where $y_{pol}(s)$ is the lateral polymorphic displacement in the $y$
direction. Integrating over the rotational zero mode readily implies
$\left\langle y_{pol}\left( s\right) \right\rangle =0$ (see Eq. \ref{lpol}).
From Eq. \ref{ypolsquare} one can writ
\[
\left\langle y_{pol}^{2}\left( s\right) \right\rangle =\int_{0}^{s}\int
_{0}^{s}G(s_{1},s_{2})ds_{1}ds_{2
\]
with the angular correlation function $G(s_{1},s_{2})=\left\langle
\theta_{y,pol}(s_{1})\theta_{y,pol}(s_{2})\right\rangle $ given by the
integration over the zero mode
\begin{equation}
G(s_{1},s_{2})=\int_{0}^{2\pi}\frac{d\phi_{0}}{2\pi}G_{0}(s_{1},s_{2},\phi
_{0}) \label{G
\end{equation}
of the angular correlation function at fixed value of $\phi_{0}$, i.e.,
$G_{0}(s_{1},s_{2},\phi_{0})=\left\langle \theta_{y,pol}(s_{1})\theta
_{y,pol}(s_{2})\right\rangle |_{\phi_{0}}$. This last expression, from the
relation $\theta_{y,pol}(s)=\kappa_{0}\int_{0}^{s}\sin\left( \widetilde{\phi
}\left( s^{\prime}\right) +q_{0}s^{\prime}+\phi_{0}\right) ds^{\prime}$
(cf. Eq. \ref{Aphi}) is explicitly given by
\begin{align}
G_{0}(s_{1},s_{2},\phi_{0}) & =\kappa_{0}^{2}\int_{0}^{s_{1}}\int_{0
^{s_{2}}\left\langle \sin\left( \widetilde{\phi}\left( s\right)
+q_{0}s+\phi_{0}\right) \right. \nonumber\\
& \left. \sin\left( \widetilde{\phi}\left( s^{\prime}\right)
+q_{0}s^{\prime}+\phi_{0}\right) \right\rangle |_{\phi_{0}}dsds^{\prime}.
\end{align}
After integration over $\phi_{0}$ and using the known result $\left\langle
\cos\left( \widetilde{\phi}\left( s_{1}\right) -\widetilde{\phi}\left(
s_{2}\right) \right) \right\rangle =e^{-\left\vert s_{1}-s_{2}\right\vert
/2l_{\phi}}$ which results from the WLC type probability distribution of the
field $\widetilde{\phi}$, i.e., $P[\widetilde{\phi}]\sim\exp(-\frac{l_{\phi
}{2}\int_{0}^{L}ds\widetilde{\phi}^{\prime2})$ one obtains the rotational
invariant correlation function in the for
\begin{equation}
G(s_{1},s_{2})=\frac{\kappa_{0}^{2}}{2}\int_{0}^{s_{1}}\int_{0}^{s_{2
}e^{-\frac{\left\vert s-s^{\prime}\right\vert }{2l_{\phi}}}\cos\left(
q_{0}(s-s^{\prime})\right) dsds^{\prime} \label{angular
\end{equation}
Computation of the integrals in Eq. \ref{angular} gives finally the following
expression for the polymorphic contribution of the transverse displacement
\begin{align}
\left\langle y_{pol}(s)^{2}\right\rangle & =\frac{2\kappa_{0}^{2}l_{\phi
}{3(1+4l_{\phi}^{2}q_{0}^{2})^{4}}\left\{ P_{1}-e^{-\frac{s}{2l_{\phi}}
P_{2}\cos\left( q_{0}s\right) \right. \nonumber\\
& \left. +e^{-\frac{s}{2l_{\phi}}}P_{3}\sin(q_{0}s)\right\} \label{lateral
\end{align}
with ${\small P}_{1}(s){\small =24l}_{\phi}^{3}\left( 1-6x+x^{2}\right)
{\small -3l}_{\phi}\left( 1+x-x^{2}-x^{3}\right) {\small s}^{2}$
${\small +}\left( 1+3x+3x^{2}+x^{3}\right) {\small s}^{3},$ ${\small P
_{2}(s){\small =24l}_{\phi}^{3}\left( 1-6x+x^{2}\right) $
${\small +12l}_{\phi}^{2}\left( 1-2x-3x^{2}\right) {\small s}$ and
${\small P}_{3}(s){\small =192l}_{\phi}^{4}{\small q}_{0}{\small (1-x)}$
${\small +24l}_{\phi}^{3}{\small q}_{0}{\small (3+2x-x}^{2}{\small )s}$ where
we have introduced the notation $x=4l_{\phi}^{2}q_{0}^{2}.$
From Eq. \ref{lateral}, we get the polymorphic persistence length
$l_{pol}^{\ast}(s)$ defined in Eq. \ref{lpol1}, and in turn the global
persistence length $l_{p}^{\ast}(s)$ depicted in Fig. 13. Its physical
interpretation is discussed in the main text.
We now consider the second definition of the persistence length $l_{p}\left(
s-s^{\prime}\right) =\left\vert s-s^{\prime}\right\vert /V(s-s^{\prime})$.
From Eq. \ref{angular}, the angular variance $V_{pol}$ can easily be
evaluated
\begin{align}
V_{pol}(s) & =\frac{2\kappa_{0}^{2}l_{\phi}}{1+4l_{\phi}^{2}q_{0}^{2
}\left( s-\frac{2l_{\phi}\left( 1-4q_{0}^{2}l_{\phi}^{2}\right)
}{1+4l_{\phi}^{2}q_{0}^{2}}\right) \nonumber\\
& +\frac{4\kappa_{0}^{2}l_{\phi}^{2}e^{-\frac{s}{2l_{\phi}}}}{\left(
1+4l_{\phi}^{2}q_{0}^{2}\right) ^{2}}\left( \left( 1-4q_{0}^{2}l_{\phi
^{2}\right) \cos\left( q_{0}s\right) \right. \nonumber\\
& \left. -4q_{0}l_{\phi}\sin\left( q_{0}s\right) \right) \label{C3
\end{align}
The resulting persistence length $l_{p}$ (depicted in Fig. 13) shows a rich
behavior similar to the persistence length $l_{p}^{\ast}\left( s\right) $
but displays a distinct functional form from the latter. However as expected,
both curves reach the same asymptotic value at very short and very long MT lengths.
\begin{figure}[ptb]
\begin{center}
\includegraphics[
height=2.6in]{Fig13.eps}
\end{center}
\caption{Different definitions of the persistence length can deviate from each
other for a polymorphic chain. The "clamped persistence length" $l_{p}^{\ast}$
(thick line) and the "tangent persistence length" $l_{p}$ (thin line) (for
$l_{B}=10mm,$ $l_{\phi}=50\mu m,$ $\kappa_{0}=0.03\mu m^{-1}$ and
$q_{0}=0.7\mu m^{-1}$).
\end{figure}
\subsection{D. Zero mode dynamics}
The evolution of the zero mode $\phi_{0}\left( t\right) $ is given by Eq.
\ref{phiotime}
\begin{equation}
\frac{d}{dt}\phi_{0}\left( t\right) =\frac{1}{\xi_{tot}}L^{-1}\int_{0
^{L}\Gamma_{\phi}\left( s,t\right) ds \label{dphiodt
\end{equation}
with a friction constant $\xi_{tot}=\xi_{int}+\xi_{ext}$ where $\xi_{ext}$ is
given by Eq. \ref{Xiexternal}. The correlation function of the thermal white
$\Gamma_{\phi}\left( s,t\right) $ noise is $\left\langle \Gamma_{\phi
}\left( s,t\right) \Gamma_{\phi}\left( s^{\prime},t^{\prime}\right)
\right\rangle =D\delta(s-s^{\prime})\delta(t-t^{\prime})$ with a coefficient
$D$ that can be determined in the following manner. Notice first that
$\phi_{0}$ performs a \ free Brownian motion and its quadratic fluctuations
necessarily satisfy the relation $\left\langle \left( \phi_{0}\left(
t\right) -\phi_{0}\left( 0\right) \right) ^{2}\right\rangle =\frac
{2k_{B}T}{L\xi_{tot}}t$. On another hand integrating Eq. \ref{dphiodt}
\begin{equation}
\phi_{0}\left( t\right) -\phi_{0}\left( 0\right) =\xi_{tot}^{-1
{\displaystyle\int\nolimits_{0}^{t}}
\left( \frac{1}{L}\int_{0}^{L}\Gamma\left( s,t^{\prime}\right) ds\right)
dt^{\prime} \label{phiT2
\end{equation}
and averaging over the white noise the quadratic phase fluctuations one
obtains : $\left\langle \left( \phi_{0}\left( t\right) -\phi_{0}\left(
0\right) \right) ^{2}\right\rangle =\frac{D}{\xi_{tot}^{2}L}t,$ from which
we readily deduce $D=2\xi_{tot}k_{B}T$ - as expected from the fluctuation
dissipation theorem.
The relaxation time is generally given from the time correlation function
$<y_{pol}(s,0)y_{pol}(s,t)>$\ with the lateral position $y_{pol
(s,t)=\frac{\kappa_{0}}{q_{0}^{2}}(sq_{0}\cos\left( \phi_{0}\left( t\right)
+\alpha\right) +\sin\left( \phi_{0}\left( t\right) +\alpha\right)
-\sin\left( q_{0}s+\phi_{0}\left( t\right) +\alpha\right) )$\ obtained
from Eq. \ref{Polydeplacement} with $l_{\phi}>>s$. The average must first take
into account all statistically equivalent values of angular orientations
$\alpha\in\left[ 0,2\pi\right] ,$\ such that $\left\langle y_{pol
(s,0)y_{pol}(s,t)\right\rangle =\int\nolimits_{0}^{2\pi}\left\langle
y_{pol}(s,0)y_{pol}(s,t)\right\rangle _{\alpha}\frac{d\alpha}{2\pi}$\ and we
obtain
\begin{equation}
\left\langle y_{pol}(s,0)y_{pol}(s,t)\right\rangle =\left\langle y_{pol
^{2}(s)\right\rangle \left\langle \cos(\phi_{0}\left( t\right) -\phi
_{0}\left( 0\right) )\right\rangle
\end{equation}
with $<y_{pol}^{2}(s)>=\frac{\kappa_{0}^{2}}{q_{0}^{2}}\left( \frac{s^{2}
{2}+\frac{1-\cos(q_{0}s)}{q_{0}^{2}}-\frac{s\sin(q_{0}s)}{q_{0}}\right)
$\ corresponding to the static result Eq. \ref{lateral} in the limit
$s/l_{\phi}<<1$. With $\ref{phiT2}$\ defining a simple Gaussian random walk
processes one straightforwardly obtain
\begin{equation}
\left\langle \cos(\phi_{0}\left( t\right) -\phi_{0}\left( 0\right)
)\right\rangle =e^{-t/\tau
\end{equation}
with the relaxation time given by
\begin{equation}
\tau=L\frac{\xi_{tot}}{k_{B}T}.
\end{equation}
\subsection{E. Comment on MT surface attachment and the robustness of
"wobbling"}
Throughout this work we have assumed that the free rearrangement of the
polymorphic lattice states is not significantly hindered by the covalent
surface attachment of the MT, as e.g. performed in \cite{Pampaloni} and
\cite{Taute}. This assumption is integral to the "wobbling" motion and in turn
to understanding the static and dynamic data scaling. It therefore deserves a
closer consideration.
In the experiments by Pampaloni \cite{Pampaloni} et al. and Taute et al.
\cite{Taute} the adsorbed MT part is attached to a gold (electron microscopy
grid) surface via thiol groups. It is likely that $\approx$ 1 - 2
protofilaments will establish localized chemical contacts with the gold
microgrid. While a substantial perturbation of the dimer like e.g.
denaturation appears unlikely, it is unclear to what extent this procedure
will perturb the inner (polymorphic) dynamics of the entire tubulin dimer
units. In principle one can anticipate two plausible scenarios that would to a
varying degree interfere with the free "wobbling" motion:
S1) Due to high cooperativity (large coupling $J$) the polymorphic state
transition can propagate within a certain penetration depth into the adsorbed
(straight-planar) MT section.
S2) The cooperativity is too weak to compete with the constraints imposed by
the surface (including chemical perturbations) and the polymorphic transition
does not propagate into the straight adsorbed MT section.
In both cases we have a non vanishing deflection angle between a forced
(adsorbed) planar section and the free helical section direction - causing
effectively the characteristic MT "kink" at the surface interface. However the
rotational mobility of this "kink" (wobbling mode) which is integral to our
theory will be affected in slightly different manner.
If in case S1 in the adsorbed section the polymorphic order parameter P can
rearrange to some extent (by switching the monomer states without causing a
detectable deformation) except for possibly in the few surface interacting
dimers, then the effects of the "wobbling" motion will be hindered only mildly
in the following sense. To retrieve the anomalous lateral fluctuations it is
indeed enough for the wobbling angle $\phi_{0}$ to move freely in a certain
non-vanishing angular interval. A single complete or multiple rotations of the
order parameter $\vec{P}$ are not strictly necessary for the "hinge" effect -
and they are in fact equivalent in lateral projection (as in experiment) to
the motion of the wobbling angle $\phi_{0}$ in the smaller interval
$[-\pi/2,+\pi/2]$. Note that even smaller intervals than that will lead to a
similar phenomenology (in particular dynamic and static variable scalings with
length). Thus the conical hinge-like motion is in a sense robust with respect
to a limited local rotational hindrance perturbation in the adsorbed region.
In the scenario S2 the situation is somehow simpler as the polymorphic
dynamics of the adsorbed region is not involved in the process (the
polymorphic order parameter vanishes there: $\vec{P}=0$). Wobbling is realized
through a coherent rearrangement of the free MT section alone - without a
strong coupling to the adsorbed region.
Although both attachment scenarios S1 and S2 appear to some extent plausible,
at present it is difficult to make reliable statements about their respective
likelihood. In fact only a posteriori we can cautiously state that based on
the experimental static and dynamic measurement evidence: the chain "wobbles"
to a high enough extent to display the effects that we observe.
|
1,108,101,565,446 | arxiv |
\section{Introduction}
Today, algorithmic systems driven by large amounts of data
are increasingly being used in all aspects of life.
Often, such systems are being used to assist, or, even replace
human decision-making.
This increased dependence on algorithms has given rise to the field of algorithmic fairness, where the goal is to ensure that algorithms do not exhibit biases towards specific individuals, or groups of users (see e.g., \cite{fairness-study} for a survey).
We also live in a connected world where networks, be it, social, communication,
interaction, or cooperation networks, play a central role.
However, surprisingly, fairness in networks
has received less attention.
Link analysis algorithms, such as Pagerank~\cite{pagerank}, take a graph as input and use the structure of the graph to determine the relative importance of its nodes.
The output of the algorithms is a numerical weight for each node that reflects its importance. The weights are
used to produce a ranking of the nodes, but also as input features in a variety of
machine learning algorithms including classification~\cite{spam-classifier}, and
search result ranking~\cite{pagerank}.
Pagerank performs a random walk on the input graph, and weights the nodes according to the stationary probability distribution of this walk.
At each step, the random walk restarts with probability $\gamma$, where
the restart node is selected according to a``jump'' distribution vector $\mathbf{v}$.
Since its introduction
in the Google search engine, Pagerank has been the
cornerstone algorithm in several applications (see, e.g., \cite{pagerank-survey}).
Previous research on the fairness of centrality measures has considered only degrees and found biases that arise as a network
evolves
\cite{glass-ceiling,glass-ceiling-recommend}, or has studied general notions of fairness in graphs based on the premise that similar nodes should get similar outputs \cite{inform}.
In this work, we focus on the fairness of the Pagerank algorithm.
As in previous research, we view fairness as lack of discrimination against a
protected group defined by the value of a sensitive attribute, such as, gender, or race \cite{fairness-study}.
We operationalize this view by saying that a link analysis algorithm is \textit{$\phi$-fair}, if the fraction of the total weight allocated to the members of the protected group is $\phi$.
The value of $\phi$ is a parameter that can be used to implement different fairness policies. For example, by setting $\phi$ equal to the ratio of the protected nodes in the graph, we ask that the protected nodes have a share in the weights proportional to their share in the population, a property also known as demographic parity
\cite{fairness-awarness}.
We also consider \emph{targeted} fairness, where we focus on a specific subset of nodes to which we want to allocate weights in a fair manner.
We revisit Pagerank through the lens of our fairness definitions,
and we consider the problem of defining families of Pagerank algorithms that are fair.
We also define the \emph{utility loss} of a fair algorithm as the difference between its output and the output of the original Pagerank algorithm, and we pose the problem of achieving fairness while minimizing utility.
We consider two approaches for achieving fairness.
The first family of algorithms we consider is the \emph{fairness-sensitive} Pagerank family which exploits the jump vector $\mathbf{v}$.
There has been a lot of work on modifying the jump vector to obtain variants of Pagerank biased towards a specific set of nodes.
The topic-sensitive Pagerank algorithm \cite{topic-sensitive} is such an example, where the probability is assigned to nodes of a specific topic.
In this paper, we take the novel approach of using the jump vector to achieve $\phi$-fairness.
We determine the conditions under which this is feasible
and formulate the problem of finding the jump vector that achieves $\phi$-fairness while minimizing utility loss
as a convex optimization problem.
Our second family of algorithms takes a microscopic view of fairness by looking at the behavior of each individual node in the graph.
Implicitly, a link analysis algorithm assumes that links in the graph correspond to endorsements between the nodes. Therefore, we can view each node, as an agent that \emph{endorses} (or \emph{votes for}) the nodes that it links to. Pagerank defines a process that takes these individual actions of the nodes and transforms them into a global weighting of the nodes.
We thus introduce, the \textit{locally fair PageRank algorithms}, where each individual node acts fairly by distributing its own pagerank to the protected and non-protected groups according to the fairness ratio $\phi$.
Local fairness defines a dynamic process that can be viewed as a \textit{fair random walk}, where \emph{at each step} of the random walk (not only at convergence), the probability of being at a node of the protected group is $\phi$.
In our first locally fair PageRank algorithm, termed the \textit{neighborhood locally fair} Pagerank algorithm, each node distributes its pagerank fairly among its immediate neighbors, allocating a fraction $\phi$ to the neighbors in the protected group, and $1-\phi$ to the neighbors in the non-protected group.
The \textit{residual-based locally fair} Pagerank algorithms generalizes this idea.
Consider a node $i$ that has less neighbors in the protected group than $\phi$. The node distributes an equal portion of its pagerank to each of its neighbors and a residual portion $\delta(i)$
to members in the protected group but not necessarily in its own neighborhood.
The residual is allocated based on \textit{a residual redistribution policy}, which allows us to control the fairness policy.
In this paper, we exploit a residual redistribution policy that minimizes the utility loss.
We then define a stronger fairness requirement, termed universal personalized fairness, that asks that the derived personalized pageranks of all nodes are fair. We prove that the locally fair algorithms achieve also universal personalized fairness.
Surprisingly, the locally fair algorithms are the \emph{only} family of algorithms with this property. Thus, we show that an algorithm is locally fair, if and only if, it is universally personalized fair.
We use real and synthetic datasets to study the conditions under which {{Pagerank}} and personalized {{Pagerank}} are fair.
We also evaluate both quantitatively and qualitatively the output of our fairness-aware algorithms.
In summary, in this paper, we make the following contributions:
\begin{itemize}
\item We initiate a study of fairness for the {{Pagerank}} algorithm.
\item We propose the fairness-sensitive Pagerank family that modifies the jump vector so as to achieve fairness, and the locally fair Pagerank family that guarantees that individually each node behaves in a fair manner.
\item We prove that local fairness implies universal personalized fairness and also that this is the only family of algorithms with this property, establishing an equivalence between local fairness and universal personalized fairness.
\item We perform experiments on several datasets to study the conditions under which Pagerank unfairness emerges and evaluate the utility loss for enforcing fairness.
\end{itemize}
The remainder of this paper is structured as follows.
In Section \ref{sec:definitions}, we provide definitions of fairness and we formulate our problems. In Sections \ref{sec:fairness-sensitive} and
\ref{sec:local-fair}, we introduce the fairness sensitive and
the locally fair families of {{Pagerank}} algorithms.
In Section \ref{sec:universal}, we discuss personalized fairness and we show an equivalence between local and universal personalized fairness. The results of our experimental evaluation are presented in Section \ref{sec:experiments}. Finally, we present related research in Section \ref{sec:related-work} and our conclusions in Section \ref{sec:conclusions}.
\section{Definitions}
\label{sec:definitions}
In this section, we first present background material and then we define Pagerank fairness.
\subsection{Preliminaries}
The {{Pagerank}} algorithm~\cite{pagerank} pioneered link analysis for weighting and ranking the nodes of a graph. It was popularized by its application in the Google search engine, but it has found a wide range of applications in different settings~\cite{pagerank-survey}. The algorithm takes as input a graph $G = (V,E)$, and produces a scoring vector, that assigns a weight to each node $v \in V$ in the graph. The scoring vector is the stationary distribution of a random walk on the graph $G$.
The {{Pagerank}} random walk is a random walk with restarts. It is parameterized by the value $\gamma$, which is the probability that the random walk will restart at any step.
The node from which the random walk restarts is selected according to the jump vector $\mathbf{v}$, which defines a distribution over the nodes in the graph.
Typically, the jump probability is set to $\gamma = 0.15$, and the jump vector is set to the uniform vector.
The ``organic'' part of the random walk is governed by the transition matrix $\mathbf{P}$, which defines the transition probability $P[i,j]$ between any two nodes $i$ and $j$. The transition matrix is typically defined as the normalized adjacency matrix of graph $G$. Special care is required for the sink nodes in the graph, that is, nodes with no outgoing edges. In our work, we adopt the convention that, when at a sink node, the random walk performs a jump to a node chosen uniformly at random~\cite{pagerank-survey}. That is, the corresponding zero-rows in the matrix $\mathbf{P}$ are replaced by the uniform vector.
The Pagerank vector $\mathbf{p}$ satisfies the equation:
\begin{equation}
\mathbf{p}^T= (1-\gamma) \mathbf{p}^T \mathbf{P} + \gamma \, \mathbf{v}^T
\label{eqn:PR}
\end{equation}
It can be computed either by solving the above equation, or by iteratively applying it to any initial probability vector.
The Pagerank algorithm is fully defined by the three parameters we described above: the transition matrix $\mathbf{P}$, the restart probability $\gamma$, and the restart (or jump) vector $\mathbf{v}$. Different settings for these parameters result in different algorithms.
Given a graph $G = (V,E)$, let $\mathcal{PR}(G)$ denote the family of all possible Pagerank algorithms on graph $G$. Each algorithm in $\mathcal{PR}(G)$ corresponds to a triplet $(\mathbf{P}(G), \gamma, \mathbf{v}(G))$ for the parameters of the random walk. This is a very general family that essentially includes all possible random walks defined over the nodes of graph $G$.
We will refer to the algorithm that uses the typical settings as the \emph{original} Pagerank algorithm, $\textsc{OPR}$, and use $\mathbf{p}_O$ to denote its pagerank vector.
Several variations of the original Pagerank algorithm have been proposed, that modify the above parameters to achieve different goals~\cite{pagerank-survey}.
We are interested in defining \emph{fair} Pagerank algorithms.
\subsection{Fair Pagerank}
We focus on graphs where a set of nodes defines a protected group based on the value of some protected attribute.
For example, in the case of social, collaboration, or citation networks where each node is a person, protected attributes may correspond to gender, age, race, or religion. In the following for simplicity, we assume binary such attributes, but our algorithms can be extended for the general case.
We consider two types of nodes, red and blue nodes, and the corresponding groups denoted $R$ and $B$ respectively. Group $R$ is the protected group.
We denote with $r = \frac{|R|}{n}$, and $b = \frac{|B|}{n}$, the fraction of nodes that belong to the red and blue group respectively.
Let $\textsc{PR}\in\mathcal{PR}(G)$ be a Pagerank algorithm on graph $G$. We will use $\textsc{PR}(u)$ to denote the pagerank mass that $\textsc{PR}$ assigns to node $u$, and, abusing the notation, $\textsc{PR}(R)$ to denote the total pagerank mass that $\textsc{PR}$ assigns to the nodes in the red group (for the blue group, $\textsc{PR}(B) = 1-\textsc{PR}(R))$.
We will say that $\textsc{PR}$ is \emph{fair}, if it assigns weights to each group according to a specified ratio $\phi$.
\begin{definition} [Pagerank Fairness]
Given a graph $G = (V,E)$ containing the protected group $R\subseteq V$, and a value $\phi \in (0,1)$, a Pagerank algorithm $\mathrm{PR} \in \mathcal{PR}(G)$ is $\phi$-fair on graph $G$, if $\mathrm{PR}(R) = \phi$.
\end{definition}
The ratio $\phi$ may be specified so as to implement specific affirmative action policies,
or other fairness enhancing interventions.
For example, $\phi$ may be set in accordance to the 80-percent rule advocated by the US
Equal Employment Opportunity Commission (EEOC), or some other formulation of disparate impact \cite{disparate-impact}.
Setting $\phi$ = $r$, we ask for a fair Pagerank algorithm that assigns weights proportionally to the sizes of the two groups.
In this case, fairness is analogous to demographic parity, i.e., the requirement that the demographics of those receiving a
positive outcome are identical to the demographics of the
population as a whole \cite{fairness-awarness}.
Our goal is to define fair Pagerank algorithms. We say that a \emph{family} of Pagerank algorithms $\textsl{FPR} \subseteq \mathcal{PR}(G)$ is $\phi$-fair if all the Pagerank algorithms in the family are $\phi$-fair. The first problem we consider is to find such families of algorithms.
\begin{problem}
Given a graph $G = (V,E)$ containing a protected group of nodes $R \subseteq V$, and a value $\phi \in (0,1)$, define a family of algorithms $\textsl{FPR} \subseteq \mathcal{PR}(G)$ that is $\phi$-fair.
\end{problem}
We can achieve fairness by modifying the parameters of the Pagerank algorithm.
For the following, we assume the jump probability $\gamma$ to be fixed, and we only consider modifications to the transition matrix $\mathbf{P}$ and the jump vector $\mathbf{v}$. A $\phi$-fair family of algorithms is defined by a specific process, parameterized by $\phi$, for defining $\mathbf{P}$ and $\mathbf{v}$.
A fair Pagerank algorithm will clearly output a different weight vector than the original Pagerank algorithm.
We assume that the weights of the original Pagerank algorithm carry some \emph{utility}, and use these weights to measure the \emph{utility loss} for achieving fairness. Concretely, if $\mathbf{f}$ is the output of a fair Pagerank algorithm $\textsc{FPR}$ and $\mathbf{p}_O$ is the output of the original Pagerank algorithm $\textsc{OPR}$, we define the utility loss of $\textsc{FPR}$ as: $L(\textsc{FPR}) = L(\mathbf{f},\mathbf{p}_O) = \|\mathbf{f}-\mathbf{p}_O\|^2$.
The second problem we consider is finding a fair algorithm that minimizes the utility loss.
\begin{problem}
Given a $\phi$-fair family of algorithms $\textsl{FPR} \subset \mathcal{PR}(G)$, find an algorithm $\mathrm{PR}$ $\in \textsl{FPR}$ that minimizes the utility loss $L(\mathrm{PR})$.
\end{problem}
Finally, we introduce an extension of the fairness definition, termed \emph{targeted fairness}, that asks for a fair distribution of
weights among a specific set of nodes $S$ that is given as input. The subset $S$ contains a protected group of nodes $S_R$.
Targeted fairness asks that a fraction $\phi$ of the pagerank mass that $\textsc{PR}$ assigns to $S$ goes to the protected group $S_R$.
For example, assume a co-authorship graph $G$, where $S$ is the set of authors of a particular male-dominated field. We want to allocate to the female authors $S_R$ in this field a fraction $\phi$ of the total pagerank allocated to $S$.
\begin{definition} [Targeted Fairness]
Given a graph $G = (V,E)$, a subset of nodes $S\subseteq V$ containing a protected group $S_R\subseteq S$, and a value $\phi \in (0,1)$, a Pagerank algorithm $\mathrm{PR}\in \mathcal{PR}(G)$, is targeted $\phi$-fair on the subset $S$ of $G$, if $\mathrm{PR}(S_R) = \phi \mathrm{PR}(S)$.
\end{definition}
The two problems we defined above can also be defined for targeted fairness.
\section{Fairness Sensitive PageRank}
\label{sec:fairness-sensitive}
Our first family of algorithms achieves fairness by keeping the transition matrix $\mathbf{P}$ fixed and changing the jump vector $\mathbf{v}$ so as to meet the fairness criterion. We denote this family of algorithms as the \emph{Fairness-Sensitive Pagerank} (\textsl{FSPR}) algorithms.
\subsection{The \textsl{FSPR} Algorithm}
First, we note that that pagerank vector $\mathbf{p}$ can be written as a linear function of the jump vector $\mathbf{v}$.
Solving Equation (\ref{eqn:PR}) for $\mathbf{p}$,
we have that $\mathbf{p}^T = \mathbf{v}^T\mathbf{Q}$, where
\[
\mathbf{Q} = \gamma \left[\mathbf{I} - (1-\gamma)\mathbf{P} \right]^{-1}
\]
Note that if we set $\mathbf{v} = \mathbf{e}_j$, the vector with $\mathbf{e}_j[j] = 1$ and zero elsewhere, then $\mathbf{p}^T = \mathbf{Q}_{j}^T$, the $j$-th row of matrix $\mathbf{Q}$. Therefore, the row vector $\mathbf{Q}_{j}^T$ corresponds to the personalized pagerank vector of node $j$. The pagerank vector $\mathbf{p}$ is a linear combination of the personalized pagerank vectors of all nodes, as defined by the jump vector.
Let $\mathbf{p}[R]$ denote the pagerank mass that is allocated to the nodes of the protected category.
We have that
$$
\mathbf{p}[R] = \sum_{i\in R} \left(\mathbf{v}^T \mathbf{Q} \right) [i]
= \mathbf{v}^T \left( \sum_{i\in R} \mathbf{Q}_{i} \right)
= \mathbf{v}^T \mathbf{Q}_R
$$
where $\mathbf{Q}_{i}$ is the $i$-th column of matrix $\mathbf{Q}$, and $\mathbf{Q}_R$ is the vector that is the sum of the columns of $\mathbf{Q}$ in the set $R$. $\mathbf{Q}_R[j]$ is the total personalized pagerank that node $j$ allocates to the red group.
For the algorithm to be fair, we need $\mathbf{p}[R] = \phi$.
Thus, our goal is to find a jump vector $\mathbf{v}$ such that $\mathbf{v}^T \mathbf{Q}_R = \phi$. Does such a vector always exist? We prove the following:
\begin{lemma}
\label{lemma:solution-feasibiliy}
Given the vector $\mathbf{Q}_R$, there exists a vector $\mathbf{v}$ such that $\mathbf{v}^T \mathbf{Q}_R = \phi$, if and only if, there exist nodes $i,j$ such that $\mathbf{Q}_R[i] \leq \phi$ and $\mathbf{Q}_R[j] \geq \phi$
\end{lemma}
\begin{proof}
We have
$
\mathbf{p}[R]= \mathbf{v}^T \mathbf{Q}_R = \sum_{j = 1}^N \mathbf{v}[j]\mathbf{Q}_R[j]
$,
with $0 \leq \mathbf{v}[j] \leq 1$. If $\mathbf{v}^T \mathbf{Q}_R = \phi$, there must exist $i,j$ with $\mathbf{Q}_R[i] \leq \phi$, and $\mathbf{Q}_R[j] \geq \phi$.
Conversely, if there exists two such entries $i,j$, then we can find values $\pi_i$ and $\pi_j$, such that
$\pi_i \mathbf{Q}_R[i] + \pi_j \mathbf{Q}_R[j] = \phi$ and $\pi_i + \pi_j = 1$.
\end{proof}
\noindent {\bf Complexity.} Note that there is a very efficient way to compute the personalized pagerank, $\mathbf{Q}_R[j]$, that node $j$ allocates to the red group. We can add a red and a blue absorbing node in the random walk and estimate the probability for each node $j$ to be absorbed by the corresponding absorbing node. This can be done in the time required for running Pagerank. Thus, it is possible to compute the $\mathbf{Q}_R$ vector without doing matrix inversion to compute $\mathbf{Q}$.
\subsection{Minimizing Utility Loss}
\label{sec:optimization-fair-sensitive}
An implication of Lemma~\ref{lemma:solution-feasibiliy} is that, in most cases, there are multiple jump vectors that give a fair pagerank vector. We are interested in the solution that minimizes the utility loss.
To solve this problem we exploit the fact that the utility loss function $L(\mathbf{p}_\mathbf{v},\mathbf{p}_O) = \|\mathbf{p}_\mathbf{v} - \mathbf{p}_O\|^2$, where $\mathbf{p}_\mathbf{v}$ is the fair pagerank vector and $\mathbf{p}_O$ the original vector, is convex. We also can express the fairness requirement as a linear constraint. Thus, we define the following convex optimization problem.
\begin{equation*}
\begin{aligned}
& \underset{\mathbf{x}}{\text{minimize}} & & \|\mathbf{x}^T \mathbf{Q} - \mathbf{p}_O\|^2 \\
& \text{subject to} & & \mathbf{x}^T \mathbf{Q}_R = \phi\\
& & & \sum_{i = 1}^{n} \mathbf{x}[i] = 1 \\
& & & 0 \leq \mathbf{x}[i] \leq 1, \; i = 1, \ldots, n \\
\end{aligned}
\end{equation*}
This problem can be solved using standard convex optimization solvers.
In our work, we used the CVXOPT software package\footnote{https://cvxopt.org/}.
The complexity of the algorithm is dominated by the computation of matrix $\mathbf{Q}$ which requires a matrix inversion. We can speed up this process by exploiting the fact that the rows of $\mathbf{Q}$ are personalized pagerank vectors, which can be computed (in parallel) by performing multiple random walks. We can improve performance further using approximate computation, e.g., \cite{approximatepr}.
\subsection{Targeted Fairness \textsl{FSPR} Algorithm}
We can formulate a similar convex optimization problem for the targeted fairness problem.
Let $\mathbf{Q}_S = \sum_{i\in S} \mathbf{Q}_i$ be the sum of columns of $\mathbf{Q}$ for the nodes in $S$, and $\mathbf{Q}_{S_R} = \sum_{i\in S_R} \mathbf{Q}_i$be the sum of columns of $\mathbf{Q}$ for the nodes in $S_R$. We define a convex optimization problem that is exactly the same as in Section~\ref{sec:optimization-fair-sensitive}, except for the fact that we replace the constraint $\mathbf{x}^T \mathbf{Q}_R = \phi$ with the constraint $\mathbf{x}^T \mathbf{Q}_{S_R} = \phi \mathbf{x}^T \mathbf{Q}_S $.
\section{Locally Fair PageRank}
\label{sec:local-fair}
Our second family of fair Pagerank algorithms, termed \emph{Locally Fair Pagerank}
(\textsl{LFPR}), takes a microscopic view of fairness,
by asking that \textit{each individual node} acts fairly, i.e., each node distributes its own pagerank to red and blue nodes fairly.
In random walk terms, local fairness defines a dynamic process
that can be viewed as a random walk that is
fair at each step, and not just at convergence.
The \textsl{LFPR} contains all Pagerank algorithms, where all rows of the transition matrix $\mathbf{P}$ are $\phi$-fair vectors. That is, for every node $i \in V$, $\sum_{j \in R} P[i,j] = \phi$. Also, the jump vector $\mathbf{v}$ is $\phi$-fair: $\sum_{j \in R} \mathbf{v}[j] = \phi$.
We now consider specific algorithms from the family of locally fair algorithms.
\subsection{The Neighborhood \textsl{LFPR} Algorithm}
We first consider a node that treats its neighbors fairly by allocating a fraction $\phi$ of its pagerank to its red neighbors
and the remaining $1 - \phi$ fraction to its blue neighbors.
In random walk terms, at each node the probability of transitioning to a red neighbor is $\phi$ and the probability of transitioning to a blue neighbor $1-\phi$.
Formally, we define the \textit{neighborhood locally fair pagerank} ({{\sc LFPR$_N$}}) as follows.
Let $out_R(i)$ and $out_B(i)$ be the number of outgoing edges from node $i$ to red nodes and blue nodes respectively. We define $\mathbf{P}_R$ as the stochastic transition matrix that handles transitions to red nodes, or random jumps to red nodes
if such links do not exist:
\[
\mathbf{P}_R[i,j] = \left \{
\begin{tabular}{cl}
$\frac{1}{out_R(i)}$, & if $(i, j)$ $\in$ E and $j$ $\in$ $R$ \\
$\frac{1}{|R|}$, & if $out_R(i) = 0$ and $j$ $\in$ $R$ \\
0, & otherwise \\
\end{tabular}
\right.
\]
The transition matrix
$\mathbf{P}_B$ for the blue nodes is defined similarly. The transition matrix $\mathbf{P}_N$ of the {{\sc LFPR$_N$}} algorithm is defined as:
\begin{center}
$\mathbf{P}_N = \phi \, \mathbf{P}_R + (1 - \phi) \, \mathbf{P}_B
$
\end{center}
We also define a $\phi$-fair jump vector $\mathbf{v}_N$ with $\mathbf{v}_N[i]$ = $\frac{\phi}{|R|}$, if $i \in R$, and
$\mathbf{v}_N[i]$ = $\frac{1-\phi}{|B|}$, if $i \in B$.
The neighborhood locally-fair pagerank vector $\mathbf{p}_N$ is defined as:
\[
\mathbf{p}_N^T = (1 - \gamma) \mathbf{p}_N^T \mathbf{P}_N + \gamma \, \mathbf{v}_N^T
\]
\subsection{The Residual-based \textsl{LFPR} Algorithms}
We consider an alternative fair behavior for individual nodes.
Similarly to the {{\sc LFPR$_N$}} algorithm, each node $i$ acts fairly by respecting the $\phi$ ratio when distributing its own pagerank to red and blue nodes. However, now node $i$
treats its neighbors the same, independently of their color and assigns to each of them the same portion of its pagerank. When a node is in a ``biased'' neighborhood, i.e., the ratio of its red neighbors is different than
$\phi$, to be fair, node $i$ distributes only a fraction of its pagerank to its neigbors, and the remaining portion of its pagerank
to nodes in the underrepresented group. We call the remaining portion \emph{residual} and denote it by $\delta(i)$.
How $\delta(i)$ is distributed to the underrepresented group is determined by a \textit{residual policy}.
Intuitively, this corresponds to a fair random walker that upon arriving at a node $i$, with probability 1-$\delta(i)$ follows one of $i$'s outlinks and with probability $\delta(i)$
jumps to one or more node belonging to the group that is locally underrepresented.
We now describe the algorithm formally.
We divide the (non sink) nodes in $V$ into two sets, $L_R$ and $L_B$, based on the fraction of their red and blue neighbors.
Set $L_R$ includes
all nodes $i$ such that
$out_R(i)/out(i) < \phi$, where $out(i)$ the out-degree of node $i$,
that is, the nodes for which the ratio of red nodes in their neighborhoods is smaller than the required $\phi$ ratio. These nodes have a residual that needs to be distributed to red nodes.
Analogously, $L_B$ includes
all nodes $i$ such that
$out_R(i)/out(i) \geq \phi$,
that have a residual to be distributed to blue nodes.
Consider a node $i$ in $L_R$.
To compute $\delta_R(i)$ note that each neighbor of $i$ gets a fraction $\rho_R(i) = \frac{1-\delta_R(i)}{out(i)}$ of $i$'s pagerank. The residual $\delta_R(i)$ of $i$'s pagerank goes to red nodes. In order for node $i$ to be $\phi$-fair, we have:
\begin{equation}
\frac{1-\delta_R(i)}{out(i)}out_R(i) + \delta_R(i)= \phi \label{eq:excess1}
\end{equation}
Solving for $\delta_R(i)$ and using the fact that $out(i) = out_R(i)+out_B(i)$ we get $\delta_R(i) = \phi - \frac{(1-\phi)\,out_R(i)}{out_B(i)}$, and $\rho_R(i) = \frac{1-\phi}{out_B(i)}$.
\iffalse
Consider a node $i$ in $L_R$.
Each neighbor of $i$ gets the same portion of $i$'s pagerank, let $\rho_R(i)$ be this portion. To attain the $\phi$ ratio, the residual $\delta_R(i)$ of $i$'s pagerank goes to the red nodes. Portions $\rho_R(i)$ and $\delta_R(i)$ must be such that:
\begin{equation}
(1- \phi) \,\, (out_R(i) \, \rho_R(i) + \delta_R(i)) = \phi \,\, (out_B(i) \, \rho_R(i)) \label{eq:excess1}
\end{equation}
\begin{equation}
out_R(i) \,\rho_R(i) + out_B(i) \, \rho_R(i) + \delta_R(i) = 1 \label{eq:excess2}
\end{equation}
From Equations (\ref{eq:excess1}) and (\ref{eq:excess2}), we get $\rho_R(i)$ = $\frac{1-\phi}{out_B(i)}$ and the residual is $\delta_R(i) = \phi - \frac{(1-\phi)\,out_R(i)}{out_B(i)}$.
\fi
Analogously, for a node $i$ in $L_B$, we get a residual $\delta_B(i) = (1 -\phi) - \frac{\phi\,out_B(i)}{out_R(i)}$ that goes to the blue nodes, and $\rho_B(i)$ = $\frac{\phi}{out_R(i)}$.
For a sink node $i$, we assume that $i$ belongs to both $L_R$ and $L_B$ with residual $\delta_R(i)$ = $\phi$ and $\delta_B(i)$ = $1- \phi$.
\vspace*{0.1in}
\noindent \textit{Example.}
Consider a node $i$ with 5 out-neighbors, 1 red and 4 blue, and let $\phi$ be $0.5$.
This is a ``blue-biased''node, that is, $i$ $\in$ $L_R$.
With the residual algorithm, each of $i$'s neighbors gets
$\rho_R(i)$ = $0.5/4 = 1/8$ portion of $i$'s pagerank, resulting in red neighbors getting $1/8$ and blue neighbors $4/8$ of i's pagerank. The residual $\delta_B(i)$ = $3/8$ goes to nodes in the red group so as to attain the $\phi$ ratio and make $i$ fair.
In terms of the random walker interpretation, a random walker that arrives at $i$, with probability $5/8$ chooses one of $i$'s outlinks uniformly at random and
with probability $3/8$ jumps to nodes in the red group. \hfill$\qed$.
Formally, we define $\mathbf{P}_L$
as follows:
\[
\mathbf{P}_L[i, j] = \left \{
\begin{tabular}{cl}
$\frac{1-\phi}{out_B(i)}$, & if $(i, j)$ $\in$ $E$ and $i \in L_R$\\
$\frac{\phi}{out_R(i)}$, & if $(i, j)$ $\in$ $E$ and $i \in L_B$ \\
$0$ & otherwise
\end{tabular}
\right.
\]
\iffals
Let $\mathbf{\delta}_R$ be the vector carrying the red residual, that is,
$\mathbf{\delta}_R[i] = \phi - \frac{(1-\phi)\,out_R(i)}{out_B(i)}$, if $i \in L_R$ and 0 otherwise.
Similarly, let $\mathbf{\delta}_B$ be the vector carrying the blue residual, that is,
$\delta_B(i) = (1 -\phi) - \frac{\phi\,out_B(i)}{out_R(i)}$, if $i \notin L_B$ and 0 otherwise.
We have a total red residual $\Delta_R = \mathbf{p}_L^T \, \mathbf{\delta}_R $ and a total blue residual
$\Delta_B = \mathbf{p}_L^T \, \mathbf{\delta}_B$, where $\mathbf{p}_L$ is the locally fair pagerank vector.
\f
$\mathbf{P}_L$ handles the transitions of nodes to their neighborhood.
To express the residual distribution policy, we introduce matrices $\mathbf{X}$ and $\mathbf{Y}$, that
capture the policy for distributing the residual to red and blue nodes respectively.
Specifically, $\mathbf{X}[i, j]$ denotes the portion of the $\delta_R(i)$ of node $i$ $\in$ $L_R$ that goes to node $j$ $\in$ $R$, and
$\mathbf{Y}[i, j]$ the portion of the $\delta_B(i)$ of node $i$ $\in$ $L_B$ that goes to node $j$ $\in$ $B$.
The locally-fair pagerank vector $\mathbf{p}_L$ is defined as:
\[
\mathbf{p}_L^T = (1 - \gamma) \mathbf{p}_L^T \, (\mathbf{P}_L + \mathbf{X} + \mathbf{Y}) + \gamma \, \mathbf{v}_N^T
\]
\noindent \textbf{Residual Distribution Policies.} The $\mathbf{X}$ and $\mathbf{Y}$ allocation matrices allow us the flexibility to
specify appropriate policies for distributing the residual.
For example,
the {{\sc LFPR$_N$}} algorithm is a special case of the residual-based algorithm, with
\vspace*{-0.05in}
\[
\mathbf{X}_N[i, j] = \left \{
\begin{tabular}{cl}
$\frac{\delta_R(i)}{out_R(i)}$, & if $i \in L_R$, $(i,j) \in E$, and $j \in R$ \\
$\frac{\delta_R(i)}{|R|}$, & if $i \in L_R$, $out_R(i) = 0$, and $j \in R$ \\
0 & otherwise \\
\end{tabular}
\right.
\]
and $\mathbf{Y}_N[i, j]$ defined analogously.
We also consider residual policies where all nodes follow the same policy in distributing their residual, that is, each red node gets the same portion of the red residuals
and each blue node the same portion and the blue residuals. In this case, the residual policy is expressed through two (column) vectors $\mathbf{x}$ and $\mathbf{y}$, with
$\mathbf{x}[j]$ = 0 if $j \in B$ and $\mathbf{y}[j]$ = 0, $j \in R$.
Each node $i \in L_R$ sends a fraction $\delta_R(i)\mathbf{x}[j]$ of its pagerank to red node $j$, while each node $i \in L_B$ sends a fraction $\delta_B(i)\mathbf{y}[j]$ of its pagerank to blue node $j$.
Let $\mathbf{\delta}_R$ be the vector carrying the red residual, and
$\mathbf{\delta}_B$ the vector carrying the blue residual.
We have:
\begin{equation*}
\mathbf{p}_L^T = (1 - \gamma)\mathbf{p}_L^T\, (\mathbf{P}_L + \mathbf{\delta}_R \, \mathbf{x}^T + \mathbf{\delta}_B \, \mathbf{y}^T) + \gamma \, \mathbf{v}_N^T.
\end{equation*}
We define two locally fair {{Pagerank}} algorithms based on two intuitive policies of distributing the residual:
\vspace{0.05in}
\noindent
The \textit{Uniform Locally Fair {{Pagerank}}} ({{\sc LFPR$_U$}}) algorithm distributes the residual uniformly. Specifically,
we define the vector $\mathbf{x}$, as $\mathbf{x}[i]$ = $\frac{1}{|R|}$ for $i \in R$,
and the vector $\mathbf{y}$, as $\mathbf{y}[i]$ = $\frac{1}{|B|}$, for $i \in B$.
\vspace{0.05in}
\noindent
The \textit{Proportional Locally Fair {{Pagerank}}} ({{\sc LFPR$_P$}}) algorithm distributes the residual proportionally to the original pagerank weights $\mathbf{p}_O$
Specifically,
we define the vector $\mathbf{x}$, as
$\mathbf{x}[i]$ = $\frac{ \mathbf{p}_O[i] }{\sum_{i \in R}\mathbf{p}_O[i]}$, for $i \in R$,
and the vector $\mathbf{y}$, as $\mathbf{y}[i]$ =
$\frac{ \mathbf{p}_O[i] }{\sum_{i \in B}\mathbf{p}_O[i]}$, for $i \in B$.
\subsection{Fairness of the \textsl{LFPR} Algorithms}
Although each node acts independently of the other nodes in the network,
this microscopic view of fairness
results in a macroscopic view of fairness. Specifically, we prove the following theorem.
\begin{theorem}
The locally fair {{Pagerank}} algorithms are $\phi$-fair. \label{theorem:local}
\end{theorem}
\iffals
\begin{proof}
We must show that $\sum_{v \in R} \mathbf{p}_N(u) = \phi$.
Since each node in the graph gives a portion $\phi$ of its pagerank to red nodes, we have
\[\sum_{v \in R} \mathbf{p}_N(u) = \sum_{v \in V} \phi \, \mathbf{p}_N(u)
\]
which proves the theorem.
\end{proof}
\fi
\begin{proof}
Let $\mathbf{e}_R$ denote the vector with 1's at the red nodes and zero at the blue nodes. The amount of pagerank that vector $\mathbf{p}_L$ gives to the red nodes can be expressed as $\mathbf{p}_L^T\mathbf{e}_R$. Let $\mathbf{P}_D$ = $\mathbf{P}_L + \mathbf{X} + \mathbf{Y}$, we have:
\[
\mathbf{p}_L^T\mathbf{e}_R= (1-\gamma)\mathbf{p}_L^T \mathbf{P}_D\mathbf{e}_R + \gamma \mathbf{v}_N^T\mathbf{e}_R
\]
By design we have that $\mathbf{v}_N^T\mathbf{e}_R = \phi$. For transition matrix $\mathbf{P}_D$, due to the local fairness property, we know that each row has probability $\phi$ of transitioning to a red node. Therefore, $\mathbf{P}_D\mathbf{e}_R = \phi\mathbf{e}$, where $\mathbf{e}$ is the vector of all ones. Note that since $\mathbf{p}_L$ is a distribution we have that $\mathbf{p}_L^T\mathbf{e} = 1$. We thus have:
$\mathbf{p}_L^T\mathbf{e}_R= (1-\gamma)\phi + \gamma\phi = \phi$.
\end{proof}
\subsection{Minimizing Utility Loss}
We now consider how to distribute the residual so as to minimize the utility loss of the locally fair Pagerank. We denote this algorithm as {{\sc LFPR$_O$}}. To this end,
we compute the $\mathbf{x}$ and $\mathbf{y}$ residual distribution vectors by formulating an optimization problem.
We can write the vector $\mathbf{p}_L$ as a function of the vectors $\mathbf{x}$ and $\mathbf{y}$ as follows:
\[
\mathbf{p}_L^T(\mathbf{x},\mathbf{y}) = \gamma \, \mathbf{v}_N^T \left[\mathbf{I} - (1-\gamma)(\mathbf{P}_L + \delta_R\, \mathbf{x}^T + \delta_B\, \mathbf{y}^T) \right]^{-1}
\]
We can now define the optimization problem of finding the vectors $\mathbf{x}$ and $\mathbf{y}$ that minimize the loss function $L(\mathbf{p}_L, \mathbf{p}_O) = \|\mathbf{p}_L(\mathbf{x},\mathbf{y}) - \mathbf{p}_O\|^2$ subject to the constraint that the vectors $\mathbf{x}$ and $\mathbf{y}$ define a distribution over the nodes in $R$ and $B$ respectively.
Since our problem is not convex, we implement a Stochastic Random Search algorithm for solving it, that looks at randomly selected directions for the gradient, and selects the one that causes the largest decrease.
We enforce the distribution constraints
by adding a penalty term
$\lambda$ $\left ( (\sum_{i = 1}^{n} \mathbf{x}_i - 1)^2 + (\sum_{i = 1}^{n} \mathbf{y}_i - 1)^2\right )$.
We enforce the positivity constraints
through proper bracketing at the line-search step.
The complexity of the algorithm is $O\left ( I\cdot K\cdot T_{PR}\right )$, where $I$ is the number of iterations, $K$ the number of random directions that are examined, and $T_{PR}$ the cost of running Pagerank. In our experiments, $I$ and $K$ are in the order of a few hundreds.
\subsection{Targeted Fairness \textsl{LFPR} Algorithms}
We can apply the locally fair algorithms
to the targeted fairness problem.
Let $S_R$ and $S_B$ be the red and blue nodes in the set $S$ respectively, and let $I_S$ be the set of in-neighbors of $S$.
The idea is that the nodes in $I_S$ should distribute their pagerank to $S_R$ and $S_B$ fairly,
such that the
portion that goes to nodes in $S_R$ is a $\phi$ fraction of the total pagerank that goes to the set $S$.
We can implement the same redistribution policies as in the case of the neighborhood local and the residual-based local fair algorithms.
We also need the (global) jump vector $\mathbf{v}$ to obey the $\phi$ ratio for the nodes in $S$. We can achieve this by redistributing the probability $|S|/n$ of the jump vector according to the $\phi$ ratio.
\section{Personalized Fairness}
\label{sec:universal}
A special case of the {{Pagerank}} algorithm is when the restart vector is defined to be the unit vector $\mathbf{e}_i$ that puts all the mass at a single node $i$. For any {{Pagerank}} algorithm $\textsc{PR} \in \mathcal{PR}$, we can define the corresponding personalized algorithm $\textsc{PR}_i$ by setting $\mathbf{v} = \mathbf{e}_i$.
The output of the algorithm $\textsc{PR}_i$ is a probability vector, where $\textsc{PR}_i(u)$ is the probability that a random walk that always restarts at node $i$ is at $u$ after infinite number of steps. We say that node $i$ allocates this probability to node $u$. Personalized random walks have found several applications in network analysis~\cite{pagerank-survey}. For example, the probability $\textsc{PR}_i(u)$ can be seen as a measure of proximity between node $i$ and node $u$, and it has been used for recommendations.
For a personalized Pagerank algorithm $\textsc{PR}_i$, we define $\textsc{PR}_i(R)$ and $\textsc{PR}_i(B)$ to be the probability that node $i$ allocates to the red and blue groups respectively. Recall that if $\mathbf{v}$ is the jump vector, then $\textsc{PR}(R) = \sum_{i\in V} \mathbf{v}[i] \textsc{PR}_i(R)$. We can think of the probability $\textsc{PR}_i(R)$, as a measure of how a specific node $i$ ``views'' the red group, while $\textsc{PR}(R)$ captures the value that the network places on the red nodes on aggregate.
We could define fairness for the $\textsc{PR}_i$ algorithm using the standard fairness definition. However, note that since the random walk jumps with probability $\gamma$ to node $i$ at every step, this immediately adds probability $\gamma$ to the category of node $i$. This probability is due to the random jump and not due to the ''organic'' random walk, and the structure of the graph. We thus subtract this probability, and we define the vector $\overline{\textsc{PR}_i}$, where $\overline{\textsc{PR}_i}(i) = \textsc{PR}_i(i)-\gamma$, and $\overline{\textsc{PR}_i}(u) = \textsc{PR}_i(u)$, for $u \neq i$. Another way to think of this is that an amount $\gamma$ of probability is reserved for restarting, and the remaining $1-\gamma$ is allocated through the random walk. This is the probability mass that we want to allocate fairly. We say that the \emph{personalized} Pagerank algorithm $\textsc{PR}_i$ is $\phi$-fair if $\overline{\textsc{PR}_i}(R) = \phi(1-\gamma)$.
Intuitively, fairness of $\textsc{PR}_i$ implies that node $i$ treats the red and blue groups fairly. For example, if we think of $\overline{\PR_i}(R)$ as a proximity measure, and $\phi = 0.5$, $\phi$-fairness implies that node $i$ is equally close to the red and blue groups. Given that this probability is often used for recommendations, this has also implications to the fairness of the recommendations.
\begin{figure}[]
\includegraphics[width = \columnwidth]{books_hist_sep2.pdf}
\caption{Histogram of personalized red ratio values.}
\label{books-histogram}
\end{figure}
Note that a Pagerank algorithm $\textsc{PR}$ may be fair, while the corresponding personalized Pagerank algorithms are not. In Figure~\ref{books-histogram}, we consider the original Pagerank algorithm $\textsc{OPR}$, and we show the histogram of the $\overline{\textsc{OPR}_i}(R)$ values for the \textsc{books} dataset (described in Section~\ref{sec:experiments}), for the red and blue nodes, in red and blue respectively. For this dataset, we have that $r = 0.47$ and $\textsc{OPR}(R)$ is 0.46. That is, the original Pagerank algorithm is almost fair for $\phi = r$. However, we observe that the distribution of the $\overline{\textsc{OPR}_i}(R)$ values is highly polarized.
The values for the red nodes (shown in red) are concentrated in the interval $[0.8,1]$, while the values for the blue nodes (in blue) are concentrated in the interval $[0,0.2]$. Thus, although the network as a whole has a fair view of the two groups, the individual nodes are highly biased in favor of their own group.
Motivated by this observation, we consider a stronger definition of fairness, where given an algorithm $\textsc{PR}$ we require that \emph{all} derivative Personalized Pagerank versions of this algorithm are fair. That is, it is not sufficient that the algorithm treats the red group fairly on aggregate, but we require that each individual node is also fair.
\begin{definition}[Universal Personalized Fairness]
Given a graph $G = (V,E)$ containing the protected group $R\subseteq V$, and a value $\phi \in (0,1)$, a Pagerank algorithm $\textsc{PR} \in \mathcal{PR}(G)$ is universally personalized $\phi$-fair on graph $G$, if for every node $i \in V$, the personalized Pagerank algorithm $\textsc{PR}_i$ is personalized $\phi$-fair.
\end{definition}
Since we want all personalized algorithms to be fair, universal personalized fairness (universal fairness for short) is a property of the transition matrix $\mathbf{P}$ of the Pagerank algorithm. Since the \textsl{FSPR} family does not change the matrix $\mathbf{P}$, it is not universally fair. We will show that the locally fair algorithms are universally fair. Furthermore, we can prove that universally fair algorithms are locally fair. Therefore, universal fairness is equivalent to local fairness.
\begin{theorem}
A Pagerank algorithm $\textsc{PR}$ is universally personalized $\phi$-fair if and only if it is locally fair.
\end{theorem}
\begin{proof}
We first prove that if an algorithm $\textsc{PR}$ is locally fair then it is personalized fair. The proof is similar to that of Theorem~\ref{theorem:local}.
Let $\mathbf{p}_i$ denote the personalized pagerank vector of algorithm $\textsc{PR}_i$. We know that
$
\mathbf{p}_i^T = (1-\gamma)\mathbf{p}_i^T\mathbf{P} + \gamma \mathbf{e}_i^T
$
where $\mathbf{e}_i$ is the vector with 1 at the position $i$, and 0 everywhere else.
The amount of probability that $\textsc{PR}_i$ allocates to the red category can be computed as $\textsc{PR}_i(R) = \mathbf{p}_i^T \mathbf{e}_R$, where $\mathbf{e}_R$ is the vector with 1 at the positions of all red nodes and 0 everywhere else. Multiplying the equation for $\mathbf{p}_i^T$ with $\mathbf{e}_R$ we have:
\[
\textsc{PR}_i(R) = (1-\gamma)\mathbf{p}_i^T\mathbf{P}\mathbf{e}_R + \gamma \mathbf{e}_i^T\mathbf{e}_R
\]
Since $\textsc{PR}$ is fair, for every row of the transition matrix $\mathbf{P}$, the probability of transitioning to a red node is $\phi$. Therefore we have that $\mathbf{P} \mathbf{e}_R = \phi\mathbf{e}$, where $\mathbf{e}$ is the vector of all 1's. Also, since $\mathbf{p}_i$ defines a distribution $\mathbf{p}_i^T\mathbf{e} = 1$. Therefore $(1-\gamma)\mathbf{p}_i^T\mathbf{P}\mathbf{e}_R = \phi(1-\gamma)$. We have:
\[
\textsc{PR}_i(R) = \phi(1-\gamma) + \gamma\mathbf{e}_i^T \mathbf{e}_R
\]
The value of the second term $\gamma\mathbf{e}_i^T \mathbf{e}_R$ depends on whether the node $i$ is red or blue. If $i$ is blue, $\gamma\mathbf{e}_i^T \mathbf{e}_R = 0$, and we have $\textsc{PR}_i(R) = \phi(1-\gamma)$. If $i$ is red, $\gamma\mathbf{e}_i^T \mathbf{e}_R = \gamma$, and thus $\textsc{PR}_i(R) = \phi(1-\gamma) + \gamma$, which proves our claim.
For the opposite direction we make use of the fact that the pagerank vector can be written as $\mathbf{p}^T = \mathbf{v}^T\mathbf{Q}$, where $\mathbf{v}$ is the jump vector and $\mathbf{Q} = \gamma\left[\mathbf{I} - (1-\gamma)\mathbf{P}\right]^{-1}$.
%
%
From Section~\ref{sec:fairness-sensitive} we know that
the $i$-th row of matrix $\mathbf{Q}$ is equal to the personalized pagerank vector $\mathbf{p}_i$. The product $\mathbf{r} = \mathbf{Q}\mathbf{e}_R$ is a vector where $\mathbf{r}[i] = \textsc{PR}_i(R)$. We have assumed that the $\textsc{PR}$ algorithm is universally personalized $\phi$-fair. Therefore $\mathbf{r}[i] = \phi(1-\gamma) + \gamma$ if $i$ is red, and $\mathbf{r}[i] = \phi(1-\gamma)$ if $i$ is blue. That is, $\mathbf{r} = \phi(1-\gamma)\mathbf{e} + \gamma\mathbf{e}_R$. Using the fact that $\mathbf{r} = \mathbf{Q}\mathbf{e}_R$, and that $\mathbf{Q}\mathbf{e} = \mathbf{e}$, since $\mathbf{Q}$ is stochastic, we have the following derivations:
\begin{align}
\mathbf{Q}\mathbf{e}_R & = \phi(1-\gamma)\mathbf{Q}\mathbf{e} + \gamma\mathbf{e}_R \nonumber\\
\mathbf{Q}^{-1}\mathbf{Q}\mathbf{e}_R & = \phi(1-\gamma)\mathbf{Q}^{-1}\mathbf{Q}\mathbf{e} + \gamma\mathbf{Q}^{-1}\mathbf{e}_R \nonumber\\
\mathbf{e}_R & = \phi(1-\gamma)\mathbf{e}
+ \gamma\frac1\gamma \left(\mathbf{I} - (1-\gamma)\mathbf{P} \right)\mathbf{e}_R \nonumber\\
\mathbf{e}_R & = \phi(1-\gamma)\mathbf{e} + \mathbf{e}_R - (1-\gamma) \mathbf{P} \mathbf{e}_R \nonumber\\
\mathbf{P} \mathbf{e}_R & = \phi \mathbf{e} \nonumber
\end{align}
The last equation means that the probability of transitioning from any node in the graph to a red node is $\phi$, which proves our claim.
\end{proof}
The theorem holds also when we consider targeted fairness. We can prove that an algorithm is universally personalized targeted fair, if and only if it is locally fair. We omit the details of the proof due to lack of space.
\section{Experimental Evaluation}
\label{sec:experiments}
Our goal is to evaluate Pagerank fairness in different kinds of networks, identify the conditions under which Pagerank unfairness emerges and evaluate the effect of the proposed fair Pagerank algorithms. Previous research has shown that homophily and size imbalance may lead to degree unfairness \cite{glass-ceiling,glass-ceiling-recommend,xyz}. Is this the case for Pagerank unfairness?
Specifically, we address the following three research questions:
\vspace*{0.03in}
\noindent \textbf{RQ1:} Under which conditions are the original Pagerank and personalized Pagerank algorithms fair?
\noindent \textbf{RQ2:} What is the utility loss incurred by the proposed fair Pagerank algorithms in different networks?
\noindent \textbf{RQ3:} What are the qualitative characteristics of the proposed fair Pagerank algorithms?
\begin{figure}[]
\centering
{\includegraphics[width = 0.4\textwidth]{real_violinplots.pdf}}
\caption{Distribution of the red personalized pagerank of the red and blue nodes for the real datasets, $\phi$-fairness when the red pagerank is $r$ (\sc{books} $r$ = 0.47, \sc{blogs} $r$ = 0.48, \sc{dblp} $r$ = 0.17, and \sc{twitter} $r$ = 0.61).}
\label{fig:real-violin-plots}
\end{figure}
\vspace*{0.03in}
\noindent \textbf{Datasets.}
We use both real and synthetic datasets. Our real datasets are the following:
\begin{itemize}
\item {\sc{books}}: A network of books about US politics where edges between books represented co-purchasing\footnote{\url{http://www-personal.umich.edu/~mejn/netdata/}}.
\item {\sc{blogs}}: A directed network of hyperlinks between weblogs on US politic \cite{blogs-dataset}.
\item {\sc{dblp}}: An author collaboration network constructed from DBLP including a subset of data mining and database conferences.
\item {\sc{twitter}}: A political retweet graph from \cite{nr}.
\end{itemize}
The characteristics of the real datasets are summarized in Table \ref{table:real}.
To infer the gender in {\sc{dblp}}, we used the python \textit{gender guesser} package\footnote{\url{https://pypi.org/project/gender-guesser/}}.
Regarding $homophily$,
we measure for each group, the percentage of the edges that are \textit{cross-edges} that is they point to nodes of the other group divided by the expected
number of such edges. We denote these quantities as $\textit{cross}_R$ and $\textit{cross}_B$.
Values significantly smaller than 1 indicate that the corresponding group exhibits homophily \cite{network-book}.
Synthetic networks are generated using the biased preferential attachment model introduced in \cite{glass-ceiling}. The graph evolves over time as follows.
Let $G_t = (V_t, E_t)$ and $d_t(v)$ denote the graph and the degree of node $v$ at time $t$, respectively.
The process starts with an arbitrary connected graph $G_0$, with $n_0 \, r$ red and $n_0 \,(1 - r)$ blue nodes.
At time step $t + 1$, $t > 0$, a new node $v$ enters the graph. The color of $v$ is red with probability $r$ and blue with probability $1-r$. Node $v$ chooses to connect with an existing node $u$ with probability
$\frac{d_t(u)}{\sum_{w \in G_{t} d_t(w)}}$. If the color of the chosen node $u$ is the same with the color of the new node $v$, then an edge between them is inserted with probability $\alpha$; otherwise an edge is inserted with probability
$1-\alpha$. If no edge is inserted, the
process of selecting a neighbor for node $v$ is repeated until an edge is created.
\begin{figure*}[]
\centering
\subfigure[{symmetric, $r$ = 0.3}]{
{\includegraphics[width = 0.22\textwidth]{sym_varyHom_size_03_utility_plot.pdf}}
}
\subfigure[{asymmetric, $r$ = 0.3}]{
{\includegraphics[width=0.22\textwidth]{asym_varyHom_size_03_utility_plot4.pdf}}
}
\subfigure[{symmetric, $r$ = 0.5}]{
{\includegraphics[width = 0.22\textwidth]{sym_varyHom_size_05_utility_plot.pdf}}
}
\subfigure[{asymmetric, $r$ = 0.5}]{
{\includegraphics[width=0.22\textwidth]{asym_varyHom_size_05_utility_plot1.pdf}}
}
\caption{Utility loss for synthetic networks, $\phi$ = 0.5.}
\label{fig:synth-utility-homo}
\end{figure*}
\begin{figure}[]
\centering
\subfigure[{$r$ = 0.3}]{
{\includegraphics[width = 0.22\textwidth]{sym_varyPhi_size_03_utility_plot.pdf}}
}
\subfigure[{$r$ = 0.5}]{
{\includegraphics[width = 0.22\textwidth]{sym_varyPhi_size_05_utility_plot4.pdf}}
}
\caption{Utility loss for the synthetic datasets, $\alpha$ = 0.5.}
\label{fig:synth-utility-phi}
\end{figure}
\begin{figure*}[]
\centering
\subfigure[{\sc{books}}]{
{\includegraphics[width = 0.22\textwidth]{books_utility_plot.pdf}}
}
\subfigure[{\sc{blogs}}]{
{\includegraphics[width = 0.22\textwidth]{blogs_utility_plot.pdf}}
}
\subfigure[{\sc{dblp}}]{
{\includegraphics[width = 0.22\textwidth]{dblp_course_utility_plot.pdf}}
}
\subfigure[{\sc{twitter}}]{
{\includegraphics[width = 0.22\textwidth]{twitter_utility_plot2.pdf}}
}
\caption{Utility loss for the real networks, (original red pagerank, \sc{books}: 0.48, \sc{blogs}: 0.33, \sc{dblp}: 0.16, and \sc{twitter}: 0.57).}
\label{fig:real-utility}
\end{figure*}
Parameter $r$ controls the group size imbalance.
Parameter $\alpha$ controls the level of homophily: $\alpha < 0.5$ corresponds to heterophily,
$\alpha = 0.5$ to neutrality and $\alpha > 0.5$ to homophily.
We consider: (a) a symmetric case, where $\alpha$ is the same for both groups and (b)
an asymmetric case, where we set $\alpha$ = 0.5 for the blue group, making it neutral, and vary $\alpha$ for the red group.
The datasets and code are available in GitHub\footnote{\url{https://github.com/SotirisTsioutsiouliklis/FairLaR}}.
\subsection{When is Pagerank Fair?}
We study the conditions under which the original Pagerank and personalized Pagerank
algorithms are fair. We assume that the algorithms are fair, if they respect demographic parity, that is, if each group gets pagerank equal to its ratio in the overall population ($\phi$ = $r$). For brevity, we shall call the (personalized) pagerank allocated to red nodes
\textit{red (personalized) pagerank} and the (personalized) pagerank allocated to blue nodes \textit{blue (personalized) pagerank} .
First, we study homophily and size imbalance using synthetic datasets.
In Figure \ref{fig:local-synthetic-all}(a), we report the red pagerank
for the symmetric and in Figure \ref{fig:local-synthetic-all}(b) for the asymmetric case.
Fairness corresponds to the identity line (red pagerank = $r$).
Values above the line indicate unfairness towards the blue group, while values below the line unfairness towards the red group.
We also plot the distribution of the red personalized pagerank
in Figures \ref{fig:sym-violin-plots} and \ref{fig:asym-violin-plots}
for the symmetric and asymmetric case respectively.
To test whether the red personalized pagerank of a node depends on its color,
we plot two distributions, one for the red personalized pagerank of the red nodes and one for the red personalized pagerank of the blue nodes.
Distributions are plotted in the form of violin plots. Personalized pagerank fairness corresponds to the case in which the two distributions overlap, with their mean on value $r$.
Note that when a group is homophilic, it is expected that a large part of its personalized pagerannk goes to nodes of its own color, e.g., red homophilic nodes have large red personalized pageranks.
\noindent \textbf {Symmetric case} (Figures \ref{fig:local-synthetic-all}(a) and \ref{fig:sym-violin-plots}): When both groups are homophilic ($\alpha$ = 0.7, 0.9), the nodes tend to form two clusters, one with red nodes and one with blue nodes
sparsely connected to each other.
This leads to an almost fair pagerank (with a very slight unfairness towards the minority group), but highly unfair personalized pageranks.
On the contrary, when there is
heterophily ($\alpha = 0.1, 0.3$),
there are no clusters, nodes tend to connect with nodes of the opposite color, and the larger group favors the smaller one. In this case, the pagerank and personalized pageranks of both the blue and the red nodes are all unfair towards the majority group. This is especially evident when the imbalance in size is large (small $r$).
\noindent \textbf {Asymmetric case} (Figures \ref{fig:local-synthetic-all}(b) and \ref{fig:asym-violin-plots}):
When the red nodes are homophilic ($\alpha$ = 0.7, 0.9), the red group keeps the pagerank to itself. As a result both pagerank and personalized pageranks are unfair towards the blue nodes, especially for larger $r$.
When the red nodes are heterophilic ($\alpha$ = 0.1, 0.3), the red group favors the blue group, and as a result, both the pagerank and the personalized pagerank are unfair towards the red nodes, especially for larger $r$.
Thus, independently of the size $r$,
pagerank is unfair to the blue (the neutral) group in case of homophily, and unfair to the red group in the case of heterophily.
\noindent \textbf {Universal fairness:} The only case when both pagerank and personalized pageranks are all fair is in a neutral network ($\alpha = 0.5$) with same-size groups ($r$ = 0.5) (middle violin plots in Figures
\ref{fig:sym-violin-plots}(c) and \ref{fig:asym-violin-plots}(c)).
\noindent \textbf {Real datasets:}
For the real datasets, we report the red pagerank in Table \ref{table:real} ($p_R$ value) and plot the distributions of the red personalized pagerank of the blue and red nodes in Figure \ref{fig:real-violin-plots}.
For {\sc books}, there is no size imbalance and there is strong symmetric homophily leading to fair pagerank and highly unfair personalized pageranks.
For {\sc blogs}, there is no size imbalance, but the blue group is slightly more homophilic, which leads both to unfairness in pagerank for the red group, and unfairness of the personalized pagerank of the blue nodes towards the red ones.
For {\sc dblp}, we have large size imbalance with the red group being the minority but almost neutral behavior in terms of homophily, leading to an almost fair pagepank, and the majority of personalized pageranks being fair.
Finally, for {\sc twitter}, the red group is larger than the blue group but less homophilic which leads to a slight unfairness towards the red group for both pagerank and personalized pageranks.
\subsection{What is the Utility Loss for Fairness?}
We now look into the utility loss
for achieving fairness.
We can view utility loss for each network as a measure of the cost we have to pay to achieve $\phi$-fairness for this network.
First,
to assess the utility loss of our algorithms in absolute terms
we compute a lower bound for the utility loss.
\noindent \textbf{Lower Bound.}
We compute a lower bound on the utility loss, by constructing the probability vector $\mathbf{w}$ that is $\phi$-fair, and it has the minimum utility loss compared to the original pagerank vector $\mathbf{p}_O$.
Note that vector $\mathbf{w}$ is not necessarily attainable by any Pagerank algorithm in $\mathcal{PR}$.
To compute $\mathbf{w}$, we start with $\mathbf{p}_O$ and we redistribute the probability between the two groups to make it fair. Let $\mathbf{p}_O(R)$ be the probability assigned to the red group by $\mathbf{p}$. Without loss of generality, assume that $\mathbf{p}_O(R)< \phi$, and let $\Delta = \phi-\mathbf{p}_O(R)$. To make the vector fair, we need to remove $\Delta$ probability mass from the nodes in $B$, and redistribute it to the nodes in $R$. It is easy to show that to minimize the loss, the optimal redistribution will remove uniformly $\Delta/|B|$ probability from all nodes in $B$, and add uniformly $\Delta/|R|$ to all nodes in $R$.
This follows from the fact that among all distribution vectors the one with the smallest length is the uniform one. However, this process does not guarantee that the resulting vector will not have negative entries, since some blue nodes may have probability less than $\Delta/|B|$. Let $\beta$ be the smallest non-zero such probability of any blue node. Our algorithm transfers $\beta$ probability from all the non-zero blue nodes to the red nodes, and then recursively applies the same procedure for the residual amount of probability that has not been transferred. It is not hard to show that this process will produce a fair vector with the minimum utility loss with respect to $\mathbf{p}_O$.
Figures \ref{fig:synth-utility-homo} and \ref{fig:synth-utility-phi} report the utility loss for selected synthetic networks
and Figure \ref{fig:real-utility} for different values of
$\phi$ for the real datasets. $LB$ is the
lower bound on utility loss.
\noindent \textbf{Effect of $\phi$:} In all cases, loss increases as the requested $\phi$ deviates from the red pagerank originally assigned to the red group
(Figures \ref{fig:synth-utility-phi} and \ref{fig:real-utility}).
\noindent \textbf{\textsl{FSPR}:} \, In some cases {{\sc FSPR}} incurs high utility loss and, occasionally, it is even unable to find an appropriate solution.
{{\sc FSPR}} achieves fairness by changing the jump vector of the Pagerank algorithm.
The overall pagerank vector is a linear combination of the personalized pagerank vectors, with the jump vector providing the coefficients of this linear combination.
{{\sc FSPR}} increases the jump probability for the vectors that favor the group it wants to promote and takes away probability from the vectors that favor the other group. However, when there are few, or no appropriate such vectors, {{\sc FSPR}} is forced to make extreme choices (assign a lot of probability to a few vectors) thus incurring high utility loss, or it is unable to find any solution.
There are several such examples in our datasets.
For the {\sc dblp} dataset (Figure \ref{fig:real-utility}(c)), for small values of $\phi$ ($\phi \leq 0.3$), the utility loss of {{\sc FSPR}} is close to the optimal, but for $\phi \geq 0.4$, it skyrockets. Looking at Figure \ref{fig:real-violin-plots}, we see that there are very few personalized pagerank vectors with red pagerank larger than 0.4. As a result, for $\phi \geq 0.4$, {{\sc FSPR}} is forced to allocate all the probability of the jump vector to these nodes, leading to high utility loss.
It is also interesting to observe the effect of homophily, or lack of, on the utility loss of {{\sc FSPR}}. In Figure \ref{fig:synth-utility-homo}(a) and (b), utility loss peaks when the network is neutral ($\alpha = 0.5$). In this case, there is no personalized pagerank vector that strongly favors one group over the other.
\noindent \textbf{\textsl{LFPR}:} \, Overall, for the locally fair family of algorithms, the utility loss function is smoother, avoiding high peaks.
The locally fair algorithms are directly affected by homophily, since this determines the composition of the neighborhoods of the nodes. As we deviate from neutrality, the loss increases (Figure \ref{fig:synth-utility-homo}).
This holds especially for the {{\sc LFPR$_N$}} algorithm.
This can be seen very clearly in {\sc books} (Figure \ref{fig:real-utility}(a)), where {{\sc FSPR}} almost achieves the lower bound, while {{\sc LFPR$_N$}} incurs high utility loss because of {\sc books} being very homophilic.
The utility loss of {{\sc LFPR$_U$}} and {{\sc LFPR$_P$}} follows in general the same trend as {{\sc LFPR$_N$}}.
Finally, {{\sc LFPR$_O$}} redistributes any residual Pagerank so that the
utility loss is optimized and in many cases its utility loss is close to the lower bound ($LB$).
\noindent \textbf{Summary:}
{{\sc FSPR}} works well when there are enough unfair nodes (i.e., nodes with unfair personalized pageranks), as it can distribute the jump probability to them to achieve fairness.
On the contrary, locally fair algorithms have high utility loss when there are many unfair nodes. {{\sc LFPR$_N$}} is the most sensitive to homophily.
The utility loss of {{\sc LFPR$_N$}} can be seen as a measure of local unfairness.
Overall, the locally fair algorithms are more stable than {{\sc FSPR}}. {{\sc LFPR$_O$}} works very well in terms of utility loss and in many cases it achieves utility close to the lower bound.
\subsection{Qualitative Evaluation}
In this section, we provide some qualitative experiments, to better understand the properties of the fair algorithms.
\begin{figure}[t]
\centering
\subfigure[Original {{Pagerank}}]{
{\includegraphics[width = 0.175\textwidth]{books_05_pagerank.png}}
}
\subfigure[{{\sc LFPR$_N$}}]{
{\includegraphics[width = 0.175\textwidth]{books_05_lfprNei.png}}
}
\subfigure[{{\sc FSPR}}]{
{\includegraphics[width = 0.175\textwidth]{books_05_sensitive.png}}
}
\subfigure[Jump vector for {{\sc FSPR}}]{
{\includegraphics[width = 0.175\textwidth]{books_05_jumpV.png}}
}
\caption{Visualization of the {\sc book} dataset.}
\label{fig:vis-books}
\end{figure}
\begin{figure}[t]
\centering
\subfigure[Original {{Pagerank}}]{
{\includegraphics[width = 0.22\textwidth]{dblp-course-05-pagerank.pdf}}
}
\subfigure[{{\sc LFPR$_N$}}]{
{\includegraphics[width = 0.22\textwidth]{dblp-course-05-lfprNei.pdf}}
}
\subfigure[{{\sc FSPR}}]{
{\includegraphics[width = 0.22\textwidth]{dblp-course-05-sensitive.pdf}}
}
\subfigure[Jump vector for {{\sc FSPR}}]{
{\includegraphics[width = 0.22\textwidth]{dblp-course-05-jumpV.pdf}}
}
\caption{Visualization of the {\sc dblp} dataset.}
\label{fig:vis-dblp}
\end{figure}
\noindent \textbf{Visualization.}
In Figures \ref{fig:vis-books} and \ref{fig:vis-dblp}, we visualize the results of the algorithms for the {\sc books} and the {\sc dblp} dataset respectively, for $\phi$ = 0.5. Red nodes are colored red, and blue nodes are colored blue. Their size depends on the value of the quantity we visualize. We visualize the pagerank values for the original Pagerank, {{\sc FSPR}} and {{\sc LFPR$_N$}} algorithms, and the jump vector probabilities for {{\sc FSPR}}.
For {\sc books}, where the original red pagerank is close to $\phi$,
{{\sc FSPR}} is very similar to the original Pagerank algorithm.
{\sc books} is homophilic and the jump vector assigns rather uniform weights
to almost all nodes.
On the other hand, {{\sc LFPR$_N$}} promotes heavily nodes connecting the two opposite groups, i.e., nodes that are minorities in their neighborhoods.
We observe a different result in {\sc dblp}, where $\phi$ is much larger than
the original red pagerank.
{{\sc LFPR$_N$}} distributes weights broadly in the red community, while {{\sc FSPR}} is much more aggressive. This is especially evident in the jump vector which promotes a few nodes in the periphery.
\begin{table}[ht]
\centering
\caption{Top-10 authors with $\phi$ = 0.3; the number in parenthesis is the position of the author in the original {{Pagerank}} ($\textsc{OPR}$) (female authors in bold).
}
\footnotesize{
\begin{tabular}{l l l}
\toprule
$\textsc{OPR}$ & {\sc FSPR} & {\sc LFPR$_N$} \\
\midrule
C. Faloutsos & C. Faloutsos (1) & C. Faloutsos (1) \\
G. Weikum & G. Weikum (2) & G. Weikum (2) \\
P. S. Yu & P. S. Yu (3) & \textbf{J. Widom} (38) \\
M. Stonebraker & M. Stonebraker (4) & M. Stonebraker (4) \\
M. J. Franklin & M. J. Franklin (5) & P. S. Yu (3)\\
H. Garcia-Molina & H. Garcia-Molina (6) & \textbf{S. T. Dumais} (28) \\
D. Kossmann & D. Kossmann (7) & \textbf{M. Lalmas} (27) \\
W. Lehner & \textbf{E. A. Rundensteiner} (22) & P. Serdyukov (17) \\
M. J. Carey & R. Agrawal (11) & \textbf{E. Bertino} (25) \\
M. de Rijke & W. Lehner (8) & \textbf{E. A. Rundensteiner} (22) \\
\bottomrule
\end{tabular}
\label{table:case-study}
}
\end{table}
\noindent
\textbf{Anecdotal Examples.}
We will use the {\sc dblp} dataset for a qualitative evaluation of the results of the algorithms. Recall that for this dataset, women are the minority with $r$ = 0.17, and the original red pagerank is equal to 0.16.
To achieve a fairer representation of women, we apply the {{\sc LFPR$_N$}} and {{\sc FSPR}} algorithms with $\phi$ = 0.3. In Table \ref{table:case-study}, we present the first 10 authors for each algorithm.
In the original Pagerank algorithm $\textsc{OPR}$, there is no female author in the top-10 (the first one appears in position 22). {{\sc FSPR}} achieves fairness but also minimizes utility loss, so the result is fair but also close to that of $\textsc{OPR}$. {{\sc LFPR$_N$}} asks that all authors
have $\phi$-fair inter-gender collaborations resulting
in a large number of female authors appearing in the top-10 positions.
\begin{table}[ht]
\centering
\caption{Top-3 female authors by conference, $\phi$ = 0.3.}
\footnotesize {
\begin{tabular}{lll}
\toprule
&SIGIR&SIGMOD\\
\midrule
\multirow{3}{*}{{\sc FSPR}}&
Mounia Lalmas & Elke A. Rundensteiner \\
&Susan T. Dumais & Elisa Bertino \\
&Juliana Freire & Tova Milo \\
\midrule
\multirow{3}{*}{{\sc LFPR$_N$}}&
Susan T. Dumais & Jennifer Widom \\
& Mounia Lalmas & Elke A. Rundensteiner \\
& Emine Yilmaz & Fatma Ozcan \\
\bottomrule
\end{tabular}
}
\label{table:case-study-conf}
\end{table}
We also use {\sc dblp} to study the targeted fair Pagerank algorithms.
In this case, we want to enforce fair behavior towards authors in specific conferences. We consider two such conferences, SIGIR and SIGMOD, and select $S$ to include authors of each one of them. In Table \ref{table:case-study-conf}, we show the top-3 women authors for each conference according to our algorithms. We observe that the algorithms produce different results depending on the conference, each time promoting women that are authorities in their respective fields, such as, Suzan Dumais when $S$ is SIGIR, and Jennifer Widom when $S$ is SIGMOD.
\section{Related Work}
\label{sec:related-work}
\noindent \textbf{Algorithmic fairness.} Recently, there has been increasing interest in algorithmic fairness, especially in the context of machine learning.
Fairness is regarded as the lack of discrimination on the basis of some protective attribute.
Various definition of fairness having proposed especially for classification \cite{fairness-awarness,fairness-study,fairness-measures,bias-fairness-survey}. We use a group-fairness definition, based on parity.
Approaches to handing fairness can be classified as \textit{pre-processing}, that modify the input data, \textit{in-processing}, that modify the algorithm and \textit{post-processing} ones, that modify the output. We are mostly interested in in-processing techniques.
There is also prior work on fairness in ranking~\cite{fai*r,Julia17,Biega18,pair-wise}.
All of these works consider ranking as an ordered list of items, and use different rules for defining and enforcing fairness that consider different prefixes of the ranking~\cite{fai*r,Julia17}, pair-wise orderings~\cite{pair-wise}, or exposure and presentation bias~\cite{exposure-ranking,Biega18}.
Our goal in this paper is not to propose a new definition of ranking fairness, but rather to initiate a study of fairness in link analysis.
A distinguishing aspect of our approach is that we
take into account the actual Pagerank weights of the nodes, not just their ranking.
Furthermore, our focus in this paper, is to design in-processing algorithms that incorporate fairness in the inner working of the Pagerank algorithm. We present a post-processing approach as a means to estimate a lower bound on the utility loss. None of the previous approaches considers ranking in networks, so the proposed approaches are novel.
\noindent \textbf{Fairness in networks.}
There has been some recent work on network fairness in the context of graph embeddings \cite{fair-embedding,filter-bubbles,fair-walk}.
The work in \cite{fair-embedding} follows an in-processing approach
that extends the learning function with regulatory fairness enforcing terms, while the work in \cite{filter-bubbles} follows a post-processing approach so as to promote link recommendations to nodes belonging
to specific groups. Both works are not related to our approach.
The work in \cite{fair-walk} extends the node2vec graph embedding method by modifying the
random walks used in node2vec with fair walks, where nodes are partitioned into groups and each group is given the same probability of being selected when a node makes a transition. The random walk
introduced in \cite{fair-walk} has some similarity with the
random walk interpretation of {{\sc LFPR$_N$}}. It would
be interesting to see, whether our extended residual-based algorithms could be utilized also in the context of graph embeddings, besides its use in link analysis.
There are also previous studies on the effect of
homophily, preferential attachment and imbalances in group sizes.
It was shown that the combination of these three factors leads to
uneven degree distributions between groups \cite{glass-ceiling}.
Recent work shows that this phenomenon is exaggerated by many link recommendation algorithms \cite{glass-ceiling-recommend}. Evidence of inequality between degree distribution of minorities and majorities was also found in many real networks \cite{homophily-ranking}.
Our work extends this line of research by looking at Pagerank values instead of degrees.
Along this lines, recent work studies in depth how homophily and size imbalance can affect the visibility
that different groups get in link recommendations, i.e, how often nodes in each group get recommended \cite{xyz}.
Very recent work also looks at graph mining algorithms in general from the perspective of individual fairness, where the goal is to produce a similar output for similar nodes \cite{inform}.
Finally, there is previous work on diversity in network ranking.
The goal is to find important nodes that also maximally cover the nodes in the network~\cite{grasshoper,divrank}.
Our problem is fundamentally different, since we look for scoring functions that follow a parity constraint.
\section{Conclusions}
\label{sec:conclusions}
In this paper, we initiate a study of fairness for Pagerank.
We provide definitions of fairness, and we propose two approaches for achieving fairness: one that modifies the jump vector, and one that imposes a fair behavior per node. We prove that the latter is equivalent to a stronger notion of fairness that also guarantees personalized Pagerank fairness. We also consider the problem of attaining fairness while minimizing the utility loss of Pagerank. Our experiments demonstrate the behavior of our different algorithms.
There are many direction for future work. First, we would like to study the role of
$\gamma$, i.e., the jump probability. Then, it would be interesting to explore other notions of Pagerank fairness, besides $\phi$-fairness, for instance ones based on
rank-aware fairness \cite{edbt-tut}. Furthermore, we plan to explore further applications of the theory behind our fair Pagerank algorithms to derive novel notions of fair random walks. |
1,108,101,565,447 | arxiv | \section{Introduction}
\label{sec:intro}
\begin{comment}
\textcolor{red}{I am doing this paper to show the construction of the eft using theoretical tools, and showing its connections to current used frameworks.}
\end{comment}
The recent detection of gravitational waves (GWs) from various coalescing binaries \cite{Abbott:2016blz, TheLIGOScientific:2017qsa,LIGOScientific:2017ync,LIGOScientific:2017zic,LIGOScientific:2021qlt} consisting of compact objects, such as black holes (BHs) and neutron stars (NSs), have opened up the possibility to test fundamental physics in the strong regime of gravity. With upcoming sensibility upgrades in current GW detectors, and future earth \cite{Punturo:2010zz,Maggiore:2019uih} and space based detectors \cite{Barausse:2020rsu}, the era of high precision gravity is arriving, and with it the potential of great discoveries.
One of the key potentials is to probe the internal structure of the compact objects by matching the coefficients of the theory with GW observations \cite{Flanagan:2007ix, Cardoso:2017cfl}. Thus the need to develop accurate theoretical models that describe these extended objects, taking into account for the different effects that can play a role in the waveform, such as the spin, charge, and their internal structure due to its extended nature.
In the EFT for extended objects \cite{Goldberger:2004jt}, where non-rotating objects are introduced, they are described as worldline point particles, with higher order operators taking into account for their properties and finite-size structure, which are accompanied with coefficients that encapsulates their internal structure. Since the introduction of compact objects as an EFT \cite{Goldberger:2004jt, Goldberger:2005cd}, the framework has received a big flush of improvements. Some of the most relevant extensions were the inclusion of spin \cite{Porto:2005ac,Delacretaz:2014oxa, Levi:2015msa,Goldberger:2020fot, Liu:2021zxr}, charge \cite{Patil:2020dme}, spin-tidal effects \cite{Porto:2005ac, Levi:2014gsa}, dissipation \cite{Goldberger:2005cd, Porto:2007qi,Goldberger:2020fot}, dynamical oscillations \cite{Steinhoff:2016rfi} and gravitational wave radiation \cite{Goldberger:2009qd}. Moreover, the finding of new methods that simplifies the computations in perturbative approximations, such as in the Post-Newtonian (PN) \cite{Levi:2018nxp, Kuntz:2020gan} and Post-Minkowskian (PM) \cite{Goldberger:2009qd,Kalin:2020mvi} expansions, have pushed the field forward to obtain higher and higher order corrections to the dynamics of binary compact objects, leading to an unprecedented accuracy \cite{Levi:2008nh, Porto:2008jj, Porto:2008tb, Goldberger:2009qd, Levi:2010zu, Porto:2010tr, Foffa:2011ub,Ross:2012fc, Levi:2015msa, Levi:2016ofk, Foffa:2019yfl, Foffa:2019hrb, Levi:2019kgk,Levi:2020lfn,Levi:2020kvb,Levi:2020uwu,Kalin:2020lmz, Kalin:2020fhe,Dlapa:2021vgp,Cho:2022syn,Kalin:2022hph}.
Nevertheless, certain ambiguities arises from different developed EFTs, i.e. on spinning objects, and an effective theory that unifies the different developed tools for the description of compact objects is still lacking. Therefore, the purpose of this work is to develop an EFT theory using modern theoretical tools, that can incorporate all the ingredients necessary to describe the most general compact object allowed in a theory of gravity as GR with classical electrodynamics, which can be charged and spinning. By exploring the mathematical structure of such theory, we elucidate on the description of compact objects as an EFT.
Using the coset construction \cite{Coleman:1969, Callan:1969sn,Volkov:1973vd, Ivanov:1975zq}, and following the prescription to describe spinning extended objects in \cite{Delacretaz:2014oxa}, we propose an EFT that can describe the relativistic dynamics of compact objects without introducing additional degrees of freedom. This theory, which is worldline re-parameterization invariant, can naturally be used to obtain both the PN and PM expansion. This approach allows us to disentangle multiple features of spinning objects, like the relativistic angular velocity, and the Spin Supplementary Conditions. The developed EFT, which is an extension of \cite{Delacretaz:2014oxa,Endlich:2015mke}, contains well motivated differences, which are discussed in detail. The predictability of our theory is shown in a forthcoming work.
To describe extended objects which are charged and spinning, we first develop an effective theory of gravity with classical electrodynamics. General Relativity as an effective theory can be derived using the coset construction by weakly gauging the space-time symmetry group of gravity, the Poincaré group ISO(3,1), and realizing translations non-linearly \cite{Ivanov:1981wn}. With this procedure one can derive Einstein's vierbein theory of curved space-time, which is a generalization of the theory of General Relativity that is independent of a coordinate frame, and which was introduced by Einstein \cite{Einstein:1928} in order to unify gravity and electromagnetism. Therefore, to the procedure carried out in \cite{Delacretaz:2014oxa}, we incorporate the internal $U(1)$ symmetry of classical electrodynamics, to derive an effective Einstein-Maxwell theory. Once the underlying theory is developed, we proceed to describe compact objects.
\begin{comment}
In this theory, BHs are constrained by the no hair theorem, which states that a BH can be described by only three parameters, its mass, spin and charge, behaving effectively as a point particle \cite{Arkani-Hamed:2019ymq,Moynihan:2019bor}. In this sense, we derive the leading order effective action for a charged spinning compact object. In the EFT framework, we treat compact objects as point particles, with their additional effects and internal structure encoded as higher order corrections in the action, which are made out of the allowed invariant operators of the theory. These operators are accompanied by coefficients which encapsulates the properties and internal structure of the objects.
\end{comment}
By identifying the symmetry breaking pattern of a charged spinning extended object, we derive the covariant building blocks that are used to construct the effective action. Due to the Goldstone's theorem \cite{Goldstone:1961eq}, the symmetry breaking pattern of such extended object in curved space-time implies the existence of a Nambu-Goldstone field, whose covariant derivative encodes the angular velocity in its spatial components, and the acceleration in the temporal-spatial ones \cite{Delacretaz:2014oxa}. As such elements are derived from a symmetry reasoning, this yields a very well motivated theory to describe spinning extended objects. This is one of the main differences of our effective theory with others, that all covariant quantities are derived using symmetry principles. Although from construction the effective action corresponds to the low energy dynamics of the theory, which implies a theory for slowly spinning objects, we show that compact objects can be considered as "slowly" spinning.
\begin{comment}
This is due to the fact that compact objects are described classically, and in this regime, the objects can acquire a slow spin. Therefore, this model can be safely used to model any astrophysical source.
\end{comment}
\begin{comment}
In this work we go further and consider higher order spin corrections, include electromagnetic charge in the description, and consider the relativistic internal structure. By deriving in detail the building blocks of the effective theory and its constraints, we show the construction of the tower of invariant operators to form the effective action to all orders. We show that the spin orbit coupling, generally obtained by introducing the canonical spin and Legendre transforming the action \cite{Porto:2005ac}, can instead be naturally constructed by coupling the gauge field from the Lorentz transformations with the angular velocity. Moreover, we show that the spin-acceleration correction in \cite{Levi:2015msa}, which is necessary to obtain the correct rules for the perturbative expansion of the dynamics, is encoded in higher order boson couplings. On the electrodynamics, we have considered the electromagnetic charge U(1) symmetry, as an internal symmetry in the group parameterization, U(1)$\times$ISO(3,1), which allows us to derive the Einstein-Maxwell action. Then, by identifying the symmetry breaking pattern for a charged spinning point particle under a U(1) symmetry, which corresponds to the eigenstate of the charge and does note break U(1), we derive the corrections to the point particle due to charge and spin.
\end{comment}
Therefore, with the derived building blocks, we build the leading order invariant operators that are needed to describe a charged spinning extended object, such as a pulsar. We discuss in detail each of the built operators and compare them to the literature. The constructed effective action is the one of a massive point particle, with higher order corrections made out of the constructed operators describing the different effects: spin, charge, tides, polarization and dissipation. The coefficients appearing in front of the operators are free parameters to be fixed by the full theory or observations. Beyond the discussion on the spin-orbit coupling, for which we use a specific value of the coefficients, we leave the coefficients undetermined, to be matched in future work.
The constructed effective action has multiple applications in the description of the coalescence of binaries. On the PN expansion, as a perturbative series in terms of the expansion parameter $v/c <1$, with $v$ the relative velocity of the binary, and on the PM expansion, in which one expands over the gravitational constant, $G$, and which encodes the PN expansion to all orders in $v$, to a given order in $G$. The connection between the different expansions is shown in a future work, where the dynamics to lowest order due to all different effects are obtained. Finally, the derived building blocks could be used in the Worldline Quantum Field Theory (WQFT) formalism \cite{Mogull:2020sak}, and to construct the effective one body (EOB) framework.
\begin{comment}
On size effects, for which we also leave coefficients undetermined, we take into account first static tidal effects and polarization, and then generalize to dynamical ones, including dissipative effects. Therefore, incorporating all necessary effects to completely characterize a compact object that can be found in our universe, i.e. a pulsar.
\end{comment}
In section \ref{sec:coset}, we start with a very brief review of the basic ingredients of the coset construction to derive our effective action. In section \ref{sec:EFTgrav}, we derive the underlying effective theory, which is Einstein-Maxwell in the vierbein formalism. Then, in \ref{sec:gravity}, derive the covariant building blocks and constraints to build up the worldline effective action of a charged spinning compact object in curved space-time. We start by considering massive point-like objects, then with charge, and then include the spin. Then we consider field-field interactions, taking into account spin-gravity, spin-electro and gravity-electro effects. On size effects, we first take into account static tidal effects and polarization, which are conservative effects. Then we consider dissipation, and generalize to consider dynamical tides with dissipation included. After constructing all relevant invariant operators to lowest order, we build an effective action that describes the most general compact object in a Maxwell-Einstein effective theory. Finally in section \ref{sec:discussion}, we conclude.
\section{Basics of the Coset Construction}
\label{sec:coset}
We start with the very basics of the coset construction to develop this paper. A brief but more comprehensive review can be found in \cite{Ogievetsky:1974,Delacretaz:2014oxa,Penco:2020kvy}. We use the notation as in \cite{Delacretaz:2014oxa} to consider the breaking of internal \cite{Callan:1969sn, Coleman:1969} and space-time symmetries \cite{Volkov:1973vd, Ivanov:1981wn} alike.
\begin{comment}
Any state other than vacuum will break at least some of the symmetries, and by correctly identifying the pattern, it is possible to derive the covariant building blocks that transform correctly under the relevant symmetries. These building blocks are then used to form invariant operators, to build an effective action.
\end{comment}
The coset construction is a very general technique from the EFT framework that can be used whenever there is a symmetry breaking. The breaking of symmetries implies the existence of additional degrees of freedom, known as Nambu-Goldstone bosons or simply as Goldstone fields.\footnote{Goldstone theorem \cite{Goldstone:1961eq} implies the existence of a Goldstone field for each broken internal symmetry, but for the case in which space-time symmetries are broken, there can be a mismatch on the number of degrees of freedom and broken symmetries, for which additional constraints are needed. See the Inverse Higgs constraint below.} The coset construction is then used to derive building blocks for the Goldstone fields that transform correctly under the relevant symmetries, blocks that can be used to build up invariant operators to form an effective action. Any state other than vacuum breaks at least some of the symmetries, and by appropriately identifying the pattern of the symmetry breaking, we can use it as a guide to derive the effective action. Within this approach, the coefficients that appear in front of the invariant operators are treated as free parameters to be fixed by a matching procedure to the full theory or to observations.
\begin{comment}
Therefore, we are interested to know what was the full symmetry group G of the EFT, and what subgroup H was realized non-linearly, parameterized by the coset, G/H.
\end{comment}
We can formulate an EFT using the symmetry breaking pattern as the only input, knowing the full symmetry group G that is broken, and the
subgroup H that is non-linearly realized \cite{Penco:2020kvy}. If the group is broken, G $\rightarrow$ H, due to a spontaneous symmetry breaking, the coset recipe \cite{Delacretaz:2014oxa,Penco:2020kvy} tells us that we can classify the generators into three categories:
\begin{flalign}
\begin{split}
P_{a} &= \mathrm{generators \; of \; unbroken \; translations},\\
T_A &= \mathrm{generators \; of \; all \; other \; unbroken \; symmetries}, \\
X_{\alpha} &= \mathrm{generators \; of \; broken \; symmetries},
\end{split}
\end{flalign}
\noindent where the broken generators, $X_{\alpha}$, and the unbroken ones, $T_A$, can be of space-time symmetries, as well as of internal ones. Whenever the set of generators for broken symmetries is non-zero, some Goldstone fields will arise. The broken symmetries and the unbroken translations are realized non-linearly on the Goldstone bosons \cite{Delacretaz:2014oxa}.
Following the coset recipe \cite{Delacretaz:2014oxa,Penco:2020kvy}, we do a local parameterization of the coset, G/H$_0$, with $H_0$, the subgroup of $H$ generated by the unbroken generators, $T$'s. The coset is parameterized as
\begin{flalign}
g (x, \pi) = e^{iy^{a}(x) P_{a}} e^{i \pi^{\alpha}(x) X_{\alpha}},
\label{eq:gcoset}
\end{flalign}
\noindent where the factor, $e^{iy^a (x) P_a}$, describes a translation from the origin of the coordinate system to the point, $x_a$, at which the Goldstone fields, $\pi^{\alpha} ( x )$, are evaluated. This factor ensures that the $\pi$’s transform correctly under spatial translation. The group element $g$, which is generated by the $X$'s and the $P$'s, is known as the coset parameterization. For the case of flat space-time, the translation is simply parameterized by, $e^{ix^a P_a}$, with $y(x) \equiv x$. To obtain building blocks that depend on the Goldstone bosons and that have simple transformation rules, we couple them through their derivatives \cite{Penco:2020kvy}.
We introduce a very convenient quantity that is an element of the algebra of G, the Maurer-Cartan form, $g^{-1} \partial_{\mu} g$, which can be written as a linear combinations of all the generators \cite{Delacretaz:2014oxa},
\begin{equation}
g^{-1} \partial_{\mu} g = ( e_{\mu}^{\; \; a} P_a + \nabla_{\mu} \pi^{\alpha} X_{\alpha} + C_{\mu}^{\; \; B} T_B).
\label{eq:mauren}
\end{equation}
\noindent The coefficients $e_{\mu}^{a}$, $\nabla_{\mu} \pi^{\alpha}$ and $C_{\mu}^B$, in general are non-linear functions of the Goldstones, and are basic ingredients of the effective theory, with $\nabla_{\mu} \pi^{ \alpha} = e_{\mu}^{\; \;a} \nabla_{a} \pi^{\alpha}$ and $C_{\mu}^{\; \; B} = e_{\mu}^{\; \; a} C_{a}^{\;\; B}$. The explicit expression of the aforementioned building blocks can be obtained using the algebra of the group $G$.
We can use the coefficients of the unbroken symmetries, $C$'s, and its operators, $T$'s, to define the covariant derivative,
\begin{flalign}
\nabla_a \equiv (e^{-1})_{a}^{\; \mu} (\partial_{\mu} + i C_{\mu}^B T_B),
\label{eq:covDcoset}
\end{flalign}
\noindent which can be used to define higher covariant derivatives on the Goldstone fields, as well as on the building blocks that transform linearly under the unbroken group. Then, by considering all the allowed contractions of the building blocks including the covariant derivative, it is possible to build up invariant operators under the full symmetry group $G$, and construct the effective action.
In gauge symmetries, it is necessary to promote the partial derivative to a covariant one in the Maurer-Cartan form, $\partial_{\mu} \rightarrow D_{\mu}$ \cite{Delacretaz:2014oxa}. Consider the gauged generator, $E_I$, from a subgroup, $G' \subseteq G$, with corresponding gauge field, $w^{I}_{\mu}$. Thus, by replacing the partial derivative with a covariant one, we obtain the modified Maurer-Cartan form,
\begin{equation}
g^{-1} \partial_{\mu} g \rightarrow g^{-1} D_{\mu} g = g^{-1} (\partial_{\mu} + i w^{I}_{\mu} E_{I} )g
\label{eq:maurenmod}.
\end{equation}
\noindent This modification of the Maurer-Cartan form can also be written as a linear combination of the generators as in eq. (\ref{eq:mauren}), with a new building block made up of the gauge field, $w_{\mu}^{I}$, accompanying the gauged generator, $E_I$. Now the building blocks can also depend on the gauge fields. The modified Maurer-Cartan form, $ g^{-1} D_{\mu} g $, is invariant under local transformations, and its explicit components can be obtained using the commutation relations of the generators.
\subsection*{Inverse Higgs Constraint}
The Goldstone’s theorem \cite{Goldstone:1961eq}, which states that a Goldstone mode exists for each broken generator, is only valid for internal symmetries. If space-time symmetries are spontaneously broken, there can be a mismatch in the number of broken generators and the number of bosons \cite{Low:2001bw}. Nevertheless, we can preserve all the symmetries by imposing additional local constraints, which can be solved to write down some of the Goldstone’s modes in terms of others \cite{Penco:2020kvy}. Using the inverse Higgs constraint \cite{Ivanov:1975zq}, we can set to zero one or more of the coset covariant derivatives, whenever $X$ and $X'$, are two multiplets of the broken generator, such that the commutators of the unbroken translations, $P$, and the broken generator, $X'$, yields a different broken generator, $X$: $\; [P, X'] \supset X$. If this is the case, we can set some of the covariant derivatives of the Goldstones to zero.
\begin{comment}
By imposing all possible inverse Higgs constraints, one obtains the only relevant building blocks.
\end{comment}
\section{Effective Theory of Gravity}
\label{sec:EFTgrav}
\begin{comment}
Before constructing the effective action for a compact object, we review how a theory of gravity can be derived using symmetries as the only input. the coset construction as in \cite{Ivanov:1981wn,Delacretaz:2014oxa}, where a frame independent generalization of general relativity, known as Einstein's vierbein field theory, is derived. This is a theory that can naturally incorporate spinning objects. \Irv{Then, to extend the work in \cite{Delacretaz:2014oxa}, we include an internal U(1) gauge symmetry, to describe electrodynamics in such effective theory of gravity, which allows us to derive the Einstein-Maxwell action in the vierbein formalism. } Once the underlying theory of gravity has been developed, then we construct the action for a charged spinning compact object in the effective theory of general relativity.
\end{comment}
There are two symmetries to consider in a theory of gravity as General Relativity: Poincaré symmetry, and diffeomorphisms invariance or worldline reparameterization. The former, determined by the Poincaré group, $G=ISO(3,1)$, contains the generators for translations, $P_a$, and Lorentz transformations, $J_{ab}$, with their corresponding gauge fields, $\breve{e}_{\mu}^{a}$ and $\breve{\omega}_{\mu}^{ab}$.
By considering the principal bundle, $P(M,G)$, with base manifold, $M$, and structure group, $G$, when weakly gauging the Poincaré group and realizing translations non-linearly, Poincaré and diffeomorphisms are separated. The coordinates, $x^{\mu}$, describing the position on the considered manifold are not affected by the local Poincaré group but transformed under diffeomorphisms, while matter fields are realized as sections of their respective fiber bundle \cite{Delacretaz:2014oxa}. The local Poincaré transformations act along the fiber, while diffeomorphisms can be considered as relabeling the points on the manifold.
To derive an effective theory of gravity, we proceed to gauge the Poincaré group and realize translations non-linearly \cite{Delacretaz:2014oxa}. The coset, $ISO(3,1)/SO(3,1)$, is parameterized by
\begin{equation}
g = e^{i y^a (x) P_a},
\label{eq:gravparam}
\end{equation}
\noindent which ensures the non-linear realization of translations \cite{Ivanov:1981wn}. From this coset parameterization, the covariant Maurer-Cartan form, expressed as a linear combination of the generators of the theory, reads
\begin{flalign}
\begin{split}
g^{-1} D_{\mu} g &= e^{-i y^a(x) P_a} \left( \partial_{\mu} + i \breve{e}_{\mu}^{\;\; a} P_a + \frac{i}{2} \breve{\omega}_{\mu}^{ab} J_{ab} \right) e^{iy^a (x) P_a} \\
&= i e_{\mu}^{\;\;a} P_a + \frac{i}{2} \omega_{\mu}^{ab} J_{ab},
\label{eq:maurercartangravity}
\end{split}
\end{flalign}
\noindent where we have used the commutation relation rules of the symmetries (See appendix \ref{app:symmetries}).
From the linear combination of the generators in eq. (\ref{eq:maurercartangravity}), we can extract the first building blocks of the theory,
\begin{flalign}
e_{\mu}^{\;\; a} &= \breve{e}_{\mu}^{a} + \partial_{\mu} y^a + \breve{\omega}_{\mu \; b}^{a} y^b, \label{eq:efield}\\
\omega^{ab}_{\mu} &= \breve{\omega}^{ab}_{\mu}. \label{eq:spinfield}
\end{flalign}
\noindent The field, $e^{a}_{\mu}$, is the vierbein, which defines the metric as $g_{\mu \nu} = \eta_{ab} e_{\mu}^a e_{\nu}^b$. It can be used to build up the invariant element, $\mathrm{d}^4 x \, \mathrm{det}\,e$, as well as to change from orthogonal frame, i.e. $V_{\mu} = e_{\mu}^{\; b} V_{b}$, with $V_a$, a vector field. The field, $\omega_{\mu}^{ \; ab}$, is the spin connection.
We can introduce the covariant derivative for matter fields, using the coefficients from the unbroken Lorentz generators,
\begin{equation}
\nabla^{g}_{a} = (e^{-1})_a^{\;\;\mu}(\partial_{\mu} + \frac{i}{2} \omega_{\mu}^{bc} J_{bc}),
\label{eq:cdg}
\end{equation}
\noindent where the upper index, $g$, denotes gravity. The only required ingredients to describe the non-linear realizations of translations and the local transformations of the Poincaré group, is the covariant derivatives and the vierbein \cite{Delacretaz:2014oxa}.
Moreover, we can extract building blocks from the gauge field strengths. The curvature invariants are obtained from the covariant commutator of the derivative in the Maurer-Cartan form,
\begin{flalign}
\begin{split}
g^{-1} [D_{\mu}, D_{\nu}] g &= i T^{a}_{\mu \nu} P_a + \frac{i}{2} R^{ab}_{\mu \nu} J_{ab},\\
&= i \left(\partial_{\mu} e^{a}_{\nu} - \partial_{\nu} e^{a}_{\mu} + e^{}_{\mu b} \omega^{ab}_{\nu} - e^{}_{\nu b} \omega^{ab}_{\mu}\right)P_a\\
& \;\;\; + \frac{i}{2} \left( \partial_{\mu} \omega^{ab}_{\nu} -\partial_{\nu} \omega_{\mu}^{ab} + \omega^{a}_{\mu c}\omega^{cb}_{\nu} - \omega^{a}_{\nu c} \omega^{cb}_{\mu} \right) J_{ab},
\label{eq:invariantgravity}
\end{split}
\end{flalign}
\noindent with $T^{a}_{\mu \nu} = \breve{T}^{a}_{\mu \nu} + \breve{R}^{ab}_{\mu \nu} y_b$, and $R^{ab}_{\mu \nu} = \breve{R}^{ab}_{\mu \nu}$, the covariant torsion and Riemann tensor respectively. The covariant quantities have been defined in this way, from $[D_{\mu}, D_{\nu}] = i \breve{T}^{a}_{\mu\nu} P_a + \frac{i}{2} \breve{R}^{ab}_{\mu \nu } J_{ab}$, such that by construction, $T^{a}_{\mu}$ and $R^{ab}_{\mu \nu}$ transform independently under the local transformations \cite{Delacretaz:2014oxa}.
\begin{comment}
in eq. (\ref{eq:commutatorflat})
\end{comment}
As we are interested in a gravitational theory as General Relativity, we set the torsion tensor to be zero. From the vanishing of the torsion tensor, one can obtain an equation for the spin connection in terms of the vierbein,
\begin{equation}
\omega_{\mu}^{ab} (e) = \frac{1}{2} \left\{ e^{\nu a} (\partial_{\mu} e_{\nu}^{\;\;b} - \partial_{\nu} e_{\mu}^{\;\;b}) + e_{\mu c} e^{\nu a} e^{\lambda b} \partial_{\lambda} e_{\nu}^{\;\;c} - (a \leftrightarrow b)\right\}.
\label{eq:spinc}
\end{equation}
\noindent Then, eq. (\ref{eq:invariantgravity}),
\begin{flalign}
\begin{split}
g^{-1} [D_{\mu}, D_{\nu}] g = \frac{i}{2} R^{ab}_{\mu \nu} \left(\omega_{\rho}^{cd} (e)\right) J_{ab}.
\end{split}
\end{flalign}
\begin{comment}
\noindent From the expression for the torsion in eq. (\ref{eq:invariantgravity}), one can read the Christoffel symbols, $T_{\mu \nu}^a = \Gamma^{a}_{\mu \nu} - \Gamma_{\nu \mu}^a$. Therefore,
\begin{flalign}
\Gamma_{\mu \nu}^a = \partial_{\mu} e^{\;a}_{\nu} - e_{\nu b} \omega_{\mu}^{ab},
\label{eq:cristoffele}
\end{flalign}
\noindent which is a function of the vierbein only. From eq. (\ref{eq:cristoffele}), the spin connection can be expressed in terms of the vierbein and the Christoffel symbol as well.
\end{comment}
The Riemann tensor, $R^{ab}_{\mu \nu}$, is used to build up a Lagrangian that describes our gravitational theory. By considering the lowest order correction, we build
\begin{equation}
\mathcal{S} = \int \mathrm{d}^4 x \; \mathrm{det} \; e \; \alpha R \; + .\;.\;. = \int \mathrm{d}^4 x \; \mathrm{det} \; e \; \frac{1}{16 \pi G} R \;+ .\;.\;. ,
\label{eq:generalactiongrav}
\end{equation}
\noindent with $R = R^{ab}_{\; \; \; ab} = e^{\mu}_{\;a} e^{\nu}_{\;b} R^{ab}_{\; \; \; \, \mu \nu}$, the low energy term of the effective theory of gravity. We have matched the coefficient $\alpha = (16 \pi G)^{-1}$ in eq. (\ref{eq:generalactiongrav}), from the well known theory of General Relativity, to obtain the Einstein-Hilbert action in the vierbein formalism \cite{Ivanov:1981wn}. The ellipsis stands for higher order corrections made out of the Riemann tensor, describing a more fundamental theory of gravity.
\begin{comment}
One can easily obtain the well known gravitational action by changing to the space-time indices:
\begin{equation}
\mathcal{S} = \int \sqrt{-g} \; \mathrm{d}^4 x \frac{1}{16 \pi G } R ,
\label{eq:generalactiongravEins}
\end{equation}
\noindent with $R = R^{\mu \nu}_{\;\;\; \mu \nu} = e^{\mu}_{\;a} e^{\nu}_{\;b} R^{ab}_{\; \; \; \mu \nu} $, and where we have used $g_{\mu \nu} = e_{\mu}^a e_{\nu}^b \eta_{ab}$.
\end{comment}
Given that we aim to describe not only black holes but neutron stars as well, and that it is well know that they can possess very strong magnetic fields, we add the internal U(1) symmetry of classical electromagnetism, with gauge field, $\breve{A}_{\mu}$, and generator $Q$. The generator of the charge, $Q$, is encoded as a time translation, $\bar{P}_0 = P_0 + Q$, which ensures the correct transformation rules for the gauge field under the $U(1)$ symmetry. Therefore, by considering the symmetry group, $ G = U(1) \times ISO(3,1)$, we can proceed with the coset construction as before. The coset parameterization now reads,
\begin{equation}
g = e^{i y^a (x) \bar{P}_a},
\label{eq:gravparamele}
\end{equation}
\noindent with $\bar{P}_a = (\bar{P}_0, P_i)$.
Then, the Maurer-Cartan form from the coset parameterization (\ref{eq:gravparamele}), reads
\begin{flalign}
\begin{split}
g^{-1} D_{\mu} g &= e^{-i y^a(x) \bar{P}_a} \left( \partial_{\mu} + i \breve{A}_{\mu}^{} Q + i \breve{e}_{\mu}^{\;\; a} P_a + \frac{i}{2} \breve{\omega}_{\mu}^{ab} J_{ab} \right) e^{iy^a(x) \bar{P}_a} \\
&= \partial_{\mu} + i A_{\mu} Q + i e_{\mu}^{\;\;a} P_a + \frac{i}{2} \omega_{\mu}^{ab} J_{ab}.
\label{eq:maurercartangravityele}
\end{split}
\end{flalign}
\noindent Including the $U(1)$ symmetry, we obtain a new building block,
\begin{flalign}
A_{\mu} &= \breve{A}_{\mu} + \partial_{\mu} y^0,
\end{flalign}
\noindent which transforms as expected, $ \partial_{\mu}y^0 = \partial_{\mu} \xi (y)$. The rest of the building blocks, eq. (\ref{eq:efield}) and eq. (\ref{eq:spinfield}), remain unchanged given that we are considering $U(1)$ as an internal symmetry.
The coefficients from the unbroken $U(1)$ gauge field can be used as well to define the covariant derivative for charged fields,
\begin{equation}
\nabla^{q}_{a} = (e^{-1})_a^{\;\;\mu}(\partial_{\mu} + iA_{\mu}Q).
\label{eq:cdq}
\end{equation}
\noindent Moreover, a new gauge field strength is obtained,
\begin{flalign}
\begin{split}
g^{-1} [D_{\mu}, D_{\nu}] g &= i F_{\mu \nu} Q + i T^{a}_{\mu \nu} P_a + \frac{i}{2} R^{ab}_{\mu \nu} J_{ab},
\label{eq:invariantgravityele}
\end{split}
\end{flalign}
\noindent with $F_{\mu \nu} = \partial_{\mu} A^{}_{\nu} - \partial_{\nu} A^{}_{\mu}$, the electromagnetic stress field tensor. The torsion and Riemann tensor in eq. (\ref{eq:invariantgravityele}), remain unchanged, and one can proceed to construct the effective theory as before.
After imposing the vanishing of the torsion tensor, we are now able to use both, $R^{ab}_{\;\;\;cd}$ and $F_{a b}$, as building blocks to construct the effective action. To lowest order,
\begin{equation}
\mathcal{S}_{eff} = \int \mathrm{d}^4 x \; \mathrm{det} \; e \; \left\{ -\frac{1}{4 \mu_0} F_{a b} F^{a b} + \frac{1}{16 \pi G} R + .\;.\;. \right\},
\label{eq:generalactionelectrograv}
\end{equation}
\noindent where the coefficient from the first term has been matched from Maxwell's action in curved space-time, with $\mu_0$, the magnetic permeability of vacuum. Eq. (\ref{eq:generalactionelectrograv}), is the Einstein-Maxwell action in the vierbein formalism. Beyond lowest order, one could have the higher order correction \cite{Camanho:2014apa},
\begin{flalign}
\mathcal{S}_{} = \int \mathrm{det} \; e \; \mathrm{d}^4 x \; \alpha_{gq} R^{ab}_{\;\;\;cd} F_{ab} F^{cd} + .\;.\;..
\label{eq:bulkgravielectro}
\end{flalign}
\begin{comment}Furthermore, one can also recover the Christoffel symbol, eq. (\ref{eq:cristoffele}), in terms of the metric via, $\Gamma^{\alpha}_{\mu \nu} = e_{\;a}^{\alpha} \Gamma^{a}_{\mu \nu}$.
\end{comment}
\begin{comment}
\subsection{Extensions to General Relativity}
As mentioned above, the Einstein-Hilbert action, which is made out of the Riemann tensor, is the lowest order correction made out of our derived building blocks, given that Einstein's equations in vacuum implies that the Ricci tensor, $R_{ab}$, and the Ricci scalar, $R$, vanishes to lowest order. Within our construction, it is straightforward to consider contributions coming from a more complete gravitational theory. These contributions can be taken into account systematically, order by order, considering higher order corrections made out of the Riemann tensor. This type of modified theories are usually known as $f(R)$ theories, and there exists a vast literature on these theories, for which a review on them is beyond our scope. Nevertheless, a construction of such theories from an EFT perspective has been carried out in \cite{Endlich:2017tqa}, which agrees with our view point.
The theory developed in \cite{Endlich:2017tqa} is of great interest given that it is consistent with current gravity tests, does not introduce new light degrees of freedom, and it agrees with widely accepted physical principles such as locality, causality and unitarity. In \cite{Endlich:2017tqa}, the modifications of general relativity are associated to states that are heavier than the curvature scale of the compact objects. A crucial ingredient is the cutoff energy scale $\Lambda_c$, that sets the validity regime of the theory. For energies above the cutoff, information about the UV is needed. Below the cutoff, The EFT is universal and under perturbative control through the expansion parameter $E/\Lambda_c$. Gravity modifications can be suppressed unless the scale of the space-time curvature and the energy scale $\Lambda_c$ are of the same order. Given that compact objects are sources of large curvatures, binary mergers are the perfect environments for gravity modifications to arise.
We point out the considered modifications in \cite{Endlich:2017tqa}, which are higher order corrections made out of the Riemann tensor, suppressed by an energy scale $\Lambda$. After building the Einstein-Hilbert action, the next term to consider is Riemann squared,
\begin{flalign}
\mathcal{S}_{R^2} \propto \int \mathrm{d}^4x\, \mathrm{det}e \, R_{abcd} R^{abcd},
\end{flalign}
\noindent which can be reduced to terms made out of $R$ and $R_{ab}$ after integration by parts, and therefore vanishes due to Einstein's equations in vacuum. Therefore, the next possible corrections arises from cubic terms \cite{Endlich:2017tqa}
\begin{flalign}
\mathcal{S}_{R^3} = \int \mathrm{d}^4 x\, \mathrm{det} e \, \left( c_{3} \frac{R_{abcd}R^{ab}_{\;\;\;ef}R^{efcd}}{\Lambda^4} + \tilde{c}_3 \frac{\tilde{R}_{abcd} R^{ab}_{\;\;\;ef} R^{efcd}}{\tilde{\Lambda}^4} \right),
\label{eq:gcubic}
\end{flalign}
\noindent and from quartic terms,
\begin{flalign}
\mathcal{S}^{}_{R^4} = \int \mathrm{d}^4 x\, \mathrm{det} e \, \left( c_{4}\frac{(R_{abcd} R^{abcd})^2}{\Lambda^6} + \tilde{c}_{4} \frac{(\tilde{R}_{abcd} R^{abcd})^2}{\tilde{\Lambda}_{}^6} + .\;.\;. \right),
\label{eq:gquartic}
\end{flalign}
\noindent with $\tilde{R}_{abcd} = \epsilon^{\;\;\;ef}_{ab} R_{efcd}$, and with $\Lambda$ and $\tilde{\Lambda}$, the energy scales that suppress the higher order corrections. For these modifications to show up in binary interactions, $R^3 \sim \Lambda^4$ and $R^4 \sim \Lambda^6$, and therefore, $\Lambda$'s $\sim O(\mathrm{km}^{-1})$ \cite{Endlich:2017tqa}.
For the details on the derivation of the higher order corrections in eq. (\ref{eq:gcubic}) and eq. (\ref{eq:gquartic}), we refer the reader to \cite{Endlich:2017tqa}. What is important to point out here, is that the derived effective action that describes a compact object in the next section, is valid for such theory, and generally for any other $f(R)$ theory of gravity which does not involve new degrees of freedom than those from the theory of general relativity. Gravity extensions with extra light degrees of freedom are considered in a future work.
\Irv{Talk also about scalar-tensor field theory}
\end{comment}
\section{Compact Objects in Effective Field Theory}
\label{sec:gravity}
\begin{comment}
The description of extended objects in the EFT framework was first introduced in a seminal work in \cite{Goldberger:2004jt, Goldberger:2005cd} for the non-spinning case. Then, spinning extended objects were introduced in \cite{Porto:2005ac}, and later developed \cite{Levi:2008nh}. The effective theory for spinning extended objects derived using the coset construction was introduced in \cite{Delacretaz:2014oxa}, but it was not further developed to take into account for the dynamics of compact objects as in \cite{Porto:2005ac, Levi:2015msa}. Recently, new developments of spinning extended objects have been introduced in \cite{Goldberger:2020fot}, with a different construction from the aforementioned ones. On the other hand, although BH electrodynamics was introduced in \cite{Goldberger:2005cd}, it was not until \cite{Patil:2020dme} that charge was considered in the EFT for extended objects to obtain the dynamics for non-spinning charged BHs.
\end{comment}
Since the introduction of the description of extended objects in the EFT framework \cite{Goldberger:2004jt}, a considerable number of extensions and improvements to the framework have been developed. A thoroughly review is beyond our scope, but we will refer to the relevant literature when build the EFT. We refer the reader to \cite{Porto:2016pyg, Levi:2018nxp, Goldberger:2022ebt} for a more complete treatment on compact binary dynamics.
The effective theory we develop is based on \cite{Delacretaz:2014oxa}, where an effective description for spinning extended objects in curved space-time is derived using the coset construction. In \cite{Endlich:2015mke}, the model is used in the Newtonian regime, which incorporates tidal and dissipative effects. Therefore, this work is an extension of \cite{Delacretaz:2014oxa, Endlich:2015mke}, and it describes general relativistic charged and spinning massive point particles, with higher order corrections due to the finite-size structure.
\begin{comment}
In the following, we will use the coset construction and extend the work on spinning extended objects in \cite{Delacretaz:2014oxa}, to consider the description of compact objects and their interactions, using all the other EFTs \cite{Porto:2005ac, Levi:2015msa, Goldberger:2020fot}, as a guideline for the development of our effective theory. Furthermore, we include the U(1) gauge symmetry of classical electromagnetism into the coset parameterization, to describe charged spinning extended objects. On the size effects due to the nature of the extended compact objects, we discuss in detail all relevant operators that play a role in the dynamics, and consider dynamical tidal effects, which are standard for modeling the dynamics of NSs, i.e. \cite{Flanagan:2007ix}.
\end{comment}
\subsection{Charged Spinning Compact Objects}
An invariant action for an extended object can be constructed by identifying the symmetry breaking pattern that such object generates.
In order to describe a charged spinning extended object, we need to consider the full symmetry group before being broken, $G = U(1)\times S \times ISO(3,1)$, with $S$, the internal symmetry of a spinning extended object which characterizes the low energy dynamics \cite{Delacretaz:2014oxa}. One can choose a coset parameterization such that the internal symmetry of the spinning extended object is unbroken, and only the full local Poincaré group is broken.
Moreover, in the comoving frame of the object, the state of a charged point like object is an eigenstate of the charge and does not break the U(1) symmetry. Although in principle a neutron star can have its own charge internal symmetries, such as the ones of a superfluid \cite{Delacretaz:2014oxa, Nicolis:2013lma}, given that in our effective approach we consider an extended object as a point particle with their properties encoded in higher order operators, we leave the charge symmetry unbroken.
\begin{comment}
It is worth noting that in principle, as a first good approximation, one can consider a NS as a superfluid. The symmetry breaking pattern of a superfluid is well known , which breaks global $U(1)$ charge Q. One may ask whether or not, in order to describe NSs, the breaking of the $U(1)$ symmetry is needed. Because we consider an extended object as a point particle, with its properties encoded in higher order operators which are accompanied by coefficients that encapsulates the internal structure, we leave the global charge unbroken.
\end{comment}
\begin{comment}
Nevertheless, it remains as an open question if NS have a different symmetry pattern to the one considered here, and whether or not the coset can parameterize it to provide a better description.
\end{comment}
\begin{comment}
\Irv{given that the generated electromagnetic field in the rotating frame of the particle, is the same as if it would be nonspinning}.
\end{comment}
\subsubsection*{Symmetry Breaking Pattern}
In the comoving frame, the group $G$ is broken into a linear combination of internal rotations, $S_{ij}$, and spatial rotations, $J_{ij}$, such that the symmetry breaking pattern for a charged spinning extended object reads,
\begin{flalign}
\begin{split}
\mathrm{Unbroken \; generators} &=
\begin{cases}
&\bar{P}_0 = P_0 + Q \;\;\;\;\;\;\;\;\;\, \mathrm{time \; translations},\\
&\bar{J}_{ij} \;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\; \;\;\;\;\; \mathrm{internal \; and \; space-time \; rotations.}
\end{cases}\\
\mathrm{Broken \; generators} &=
\begin{cases}
&P_i \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mathrm{spatial \; translations} , \\
&J_{ab}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\, \mathrm{boosts \; and \; rotations},
\end{cases}
\end{split}
\end{flalign}
\noindent with, $\bar{J}_{ij}$, the sum of the internal and space-time rotations \cite{Delacretaz:2014oxa}. We consider a spherical extended object at rest, for which $S_{ij}$ are the generators of the internal $SO(3)$ group. Translations are non-linearly realized and the local Poincaré and U(1) transformations are considered to take place along the fiber.
\begin{comment}
, such that $ \bar{J}_{ij} = S_{ij} + J_{ij}$
\end{comment}
A spinning extended object breaks the full local Lorentz group. Therefore, the coset parameterization,
\begin{equation}
g = e^{i y^a \bar{P}_a} e^{i \alpha_{ab} J^{ab}/2} = e^{i y^a \bar{P}_a} e^{i \eta^j J_{0j}} e^{i \xi_{ij} J_{ij}/2} = e^{i y^a \bar{P}_a} \bar{g}^{},
\label{eq:cosetcspp}
\end{equation}
\noindent which implies a correspondence between the Goldstone fields, $\alpha_{ab}$ and $\eta_i$, $\xi_{ij}$, due to the breaking of boosts and rotations, respectively. The parameterization in eq. (\ref{eq:cosetcspp}), allows us to obtain the Maurer-Cartan form without the need to specify the explicit unbroken generators of rotations. The residual symmetry, SO(3), requires all spatial indices to be contracted in an SO(3) invariant manner \cite{Delacretaz:2014oxa}
\subsubsection*{The Building Blocks}
The relevant degrees of freedom can be identified from the Maurer-Cartan form, projected onto the worldline of the object,
\begin{flalign}
\begin{split}
\dot{x}^{\mu} g^{-1} D_{\mu} g =& \; \dot{x}^{\mu} g^{-1} (\partial_{\mu} + i \breve{A}_{\mu} Q + i \breve{e}_{\mu}^{\;\; a} P_a + i \breve{\omega}_{\mu}^{\;\; ab} J_{ab} ) g \\
=& \; \dot{x}^{\mu} \bar{g}^{-1}(\partial_{\mu} + i A_{\mu} Q + i e_{\mu}^{\;\;a} P_a + \frac{i}{2} \omega_{\mu}^{ab} J_{ab}) \bar{g} \\
=& \; ie( P_0 + AQ + \nabla \pi^i P_i + \frac{1}{2} \nabla \alpha_{cd} J^{cd}).
\end{split}
\end{flalign}
\noindent The building blocks of the low energy dynamics are,
\begin{flalign}
e &= \dot{x}^{\mu} e_{\mu}^{\; \; a} \Lambda_a^{\;\; 0}, \\
A &= e^{-1} \dot{x}^{\mu} A_{\nu}^{} \Lambda_{\;\; \mu}^{\nu}, \label{eq:vierbein} \\
\nabla \pi^i &= e^{-1} \dot{x}^{\mu} e_{\mu}^{\; \; a} \Lambda_a^{\;\; i}, \\
\nabla \alpha^{ab} &= e^{-1} \left( \Lambda_c^{\;\; a} \dot{\Lambda}^{cb} + \dot{x}^{\nu} \omega_{\nu}^{cd} \Lambda_c^{\;\; a} \Lambda_d^{\;\; b} \right),
\label{eq:spinbuilding}
\end{flalign}
\noindent where $\dot{x}^{\mu} = \partial_{\sigma} x^{\mu}$, is the four velocity with $\sigma$ is the worldline parameter that traces out the trajectory of the particle. The $\Lambda$'s, are the Lorentz transformations parameterized by $\alpha$, or equivalently by $\eta$ and $\xi$.
In the breaking of space-time symmetries, one can impose the inverse Higgs constraint to remove some of the Goldstones \cite{Ivanov:1975zq}. Given that the commutator between the unbroken time translations and boosts gives broken spatial translations, $[K_i, P_0] = i P_i$, we set to zero the covariant derivative of the Goldstone,
\begin{equation}
\nabla \pi^i = e^{-1} (\dot{x}^{\nu} e_{\nu}^{a} \Lambda_{a}^{\; \; i} ) = e^{-1} (\dot{x}^{\nu} e_{\nu}^{0} \Lambda_{0}^{\; \; i} + \dot{x}^{\nu} e_{\nu}^{j} \Lambda_{j}^{\; \; i} ) = 0.
\label{eq:higgs-constraint}
\end{equation}
\noindent From this constraint, one can define the velocity \cite{Delacretaz:2014oxa}
\begin{flalign}
\beta^i = \frac{\dot{x}^{\mu} e_{\mu}^i}{\dot{x}^{\nu} e_{\nu}^0} = \frac{\dot{x}^i}{e},
\end{flalign}
\noindent which imply that $\Lambda^a_{\;\;0} = \dot{x}^a$. Therefore, $\beta^i$ parameterize the the boost necessary to get into the particle's rest frame.
The physical interpretation of the inverse Higgs constraint can be seen as well if we rewrite \cite{Delacretaz:2014oxa},
\begin{comment}
\begin{flalign}
\begin{split}
e =& \sqrt{(e \nabla \pi^i)^2 - (\dot{x}^{\nu} e_{\nu}^{\;\; a}\Lambda_{a}^{\;\; c} \dot{x}^{\mu} e_{\mu}^{b} \Lambda_{bc})}\\
=& \sqrt{-\eta_{ab} e_{\nu}^{\;\; a} e_{\mu}^{\;\; b} \dot{x}^{\nu} \dot{x}^{\mu}} = \sqrt{- g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}} = \frac{\mathrm{d} \tau}{\mathrm{d} \sigma},
\end{split}
\label{eq:E}
\end{flalign}
To obtain last equation, we have imposed the inverse Higgs constraint, and the property of the boost matrices, $\Lambda_{a}^{\;\; b} \Lambda^{a}_{\;\; c} = \delta^{b}_{\;\;c}$.
\end{comment}
\begin{flalign}
\begin{split}
e =& \sqrt{\dot{x}^{\mu} e_{\mu}^{\; \; a} \Lambda_a^{\;\; 0}\dot{x}^{\nu} e_{\nu}^{\; \; b} \Lambda_{b0}^{\;\; }}= \sqrt{(e \nabla \pi^i)^2 - (\dot{x}^{\nu} e_{\nu}^{\;\; a}\Lambda_{a}^{\;\; c} \dot{x}^{\mu} e_{\mu}^{b} \Lambda_{bc})}\\
=& \sqrt{-\eta_{ab} e_{\nu}^{\;\; a} e_{\mu}^{\;\; b} \dot{x}^{\nu} \dot{x}^{\mu}} = \sqrt{- g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}} = \frac{\mathrm{d} \tau}{\mathrm{d} \sigma},
\end{split}
\label{eq:E}
\end{flalign}
\noindent with $\sigma$ the worldline parameter, and $\tau$ the proper time measured in the particle's rest frame. To obtain last equation, we have imposed the inverse Higgs constraint, and the property of the Lorentz matrices, $\Lambda_{a}^{\;\; b} \Lambda_{c}^{\;\; a} = \delta^{b}_{\;\;c}$. We identify the building block $e$, as the einbein.
Therefore, with eq. (\ref{eq:E}), we can rewrite the inverse Higgs constraint in a way that makes manifest its physical interpretation of \cite{Delacretaz:2014oxa}. For rotations, $\Lambda^0_{\;\;a} (\xi) = \delta^0_a$ and $\Lambda^i_{\; j} (\xi) = \mathcal{R}^i_{\;\;j} (\xi)$, with $\mathcal{R}(\xi)$ an $SO(3)$ matrix, such that the constraint (\ref{eq:higgs-constraint}), now reads
\begin{equation}
u^a \Lambda_a^{\;\; i} (\eta) \mathcal{R}_{i}^{\;\; j} (\xi) = 0,
\end{equation}
\noindent with, $u^a = e_{\mu}^{\; \; a} \partial_{\tau} x^{\mu}$, the velocity measured in the local co-moving frame defined by the vierbein. Given that the matrix $\mathcal{R}^{\;\;i}_j (\xi)$ is invertible, we obtain
\begin{equation}
u^a \Lambda_a^{\;\; i} (\eta) = 0.
\end{equation}
These quantities now have a clear geometrical interpretation: the set of local Lorentz vectors,
\begin{equation}
{\hat{n}^a_{\;\; (0)} \equiv u^a = \Lambda^a_{\;\; 0} (\eta) \;, \;\;\;\; \hat{n}^{a}_{(i)} \equiv \Lambda^a_{\;\; i} (\eta) },
\label{eq:neta}
\end{equation}
\noindent define an orthonormal local basis with respect to the local flat metric, $\eta_{a b}$, in the frame that is moving with the particle's trajectory \cite{Delacretaz:2014oxa}. One can also define the orthonormal basis in terms of the space-time vectors, $\hat{n}^{\mu}_{\;(b)} \equiv e^{\mu}_{a} \hat{n}^a_{(b)}$, with respect to the full metric, $g_{\mu \nu}$. Moreover, an additional set of orthonormal vectors is obtained \cite{Delacretaz:2014oxa},
\begin{equation}
\hat{m}^b_{\;\; (a)} \equiv \Lambda^b_{\; \; a} (\alpha) = \Lambda^b_{\; \; c} (\eta) \mathcal{R}^c_{\; \; a} (\xi),
\label{eq:orthom}
\end{equation}
\noindent with the zeroth vector, $\hat{m}^b_{\;\;(0)} = \hat{n}^b_{\;\;(0)} = u^b$, coinciding. The rest of the vectors differs by a rotation, $\mathcal{R} (\xi)$. Therefore, the set of vectors in eq. (\ref{eq:orthom}), contain the information about the rotation, parameterized by the degrees of freedom of $\xi$.
With these identifications, in the co-moving and co-rotating frame of the particle, the covariant derivatives of the Goldstone, $\nabla \alpha^{0i}$, can be expressed as \cite{Delacretaz:2014oxa}
\begin{equation}
\nabla \alpha^{0i} = \mathcal{R}_{j}^{\; \; i} (\xi) \Lambda_{a}^{\;\; j} (\eta) (\partial_{\tau} u^a + u^{\mu} \omega_{\mu\;\;c}^{\;\;a} u^c) = \Lambda_{a}^{\; \; i} (\alpha) u^{\mu} \nabla_{\mu} u^a = \Lambda_{a}^{\; \; i} (\alpha) e_{\mu}^a a^{\mu}
\label{eq:temporalspinalpha}
\end{equation}
\noindent which is the acceleration projected into the orthonormal basis defined by the $\hat{m}$'s. In the co-moving frame, $\nabla \alpha^{0i} = 0$, by definition. The rest of the covariant derivatives of the Goldstone, reads \cite{Delacretaz:2014oxa}
\begin{equation}
\nabla \alpha^{ij} = \Lambda^{\;\;i}_{k} (\alpha) (\eta^{kl} \partial_{\tau} + \omega_{\mu}^{kl} u^{\mu}) \Lambda_{l}^{\;\; j} (\alpha)= \Omega^{ij} (\tau),
\end{equation}
\noindent which is the angular velocity of the object in the co-moving frame.
\begin{comment}
\end{comment}
\begin{comment}
In the absence of external forces, this building block is zero. Nevertheless, it must be considered to build up invariant operators when an external force, such as the one from an external gravitating object, or from the charge of another object, is strong enough to make this building block relevant.
\end{comment}
\begin{comment}
In order to take into account for all possible operators that contribute to the dynamics, we need to consider the electromagnetic and Riemann tensor obtained as the curvature invariants \cite{Goldberger:2004jt,Goldberger:2005cd}.
\end{comment}
The gauge field strengths can be used as well as building blocks in the worldline of the particle. To use these operators, we define the transformation \cite{Delacretaz:2014oxa},
\begin{equation}
R_{abcd} = g^{-1}_{L} \; \tilde{R}^{} = (\Lambda^{-1} )_{a}^{\;\; e} (\Lambda^{-1} )_{b}^{\;\; f} (\Lambda^{-1} )_{c}^{\;\; g} (\Lambda^{-1})_{d}^{\;\; h} \tilde{R}^{}_{efgh},
\label{eq:Riemannproper}
\end{equation}
\noindent with, $g_{L}$, the Lorentz part of the parameterization in eq. (\ref{eq:cosetcspp}), and $R_{abcd}$, the Riemann tensor in the local rest frame of the object. To describe induced moments on the worldline, one works instead with the Weyl tensor, $W_{abcd}$ which has the physical content \cite{Goldberger:2004jt}. The Weyl tensor is obtained by subtracting out various traces from the Riemann tensor. The electromagnetic stress tensor transformation,
\begin{equation}
F_{ab} = (\Lambda^{-1} )_{a}^{\;\; c} (\Lambda^{-1} )_{b}^{\;\; d} \tilde{F}^{}_{cd}.
\label{eq:Fproper}
\end{equation}
\begin{comment}
\begin{equation}
W_{abcd} = R_{abcd} + \frac{1}{2} (R_{ad}g_{bc} - R_{ac}g_{bd} + R_{bc}g_{ad} - R_{bd}g_{ac}) + \frac{1}{6} R (g_{ac} g_{bd} - g_{ad} g_{bc}).
\end{equation}
\end{comment}
\begin{comment}
\noindent The Weyl tensor contains the tidal force exerted on an extended particle that is moving along the worldline, taking into account for how the shape of the body is distorted.
\end{comment}
\begin{comment}
The Weyl tensor measures the curvature of the space-time and contains the tidal force exerted on an extended particle that is moving along the worldline,
\end{comment}
\begin{comment}
with $\tilde{A}_{\mu}^{} = A_{\nu}^{} \Lambda_{\;\; \mu}^{\nu}$.
\end{comment}
\subsubsection*{Massive Objects}
We start by considering the einbein, $e$. The action that can be built with this building block is simply \cite{Delacretaz:2014oxa}
\begin{flalign}
\begin{split}
\mathcal{S} =& - n_e \int \mathrm{d} \sigma \, e
= - m \int \mathrm{d} \sigma \sqrt{- g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}},
\label{eq:pp}
\end{split}
\end{flalign}
\noindent where the coefficient, $n_e$, has been identified as the mass of the object, $m$. Eq. (\ref{eq:pp}), which is invariant under re-parameterizations of the particle's trajectory, is the usual point particle action that can be expanded around a small gravitational perturbation $h_{\mu \nu} = g_{\mu \nu} - \eta_{\mu \nu}$, and in powers of the velocity parameter, $v$. Nevertheless, this equation for the point particle has a disadvantage, that when expanding around $h_{\mu \nu}$, it will generate an infinite number of powers of $h$.
This can be fixed by recasting the point particle action in a Polyakov form \cite{Green:1987sp},
\begin{flalign}
\mathcal{S}_{} = - \frac{1}{2}\int \mathrm{d}\sigma e \left(m^2 - e^{-2}g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} \right).\label{eq:Poly}
\end{flalign}
\noindent which no longer has a square root on the metric. This action now linear in the metric, implies a one point function only. Non-linearities to the point particle are encoded in the einbeins \cite{Kuntz:2020gan}.
The physical interpretation of the inverse einbein become manifest when integrating it out from the effective action, $\delta \mathcal{S} / \delta e = 0$. It leads to the mass-shell constraint
\begin{flalign}
g_{\mu \nu } \dot{x}^{\mu} \dot{x}^{\nu} + e^2 m^2 = 0.
\end{flalign}
\noindent In the proper frame of the particle, $ \dot{x}^2 = -1$, and the einbein, $1/e = m$, is the mass. Sitting outside the rest frame, we can parameterize the velocity $\dot{x}^{\mu} = (1, v^i)$, and the inverse einbein is now the relativistic mass,
\begin{flalign}
\frac{1}{e} = \frac{m}{\dot{x}^2} = \frac{m}{\sqrt{1 - v^2}} = m \gamma,
\label{eq:einbeinmass}
\end{flalign}
\noindent where we have considered the special relativistic limit. Therefore, in $\beta^i$, the einbein parameterizes the boost to the particle's rest frame.
The definition of the Polyakov action used in (\ref{eq:Poly}), gives a transparent construction of the action in terms of invariant quantities using the relativistic momentum, $p^{\mu} = \dot{x}^{\mu}/e = m \gamma \dot{x}^{\mu} $. The inverse einbein (\ref{eq:einbeinmass}), can be seen as the energy of the particle. Substituting the explicit value of the einbein (\ref{eq:einbeinmass}) in the effective action (\ref{eq:Poly}), one recovers the point particle action in eq. (\ref{eq:pp}). Nontheless, with such definition of the Polyakov action, each einbein is mass dependent and we would need to construct invariant operators that are supressed by mass factors of the order of the number of einbeins that the operator posses.
This is remediated by using the definition of the Polyakov action currently used in the EFT for compact objects \cite{Kuntz:2020gan,Kalin:2020mvi,Mogull:2020sak},
\begin{flalign}
\mathcal{S}_{} = - \frac{m}{2}\int \mathrm{d}\sigma e \left(1 - \frac{1}{e^2} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} \right).\label{eq:Polynom}
\end{flalign}
\noindent Integrating out the einbein, the constraint now leads to the einbein derived in (\ref{eq:E}),
\begin{flalign}
e^2 = - g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu},
\label{eq:onshellconstr}
\end{flalign}
\noindent from which one recovers the point particle action in (\ref{eq:pp}) as well. Now the on-shell condition implies that $e^2 = 1$.
Therefore, the only difference between using eq. (\ref{eq:Poly}) and (\ref{eq:Polynom}), is how we parameterize our invariant quantities. We choose to use eq. (\ref{eq:Polynom}), such that the einbein has no dependence on the mass, and the relativistic momentum is defined as
\begin{flalign}
p^{\mu} = m \frac{\dot{x}^{\mu}}{e}.
\end{flalign}
\noindent With such parameterization, in the special relativistic limit, $1/e = \gamma$, and $\beta^i$, is the relativistic velocity which is worldline re-parameterization covariant.
By choosing eq. (\ref{eq:Polynom}), the Polyakov action can be exploited only through the covariant quantity, $\dot{x}^{\mu}/e$.
\subsubsection*{Charged Point Particles}
Now we consider the building block, $A$. The effective action of a massive charged point particle reads
\begin{flalign}
\begin{split}
\mathcal{S} =& \int \mathrm{d} \sigma e ( - m + n_A A ) = \int \mathrm{d} \sigma e( -m + q \frac{\dot{x}^{\mu}}{e} A_{\mu} ) = - m \int \mathrm{d} \sigma e + q \int \mathrm{d} \sigma \dot{x}^{\mu} A_{\mu} ,
\label{eq:ppchg}
\end{split}
\end{flalign}
\noindent where we have matched the coefficient, $n_A = q$, from the action of a charged point particle \cite{Goldberger:2004jt,Patil:2020dme} with net charge, $q$. The new correction due to charge is invariant under worldline re-parameterizations as well, and it is independent of the einbein. Therefore, in eq. (\ref{eq:ppchg}), we could re-writte the point particle term in a Polyakov form, and the on-shell constraint, (\ref{eq:onshellconstr}), would remain the same.
\subsubsection*{Spinning Objects}
Now we include the building block, $\nabla \alpha^{ab}$, neglecting charge for simplicity. To lowest order, the effective action for a spinning object reads,
\begin{equation}
\mathcal{S} = \int \mathrm{d} \sigma e \left(-m + n_{\alpha} \nabla \alpha_{ab} \nabla \alpha^{ab} + .\;.\;. \right),
\end{equation}
\noindent where a term linear in $\nabla \alpha$ has been discarded by time reversal symmetry, and we have considered spherical objects at rest \cite{Delacretaz:2014oxa}. The ellipses denotes higher order corrections made out of the Goldstone field.
In contrast to \cite{Delacretaz:2014oxa}, where boosts and rotations are treated as independent building blocks, we propose that the natural covariant object to work with when describing relativistic spinning objects, is the covariant derivative of the Goldstone boson. Therefore, we define the relativistic angular velocity through,
\begin{flalign}
\nabla \alpha^{ab} = e^{-1} \Omega^{ab} (\sigma) = e^{-1} \Lambda^{\;\;a}_{c} (\eta^{cd} \partial_{\sigma} + \omega_{\mu}^{cd} \dot{x}^{\mu}) \Lambda_{d}^{\;\; b}.
\end{flalign}
\noindent The implications of this definition for the relativistic angular velocity, is that $\Omega^{0b} = a^{b}$. This is in agreement with the fact that $ \Omega^{0b}$ is a gauge choice \cite{Steinhoff:2015ksa}, and that in the rest frame of the particle, $\Omega^{0b} = a^b = 0 $.
\begin{comment}
To characterize the rotation of a spherical rigid object, only two parameters are needed: the mass, $m$, and moment of inertia, $I$.
\end{comment}
Comparing our action to the one of a relativistic spinning point particle \cite{HANSON1974498}, we can match the coefficient, $n_{\Omega} = I/4$ \cite{Delacretaz:2014oxa}, with $I$ the moment of inertia. The relativistic action for a spinning point particle in curved space-time,
\begin{equation}
\mathcal{S} = \int \mathrm{d} \sigma e \left(-m + \frac{I}{4} \nabla \alpha^{ab} \nabla \alpha_{ab} + .\;.\;. \right) =\int \mathrm{d} \sigma e \left(-m + \frac{I}{4} e^{-2} \Omega_{ab} \Omega^{ab} + .\;.\;. \right).
\label{eq:spinningLO}
\end{equation}
\noindent The correction to the point particle due to rotation is worldline re-parameterization invariant. To lowest order on the second term, the einbein can be set to one, and one recovers the usual correction of the angular velocity \cite{HANSON1974498,Porto:2005ac}. But in contrast to any other effective action for spinning objects, beyond lowest order we have einbeins which are in general are different from one.
\begin{comment}
Another approach would be to consider the effective action
\begin{equation}
\mathcal{S} =- \frac{m}{2}\int \mathrm{d}\sigma \left(e - \frac{1}{e} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} \right) + \frac{I}{4} \int \mathrm{d} \sigma \frac{1}{e} \Omega_{ab} \Omega^{ab} + .\;.\;. .
\label{eq:spinningLO2}
\end{equation}
\noindent Integrating out the einbein, we obtain the constraint
\begin{flalign}
\begin{split}
e^2 = - g_{\mu \nu }\dot{x}^{\mu} \dot{x}^{\nu} - \frac{I}{2m} \Omega^2.
\end{split}
\end{flalign}
\noindent Setting the angular velocity to zero, we recover the expected on-shell constraint. The combination, $I\Omega^2/m$ is dimensionless. Considering the special relativistic case, we obtain the einbein to lowest order in the angular velocity,
\begin{flalign}
e = - \sqrt{1 - \left(v^2 + \frac{I \Omega^2}{2m}\right)}.
\label{eq:einbeinspin}
\end{flalign}
\noindent which in principle can be expanded in terms of a small expansion parameter as well.
\end{comment}
There exist a tower of operators made out of the Goldstone boson in eq. (\ref{eq:spinningLO}). Some of the corrections that can play a role in the evolution of the angular velocity of the object are \cite{Delacretaz:2014oxa,Endlich:2015mke,HANSON1974498},
\begin{flalign}
\begin{split}
\mathcal{S} =& \int \mathrm{d} \sigma e \left(-m + \frac{I}{4} e^{-2} \Omega_{ab} \Omega^{ab} + n_{\Omega^2} e^{-4}(\Omega_{ab} \Omega^{ab})^2 + e^{-4} n_{\Omega,u}\Omega_{ac} \Omega^{cb}\dot{x}^a \dot{x}_b + .\;.\;. \right).
\label{eq:spinningNLO}
\end{split}
\end{flalign}
\noindent For a compact object, the frequency of the normal modes, $\omega_{N} \sim c/\ell$, with $\ell$ the radius of the extended object. The tower of higher order operators made out of the angular velocity is valid in perturbation theory as long as the rotational frequency is much less than the speed of sound of the material $c_{s}$ \cite{Delacretaz:2014oxa}, which for relativistic objects, $c_{s} \sim c$.
For a rotating extended object in the classical limit, the angular velocity is constrained by the dimensionless spin,
\begin{equation}
\chi = \frac{ J}{G M^2} = \frac{ I \Omega}{G M^2},
\label{eq:chi}
\end{equation}
\noindent which $\chi \leq 1$. Using the well constrained properties of compact objects \cite{LIGOScientific:2018mvr,Abbott:2020gyp}, one can realize that the expansion over the parameter, $\Omega/c$, is under perturbative control as long as we are considering similar properties to those so far detected by the LIGO-Virgo observatories.
\begin{comment}
\noindent The gravitational dynamics of spinning compact objects is treated in detail in an upcoming work, where the einbein in eq. (\ref{eq:einbeinspin}) is explored, as well as the corrections from eq. (\ref{eq:spinningNLO}).
\begin{equation}
\mathcal{S} = \int \mathrm{d} \tau \left\{ -m + \frac{I}{4} \Omega_{ab} \Omega^{ab} + \frac{J}{8} (\Omega_{ab} \Omega^{ab})^2 + n_{\alpha,u} \Omega_{ac} \Omega^{cb} \frac{u^a u_b}{u^2} + .\;.\;. \right\}.
\end{equation}
\end{comment}
\begin{comment}
To eq. (), Finally, before proceeding to consider the spin gravitational coupling and other corrections, we recall some of the spin/angular velocity corrections that can play a role in the evolution of the angular velocity of the object \cite{Delacretaz:2014oxa,Endlich:2015mke,HANSON1974498},
\begin{equation}
\mathcal{S} = \int \mathrm{d} \tau \left\{ -m + \frac{I}{4} \Omega_{ab} \Omega^{ab} + \frac{J}{8} (\Omega_{ab} \Omega^{ab})^2 + n_{\alpha,u} \Omega_{ac} \Omega^{cb} \frac{u^a u_b}{u^2} + .\;.\;. \right\}.
\end{equation}
\end{comment}
\subsubsection*{Spin Supplementary Condition}
Since the inverse Higgs constraint implies that $e^{-1} (\dot{x}^{a} \Lambda_{a}^{\; \; i} ) = 0$, then the Goldstone boson is orthogonal to the four velocity as well, $ \dot{x}_a \nabla \alpha^{ab} = 0$. This has a direct implication on what is known as the Spin Supplementary Condition. Expanding the components of the Goldstone field, we obtain
\begin{flalign}
\begin{split}
e^{-1} \dot{x}_a \nabla \alpha^{ab} =& -e^{-1} \nabla \alpha^{0b} + e^{-1} \dot{x}_i \nabla \alpha^{ib} = e^{-2} \dot{x}_{\mu} \nabla^{\mu} \dot{x}^{b} + e^{-2} \dot{x}_i \Omega^{ib} = 0.
\end{split}
\end{flalign}
\begin{comment}
\frac{1}{e^2} a^{b} + \frac{\dot{x}_i \Omega^{ib}}{e^2}
\end{comment}
Removing the einbeins, we are left with the constraint,
\begin{flalign}
\begin{split}
\dot{x}_a \Omega^{ab} =& \dot{x}_0 \Omega^{0b} + \dot{x}_i \Omega^{ib} = \dot{x}_{\mu} \nabla^{\mu} \dot{x}^{b} + \dot{x}_i \Omega^{ib} = 0,
\end{split}
\end{flalign}
\noindent which is dependent on the chosen worldline, $x^{\mu}$. This is the analog of the spin supplementary condition (SSC), written in a generic fashion and in terms of the angular velocity. The condition is fixed when choosing a worldline to describe the dynamics, but it is needed as well to choose a frame to describe the rotation.
In the rest frame of the particle, $\Omega^{0b} = 0$, and $\dot{x}^i = 0$. Therefore, we end up with the constraint, $\dot{x}_a \Omega^{ab} = 0$, known as the Covariant SSC. In this frame, $\sigma = \tau$, we have that $\partial_{\tau} x_0 = u_0 = \sqrt{-u^2}$, which is an invariant quantity. Thus, we can replace, $\dot{x}_0 = \sqrt{-\dot{x}^2}$. Then, to describe the dynamics on another worldline with a different set of local orthonormal basis, one transforms the local indices with Lorentz matrices, $\tilde{\Omega}^{ab} = \Lambda^a_{\;\;c} \Lambda^b_{\;\;d} \Omega^{cd}$ and choose a different einbein or worldline parameter. A useful parameterization for the velocity is, $\dot{x}^a = \partial_{t} x^a = (1, v^i) $, with $\sigma = t$, the coordinate time. In this frame, the SSC condition,
\begin{flalign}
\begin{split}
\dot{x}_a \tilde{\Omega}^{ab} =& \; e \tilde{\Omega}^{0b} + \dot{x}_i \tilde{\Omega}^{ib} = 0 \\
\label{eq:ssccoordinate}
\end{split}
\end{flalign}
\noindent is the analog to the Canonical SSC \cite{Steinhoff:2015ksa,Levi:2015msa}, with $e = \sqrt{-\dot{x}^2}$, and $\tilde{\Omega}^{ab}$, analog to the canonical spin \cite{Levi:2015msa}. Although to lowest order, in eq. (\ref{eq:ssccoordinate}), $\tilde{\Omega}^{0b} = 0$, when working outside the co-rotating frame of the particle, one should consider the SSC in eq. (\ref{eq:ssccoordinate}), to remove the temporal components of the Goldstone boson accordingly.
\subsubsection*{Field-Field Interactions}
From an EFT point of view, we should consider all possible combinations between fields that are allowed by the symmetries. For a charged spinning object, a first possible operator to consider, even in the absence of gravity, is the angular velocity or the spin coupled with the electromagnetic tensor. Such Spin-Field coupling,
\begin{flalign}
\mathcal{S} = n_{\Omega,q} \int \mathrm{d} \sigma e \; \nabla \alpha_{ab} F^{ab} = n_{\Omega,q} \int \mathrm{d} \sigma \; \Omega_{ab} F^{ab} ,
\end{flalign}
\noindent corresponds to a Pauli interaction term \cite{Pauli:1941zz}, which needs to be taken into account \cite{Skagerstam:1981xp, vanHolten:1990we}. The gauge field of the unbroken U(1) generator, $A_{a}$, enters into the dynamics, first with the invariant combination, $e^{-1} A_{\mu} \dot{x}^{\mu}$, and then, through the covariant quantity, $F_{ab}$.
Therefore, on the gravitational side, we can use the coefficients from the unbroken Lorentz generators (from the bulk theory) with the four velocity. The covariant building block made out of the spin connection is, $e^{-1} \omega_{\mu}^{ab} \dot{x}^{\mu}$. Therefore, we can build up the correction,
\begin{flalign}
\mathcal{S} = n_{\Omega,g} \int \mathrm{d} \sigma e \; \left( \nabla \alpha_{ab} e^{-1}\omega^{ab}_{\mu} \dot{x}^{\mu} \right) = \int \mathrm{d} \sigma e^{-1} \; n_{\Omega,g} \Omega_{ab} \omega^{ab}_{\mu} \dot{x}^{\mu} .
\label{eq:spino}
\end{flalign}
\noindent Comparison to the full theory shows that the coefficient $n_{\Omega,g} = I/2$, with $I$ the moment of inertia, and therefore the spin is defined as $S^{ab} = I \Omega^{ab}$. To lowest order with $e = 1$, one recovers the well known results in both the PN and PM expansions for the Spin-Orbit coupling in curved space-time. This is a crucial difference of our EFT with respect to others, to allow the covariant quantity, $e^{-1} \omega_{\mu}^{ab} \dot{x}^{\mu}$ to be a building block. Therefore, in eq. (\ref{eq:spino}), there is an inverse einbein in the spin-orbit correction, for which beyond lowest order, the equations of motion will differ from to the ones obtained using conjugate variables, i.e. \cite{Porto:2005ac,Levi:2015msa,Liu:2021zxr}.
Another implication is that the complete Spin-Orbit correction is described by two terms,
\begin{flalign}
\mathcal{S} = \int \mathrm{d} \sigma e \; \left( n_{\Omega,g} e^{-2} \Omega_{ab} \omega^{ab}_{\mu} \dot{x}^{\mu} + n_{\Omega, g, u} e^{-4} \Omega_{ab} \dot{x}^a \omega_{\mu}^{cb} \dot{x}^{\mu} \dot{x}_c \right).
\label{eq:spino2}
\end{flalign}
\noindent This extra term comes from the possible combinations of the angular velocity with the spin connection and the four velocity. The extra term does not spoil the effective theory. To lowest order, in the PM expansion the extra term is removed by the corresponding SSC, recovering the results in \cite{Liu:2021zxr} following their procedure. In the PN expansion, this term can be seen as the contribution from the relativistic acceleration in \cite{Levi:2015msa}, which plays a role on the Spin-Orbit dynamics of the PN expansion.
\begin{comment}
Following our reasoning we can keep looking for corrections that are made out of our building blocks. A tentative operator would be, $\Omega_{ab} B^{ab}$, with $B^{ab}$, the magnetic component of the Riemann tensor. Nevertheless, although this correction is parity invariant, it does not satisfy reversal symmetry. Therefore, we are not able to find any other corrections, other than the ones in eq. (\ref{eq:spino}), that enters into the spin-orbit. But a natural question comes out: Can we form higher order corrections our of the $\Omega^{ab}$ and $\omega^{ab}_{\mu}$?.
\end{comment}
The next set of corrections that can be built with the spin connection and the angular velocity that satisfies parity and time reversal symmetry,
\begin{flalign}
\begin{split}
\mathcal{S} = \int \mathrm{d} \sigma e \left( n_{\omega^2 \Omega^2} e^{-2} \omega_{\nu}^{ab} \omega^{\nu cd} \Omega_{ab} \Omega_{cd} + n_{R \Omega^2} e^{-4} R_{abcd} \Omega^{ab} \Omega^{cd} \right),
\end{split}
\end{flalign}
\noindent with the Riemann tensor, $R_{abcd} = R_{abcd} (\omega_{\mu}^{ab})$. Both of these corrections are considered in \cite{Vines:2016unv} in the Hamiltonian formalism, and the second term is considered as well in \cite{Bern:2020buy} and as the lowest order correction in \cite{Bern:2022kto} from a tower of correction made out of the the covariant derivative and the projected spin tensor, tensor which is defined below. Another possible operator that can be built is
\begin{flalign}
\mathcal{S} = \int \mathrm{d} \sigma e \; n_{\Omega^2 R} e^{-4} \Omega^{ab} \Omega^{cd} \dot{x}_{a} R_{bfcd} \dot{x}^f,
\label{eq:spinR}
\end{flalign}
\noindent correction that appears in the Routhian of \cite{Liu:2021zxr}, and without going into details into the Routhian formalism, we suggest that this is a term to consider when building the EFT, given that is built out of the building blocks and is allowed by the symmetries.
We can couple the electromagnetic field tensor as well with the Lorentz gauge field, and build the Electro-Gravity couplings,
\begin{flalign}
\mathcal{S} = \int \mathrm{d} \sigma e \; \left(n_{F,g} e^{-1} F_{ab} \omega^{ab}_{\mu} \dot{x}^{\mu} + n_{F,g,u} e^{-3} F_{ab} \dot{x}^{b} \omega^{ca}_{\mu} \dot{x}^{\mu} \dot{x}_{c} + .\;.\;. \right).
\end{flalign}
\noindent In general we should couple all of the covariant operators in all the possible ways that are allowed by the symmetries. By including all possible operators there may be redundancies via the equations of motion, but even if all of them are included, one will get the same answer for anything physical.
\subsubsection*{Extended Objects}
We now describe the properties of an extended object, beyond the point particle approximation. Invariant operators can be built to account for the tidal deformation of the compact object using the electric and magnetic components of the electromagnetic and Weyl tensor in the worldline \cite{Goldberger:2004jt}. It is useful to work with multipole expansions at the level of the action \cite{Ross:2012fc}, where induced multipole moments are organized in irreducible representations of the rotation SO(3) group. This makes the task of making invariant operators more transparent and ensures that multipole moments are not mixed.
The projected electromagnetic and Weyl tensor are used to define their electric, $E_{a .\,. \,.}$ and magnetic, $B_{a .\,. \,.}$, components. Size effects are made out of the curvature,
\begin{flalign}
\begin{split}
E^{}_{ab} =& e^{-2} W_{acbd} \dot{x}^c \dot{x}^d, \\
B_{ab}^{} =& \epsilon_{cdea} e^{-2} W^{cd}_{\;\;\;\;fb} \dot{x}^e \dot{x}^f.
\label{eq:GravityEM}
\end{split}
\end{flalign}
\begin{comment}
B_{ab}^{} =& \frac{1}{2} \epsilon_{cdea} e^{-2} W^{cd}_{\;\;\;\;fb} \dot{x}^e \dot{x}^f.
\end{comment}
\noindent The polarization is accounted by the electromagnetic tensor,
\begin{flalign}
\begin{split}
E^{}_{a} =& e^{-1 }F_{ab} \dot{x}^b, \\
B_a^{} =& \epsilon_{abcd} e^{-1} F^{bc} \dot{x}^d.
\label{eq:ElectroEM}
\end{split}
\end{flalign}
\noindent These operators, which are traceless and transverse to $\dot{x}^a$, are covariant under worldline re-parameterizations. They couple to all other derived building blocks, in the possible ways allowed by the symmetries, to construct higher order operators.
\begin{comment}
The electric and magnetic components of the electromagnetic tensor, $E^{}_{a}$ and $B_a^{} $, and the electric and magnetic components of the Weyl tensor, $E^{}_{ab}$ and $B_{ab}^{} $, are extensively used as building blocks of the multipole moments.
\end{comment}
We can also build spin-induced multipoles \cite{Porto:2005ac,Levi:2014gsa}, using the angular velocity as the Pauli-Lubanski spin tensor,
\begin{flalign}
e^{-2} \Omega_a = \frac{1}{2} \epsilon_{abcd} e^{-2} \Omega^{cd} \dot{x}^{b}.
\label{eq:PaulL}
\end{flalign}
\noindent In the proper frame of the object,
\begin{equation}
\Omega_i = - \frac{1}{2} \epsilon_{ijk} \Lambda_{a}^{\;\; j}(\eta^{ab} \partial_{\tau} + \omega_{\mu}^{ab} u^{\mu})\Lambda_{b}^{\;\;k}.
\label{eq:spinepsilon}
\end{equation}
Considering the covariant operators and their corresponding covariant derivatives, to lowest order we build
\begin{flalign}
\begin{split}
\mathcal{O}(\Omega,E,B) &=
\begin{cases}
& e^{-4} E^{ab} E_{ab},\, e^{-4} B^{ab} B_{ab} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mathrm{Gravity},\\
& e^{-6} \Omega^a \Omega^b E_{ab},\, e^{-9} \Omega^a \Omega^b \Omega^c \nabla_c B_{ab} \,\;\;\;\;\;\;\;\;\;\;\; \mathrm{Spin-Gravity},\\
& e^{-2} E_{}^{a} E_{a}, \, e^{-2} B_{}^{a} B_{a} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\;\;\; \;\; \mathrm{Electromagnetic,}\\
& e^{-3} \Omega^a B_a, \; e^{-6} \Omega^a \Omega^b \nabla_b E_a \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\; \mathrm{Spin - Electro,}\\
\end{cases}
\end{split}
\label{eq:operators}
\end{flalign}
\noindent which are parity and time reversal invariant. The operators from the first \cite{Goldberger:2005cd}
and second line \cite{Porto:2005ac, Levi:2015msa} in (\ref{eq:operators}), encode the tidal deformation due to an induced multipole. The rest of the operators takes into account for the polarization of the extended object. The ones from the third line were first considered in \cite{Goldberger:2004jt}, and from the fourth line we have not found a reference yet. The latter could play a crucial role in the description of pulsars: neutron stars with considerable spin and strong magnetic field. In the effective action, each of the operators are accompanied by a Wilson coefficient that encodes the internal structure of the compact object. The operators in eq. (\ref{eq:operators}), contains linear corrections due to the tidal deformation and polarization. Non-linear effects \cite{Bern:2020uwk} can be considered using the blocks from eq. (\ref{eq:GravityEM}) and eq. (\ref{eq:ElectroEM}).
The covariant derivative can be used in the Goldstone's building blocks, and therefore we should build terms made out of $\nabla_{b} \Omega_{a}$ as well. Moreover, in principle one can couple two different multipole expansions, respecting parity invariance and time reversal symmetry. From such considerations, we build the following operators,
\begin{flalign}
\begin{split}
\mathcal{O}(\Omega,E,B) &=
\begin{cases}
& e^{-4} \nabla_{a} E_{b} E^{ab},\, e^{-4} \nabla_{a} B_{b} B^{ab},\,e^{-5} \nabla_b \Omega_a B^{ab},\, e^{-5} \Omega_{a} B_b E^{ab} \;.\;.\;..\\
\end{cases}
\end{split}
\label{eq:operatorsGEM}
\end{flalign}
\noindent Although to lowest order, $\nabla_b \Omega_a = 0$, the spin evolves through time due to spin-size effects even in the Newtonian limit \cite{Endlich:2015mke}, and therefore one should carefully treat such term.
\begin{comment}
\begin{flalign}
\nabla_a \nabla_b E_{cd} E^{ab} E^{cd}, \;\; \nabla_a E_{b} \nabla_c E_{d} E^{ab} E^{cd}, \;\; E_a E_b E^{ab} \;\; B_{ab} \omega_{\mu}^{ab}, \;\; E_a B_b \omega_{\mu}^{ab}, B_{a} B_{b} E^{a b}, E_{a} B_{b} B^{a b}
\end{flalign}
These are parity invariant, looks possible. Is there a possibility to generalize this couplings? \textcolor{red}{We are trying to consider all possible terms}
\end{comment}
It should be noted that size effects can be seen as encoded in a composite operator $Q^{ab}$ \cite{Goldberger:2005cd}, which we discuss in detail below. By considering the tidal deformation as a dynamical operator, we can consider considering dissipative effects and dynamical tides.
\begin{comment}
One should build all the couplings allowed by the symmetries. For instance, the combinations, $\Omega_{a} B_{b} E^{ab}\, \mathrm{and} \, \Omega_a E_b B^{ab}$, although they respect parity invariance, these operators contains mixed multipole moments and are not allowed \textcolor{red}{(?)}.
\end{comment}
\subsubsection*{Dynamical Operators}
One can introduce composite dynamical operators, $Q_{ab} (\sigma)$ and $Q_{a} (\sigma)$, to account for the tidal and polarization response function respectively. The simplest operators that can be built are \cite{Goldberger:2005cd}
\begin{flalign}
\begin{split}
\mathcal{O} (E) &=
\begin{cases}
& e^{-4} Q^{ab} (\sigma) E_{ab} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \textrm{Gravity} ,\\
& e^{-2} Q^{a} (\sigma) E_{a} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\, \textrm{Electro} ,
\label{eq:dynoper}
\end{cases}
\end{split}
\end{flalign}
\noindent with $Q^{ab}$ and $Q^a$, composed of the worldline degrees of freedom, $E^{ab}$ and $E^a$ respectively. The covariant building blocks with magnetic like parity, $B^{ab}$ and $B^{a}$, are used as well to construct invariant dynamical operators \cite{Goldberger:2005cd}.
The operators, $Q^{ab}$ and $Q^{a}$, are dependent on the internal degrees of freedom of the compact object in an unspecified way, but which explicit form is not necessary to obtain the dynamics \cite{Goldberger:2005cd, Goldberger:2020fot}. The dynamics of the system can be obtained using the in-in closed time path \cite{Jordan:1986ug}, a formalism that allows us to treat the tidal response function with dissipative effects in a time asymmetric fashion \cite{Goldberger:2005cd}. The internal structure is encoded in the Wilson coefficients coming from the response function.
The interactions of the mass quadrupole $Q^{ab}$, up to linear order, can be described in terms of an invertible Hermitian linear operator $\mathcal{O}^{ab}_{\;\;\;cd}$,
\begin{flalign}
\mathcal{O} &=
\begin{cases}
& e^{-4} Q_{ab} \mathcal{O}^{ab}_{\;\;\; cd} Q^{cd},
\end{cases}
\end{flalign}
\noindent The effective action that describes the dynamics of the dynamical variable $Q^{ab}$,
\begin{flalign}
\mathcal{S} = \int d\sigma e \left( e^{-4}E_{ab}Q^{ab} + e^{-4} Q_{ab} \mathcal{O}^{ab}_{\;\;\;cd} Q^{cd}\right),
\label{eq:Qdynamics}
\end{flalign}
\noindent leads to the equations of motion for $Q^{ab}$,
\begin{flalign}
\mathcal{O}^{ab}_{\;\;\; cd} Q^{cd} = -\frac{1}{2}E^{ab}.
\end{flalign}
\noindent The Green's function for $\mathcal{O}^{ab}_{\;\;\;cd}$, is the solution to the above equation with the delta function as a source, $\mathcal{O}^{ab}_{\;\;\; cd} G^{\;\;\; cd}_{ab} = \delta(\sigma - \sigma^{\prime})$, such that the Green's function can be given in terms of the inverse operator, $ \mathcal{O}^{-1 \; ab}_{\;\;\;\;\;\,\;\;\;\;cd}$.
\begin{comment}
\begin{flalign}
G^{ab}_{\;\;\;cd} = \mathcal{O}^{-1 \, ab}_{\;\;\;\;\;\;\;\;cd}\delta(\sigma - \sigma^{\prime}).
\end{flalign}
\end{comment}
\begin{comment}
The field that arises from our actual source is given by integrating the Green's function over the source $E^{ab}(\tau)$:
\begin{flalign}\label{Q_operator}
Q^{ab} &= \int \mathrm{d} \sigma^{\prime} G^{ab}_{cd}(\sigma, \sigma^{\prime})E^{cd}(\sigma^{\prime})\nonumber\\
& = \mathcal{O}^{-1 \; ab}_{cd} E^{cd},
\end{flalign}
\end{comment}
The expectation value, $\braket{Q^{ab}(\sigma)}$, is obtained by integrating the Green's function over the source $E^{ab}$, which is the expectation value in the initial state of the internal degrees of freedom. The requirement that the external field is zero at the initial state of the interaction requires to choose the retarded Green's function. Considering the linear response in a weak external field, the in-in formalism implies the expectation value \cite{Goldberger:2020fot}
\begin{comment}
, $G^{\;ab}_{R\; cd}(\sigma, \sigma^{\prime}) = 0$ for $\sigma^{\prime} > \sigma$
\end{comment}
\begin{flalign}
\braket{Q^{ab} (\sigma)} = \int \mathrm{d}\sigma' e' G^{ab,cd}_{R} (\sigma - \sigma') e'^{-2} E_{cd} (\sigma') + O\left(e^{-4} E^2\right),
\end{flalign}
\noindent with retarded Green's function,
\begin{comment}
\noindent where the expectation values of the retarded Green's function,
\end{comment}
\begin{flalign}
G^{ab,cd}_{R} (\sigma - \sigma') = -i \theta (\sigma -\sigma') \braket{[Q^{ab}(\sigma),Q^{cd}_{}(\sigma')]},
\label{eq:greensr}
\end{flalign}
\begin{comment}
\begin{flalign}
G^{ab,cd}_{R} (\tau - \tau') = - i \theta (\tau -\tau') \braket{[Q^{ab}(\tau),Q^{cd}_{}(\tau')]},
\label{eq:greensrQ}
\end{flalign}
\end{comment}
\noindent The Green's function provides the response function of the quadrupole under external gravitational forces.
By considering low frequencies, from which we assume that the degrees of freedom from the operator, $Q^{ab}$, are near equilibrium, the time ordered two point correlation function imply that the Fourier transform, $G_R$, must be an odd, analytic function of the frequency, $\omega > 0$. Given that the operator $Q_{ab}$ is a symmetric trace-free tensor, the Green's function is projected with the symmetric trace-free projection operator. In terms of the worldline parameter, $\omega \sim \partial_{\sigma}$, the retarded correlation function \cite{Goldberger:2005cd}
\begin{equation}
G^{ab,cd}_R (\sigma) \simeq F(\sigma) \left( \delta^{ac} \delta^{bd} + \delta^{ad} \delta^{cb} - \frac{2}{3} \delta^{ab} \delta^{cd} \right),
\label{eq:retarded2}
\end{equation}
\noindent with $F (\sigma)$,
\begin{flalign}
F(\sigma) = n_{g}+i c_g \frac{\mathrm{d}}{\mathrm{d}\sigma}+ n_{g}^{\prime} \frac{\mathrm{d}^2}{\mathrm{d}\sigma^2}+.\;.\;..
\label{eq:responsefunc}
\end{flalign}
\begin{comment}
\noindent Note that, in contrast to \cite{Goldberger:2005cd}, we have absorbed the $1/2$ factor appearing in front of (\ref{eq:retarded2}) into the dissipative coefficient.
The coefficient for dissipative effects, $c_g \geq 0$.
(\textcolor{blue}{Check prefactor and normalization in the response function}).
\end{comment}
Considering the response of the interaction to be nearly instantaneous, the expectation value of the dynamical operator encoding the
tidal response function in the adiabatic approximation reads \cite{Goldberger:2005cd,Goldberger:2020fot}
\begin{flalign}
\braket{Q_{ab} (\sigma)} \simeq n_{g} E_{ab} + ic_{g} \frac{\mathrm{d}}{\mathrm{d}\sigma} E_{ab} + n_{g}^{'} \frac{\mathrm{d}^2}{\mathrm{d}\sigma^2} E_{ab} + \,.\;\;.\;\;.\,.
\label{eq:responseQ1}
\end{flalign}
\noindent From the first term in the expansion, it yields the lowest order operator built for the tidal deformation in eq. (\ref{eq:operators}). The second one accounts for dissipative effects to lowest order, and the third for dynamical oscillations in the quasi-static limit. The Wilson coefficients appearing in eq. (\ref{eq:responseQ1}), encode the internal structure.
On the electromagnetic side, an analog procedure can be taken, for which retarded correlation function, \cite{Goldberger:2005cd}
\begin{equation}
G^{ab}_R (\sigma) \simeq \left(n_q + i c_q \frac{\mathrm{d}}{\mathrm{d}\sigma} + .\;.\;. \right)\delta^{ab},
\label{eq:retardedq}
\end{equation}
\noindent Thus, the dynamical operator to account for the polarizability in the quasi-static limit reads
\begin{flalign}
\braket{ Q_{a} (\sigma)} \simeq n_q e^{-1} E_a + ic_{q} \frac{\mathrm{d}}{\mathrm{d}\sigma} e^{-1}E_{a} + \,.\;\;.\;\;.\,.
\label{eq:responseP}
\end{flalign}
\subsubsection*{Dissipative Effects}
In the EFT description of extended objects, dissipative effects arise due to the existence of gapless modes that are localized on the worldline of the particle, taking into account for the energy and momentum loss from the interaction with external sources \cite{Goldberger:2005cd}.
These effects occur due to a relative time-dependence between the object and its tidal environment, even if it is not rotating. If the extended object is spinning, in the co-rotating frame the external environment rotates at the frequency of the angular velocity, generating spin dependent dissipative effects.
After dissipation in a worldline EFT description for a static compact object was introduced \cite{Goldberger:2004jt}, dissipative effects for spinning extended objects in EFT were incorporated in \cite{Porto:2007qi} for the PN expansion in phase space. Using the model from the coset \cite{Delacretaz:2014oxa} in the Newtonian limit \cite{Endlich:2015mke}, spin dissipative effects were obtained from the effective action when considering a modified variation that takes into account for non-conservative effects \cite{Galley:2012hx}. Dissipation for maximally spinning extended objects were introduced in \cite{Goldberger:2020fot}, which upon taking the Newtonian limit, recovers the results from \cite{Endlich:2015mke}. We generalize \cite{Endlich:2015mke} for the relativistic case.
These large number of degrees of freedom can be encoded in operators allowed by the symmetries of the object as shown above. Given the imaginary dependence of the dissipative effects, and in order to extract the dynamics using a modified variation, it is useful to separate the conservative from the non-conservative part from the tidal response function. Therefore, from the operator in eq. (\ref{eq:responseP}), we extract the imaginary part, which to lowest order \cite{Goldberger:2005cd, Endlich:2015mke}:
\begin{flalign}
\begin{split}
\mathcal{O}(E) &=
\begin{cases}
& e^{-2} \mathcal{D}^a (\sigma) E_{a} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mathrm{Electro},\\
& e^{-4} \mathcal{D}^{ab} (\sigma) E_{ab} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \mathrm{Gravity},
\label{eq:dissoper}
\end{cases}
\end{split}
\end{flalign}
\noindent with the covariant quantities, $ \mathcal{D}^{a}(\sigma)$ and $ \mathcal{D}^{ab}(\sigma)$, composite dynamical operators made out of $e^{-1} E^{a}$ and $e^{-2} E^{ab}$ respectively, encoding the dissipative degrees of freedom. The covariant building blocks with magnetic like parity, $B^{a}$ and $B^{ab}$, are used as well to construct invariant operators \cite{Goldberger:2005cd, Endlich:2015mke, Goldberger:2020fot}.
Therefore, the non-conservative expectation value of the gravitational dissipation in the quasi-static limit,
\begin{flalign}
\braket{\mathcal{D}_{ab} (\sigma)} \simeq ic_{g} \frac{\mathrm{d}}{\mathrm{d}\sigma} e^{-2} E_{ab} + \,.\;\;.\;\;.\,.
\label{eq:response}
\end{flalign}
\noindent Once we have the effective action, the non-conservative dynamics can be obtained from the modified variation \cite{Galley:2012hx}
\begin{equation}
\delta \mathcal{S} + i \int \mathrm{d} \sigma \mathrm{d}\sigma' \delta E_{ab} (\sigma) G^{ab,cd}_R (\sigma - \sigma') E_{cd}(\sigma') = 0.
\label{eq:modified}
\end{equation}
\noindent On the electromagnetic dissipation an analog procedure is taken.
For a non-spinning black hole, dissipation takes into account for the absorption of electromagnetic and gravitational waves. For a non-spinning neutron star, dissipative effects can be generated due to the internal viscosity of the star \cite{Goldberger:2005cd}, or due to a time asymmetry in the formation of a tidal bulge \cite{Hut1981}. When one is dealing with spinning extended objects, the spin has a time dependence between the object and its environment, which contributes to dissipation \cite{Endlich:2015mke}.
\begin{comment}
Each of the shown coefficients in eq. (\ref{eq:responsefunc}) encodes information about the internal structure of the compact object. They are discussed when building the effective action.
\end{comment}
\begin{comment}
\Irv{State of the art numerical experiments suggests that, the energy loss during a gravitational encounter, is not due to the viscosity of the object. Therefore, dissipative effects arise , which is due to the fact that the star is an extended object}. \Irv{Expand on this}.
\noindent The dissipative effects in the quasi-static limit, eq. (\ref{eq:response}), are a good approximation for describing a BH, given that their Love numbers vanishes \cite{Hui:2021vcv,Chia:2020yla}. Nevertheless, for a NS, which is subjected to tidal deformations (non-zero love numbers), one should go beyond the quasi-static approximation.
\end{comment}
\subsubsection*{Dynamical Oscillations}
One can use the composite operator, $Q_{ab} (\sigma)$, in eq. (\ref{eq:responseQ1}), to account for the dynamical tidal response in the adiabatic approximation, where the last term is a dynamical correction in the quasi-static limit. For black holes, eq. (\ref{eq:responseQ1}) might describe them accurately given that the coefficients $n_g$ and $n_g^{'}$ vanishes \cite{Poisson:2004cw,Hui:2021vcv,Chia:2020yla}. Deviations from a more fundamental theory of gravity might imply the existence of a non-zero coefficients \cite{Cardoso:2018ptl}, for which then eq. (\ref{eq:responseQ1}) can be used to test gravity.
For neutron stars, the adiabatic approximation is not enough. It is possible to treat $Q^{ab}$ as an independent degree of freedom \cite{Steinhoff:2016rfi}, and build the effective action
\begin{flalign}
\begin{split}
\mathcal{S} =& \int d\sigma e \left( .\;.\;. + e^{-4}E_{ab}Q^{ab} + e^{-4} Q_{ab} F^{-1} (\sigma) Q^{ab}\right)\\
=& \int d\sigma e \left( .\;.\;. + e^{-4}E_{ab}Q^{ab} + n_o e^{-4} Q_{ab}Q^{ab} + n_o^{\prime} e^{-4} \dot{Q}_{ab}\dot{Q}^{ab} + c_o e^{-4}\dot{Q}_{ab} Q^{ab}\right).
\end{split}
\end{flalign}
\noindent The third and fourth term in the last line correspond to the dynamical internal oscillations of the compact object, induced by an external tidal force. The last term is not time-reversal invariant and therefore a non-conservative effect. The perturbative dynamics of the binary in terms of $Q^{ab}$ can be obtained from the effective action \cite{Steinhoff:2016rfi,Steinhoff:2021dsn}.
\subsection{The Effective Action}
Collecting the constructed invariant operators, we can build an effective action that describes a compact object. There are some differences in describing the two currently detected compact objects, black holes and neutron stars, which is reflected in the values for the Wilson coefficients. Despite their differences, we can build a generic effective action. To lowest order, a compact object that is charged and spinning is described by the effective action in the rest frame,
\begin{flalign}
\begin{split}
\mathcal{S}_{eff} =& \int \mathrm{d}\tau \big(- \frac{m}{2}\left( 1 - g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} \right) + q \dot{x}^{\mu} A_{\mu} + \frac{I}{4} \Omega_{ab} \Omega^{ab} \\
& \;\;\;\;\;\;\;\;\;\;\;\; + n_{\Omega,q} \Omega_{ab} F^{ab} + n_{\Omega,g} \Omega_{ab} \omega^{ab}_{\mu} \dot{x}^{\mu} + n_{q,g} F_{ab} \omega^{ab}_{\mu} \dot{x}^{\mu}\\
& \;\;\;\;\;\;\;\;\;\;\;\; + n_{\Omega,B_q} \Omega^a B_a + n_{\Omega,E_g}\Omega^a \Omega^b E_{ab} + n_{E_q} E_{}^{a} E_{a} \\
& \;\;\;\;\;\;\;\;\;\;\;\; + n_{E_g} E^{ab} E_{ab} + i c_{E_g} E^{ab} \dot{E}_{ab} + n^{'}_{E_g} \dot{E}^{ab} \dot{E}_{ab} +.\;.\;. \; \big) \\
&\;\;\;\;\; + \mathcal{S}_{0}, \\
\end{split}
\label{eq:effectivetheory}
\end{flalign}
\noindent in the quasi-static limit, with the ellipsis representing all other operators including the ones built in this work, and all others that can be constructed and are allowed by the symmetries. The interaction action, $\mathcal{S}_{0}$, is the Einstein-Maxwell action, eq. (\ref{eq:generalactionelectrograv}).
\begin{comment}
\begin{flalign}
\mathcal{S}_0 = \int \mathrm{det} \; e \; \mathrm{d}^4 x \left\{ -\frac{1}{4 \mu_0} F_{a b} F^{a b} + \frac{1}{16 \pi G} R + .\;.\;. \right\}.
\end{flalign}
\end{comment}
The first line of eq. (\ref{eq:effectivetheory}) describes a massive charged spinning point particle, with their coefficients matched to the well known theories. The second line describes the interaction between the different degrees of freedom, which includes the spin-orbit correction to lowest order. The third and fourth line are multipole moments taking into account for the tidal deformation and the polarization. In the fourth line we have included the leading order dynamical tidal effects in the quasi-static limit, including dissipation.
The rest of the coefficients are left unmatched, which will be done in a future work. The action describing the charged spinning compact object lives in the worldline, while $\mathcal{S}_0$ lives in the bulk.
\begin{comment}
The effective theory in eq. (\ref{eq:effectivetheory}), parameterize any compact object that could possibly exist under the currently known underlying physics
\end{comment}
\section{Conclusions}
\label{sec:discussion}
We have constructed an EFT that describes compact objects as point particles with higher order corrections made out of the allowed couplings between the derived covariant building blocks. It is based in the EFT for spinning extended objects introduced in \cite{Delacretaz:2014oxa}, which is derived using the powerful method of the coset construction, that allows us to build effective theories from the symmetry breaking pattern as the only input. The developed worldline effective theory, describes compact objects that are characterized by their mass, spin and charge, as well as their finite-size structure, with an underlying effective Einstein-Maxwell bulk theory. The EFT is described in the vierbein formalism.
The development of the framework brings various advantages. It allows to obtain relativistic dynamics without going to phase space, thus being a Lagrangian framework. It incorporates multiple different developed tools in the EFT for extended objects into a single framework, without the need of introducing additional degrees of freedom other than the ones derived from the breaking of symmetries, and the one needed to take into account for dynamical oscillations beyond the quasi-static limit. It provides a connection between the different developed theories for spinning objects, shedding light onto a common framework to describe the spin dynamics. It allows to build invariant operators with explicit dependence on the einbeins, which then have implications on the obtention of the dynamics using the Polyakov action, providing new perturbative approaches to obtain the dynamics.
Moreover, it provides a transparent connection between the different inertial frames from which the dynamics are described, as naturally expected from the vierbein formalism. Therefore, the effective action in eq. (\ref{eq:effectivetheory}) is valid to obtain both the PN and PM expansion, and one can simply change from one frame to another by choosing the appropriate worldline parameter $\sigma$ and worldline parameterization $\dot{x}^{\mu}$, or in other words, by choosing an einbein, $e$. The dynamics of binary systems using the derived effective action are obtained in an upcoming work, where the vierbein formalism with the use of einbeins is exploited.
\begin{comment}
\textcolor{red}{Needs to be updated} In this work we have reviewed and extended the model for spinning extended objects introduced in \cite{Delacretaz:2014oxa}, which is derived using the coset construction \cite{Callan:1969sn,Ivanov:1981wn}, a very powerful method that allows us to construct an effective theory from the symmetry breaking pattern as the only input. In this approach, a spinning extended object whose ground state breaks space-time symmetries, is coupled to a gravitational theory formulated as a gauge theory with local Poincaré symmetry and translations being non-linearly realized. We have included the internal structure \cite{Goldberger:2004jt, Goldberger:2005cd, Endlich:2015mke} and electromagnetic charge \cite{Goldberger:2005cd,Patil:2020dme}, such that we describe charged spinning extended objects, the most general extended object allowed in a theory of gravity such as general relativity.
We have derived the covariant building blocks of the effective theory, to build up invariant operators to form an action. We built our underlying theory and matched the coefficients to the full known theory, to obtain the Einstein-Maxwell action in the vierbein formalism. Then, by recognizing the symmetry breaking pattern of a charged spinning extended object, we have built the leading order invariant operators that are allowed by the symmetries, to describe it as a worldline point particle with its properties and internal structure encoded in higher order corrections in the action. Such corrections take into account for the basic necessary ingredients to completely describe an extended object in an effective theory of gravity as general relativity. By matching the coefficients of the effective action from the literature, we have described charged spinning compact objects, such as BHs and NSs.
Although this effective theory for spinning extended objects \cite{Delacretaz:2014oxa} by construction is a low energy description of the dynamics, compact objects which are described classically, fit into the description of "slowly" spinning \cite{Martinez:2020loq}. We have shown the equivalence of our effective theory to the ones currently used to obtain state of the art perturbative results of the binary dynamics \cite{Porto:2005ac,Levi:2015msa}, with the advantage that the covariant building blocks to construct the tower of invariant operators to all orders have been derived. Therefore, our work complements the aforementioned theories for spinning extended objects, and lays on the foundations for a full description of the possible compact objects that can exists.
The most direct application of our derived action is on the PN expansion \cite{Martinez:2022vnx}, where we have shown that our theory reproduces the well known results for spinning \cite{Levi:2015msa} and charged \cite{Patil:2020dme} extended objects. Moreover, novel results in the PN expansion have been derived on the internal structure of charged spinning compact objects \cite{Martinez:2022vnx}.
\end{comment}
\acknowledgments
I.M. is very thankful to A. Weltman, R. Penco, I. Rothstein, J. Steinhoff, A. Luna, T. Hinderer, S. Mougiakakos, G. Mogull, G. Creci, I. Meijer for the many enlightening conversations and discussions. I.M. gratefully acknowledge support from the University of Cape Town Vice Chancellor's Future Leaders 2030 Awards programme which has generously funded this research, support from the South African Research Chairs Initiative of the Department of Science and Technology and the NRF. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, and from the Educafin-JuventudEsGto Talentos de Exportacion programme.
|
1,108,101,565,448 | arxiv | \section{Introduction}
\label{Introduction}
Chemical reaction calculations have a wide range of applications in the materials developments, such as catalysts~\cite{Reiher2017-jt,Norskov2009-fr} and batteries~\cite{Rice2021-si}.
The chemical reaction speed is determined by the reaction rate constant $k$ which depends on the activation energy $E_a$ in the form of an exponent as $k \propto \mathrm{exp}(E_a/RT)$, where $R$ is the gas constant and $T$ is the temperature.
Hence, calculating $E_a$ with high accuracy is required for predicting chemical reactions.
The examples of the reaction path optimization methods for obtaining $E_a$ include the quadratic synchronous transit method~\cite{Govind2003-ni}, intrinsic reaction coordinate method~\cite{Fukui1981-gp}, dimer method~\cite{Henkelman1999-sh}, and nudged elastic band (NEB) method~\cite{Jonsson1998-gi,Henkelman2000-sp,Henkelman2000-zo}.
The NEB method searches for a minimum energy path (MEP), which has the smallest energy maximum for the reaction path connecting a reactant and a product on a potential energy surface given in reaction coordinates (e.g., bond lengths and angles).
Concretely, the method searches the reaction path on the potential energy surface by generating several intermediate states, called “images”, between the reaction coordinates corresponding to a reactant (initial state) and a product (final state).
The adjacent images are connected by virtual springs along the reaction path to maintain an equal distance between the images during the optimization, where the optimization of the intermediate images is performed so that the maximum energy of the images becomes smaller.
Here, the ground-state energy on each image is calculated by using an electronic structure calculation method such as the density functional theory (DFT).
$E_a$ is obtained from the optimized reaction path.
While the NEB method is widely used in chemical reaction path calculations, there are mainly two obstacles to the NEB method for accurately obtaining $E_a$.
The first obstacle is a lack of accuracy in the ground-state energy calculation.
The common electronic structure calculation methods such as the DFT may lack the accuracy required for predicting chemical reactions when the electronic correlation of the system is essential to the calculation~\cite{Weymuth2014-le}.
The second obstacle is a slow convergence of path optimization. Since the images interact with each other in the NEB method, the convergence speed of the optimization depends on the handling procedure of the interactions, and thus we sometimes struggle with slow convergence, especially in a complex chemical reaction path~\cite{Henkelman2000-zo,Asada2018-ix,Galvan2008-at}.
Meanwhile, a quantum computer has attracted considerable attention as it can perform some tasks more efficiently than a classical computer, and the quantum computer has the potential to solve the above obstacles.
For example, the quantum algorithm, called variational quantum eigensolver (VQE)~\cite{Peruzzo2014-kp,Cao2019-qa,McArdle2020-fz}, is expected to solve the first obstacle.
The VQE is an algorithm for obtaining the ground-state energy by repeating the generation of a (variational) wave function by using a parameterized quantum circuit and the update of the parameters on a classical computer.
In the VQE, the multipartite quantum entanglement between the qubits is expected to enable us to calculate the wave functions that cannot be represented by classical computers efficiently.
In addition, while current quantum computers, called noisy intermediate-scale quantum (NISQ) devices~\cite{Preskill2018-sc}, are not free from noises in quantum gate operations, the results of the VQE are robust to physical noise since only short depth quantum circuits are used.
The VQE has been conducted for the calculation of molecules by using NISQ devices~\cite{Peruzzo2014-kp,Shen2017-yt}.
Furthermore, a quantum computer is expected to solve the second obstacle also since it has been confirmed that the multipartite entanglements in quantum circuits affect the convergence speed of the objective function in optimization tasks~\cite{Abbas2021-ki,Patti2021-jc}.
However, despite these circumstances, a quantum algorithm for obtaining $E_a$ has not been performed.
In this study, we propose a quantum algorithm for chemical reaction path optimization.
In our algorithm, quantum circuits can be used not only for the ground-state calculation but also for the chemical reaction path generation.
The ground state of each image is calculated by using the VQE (or the exact diagonalization, ED, for comparison).
We demonstrated our quantum algorithm by applying it to \ce{H2 + H -> H + H2} reaction.
We found that the proposed algorithm correctly optimizes the reaction path and accurately obtains $E_a$ from the reaction path.
We also examined the dependencies of the convergence behavior of the path optimization on the entanglers in the path generating quantum circuits and confirmed that the existence of the entanglers accelerates the path optimization, especially when the number of images is large.
The results show that the proposed quantum algorithm expects to accelerate a chemical reaction path optimization and obtain accurate activation energy compared to the classical algorithms.
\section{Method} \label{Method} We first give an overview of the NEB method and then explain the proposed algorithm for the reaction path optimization.
\subsection{Overview of the NEB method}
\label{Sec: Overview of the NEB method}
We first generate $N_{image}$ images between the reaction coordinate of the reactant and that of the product on the potential energy surface.
Then, we create a reaction path by connecting adjacent images using virtual springs to obtain the force on each image in the reaction path.
The evaluation value $\bar{F}$ is defined as the average of the norms of the forces on each image,
\begin{equation}
\begin{aligned}
\bar{F}=\frac{1}{N_{image}} \sum_{i=1}^{N_{image} }|\mathbf{F}_{i}|,
\label{Eq: Fbar}
\end{aligned}
\end{equation}
where $\mathbf{F}_i$ is the force on the $i$-th image ($i=1,2, \dots, N_{image}$).
$\mathbf{F}_i$ is composed of the spring force $\mathbf{F}_i^S$ and the gradient for the potential energy surface $\boldsymbol{\nabla}E(\mathbf{R}_i)$,
\begin{equation}
\begin{aligned}
\mathbf{F}_{i}=\mathbf{F}_i^S|_{\|}-\boldsymbol{\nabla} E(\mathbf{R}_{i})_{\perp},
\end{aligned}
\end{equation}
where $\mathbf{R}_i$ is the reaction coordinate in the $i$-th image, and $E(\mathbf{R}_i)$ is the energy of the system specified by $\mathbf{R}_i$. $\mathbf{F}_i^S|_{\|}$ is the tangential component for the path of $\mathbf{F}_i^S$, and $\boldsymbol{\nabla} E(\mathbf{R}_{i})_{\perp}$ is the tangential perpendicular component of $\boldsymbol{\nabla} E(\mathbf{R}_{i})$,
\begin{equation}
\begin{aligned}
\left.\mathbf{F}_{i}^{S}\right|_{\|} &=\left(\mathbf{F}_{i}^{S} \cdot \hat{\boldsymbol{\tau}}_{i}\right) \hat{\boldsymbol{\tau}}_{i}\\
&=K\left(\left|\mathbf{R}_{i+1}-\mathbf{R}_{i}\right|-\left|\mathbf{R}_{i}-\mathbf{R}_{i-1}\right|\right) \hat{\boldsymbol{\tau}}_{i}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\boldsymbol{\nabla} E(\mathbf{R}_{i})|_{\perp}=\boldsymbol{\nabla} E(\mathbf{R}_{i})-\boldsymbol{\nabla} E(\mathbf{R}_{i}) \cdot \hat{\boldsymbol{\tau}}_{i},
\label{Eq: Grad E}
\end{aligned}
\end{equation}
where ${\hat{\boldsymbol{\tau}}}_i$ is the tangent vector, and $K$ is the spring constant.
${\hat{\boldsymbol{\tau}}}_i$ is defined as
\begin{equation}
\begin{aligned}
\hat{\boldsymbol{\tau}}_i = \begin{cases} \hat{\boldsymbol{\tau}}_i^+ & \mathrm{if}\ E(\mathbf{R}_{i+1})>E(\mathbf{R}_{i})>E(\mathbf{R}_{i-1}) \\\hat{\boldsymbol{\tau}}_i^- & \mathrm{if}\ E(\mathbf{R}_{i+1})<E(\mathbf{R}_i)<E(\mathbf{R}_{i-1}) \end{cases},
\label{eq: Tangent 1}
\end{aligned}
\end{equation}
else if $E(\textbf{R}_{i+1})<E(\textbf{R}_{i})>E(\textbf{R}_{i-1})$ or $E(\textbf{R}_{i+1})>E(\textbf{R}_{i})<E(\textbf{R}_{i-1})$
\begin{equation}
\begin{aligned}
&\hat{\boldsymbol{\tau}}_i = \begin{cases} \hat{\boldsymbol{\tau}}_i^+\Delta E_i^{\mathrm{max}}+\hat{\boldsymbol{\tau}}_i^- \Delta E_i^{\mathrm{min}} & \mathrm{if}\ E(\mathbf{R}_{i+1})>E(\mathbf{R}_{i-1}) \\\hat{\boldsymbol{\tau}}_i^+\Delta E_i^{\mathrm{min}}+\hat{\boldsymbol{\tau}}_i^- \Delta E_i^{\mathrm{max}} & \mathrm{if}\ E(\mathbf{R}_{i+1})<E(\mathbf{R}_{i-1}) \end{cases},
\label{eq: Tangent 2}
\end{aligned}
\end{equation}
where $\hat{\boldsymbol{\tau}}_i^+ =\mathbf{R}_{i+1}-\mathbf{R}_{i} $, $\hat{\boldsymbol{\tau}}_i^- = \mathbf{R}_{i}-\mathbf{R}_{i-1}$, $\Delta E_i^{\mathrm{max}}=\mathrm{max}(|E(\mathbf{R}_{i+1})-E(\mathbf{R}_{i})|,|E(\mathbf{R}_{i-1})-E(\mathbf{R}_{i})|)$, and $\Delta E_i^{\mathrm{min}}=\mathrm{min}(|E(\mathbf{R}_{i+1})-E(\mathbf{R}_{i})|,|E(\mathbf{R}_{i-1})-E(\mathbf{R}_{i})|)$.
${\hat{\boldsymbol{\tau}}}_i$ needs to be normalized.
Note that the tangential perpendicular component of $\mathbf{F}_i^S$, $\mathbf{F}_i^S|_{\perp}$, and the tangential component of $\boldsymbol{\nabla} E(\mathbf{R}_{i})$, $\boldsymbol{\nabla} E(\mathbf{R}_{i})_{\|}$, are not used in the NEB method to prevent the shifting of the image from the MEP and to keep the distance between the images equal, respectively~\cite{Jonsson1998-gi}.
In addition, ${\hat{\boldsymbol{\tau}}}_i$ in Eqs.~(\ref{eq: Tangent 1}) and (\ref{eq: Tangent 2}) are defined for preventing the path from oscillating during the convergence process~\cite{Henkelman2000-sp}.
$N_{image}$ was set to be three, five, or seven, and $K$ was set to be 0.1 Hartree/Å$\mathrm{^2}$.
\subsection{Optimization of the reaction path by using the path generation through quantum circuits}
\label{Sec: Optimization of reaction path}
Next, we explain the detail of the proposed algorithm.
Figure \ref{fig: Overview} shows a calculation flow of the proposed quantum algorithm for obtaining $E_a$.
The outline of the calculation is shown in the following four steps, and the capital letters in each step correspond to the letters in Fig.~\ref{fig: Overview}.
\begin{enumerate}[Step A.]
\item Generate the reaction path (the geometries) using a parameterized quantum circuit ($\boldsymbol{\theta}$ is a set of the parameters).
\item Calculate the ground-state energy $E(\mathbf{R}_{i})$ and the gradient $\boldsymbol{\nabla} E(\mathbf{R}_{i})$ and obtain the evaluation value $\bar{F}$.
\item Calculate the gradient for $\boldsymbol{\theta}$ of $\bar{F}$, $\boldsymbol{\nabla}_{\boldsymbol{\theta}}\bar{F}$, and update $\boldsymbol{\theta}$.
\item Repeat steps A to C until the termination condition is satisfied.
After the termination, $E_a$ is obtained from the optimized reaction path.
\end{enumerate}
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{qNEB_Fig1_1col.png}
\caption{The calculation flow of optimizing a reaction path and obtaining the activation energy $E_a$ by using the path generation though quantum circuits.}
\label{fig: Overview}
\end{figure}
The details for each step are as follows.
\begin{center}
\emph{Step A. Reaction path generation method by using quantum circuits}
\end{center}
Figure~\ref{fig: Initial reaction path}(a) show the depiction of the reaction path, and we assume the reaction of H\textsubscript{2} + H → H + H\textsubscript{2} with one-dimensional system and $N_{image}=3$.
Each row corresponds to any of the initial state (IS), the image, or the final state (FS).
$(N_{image} - 1)/2 = 1$ image is interpolated each between the IS and the intermediate point (IMP) and between the IMP and the FS.
In each row, the three hydrogen atoms are named H\textsubscript{A}, H\textsubscript{B}, and H\textsubscript{C} from left to right, and the position of H\textsubscript{A} is defined as the origin.
The distance between H\textsubscript{A} and H\textsubscript{B} (H\textsubscript{B} and H\textsubscript{C}) is described as $\text{R}_{\text{AB}}$ ($\text{R}_{\text{BC}}$).
We assumed $\text{R}_{\text{AB}}$ and $\text{R}_{\text{BC}}$ as the reaction coordinates on the potential energy surface.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{qNEB_Fig2_1col.png}
\caption{Initial reaction path for the H\textsubscript{2} + H → H + H\textsubscript{2} reaction and the quantum circuit used for generating the path.
We show the example of $N_{image} = 3$.
(a) Depiction of the reaction path.
Atoms with fixed (unfixed) positions in the reaction path optimization calculations are shown in black (red).
(b) Details of the circuit.
The upper panel show whole circuit implementation.
The lower panel show details of $U_{g}(\boldsymbol{\theta})$.
The block surrounded by the dashed line was repeated $D_{g}$ times.}
\label{fig: Initial reaction path}
\end{figure}
First, we explain encoding the single image into a circuit.
The distances in the Cartesian coordinates are divided by the reference length $\text{R}_{\text{ref}}\left( \geq \text{R}_{\text{AB}} + \text{R}_{\text{BC}} \right)$ to convert them to the fractional coordinates of $[0, 1]$.
Specifically, the coordinates of H\textsubscript{A}, H\textsubscript{B}, and H\textsubscript{C} are converted from $0$, $\text{R}_{\text{AB}}$, and $\text{R}_{\text{AB}} + \text{R}_{\text{BC}}$ to $0$, $\text{R}_{\text{AB}}/\text{R}_{\text{ref}}$, and $( \text{R}_{\text{AB}} + \text{R}_{\text{BC}} )/\text{R}_{\text{ref}}$, respectively.
Here, $\text{R}_{\text{ref}}$ was set to be 6.00 Å.
Then, the fractional coordinate is encoded to the wave function as $R_y(2\arccos(r))\ket{0}$, where $R_{y}(\varphi)$ is $R_y$ gate with a rotation angle $\varphi$, and $r$ is a fractional coordinate.
Specifically, the corresponding states are respectively $R_y(0)\ket{0} = \ket{0}$,
$R_y(2\arccos( \sqrt{\text{R}_{\text{AB}}/\text{R}_{\text{ref}}}))\ket{0} = \sqrt{\text{R}_{\text{AB}}/\text{R}_{\text{ref}}}\ket{0} + \sqrt{1 - \text{R}_{\text{AB}}/\text{R}_{\text{ref}}}\ket{1}$,
and $R_y(2\arccos( \sqrt{( \text{R}_{\text{AB}} + \text{R}_{\text{BC}})/\text{R}_{\text{ref}}} ))\ket{0} = \sqrt{( \text{R}_{\text{AB}} + \text{R}_{\text{BC}})/\text{R}_{\text{ref}}}\ket{0} + \sqrt{1 - ( \text{R}_{\text{AB}} + \text{R}_{\text{BC}})/\text{R}_{\text{ref}}}\ket{1}$.
We can encode the single image into a circuit by arranging the wave functions, i.e., $\ket{\Psi_i} = R_y(0)\ket{0} \otimes R_y(2\arccos( \sqrt{\text{R}_{\text{AB}}/\text{R}_{\text{ref}}}))\ket{0} \otimes R_y(2\arccos( \sqrt{( \text{R}_{\text{AB}} + \text{R}_{\text{BC}})/\text{R}_{\text{ref}}} ))\ket{0}$.
We can also encode the path by arranging the image-encoded states $\ket{\Psi_i}$ on a circuit, i.e., $\bigotimes_{i=1}^{N_{image}}\ket{\Psi_i}$.
However, it is not necessary to encode the geometrical information of the atoms of which the coordinates are fixed in the NEB calculation.
As shown in Figure~\ref{fig: Initial reaction path}(a), the IS, the FS, and the positions of H\textsubscript{A} are fixed (black in the Figure).
Since 6 unfixed atoms (red in the Figure) are present among a total of 15 atoms in the path including the IS and the FS, the number of qubits for obtaining the path-encoded state $\ket{\psi_{enc}}$ can be reduced from 15 to 6.
We note that the number of qubits required for the path generation of an $N_{dim}$-dimensional system is $N_{dim}N_{UFA} \sim O( N_{dim}N_{atom}N_{image})$, where $N_{UFA}$ is the number of the unfixed atoms, and $N_{atom}$ is the number of atoms in a image.
The circuit implementation for generating the path is shown in Fig.~\ref{fig: Initial reaction path}(b).
The number assigned to each atom in red in Fig.~\ref{fig: Initial reaction path}(a) corresponds to the qubit index $m$ $(m = 1,\ 2, \dots, 6)$ in Fig.~\ref{fig: Initial reaction path}(b).
We create the wave functions of the entangled geometrical information, $\ket{\psi(\boldsymbol{\theta})}$, by operating unitary gates with parameters $U_{g}(\boldsymbol{\theta})$ on $\ket{\psi_{enc}}$, i.e., $\ket{\psi(\boldsymbol{\theta})} = U_{g}(\boldsymbol{\theta})\ket{\psi_{enc}}$.
The quantum entanglement expects to allow to take a different behavior of path optimization convergence compared with the classical algorithms (see results in Sec.~\ref{sec: Convergence of the evaluation value} and Appendix~\ref{sec: Futher study of convergence}).
The parameterized unitary gates $U_{g}(\boldsymbol{\theta})$ were implemented based on Hardware efficient ansatz~\cite{Kandala2017-lh}.
All initial $\boldsymbol{\theta}$ were set to zero, and the depth of the path generating circuit $D_{g}$ (lower panel of Fig.~\ref{fig: Initial reaction path}(b)) was set to be two.
Then, the path (the geometrical information) are extracted from the wave function.
We make the probability of measuring $m$-th qubit of the circuit, $P_m$, correspond to the fractional coordinate of the $m$-th atom,
\begin{equation}
\begin{aligned}
P_m = |\langle 0|_{m}|\psi(\boldsymbol{\theta})\rangle|^{2},
\label{Eq: Frac}
\end{aligned}
\end{equation}
where $\bra{0}_{m}$ represents $\bra{0}$ acting on the $m$-th qubit.
We mention that in the definition in Eq.~(\ref{Eq: Frac}), $P_m$ corresponds to the initial fractional coordinate of the $m$-th atom when $U_g(\boldsymbol{\theta}) = I$, where $I$ is the identity operator.
All quantum circuit simulations in this work were classically implemented by using Blueqat package~\cite{Mochizuki2019-oa}.
We note two points about quantum advantages in the path generation method. Firstly, the expressible reaction path by the circuit does not change with or without the entanglement (i.e., the $\text{CZ}$ gate) since all the geometries can be represented without $U_g(\boldsymbol{\theta})$ by changing the rotation angle of $R_{y}$ gate in $\ket{\psi_{enc}}$.
Our expectation for quantum entanglement in the path generating circuit is to improve the expressive power of the reaction path update, not of the reaction path itself.
Secondly, in this study, we do not discuss the quantum advantage of encoding the path information to quantum circuits in this study although the efficient encoding of classical data to a quantum circuit is itself a remarkable research topic~\cite{Schuld2018-vy}.
\begin{center}
\emph{Step B. Energy, gradient, and evaluation value calculation}
\end{center}
The values of $E(\mathbf{R}_{i})$ and $\boldsymbol{\nabla}E(\mathbf{R}_{i})$ are obtained by calculating the ground-state energy at the geometries corresponding to $\mathbf{R}_{i}$ and around $\mathbf{R}_{i}$.
$\bar{F}$ is obtained by substituting the values $\mathbf{R}_{i}$, $E(\mathbf{R}_{i})$, and $\boldsymbol{\nabla}E(\mathbf{R}_{i})$ into Eqs.~(\ref{Eq: Fbar})-(\ref{eq: Tangent 2}).
The computational details are as follows.
The Hamiltonians used in the ground-state calculations were created using the OpenFermion package~\cite{McClean2020-zp}.
The molecular wave functions of the converged Hartree-Fock calculations by STO-3G were used, and the number of spin orbitals was six, i.e., two per single hydrogen atom.
The ground-state calculation was performed by using the VQE and the ED.
Figure~\ref{fig: Quantum circuit} shows the quantum circuit used in the ground-state calculation of the VQE.
The optimization in the VQE was based on a gradient-free method, Rotoselect~\cite{Ostaszewski2021-yk}.
In the Rotoselect, not only the parameters $\boldsymbol{\lambda}$ but also the types of the rotation gates in the circuit are optimized.
The initial parameters of $\boldsymbol{\lambda}$ were set to be random numbers of $[0.0, 0.1)$, and the initial rotation gates were set to be $R_{x}$.
The depth of the quantum circuit $D_{e}$ was set to be five.
The energy evaluation of the VQE was repeated until the energy difference for the VQE iteration became less than $10^{-4}$ Hartree.
$\boldsymbol{\nabla}E(\mathbf{R}_{i})$ was obtained by using the central difference method, and the difference value for each reaction coordinate was set to 0.1 Å.
\begin{figure}[!t]
\includegraphics[width=1\columnwidth]{qNEB_Fig3_1col.png}
\caption{Quantum circuit used in the VQE.
The number of qubits (= spin orbitals) were six.
The unitary gate $U_{e}(\boldsymbol{\lambda})$ (left panel) was composed of blocks of $R_{l}$ $(l = x,y,z )$ and $\text{CNOT}$ gates within the dashed lines (right panel), and the block surrounded by the dashed line was repeated $D_{e}$ times.}
\label{fig: Quantum circuit}
\end{figure}
\begin{center}
\emph{Step C. Calculation of the parameter gradient of the evaluation value and parameter update}
\end{center}
By performing steps A and B, we calculate $\boldsymbol{\nabla}_{\boldsymbol{\theta}}\bar{F}$ and update $\boldsymbol{\theta}$. $\boldsymbol{\nabla}_{\boldsymbol{\theta}}\bar{F}$ was obtained by using the central difference method, and the parameter difference value was set to be 0.001.
$\boldsymbol{\theta}$ was updated using Adam~\cite{Kingma2014-og} with a learning rate of 0.01.
\begin{center}
\emph{Step D. Activation energy evaluation}
\end{center}
We set the termination condition as 100 iterations.
In the optimized path, $E_{a}$ is defined as the difference between the highest energy among all images and the energy of the IS.
\section{Results}
\label{sec: Results}
\begin{figure*}[!ht]
\includegraphics[width=0.8\textwidth]{qNEB_Fig4_2col.png}
\caption{Results of reaction path optimization.
(a) and (b) [(c) and (d)] are the results for $N_{image} = 3$ [$5$].
(a) and (c) are reaction paths on the potential surface.
Each mark corresponds to any of the IS, an image, the FS, and a saddle point state.
The reaction proceeds from the IS at the upper left to the FS at the lower right.
(b) and (d) show the energies on each reaction path.
The types of marks and lines in (b) and (d) correspond to those in (a) and (c), respectively.
The reaction proceeds from the IS on the left to the FS on the right.
The length of each reaction path was normalized to compare the paths of different lengths.}
\label{fig: Results of reaction}
\end{figure*}
\begin{table*}[!ht]
\begin{center}
\caption{List of obtained values for $N_{image} = 3$ and $5$ after the optimization termination.
When the values of $N_{image} = 3$ and $5$ were different, the result for 5 images was written in parentheses.}
\label{tbl: List of Fbar}
\begin{tabular}{c c c c c c} 3 Images (5 Images) & $\bar{F}$ (Ha/Å) & $\text{R}_{\text{AB}}$ (Å) & $\text{R}_{\text{BC}}$ (Å) & $E_{a}$ (mHa) & $|\Delta_{saddle}|$ (mHa) \\ \hline
Initial path & 0.17 (0.12) & 0.73 & 0.73 & 83 & 50 \\ Eval\_VQE & 0.02 (0.03) & 0.82 (0.91) & 1.09 (1.01) & 38 (41) & 5 (8) \\
Eval\_ED & 0.00 (0.00) & 0.94 (0.87) & 0.95 (1.04) & 33 (31) & 0 (2) \\
Saddle Point & - & 0.94 & 0.94 & 33 & - \\
\end{tabular}
\end{center}
\end{table*}
\begin{figure*}[!ht]
\includegraphics[width=0.8\textwidth]{qNEB_Fig5_2col.png}
\caption{The evaluation value for each iteration.
(a) and (b) [(c) and (d)] are the results for $N_{image} = 3$ [$5$].
(b) and (d) are enlarged views of the vertical axes in (a) and (c), respectively.
}
\label{fig: Evaluation value}
\end{figure*}
\subsection{Numerical results of the proposed algorithm}
\label{sec: Execution results}
Figure~\ref{fig: Results of reaction}(a) and (c) shows the results of the optimized reaction paths of $N_{images} =$ 3 and 5, respectively, on the potential energy surface.
In both Figures, the paths obtained by the NEB method by using the VQE [Eval\_VQE, green line in the Figure] and the ED [Eval\_ED (CZ), red line] become closer to the saddle point [Saddle point (Exact), blue cross mark] and smoother than initial reaction path [Initial path, black dotted line].
Furthermore, as shown in Figs.~\ref{fig: Results of reaction}(b) and (d), which are the energies on the reaction paths corresponding to Figs.~\ref{fig: Results of reaction}(a) and (c), respectively, the maximum energies in the reaction paths for both the VQE and the ED are closer to the saddle point energy than that in the initial reaction path.
Thus, we found that the reaction path is correctly optimized by the path generation method using quantum circuits.
In addition, the VQE results show that quantum computers are useful for optimizing the reaction path since $E_a$ is accurately obtained even when the quantum circuit is used for both the path generation and ground-state calculation.
We mention the effect of the convergence condition of the VQE.
Table~\ref{tbl: List of Fbar} shows the obtained values, where $|\Delta_{saddle}|$ is an absolute value of the difference between the exact activation energy and the calculated one.
Although the values of $|\Delta_{saddle}|$ in both the VQE and the ED were sufficiently small compared to that in the initial reaction path, the values in the VQE (5 and 8 mHartree in $N_{image}$ = 3 and 5, respectively) were larger than the chemical accuracy (about 1.6 mHartree).
However, the accuracy is improved by tightening the convergence condition in the VQE.
We performed the VQE in the geometry on the saddle point and found that the absolute value of the difference from the exact value decreased from 4.5 mHartree to 0.031 mHartree when the convergence condition was changed from the $10^{-4}$ Hartree (default value) to $10^{-6}$ Hartree.
Hence, tighter computational conditions than the default may need to be imposed in practical applications of the proposed method.
Note that we choose the default value in order to save the calculation time since multiple numbers of ground-state calculations were needed for the NEB calculation.
For example, the numbers of performed ground-state calculations per NEB iteration in $N_{image} = 3$ and $5$ were 1225 and 2835, respectively.
In general, the number of calculations is $O( N_{image}^{2}D_{g}N_{atom}^{2}N_{dim}^{2})$, where $O( N_{image}N_{atom}N_{dim})$ and $O( N_{image}D_{g}N_{atom}N_{dim})$ come from calculating $\bar{F}$ and updating $\boldsymbol{\theta}$, respectively.
\subsection{Convergence of the evaluation value}
\label{sec: Convergence of the evaluation value}
Figures~\ref{fig: Evaluation value}(a) and (c) show $\bar{F}$ for each iteration in $N_{image}$ = 3 and 5, respectively.
In both cases, $\bar{F}$ tends to converge as the number of iterations increases.
The use of the ED for the ground-state calculation (red line) converged faster than that of the VQE (green line), indicating that the accuracy of the ground-state calculation affects the convergence speed.
Finally, we discuss the effect of the entanglers ($\text{CZ}$ gates) on the convergence.
The purple dotted lines in Fig.~\ref{fig: Evaluation value} show the convergence when excluding $\text{CZ}$ gate from $U_{g}(\boldsymbol{\theta})$ in Fig.~\ref{fig: Initial reaction path}(b), where the ground-state calculation is performed on the ED.
Figures~\ref{fig: Evaluation value}(b) and (d) are enlarged vertical axes of Figs.~\ref{fig: Evaluation value}(a) and (c), respectively.
We found that the value with $\text{CZ}$ gates tends to be lower than that without $\text{CZ}$ gates after about 50 iterations.
In addition, from the average results of 10 initial reaction paths (in $N_{image}=3, 5,$ and $7$) generated by slightly changing the atomic positions from Fig.~\ref{fig: Initial reaction path}(a), we confirmed that the trend for CZ gates become clearer as $N_{image}$ increases (see Appendix~\ref{sec: Futher study of convergence}).
Therefore, the entanglement between images would accelerate the convergence of reaction path optimization.
\section{Conclusion}
\label{conclusion}
In this study, we proposed a quantum algorithm for chemical reaction path optimization.
In our algorithm, we can use the quantum circuit at two steps in the calculation flow: one is the calculation of the ground-state energy, and the other is the generation of the reaction path.
In the path generating method, the coordinates of each atom in each image of the initial reaction path are encoded to a quantum circuit through the rotation angle of $R_{y}$ gate.
The encoded state is entangled by using one- and two-qubit gates, and the geometrical information is extracted from the measurement value.
The nudged elastic band (NEB) method was used for the reaction path optimization.
We applied the algorithm to optimize the reaction path for the H\textsubscript{2}+H → H + H\textsubscript{2} reaction, where the variational quantum eigensolver (VQE) or the exact diagonalization (ED) were used in the ground-state calculation.
We found that the reaction paths were accurately optimized, and the activation energy that is close to the saddle point energy was obtained in both the VQE and the ED cases.
In addition, we confirmed that the entanglement of quantum circuits leads to faster convergence of the NEB method, especially when the number of images is large.
The results show that the proposed quantum algorithm paves the way to more accurate and faster chemical reaction path optimization.
\section{Acknowledgement} The author would like to thank Tomofumi Tada for useful discussion about the study and for providing computational resources, Takuya Nakao for providing information about the NEB method and for discussing the NEB method in the manuscript, Suguru Endo for discussing the VQE in the manuscript, Asahi Chikaoka for providing information about Rotoselect, Taishi Mizuguchi for exchanging ideas about quantum computing, and Blueqat Co., Ltd.~for various information on quantum computers and quantum simulators.
\emph{Note added} During the update of this manuscript, several studies of the quantum algorithm for chemical reaction were reported~\cite{Parrish2021-mt,Omiya2022-sq,Yalouz2022-qq}.
\bibliographystyle{apsrev4-1}
|
1,108,101,565,449 | arxiv | \section{Introduction}\label{sec:intro}
Denmark is one of the most digitized countries in Europe~\cite{desi2019}.
After implementation of the 2011 Danish e-government strategy~\cite{danmark2011digital}, citizens interact with the government mostly through e-government systems, such as, self-service web-applications and websites.
Examples for such systems are \toolname{Sundhed.dk}\citeurl{https://www.sundhed.dk/} (health.dk) or \toolname{Borger.dk}\citeurl{https://www.borger.dk} (citizen.dk).
These e-government systems are usually implemented as closed-source systems by private contractors and often in a combination of C\raisebox{0.5ex}{\tiny\textbf{\#}}\xspace, Java, JavaScript, etc.
Besides application of modern programming languages, contemporary software development relies to a large degree on \emph{code reuse}.
Software is usually build with the help of reusable components, which we call \emph{packages} in this paper.
Programming language ecosystem emerge usually around specific package managers, which download and setup needed dependencies from remote package registries, such as, \toolname{NPM}, \toolname{Maven Central}, \toolname{PyPI}, \toolname{RubyGems}, etc.
Code reuse via dependency on external packages is also encouraged by software quality models and tools, such as the SIG/TÜViT Evaluation Criteria for Trusted Product Maintainability~\cite{sig10criteria}.
It recommends using \emph{``libraries and frameworks over 'homegrown' implementations of standard functionality''}~\cite{visser2016building} for example to keep the size of software under own maintenance small.
However the advantages of code reuse, e.g., increased software quality or developer productivity~\cite{basili1996reuse,lim1994effects,mohagheghi2004empirical} are attended by drawbacks, such as, increased maintenance cost or legal issues
~\cite{orsila2008update,vendome2018assisting,mathur2012empirical,duan2017identifying}.
This paper focuses on license incompatibilities, which are examples of such legal issues.
For instance, in March 2020 it was reported that \numprint{577148} software projects which are build on top of \gls{rails} face license issues\citeurl{https://www.theregister.com/2021/03/25/ruby_rails_code/}.
An indirect dependency of the web-application framework \toolname{rails} (see \toolname{mimemagic} in \autoref{fig:deps}) switched its license from the permissive \license{MIT} to the more restrictive \license{GPL-2.0}\xspace.
The restrictive strong copyleft \gls{gpl} is viral in that all work that is distributed and that includes code under \gls{gpl} has to adopt the same --or a compatible-- license.
Consequently, all dependents of \toolname{mimemagic}, i.e., \toolname{rails} and all applications build with it, would have to either switch to \gls{gpl} too or they would all have to replace the indirect dependency \toolname{mimemagic}, with a suitably licensed package.
Such a small \license{GPL}\xspace licensed package hidden in a dependency network would pose a legal issue for commercial vendors that create and ship software on top of \toolname{rails}.
Usually they are not interested in open-sourcing their products on distribution, which is a requirement of the \gls{gpl}.
The younger \gls{agpl}\citeurl{https://tldrlegal.com/license/gnu-affero-general-public-license-v3-(agpl-3.0)} is even more restrictive on when to open-source software that reuses correspondingly licensed packages.
It prescribes that software has to be made publicly available when it accessed via a network and when it relies on \gls{agpl} licensed components.
Consequently, if one of Denmark's online e-government systems had a direct or indirect dependency to an \gls{agpl} licensed package, either it would have to be open-sourced or the respective dependency would have to be replaced with software under a compatible license.
To understand to which degree license incompatibilities pose an issue in seven major software ecosystems (\toolname{Cargo}\xspace (Rust), \toolname{Maven}\xspace (Java and other JVM languages), \toolname{NPM}\xspace (mostly JavaScript), \toolname{NuGet}\xspace (C\raisebox{0.5ex}{\tiny\textbf{\#}}\xspace and other .Net languages), \toolname{Packagist}\xspace (PHP), \toolname{PyPI}\xspace (Python), \toolname{RubyGems}\xspace (Ruby)), we investigate empirically and quantitatively the following research questions on the \emph{Libraries.io} dataset~\cite{katz2020_3626071}.
\begin{description}
\item[RQ1] Which licenses are mainly used in various software ecosystems?
\item[RQ2] How many direct license incompatibilities exist in various ecosystems?
\item[RQ3] How many direct license incompatibilities are caused by \gls{agpl} licenses?
\item[RQ4] What is the impact of indirect \gls{agpl} license incompatibilities?
\end{description}
Our goal is to investigate how many cases similar to the motivational \toolname{rails} example exist in various ecosystems and to assess if there is a risk for danish online e-government systems.
Our results show that permissive licenses, such as, \license{MIT}\xspace and \license{Apache}\xspace are most frequent in the studied ecosystems.
The number of direct license incompatibilities is not negligible, ranging from low 2.3\% in \toolname{Cargo}\xspace over 4.7\%
, 5.2\%, 6.2\%, 9.1\%, and 9.1\% in \toolname{NuGet}\xspace, \toolname{NPM}\xspace, \toolname{RubyGems}\xspace, \toolname{Maven}\xspace, and \toolname{Packagist}\xspace respectively to a large 20.8\% in \toolname{PyPI}\xspace.
Only a low amount of direct license incompatibilities are caused by \license{AGPL}\xspace licensed packages (max. 0.04\% in \toolname{PyPI}\xspace).
Lastly, a whopping 6.62\% of \toolname{Maven}\xspace packages are violating the \license{AGPL}\xspace license of an indirect dependency.
However, only a few packages are responsible for this large number of indirect license incompatibilities in \toolname{Maven}\xspace.
In the remainder, we describe the impact of license incompatibilities and potential resolutions more detailed in \autoref{sec:background}, we detail the our experiment setup in \autoref{sec:method}, we describe our results in \autoref{sec:results}, and we discuss our results in \autoref{sec:discussion} and \autoref{sec:conclusions}.
\section{Background}\label{sec:background}
\begin{figure}[b]
\vspace{-1em}
\centering
\vspace{0.1em}
\includegraphics[width=\linewidth]{images/deps.png}
\vspace{-1em}
\caption{Direct and indirect dependencies of \gls{rails} with a license incompatibility between \license{MIT}\xspace and \license{GPL-2.0}.}
\label{fig:deps}
\end{figure}
\textbf{License Incompatibility Example:}
\autoref{fig:deps} illustrates an excerpt of the dependencies of the web-framework Ruby on Rails (\toolname{rails}) before Mar. 24th 2021.
Via the package manager \toolname{RubyGems}, \toolname{rails} depends indirectly (by \toolname{activestorage} and \toolname{marcel}) on \toolname{mimemagic}, a library for detecting MIME types of files.
\toolname{mimemagic} in turn depends on a single artifact from the \toolname{xdg-shared-mime-info}\citeurl{https://github.com/freedesktop/xdg-shared-mime-info} project.
That artifact, is an XML file\citeurl{https://github.com/freedesktop/xdg-shared-mime-info/blob/master/data/freedesktop.org.xml.in} that serves as database of MIME types, which is copied from the original project and reused in \toolname{mimemagic}.
The XML file and the entire \toolname{xdg-shared-mime-info} are licensed under \emph{GPL-2.0} whereas \toolname{mimemagic} is under \emph{MIT} license.
These two licenses are incompatible, since \emph{GPL-2.0} is a so-called \emph{"viral"} license that would require dependents to open-source their code under the same license, which the \emph{MIT} license does not require.
After being informed about the license incompatibility\citeurl{https://github.com/mimemagicrb/mimemagic/issues/97}, the \toolname{mimemagic} maintainer decides to resolve the license incompatibility by releasing new versions of \toolname{mimemagic} that switches license from \emph{MIT} to \emph{GPL-2.0}\citeurl{https://github.com/mimemagicrb/mimemagic/commit/c0f7b6b21a192629839db87612794d08f9ff7e88}, the same as the license of the required artifact.
Additionally, the maintainer decides to remove all previous releases with an incompatible license.
This change however, caused that the licenses of the web-framework \toolname{rails} and its dependency \toolname{mimemagic} are now incompatible with each other\citeurl{https://github.com/rails/rails/issues/41757}.
Essentially they were already before the license change, since \toolname{rails} depends on \toolname{xdg-shared-mime-info} indirectly, which would also require it to be under license compatible to \emph{GPL-2.0}.
The rails community discussed now how to resolve the license incompatibility issue on their side\citeurl{https://github.com/rails/rails/issues/41750}.
Since many commercial closed-source projects are built on top of \toolname{rails}, changing license to \emph{GPL-2.0} is not an option.
Developers discuss if they can replace the dependency to \toolname{mimemagic} with another suitably licensed project.
Finally, after consulting the \toolname{mimemagic} developers, the latter decide to license their project again under \emph{MIT}, to remove the artifact that generated the issue from their sources, and to describe to developers how to obtain a copy of that artifact at runtime so that \toolname{mimemagic} can import it from the user's filesystem.
\textbf{Effects of License Incompatibilities:}
Imagine an e-government system that is built on top of \toolname{rails} by a third-party vendor and that \toolname{rails} were licensed under \license{GPL}\xspace.
When the vendor hosts this system for a public sector institution, then it does not matter if any dependency is under \license{GPL}\xspace, since it only prescribes open-sourcing a system on distribution.
That is, if the same vendor delivers the system to another party for hosting, the \license{GPL}\xspace prescribes that the sources of the system are made publicly available.
Imagine now, that \toolname{rails} or its indirect dependency \toolname{mimemagic} were under \license{AGPL}\xspace.
That would mean that as soon as the system is publicly accessible, its sources would have to be made publicly available too, no matter which party is hosting the system.
That is the motivation for this work.
Usually, Danish e-government systems are closed source.
However, since they are most often web-applications and since they usually rely on third-party packages from various ecosystems, it is important to access how likely the license of a dependency would create an issue for online e-government systems.
\textbf{Terminology:}
Various ecosystems and package managers call reusable components differently.
For example, in the \toolname{Maven}\xspace realm reusable components are called \emph{Jars}, in \toolname{Ruby} they are calle \emph{Gems}, in \toolname{Cargo}\xspace they are called \emph{Crates} and in other ecosystems, e.g., \toolname{PyPI}\xspace and \toolname{NPM}\xspace they are just called \emph{packages}.
Irrespective the ecosystem, we call reusable components that are distributed via package managers uniformly \emph{packages}.
We call a dependency between two packages that are connected via a dependency relation in the dependency network a \emph{direct} dependency.
Contrary two packages possess an \emph{indirect} dependency if the required package is not directly declared by the dependent package but by an intermediate.
We call the act of making the sources of a software system publicly available under a corresponding license \emph{open-sourcing}.
Open-source software licenses fall into two major categories: \emph{permissive} and \emph{protective} licenses~\cite{kapitsaki2015insight}.
The latter are often called \emph{copyleft} licenses.
Permissive licenses restrict minimally how software can be used, modified, or redistributed.
Examples of such licenses are the \license{MIT}\xspace, \license{Apache-2.0}, \license{BSD-3-Clause}, \license{ISC}, or the \license{Unlicense}.
Permissive licenses permit relicensing of derivative work and allow for use of software also in proprietary software that is distributed.
There are two kinds of \emph{protective} licenses: \emph{weakly} and \emph{strongly} protective licenses.
Examples of weakly protective licenses are the \license{LGPL-3.0} or \license{MPL-2.0} and examples of strongly protective licenses are the \license{GPL-3.0} or \license{AGPL-3.0}. %
Strongly protective licenses prescribe to open source derived software and to distribute it under the same license\citeurl{https://www.gnu.org/licenses/copyleft.html}.
Software under a weak protective license can be redistributed under another license, as long as that software is not modified and software that reuses such does not need to be open-sourced\citeurl{https://www.gnu.org/licenses/license-compatibility.html}.
\section{Method}\label{sec:method}
\begin{figure}[b]
\vspace{-1em}
\centering
\vspace{0.1em}
\includegraphics[width=\linewidth]{images/setup.png}
\vspace{-1em}
\caption{Experiment setup.}
\label{fig:setup}
\end{figure}
\autoref{fig:setup} illustrates the dataflow in our study setup.
A couple of \toolname{Bash}, \toolname{Python}, and \toolname{SWI Prolog} scripts automatically process and analyze the dependency networks of various software ecosystems and generate the results in the following section (\autoref{sec:results}).
A reproduction kit containing all these scripts and configurations is available online\citeurl{https://github.com/HelgeCPH/license_compat_study/}.
Our study is based on the \emph{Libraries.io} dataset~\cite{katz2020_3626071}. %
For various ecosystems, that dataset contains names, licenses, and dependency links of all registered packages.
Note, that the dataset provides only the license of the most \emph{current} release of a package, i.e., not the licenses of earlier versions of a package.
License information per package is stored via \gls{spdx} identifiers\citeurl{https://spdx.org/licenses/} --or a comma separated list of these--, which we use for this study.
We focus our study on the seven ecosystems \toolname{Cargo}\xspace, \toolname{Maven}\xspace, \toolname{NPM}\xspace, \toolname{NuGet}\xspace, \toolname{Packagist}\xspace, \toolname{PyPI}\xspace, and \toolname{RubyGems}\xspace since \emph{a)} these ecosystems are most relevant for Danish e-government systems and \emph{b)} the \emph{Libraries.io} dataset contains dependency links for these ecosystems\footnote{It does not necessarily contain dependency links for all other ecosystems in the dataset}.
The \emph{filter} action in \autoref{fig:setup} illustrates that our scripts download the \emph{Libraries.io} dataset (more than 20GB in size) and extract all information relating to packages from the seven ecosystems
and their dependencies.
Furthermore, the available data is reduced to retain only package identifiers, names, and license information.
We keep only those dependency links that are labeled as \emph{compile}- or \emph{runtime}-dependency in the original dataset since such dependencies may create license incompatibilities\footnote{Other dependencies types, e.g., \emph{test}, \emph{development} dependencies, etc. usually do not create a license incompatibility since they are not distributed with the resulting system.}.
Since the \emph{Libraries.io} dataset contains license information only on a package level, we reduce the dependencies between versions of packages to logical dependencies.
That is, we reduce the directed multigraph from the input dataset into a directed graph, e.g., as soon as one version of a package eventually depended on a version of another package, a single dependency link is retained.
The filtered data is stored for all seven ecosystems in two CSV files, one for the packages and another for the dependency links.
In a next step (illustrated as \emph{convert} action in \autoref{fig:setup}), these two CSV files are converted into \toolname{SWI Prolog} knowledge bases.
We do that, since license compatibility is a constraint satisfaction problem, which can be efficiently encoded and resolved with \toolname{Prolog} programs.
Manually, we create a set of license incompatibility rules as facts in a \toolname{Prolog} knowledge base.
\autoref{lst:prolog} shows \toolname{Prolog} facts that declare \license{AGPL-3.0} incompatible with the respectively stated license.
These \toolname{Prolog} facts are translated from the license incompatibility matrix from Kamocki et al.'s~\cite{kamocki2016public} \toolname{Public License Selector}\citeurl{https://github.com/ufal/public-license-selector/blob/97a7af0a7af00829bf43958669c79334cf77015c/src/definitions.coffee\#L257}.
During translation, we map the used license identifiers to the respective \gls{spdx} identifiers.
\begin{figure}[t]
\begin{lstlisting}[language=prolog, caption={Excerpt of \toolname{Prolog} license incompatibility rules.},label={lst:prolog}]
incompatible('AGPL-3.0', 'LGPL-2.1').
incompatible('AGPL-3.0', 'LGPL-2.1-only').
incompatible('AGPL-3.0', 'CDDL-1.0').
incompatible('AGPL-3.0', 'GPL-2.0').
incompatible('AGPL-3.0', 'GPL-2.0-only').
incompatible('AGPL-3.0', 'AGPL-1.0').
incompatible('AGPL-3.0', 'AGPL-1.0-only').
\end{lstlisting}
\end{figure}
\begingroup
\setlength{\tabcolsep}{2pt}
\begin{table*}[t]
\centering
\caption{Descriptive statistics of licenses and incompatibilities.}
\begin{tabular}{lrrrrrr}
\toprule
{} & $\lvert \text{Packages} \rvert$ & $\lvert \text{Dependencies} \rvert$ & $\lvert \text{Packages}_{\text{discon}} \rvert$ & $\lvert \text{Packages}_{\text{con}} \rvert$ & $\lvert \text{Incompatibilities} \rvert$ & Incompatibilities\% \\
\midrule
Cargo & \numprint{35635} & \numprint{19968} & \numprint{5918} & \numprint{29717} & \numprint{453} & \numprint{2.3}\% \\
Maven & \numprint{184871} & \numprint{426804} & \numprint{101788} & \numprint{83083} & \numprint{39002} & \numprint{9.1}\% \\
NPM & \numprint{1275011} & \numprint{4927446} & \numprint{912541} & \numprint{362470} & \numprint{257593} & \numprint{5.2}\% \\
NuGet & \numprint{199447} & \numprint{488385} & \numprint{145996} & \numprint{53451} & \numprint{22949} & \numprint{4.7}\% \\
Packagist & \numprint{313278} & \numprint{473083} & \numprint{162630} & \numprint{150648} & \numprint{43196} & \numprint{9.1}\% \\
PyPI & \numprint{231690} & \numprint{152779} & \numprint{49197} & \numprint{182493} & \numprint{31816} & \numprint{20.8}\% \\
Rubygems & \numprint{161608} & \numprint{276580} & \numprint{108552} & \numprint{53056} & \numprint{17267} & \numprint{6.2}\% \\
\bottomrule
\end{tabular}
\label{tab:stats}
\end{table*}
\endgroup
\begin{figure}[b]
\vspace{-1em}
\centering
\vspace{0.1em}
\includegraphics[width=0.7\linewidth]{images/example_deps.png}
\vspace{-1em}
\caption{Exemplary dependency network.}
\label{fig:xmpl}
\end{figure}
To study RQ1 (\emph{Which licenses are mainly used in various software ecosystems?}), we just count frequencies of licenses per ecosystem and compute their relative frequency compared to the total number of packages in the relative ecosystem.
To investigate RQ2 (\emph{How many direct license incompatibilities exist in various ecosystems?}), we let the \toolname{Prolog} program compare all packages between which a dependency link exists.
For each such pair of nodes, the program checks if a license incompatibility rule --as in \autoref{lst:prolog}-- holds true.
For RQ3 (\emph{How many direct license incompatibilities are caused by \gls{agpl} licenses?}), we do the same as for RQ2 only that we restrict the analysis to dependency links that point to a package with any \gls{agpl} license, i.e., \license{APGL-1.0}, \license{APGL-2.0}, etc.
As mentioned above, packages in the \emph{Libraries.io} dataset can have more than one license assigned as a comma separated list of \gls{spdx} identifiers.
The semantics of a comma in a list of licenses is not specified in the dataset description\citeurl{https://libraries.io/data\#projectFields}.
We interpret such lists to prescribe a disjunction of possible licenses, i.e., a list of possible licenses to choose from depending on the use case.
Therefore, when searching for license incompatibilities, we analyze the cross-product of license combinations and record a license incompatibility only when all possible license pair combinations are incompatible.
Or vice versa, if one combination allows for a compatible use of a package, we do not mark the dependency as incompatible.
For investigation of RQ4 (\emph{What is the impact of indirect \gls{agpl} license incompatibilities?}), we want to incorporate the transitive structure of dependency relations.
Since license incompatibilities can propagate through a dependency network, as in the motivating example with \toolname{rails} and its indirect dependency \toolname{mimemagic} (\autoref{sec:background}), we identify all those packages that are under an \gls{agpl} and for which a dependent possesses an incompatible license.
Thereafter, we identify all packages that are on a path that ends in a package with a direct dependency to an \gls{agpl} licensed package with a license that is incompatible to an \gls{agpl} license.
Consider an exemplary dependency network as in \autoref{fig:xmpl}.
Let us assume that all packages (nodes) in that network are under \license{MIT}\xspace license, except of packages \emph{i}, which is under \license{AGPL-3.0}.
For RQ1, we would report that $87.5\%$ of all licenses in the ecosystem are \license{MIT}\xspace and that $12.5\%$ of all licenses are \license{AGPL-3.0}.
The \license{MIT}\xspace license is incompatible with \license{AGPL-3.0}.
Consequently, there is a direct license incompatibility between package \emph{e} and package \emph{i}.
Therefore, we would report for RQ2 and RQ3 respectively, that there are \numprint{1} direct license incompatibility, i.e., $12.5\%$ of all dependency links point to a package where the dependency has a license that is incompatible with the dependent.
The impact of the incompatibility with \emph{i}'s \license{AGPL-3.0} is quite large since packages \emph{a}, \emph{b}, \emph{c}, and \emph{d} depend directly or indirectly on package \emph{e}.
That is, $50\%$ of the ecosystem depend on a package that reuses a package with a license that is incompatible to the respective own license.
By transitivity, they have an incompatible license too.
\begingroup
\setlength{\tabcolsep}{2pt}
\begin{table*}[t]
\centering
\caption{Descriptive statistics of licenses and incompatibilities.}
\begin{tabular}{lrrrr}
\toprule
{} & $\lvert \text{Incompatibilites}_{\text{AGPL}} \rvert$ & $\text{Incompatibilities}_{\text{AGPL}}\text{\%}$ & $\lvert \text{Packages}_{\text{affected}} \rvert$ & $\text{Packages}_{\text{affected}}\text{\%}$ \\
\midrule
Cargo & \numprint{0} & \numprint{0.00}\% & \numprint{0} & 0.00\% \\
Maven & \numprint{148} & \numprint{0.03}\% & \numprint{12236} & 6.62\% \\
NPM & \numprint{777} & \numprint{0.02}\% & \numprint{30377} & 2.38\% \\
NuGet & \numprint{26} & \numprint{0.01}\% & \numprint{52} & 0.03\% \\
Packagist & \numprint{66} & \numprint{0.01}\% & \numprint{120} & 0.04\% \\
PyPI & \numprint{67} & \numprint{0.04}\% & \numprint{109} & 0.05\% \\
Rubygems & \numprint{49} & \numprint{0.02}\% & \numprint{53} & 0.03\% \\
\bottomrule
\end{tabular}
\label{tab:stats2}
\end{table*}
\endgroup
\section{Results}\label{sec:results}
\begin{figure}[b]
\centering
\includegraphics[width=\linewidth]{images/license_dist.png}
\caption{Distribution of most common licenses per ecosystem.}\label{fig:licenses}
\end{figure}
\subsubsection{RQ1: Which licenses are mainly used in various software ecosystems?}\label{sec:rrq1}
\autoref{fig:licenses} illustrates the share of the most common licenses per ecosystem.
For clarity, we plot only the seven most common licenses per ecosystem.
A plethora of less frequent licenses --some of which are only used by a single package-- occupy the remaining percentages in \autoref{fig:licenses}. %
The \license{MIT}\xspace license is most frequent in \toolname{Cargo}\xspace ($\approx54\%$), \toolname{NPM}\xspace ($\approx56\%$), \toolname{Packagist}\xspace ($\approx64\%$), \toolname{PyPI}\xspace ($\approx38\%$), and \toolname{RubyGems}\xspace ($\approx64\%$).
For \toolname{Maven}\xspace, the \license{Apache-2.0} license is most common ($\approx41\%$) with a considerable share in \toolname{Cargo}\xspace ($\approx28\%$) too.
In the other five ecosystems the \license{Apache-2.0} license is less frequent with ca. $8\%$ in \toolname{PyPI}\xspace, ca. $7\%$ in \toolname{NuGet}\xspace, and ca. $4\%$ in \toolname{NPM}\xspace, \toolname{RubyGems}\xspace, and \toolname{Packagist}\xspace respectively.
The only ecosystem in this study with a major share of \license{ISC} licensed packages is \toolname{NPM}\xspace where around a quarter of all packages carry it.
Generally, permissive licenses are most common in the ecosystems.
Except of \toolname{NuGet}\xspace, where the majority of packages do not carry any license, the majority of packages are permissively licensed.
A considerable amount of packages do not carry a license at all.
In \toolname{NuGet}\xspace the majority of packages ($\approx63\%$) do not carry a license.
The amount of such packages is lower in the other six ecosystems: \toolname{Maven}\xspace ($\approx20\%$), \toolname{RubyGems}\xspace ($\approx24\%$), \toolname{PyPI}\xspace ($\approx14\%$), \toolname{Packagist}\xspace ($\approx9\%$), and \toolname{NPM}\xspace ($\approx7\%$).
In \autoref{fig:licenses} and in the \emph{Libraries.io} dataset packages without assigned license are labeled as \emph{``None''}.
Package without license are essentially not meant for reuse. %
Except of \toolname{PyPI}\xspace, strongly protective license are quite rare in the ecosystems.
\toolname{PyPI}\xspace sports the highest share of strongly protective licenses.
There, \license{GPL-3.0} and \license{AGPL-3.0} account together for $\approx9.2\%$ ($\approx6.1\%$ and $\approx3.1\%$ respectively).
In \toolname{Packagist}\xspace \license{GPL-2.0} and \license{GPL-3.0} are the common strongly protective licenses, together with a share of only $\approx4.2\%$.
For the remaining ecosystems, these numbers are even lower:
in \toolname{Maven}\xspace $\approx3.0\%$ of packages are \license{GPL-3.0} licensed,
in \toolname{Cargo}\xspace these are $\approx2.3\%$,
in \toolname{RubyGems}\xspace packages with \license{GPL-2.0} or \license{GPL-3.0} together account for $\approx1.8\%$,
and \toolname{NPM}\xspace and \toolname{NuGet}\xspace have $\approx1.0\%$ of packages under \license{GPL-3.0} respectively.
Only in \toolname{PyPI}\xspace a noticeable share of packages is licensed under a version of \gls{agpl} (3.1\%, i.e., \numprint{7307} packages).
For the other ecosystems, this share is below one percent (\toolname{Cargo}\xspace: $0.78\%$, \toolname{Maven}\xspace: $0.74\%$, \toolname{NPM}\xspace: $0.24\%$, \toolname{NuGet}\xspace: $0.1\%$, \toolname{Packagist}\xspace $0.21\%$, and \toolname{RubyGems}\xspace: $0.18\%$).
\begin{figure*}[t]
\vspace{-1em}
\centering
\vspace{0.1em}
\includegraphics[width=0.8\linewidth]{images/most_affected2.png}
\vspace{-1em}
\caption{Incompatibilities with \license{AGPL} licensed packages in \toolname{Maven}\xspace that pose major issues.}
\label{fig:affected}
\end{figure*}
\subsubsection{RQ2: How many direct license incompatibilities exist in various ecosystems?}
\autoref{tab:stats} lists descriptive statistics for the seven studied ecosystems.
Column ``$\lvert \text{Packages} \rvert$'' shows the total number of packages, ``$\lvert \text{Dependencies} \rvert$'' shows the total number of logical dependency links in the dependency network (see \autoref{sec:background}), ``$\lvert \text{Packages}_{\text{discon}} \rvert$'' shows the total number of packages that neither depend on another package nor are required by any other package, ``$\lvert \text{Packages}_{\text{con}} \rvert$'' shows the number of packages that are connected as dependencies or dependents in the dependency network, and ``$\lvert \text{Incompatibilities} \rvert$'' and ``Incompatibilities\%'' show the absolute and relative amount of dependency links where the license of the dependency is incompatible with the license of the dependent package.
Obviously, \toolname{Cargo}\xspace is the smallest ecosystem with \numprint{35635} packages of which \numprint{29717} (ca. 83\% of all packages) contribute to the dependency network.
\toolname{NPM}\xspace is the largest ecosystem with \numprint{1275011} packages, where \numprint{912541} (ca. 72\% of all packages) are linked in the dependency network.
The sizes of the other five ecosystems are in between these two extremes, ranging from \numprint{161608} packages in \toolname{RubyGems}\xspace to \numprint{313278} in \toolname{Packagist}\xspace.
The share of packages that are connected in the dependency network in the other ecosystems ranges from a meager 27\% in \toolname{NuGet}\xspace over 33\% (\toolname{RubyGems}\xspace), 45\% (\toolname{Maven}\xspace), and 48\% \toolname{Packagist}\xspace to 79\% in \toolname{PyPI}\xspace.
\toolname{Cargo}\xspace sports the lowest amount of license incompatibilities with \numprint{2.3}\% of all dependencies that link packages with incompatible licenses (corresponding to \numprint{453} incompatibilities dependency links).
In \toolname{NuGet}\xspace these are \numprint{4.7}\% (\numprint{22949}).
In absolute numbers, \toolname{NPM}\xspace's \numprint{257593} license incompatibilities are the largest of all ecosystems.
But due to its total size, only \numprint{5.2}\% of all dependency links connect packages with incompatible licenses.
In the remaining ecosystems, the share of incompatible licenses raises from \numprint{6.2}\% in \toolname{RubyGems}\xspace (\numprint{17267}) over \numprint{9.1}\% in both \toolname{Maven}\xspace (\numprint{39002}) and \toolname{Packagist}\xspace (\numprint{43196}) to a staggering \numprint{20.8}\% in \toolname{PyPI}\xspace (\numprint{31816}).
\subsubsection{RQ3: How many direct license incompatibilities are caused by \gls{agpl} licenses?}
\autoref{tab:stats2} lists the number of dependency links between packages where the dependency is under an \license{AGPL}\xspace license and the dependent is under an incompatible license (column $\lvert \text{Incompatibilites}_{\text{AGPL}} \rvert$ lists absolute numbers and $\text{Incompatibilities}_{\text{AGPL}}\text{\%}$ lists the ratio compared to all dependency links).
The only ecosystem without any license incompatibilities ``caused'' by dependencies to \license{AGPL}\xspace licensed packages is \toolname{Cargo}\xspace.
\toolname{NPM}\xspace and \toolname{Maven}\xspace have the highest absolute amount of such license incompatibilities with \numprint{777} and \numprint{148} respectively.
However, in general the ratio of incompatibilities caused by packages that incorrectly depend on \license{AGPL}\xspace licensed packages is low.
It is consistently below half per mill.
The highest relative share of incompatibilities caused by \license{AGPL}\xspace licensed packages have \toolname{PyPI}\xspace (0.04\%) and \toolname{Maven}\xspace (0.03\%).
Remember however, that \toolname{PyPI}\xspace is the ecosystem with the highest amount of \gls{agpl} licensed packages in this study, see \autoref{sec:rrq1}.
\subsubsection{RQ4: What is the impact of indirect \gls{agpl} license incompatibilities?}
Besides absolute and relative numbers of direct license incompatibilities that are caused by incorrectly depending on \license{AGPL}\xspace licensed packages, \autoref{tab:stats2} lists the number of packages that are affected by such an incompatibility (see columns ``$\lvert \text{Packages}_{\text{affected}} \rvert$'' for absolute and ``$\text{Packages}_{\text{affected}}\text{\%}$'' for relative numbers).
All packages that either directly or indirectly depend on a package with a license incompatibility are \emph{affected} by it.
The absolute and relative numbers of packages affected by an incompatibility with an \license{AGPL}\xspace license are non-existent in \toolname{Cargo}\xspace and very low for \toolname{NuGet}\xspace and \toolname{RubyGems}\xspace (0.03\% with 52 and 53 affected packages respectively), for \toolname{Packagist}\xspace (0.04\%), and for \toolname{PyPI}\xspace (0.05\%).
For \toolname{NPM}\xspace, the share of packages that are affected by a license incompatibility with an \license{AGPL}\xspace licensed package is 2.38\%, which corresponds to \numprint{30377} packages in total.
However, these affected packages are fairly wide spread over the ecosystem.
In \toolname{NPM}\xspace none of the \license{AGPL}\xspace licensed dependencies with an incompatibly licensed dependent affects more than thousand other packages.
That is, it is not a few \license{AGPL}\xspace licensed packages that are dependencies of many other packages with an incompatible license.
To the contrary \toolname{Maven}\xspace is the ecosystem with the highest relative amount of \license{AGPL}\xspace-based license incompatibilities (6.62\%) with in total \numprint{12236} affected packages.
Here, a low number of packages that incorrectly depend on a package under \license{AGPL}\xspace license affects a large amount of other packages.
\autoref{fig:affected} illustrates all those packages from the \toolname{Maven}\xspace ecosystem (green, blue, and orange nodes on top) that depend on an \license{AGPL}\xspace licensed package (bottom nodes in pink) and that affect more than one thousand packages.
Sizes of nodes are adjusted to the \emph{PageRank}\xspace of the respective package in the \toolname{Maven}\xspace dependency network.
The larger a node the more central it is in the \toolname{Maven}\xspace ecosystem, i.e., the more packages depend either directly or indirectly on the respective package.
The three packages from the Spring framework\citeurl{https://spring.io/projects/spring-framework} (\toolname{org.springframework:spring-webmvc}, \toolname{org.springframework:spring-context-support}, \toolname{org.springframework:spring}) are all quite central to the dependency network.
They are respectively ranked to be the 171\textsuperscript{th}, 351\textsuperscript{st}, and 352\textsuperscript{nd} most central (important) nodes in the dependency network.
Less central but still fairly important are the \toolname{jfree}, \toolname{jasperreports}, \toolname{grails}, and \toolname{xhtmlrenderer} packages.
All of these depend on only three packages \toolname{com.lowagie.itext}, \toolname{com.itextpdf:itextpdf}, and \toolname{com.itextpdf-itext-pdfa}.
Each of these dependents affects more than \numprint{11500} other packages.
The two \toolname{neo4j} packages \toolname{org.neo4j:neo4j-kernel} and \toolname{org.neo4j:neo4j} are less central than the \toolname{springframework} packages in the dependency graph (ranks 465 and 761 respectively) but still, they affect more than \numprint{1300} packages each.
Interestingly, for these two packages the license incompatibility is caused by three packages (\toolname{org.neo4j:neo4j-remote-graphdb} and \toolname{org.neo4j:neo4j-lucene-upgrade}, \toolname{org.neo4j:neo4j-online-backup} respectively), which are created by the same organization.
Consequently, it is not too unlikely in \toolname{Maven}\xspace to indirectly depend on a package that has a license incompatibility with an \license{AGPL}\xspace licensed package.
Furthermore, in the \toolname{Maven}\xspace ecosystem we find a case similar to the example in the introduction, see \autoref{sec:intro}
Interestingly, like with \gls{rails} the Spring framework is also a web-development framework.
\section{Discussion}\label{sec:discussion}
As sketched in \autoref{sec:background}, there are multiple ways of resolving a license incompatibilities.
For example, a restrictive license of a dependency can be changed by the respective project owners to a more permissive license that is compatible to those of the dependents.
Alternatively, the dependent project changes its license from a more permissive to a more restrictive license that is compatible to the one of its dependency or it replaces the dependency all together with another suitably licensed package.
Not all license incompatibilities are necessarily an issue.
For example, imagine a system that relies for operation on the monitoring tool \toolname{Prometheus}\citeurl{https://prometheus.io/}, which is licensed under the \license{Apache-2.0} license, and the dashboard visualization tool \toolname{Grafana}, which is licensed under the \license{AGPL-3.0} license.
Such system, does not have to be licensed under \license{AGPL-3.0} and be provided as open-source, since it is not derived work from \toolname{Grafana} but rather reuses a certain web-API during operation.
However, if this system was derived from \toolname{Grafana}, e.g., a modified version of it that is used to display data via a web-application, it would have to be open-sourced under a respective license.
These semantic differences of the purpose of package reuse cannot be inferred via static analysis, i.e., they are not represented in the results of this study.
The purpose of this study is only to access the size of a potential issue and to indicate potential refactoring candidates.
License incompatibilities as described in this paper may not be an issue in non-US American/Anglo-Saxon legislations.
For example, a French court ruled recently that \license{GPL}\xspace copyright claims are invalid in France and may only be enforced via contractual disputes\citeurl{https://thehftguy.com/2021/08/30/french-appeal-court-affirms-decision-that-copyright-claims-on-gpl-are-invalid-must-be-enforced-via-contractual-dispute/}
Additionally, it is unclear if licenses, such as, the \license{GPL}\xspace are actually a copyright or a contract in non US legislations~\cite{szattler2007gpl,guadamuz2004viral}.
\textit{Threats to validity:}
The quality of our results depend on the accuracy of dependency link information in the \emph{Libraries.io} dataset, which we trust to be accurate.
We are no lawyers.
That is, all reported results assume that the license incompatibility model, which we adapt from Kamocki et al.'s~\cite{kamocki2016public}, accurately captures incompatible licenses.
Likely this model is coarse grained, since license incompatibility also depends on the use case of software that reuses other packages.
Our model is use case agnostic and therefore might be too strict.
We rely on the latest available \emph{Libraries.io} dataset~\cite{katz2020_3626071}, which was released on Jan. 12th, 2020.
The dataset contains license information only statically on package level.
That is, for each package only one or more licenses are given.
However, a package might change its license(s) over time.
For example, versions of \toolname{com.lowagie:itext} below version 5.0.0 were released under the more permissive \license{MPL} or \license{LGPL}\citeurl{https://en.wikipedia.org/wiki/IText} and first with version 5.0.0 the license was switched to \license{AGPL-3.0}.
Since the most current, this is the only license that is recorded for the package.
Consequently, our results in \autoref{sec:results} may be overestimating \license{AGPL}\xspace/based license incompatibilities, since certain versions of packages that in our study exhibit license incompatibilities may depend on suitably licensed earlier versions.
The dependency links available in the \emph{Libraries.io} dataset are a lower bound approximation of the ``real'' dependency links in the ecosystems.
The dataset only includes those dependency links that are statically declared in the package metadata.
For example, \toolname{PyPI}\xspace and \toolname{Maven}\xspace packages can execute code during installation of packages, which can be used to install other dependencies on top of those that are statically declared.
These are not subject of this study.
The \emph{Libraries.io} dataset provides package licenses as comma separated lists of one or more licenses.
The semantics of such commas in a list of licenses is not specified in the dataset description\textsuperscript{16}.
Thus, we interpret that lists prescribe a disjunction of possible licenses, i.e., a list of possible licenses to choose from depending on the use case.
When searching for license incompatibilities, we analyze the cross-product of license combinations and record a license incompatibility only when all possible license pair combinations are incompatible.
That might be an overly permissive interpretation, since some projects want to express via multiple licenses that the provided sources are made up by source under the given licenses, i.e., a conjunction of these.
\section{Future Work}\label{sec:future}
In future, we plan to execute the experiment again on another dataset, which captures licenses per versions of packages and not per package in general.
This will allow us to assess if the reported high impact of license incompatibilities with a few \license{AGPL}\xspace licensed packages in \toolname{Maven}\xspace is an artifact or if it adequately represents reality.
We are currently working on creating such an updated dataset ourselves, since the \emph{Libraries.io} dataset appears to be stale\citeurl{https://github.com/librariesio/libraries.io/issues/2744}.
Furthermore, working on more recent data will allow us to appropriately describe the current impact of \license{AGPL}\xspace licensed packages.
In our results, the amount of license incompatibilities that are caused by incorrect use of \license{AGPL}\xspace licensed packages is quite low or negligible for all ecosystems except \toolname{Maven}\xspace, see \autoref{sec:results}.
Likely, the reason for this is that the \license{AGPL}\xspace becomes increasingly popular first recently\citeurl{https://www.synopsys.com/blogs/software-security/using-agpl-adoption/}, i.e., after the latest release of the \emph{Libraries.io} dataset from Jan. 12th, 2020.
\section{Conclusions}\label{sec:conclusions}
In this paper, we investigate empirically and quantitatively to which degree license incompatibilities pose an issue in the seven major software ecosystems \toolname{Cargo}\xspace, \toolname{Maven}\xspace, \toolname{NPM}\xspace, \toolname{NuGet}\xspace, \toolname{Packagist}\xspace, \toolname{PyPI}\xspace, and \toolname{RubyGems}\xspace.
Our results (\autoref{sec:results}) show that in most ecosystems the permissive \license{MIT}\xspace and \license{Apache}\xspace licenses are most common, only in \toolname{NuGet}\xspace it is most common that packages do not carry any license.
\toolname{PyPI}\xspace is the only ecosystem with a noteworthy amount of packages under \license{AGPL-3.0} but only 3.1\% of all packages carry it.
The amount of license incompatibilities between packages that are directly connected via dependency links differs across the ecosystems.
The lowest amount of dependency links that cause a license incompatibility can be found in \toolname{Cargo}\xspace 2.3\% (453 dependencies).
This number increases from 4.7\% (\numprint{22949}) in \toolname{NuGet}\xspace over 5.2\% (\numprint{257593}), 6.2\% (\numprint{17267}), 9.1\% (\numprint{39002}), and 9.1\% (\numprint{43196}) in \toolname{NPM}\xspace, \toolname{RubyGems}\xspace, \toolname{Maven}\xspace, and \toolname{Packagist}\xspace to 20.8\% (\numprint{31816}) in \toolname{PyPI}\xspace.
However, only a low amount of direct license incompatibilities are caused by packages that depend on \license{AGPL}\xspace licensed packages but possess an incompatible license themselves (max. 0.04\% in \toolname{PyPI}\xspace).
When studying the amount of packages that are either directly or indirectly affected by a license incompatibility with \license{AGPL}\xspace licensed packages, we find that 6.62\% (\numprint{12236}) of all \toolname{Maven}\xspace packages are affected but that only a few packages are causing such indirect license incompatibilities.
\subsection{Implications for Practitioners}
Our results illustrate that license incompatibilities in general and license incompatibilities caused by inappropriately depending on \license{AGPL}\xspace licensed packages are more frequent in some ecosystems compared to others.
Consequently, we recommend that practitioners not only check direct dependencies for compatible licenses but also indirect dependencies especially when working with \toolname{Maven}\xspace and \toolname{PyPI}\xspace.
Some package managers, e.g., \toolname{mvn} with the \toolname{dependency:tree} or \toolname{npm} with the \toolname{list} command, allow to inspect direct and indirect dependencies simultaneously.
In other package managers, such as, \toolname{pip}, dependency trees can be inspected via tools like \toolname{pipdeptree}\citeurl{https://github.com/naiquevin/pipdeptree}.
On top of that, there exist tools that can be integrated to the development process to support developers in creating software free of license incompatibilities, e.g., the IDE plugin \toolname{Sorrel}~\cite{pogrebnoy2021sorrel}.
\subsection{Implications for Researchers}
We only studied license incompatibilities within ecosystems.
Modern applications, in particular web-applications, typically reuse code from multiple ecosystems.
It remains an open question to which degree software projects from various domains are affected by license incompatibilities within ecosystems or to which degree they introduce even new incompatibilities on top of existing ones.
\bibliographystyle{IEEEtran}
|
1,108,101,565,450 | arxiv |
\section*{Background}
Influenza, short for flu, is an acute respiratory infection caused by flu viruses. Flu circulates in all over the world. The worldwide infection places a substantial burden on people's health every year. According to World Health Organization(WHO)’s report, flu is estimated to result in about 3 to 5 million cases of severe illness, and about 290 000 to 650 000 deaths. Accurately forcasting of influenza outbreaks could help taking appropriate actions, such as school closure, to prevent or reduce flu illness.
Regardless of the characteristic of its worldwide circulation of flu, most previous studies have focused on regional prediction of flu outbreaks \cite{wang2016regional,kane2014comparison,malik2017distressed,wu2017time} for two probable reasons. First, different locations in one country or one region, to some extent, share similar geo-locational characteristics, such as humidity and temperature. Flu virus shows a sensitivity to temperature and humidity. As a result, predicting flu outbreaks of one country or one region is considered reasonable and approachable. Second, flu virus transmission is believed to occur mostly over relatively short distances. Usually, flu virus is spread through the air from coughs or sneezes. When an infected person coughs or sneezes, droplets containing viruses (infectious droplets) are dispersed into the air and can spread up to one meter, and infect persons in close proximity who breathe these droplets in.
However, one fast-growing risk group, travelers, is neglected from these overviews. Several changes in our globalizing world contribute to the growing influence of the traveller group: (i) steady increase in total travel volume worldwide, (ii) advent of mass-tourism and (iii) increasing numbers of immune-compromised and elderly travelers. International sporting events and festivals as well as traveling by airplane or cruise ship could facilitate flu virus transmissionand therefore global spread of influenza \cite{goeijenbier2017travellers}. The study in \cite{he2015global} shows that flu outbreaks correlate with each other in all countries around the world. The methodology of considering the correlation could help forecast the flu outbreaks.
Furthermore, forecasting a longer-term flu outbreak, and knowing its outbreak trend more accurately could help hospitals, clinics, and pharmaceutical companies to better prepare for annual flu outbreaks. First, manufacturing flu vaccine is a challenging work. According to WHO’s report, vaccination is the most effective way to prevent the disease. During 2015-2016 flu seasons, flu vaccine prevented an estimated 5.1 million illnesses, 2.5 million medical visits, 71,000 hospitalizations, and 3,000 pneumonia \& influenza (P\&I) deaths. The problem is that flu virus undergoes high mutation rates and frequent genetic re-assortment (combination and rearrangement of genetic material). This characteristic of flu complicates the procedure of flu vaccines production. In Februaries, World Health Organization (WHO) assesses the strains of flu virus that are most likely to be circulating over the following winter. Then, vaccine manufacturers produce flu vaccines in a very limited time. Usually, the first batch of vaccine is unavailable until September. As a result, in an extremely limited time, manufacturers have to prepare enough vaccines \cite{gerdil2003annual,lubeck1980antigenic}. Second, beds assignment to flu patients is another challenging task due to the limited capacity of hospital beds, time-dependencies of bed request arrivals, and unique treatment requirements of flu patients. Flu seasons vary in timing, severity, and duration from one season to another. Therefore, flu hospitalization also varies greatly by sites and time in each season \cite{puig2014first}. Predicting a sequence of values in future,namely,the multi-step predication of flu outbreaks should cause concern.
Therefore, we highlight the importance of developing global methodologies to perform multi-step prediction of worldwide influenza outbreaks. Nonetheless, not many past studies focused on multistep prediction of influenza outbreaks. The probable reason could be that multistep prediction usually results in poor accuracy due to some insuperable problems, such as error accumulation, etc. \cite{zhang2013iterated,akhlaghi2017adaptive}. One compromising method is that one can aggregate raw data to a larger time unit and then use the single-step prediction to replace multi-step prediction. For instance, if raw data is weekly based, we can aggregate weekly values to monthly values and then perform single-step prediction of the total value of the coming month (roughly around four weeks). However the aggregation hinders us from understanding the internal variation during the coming four weeks.
In this study, we performed multi-step prediction by leveraging Long Short Term Memory (LSTM). The LSTM is a special kind of RNN. In theory, the complex structure (layers and gated cells) enables LSTM to learn long-term dependencies \cite{hochreiter1997long}, simulate nonlinear function, and refine time-series prediction very well \cite{gers1999learning}.
\section*{Methods}
As shown in the Figure 1, to perform spatio-temporal flu prediction based on historical data, firstly, we scraped flu data of all the 155 countries from the FluNet, a global web-based tool for flu virological surveillance in WHO. We selected 23 countries as features since other countries have N/As in their flu data. We selected spatio-temporal related features. Then, we send those features into a model combined with LSTM and fully connected layers. Finally, the model predicts the flu data of the 1-, 2-, 3-, and 4-week ahead with other countries' flu data. To compare the results, we also predicts the flu data of the 1-, 2-, 3-, and 4-week ahead without other countries' flu data. The following subsections presents the details.
\subsection*{ Data acquisition}
FluNet is a global web-based tool for flu virological surveillance \cite{fluNet}. The data at country level are available and updated weekly. From FLuNet, we collected the flu data of 155 countries around the world from the 1st week of 2010 to the 18th week of 2018. We select 23 countries, the flu data of which have no NAs. The 23 countries are Australia, Brazil, Cambodia , China , Egypt , French Guiana , Ghana , Indonesia , Iran , Iraq , Ireland , Japan , Netherlands , Nicaragua, Niger, Norway, Panama, Poland, Republic of Korea, Russia, United Kingdom of Great Britain and Northern Ireland (UK), United States of America (USA).
\subsection*{ Feature Selection}
The features are selected or generated considering the spatital and temporal influence of flu outbreaks.
\subsubsection*{Temporal factors}
In temperate climates, flu outbreaks occur mainly during winter; while in tropical regions, flu outbreaks occur throughout the year. Considering the possible one-year long period of flu outbreaks our previous studies compared the performance of the time lags of 2, 4, 9, 13, 26, and 52 week, and found that 52 weeks lead to the best accuracy \cite{zhang2017comparative}. The temperature changes could affect flu virus, and people tend to get illness. Therefore we construct the temporal factors with three kinds of data: the original data of the past 52 weeks; the first order difference; the mean, median, standard deviations(std), maximum, and minimum of windows, the length of which are 1, 2, 3, 4, 9, 13, 26, 52 weeks.
\subsubsection*{Spatial factors}
Considering the global spread of influenza and the correlation between countries, we use the historical flu data of another above-mentioned 22 countries as the prediction features when predicating one country. Therefore when we are predicating the flu outbreaks of one country, the other countries could affect the outcome by adjusting their weight parameters. By this way, we get another 1,144 (22 times 52) features.
\subsection*{Multi-step Prediction}
There are two types of prediction of flu outbreaks: (a) single-step prediction: predicting the coming value in future by analyzing observed values in the past; and (b) multistep prediction: predicting a sequence of values in future by analyzing observed values in the past. The idea (a) tend to accumulate the errors induced in the previous steps to future predictions. In this study we use multiple single-output prediction (MSOP) to implement multi-step prediction.
MSOP predicts the coming several values by the same past values. In other words, when predicting $X_{t+p(p>=2)}$, MSOP jumps $X_{t+p-1 (p>=2)}$,$X_{t+p-2 (p>=2)}$, …, and $X_{t+1}$. Formula 3 explains the algorithm of MSOP. Its flow are presented in Figure 2.
\begin{equation}
\begin{split}
X_{t+1}(predicted)& =LSTM\_MODEL\_\#01[X_{t}(observed), X_{t-1}(observed), …, X_{t-51}(observed)]\\
X_{t+2}(predicted)& =LSTM\_MODEL\_\#02[X_{t}(observed), X_{t-1}(observed), …, X_{t-51}(observed)]\\
X_{t+3}(predicted)& =LSTM\_MODEL\_\#03[X_{t}(observed), X_{t-1}(observed), …, X_{t-51}(observed)]\\
X_{t+4}(predicted)& =LSTM\_MODEL\_\#04[X_{t}(observed), X_{t-1}(observed), …, X_{t-51}(observed)]\\
\end{split}
\end{equation}
As shown in Figure 2, to predict $x_{t+1}$, we train a model by using $X_{t}, X_{t-1}, X_{t-2}, …, X_{t-52}$ as features. To predict $X_{t+2}$, we train another model by still using $X_{t}, X_{t-1}, X_{t-2}, …, X_{t-52}$ as features. Although we use the same feature space in these two models, the two models are trained differently with different responses ($x_{t+1} and x_{t+2}$). The research in \cite{zhang2017comparative} shows the 3-layered LSTM is efficient enough in predicting flu outbreaks.
\subsection*{ Metrics}
Because the population of some countries is quite small and only 1 or 2 flu patients every week are reported, the study on those countries is insignificant.
We predicted the flu data of the coming weeks in countries with a large population. We selected Australasia, Brazil, China, Japan, UK, and USA when considering population and location.
We investigated the distribution of the flu data, and found that it was non-normal distribution. In our opinion, comparing models’ accuracy by Mean Absolute Percentage Error (MAPE, as shown in the Formula \ref{mape}) reflects the difference based on the median, while comparing models’ accuracy by Root Mean Square Error (RMSE) is based on means. Therefore, we used MAPE as a metrics to compare predicting the accuracy of models.
\begin{equation}
MAPE=\frac{1}{n}\sum _{t=1}^{{n}_{x}}\left|\frac{F_{t}-A_{t}}{A_{t-1}}\right|\label{mape}
\end{equation}
\section*{Results}
Table 1 presents the MAPEs of RF, SVM, and LSTM models with and without other countries’ flu data. For example, when forecasting China’s flu data of 1-, 2-, 3-, and 4-week ahead, the MAPEs of the LSTM models with other countries’ flu data are 13.1\%, 19.8\%, 26.7\%, 36.2\%; while the MAPEs of the LSTM models without other countries’ flu data are 12.5\%, 20.2\%, 29.0\%, and 36.7\%.
Figure 3 compares the MAPEs of SVM, RF, LSTM models of predicting flu data of the 1-, 2-, 3-, and 4-week ahead with other countries' flu data. In most cases, the LSTM models achieved the lowest MAPEs.
Alike, Figure 4 compares the MAPEs of SVM, RF, LSTM models of predicting flu data of the 1-, 2-, 3-, and 4-week ahead without other countries' flu data. In most cases, the LSTM models achieved the lowest MAPEs.
Figure 5 compares the MAPEs of the LSTM models with and without other countries’ flu data. As for countries in Southern hemisphere, i.e. Australia and Brazil, the MAPEs of predicting flu data of the 1-, 2-, 3-, and 4-week ahead with other countries are higher than those of predicting without other countries. For countries in Northern hemisphere, i.e. China, Japan, UK, and USA, the MAPEs of predicting flu data of the 2-4 weeks ahead with other countries are lower than those of predicting without other countries. Interestingly, when predicting flu data of the 1 week ahead, the MAPEs of predicting with other countries are usually higher than those of predicting without other countries, except for UK.
\section*{Discussion}
We found, in southern hemisphere (Australia and Brazil), the MAPEs of predicting flu data with other countries are higher than the MAPEs without other countries. The probable reasons are the southern hemisphere’s countries have totally different flu seasons since their winters are in June, July and August. And the countries selected in this study are mostly in Northern hemisphere and their flu data are barely correlated to the flu data of southern hemisphere’s countries. In addition, Australia is geographically isolated from other countries. As for Northern hemisphere, the MAPEs of predicting flu data of the 2-4 weeks ahead with other countries are lower than those without other countries. That is because of high correlations among flu data of Northern hemisphere’s countries. However, the MAPEs of predicting flu data of the 1 week ahead with other countries are lower than those without other countries. That is probably because flu infection in other countries does not impact the of flu infection in target countries in one week ahead because of geographical distance.
The best MAPEs of LSTM models achieved were still very high because we used the flu data in 2017-2018 as a testing set. The 2017-2018 flu season, a pandemic-alike season, is quite different from and seriously heavier the past few seasons. And using other machine learning metrologies, such as SVR and RF, result in higher MAPEs. In this study, we used only historical values. To some extent, historical values are a reflection of all possible related factors. However, one might say other features, such as temperature and humidity, could help predict more accurately, especially at turning points. For one thing, when we predict future values, we have to use predicted data, e.g. weather forecast. The predicting error of predicted data could intensively enlarge the predicting error in further steps. For another, how to express one country’s weather could be another problem if the country has a large area and population. A possible solution could be using two convolutional neural networks to extract features of weather and population of the whole country.
\section*{Conclusions}
In this study, we performed the spatio-temporal multi-step prediction of influenza outbreaks.The methodology considering the spatio-temporal features improves the multi-step prediction of flu outbreak. We compared the MAPEs of SVM, RF, LSTM models of predicting flu data of the 1-4 week(s) ahead with and without other countries' flu data. We found the LSTM models achieved the lowest MAPEs in most cases. As for countries in Southern hemisphere, the MAPEs of predicting flu data with other countries are higher than those of predicting without other countries. For countries in Northern hemisphere, the MAPEs of predicting flu data of the 2-4 weeks ahead with other countries are lower than those of predicting without other countries; and the MAPEs of predicting flu data of the 1-weeks ahead with other countries are higher than those of predicting without other countries, except for UK.
\begin{backmatter}
\section*{Competing interests}
The authors declare that they have no competing interests.
\bibliographystyle{bmc-mathphys}
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1,108,101,565,451 | arxiv | \section{Introduction and statement of the main results}
Let $M$ be any $\sigma$-finite von Neumann algebra and $\varphi \in M_*$ any faithful state. Following \cite{Co80, Ha85}, we define the \emph{bicentralizer} of $M$ with respect to $\varphi$ by
$$\mathord{\text{\rm B}}(M, \varphi) = \left\{ x \in M \mid x a_{n} - a_{n} x \to 0 \text{ strongly}, \forall (a_{n})_{n} \in \mathord{\text{\rm AC}}(M, \varphi)\right\}$$
where $$\mathord{\text{\rm AC}}(M, \varphi) = \left \{ (a_{n})_{n} \in \ell^{\infty}(\mathbf{N}, N) \mid \lim_{n} \|a_{n} \varphi - \varphi a_{n}\| = 0\right\}$$ is the {\em asymptotic centralizer} of $\varphi$. One of the most famous open problems in the theory of type ${\rm III}$ factors is Connes' bicentralizer problem asking whether for any type ${\rm III}_1$ factor $M$ with separable predual and any faithful state $\varphi \in M_*$, we have $\mathord{\text{\rm B}}(M, \varphi)=\mathbf{C} 1$. This question was solved affirmatively by Haagerup in \cite{Ha85} for \emph{amenable} $M$, thus settling the problem of the classification of amenable factors with separable predual (see \cite{Co75b, Co85}).
Nowadays, the bicentralizer problem is still of premium importance. Indeed by \cite[Theorem 3.1]{Ha85}, for any type ${\rm III_{1}}$ factor $M$ with separable predual, $M$ has trivial bicentralizer if and only if there exists a faithful state $\varphi \in M_{\ast}$ with an irreducible centralizer, meaning that $(M_{\varphi})' \cap M = \mathbf{C} 1$. Then by \cite[Theorem 3.1]{Ha85} and \cite[Theorem 3.2]{Po81}, $M$ has trivial bicentralizer if and only if there exists a maximal abelian subalgebra $A \subset M$ that is the range of a normal conditional expectation (see \cite[Question]{Ta71} where the problem of finding such maximal abelian subalgebras is mentioned). For these reasons, Connes' bicentralizer problem appears naturally when one tries to use Popa's deformation/rigidity theory in the type ${\rm III}$ context (see for instance \cite[Theorem C]{HI15}). The bicentralizer problem is known to have a positive solution for particular classes of nonamenable type ${\rm III}_{1}$ factors: factors with a Cartan subalgebra; Shlyakhtenko's free Araki--Woods factors (\cite{Ho08}); (semi-)solid factors (\cite{HI15}); free product factors (\cite{HU15}). However, the bicentralizer problem is still wide open for arbitrary type ${\rm III}_1$ factors.
In his attempt to solve the bicentralizer problem, Connes observed that for any type ${\rm III}_1$ factor $M$, the bicentralizer $\mathord{\text{\rm B}}(M,\varphi)$ does not depend on the choice of the state $\varphi$ up to a canonical isomorphism. In around 2012--2013, Haagerup found out that the idea of Connes' isomorphism (denoted by $\beta_{\psi,\varphi}$ below) can be enhanced to construct a canonical flow ($u$-continuous action) $\beta^{\varphi}\colon \mathbf{R}_+^*\curvearrowright \mathord{\text{\rm B}}(M,\varphi)$ with interesting properties. This flow was independently discovered by Marrakchi and this was the starting point of our joint research.
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras {\em with expectation}, meaning that there exists a faithful normal conditional expectation $\mathord{\text{\rm E}}_{N} : M \to N$. Following \cite[Definition 4.1]{Ma03}, we define the {\em relative bicentralizer} $\mathord{\text{\rm B}}(N \subset M, \varphi)$ of the inclusion $N \subset M$ with respect to the faithful state $\varphi \in N_*$ by
$$\mathord{\text{\rm B}}(N \subset M, \varphi) = \left\{ x \in M \mid x a_{n} - a_{n} x \to 0 \text{ strongly}, \forall (a_{n})_{n} \in \mathord{\text{\rm AC}}(N, \varphi)\right\}.$$
Observe that we always have $N' \cap M \subset \mathord{\text{\rm B}}(N\subset M, \varphi) \subset (N_{\varphi})' \cap M$. When $N = M$, we simply have $\mathord{\text{\rm B}}(N \subset M,\varphi)=\mathord{\text{\rm B}}(M, \varphi)$.
Our first main result deals with the construction of the canonical flow on the relative bicentralizer $\mathord{\text{\rm B}}(N \subset M, \varphi)$.
\begin{letterthm} \label{thm: Connes' isomorphism and bicentralizer flow}
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation. Assume that $N$ is a type ${\rm III}_1$ factor. Then the following assertions hold:
\begin{itemize}
\item [$(\rm i)$] For every pair of faithful states $\varphi, \psi \in N_*$, there exists a canonical isomorphism $$\beta_{\psi, \varphi} : \mathord{\text{\rm B}}(N \subset M,\varphi) \rightarrow \mathord{\text{\rm B}}(N \subset M,\psi)$$ characterized by the following property: for any uniformly bounded sequence $(a_n)_{n \in \mathbf{N}}$ in $N$ and any $x \in \mathord{\text{\rm B}}(N \subset M,\varphi)$, we have
$$ \|a_n \varphi - \psi a_n\| \to 0 \quad \Rightarrow \quad a_nx-\beta_{\psi, \varphi}(x)a_n \to 0 \; \ast\text{-strongly}. $$
\item [$(\rm ii)$] There exists a canonical flow $$\beta^{\varphi}: \mathbf{R}^*_+ \curvearrowright \mathord{\text{\rm B}}(N \subset M,\varphi)$$ characterized by the following property: for any uniformly bounded sequence $(a_n)_{n \in \mathbf{N}}$ in $N$, any $x \in \mathord{\text{\rm B}}(N \subset M,\varphi)$ and any $\lambda > 0$, we have
$$ \|a_n \varphi -\lambda\varphi a_n\| \to 0 \quad \Rightarrow \quad a_nx-\beta_\lambda^{\varphi}(x)a_n \to 0 \; \ast\text{-strongly}.$$
\item [$(\rm iii)$] We have $\beta_{\varphi_3,\varphi_2} \circ \beta_{\varphi_2, \varphi_1}=\beta_{\varphi_3, \varphi_1}$ for every faithful state $\varphi_i \in N_*$, $i \in \{1,2,3\}$, and $\beta_\lambda^{\psi} \circ \beta_{\psi, \varphi} = \beta_{\psi, \varphi} \circ \beta_\lambda^{\varphi}$ for every pair of faithful states $\psi, \varphi \in N_*$ and every $\lambda > 0$.
\item [$(\rm iv)$] For every pair of faithful states $\psi, \varphi \in N_*$ and every $\lambda > 0$, we have
$$\mathord{\text{\rm E}}_{N' \cap M}^{\psi} \circ \beta_{\psi, \varphi} = \mathord{\text{\rm E}}_{N' \cap M}^{\varphi}= \mathord{\text{\rm E}}_{N' \cap M}^{\varphi} \circ \beta_{\lambda}^{\varphi} $$
where $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi} : M \rightarrow N' \cap M$ is the unique normal conditional expectation such that $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(x)=\varphi(x)1$ for all $x \in N$.
\end{itemize}
\end{letterthm}
The proof of Theorem \ref{thm: Connes' isomorphism and bicentralizer flow} uses ultraproduct von Neumann algebras \cite{Oc85, AH12} and relies on Connes--St\o rmer transitivity theorem \cite{CS76} and the fact that any $\lambda > 0$ is an approximate eigenvalue for the faithful state $\varphi \in N_{\ast}$ (see Lemma \ref{lem: family of partial isometries}).
The meaning of the compatibility relations given in item $(\rm iii)$ is that the W$^*$-dynamical system $(\mathord{\text{\rm B}}(N \subset M, \varphi), \beta^\varphi)$ does not depend on the choice of $\varphi \in N_\ast$ up to the canonical isomorphism $\beta_{\psi, \varphi}$. Thus, $(\mathord{\text{\rm B}}(N \subset M, \varphi), \beta^\varphi)$ is an invariant of the inclusion $N \subset M$. We call it the \emph{relative bicentralizer flow} of the inclusion $N \subset M$. When $N=M$, we simply call it the \emph{bicentralizer flow} of $M$. In this paper, we study this invariant and we relate it to some structural properties of the inclusion $N \subset M$.
\subsection*{Self-bicentralizing factors}
For any von Neumann algebra $M$ and any faithful state $\varphi \in M_{\ast}$, we have $\mathord{\text{\rm B}}(\mathord{\text{\rm B}}(M,\varphi), \varphi|_{\mathord{\text{\rm B}}(M,\varphi)})=\mathord{\text{\rm B}}(M,\varphi)$ and if $M$ is a type ${\rm III}_1$ factor then $\mathord{\text{\rm B}}(M,\varphi)$ is either trivial or a type ${\rm III}_1$ factor. Therefore, the bicentralizer problem reduces to the following question: does there exist a type ${\rm III}_1$ factor $M$ which satisfies $M=\mathord{\text{\rm B}}(M,\varphi)$ for some faithful state $\varphi \in M_{\ast}$? We call such a state $\varphi$ a \emph{bicentralizing} state on $M$. If such a factor exists, by \cite{Ha85}, it must be nonamenable and by \cite{HI15}, it must be McDuff, that is, $M\cong M \mathbin{\overline{\otimes}} R$ where $R$ is the hyperfinite type ${\rm II}_1$ factor. In our next result, we use Theorem \ref{thm: Connes' isomorphism and bicentralizer flow} to further understand the structure of these mysterious self-bicentralizing type ${\rm III}_1$ factors. We show that the bicentralizing state is unique up to conjugacy by a unique approximately inner automorphism. We also show that their automorphism group splits as a semi-direct product. Finally, we relate the period of the bicentralizer flow to the tensorial absorption of Powers factors.
\begin{letterthm} \label{thm: self-bicentralizing}
Let $M$ be any type ${\rm III}_1$ factor with separable predual and with a bicentralizing state $\varphi \in M_{\ast}$. Then the following properties hold:
\begin{itemize}
\item [$(\rm i)$] Let $\Delta(M)$ be the set of all bicentralizing states of $M$. Then the map
$$\overline{\mathord{\text{\rm Inn}}}(M) \ni \alpha \mapsto \alpha(\varphi) \in \Delta(M)$$ is a homeomorphism and its inverse is given by
$$ \Delta(M) \ni \psi \mapsto \beta_{\psi, \varphi} \in \overline{\mathord{\text{\rm Inn}}}(M).$$
\item [$(\rm ii)$] Define $\mathord{\text{\rm Aut}}_\varphi(M) = \{\alpha \in \mathord{\text{\rm Aut}}(M) \mid \alpha(\varphi) = \varphi\}$ and consider the conjugation action $\mathord{\text{\rm Aut}}_\varphi(M) \curvearrowright \overline{\mathord{\text{\rm Inn}}}(M)$. Then the natural homomorphism
$$ \iota : \overline{\mathord{\text{\rm Inn}}}(M) \rtimes \mathord{\text{\rm Aut}}_{\varphi}(M) \ni (g,h)\mapsto g \circ h \in \mathord{\text{\rm Aut}}(M)$$
is an isomorphism of topological groups. In particular, $\sigma_t^{\varphi} \notin \overline{\mathord{\text{\rm Inn}}}(M)$ for all $t \neq 0$.
\item [$(\rm iii)$] For every $0<\lambda<1$, we have
$$ M\cong M\overline{\otimes}R_{\lambda} \; \Leftrightarrow \; \beta_\lambda^{\varphi}=\text{\rm id} \; \Leftrightarrow \; \beta_\lambda^{\varphi} \in \overline{\mathord{\text{\rm Inn}}}(M).$$
In particular, we have $M\cong M\overline{\otimes}R_{\infty}$ if and only if the bicentralizer flow $\beta^{\varphi}$ is trivial.
\item [$(\rm iv)$] For every $\lambda > 0$, the automorphism $\beta_\lambda^{\varphi} \odot \text{\rm id}$ of $M \odot M^{\mathord{\text{\rm op}}}$ extends to the $\mathord{\text{\rm C}}^{*}$-algebra $\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)$ generated by the standard representation of $M \odot M^{\mathord{\text{\rm op}}}$ on $\mathord{\text{\rm L}}^{2}(M)$.
\end{itemize}
\end{letterthm}
In Connes' strategy to prove the uniqueness of the amenable type ${\rm III}_1$ factor \cite{Co85}, a crucial step was to show that $\sigma_t^{\varphi}\in \overline{\rm{Inn}}(M)$ for every $t \in \mathbf{R}$. The amenability of $M$ implies that $\mathord{\text{\rm C}}^*_{\lambda . \rho}(M)=M \otimes_{\min} M^{\mathord{\text{\rm op}}}$ and so any automorphism of $M$ satisfies the property $(\rm iv)$ above. Connes conjectured \cite[Section IV]{Co85} that for any type ${\rm III}_1$ factor $M$, any automorphism satisfying the property $(\rm iv)$ above must be approximately inner. It is in trying to prove this conjecture that he encountered the bicentralizer problem. As we see from item $(\rm iii)$ and item $(\rm iv)$ above, if this conjecture is true then the bicentralizer flow $\beta^\varphi$ must be trivial.
\subsection*{Irreducible hyperfinite subfactors in inclusions of type ${\rm III}$ factors}
Let $N \subset M$ be any irreducible inclusion of factors with separable predual and with expectation. In \cite{Po81}, Popa proved that if $N$ is \emph{semifinite}, then there exists a hyperfinite subfactor with expectation $P \subset N$ such that $P' \cap M=\C1$. We extend this theorem to the case when $N$ is a type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$ in Theorem \ref{thm: hyperfinite subfactor III_lambda}. In the case when $N$ is a type ${\rm III}_1$ factor, we relate this question to the ergodicity of the relative bicentralizer flow.
\begin{letterthm} \label{thm: hyperfinite subfactor III_1}
Let $N \subset M$ be any inclusion of von Neumann algebras with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_1$ factor. Let $\varphi \in N_*$ be any faithful state. The following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta^{\varphi}}=N' \cap M$.
\item [$(\rm ii)$] There exists a hyperfinite subfactor with expectation $P \subset N$ such that $P' \cap M=N' \cap M$.
\end{itemize}
We can always choose $P = R_{\infty}$ to be the hyperfinite type ${\rm III}_1$ factor.
\begin{itemize}
\item We can moreover choose $P = R_{\lambda}$ to be the hyperfinite type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$ if and only if $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta^{\varphi}_\lambda}=N' \cap M$.
\item We can moreover choose $P = R$ to be the hyperfinite type ${\rm II}_1$ factor if and only if $\mathord{\text{\rm B}}(N \subset M, \varphi)=N' \cap M$.
\end{itemize}
\end{letterthm}
The proof of Theorem \ref{thm: hyperfinite subfactor III_1} generalizes the methods developed by Popa in \cite[Theorem 3.2]{Po81} and Haagerup in \cite[Theorem 3.1]{Ha85}.
Let us mention that by \cite[Theorem 2]{Po83} (see also \cite[Theorem 5.1]{Lo83} for the infinite case), any factor $M$ with separable predual possesses an irreducible hyperfinite subfactor that is typically {\em not} the range of a normal expectation. In contrast, constructing an irreducible hyperfinite subfactor that is the range of a normal expectation is a more subtle problem. In \cite[Corollaries 1.3 and 3.4]{Po85}, Popa shows that any type ${\rm III_{\lambda}}$ factor $M$ with separable predual with $0 \leq \lambda < 1$ possesses an irreducible hyperfinite subfactor with expectation. Our Theorem \ref{thm: hyperfinite subfactor III_1} shows in particular that a type ${\rm III_{1}}$ factor $M$ with separable predual possesses an irreducible hyperfinite subfactor with expectation if and only if the bicentralizer flow is ergodic.
Following \cite{Co72, Co74}, a $\sigma$-finite von Neumann algebra $Q$ is {\em almost periodic} if $Q$ possesses an {\em almost periodic} state, that is, a faithful normal state for which the corresponding modular operator is diagonalizable. By \cite{Co72, Co74}, any $\sigma$-finite type ${\rm III_{\lambda}}$ factor with $0 \leq \lambda < 1$ is almost periodic. When $N \subset M$ is an irreducible inclusion of factors with separable predual and with expectation, a sufficient condition for the relative bicentralizer flow $\beta^{\varphi} : \mathbf{R}^{*}_{+} \curvearrowright \mathord{\text{\rm B}}(N \subset M, \varphi)$ to be ergodic is the existence of an almost periodic subfactor with expectation $Q \subset N$ such that $Q' \cap M = \mathbf{C} 1$. Using Theorem \ref{thm: hyperfinite subfactor III_1}, we derive the following application that gives a partial solution to \cite[Problem 4]{Po85}.
\begin{letterapp}\label{app-almost-periodic}
Let $M$ be any ${\rm III_{1}}$ factor with separable predual. Assume that there exists an irreducible almost periodic subfactor with expectation $Q \subset M$.
Then there exists an irreducible hyperfinite subfactor with expectation $P \subset M$.
\end{letterapp}
We point out that it is unclear whether we can choose $P$ as a subfactor of $Q$. We can do so if $Q$ possesses an almost periodic faithful state $\varphi \in Q_\ast$ such that its centralizer $Q_\varphi$ is a type ${\rm II_1}$ factor (see Theorem \ref{thm: hyperfinite subfactor III_lambda}). However, when $Q$ is a type ${\rm III_0}$ factor, no such almost periodic state exists on $Q$ and so we really need to exploit the ergodicity of the relative bicentralizer flow to construct the hyperfinite subfactor $P \subset N$.
A sufficient condition for an inclusion of factors $N \subset M$ to be irreducible is the existence of an abelian subalgebra $A \subset N$ that is maximal abelian in $M$. One of Kadison's well-known problems in \cite{Ka67} asks whether the converse is true. We will say that an irreducible inclusion of factors with expectation $N \subset M$ satisfies {\em Kadison's property} if there exists an abelian subalgebra with expectation $A \subset N$ that is maximal abelian in $M$. Popa proved in \cite[Theorem 3.2]{Po81} that any irreducible inclusion $N \subset M$ with separable predual and with expectation such that $N$ is semifinite satisfies Kadison's property.
Combining Theorem \ref{thm: hyperfinite subfactor III_1} with \cite[Theorem 3.2]{Po81}, we obtain the following characterization:
\begin{lettercor} \label{kadison bicentralizer}
Let $N \subset M$ be any irreducible inclusion of factors with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_1$ factor. Then the following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] $\mathord{\text{\rm B}}(N \subset M, \varphi)=\C1$ for some (or any) faithful state $\varphi \in N_*$.
\item [$(\rm ii)$] The inclusion $N \subset M$ satisfies Kadison's property.
\end{itemize}
\end{lettercor}
In the case when $N \subset M$ has finite index, Corollary \ref{kadison bicentralizer} follows from \cite[Theorem 4.2]{Po95}. In order to find new examples of inclusions $N \subset M$ that satisfy Kadison's property, we will prove a relative bicentralizer theorem for {\em discrete} inclusions.
\subsection*{Relative bicentralizers of discrete inclusions}
Following \cite[Definition 1.1]{Po98}, we say that an inclusion of von Neumann algebras $Q \subset P$ satisfies the {\em weak relative Dixmier property} if for every $x \in P$, we have
$$\mathcal K_{Q}(x) \cap (Q' \cap P) \neq \emptyset$$
where $\mathcal K_{Q}(x) = \overline{\mathord{\text{\rm co}}}^{w} \left \{uxu^{*} \mid u \in \mathcal U(Q)\right \}$. Recall that $Q$ is amenable if and only if the inclusion $Q \subset \mathbf B(H)$ satisfies the weak relative Dixmier property for some (or any) unital faithful normal representation of $Q$ on $H$ (see \cite{Sc63}). In particular, any inclusion $Q \subset P$ with $Q$ amenable satisfies the weak relative Dixmier property.
In \cite{Ha85}, a connection was established between the bicentralizer problem and the weak relative Dixmier property. Indeed, by \cite[Theorem 3.1]{Ha85}, a type ${\rm III}_1$ factor with separable predual has trivial bicentralizer if and only if the inclusion $M_\psi \subset M$ satisfies the weak relative Dixmier property for some (or any) dominant weight $\psi$ on $M$. This solved in particular the bicentralizer problem for amenable $M$.
Following \cite[Definition 3.7]{ILP96}, an inclusion of von Neumann algebras $N \subset M$ with separable predual and with expectation is {\em discrete} if the inclusion $N \subset \langle M, N \rangle = (JNJ)' \cap \mathbf B(\mathord{\text{\rm L}}^{2}(M))$ is with expectation (this is indeed equivalent to \cite[Definition 3.7]{ILP96} thanks to \cite[Theorem 6.6]{Ha77}).
\begin{example}
Here are some fundamental examples of discrete inclusions of factors with separable predual.
\begin{itemize}
\item [$(\rm i)$] The inclusion $N \otimes \mathbf{C} 1 \subset N \mathbin{\overline{\otimes}} Q$, where $N, Q$ are any factors.
\item [$(\rm ii)$] Any finite index inclusion $N \subset M$ (see \cite{Jo82, PP84, Ko85}).
\item [$(\rm iii)$] The inclusion $N \subset N \rtimes \Gamma$, where $\Gamma$ is any countable discrete group, $N$ is any factor and $\Gamma \curvearrowright N$ is any outer action (see \cite[Section 3]{ILP96}).
\item [$(\rm iv)$] The inclusion $M^{\mathbf G} \subset M$, where $\mathbf G$ is any compact second countable group, $M$ any factor and $\alpha : \mathbf G \curvearrowright M$ any {\em minimal} action, meaning that $\alpha$ is faithful and $(M^{\mathbf G})' \cap M = \mathbf{C} 1$ (see \cite[Section 3]{ILP96}).
\end{itemize}
\end{example}
For discrete inclusions of von Neumann algebras $N \subset M$ where $N$ is a type ${\rm III_{1}}$ factor, we obtain the following {\em relative bicentralizer theorem} that generalizes Haagerup's result \cite[Theorem 3.1]{Ha85} (when $N = M$) and Popa's result \cite[Theorem 4.2]{Po95} (when $N \subset M$ has finite index).
\begin{letterthm}\label{thm-relative-bicentralizer}
Let $N \subset M$ be any inclusion of von Neumann algebras with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_1$ factor. Consider the following assertions:
\begin{itemize}
\item [$(\rm i)$] For some (or any) faithful state $\varphi \in N_{\ast}$, we have $\mathord{\text{\rm B}}(N\subset M, \varphi) = N' \cap M$.
\item [$(\rm ii)$] For some (or any) dominant weight $\psi$ on $N$, we have $(N_{\psi})' \cap M = N' \cap M$ and the inclusion $N_{\psi} \subset M$ satisfies the weak relative Dixmier property.
\end{itemize}
Then $(\rm i) \Rightarrow (\rm ii)$. If the inclusion $N \subset M$ is discrete, then $(\rm ii) \Rightarrow (\rm i)$.
\end{letterthm}
Let us point out that Theorem \ref{thm-relative-bicentralizer} gives a positive answer to \cite[Further Remarks 4.9 (2)]{Po95}. Moreover, by combining Theorem \ref{thm-relative-bicentralizer} with Corollary \ref{kadison bicentralizer} and a generalization of Connes--Takesaki relative commutant theorem (see Theorem \ref{thm-relative-connes-takesaki}), we solve Kadison's problem for all discrete irreducible inclusions $N \subset M$ where $N = R_{\infty}$ is the hyperfinite type ${\rm III_{1}}$ factor. This answers a question raised in \cite[Problem 6]{HP17}.
\begin{lettercor}\label{cor-characterization}
Let $N \subset M$ be any discrete irreducible inclusion of factors with separable predual and with expectation. Assume that $N = R_{\infty}$ is the hyperfinite type ${\rm III_{1}}$ factor. Then the following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] The inclusion of continuous cores $\mathord{\text{\rm c}}(N) \subset \mathord{\text{\rm c}}(M)$ is irreducible, i.e.\ $\mathord{\text{\rm c}}(N)' \cap \mathord{\text{\rm c}}(M) = \mathbf{C} 1$.
\item [$(\rm ii)$] The inclusion $N \subset M$ satisfies Kadison's property.
\end{itemize}
\end{lettercor}
We emphasize the fact that condition $(\rm i)$ in Corollary \ref{cor-characterization} can be concretely checked in many situations. In particular, for crossed products by discrete groups, combining Corollary \ref{cor-characterization} with \cite[Proposition 5.4]{HS88}, we obtain:
\begin{letterapp}\label{app-crossed-product}
Let $N = R_{\infty}$ be the hyperfinite type ${\rm III}_1$ factor. Let $\Gamma$ be any countable discrete group and $\alpha : \Gamma \curvearrowright N$ any outer action. Then the following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] The automorphism $\alpha(g) \circ \sigma_t^\varphi$ is outer for all $g \in \Gamma \setminus \{e\}$ and all $t \in \mathbf{R}$.
\item [$(\rm ii)$] The factor $N \rtimes \Gamma$ is of type ${\rm III}_1$.
\item [$(\rm iii)$] The inclusion $N \subset N \rtimes \Gamma$ satisfies Kadison's property.
\end{itemize}
\end{letterapp}
In particular, the above conditions are satisfied by all Bernoulli actions $\Gamma \curvearrowright \mathbin{\overline{\bigotimes}}_{\Gamma} (R_{\infty}, \phi)$ where $\Gamma$ is an arbitrary countable discrete group and $\phi \in (R_{\infty})_{\ast}$ is an arbitrary faithful state. This result is new in the case when $\phi$ is ergodic, that is, when $(R_{\infty})_{\phi} = \mathbf{C} 1$.
For minimal actions of compact second countable groups, combining Corollary \ref{cor-characterization} with \cite[Corollary 5.14]{Iz01}, we obtain:
\begin{letterapp}\label{app-minimal-action}
Let $M = R_{\infty}$ be the hyperfinite type ${\rm III}_1$ factor. Let $\mathbf G$ be any compact connected semisimple Lie group and $\alpha : \mathbf G \curvearrowright M$ any minimal action.
Then the inclusion $M^{\mathbf G} \subset M$ satisfies Kadison's property.
\end{letterapp}
\subsection*{Bicentralizers of tensor product factors}
It is straightforward to see that if $M$ and $N$ have trivial bicentralizer, then $M \mathbin{\overline{\otimes}} N$ also has trivial bicentralizer (see Proposition \ref{tensor-formula}). The following result provides a partial converse.
\begin{letterthm} \label{tensor_products_trivial}
Let $M$ be any $\sigma$-finite type ${\rm III}_1$ factor. Suppose that there exists a $\sigma$-finite factor $N$ such that $M \mathbin{\overline{\otimes}} N$ has trivial bicentralizer. Then $M \mathbin{\overline{\otimes}} R_\infty$ has trivial bicentralizer.
If $N$ is a type ${\rm III}_\lambda$ factor for $0 < \lambda < 1$, then $M \mathbin{\overline{\otimes}} R_\lambda$ has trivial bicentralizer. If $N$ is semifinite, then $M$ has trivial bicentralizer.
\end{letterthm}
We already mentioned the importance of the bicentralizer problem in the framework of Popa's deformation/rigidity theory for type ${\rm III}$ factors. In this respect, Theorem \ref{tensor_products_trivial} has a direct application to Unique Prime Factorization results. Indeed, we can remove all the assumptions on the unknown tensor product decomposition in \cite[Theorem B]{HI15} to obtain the following W$^{*}$-rigidity result.
\begin{letterapp}\label{UPF}
Let $m,n \geq 1$ be any integers. For each $1 \leq i \leq m$, let $M_i$ be a nonamenable factor in the class $\mathcal{C}_{\rm (AO)}$. For each $1 \leq j \leq n$, let $N_j$ be any non type ${\rm I}$ factor and suppose that
$$M= M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m = N_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} N_n.$$
Then there exists a surjection $\sigma : \{1,\dots,m\} \rightarrow \{1, \dots, n\}$, a family of type ${\rm I}$ factors $F_1, \dots, F_n$, and a unitary $u \in M \mathbin{\overline{\otimes}} F_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} F_n$ such that for all $1 \leq j \leq n$, we have
$$ u(F_j \mathbin{\overline{\otimes}} N_j)u^*=F_j \mathbin{\overline{\otimes}} \overline{\bigotimes}_{ i \in \sigma^{-1}(j)} M_i.$$
In particular, for all $1 \leq j \leq n$, the factor $N_j$ is stably isomorphic to $\overline{\bigotimes}_{ i \in \sigma^{-1}(j)} M_i$.
\end{letterapp}
\subsection*{Open question}
Let $N \subset M$ be any irreducible inclusion of factors with separable predual and with expectation $\mathord{\text{\rm E}}_N : M \rightarrow N$. Assume that $N$ is a type ${\rm III}_1$ factor. On the one hand, by Theorem \ref{thm: Connes' isomorphism and bicentralizer flow}, one can associate with the inclusion $N \subset M$ a canonical state preserving W$^*$-dynamical system that we called the {\em relative bicentralizer flow}
$$ (\mathord{\text{\rm B}}(N \subset M, \varphi), \beta^\varphi , (\varphi \circ \mathord{\text{\rm E}}_N) |_{ \mathord{\text{\rm B}}(N \subset M, \varphi)} ).$$
This state preserving W$^*$-dynamical system does not depend on the choice of the faithful state $\varphi \in N_*$ up to isomorphism. On the other hand, one can associate with the inclusion $N \subset M$ yet another canonical state preserving W$^*$-dynamical system that we call the \emph{relative flow of weights}
$$ ( (N_\psi)' \cap M, \theta^\psi, \mathord{\text{\rm E}}_N |_{(N_\psi)' \cap M} )$$
where $\psi$ is any dominant weight on $N$ and $\theta^\psi : \mathbf{R}^*_+ \curvearrowright (N_\psi)' \cap M$ is the flow given by $\theta^\psi_\lambda(x)=uxu^*$ for all $x \in (N_\psi)' \cap M$ and any unitary $u \in N$ such that $u\psi u^*=\lambda \psi, \; \lambda > 0$. This state preserving W$^*$-dynamical system does not depend on the choice of $\psi$ up to isomorphism and it is ergodic (see Subsection \ref{subsection:flow} for further details).
There is a striking analogy between the relative bicentralizer flow and the relative flow of weights. Moreover, Theorem \ref{thm-relative-bicentralizer} shows that they are indeed closely related. Therefore, one is naturally led to ask the following question.
\begin{question}
Are the relative bicentralizer flow and the relative flow of weights isomorphic? More precisely, does there exist an isomorphism
$$\pi : \mathord{\text{\rm B}}(N \subset M, \varphi) \rightarrow (N_\psi)' \cap M$$
such that $\theta^\psi =\pi \circ \beta^\varphi\circ \pi^{-1}$ and $\mathord{\text{\rm E}}_N(\pi(x))=\varphi(\mathord{\text{\rm E}}_N(x))1$ for all $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$?
\end{question}
Note that the above question is equivalent to the bicentralizer problem when $N=M$. Therefore, in this generality, it is out of reach at the present time. Nevertheless, in Proposition \ref{prop:open-question}, we provide examples for which the above question has a positive solution.
We also point out that since the relative flow of weights is always ergodic, it is reasonable to think that the same is true for the relative bicentralizer flow. In particular, when $N=M$, we have at the same time a good reason to believe that the bicentralizer flow is ergodic and we also have a good reason to believe that it must be trivial (see the discussion after Theorem \ref{thm: self-bicentralizing}). In other words, we strongly believe that the bicentralizer problem has a positive solution for all type ${\rm III}_1$ factors with separable predual.
\subsection*{Acknowledgments} Cyril Houdayer is grateful to Sorin Popa for thought-provoking discussions that took place in Orsay in the Spring of 2017, and later inspired the idea of the relative bicentralizer theorem for discrete inclusions (Theorem \ref{thm-relative-bicentralizer}). He also thanks him for mentioning the reference \cite{Po85} in connection with Theorem \ref{thm: hyperfinite subfactor III_1}.
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\section{Preliminaries}
\subsection{Standard form}
For any von Neumann algebra $M$, we denote by $(M, \mathord{\text{\rm L}}^2(M), J, \mathord{\text{\rm L}}^{2}(M)^{+})$ its \emph{standard form} \cite{Ha73}. Recall that $\mathord{\text{\rm L}}^2(M)$ is naturally endowed with the structure of a $M$-$M$-bimodule: we will simply write $x \xi y = x Jy^*J \xi$ for all $x, y \in M$ and all $\xi \in \mathord{\text{\rm L}}^2(M)$. For any positive functional $\varphi \in M_\ast^+$, there exists a unique vector $\xi \in \mathord{\text{\rm L}}^{2}(M)^{+}$ such that $\varphi(x)=\langle x \xi, \xi \rangle$ for every $x \in M$. We denote this vector by $\xi_\varphi \in \mathord{\text{\rm L}}^{2}(M)^{+}$. We then have $\|x\|_\varphi = \|x \xi_\varphi\|$ for all $x \in M$.
\subsection{Ultraproduct von Neumann algebras}
Let $M$ be any $\sigma$-finite von Neumann algebra, $I$ any directed set and $\omega$ any cofinal ultrafilter on $I$ (cofinal means that $\{ j \in I \mid j \geq i \} \in \omega$ for every $i \in I$). Let $A=(M,\| \cdot \|_\infty)^\omega$ be the ultraproduct Banach space of $M$ with respect to $\omega$. Then $A$ is naturally a $\mathord{\text{\rm C}}^*$-algebra but it is not a von Neumann algebra in general. Let $A^{**}$ be the bidual of $A$ which is a von Neumann algebra. Let $(M_*)^\omega$ be the ultraproduct Banach space of $M_*$. Then $(M_*)^\omega$ can be naturally identified with a closed subspace of $A^*$ via the embedding
\[ (\varphi_i)^\omega \mapsto \left( (x_i)^\omega \mapsto \lim_{i \rightarrow \omega} \varphi_i(x_i) \right) \]
The orthogonal of $(M_*)^\omega$ in $A^{**}$ defined by
\[ \mathfrak{J}=\{ x \in A^{**} \mid \forall \varphi \in (M_*)^\omega, \; \varphi(x)=0 \} \]
is a weak$^{*}$ closed ideal in the von Neumann algebra $A^{**}$ which means that the quotient $M^\omega_{GR} = A^{**}/\mathfrak{J}$ is a von Neumann algebra. It is called the \emph{Groh--Raynaud ultraproduct} of $M$ with respect to $\omega$. By construction, the predual of $M^\omega_{GR}$ is exactly $(M_*)^\omega$ and $M^\omega_{GR}$ contains the ultraproduct Banach space $A=(M,\| \cdot \|_\infty)^\omega$ as a dense $\mathord{\text{\rm C}}^*$-subalgebra. The $\ast$-homomorphism $M \to M^\omega_{GR} : x \mapsto x^\omega$ is not normal in general and so $M$ is not a von Neumann subalgebra of $M^\omega_{GR}$. The von Neumann algebra $M^\omega_{GR}$ is very large (not separable and not even $\sigma$-finite in general). The main interest in this ultraproduct comes from the fact that, as explained in \cite{AH12}, there is a natural identification $\mathord{\text{\rm L}}^2(M_{GR}^\omega)=\mathord{\text{\rm L}}^2(M)^\omega$.
Choose a faithful state $\varphi \in M_*$. Then we have $\varphi^\omega \in (M^\omega_{GR})_*^+$ but $\varphi^{\omega}$ is not faithful in general. Let $e$ be the support of $\varphi^{\omega}$ in $M^\omega_{GR}$. The projection $e$ does not depend on the choice of $\varphi$ and the corner $e(M^\omega_{GR})e$ coincides with the \emph{Ocneanu ultraproduct} of $M$ with respect to $\omega$ \cite{Oc85, AH12}. It is simply denoted by $M^{\omega}$. For every $x \in M^{\omega}$, we can find $(x_i)^{\omega} \in (M, \| \cdot \|_\infty)^{\omega}$ such that $x=(x_i)^{\omega}e=e(x_i)^{\omega}$, and in that case, when no confusion is possible, we will abuse notation and write $x=(x_i)^{\omega}$. By construction, we have $(M^\omega)_*=e(M_*)^\omega e$ and $\mathord{\text{\rm L}}^2(M^\omega)=e(\mathord{\text{\rm L}}^2(M)^\omega) e$.
We will need the following lemma which allows us to lift matrix units from the Ocneanu ultrapower of a factor to the factor itself.
\begin{lem} \label{lift_matrix}
Let $M$ be any $\sigma$-finite factor. Let $\theta : F \rightarrow M^{\omega}$ be any $*$-homomorphism where $F$ is a finite dimensional factor and $\omega \in \beta (\mathbf{N}) \setminus \mathbf{N}$ is any nonprincipal ultrafilter. Then there exists a sequence of $*$-homomorphisms $\theta_n : F \rightarrow M$ such that $\theta=(\theta_n)^{\omega}$.
Moreover, if $\mathord{\text{\rm E}} : M^\omega \rightarrow \theta(F)' \cap M^\omega$ is a conditional expectation, then for any faithful state $\varphi \in M_*$ and any $x=(x_n)^\omega \in M^\omega$, we have $\varphi^\omega \circ \mathord{\text{\rm E}}=(\varphi \circ \mathord{\text{\rm E}}_n)^\omega$ and $\mathord{\text{\rm E}}(x)=(\mathord{\text{\rm E}}_n(x_n))^\omega$, where $\mathord{\text{\rm E}}_n : M \rightarrow \theta_n(F)' \cap M$ is the unique conditional expectation such that $\mathord{\text{\rm E}}_n \circ \theta_n = \mathord{\text{\rm E}} \circ \theta$.
\end{lem}
\begin{proof}
The first part of the lemma is proved in \cite[Proposition 1.1.3]{Co75a} for the asymptotic centralizer $M_\omega$ but the same proof also works for the ultrapower $M^{\omega}$.
For the second part of the lemma, put $\psi(x)1 = (\mathord{\text{\rm E}} \circ \theta)(x)$ for every $x \in F$. Then $\psi$ is a state on $F$ and the explicit formula for the conditional expectation $\mathord{\text{\rm E}} : M^\omega \rightarrow \theta(F)' \cap M^\omega$ is given by
$$\forall x \in M^\omega, \quad \mathord{\text{\rm E}}(x)=\sum_{1\leq i,j,k \leq n} \psi(e_{ij})\theta(e_{ki})x\theta(e_{jk})$$
where $(e_{ij})_{1 \leq i,j \leq n}$ is a matrix unit for $F$. Then we also have
$$\forall x \in M^\omega, \quad \mathord{\text{\rm E}}_n(x)=\sum_{1\leq i,j,k \leq n} \psi(e_{ij})\theta_n(e_{ki})x\theta_n(e_{jk}).$$
This finishes the proof.
\end{proof}
The following well-known fact about ultralimits will be used repeatedly.
\begin{lem}\label{lem: ultralimit test}
Let $X$ be any Hausdorff space and $(x_n)_{n=1}^{\infty}$ any sequence in $X$. If there exists $x\in X$ such that $ \lim_{n\to \omega}x_n=x$ for every $\omega\in \beta (\mathbf{N})\setminus \mathbf{N}$, then $ \lim_{n\to \infty}x_n=x$.
\end{lem}
\subsection{Iterated ultraproducts}
We shall consider the iterated ultraproduct of von Neumann algebras. We will see that this procedure helps us to show that some sequence in a von Neumann algebra $M$, which defines an element in some ultrapower $M^{\omega}$, actually converges to an element, which is independent of the choice of an ultrafilter $\omega$.
Let $I,J$ be any directed sets and $\mathcal{U},\mathcal{V}$ any cofinal ultrafilters on $I$ and $J$, respectively. Then the {\it product ultrafilter}, denoted by $\mathcal{U}\otimes \mathcal{V}$, is a cofinal ultrafilter on $I\times J$ (with the partial ordering $(i,j)\le (i',j')$ if and only if $i\le i'$ and $j\le j'$) given by
\[\mathcal{U}\otimes \mathcal{V}=\{A\subset I\times J \mid \{i\in I \mid \{j\in J \mid (i,j)\in A\}\in \mathcal{V}\}\in \mathcal{U}\}.\]
\begin{prop}\label{prop: iterated ultralimit}
If $(x_{i,j})_{(i,j)\in I\times J}$ is a doubly indexed net in a compact Hausdorff space $X$, then
\[\lim_{(i,j)\to \mathcal{U}\otimes \mathcal{V}}x_{i,j}=\lim_{i\to \mathcal{U}}\lim_{j\to \mathcal{V}}x_{i,j}.\]
\end{prop}
\begin{proof}
Put $x=\lim_{(i,j)\to \mathcal{U}\otimes \mathcal{V}}x_{i,j}$ and $x_i=\lim_{j\to \mathcal{V}}x_{i,j}\ (i\in I)$. Let $W$ be an open neighborhood of $x$ in $X$. Since a compact Hausdorff space is regular, there exists an open neighborhood $W_1$ of $x$ such that $x\in W_1\subset \overline{W_1}\subset W$. Then $\{(i,j)\in I\times J \mid x_{i,j}\in W_1\}\in \mathcal{U}\otimes \mathcal{V}$, whence $I_0=\{i\in I \mid \{j\in J \mid x_{i,j}\in W_1\}\in \mathcal{V}\}\in \mathcal{U}$ holds. Let $i\in I_0$. Then $B=\{j\in J \mid x_{i,j}\in W_1\}\in \mathcal{V}$. If $V$ is any open neighborhood of $x_i$, then $B'=\{j\in J \mid x_{i,j}\in V\}\in \mathcal{V}$, whence $B\cap B'\in \mathcal{V}$ holds. In particular, we can take $j\in B\cap B'$. Then $x_{i,j}\in V\cap W_1\neq \emptyset$. Since $V$ is arbitrary, this shows that $x_i\in \overline{W_1}\subset W$. Therefore $\mathcal{U}\ni I_0\subset \{i\in I \mid x_i\in W\}$, which shows that $\{i\in I \mid x_i\in W\}\in \mathcal{U}$. Since $W$ is arbitrary, we have $\displaystyle \lim_{i\to \mathcal{U}}x_i=x$.
\end{proof}
As a consequence of Proposition \ref{prop: iterated ultralimit}, we see that for any Banach space $E$, the natural isomorphism
$$ \ell^{\infty}(I \times J, E) \ni (x_{i,j})_{(i,j) \in I \times J} \mapsto ((x_{i,j})_{j \in J})_{i \in I} \in \ell^{\infty}(I , \ell^{\infty}(J, E))$$
induces an isomorphism of the ultrapowers
$$ E^{\mathcal{U} \otimes \mathcal{V}} \ni (x_{i,j})^{\mathcal{U} \otimes \mathcal{V}} \mapsto ((x_{i,j})^{\mathcal{V}})^{\mathcal{U}} \in (E^{\mathcal{V}})^{\mathcal{U}}.$$
If we apply this to $E=M_*$ where $M$ is a von Neumann algebra, we obtain the following proposition which extends \cite[Proposition 2.1]{CP12} on iterated ultrapowers of ${\rm II}_1$ factors to arbitrary $\sigma$-finite von Neumann algebras. We leave the details to the reader.
\begin{prop} \label{prop: double ultrapower}
Let $M$ be any $\sigma$-finite von Neumann algebra. There exists a natural isomorphism of the Groh--Raynaud ultrapowers
$$ \pi_{\mathrm{GR}} : M_\mathrm{GR}^{\mathcal{U} \otimes \mathcal{V}} \rightarrow (M_\mathrm{GR}^{\mathcal{V}})_\mathrm{GR}^{\mathcal{U}}$$
characterized by
$$ \pi_{\mathrm{GR}}((x_{i,j})^{\mathcal{U} \otimes \mathcal{V}})=((x_{i,j})^{\mathcal{V}})^{\mathcal{U}} \; \text{ for all } \; (x_{i,j})_{(i,j) \in I \times J} \in \ell^\infty(I \times J, M).$$
Its predual map is the isomorphism
$$ (\pi_{\mathrm{GR}})_* : (M_*)^{\mathcal{U} \otimes \mathcal{V}} \ni (\varphi_{i,j})^{\mathcal{U} \otimes \mathcal{V}} \mapsto ((\varphi_{i,j})^{\mathcal{V}})^{\mathcal{U}} \in ((M_*)^{\mathcal{V}})^{\mathcal{U}}.$$
In particular, $\pi_{\mathrm{GR}}$ restricts to an isomorphism between the Ocneanu corners
$$ \pi : M^{\mathcal{U} \otimes \mathcal{V}} \rightarrow (M^{\mathcal{V}})^{\mathcal{U}}.$$
\end{prop}
Recall that if $N \subset M$ is a von Neumann subalgebra with faithful normal conditional expectation $\mathord{\text{\rm E}}^{M}_{N} : M \to N$, then we have a natural embedding $N^{\mathcal{U}} \subset M^{\mathcal{U}}$ with faithful normal conditional expectation $\mathord{\text{\rm E}}^{M^{\mathcal{U}}}_{N^{\mathcal{U}}}=\left(\mathord{\text{\rm E}}^{M}_N \right)^{\mathcal{U}} :M^{\mathcal{U}} \to N^{\mathcal{U}}$ and we have a commuting square
\begin{align*}
\begin{array}{ccc}
N & \subset & M \\
\rotatebox{90}{$\supset$} & & \rotatebox{90}{$\supset$} \\
N^{\mathcal{U}} & \subset & M^{\mathcal{U}}\\
\end{array}
\end{align*}
where the conditional expectations satisfy
$$\mathord{\text{\rm E}}^{M^{\mathcal{U}}}_M \circ \mathord{\text{\rm E}}^{M^{\mathcal{U}}}_{N^{\mathcal{U}}} = \mathord{\text{\rm E}}^{M^{\mathcal{U}}}_{N^{\mathcal{U}}} \circ \mathord{\text{\rm E}}^{M^{\mathcal{U}}}_M =\mathord{\text{\rm E}}^{N^{\mathcal{U}}}_N \circ \mathord{\text{\rm E}}^{M^{\mathcal{U}}}_{N^{\mathcal{U}}} = \mathord{\text{\rm E}}^{M}_N \circ \mathord{\text{\rm E}}^{M^{\mathcal{U}}}_M.$$
By applying this to the inclusion $M \subset M^{\mathcal{V}}$ with the canonical faithful normal conditional expectation $\mathord{\text{\rm E}}^{M^{\mathcal{V}}}_M : M^{\mathcal{V}} \rightarrow M$, we obtain the following result.
\begin{prop} \label{prop: commuting square}
Let $M$ be any $\sigma$-finite von Neumann algebra. Then we have a commuting square
\begin{align*}
\begin{array}{cccc}
M & \subset & M^{\mathcal{V}} &\\
\rotatebox{90}{$\supset$} & & \rotatebox{90}{$\supset$}& \\
M^{\mathcal{U}} & \subset & (M^{\mathcal{V}})^{\mathcal{U}}& =M^{\mathcal{U} \otimes \mathcal{V}}\\
\end{array}
\end{align*}
where the canonical faithful normal conditional expectations satisfy
$$\mathord{\text{\rm E}}^{M^{\mathcal{U} \otimes \mathcal{V}}}_M =\mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{V}}} \circ \mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{U}}} =\mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{U}}} \circ \mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{V}}}=
\mathord{\text{\rm E}}^{M^{\mathcal{U}}}_M \circ \mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{U}}} =\mathord{\text{\rm E}}^{M^{\mathcal{V}}}_M \circ \mathord{\text{\rm E}}^{(M^{\mathcal{V}})^{\mathcal{U}}}_{M^{\mathcal{V}}}.$$
In particular, we have $M^{\mathcal{U}} \cap M^{\mathcal{V}}=M$.
\end{prop}
\subsection{The relative flow of weights}\label{subsection:flow}
Let $N \subset M$ be any irreducible inclusion of factors with separable predual and with expectation $\mathord{\text{\rm E}}_N : M \rightarrow N$. Assume that $N$ is a type ${\rm III}_1$ factor. By using \cite[Theorem ${\rm XII}$.1.1]{Ta03}, we can identify the inclusions
$$\left( N \subset M \right) = \left ( N_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \subset M_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \right)$$
where $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$ is a trace-scaling action ($\tau \circ \theta_\lambda = \lambda^{-1}\tau$ for every $\lambda > 0$) that leaves $N_{\psi} \subset M_{\psi}$ globally invariant. We denote by $(v_{\lambda})_{\lambda > 0}$ the canonical unitaries in $N$ that implement the trace-scaling action $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$.
The {\em relative flow of weights} $\theta^\psi : \mathbf{R}^*_+ \curvearrowright (N_\psi)' \cap M$ is defined by $\theta_\lambda^\psi(x) = v_\lambda x v_\lambda^{*}$ for every $x \in (N_\psi)' \cap M$ and every $\lambda > 0$. By Connes--Takesaki relative commutant theorem \cite[Chapter II, Theorem 5.1]{CT76}, ${\mathord{\text{\rm E}}_N}|_{(N_\psi)' \cap M}$ defines a faithful normal state on $(N_\psi)' \cap M$ that is invariant under the flow $\theta^\psi$.
We observe that the relative flow of weights $\theta^\psi$ does not depend on the choice of the dominant weight $\psi$ on $N$ since all dominants weights on $N$ are unitarily conjugate\cite[Chapter II, Theorem 1.1]{CT76}. We moreover observe that the relative flow of weights $\theta^\psi$ is ergodic. Indeed, if $x \in (N_\psi)' \cap M$ is invariant under $\theta^\psi$, then $x \in (N_\psi \vee \{v_\lambda \mid \lambda > 0\})' \cap M = N' \cap M$ and so $x \in \mathbf{C} 1$.
\subsection{Kadison's property}
We introduce {\em Kadison's property} for arbitrary inclusions of von Neumann algebras with expectation.
\begin{df}
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation. We say that $N \subset M$ satisfies {\em Kadison's property} if there exists an abelian von Neumann subalgebra with expectation $A \subset N \subset M$ that is maximal abelian in $M$.
\end{df}
For any inclusion $N \subset M$ that satisfies Kadison's property, we have $N' \cap M \subset A' \cap M = A \subset N$ and so there is a unique faithful normal conditional expectation $\mathord{\text{\rm E}}_N : M \to N$ (see \cite[Th\'eor\`eme 1.5.5]{Co72}). In particular, the inclusion of continuous cores $\mathord{\text{\rm c}}(N) \subset \mathord{\text{\rm c}}(M)$ is canonical. We next prove the following useful observation.
\begin{prop}\label{prop-masa}
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation $\mathord{\text{\rm E}}_{N} : M \to N$. Assume that $N \subset M$ satisfies Kadison's property. Put $M^\infty = M \mathbin{\overline{\otimes}} \mathbf B(\mathord{\text{\rm L}}^2(\mathbf{R}))$ and regard $M^\infty = \mathord{\text{\rm c}}(M) \rtimes_\theta \mathbf{R}^*_+$ where $\theta : \mathbf{R}^*_+ \curvearrowright \mathord{\text{\rm c}}(M)$ is the dual trace-scaling action. Then we have
$$\mathord{\text{\rm c}}(N)' \cap M^\infty = \mathcal Z(\mathord{\text{\rm c}}(N)).$$
In particular, if $N$ is a type ${\rm III_{1}}$ factor, then so is $M$.
\end{prop}
\begin{proof}
Let $A \subset N \subset M$ be an abelian von Neumann subalgebra with expectation that is maximal abelian in $M$. Choose a faithful state $\varphi \in M_{\ast}$ such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$ and such that $A \subset N_{\varphi}$. Regard $\left( \mathord{\text{\rm c}}(N) \subset \mathord{\text{\rm c}}(M) \right ) = \left( \mathord{\text{\rm c}}_{\varphi}(N) \subset \mathord{\text{\rm c}}_{\varphi}(M) \right )$. Put $\mathord{\text{\rm c}}(A) = \mathord{\text{\rm c}}_{\varphi}(A) = A \mathbin{\overline{\otimes}} \mathord{\text{\rm L}}(\mathbf{R})$.
Since $A \subset M$ is maximal abelian and since $\mathord{\text{\rm L}}(\mathbf{R}) \subset \mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))$ is maximal abelian, $\mathord{\text{\rm c}}(A) = A \mathbin{\overline{\otimes}} \mathord{\text{\rm L}}(\mathbf{R}) \subset M \mathbin{\overline{\otimes}} \mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R})) = M^\infty$ is maximal abelian. Then we have $\mathord{\text{\rm c}}(N)' \cap M^\infty \subset \mathord{\text{\rm c}}(A)' \cap M^\infty = \mathord{\text{\rm c}}(A) \subset \mathord{\text{\rm c}}(N)$ and so $\mathord{\text{\rm c}}(N)' \cap M^\infty = \mathcal Z(\mathord{\text{\rm c}}(N))$.
\end{proof}
\section{Structure of relative bicentralizers}
Following \cite{Co80, Ha85, Ma03}, we introduce the following terminology that we will use throughout.
\begin{df}
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation. Let $\varphi \in N_\ast$ be any faithful state. The {\em relative bicentralizer} $\mathord{\text{\rm B}}(N \subset M, \varphi)$ of the inclusion $N \subset M$ with respect to $\varphi$ is defined by
$$\mathord{\text{\rm B}}(N \subset M, \varphi) = \left\{ x \in M \mid x a_{n} - a_{n} x \to 0 \text{ strongly}, \forall (a_{n})_{n} \in \mathord{\text{\rm AC}}(N, \varphi)\right\}$$
where $$\mathord{\text{\rm AC}}(N, \varphi) = \left \{ (a_{n})_{n} \in \ell^{\infty}(\mathbf{N}, N) \mid \lim_{n} \|a_{n} \varphi - \varphi a_{n}\| = 0\right\}$$ is the {\em asymptotic centralizer} of $\varphi$.
\end{df}
Observe that we always have $N' \cap M \subset \mathord{\text{\rm B}}(N\subset M, \varphi) \subset (N_{\varphi})' \cap M$. When $N = M$, we simply have that $\mathord{\text{\rm B}}(N \subset M,\varphi) = \mathord{\text{\rm B}}(M, \varphi)$ is the usual bicentralizer.
Let us point out that in the definition of $\mathord{\text{\rm B}}(N \subset M, \varphi)$, it is enough to consider sequences in $\mathord{\text{\rm AC}}(N, \varphi)$ consisting of unitaries. Indeed, since $\mathord{\text{\rm AC}}(N, \varphi)$ is a $\mathord{\text{\rm C}}^{*}$-algebra, every element is a linear combination of at most four unitaries.
\begin{rem}
We note that $\mathord{\text{\rm B}}(N\subset M, \varphi)$ is always with (canonical) expectation in $M$. Indeed, extend $\varphi$ to a faithful normal state on $M$ by using any faithful normal conditional expectation $\mathord{\text{\rm E}}_N : M \rightarrow N$. Then $\sigma^{\varphi}$ leaves $\mathord{\text{\rm AC}}(N,\varphi)$ globally invariant and so $\mathord{\text{\rm B}}(N \subset M, \varphi)$ is also globally invariant under $\sigma^{\varphi}$. Therefore, there exists a unique $\varphi$-preserving faithful normal conditional expectation $\mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)} : M \rightarrow \mathord{\text{\rm B}}(N \subset M, \varphi)$. Since $N' \cap M \subset \mathord{\text{\rm B}}(N \subset M, \varphi)$ and since the $\varphi$-preserving faithful normal conditional expectation $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M} : M \rightarrow N' \cap M$ does not depend on the choice of $\mathord{\text{\rm E}}_N$, it follows $\mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)}$ does not depend either on the choice of $\mathord{\text{\rm E}}_N$ and is characterized by
$$ \mathord{\text{\rm E}}^{\varphi}_{N' \cap M} \circ \mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)} = \mathord{\text{\rm E}}^{\varphi}_{N' \cap M}.$$
\end{rem}
The relative bicentralizer $\mathord{\text{\rm B}}(N \subset M, \varphi)$ has the following ultraproduct interpretation.
\begin{prop} \label{ultraproduct_bicentralizer}
Let $N \subset M$ be any inclusion of $\sigma$-finite von Neumann algebras with expectation. Let $\varphi \in N_*$ be any faithful state. For any nonprincipal ultrafilter $\omega \in \beta (\mathbf{N}) \setminus \mathbf{N}$, we have
$$ \mathord{\text{\rm B}}(N \subset M,\varphi)=(N^{\omega}_{\varphi^{\omega}})' \cap M$$
and
$$ (N^{\omega}_{\varphi^{\omega}})' \cap M^{\omega} \subset \mathord{\text{\rm B}}(N \subset M,\varphi)^{\omega}.$$
\end{prop}
\begin{proof}
Extend $\varphi$ to $M$ by using any faithful normal conditional expectation from $M$ to $N$. Then for any $x \in M$, we have $x \in (N^{\omega}_{\varphi^{\omega}})' \cap M$ if and only if for every $\varepsilon > 0$, there exists $\delta > 0$ such that for every $u \in \mathcal{U}(N)$ we have
$$\|u \varphi-\varphi u\| \leq \delta \quad \Rightarrow \quad \| ux-xu \|_\varphi \leq \varepsilon. $$
This is clearly equivalent to $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$.
For the second part of the proposition, it is enough to show that for any $x \in M$ such that $\|x\|_{\infty} \leq 1$ and $\mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)}(x)=0$ and any $\delta > 0$, we can find $u \in \mathcal{U}(N)$ such that $\|u \varphi-\varphi u\| \leq \delta$ and $\|uxu^*-x\|_\varphi \geq \|x \|_\varphi$. Let $y$ be the element of minimal $\| \cdot \|_\varphi$-norm in
$$ C_\delta(x) = \overline{\rm co}^{w} \{ uxu^{*} \mid u \in \mathcal{U}(N), \| u \varphi - \varphi u \| < \delta \}.$$
Let us show that $y \in \mathord{\text{\rm B}}(N \subset M, \varphi)$. Take a sequence $v_n \in \mathcal{U}(N)$ such that $\lim_n \|v_n \varphi-\varphi v_n\|=0$. Put $\varepsilon_n= \frac12 \max(\|v_n \varphi-\varphi v_n\|, \delta)$ for every $n \in \mathbf{N}$. Take a sequence $z_n \in \mathrm{co}\{ uxu^* \mid u \in \mathcal{U}(N), \| u \varphi - \varphi u \| < \delta-\varepsilon_n \}$ such that $\|z_n-y\|_\varphi \to 0$. Then for every $n \in \mathbf{N}$, we have $v_nz_nv_n^* \in C_\delta(x)$. Note that $\|v_nz_nv_n^*-v_nyv_n^*\|_\varphi \to 0$ and $\|v_n yv_n^*\|_\varphi \to \|y\|_\varphi$, thus $\|v_nz_nv_n^*\|_\varphi \to \|y \|_\varphi$. Since $y$ is the element of minimal norm in $C_\delta(x)$, this forces $\|v_nz_nv_n^*-y \|_\varphi \to 0$ and therefore $\|v_nyv_n^*-y\|_\varphi \to 0$. This shows that $y \in \mathord{\text{\rm B}}(N \subset M, \varphi)$. Therefore, we have
$$0=\Re(\varphi(y^*x)) \in \overline{\rm co} \{\Re(\varphi(ux^*u^*x)) \mid u \in \mathcal{U}(N), \; \| u \varphi - \varphi u \| < \delta \}.$$
This implies that there exists $u \in \mathcal{U}(N)$ with $ \| u \varphi - \varphi u \| < \delta $ such that $\Re(\varphi(ux^*u^*x)) \leq \delta/2$. Thus we obtain
$$\|uxu^*-x\|_\varphi^2 = \|uxu^*\|_\varphi^2+ \|x\|_\varphi^2 - 2\Re(\varphi(ux^*u^*x)) \geq 2\|x\|_\varphi^2-2\delta.$$
Hence if we suppose that $2 \delta \leq \|x\|_\varphi^2$ (which we can always do, without loss of generality), we obtain $\|uxu^*-x\|_\varphi \geq \|x\|_\varphi$.
\end{proof}
To prove Theorem \ref{thm: Connes' isomorphism and bicentralizer flow}, we use the ultraproduct technology. The crucial ingredient is Connes--St\o rmer transitivity theorem \cite{CS76} which shows that the ultrapower of a type ${\rm III}_1$ factor has a \emph{strictly homogeneous state space} \cite[Theorem 4.20]{AH12}. Namely, for any pair of faithful normal states $\varphi, \psi$ on $M^{\omega}$, we can find $u \in \mathcal U(M^{\omega})$ such that $u\varphi u^{*}=\psi$. To construct the bicentralizer flow, we will also need the following lemma.
\begin{lem}\label{lem: family of partial isometries}
Let $M$ be any nontrivial factor with strictly homogeneous state space. Let $\varphi \in M_{\ast}$ be any faithful state. Then $M_\varphi$ is a type ${\rm II}_1$ factor and for any $\lambda > 0$, we can find a finite family $v_1, \dots, v_n$ of partial isometries in $M$ such that $v_k \varphi = \lambda \varphi v_k$ for all $k=1,\dots, n$ and $\sum_{k=1}^{n} v_k^{*}v_k=1$. If $\lambda \geq 1$, then we can take $n=1$.
\end{lem}
\begin{proof}
By \cite[Proposition 4.24]{AH12}, we know that $M_\varphi$ is a type ${\rm II}_1$ factor and by the proof of \cite[Proposition 4.22]{AH12}, we know that if $p,q \in M_{\varphi}$ are two nonzero projections, then we can find $v \in M$ such that $v^{*}v=p$, $vv^{*}=q$ and $v\varphi=\frac{\varphi(p)}{\varphi(q)}\varphi v$. If $\lambda \geq 1$, we can take $p=1$ and $q \in M_{\varphi}$ such that $\varphi(q)=\frac{1}{\lambda}$ and we obtain an isometry $v \in M$ such that $v\varphi=\lambda \varphi v$. If $\lambda \leq 1$, choose $n \geq 1$ such that $\frac1n \leq \lambda$. Then we can find a finite partition of unity $p_1, \dots, p_n$ in $M_{\varphi}$ and some projections $q_1, \dots, q_n$ (not necessarily orthogonal) such that $\varphi(p_k)=\lambda\varphi(q_k)$. Then we can find a family $v_k \in M$ such that $v_k^{*}v_k=p_k$, $v_kv_k^{*}=q
_k$ and $v_k \varphi = \frac{\varphi(p_k)}{\varphi(q_k)}\varphi v_k=\lambda \varphi v_k$ as we wanted.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm: Connes' isomorphism and bicentralizer flow}]
$(\rm i)$ Let $\omega_1,\omega_2\in \beta (\mathbf{N})\setminus \mathbf{N}$ be any nonprincipal ultrafilters. Let $u \in \mathcal U(N^{\omega_1})$ (resp.\ $v \in \mathcal U(N^{\omega_2})$) such that $u\varphi^{\omega_1}u^*=\psi^{\omega_1}$ (resp.\ $v\varphi^{\omega_2}v^*=\psi^{\omega_2}$). Then, inside $M^{\omega_2 \otimes \omega_1}$, we have $v^*u \in N^{\omega_2 \otimes \omega_1}_{\varphi^{\omega_2 \otimes \omega_1}}$. For every $x \in \mathord{\text{\rm B}}(N \subset M,\varphi)$, we have $v^{*}ux=xv^{*}u$ which means that $uxu^{*}=vxv^{*}$. Since $uxu^{*} \in M^{\omega_1}$ and $vxv^{*} \in M^{\omega_2}$, Proposition \ref{prop: commuting square} shows that $uxu^{*}=vxv^{*}$ is an element of $M$. Thus, we have shown that for every $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$, there exists an element $\beta_{\psi,\varphi}(x) \in M$ given by $\beta_{\psi, \varphi}(x)=uxu^{*}$ where $u \in N^{\omega}$ is any unitary such that $u\varphi^{\omega}u^{*}=\psi^{\omega}$ and $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ is any nonprincipal ultrafilter. In particular, if $w$ is a unitary in $N^{\omega}_{\psi^{\omega}}$, we can replace $u$ by $wu$, so that we have $\beta_{\psi, \varphi}(x)=wuxu^{*}w^{*}=w\beta_{\psi,\varphi}(x)w^{*}$. This shows that $\beta_{\psi, \varphi}(x) \in \mathord{\text{\rm B}}(N \subset M,\psi)$. Now, if $(a_n)_{n \in \mathbf{N}}$ is a uniformly bounded sequence in $N$ such that $\|a_n \varphi-\psi a_n \| \to 0$, then it defines an element $a=(a_n)^{\omega} \in M^{\omega}$ such that $a\varphi^{\omega}=\psi^{\omega}a$ and so $u^{*}a \in N^{\omega}_{\varphi^{\omega}}$. This shows that $u^{*}ax=xu^{*}a$, that is, $ax=uxu^{*}a=\beta_{\psi, \varphi}(x)a$. Since the nonprincipal ultrafilter $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ is arbitrary, by Lemma \ref{lem: ultralimit test}, we conclude that $a_n x - \beta_{\psi,\varphi}(x)a_n \to 0$ $*$-strongly as $n \to \infty$. It is straightforward to check that $\beta_{\psi, \varphi}$ is a $*$-homomorphism and that $\beta_{\varphi_3, \varphi_2} \circ \beta_{\varphi_2, \varphi_1} = \beta_{\varphi_3, \varphi_1}$ for every faithful state $\varphi_i \in N_*$, $ i \in \{1, 2, 3\}$. This shows in particular that $\beta_{\psi, \varphi} : \mathord{\text{\rm B}}(N \subset M, \varphi) \rightarrow \mathord{\text{\rm B}}(N \subset M, \psi) $ is an isomorphism with inverse $\beta_{\varphi, \psi}$. Let $\mathord{\text{\rm E}}_N: M \rightarrow N$ be any faithful normal conditional expectation and use it to extend $\varphi$ and $\psi$ to faithful normal states on $M$. Then we clearly have $\psi \circ \beta_{\psi,\varphi} = \varphi$. Since $N' \cap M$ is clearly fixed by $\beta_{\psi,\varphi}$, this implies that
$$\mathord{\text{\rm E}}_{N' \cap M}^{\psi} \circ \beta_{\psi, \varphi} = \mathord{\text{\rm E}}_{N' \cap M}^{\varphi}.$$
$(\rm ii)$ Let $\omega_1,\omega_2 \in \beta(\mathbf{N}) \setminus \mathbf{N}$ be any nonprincipal ultrafilters and $\lambda>0$. By Lemma \ref{lem: family of partial isometries}, there exists a family $v_1,\dots, v_n$ of partial isometries in $N^{\omega_1}$ such that $v_k\varphi^{\omega_1}=\lambda \varphi^{\omega_1}v_k$ for all $k \in \{1,\dots,n\}$ and $\sum_{k=1}^{n}v_kv_k^{*}=1$. Similarly, let $w_1,\dots, w_m$ a family of partial isometries in $N^{\omega_2}$ such that $w_l\varphi^{\omega_2}=\lambda \varphi^{\omega_2}w_l$ for all $l \in \{1,\dots,m \}$ and $\sum_{l=1}^{m}w_lw_l^{*}=1$. Then inside $M^{\omega_2 \otimes \omega_1}$, we have $v_k^{*}w_l \in N^{\omega_2 \otimes \omega_1}_{\varphi^{\omega_2 \otimes \omega_2}}$ for all $k\in\{1, \dots, n\}$ and all $l \in \{1, \dots, m\}$. Then for all $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$, we have
$$ xv_k^{*}w_l=v_k^{*}w_lx$$
and so
$$ v_kxv_k^{*}(w_lw_l^{*})=(v_kv_k^{*})w_lxw_l^{*}.$$
By summing over $k$ and $l$, we obtain
\begin{equation}\label{eq:flow}
\sum_{k=1}^{n}v_kxv_k^{*}=\sum_{l=1}^{m}w_l x w_l^{*}.
\end{equation}
But the left hand side of \eqref{eq:flow} lies in $M^{\omega_1}$ and the right hand side of \eqref{eq:flow} lies in $M^{\omega_2}$. Then they are both in $M$ by Proposition \ref{prop: commuting square} and the element $\beta_\lambda^\varphi(x)=\sum_{k=1}^{n}v_kxv_k^{*} \in M$ is independent of the choice of the nonprincipal ultrafilter $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ and the family $v_1, \dots, v_n \in N^{\omega}$ as above. In particular, if $u$ is a unitary in $M^{\omega}_{\varphi^{\omega}}$, then we can replace $v_k$ by $uv_k$ for all $k\in \{1, \dots, n\}$ and we obtain $\beta_\lambda^\varphi(x)=u\beta_\lambda^\varphi(x)u^*$. This shows that $\beta_\lambda^\varphi(x) \in \mathord{\text{\rm B}}(N \subset M,\varphi)$. Let $(a_n)_{n \in \mathbf{N}}$ be a uniformly bounded sequence in $N$ such that $\lim_n \|a_n\varphi-\lambda \varphi a_n \|=0$. Then it defines an element $a=(a_n)^\omega \in N^\omega$ such that $a\varphi^\omega=\lambda\varphi^\omega a$. Then we have $v_k^*a \in N^\omega_{\varphi^\omega}$ for all $k \in \{1, \dots, n\}$. Thus, for all $x \in \mathord{\text{\rm B}}(N\subset M, \varphi)$, we have $ v_k^*ax=xv_k^*a$ and so
$$ ax=\sum_{k=1}^n v_kv_k^*ax=\sum_{k=1}^n v_kxv_k^*a=\beta_\lambda^\varphi(x)a.$$
Since the nonprincipal ultrafilter $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ is arbitrary, by Lemma \ref{lem: ultralimit test}, we conclude that $a_n x - \beta_\lambda^\varphi(x)a_n \to 0$ $\ast$-strongly as $n \to \infty$.
It is straightforward to check that $\beta_\lambda^\varphi$ is a unital $*$-homomorphism for all $\lambda > 0$ and that $\beta_\lambda^\varphi \circ \beta_\mu^\varphi = \beta_{\lambda \mu}^\varphi$ for all $\lambda, \mu > 0$. This shows that $\beta^\varphi : \lambda \mapsto \beta_\lambda^\varphi$ is a one-parameter group of automorphisms of $\mathord{\text{\rm B}}(N \subset M, \varphi)$. Also, one checks easily that $\beta^{\psi}_\lambda \circ \beta_{\psi, \varphi}=\beta_{\psi, \varphi} \circ \beta^{\varphi}_\lambda$ for all faithful normal states $\varphi, \psi \in N_*$. Extend $\varphi$ to a state on $M$ by using any faithful normal conditional expectation from $M$ to $N$. Then $\beta^{\varphi}$ is $\varphi$-preserving. Indeed, for all $\lambda > 0$, we have
$$ \varphi(\beta_\lambda^{\varphi}(x))=\sum_{k = 1}^{n} \varphi^{\omega}(v_k x v_k^{*})=\sum_{k = 1}^{n} \lambda^{-1}\varphi^{\omega}(xv_k^{*}v_k )=\sum_{k = 1}^{n} \lambda^{-1}\varphi(x)\varphi^{\omega}(v_k^{*}v_k )$$
because $x$ commutes with the factor $N^{\varphi^{\omega}}$. Since $\varphi^{\omega}(v_k^{*}v_k )=\lambda \varphi^{\omega}(v_kv_k^{*} )$, we obtain
$$ \varphi(\beta_\lambda^{\varphi}(x))=\sum_{k = 1}^{n}\varphi(x)\varphi^{\omega}(v_kv_k^{*} )=\varphi(x).$$
Thus $\beta^{\varphi}_\lambda$ is $\varphi$-preserving and since $\beta^{\varphi}$ clearly fixes $N' \cap M$, we obtain
$$ \mathord{\text{\rm E}}_{N' \cap M}^{\varphi} \circ \beta_{\lambda}^{\varphi} =\mathord{\text{\rm E}}_{N' \cap M}^{\varphi}.$$
At this point, we have proved all items $(\rm i),(\rm ii), (\rm iii)$ and $(\rm iv)$. It only remains to check that $\beta^\varphi$ is indeed a flow in the sense that it is continuous with respect to the $u$-topology on $\mathord{\text{\rm B}}(N \subset M, \varphi)$. Take a sequence $\lambda_n \in \mathbf{R}^{*}_+$ such that $\lambda_n \to 1$ and $\lambda_n \leq 1$. We have to show that $\beta^{\varphi}_{\lambda_n} \to \text{\rm id}_{\mathord{\text{\rm B}}(N\subset M, \varphi)}$ with respect to the $u$-topology. Since $\beta^{\varphi}$ is $\varphi$-preserving, it is enough to show that $\beta^{\varphi}_{\lambda_n}(x) \to x$ strongly for all $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$. Let $\omega_1 \in \beta(\mathbf{N}) \setminus \mathbf{N}$ be any nonprincipal ultrafilter and pick, for every $n \in \mathbf{N}$, a co-isometry $v_n \in N^{\omega_1}$ such that $v_n \varphi^{\omega_1}=\lambda_n \varphi^{\omega_1}v_n$ (possible because $\lambda_n \leq 1$). Let $\omega_2 \in \beta (\mathbf{N}) \setminus \mathbf{N}$ be any other nonprincipal ultrafilter. Since $\lambda_n \to 1$, then $v=(v_n)^{\omega_2}$ defines a co-isometry of $N^{\omega_2 \otimes \omega_1}$ with $v\varphi^{\omega_2 \otimes \omega_1}=\varphi^{\omega_2 \otimes \omega_1}v$. Since $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$, we get $x=vxv^{*}=(v_nxv_n^{*})^{\omega_2}=(\beta_{\lambda_n}^{\varphi}(x))^{\omega_2}$. Since the nonprincipal ultrafilter $\omega_{2} \in \beta(\mathbf{N}) \setminus \mathbf{N}$ is arbitrary, by Lemma \ref{lem: ultralimit test}, we conclude that $\beta^{\varphi}_{\lambda_n}(x) \to x$ strongly as $n \to \infty$.
\end{proof}
\begin{rem}\label{rem:flow-computation}
Although we strongly believe that the bicentralizer $\mathord{\text{\rm B}}(M, \varphi)$ should always be trivial for all type ${\rm III_{1}}$ factors $M$ with separable predual, we point out that the relative bicentralizer (flow) need not be trivial in general.
\begin{itemize}
\item [$(\rm i)$] Let $N$ be any type ${\rm III_{1}}$ factor with separable predual and with trivial bicentralizer (e.g.\ $N = R_{\infty}$). Choose a faithful state $\varphi \in N_{\ast}$. Fix $\mu \in (0, 1)$, put $T = \frac{2 \pi}{- \log(\mu)}$ and define $M = N \rtimes_{\sigma_{T}^{\varphi}} \mathbf{Z}$. Extend $\varphi$ to $M$ by using the canonical conditional expectation $\mathord{\text{\rm E}}_{N} : M \to N$. Then $M$ is a type ${\rm III_{\mu}}$ factor by \cite[Lemma 1]{Co85} and the inclusion $N \subset M$ is irreducible and with expectation. Since $\mathord{\text{\rm B}}(N, \varphi) = \mathbf{C} 1$, a straightforward argument using the Fourier expansion shows that $\mathord{\text{\rm B}}(N \subset M, \varphi) = \mathord{\text{\rm L}}(\mathbf{Z})$ and the relative bicentralizer flow $\beta^{\varphi} : \mathbf{R}^{*}_{+} \curvearrowright \mathord{\text{\rm B}}(N \subset M, \varphi)$ is given by the action $\mathbf{R}^{*}_{+} \curvearrowright \mathbf{R}^{*}_{+}/\mu^{\mathbf{Z}}$.
\item [$(\rm ii)$] We can upgrade the example given in item $(\rm i)$ to obtain an irreducible inclusion of type ${\rm III_{1}}$ factors with expectation and with nontrivial bicentralizer flow. Indeed, define $\mathcal M = (M, \varphi) \ast (R_{\infty}, \psi)$ to be the free product von Neumann algebra, where $\psi \in (R_{\infty})_{\ast}$ is any faithful state. By \cite[Theorem 4.1]{Ue10}, $\mathcal M$ is a type ${\rm III_{1}}$ factor. Moreover, \cite[Corollary 3.2]{Ue10} shows that $N \subset \mathcal M$ is irreducible and with expectation. Fix $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ any nonprincipal ultrafilter. Noticing that $\mathcal M \subset (M^{\omega}, \varphi^{\omega}) \ast (R_{\infty}, \psi)$, \cite[Corollary 3.2]{Ue10} implies that
$$\mathord{\text{\rm B}}(N \subset \mathcal M, \varphi) = (N^{\omega}_{\varphi^{\omega}})' \cap \mathcal M = (N^{\omega}_{\varphi^{\omega}})' \cap \mathcal M \cap M^{\omega} = (N^{\omega}_{\varphi^{\omega}})' \cap M = \mathord{\text{\rm B}}(N \subset M, \varphi).$$
The relative bicentralizer flow $\beta^{\varphi} : \mathbf{R}^{*}_{+} \curvearrowright \mathord{\text{\rm B}}(N \subset \mathcal M, \varphi)$ is still given by the action $\mathbf{R}^{*}_{+} \curvearrowright \mathbf{R}^{*}_{+}/\mu^{\mathbf{Z}}$.
\end{itemize}
\end{rem}
\section{Self-bicentralizing factors}
To prove Theorem \ref{thm: self-bicentralizing}, we use again the ultraproduct technology. We start by proving the following result.
\begin{prop} \label{prop: approximately inner}
Let $M$ be any factor with separable predual, $\theta \in \mathord{\text{\rm Aut}}(M)$ any automorphism and $\omega \in \beta (\mathbf{N}) \setminus \mathbf{N}$ any nonprincipal ultrafilter. Consider the following properties:
\begin{itemize}
\item [$(\rm i)$] $\theta \in \overline{\mathord{\text{\rm Inn}}}(M)$.
\item [$(\rm ii)$] There exists a unitary $u \in M^\omega$ such that $\theta(x)=uxu^*$ for all $x \in M$ and $\theta(\varphi)^\omega=u\varphi^\omega u^*$ for some (or any) faithful normal state $\varphi \in M_*$.
\item [$(\rm iii)$] There exists a nonzero partial isometry $v \in M^\omega$ such that $\theta(x)v=vx$ for all $x \in M$ and $v\varphi^\omega=\theta(\varphi)^\omega v$ for some (or any) faithful normal state $\varphi \in M_*$.
\item [$(\rm iv)$] There exists a nonzero partial isometry $v \in M^\omega$ such that $\theta(x)v=vx$ for all $x \in M$.
\item [$(\rm v)$] The automorphism $\theta \odot \text{\rm id}$ of $M \odot M^{\mathord{\text{\rm op}}}$ extends to an automorphism of the $\mathord{\text{\rm C}}^*$-algebra $\mathord{\text{\rm C}}^*_{\lambda \cdot \rho}(M)$ generated by the standard representation $\lambda \cdot \rho$ of $M \odot M^{\mathord{\text{\rm op}}}$ on $\mathord{\text{\rm L}}^2(M)$.
\end{itemize}
Then we have $(\rm i) \Leftrightarrow (\rm ii) \Leftrightarrow (\rm iii) \Rightarrow (\rm iv) \Rightarrow (\rm v)$.
\end{prop}
\begin{proof}
We only prove $(\rm iv) \Rightarrow (\rm v)$ since the other implications are well-known (for the implication $(\rm iii) \Rightarrow (\rm ii)$, see the end of the proof of \cite[Theorem 1]{Co85} starting from Lemma 4). Let $v \in M^{\omega}$ be a nonzero partial isometry such that $\theta(x)v=vx$ for every $x \in M$. Note that $vv^{*} \in M' \cap M^{\omega}$ and so $\mathord{\text{\rm E}}_M(vv^{*})=\lambda \in \mathbf{R}^{*}_+$. Take $T=\sum_i x_i \otimes y_i^{\mathord{\text{\rm op}}} \in M \odot M^{\mathord{\text{\rm op}}}$. For every unit vector $\xi \in \mathord{\text{\rm L}}^{2}(M)$, we have in $\mathord{\text{\rm L}}^{2}(M^{\omega})$ the following equalities:
$$ \|\sum_i \theta(x_i)\xi y_i \|=\frac{1}{\lambda} \|v^{*}\sum_i \theta(x_i)\xi^{\omega} y_i \|=\frac{1}{\lambda} \|\sum_i x_i v^{*}\xi^{\omega} y_i \|.$$
Since $$\| T \|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)}=\| T \|_{\mathord{\text{\rm L}}^2(M)}=\| T^{\omega} \|_{\mathord{\text{\rm L}}^2(M^{\omega})}$$
we obtain
$$\|\sum_i x_i v^{*}\xi^{\omega} y_i \| \leq \| v^{*} \xi^{\omega} \| \cdot \|T\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)} =\lambda \|T\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)}.$$
Thus we have shown that
$$ \|\sum_i \theta(x_i)\xi y_i \| \leq \|T\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)}$$
for all unit vectors $\xi \in \mathord{\text{\rm L}}^{2}(M)$. This means that
$$\|(\theta \otimes \text{\rm id})(T)\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)} \leq \|T\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)}.$$
By the same reasoning, replacing $\theta$ by $\theta^{-1}$ and $T$ by $(\theta \otimes \text{\rm id})(T)$, we obtain
$$\|(\theta \otimes \text{\rm id})(T)\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)} = \|T\|_{\mathord{\text{\rm C}}^{*}_{\lambda \cdot \rho}(M)}$$
as we wanted.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm: self-bicentralizing}]
$(\rm i)$ Let $\psi$ be any bicentralizing state on $M$. Let $\omega \in \beta(\mathbf{N}) \setminus \mathbf{N}$ be any nonprincipal ultrafilter and $u \in M^{\omega}$ any unitary such that $u\varphi^{\omega} u^{*}=\psi^{\omega}$. Then, with $M=\mathord{\text{\rm B}}(M,\varphi)$ and $M=\mathord{\text{\rm B}}(M,\psi)$, the $\ast$-homomorphism $\beta_{\psi, \varphi} : M \rightarrow M$ satisfies $\beta_{\psi, \varphi}(x)=uxu^{*}$ for all $x \in M$. This shows that $\beta_{\psi, \varphi} \in \overline{\mathord{\text{\rm Inn}}}(M)$ and we have $\beta_{\psi, \varphi}(\varphi)=\psi$. Conversely, if $\alpha \in \overline{\mathord{\text{\rm Inn}}}(M)$, then we can find a unitary $u \in M^\omega$ such that $\alpha(x)=uxu^*$ for all $x \in M$ and $u\varphi^\omega u^*=\alpha(\varphi)^\omega$. This shows that $\beta_{\alpha(\varphi),\varphi}=\alpha$. Hence the map
$$ \overline{\mathord{\text{\rm Inn}}}(M) \ni \alpha \mapsto \alpha(\varphi) \in \Delta(M)$$
is a continous bijection with inverse
$$ \Delta(M) \ni \psi \mapsto \beta_{\psi, \varphi} \in \overline{\mathord{\text{\rm Inn}}}(M).$$
It only remains to show that this inverse is continous. Let $(\psi_n)_{n \in \mathbf{N}}$ be a sequence in $\Delta(M)$ which converges to $\psi \in \Delta(M)$. Let $\theta_n=\beta_{\psi_n, \varphi}$. We have to show that $\theta_n$ converges to $\beta_{\psi, \varphi}$. Since $(\theta_n(\varphi))^\omega=(\psi_n)^\omega=\psi^\omega$, $\theta=(\theta_n)^\omega$ defines an automorphism of $M^\omega$ and we just have to show that $\theta(x)=\beta_{\psi, \varphi}(x)$ for all $x \in M$. Let $\omega_0 \in \beta(\mathbf{N}) \setminus \mathbf{N}$ be any other nonprincipal utltrafilter, and for every $n \in \mathbf{N}$, take a unitary $u_n \in M^{\omega_0}$ such that $u_n \varphi^{\omega_0} u_n^*=\psi_n^{\omega_0}$. Then we have $\theta_n(x) = \beta_{\psi_n, \varphi}(x) =u_nxu_n^*$ for all $x \in M$ and all $n \in \mathbf{N}$. Let $u=(u_n)^\omega \in M^{\omega \otimes \omega_0}$. Then, by construction, we have $\theta(x)=uxu^*$ for all $x \in M$. But we have $u (\varphi^{\omega_0})^\omega u^*=(\psi_n^{\omega_0})^\omega=(\psi^{\omega_0})^\omega$ and so $uxu^*=\beta_{\psi, \varphi}(x)$ for all $x \in M$. This shows that $\theta(x)=\beta_{\psi, \varphi}(x)$ for all $x \in M$, as we wanted.
$(\rm ii)$ Since a continuous bijective homomorphism between Polish groups is automatically a homeomorphism, we just need to show that the homomorphism $\iota$ is bijective. Let $(g,h) \in \overline{\mathord{\text{\rm Inn}}}(M) \rtimes \mathord{\text{\rm Aut}}_\varphi(M)$ and assume that $g \circ h=\text{\rm id}$. Then $h=g^{-1}$ is approximately inner. Since $h(\varphi)=\varphi$, by item $(\rm i)$, we conclude that $h=\beta_{\varphi, \varphi}=\text{\rm id}$. This shows the injectivity. For the surjectivity, we just write every $\alpha \in \mathord{\text{\rm Aut}}(M)$ as $\alpha=\beta_{\alpha(\varphi), \varphi} \circ \left( \beta_{\varphi, \alpha( \varphi)} \circ \alpha \right)$ and we note that $\beta_{\alpha(\varphi), \varphi} \in \overline{\mathord{\text{\rm Inn}}}(M)$ by item $(\rm i)$ while $\beta_{\varphi, \alpha( \varphi)} \circ \alpha \in \mathord{\text{\rm Aut}}_\varphi(M)$. We conclude that $\iota$ is indeed an isomorphism of topological groups. Moreover, it shows that $\sigma_t^\varphi \in \overline{\mathord{\text{\rm Inn}}}(M)$ if and only if $\sigma_t^\varphi=\text{\rm id}$ and so $\sigma_t^\varphi \in \overline{\mathord{\text{\rm Inn}}}(M)$ if and only if $t = 0$ because $M$ is a type ${\rm III_1}$ factor.
$(\rm iii)$ Note first that $\beta_\lambda^\varphi \in \mathord{\text{\rm Aut}}_\varphi(M)$ and so $\beta_\lambda^\varphi \in \overline{\mathord{\text{\rm Inn}}}(M)$ if and only if $\beta_\lambda^\varphi=\text{\rm id}$, thanks to item $(\rm ii)$. If $M \cong M \mathbin{\overline{\otimes}} R_\lambda$ for $0 < \lambda < 1$, then we can find a co-isometry $v \in M' \cap M^\omega$ such that $v \varphi^\omega=\lambda \varphi^\omega v$. This shows that $\beta_\lambda^\varphi(x)v=vx=xv$ for all $x \in M$. Since $v$ is a co-isometry, we obtain $\beta_\lambda^\varphi=\text{\rm id}$. Conversely, take $v$ any nonzero partial isometry $v \in M^\omega$ such that $vv^*+v^*v=1$ and $v\varphi^\omega=\lambda \varphi^\omega v$. If $\beta_\lambda^\varphi=\text{\rm id}$, then we have $v \in M' \cap M^\omega$. This shows that $M$ satisfies Araki's property L$_\lambda'$ and so $M \cong M \mathbin{\overline{\otimes}} R_\lambda$ by \cite[Theorem 1.3]{Ar70}.
$(\rm iv)$ This follows from item $(\rm ii)$ in Theorem \ref{thm: Connes' isomorphism and bicentralizer flow} and Proposition \ref{prop: approximately inner}.
\end{proof}
\begin{remark}
We observe that $\Delta(M)$ is a dense $G_\delta$ subset of the set of all faithful normal states $\mathord{\text{\rm S}}_{\rm{nf}}(M)$. The density of $\Delta(M)$ follows of course from Connes--St\o rmer transitivity. Since $\overline{\rm{Inn}}(M)$ is a Polish space, so is $\Delta(M)$, and since $\mathord{\text{\rm S}}_{\rm{nf}}(M)$ is Polish as well, $\Delta(M)$ must be a $G_{\delta}$ subset.
\end{remark}
\section{Irreducible hyperfinite subfactors in inclusions of type III factors}
\subsection{Proof of Theorem \ref{thm: hyperfinite subfactor III_1}} In this section, we prove Theorem \ref{thm: hyperfinite subfactor III_1}. We first need to prove a few technical lemmas. The next result is a straightforward variation of \cite[Lemma 2.3]{Po81}.
\begin{lem} \label{convex_automorphism}
Let $M$ be any $\sigma$-finite von Neumann algebra and $\varphi \in M_{\ast}$ any faithful state. Let $G \subset \mathord{\text{\rm Aut}}_{\varphi}(M)$ be any subgroup. Let $x \in M$ be any element that satisfies $\mathord{\text{\rm E}}_{M^G}^\varphi(x)=0$ where $\mathord{\text{\rm E}}_{M^G}^\varphi : M \to M^{G}$ denotes the unique $\varphi$-preserving conditional expectation on the fixed point algebra $M^G$. Then there exists $\alpha \in G$ such that $\|x-\alpha(x)\|_\varphi \geq \|x\|_\varphi$.
\end{lem}
\begin{proof}
Without loss of generality, we may assume that $x \neq 0$. Denote by $y$ the unique element of minimal $\| \cdot \|_\varphi$-norm in the weakly closed convex hull $\mathcal C$ of $\{ \alpha(x) \mid \alpha \in G \}$. For every $\alpha \in G$, we have $\| \alpha(y)\|_\varphi=\|y \|_\varphi$ and $\alpha(y) \in \mathcal C$. By uniqueness of $y \in \mathcal C$, we obtain $\alpha(y)=y$ for every $\alpha \in G$. This shows that $y \in M^{G}$. On the other hand, we have $\mathord{\text{\rm E}}^{\varphi}_{M^{G}}(\alpha(x))=\mathord{\text{\rm E}}^{\varphi}_{M^{G}}(x)=0$ for all $\alpha \in G$. Thus, we also have $\mathord{\text{\rm E}}^{\varphi}_{M^{G}}(y)=0$ and so $y=0$. By contradiction, if $\|x-\alpha(x)\|_\varphi < \|x\|_\varphi$ for every $\alpha \in G$, then we have $\|x\|_{\varphi}^{2} - 2 \Re(\varphi(x^{*}\alpha(x))) < 0$ for every $\alpha \in G$. By taking convex combinations and weak limits and since $y = 0$, we conclude that $\|x\|_{\varphi}^{2} \leq 0$, which is a contradiction. Therefore, there exists $\alpha \in G$ such that $\|x-\alpha(x)\|_\varphi \geq \|x\|_\varphi$.
\end{proof}
For all $0 < \lambda <1$, we denote by $\tau_{\lambda}$ the canonical periodic faithful normal state on the Powers factor $R_{\lambda}$ arising from the infinite tensor product decomposition
$$(R_{\lambda}, \tau_{\lambda})=\overline{\bigotimes}_{n \in \mathbf{N}} (\mathbf M_2(\mathbf{C}), \omega_{\lambda})$$
where $\omega_{\lambda}$ is the state on $\mathbf M_2(\mathbf{C})$ given by $\omega_{\lambda}(e_{11})=\lambda\omega_{\lambda}(e_{22})=\frac{\lambda}{1+\lambda}$ and $\omega_{\lambda}(e_{12})=\omega_{\lambda}(e_{21})=0$.
\begin{lem} \label{lem: R_lambda}
Let $N$ be any factor and $\varphi \in N_*$ any faithful state such that $N_{\varphi}$ is a type ${\rm II}_1$ factor. If $S$ is a finite subset of the point spectrum of $\varphi$, then we can find a subfactor $P \subset N$, globally invariant under $\sigma^\varphi$, such that
$$ (P,\varphi|_P) \cong \overline{\bigotimes_{\lambda \in S}} (R_\lambda, \tau_\lambda).$$
\end{lem}
\begin{proof}
Let $S=\{ \lambda_1, \dots, \lambda_p\}$. Since $N_{\varphi}$ is a type ${\rm II}_1$ factor, we can find a partial isometry $v \in N$ such that $vv^{*}+v^{*}v=1$ and $v\varphi =\lambda_1 \varphi v$. Then $v$ generates a finite dimensional factor $F_1$ that is globally invariant under $\sigma^{\varphi}$ and such that $(F_1, \varphi |_{F_1}) \cong (\mathbf M_2(\mathbf{C}), \omega_{\lambda_1})$. Observe that $F_1' \cap N \cong eNe$ where $e=vv^{*} \in N_{\varphi}$ and so $(F_1' \cap N)_{\varphi}$ is again a type ${\rm II}_1$ factor and $S$ is in the point spectrum of $\varphi |_{F_1' \cap N}$. Thus, we can find $F_2 \subset F_1' \cap N$ that is globally invariant under $\sigma^{\varphi}$ and such that $(F_2, \varphi |_{F_2}) \cong (\mathbf M_2(\mathbf{C}), \omega_{\lambda_2})$. By repeating this procedure inductively, we obtain a finite dimensional factor $Q_1 \subset N$ that is globally invariant under $\sigma^{\varphi}$ and such that
$$(Q_1, \varphi |_{Q_1}) \cong \bigotimes_{k=1}^{p} (\mathbf M_2(\mathbf{C}), \omega_{\lambda_k}).$$
By repeating this procedure with $Q_1' \cap N$, we contruct a sequence of mutually commuting factors $(Q_n)_{n \in \mathbf{N}}$ that are all globally invariant under $\sigma^{\varphi}$ and such that $(Q_n, \varphi |_{Q_n}) \cong (Q_1, \varphi |_{Q_1})$ for all $n \in \mathbf{N}$. Then $P=\bigvee_{n \in \mathbf{N}} Q_n$ provides the desired factor.
\end{proof}
\begin{lem} \label{lem: maximality eigenvalue}
Let $N$ be any factor and $\varphi \in N_*$ any faithful state such that $N_{\varphi}$ is a type ${\rm II}_1$ factor. If $\lambda$ is in the point spectrum of $\varphi$, then we can find a sequence $(a_n)_{n \in \mathbf{N}}$ in $N$ with $a_n \varphi = \lambda \varphi a_n$ for all $n \in \mathbf{N}$ such that
$ \sum_{n \in \mathbf{N}} a_n^{*}a_n= \lambda$ and $\sum_{n \in \mathbf{N}} a_na_n^{*}= 1$.
\end{lem}
\begin{proof}
Take a maximal subset $A \subset N \setminus \{0\}$ such that $a\varphi=\lambda \varphi a$ for all $a \in A$ and $\sum_{a \in F} a^{*}a \leq \lambda$ and $\sum_{a \in F} aa^{*} \leq 1$. Let $s = \lambda- \sum_{a \in F} a^{*}a $ and $t=1-\sum_{a \in F} aa^{*}$. Observe that $s,t$ are two positive elements of $N_{\varphi}$ and that $\varphi(s)=\lambda \varphi(t)$. Suppose that $s \neq 0$ and so $t \neq 0$. Since $N_{\varphi}$ is diffuse, we can find $\varepsilon > 0$ and two nonzero projections $p, q \in N_{\varphi}$ such that $\varepsilon p \leq s$, $\varepsilon q \leq t$ and $\varphi(p)=\lambda \varphi(q)$. Since $\lambda$ is in the point spectrum of $\varphi$ and $N_{\varphi}$ is a type ${\rm II}_1$ factor, we can find a partial isometry $v \in N$ such that $v \varphi = \lambda \varphi v$, $v^{*}v=p$ and $vv^{*}=q$. Then $A'=A \cup \{ \sqrt{\varepsilon}v \}$ contradicts the maximality of $A$. Thus, we have $s=t=0$ and the set $A$, which is necessarily countable, provides the desired sequence.
\end{proof}
We will prove Theorem \ref{thm: hyperfinite subfactor III_1} by using inductively the following key lemma. The last part of the lemma will be useful to control the type of the hyperfinite subfactor we want to construct.
\begin{lem} \label{induction_step}
Let $N \subset M$ be any inclusion of $\sigma$-finite factors and $\varphi \in M_*$ any faithful state such that $N$ is globally invariant under $\sigma^{\varphi}$. Denote by $\mathord{\text{\rm E}}_{N' \cap M}^\varphi : M \rightarrow N' \cap M$ the unique $\varphi$-preserving conditional expectation. Assume that $N$ is a type ${\rm III}_1$ factor and that $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta^\varphi}=N'\cap M$.
Then for every $\varepsilon > 0$ and every $x \in M$ such that $\mathord{\text{\rm E}}^\varphi_{N' \cap M}(x)=0$, we can find a finite dimensional subfactor $F \subset N$ and a faithful state $\psi \in F_*$ such that
\begin{itemize}
\item [$(\rm i)$] $\| \varphi- \varphi \circ \mathord{\text{\rm E}}_{F' \cap M} \| \leq \varepsilon$
\item [$(\rm ii)$] $\| \mathord{\text{\rm E}}_{F' \cap M}(x) \|_\varphi \leq (1-2^{-13}) \|x \|_\varphi$
\end{itemize}
where $\mathord{\text{\rm E}}_{F' \cap M} : M \rightarrow F' \cap M$ is the conditional expectation induced by $\psi$.
Moreover, for any given $0 < \mu, \nu < 1$ such that $\mu/\nu \notin \mathbf{Q}$, we can choose $(F,\psi)$ so that
$$ (F,\psi) \cong (\mathbf M_2(\mathbf{C}), \tau)^{\otimes p} \otimes (\mathbf M_2(\mathbf{C}), \omega_\mu)^{\otimes q} \otimes (\mathbf M_2(\mathbf{C}), \omega_\nu)^{\otimes r} $$
where $p,q,r \in \mathbf{N}$. We can always take $q,r \geq 1$. If $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta_\mu^\varphi}=N'\cap M$, then we can take $q \geq 1$ and $r=0$. If $\mathord{\text{\rm B}}(N \subset M, \varphi)=N'\cap M$, then we can take $q=r=0$.
\end{lem}
\begin{proof}
Let $S=\{\mu, \nu\}$. If $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta_\mu^\varphi}=N'\cap M$, take $S=\{ \mu \}$. If $\mathord{\text{\rm B}}(N \subset M, \varphi)=N'\cap M$, take $S=\emptyset$. In all cases, we have $\mathord{\text{\rm B}}(N \subset M, \varphi)^{G}=N'\cap M$, where $G$ is the subgroup generated by $\beta^\varphi_s$ for $s \in S$.
Fix $\varepsilon > 0$ and $x \in M$ such that $\mathord{\text{\rm E}}^\varphi_{N' \cap M}(x)=0$. Write $x=y+z$ where $y=x-\mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)}(x)$ and $z=\mathord{\text{\rm E}}^\varphi_{\mathord{\text{\rm B}}(N \subset M, \varphi)}(x)$. By Proposition \ref{ultraproduct_bicentralizer}, we know that $\mathord{\text{\rm E}}_{(N^{\omega}_{\varphi^{\omega}})' \cap M^{\omega}}(y)=0$. Lemma \ref{convex_automorphism} yields a unitary $u \in N^\omega_{{\varphi}^\omega}$ such that $\|uyu^*-y\|_{\varphi^\omega} \geq \|y \|_\varphi$. Note that $N^\omega_{\varphi^\omega}$ is a type ${\rm II}_1$ factor. Thus, up to a small perturbation (approximate $u$ by a unitary with finite spectrum and such that all of its spectral projections have dyadic traces), we may assume that $u$ is contained in some finite dimensional factor $\mathbf M_{2^p}(\mathbf{C}) \subset N^\omega_{\varphi^\omega}$. Then, by Lemma \ref{lem: R_lambda}, we can find a subfactor $Q \subset \mathbf M_{2^p}(\mathbf{C})' \cap N^\omega$ that is globally invariant under $\sigma^{\varphi^\omega}$ and such that $$(Q,\varphi^\omega |_Q) \cong \overline{\bigotimes_{\mu \in S}}(R_\mu, \tau_\mu).$$
Put $P= \mathbf M_{2^p}(\mathbf{C}) \vee Q$. Let us show that
$$\| \mathord{\text{\rm E}}^{\varphi^\omega}_{P' \cap M^\omega}(x) \|_{\varphi^\omega} \leq (1-2^{-11}) \|x \|_\varphi.$$
Assume first that $\| y \|_\varphi \geq 2^{-4} \|x \|_\varphi$. Since $u \in P$ and $u$ commutes with $z$, we have
\begin{align*}
\|x \|_\varphi^{2}-\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega}^{2} &= \|x-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega}^{2}\\
& \geq \frac{1}{4}\| u(x-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x))-(x-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x))u \|_{\varphi^\omega}^{2}\\
& = \frac{1}{4}\| ux-xu \|_{\varphi^\omega}^{2}\\
&=\frac{1}{4}\| uy-yu \|_{\varphi^\omega}^{2} \\
&\geq \frac{1}{4}\|y\|_\varphi^2.
\end{align*}
Then $\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega}^2 \leq \|x \|_\varphi^2- \frac{1}{4} \|y \|_\varphi^2$. Since $\| y \|_\varphi \geq 2^{-4} \|x \|_\varphi$, this shows that
$$\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega}^2 \leq \|x \|_\varphi^2- \frac{1}{4} \|y \|_\varphi^2 \leq (1-2^{-10})\|x \|_\varphi^2$$
and so
$$\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega} \leq \sqrt{1-2^{-10}} \|x \|_\varphi \leq (1-2^{-11}) \|x \|_\varphi.$$
Assume next that $\| y \|_\varphi \leq 2^{-4} \|x \|_\varphi$. By Lemma \ref{convex_automorphism}, since $\mathord{\text{\rm B}}(N \subset M, \varphi)^{G}=N'\cap M$ and since $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(z)=0$, we can find some $\lambda \in \langle S \rangle$ such that $\|z-\beta_\lambda^\varphi(z)\|_\varphi \geq \|z \|_\varphi$. By construction, $\lambda$ is an eigenvalue of $\Delta_{\varphi^\omega|_P}$ and the centralizer of $\varphi^\omega|_P$ is a factor. Thus, by Lemma \ref{lem: maximality eigenvalue}, we can find a sequence $(a_n)_{n \in \mathbf{N}}$ in $P$ with $a_n \varphi^\omega=\lambda \varphi^\omega a_n$ such that $\sum_{n \in \mathbf{N}} a_n^*a_n=\lambda$ and $\sum_{n \in \mathbf{N}} a_na_n^*=1$. Then we have
\begin{align*}
\|z-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega}^{2}
&=\frac{1}{\lambda} \sum_{n \in \mathbf{N}} \|a_n(z-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) )\|_{\varphi^\omega}^{2} \\
&=\frac{1}{\lambda} \sum_{n \in \mathbf{N}} \|(\beta_\lambda^\varphi(z)-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) )a_n\|_{\varphi^\omega}^{2} \\
&=\frac{1}{\lambda} \sum_{n \in \mathbf{N}} \varphi^\omega(a_n^* \, |\beta_\lambda^\varphi(z)-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) |^2 \, a_n)\\
&=\sum_{n \in \mathbf{N}} \varphi^\omega(a_na_n^* \, |\beta_\lambda^\varphi(z)-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) |^2)\\
&= \|\beta_\lambda^\varphi(z)-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega}^{2}.
\end{align*}
Then, by the triangle inequality, we obtain $\|z-\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega} \geq \frac{1}{2}\| z - \beta_\lambda^\varphi(z) \|_\varphi \geq \frac{1}{2}\|z\|_\varphi$. This implies that $\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega}^2 \leq \frac{3}{4} \|z \|_\varphi^2$ and so $\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega} \leq \frac{7}{8} \|z \|_\varphi $. Since $\| y \|_\varphi \leq \frac{1}{16} \|x \|_\varphi$, this shows that
$$\|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(x) \|_{\varphi^\omega} \leq \|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(y) \|_{\varphi^\omega} + \|\mathord{\text{\rm E}}_{P' \cap M^\omega}^{\varphi^\omega}(z) \|_{\varphi^\omega} \leq \| y \|_\varphi + \frac{7}{8} \|z \|_\varphi \leq \frac{15}{16} \|x \|_\varphi.$$
Thus, in all cases, we have
$$\| \mathord{\text{\rm E}}^{\varphi^\omega}_{P' \cap M^\omega}(x) \|_{\varphi^\omega} \leq (1-2^{-11}) \|x \|_\varphi.$$
Now, by construction, it is clear that we can write $P$ as an increasing union of finite dimensional subfactors $(F_n)_{n \in \mathbf{N}}$ which are all globally invariant under $\sigma^{\varphi^{\omega}}$ and such that
$$(F_n, \varphi^{\omega}|_{F_n}) \cong (\mathbf M_{2^p}(\mathbf{C}),\tau) \otimes \bigotimes_{k=1}^{n} \left( \bigotimes_{\mu \in S} (\mathbf M_2(\mathbf{C}), \omega_\mu) \right).$$
We have $\lim_{n \to \infty} \| \mathord{\text{\rm E}}^{\varphi^{\omega}}_{F_n'\cap M^{\omega}}(x) \|_{\varphi^{\omega}}=\| \mathord{\text{\rm E}}^{\varphi^{\omega}}_{P'\cap M^{\omega}}(x) \|_{\varphi^{\omega}} \leq (1-2^{-11}) \|x \|_\varphi$. Thus, for $n \in \mathbf{N}$ large enough, we have $\| \mathord{\text{\rm E}}^{\varphi^{\omega}}_{F_n'\cap M^{\omega}}(x) \|_{\varphi^{\omega}} \leq (1-2^{-12}) \|x \|_\varphi$. Finally, thanks to Lemma \ref{lift_matrix}, we can find a copy $F=F_n$ inside $N$ such that
\begin{itemize}
\item [$(\rm i)$] $\| \varphi- \varphi \circ \mathord{\text{\rm E}}_{F' \cap M} \| \leq \varepsilon$
\item [$(\rm ii)$] $\| \mathord{\text{\rm E}}_{F' \cap M}(x) \|_\varphi \leq (1-2^{-13}) \|x \|_\varphi$.
\end{itemize}
where $\mathord{\text{\rm E}}_{F' \cap M} : M \rightarrow F' \cap M$ is the conditional expectation induced by $\psi=\varphi^{\omega} |_{F}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm: hyperfinite subfactor III_1}]
Denote by $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi} : M \to N' \cap M$ the unique $\varphi$-preserving conditional expectation. Let $X=\{x_n \mid n \in \mathbf{N}\}$ be a countable $*$-strongly dense subset in $\{ x \in \mathord{\text{\rm Ball}}(M) \mid \mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(x)=0 \}$. Using Lemma \ref{induction_step}, we can construct by induction a sequence $(F_n)_{n \in \mathbf{N}}$ of finite dimensional subfactors of $N$ with faithful states $\psi_n \in (F_n)_*$ that satisfy the following properties: $F_0=\C1$, $F_{n+1} \subset (R_n)' \cap N$ with $R_n=F_0 \vee F_1 \vee \cdots \vee F_n$, and
\begin{itemize}
\item [$(\rm i)$] $ \| \varphi \circ \mathord{\text{\rm E}}_{R_{n+1}' \cap M} - \varphi \circ \mathord{\text{\rm E}}_{R_{n}' \cap M} \| \leq 2^{-(n+1)}$
\item [$(\rm ii)$] $ \| \mathord{\text{\rm E}}_{R_{n+1}' \cap M}(y_n)\|_\varphi \leq (1-2^{-13}) \|y_n \|_\varphi$
\end{itemize}
where $\mathord{\text{\rm E}}_{R_n'\cap M}$ is the conditional expectation induced by $\psi_0 \otimes \psi_1 \otimes \dots \otimes \psi_n$ on $R_n$ and $y_n=\mathord{\text{\rm E}}_{R_n'\cap M}(x_n)-\mathord{\text{\rm E}}_{N' \cap M}(\mathord{\text{\rm E}}_{R_n'\cap M}(x_n))$.
Thanks to the condition $(\rm i)$, we know that the sequence of states $\varphi_n=\varphi \circ \mathord{\text{\rm E}}_{R_n' \cap M}$ is a Cauchy sequence in $M_{\ast}$ and so it converges to some state $\varphi_\infty \in M_*$. For all $n \in \mathbf{N}$, we moreover have
$$\varphi_\infty = \lim_{k} \varphi \circ \mathord{\text{\rm E}}_{R_k' \cap M} = \lim_{k} \varphi \circ \mathord{\text{\rm E}}_{R_k' \cap M} \circ \mathord{\text{\rm E}}_{R_n' \cap M} =\varphi_\infty \circ \mathord{\text{\rm E}}_{R_n' \cap M}.$$
Denote by $e$ the support projection of $\varphi_\infty$ in $M$. Then for all $n \in \mathbf{N}$, we have $e \in R_n'\cap M$ and so $\varphi_n(e)=\varphi(e)$. Thus, by letting $n \to \infty$, we obtain $1=\varphi_\infty(e)=\varphi(e)$. This means that $e=1$ and so $\varphi_\infty$ is a faithful normal state on $M$. By construction, all the algebras $R_n$ are globally invariant under $\sigma^{\varphi_\infty}$. It follows that their union generates a hyperfinite factor $P=\bigvee_n R_n$ that is also globally invariant under $\sigma^{\varphi_\infty}$ and such that
$$ (P, \varphi_\infty) \cong \mathbin{\overline{\bigotimes}}_{n \in \mathbf{N}} (F_n, \psi_n).$$
Moreover, by the last part of Lemma \ref{induction_step}, we can always make $P$ of type ${\rm III}_1$. We can make $P$ of type ${\rm III}_\lambda$ if $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta^\varphi_\lambda}=N' \cap M$ for some $0 < \lambda <1$. We can make $P$ of type ${\rm II}_1$ if $\mathord{\text{\rm B}}(N \subset M, \varphi)=N' \cap M$.
Now, let us show that $P' \cap M=N' \cap M$. Take $x \in \mathord{\text{\rm Ball}}(P' \cap M)$ with $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(x)=0$. Then we can find a subsequence $(x_{n_k})_{k \in \mathbf{N}}$ such that $x_{n_k} \to x$ in the $*$-strong topology. Since $x \in R_{n_k}' \cap M$ and since $\mathord{\text{\rm E}}_{R_{n_k}' \cap M}$ is $\varphi_{\infty}$-preserving for all $k \in \mathbf{N}$, we have $\mathord{\text{\rm E}}_{R_{n_k}' \cap M}(x_{n_k}) \to x$ in the $*$-strong topology. Since $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(x)=0$, we also have $\mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(\mathord{\text{\rm E}}_{R_{n_k}' \cap M}(x_{n_k})) \to 0$ in the $*$-strong topology. This shows that $y_{n_k} \to x$ in the $*$-strong topology. And again, since $x \in R_{n_k+1}' \cap M$, we obtain $\mathord{\text{\rm E}}_{R_{n_k+1}' \cap M}(y_{n_k}) \to x$ in the $*$-strong topology. Therefore, by taking the limit in condition $(\rm ii)$, we obtain $\|x\|_\varphi \leq (1-2^{-13}) \|x\|_\varphi$. Thus, $x=0$ as we wanted.
\end{proof}
\begin{proof}[Proof of Application \ref{app-almost-periodic}]
Denote by $\mathord{\text{\rm E}}_{Q} : M \to Q$ the (unique) faithful normal conditional expectation and choose a faithful state $\varphi \in M_{\ast}$ such that $\varphi \circ \mathord{\text{\rm E}}_{Q} = \varphi$ and such that $\varphi|_{Q}$ is almost periodic. Then $\mathord{\text{\rm B}}(N \subset M, \varphi)^{\beta^{\varphi}} \subset Q' \cap M = \C1$ and the conclusion follows from Theorem~\ref{thm: hyperfinite subfactor III_1}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{kadison bicentralizer}]
This is a consequence of Theorem \ref{thm: hyperfinite subfactor III_1} and \cite[Theorem 3.2]{Po81}.
\end{proof}
\subsection{A type ${\rm III_\lambda}$ analogue of Theorem \ref{thm: hyperfinite subfactor III_1}}
In this subsection, we prove a type ${\rm III}_\lambda$ analogue of Theorem \ref{thm: hyperfinite subfactor III_1} when $0 < \lambda < 1$. Since the proof is similar (and easier), we only sketch it.
\begin{thm} \label{thm: hyperfinite subfactor III_lambda}
Let $N \subset M$ be any inclusion of von Neumann algebras with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$ with a $T$-periodic faithful state $\varphi \in N_*$ where $T=\frac{2\pi}{- \log \lambda}$. Then there exists a hyperfinite type ${\rm III}_\lambda$ subfactor $P \subset N$ that is globally invariant under $\sigma^\varphi$ and such that $P' \cap M=N' \cap M$.
We can find a hyperfinite type ${\rm II}_1$ subfactor with expectation $P \subset N$ such that $P' \cap M=N' \cap M$ if and only if $(N_\varphi)' \cap M=N' \cap M$. In that case, we can moreover take $P \subset N_\varphi$.
\end{thm}
The proof relies on the following analogue of Lemma \ref{induction_step}.
\begin{lem}
Let $N \subset M$ be any inclusion of von Neumann algebras with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$ with a $T$-periodic faithful state $\varphi \in N_*$ where $T=\frac{2\pi}{- \log \lambda}$. Extend $\varphi$ to $M$ by using any faithful normal conditional expectation on $N$. Denote by $\mathord{\text{\rm E}}^\varphi_{N' \cap M} : M \to N' \cap M$ the unique $\varphi$-preserving conditional expectation.
Then for every $x \in M$ such that $\mathord{\text{\rm E}}^\varphi_{N' \cap M}(x)=0$, we can find a finite dimensional factor $F \subset N$ that is globally invariant by $\sigma^\varphi$ and such that
$$ \| \mathord{\text{\rm E}}^\varphi_{F' \cap M}(x) \|_\varphi \leq (1-2^{-13})\|x \|_\varphi. $$
\end{lem}
\begin{proof}
Let $B=(N_\varphi)' \cap M$. The algebra $B$ will play the role of $\mathord{\text{\rm B}}(N \subset M, \varphi)$. There exists a $\varphi$-preserving automorphism $\theta$ of $B$ such that for all $x \in B$, all $a \in N$ and all $n \in \mathbf{Z}$, we have
$$ a\varphi=\lambda^n \varphi a \quad \Rightarrow \quad ax=\theta^n(x)a.$$
Moreover, we have $B^{\theta}=N' \cap M$.
Let $x=y+z$ where $y=x-\mathord{\text{\rm E}}_B^\varphi(x)$ and $z=\mathord{\text{\rm E}}^\varphi_B(x)$. By Lemma \ref{convex_automorphism}, we can find a unitary $u \in N_{\varphi}$ such that $\|uyu^*-y\|_{\varphi} \geq \|y \|_\varphi$. Since $N_\varphi$ is a type ${\rm II}_1$ factor, up to a small perturbation (approximate $u$ by a unitary with finite spectrum and such that all of its spectral projections have dyadic traces), we may assume that $u$ is contained in some finite dimensional factor $\mathbf M_{2^p}(\mathbf{C}) \subset N_{\varphi}$. Then, by Lemma \ref{lem: R_lambda}, we can find a subfactor $Q \subset \mathbf M_{2^p}(\mathbf{C})' \cap N$ that is globally invariant under $\sigma^{\varphi}$ and such that
$$(Q,\varphi |_Q) \cong (R_\lambda, \tau_\lambda).$$
Put $P= \mathbf M_{2^p}(\mathbf{C}) \vee Q$. Let us show that
$$\| \mathord{\text{\rm E}}^{\varphi}_{P' \cap M}(x) \|_{\varphi} \leq (1-2^{-13})\|x \|_\varphi.$$
Assume first that $\| y \|_\varphi \geq 2^{-4} \|x \|_\varphi$. Then, as in Lemma \ref{induction_step}, we can show that
$$\| \mathord{\text{\rm E}}^{\varphi}_{P' \cap M}(x) \|_{\varphi} \leq (1-2^{-11})\|x \|_\varphi.$$
Assume next that $\| y \|_\varphi \leq 2^{-4} \|x \|_\varphi$. By Lemma \ref{convex_automorphism}, since $B^{\theta}=N' \cap M$ and since $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(z)=0$, we can find $k \in \mathbf{Z}$ such that $\|z-\theta^{k}(z) \|_\varphi \geq \|z \|_\varphi$. By Lemma \ref{lem: maximality eigenvalue}, we can find a sequence $a_n \in P$ with $a_n \varphi = \lambda^{k} \varphi a_n$ such that $\sum_{n \in \mathbf{N}} a_n^{*}a_n=\lambda^{k}$ and $\sum_{n \in \mathbf{N}} a_n a_n^{*}=1$. Then we conclude as in Lemma \ref{induction_step} by showing that
$$ \|z-\mathord{\text{\rm E}}^{\varphi}_{P' \cap M}(z)\|_\varphi = \| \theta^{k}(z)-\mathord{\text{\rm E}}^{\varphi}_{P' \cap M}(z)\|_\varphi.$$
This finishes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm: hyperfinite subfactor III_lambda}]
The proof of the first part of the theorem is the same as in Theorem \ref{thm: hyperfinite subfactor III_1}. It is easier though because all the finite dimensional factors we construct are globally invariant under $\sigma^{\varphi}$ and the sequence of states $(\varphi_n)_{n \in \mathbf{N}}$ is constant.
For the second part, assume that $P \subset N$ is a hyperfinite type ${\rm II}_1$ subfactor with expectation such that $P' \cap M=N' \cap M$. Observe that $P' \cap N = \C1$. Let $\psi$ be a faithful normal state on $N$ such that $P \subset N_\psi$. We have $N_{\psi}' \cap N = \mathbf{C} 1$. Since $N$ is of type ${\rm III}_\lambda$, we know that $\sigma_T^{\psi}$ is inner and so there exists a unitary $u \in N$ such that $\sigma_{T}^{\psi} = \operatorname{Ad}(u)$. Note that $u \in (N_\psi)' \cap N = \mathbf C 1$. This implies that $\psi$ is $T$-periodic. We have $(N_\psi)' \cap M = N' \cap M$ and since $\psi$ and $\varphi$ are stably unitarily conjugate (see \cite[Th\'eor\`eme 4.3.2]{Co72}), we also have $(N_\varphi)' \cap M =N' \cap M$.
\end{proof}
We derive the following consequence of Theorem \ref{thm: hyperfinite subfactor III_lambda} that is related to \cite[Problem 2]{Po85}.
\begin{cor}\label{cor:popa-problem}
Let $M$ be any type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$ with separable predual and with a $T$-periodic faithful state $\varphi \in M_*$ where $T=\frac{2\pi}{- \log \lambda}$. Then there exists an irreducible hyperfinite type ${\rm III}_\lambda$ subfactor $P \subset M$ that is globally invariant under $\sigma^\varphi$.
\end{cor}
We end this subsection with a characterization of Kadison's property for irreducible inclusions of factors $N \subset M$ with expectation and with separable predual where $N$ is a type ${\rm III_\lambda}$ factor $(0 < \lambda < 1)$.
\begin{thm}\label{thm:Kadison-III-lambda}
Let $N \subset M$ be any irreducible inclusion of factors with separable predual and with expectation. Assume that $N$ is a type ${\rm III}_\lambda$ factor $(0 < \lambda < 1)$. Let $\varphi \in N_*$ be any $T$-periodic faithful state where $T=\frac{2\pi}{- \log \lambda}$ and $\psi$ any dominant weight on $N$.
The following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] $(N_\varphi)' \cap M = \mathbf{C} 1$.
\item [$(\rm ii)$] $(N_\psi)' \cap M = \mathcal Z(N_\psi)$
\item [$(\rm iii)$] The inclusion $N \subset M$ satisfies Kadison's property.
\end{itemize}
\end{thm}
\begin{proof}
$(\rm i) \Rightarrow (\rm iii)$ This follows from \cite[Theorem 3.2]{Po81}.
$(\rm iii) \Rightarrow (\rm ii)$ This follows from Proposition \ref{prop-masa} and the fact that $\left( N \subset M \right) \cong \left( N^{\infty} \subset M^{\infty}\right)$.
$(\rm ii) \Rightarrow (\rm i)$ Since $N$ is a type ${\rm III_\lambda}$ factor, $\sigma_T^\psi$ is an inner automorphism of $N$. Let $u \in \mathcal U(N)$ be a unitary such that $\sigma_T^\psi = \operatorname{Ad}(u)$. We have $u \in (N_\psi)' \cap N$ and so $u \in \mathcal Z(N_\psi)$ by Connes--Takesaki relative commutant theorem \cite[Chapter II, Theorem 5.1]{CT76}. Choose a nonsingular positive selfadjoint operator $h$ affiliated with $\mathcal Z(N_\psi)$ such that $u = h^{{\rm i}T}$. Define the faithful normal semifinite weight $\phi$ on $N$ by the formula $\phi = \psi (h^{-1} \, \cdot \,)$. Then we have $\sigma_T^\phi = \operatorname{Ad}(h^{-{\rm i}T}) \circ \sigma_T^\psi = \text{\rm id}_N$ and so $\phi$ is $T$-periodic. We moreover have $N_\psi \subset N_\phi$. Using the assumption, we have
$$(N_\phi)' \cap M = (N_\psi)' \cap M = \mathcal Z(N_\psi) \subset N_\phi.$$
Then \cite[Th\'eor\`eme 4.2.6]{Co72} shows that $(N_\phi)' \cap M = \C1$. Since $\phi$ and $\varphi$ are stably unitarily conjugate (see \cite[Th\'eor\`eme 4.3.2]{Co72}), we also have $(N_\varphi)' \cap M =\mathbf{C} 1$.
\end{proof}
\section{Discrete inclusions}
\subsection{Proof of Theorem \ref{thm-relative-bicentralizer}}
Let $N \subset M$ be any inclusion of von Neumann algebras with separable predual and with expectation $\mathord{\text{\rm E}}_{N} : M \to N$. Assume that $N$ is a type ${\rm III}_1$ factor. Using \cite{Ta71}, we may regard the standard form $(N, \mathord{\text{\rm L}}^{2}(N), J_{N}, \mathord{\text{\rm L}}^{2}(N)^{+})$ as a substandard form of the standard form $(M, \mathord{\text{\rm L}}^{2}(M), J_{M}, \mathord{\text{\rm L}}^{2}(M)^{+})$ via the isometric embedding $$\mathord{\text{\rm L}}^{2}(N)^{+} \hookrightarrow \mathord{\text{\rm L}}^{2}(M)^{+} : \xi \mapsto (\langle \, \cdot \, \xi , \xi\rangle \circ \mathord{\text{\rm E}}_{N})^{1/2}.$$
By this embedding, we have that $\mathord{\text{\rm L}}^{2}(N)$ is an $N$-$N$-sub-bimodule of $\mathord{\text{\rm L}}^{2}(M)$ and $J_{M} |_{\mathord{\text{\rm L}}^{2}(M)} = J_{N}$ (see \cite[p.\ 317 Equation (8)]{Ta71}). Since no confusion is possible, we simply write $J = J_{M}$. Denote by $e_{N} : \mathord{\text{\rm L}}^{2}(M) \to \mathord{\text{\rm L}}^{2}(N)$ the corresponding Jones projection. Since $e_{N} \in \langle M, N\rangle = (JNJ)' \cap \mathbf B(\mathord{\text{\rm L}}^{2}(M))$, for every $x\in M$, every $y \in N$ and every $\xi \in \mathord{\text{\rm L}}^{2}(N)^{+}$, we have
\begin{equation}\label{eq:Jones}
e_{N}(x \xi y) = e_{N} \, JyJ (x\xi) = JyJ \, e_{N} (x\xi) = JyJ \, \mathord{\text{\rm E}}_{N}(x)\xi = \mathord{\text{\rm E}}_{N}(x)\xi y.
\end{equation}
We refer to \cite[Proposition A.2]{HI15} for further details.
Let $\varphi \in M_{\ast}$ be any faithful state such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$. Since the inclusion $N' \cap M \subset M$ is globally invariant under $\sigma^{\varphi}$, we denote by $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M} : M \to N' \cap M$ the unique $\varphi$-preserving conditional expectation and we may regard the standard form $(N' \cap M, \mathord{\text{\rm L}}^{2}(N' \cap M), J_{N' \cap M}, \mathord{\text{\rm L}}^{2}(N' \cap M)^{+})$ as a substandard form of the standard form $(M, \mathord{\text{\rm L}}^{2}(M), J, \mathord{\text{\rm L}}^{2}(M)^{+})$ via the isometric embedding $$\mathord{\text{\rm L}}^{2}(N' \cap M) \hookrightarrow \mathord{\text{\rm L}}^{2}(M) : x \xi_{\varphi} \mapsto x \xi_{\varphi}.$$
Observe that the corresponding Jones projection $e_{N' \cap M}^{\varphi} : \mathord{\text{\rm L}}^{2}(M) \to \mathord{\text{\rm L}}^{2}(N' \cap M)$ satisfies $J e_{N' \cap M}^{\varphi} = e_{N' \cap M}^{\varphi} J$ and $e_{N' \cap M}^{\varphi}(x \xi_{\varphi}) = \mathord{\text{\rm E}}_{N' \cap M}^{\varphi}(x)\xi_{\varphi}$ for every $x \in M$.
\begin{proof}[Proof of Theorem \ref{thm-relative-bicentralizer}]
$(\rm i) \Rightarrow (\rm ii)$ By Theorem \ref{thm: hyperfinite subfactor III_1}, there exists a hyperfinite type ${\rm II}_1$ subfactor with expectation $P \subset N$ such that $P' \cap M=N' \cap M$. Take a faithful state $\varphi \in N_*$ such that $P \subset N_\varphi$. Put $N^{\infty} = N \mathbin{\overline{\otimes}} \mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))$ and $M^{\infty} = M \mathbin{\overline{\otimes}} \mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))$. Since $N$ is a type ${\rm III}$ factor, there is a canonical $\ast$-isomorphism $\pi : M \to M^{\infty}$ such that $\pi(N) = N^{\infty}$ and $\pi \circ \mathord{\text{\rm E}}_{N} \circ \pi^{-1} = \mathord{\text{\rm E}}_{N} \otimes \text{\rm id}_{\mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))}$. Choose a faithful normal semifinite weight $\omega$ on $\mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))$ such that $[\mathord{\text{\rm D}} \omega : \mathord{\text{\rm D}} \mathord{\text{\rm Tr}}] = \lambda_{t}$ for every $t \in \mathbf{R}$. Then $\psi = \varphi \otimes \omega$ is a dominant weight on $N^{\infty}$ (see \cite[Theorem 1.3]{CT76}). The proof of \cite[Theorem 4.2, page 57]{Po95} shows that $(N^{\infty})_{\psi}' \cap M^{\infty} = (N^{\infty})' \cap M^{\infty} = N' \cap M$ (the argument does not use the facts that $N \subset M$ has finite index or $N' \cap M$ is finite dimensional). It remains to prove that the inclusion $(N^{\infty})_{\psi} \subset M^{\infty}$ satisfies the weak relative Dixmier property.
Let $x \in M^{\infty}$ be any element. Denote by $A = \{\lambda_{t} : t \in \mathbf{R}\}^{\prime\prime}$ and observe that $A \subset \mathbf B(\mathord{\text{\rm L}}^{2}(\mathbf{R}))$ is maximal abelian. Then we have $Q' \cap M^{\infty}=(N' \cap M) \mathbin{\overline{\otimes}} A$ where $Q=P \mathbin{\overline{\otimes}} A$. Since $Q$ is amenable, the inclusion $Q \subset M^{\infty}$ satisfies the weak relative Dixmier property. Therefore, we have
$$ \mathcal{K}_Q(x) \cap (N' \cap M) \mathbin{\overline{\otimes}} A \neq \emptyset.$$
Choose $z \in \mathcal K_{Q}(x) \cap \left( (N'\cap M) \mathbin{\overline{\otimes}} A \right)$ and observe that $\mathcal K_{(N^{\infty})_{\psi}}(z) \subset \mathcal K_{(N^{\infty})_{\psi}}(x)$ because $Q \subset (N^{\infty})_{\psi}$.
Regard $(N'\cap M) \mathbin{\overline{\otimes}} A \subset (N'\cap M) \vee (N^{\infty})_{\psi} = ((N^{\infty})'\cap M^{\infty}) \vee (N^{\infty})_{\psi}$. Since $N^{\infty}$ is a factor with expectation, we have $(N'\cap M) \vee (N^{\infty})_{\psi} \cong (N'\cap M) \mathbin{\overline{\otimes}} (N^{\infty})_{\psi}$. Since $(N^{\infty})_{\psi} \subset (N'\cap M) \mathbin{\overline{\otimes}} (N^{\infty})_{\psi}$ is a semifinite von Neumann subalgebra with expectation, the inclusion $(N^{\infty})_{\psi} \subset (N'\cap M) \mathbin{\overline{\otimes}} (N^{\infty})_{\psi}$ satisfies the weak relative Dixmier property (see e.g.\ \cite[Corollary 1.3]{Po98}). Since $(N^{\infty})_{\psi}' \cap \left( (N'\cap M) \vee (N^{\infty})_{\psi}\right) = N' \cap M $, we have
$$\mathcal K_{(N^{\infty})_{\psi}}(z) \cap N'\cap M \neq \emptyset.$$
Therefore, we obtain $\mathcal K_{(N^{\infty})_{\psi}}(x) \cap N'\cap M \neq \emptyset$ and so the inclusion $(N^{\infty})_{\psi} \subset M^{\infty}$ satisfies the weak relative Dixmier property.
$(\rm ii) \Rightarrow (\rm i)$ We divide the proof into a series of claims following the proof of \cite[Theorem 2.1]{Ha85}.
Firstly, we observe that using the assumption in item $(\rm i)$, we obtain the following straightforward generalization of \cite[Lemma 2.7]{Ha85} (see also \cite[Theorem 4.2, Step ${\rm III}$]{Po95}).
\begin{claim}\label{claim1}
Let $\delta > 0$, $\varphi \in M_{\ast}$ any faithful state such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$ and $x \in M$ any element such that $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(x) = 0$. Then there exists a sequence $(a_{i})_{i}$ in $\mathord{\text{\rm Ball}}(N)$ such that the following three properties hold:
\begin{itemize}
\item [$(\rm i)$] $\mathord{\text{\rm sp}}_{\sigma^{\varphi}}(a_{i}) \subset [- \delta, \delta]$ for every $i \in \mathbf{N}$;
\item [$(\rm ii)$] $\sum_{i \in \mathbf{N}} a_{i}^{\ast} a_{i} = 1$;
\item [$(\rm iii)$] $\sum_{i \in \mathbf{N}} \|a_{i} x - x a_{i}\|_{\varphi}^{2} \geq \frac12 \|x\|_{\varphi}^{2}$.
\end{itemize}
\end{claim}
Next, using Claim \ref{claim1} and the fact that the inclusion $N \subset M$ is discrete, we prove the following generalization of \cite[Lemma 2.9]{Ha85}. Let us point out that this is the main technical novelty of the proof of Theorem \ref{thm-relative-bicentralizer} compared to the proofs of \cite[Theorem 3.1]{Ha85} (where $N = M$) and \cite[Theorem 4.2]{Po95} (where $N \subset M$ has finite index).
\begin{claim}\label{claim2}
Let $\delta > 0$, $\xi \in \mathord{\text{\rm L}}^2(N)^+$ any unit $M$-cyclic vector and $\eta \in \mathord{\text{\rm L}}^{2}(M)$ any unit vector such that $J\eta = \eta$ and $\eta \in \left( (N' \cap M) \xi\right)^{\perp}$. Then there exists $a \in \mathord{\text{\rm Ball}}(N) \setminus \{0\}$ such that
\begin{equation}\label{eq:microscopic-a}
\|a\xi\|^{2} + \|a \eta\|^{2} < 8 \|a \eta - \eta a\|^{2} \quad \text{ and } \quad \|a\xi - \xi a\|^{2} < \delta \|a \eta - \eta a\|^{2}.
\end{equation}
\end{claim}
\begin{proof}[Proof of Claim \ref{claim2}]
Put $\varphi = \langle \, \cdot \, \xi , \xi\rangle \in M_{\ast}$ and $\psi = \langle \, \cdot \, \eta, \eta\rangle \in M_{\ast}$. Observe that $\varphi \in M_{\ast}$ is a faithful state such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$.
Firstly, assume that $\psi|_{N}$ is not bounded by some multiple scalar of $\varphi|_{N}$. Then the same reasoning as in \cite[Lemma 2.9, page 113]{Ha85} shows that we may choose projections $p, q \in N$ such that $\psi(p) > (16/\delta) \varphi(p)$ and $\psi(q) = (1/16) \psi(p)$. Since $N$ is a type ${\rm III}$ factor, there exists a partial isometry $v \in N$ such that $v^{*}v = p - q$ and $vv^{*} = q$. Then the same calculations in the standard form $\mathord{\text{\rm L}}^{2}(M)$ as in \cite[Lemma 2.9, pages 113-114]{Ha85} show that $a = v$ satisfies \eqref{eq:microscopic-a}.
Secondly, assume that $\psi|_{N}$ is bounded by some multiple scalar of $\varphi|_{N}$. Then the mapping $S_{0} : M \xi \to \mathord{\text{\rm L}}^{2}(M) : x \xi \mapsto \mathord{\text{\rm E}}_{N}(x) \eta$ extends to well defined bounded operator $S \in \mathbf B(\mathord{\text{\rm L}}^{2}(M))$ such that $S \xi = \eta$, $Se_{N} = S$ and $S \in N' \cap \mathbf B(\mathord{\text{\rm L}}^{2}(M))$. Define $T = J S J \in (JNJ)' \cap \mathbf B(\mathord{\text{\rm L}}^{2}(M)) = \langle M, N\rangle$. We have $T \xi = J S J \xi = J S \xi = J \eta = \eta$ and $T e_{N} = J S J e_{N} = JS e_{N} J = JSJ = T$.
Since the inclusion $N \subset M$ is discrete in the sense of \cite[Definition 3.7]{ILP96}, there exists an increasing sequence of projections $(p_{n})_{n}$ in $N' \cap \langle M, N\rangle$ such that $\widehat \mathord{\text{\rm E}}_{N}(p_{n}) \in M$ for every $n \in \mathbf{N}$ and $p_{n} \to 1$ strongly. Observe that $\widehat \mathord{\text{\rm E}}_{N}((p_{n} T)(p_{n} T)^{\ast}) = \widehat \mathord{\text{\rm E}}_{N}(p_{n} T T^{\ast} p_{n}) \leq \|T\|_{\infty}^{2} \,\widehat \mathord{\text{\rm E}}_{N}(p_{n})$ and so $(p_{n}T)^{\ast} \in \mathfrak n_{\widehat \mathord{\text{\rm E}}_{N}}$. Applying \cite[Proposition 2.2]{ILP96} to $(p_{n}T)^{*}$ and taking the adjoint, we obtain that $\widehat \mathord{\text{\rm E}}_{N}(p_{n} T e_{N}) e_{N} = p_{n} T e_{N} = p_{n} T$. Letting $x_{n} = \widehat \mathord{\text{\rm E}}_{N}(p_{n} T e_{N}) \in M$, we have $x_{n} e_{N} = p_{n} T e_{N} = p_{n} T$. It follows that $x_{n} \xi = x_{n} e_{N} \xi = p_{n} T \xi = p_{n} \eta$. We then have $$\lim_{n} \|x_{n}\|_{\varphi} = \lim_{n}\|x_{n} \xi\| = \lim_{n} \|p_{n} \eta\| = \|\eta\| = 1$$ and
$$\lim_{n} \|\mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(x_{n})\|_{\varphi} = \lim_{n} \|e^{\varphi}_{N' \cap M}(x_{n} \xi)\| = \lim_{n} \|e^{\varphi}_{N' \cap M}(p_{n} \eta)\| = \|e^{\varphi}_{N' \cap M}( \eta)\| = 0.$$
Choose $n \in \mathbf{N}$ large enough so that $\|x_{n} - \mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(x_{n})\|_{\varphi}^{2} \geq (15/16)^{2}$. Put
$$\delta_{1} = \min \left\{ (\delta/8)^{1/2}, 2^{-9/2} \|x_{n} \|_{\infty}^{-1}\right\}.$$
Applying Claim \ref{claim1} to $x = x_{n} - \mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(x_{n})$ with $\delta_{1}$, there exists a sequence $(a_{i})_{i}$ in $\mathord{\text{\rm Ball}}(N)$ such that the following three properties hold:
\begin{itemize}
\item [$(\rm i)$] $\mathord{\text{\rm sp}}_{\sigma^{\varphi}}(a_{i}) \subset [- \delta_{1}, \delta_{1}]$ for every $i \in \mathbf{N}$;
\item [$(\rm ii)$] $\sum_{i \in \mathbf{N}} a_{i}^{\ast} a_{i} = 1$;
\item [$(\rm iii)$] $\sum_{i \in \mathbf{N}} \|a_{i} x - x a_{i}\|_{\varphi}^{2} \geq \frac12 \left(\frac{15}{16}\right)^{2}$.
\end{itemize}
Observe that item $(\rm i)$ implies that $\|a_{i}\xi - \xi a_{i}\|^{2} \leq \delta_{1}^{2}\|a_{i}\xi\|^{2} \leq (\delta/8) \|a_{i} \xi\|^{2}$ for every $i \in \mathbf{N}$.
Since $p_{n} \in N' \cap \langle M, N\rangle$ and $a_{i} \in N$, we have
\begin{align}\label{eq:claim2-1}
p_{n}(a_{i} \eta - \eta a_{i}) &= a_{i} \, p_{n} \eta - p_{n} \eta \, a_{i} \\ \nonumber
&= a_{i} \, x_{n} \xi - x_{n} \xi \, a_{i} \\ \nonumber
&= (a_{i} x_{n} - x_{n} a_{i}) \xi + x_{n} (a_{i} \xi - \xi a_{i}) \\ \nonumber
&= (a_{i} x - x a_{i}) \xi + x_{n} (a_{i} \xi - \xi a_{i}).
\end{align}
The reasoning is now identical to the one in \cite[Lemma 2.9, page 112]{Ha85}. Using the triangle inequality in $\bigotimes_{\mathbf{N}} \mathord{\text{\rm L}}^{2}(M)$ and \eqref{eq:claim2-1}, we have
\begin{align*}
\left( \sum_{i \in \mathbf{N}} \|a_{i} \eta - \eta a_{i}\|^{2} \right)^{1/2} &\geq \left( \sum_{i \in \mathbf{N}} \|p_{n}(a_{i} \eta - \eta a_{i})\|^{2} \right)^{1/2} \\
&\geq \left( \sum_{i \in \mathbf{N}} \|a_{i} x - x a_{i}\|_{\varphi}^{2} \right)^{1/2} - \|x_{n}\|_{\infty} \cdot \left( \sum_{i \in \mathbf{N}} \|a_{i} \xi - \xi a_{i}\|^{2} \right)^{1/2}\\
&\geq \frac{15}{16 \sqrt{2}} - \delta_{1} \|x_{n}\|_{\infty} \\
&\geq \frac{15}{16 \sqrt{2}} - \frac{1}{16 \sqrt{2}} = \frac{7}{8 \sqrt{2}}.
\end{align*}
On the other hand, we have
\begin{equation*}
\sum_{i \in \mathbf{N}} \left( \|a_{i} \xi\|^{2} + \|a_{i} \eta\|^{2} + 8 \delta^{-1} \|a_{i}\xi - \xi a_{i}\|^{2} \right) \leq 1 + 1 + 1 = 3.
\end{equation*}
Since $8 \sum_{i \in \mathbf{N}} \|a_{i} \eta - \eta a_{i}\|^{2} \geq 49/16 > 3$ and $\sum_{i \in \mathbf{N}} \left( \|a_{i} \xi\|^{2} + \|a_{i} \eta\|^{2} + 8 \delta^{-1} \|a_{i}\xi - \xi a_{i}\|^{2} \right) \leq 3$, there exists $i \in \mathbf{N}$ such that
$$8 \|a_{i} \eta - \eta a_{i}\|^{2} > \|a_{i} \xi\|^{2} + \|a_{i} \eta\|^{2} + 8 \delta^{-1} \|a_{i}\xi - \xi a_{i}\|^{2}.$$
Thus, $a = a_{i} \in \mathord{\text{\rm Ball}}(N) \setminus \{0\}$ satisfies \eqref{eq:microscopic-a}.
\end{proof}
Using Claim \ref{claim2} and proceeding exactly as in \cite[Lemmas 2.10, 2.11, 2.12, 2.13]{Ha85}, we obtain the following straightforward generalization of \cite[Lemma 2.13]{Ha85} (see also \cite[Theorem 4.2, Step ${\rm VI}$]{Po95}).
\begin{claim}\label{claim3}
Let $\delta > 0$, $\xi \in \mathord{\text{\rm L}}^{2}(N)^{+}$ any unit $M$-cyclic vector and $\eta \in \mathord{\text{\rm L}}^{2}(M)$ any unit vector such that $J\eta = \eta$ and $\eta \in \left( (N' \cap M) \xi\right)^{\perp}$. Then there exists a nonzero projection $p \in N$ such that
\begin{equation}\label{eq:microscopic-p}
\|p\xi\|^{2} + \|p \eta\|^{2} < 2^{7} \|p \eta - \eta p\|^{2} \quad \text{ and } \quad \|p\xi - \xi p\|^{2} < \delta \|p \eta - \eta p\|^{2}.
\end{equation}
\end{claim}
Whenever $K \subset H$ is a closed subspace of a Hilbert space $H$ and $\eta \in H$ is a unit vector such that $\eta \notin K$, we define the {\em angle} between $\eta$ and $K$ by the formula
$$\angle_{K}(\eta) = \arccos(\|P_{K}(\eta)\|)$$
where $P_{K} : H \to K$ denotes the orthogonal projection.
Using Claim \ref{claim3}, we prove the following generalization of \cite[Lemma 2.14]{Ha85} (see also \cite[Theorem 4.2, Step ${\rm VII}$]{Po95}). Let us point out that unlike \cite[Theorem 4.2, Step ${\rm VII}$]{Po95}, $N' \cap M$ is not assumed to be finite dimensional. For that reason, we provide a detailed proof.
\begin{claim}\label{claim4}
Let $\delta > 0$, $\xi \in \mathord{\text{\rm L}}^{2}(N)^{+}$ any unit $M$-cyclic vector and $\eta \in \mathord{\text{\rm L}}^{2}(M)$ any unit vector such that $J\eta = \eta$ and $\eta \not\in \overline{(N' \cap M) \xi}$.
Then there exists a nonzero projection $p \in N$ such that
\begin{equation}\label{eq:microscopic-p-theta}
\|p\xi\|^{2} + \|p \eta\|^{2} < \frac{2^{10}}{\sin^{2}\theta} \|p \eta - \eta p\|^{2} \quad \text{ and } \quad \|p\xi - \xi p\|^{2} < \delta \|p \eta - \eta p\|^{2}
\end{equation}
where $\theta = \angle_{\overline{(N' \cap M)\xi}}(\eta)$.
\end{claim}
\begin{proof}[Proof of Claim \ref{claim4}]
It is sufficient to consider $\delta < 1$. Put $\varphi = \langle \, \cdot \, \xi, \xi\rangle \in M_{\ast}$ and observe that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$. Since $J e^{\varphi}_{N' \cap M} = e^{\varphi}_{N' \cap M} J$ and $J \eta = \eta$, we have $J e^{\varphi}_{N' \cap M} (\eta) = J e^{\varphi}_{N' \cap M} J (\eta) = e^{\varphi}_{N' \cap M}(\eta)$.
Put $\theta = \angle_{\mathord{\text{\rm L}}^{2}(N' \cap M)}(\eta) = \arccos(\|e^{\varphi}_{N' \cap M}(\eta)\|)$. Put $\zeta = \frac{\eta - e^{\varphi}_{N' \cap M}(\eta)}{\|\eta - e^{\varphi}_{N' \cap M}(\eta)\|}$ and observe that $\|\zeta\| = 1$, $J \zeta = \zeta$, $\zeta \perp (N' \cap M)\xi$ and $\eta = e^{\varphi}_{N' \cap M}(\eta) + \sin\theta \, \zeta$. By Claim \ref{claim3}, there exists a nonzero projection $p \in N$ such that
\begin{equation}\label{eq:claim4-1}
\|p\xi\|^{2} + \|p \zeta\|^{2} < 2^{7} \|p \zeta - \zeta p\|^{2} \quad \text{ and } \quad \|p\xi - \xi p\|^{2} < \frac14 \delta \sin^{2}\theta \, \|p \zeta - \zeta p\|^{2}.
\end{equation}
We have
\begin{equation}\label{eq:claim4-2}
p \eta - \eta p = p e^{\varphi}_{N' \cap M}(\eta) - e^{\varphi}_{N' \cap M}(\eta) p+ \sin\theta \, (p \zeta - \zeta p).
\end{equation}
Since $ e^{\varphi}_{N' \cap M}(\eta) \in \mathord{\text{\rm L}}^{2}(N' \cap M) = \overline{(N' \cap M) \xi}$, we may choose a sequence $(x_{n})_{n}$ in $N' \cap M$ such that $ e^{\varphi}_{N' \cap M}(\eta) = \lim_{n} x_{n} \xi$. Note however that $(x_{n})_{n}$ need not be uniformly bounded. Since $p \in N$ and $\xi \in \mathord{\text{\rm L}}^2(N) \subset \mathord{\text{\rm L}}^2(M)$, we have $p \xi - \xi p \in \mathord{\text{\rm L}}^{2}(N) \subset \mathord{\text{\rm L}}^{2}(M)$. Then we obtain
\begin{align}\label{eq:claim4-3}
\|p e^{\varphi}_{N' \cap M}(\eta) - e^{\varphi}_{N' \cap M}(\eta) p\|^{2} &= \lim_{n} \|p \, x_{n} \xi - x_{n}\xi \, p\|^{2} \\ \nonumber
&= \lim_{n} \| x_{n} (p \xi - \xi p)\|^{2} \quad (\text{since } px_{n} = x_{n}p, \forall n \in \mathbf{N})\\ \nonumber
&= \lim_{n} \langle x_{n}^{*}x_{n}(p \xi - \xi p), e_{N}(p \xi - \xi p)\rangle \quad (\text{since } p \xi - \xi p \in \mathord{\text{\rm L}}^{2}(N))\\ \nonumber
&= \lim_{n} \langle e_{N}(x_{n}^{*}x_{n}(p \xi - \xi p)), p \xi - \xi p\rangle \\ \nonumber
&= \lim_{n} \langle \mathord{\text{\rm E}}_{N}(x_{n}^{*}x_{n})(p \xi - \xi p), p \xi - \xi p\rangle \quad (\text{using } \eqref{eq:Jones} \text{ with } y = p \in N)\\ \nonumber
&= \lim_{n} \|x_{n}\xi\|^{2} \cdot \|p \xi - \xi p\|^{2} \quad (\text{since } \mathord{\text{\rm E}}_{N}(x_{n}^{*}x_{n}) = \varphi(x_{n}^{*}x_{n})1, \forall n \in \mathbf{N})\\ \nonumber
&= \|e^{\varphi}_{N' \cap M}(\eta)\|^{2} \cdot \|p \xi - \xi p\|^{2} \\ \nonumber
&= \cos^{2}\theta \, \|p \xi - \xi p\|^{2}.
\end{align}
Likewise, we obtain
\begin{equation}\label{eq:claim4-4}
\|p e^{\varphi}_{N' \cap M}(\eta) \|^{2} = \cos^{2}\theta \, \|p \xi \|^{2}.
\end{equation}
Combining \eqref{eq:claim4-1},\eqref{eq:claim4-2},\eqref{eq:claim4-3}, we obtain
\begin{align}\label{eq:claim4-5}
\|p\xi - \xi p\|^{2} &< \frac14 \delta \sin^{2}\theta \, \|p \zeta - \zeta p\|^{2} \\ \nonumber
&\leq \frac12 \delta \left( \|p \eta - \eta p\|^{2} + \|p e^{\varphi}_{N' \cap M}(\eta) - e^{\varphi}_{N' \cap M}(\eta) p\|^{2} \right) \\ \nonumber
&= \frac12 \delta \left( \|p \eta - \eta p\|^{2} + \cos^{2}\theta \, \|p \xi - \xi p\|^{2} \right) \\ \nonumber
&\leq \frac12 \delta \left( \|p \eta - \eta p\|^{2} + \|p \xi - \xi p\|^{2} \right) \\ \nonumber
& \leq \delta \|p \eta - \eta p\|^{2} \quad (\text{since } \delta \leq 1).
\end{align}
Using \eqref{eq:claim4-4}, we have
\begin{align}\label{eq:claim4-6}
\|p \eta\| &\leq \| p e^{\varphi}_{N' \cap M}(\eta)\| + \sin \theta \, \|p \zeta \| \\ \nonumber
&\leq \cos \theta \, \| p \xi\| + \sin \theta \, \|p \zeta \| \\ \nonumber
&\leq \left( \| p \xi\|^{2} + \|p \zeta \|^{2}\right)^{1/2}.
\end{align}
Combining \eqref{eq:claim4-1},\eqref{eq:claim4-2},\eqref{eq:claim4-3},\eqref{eq:claim4-5},\eqref{eq:claim4-6} we obtain
\begin{align*}
\|p\xi\|^{2} + \|p \eta\|^{2} &\leq 2 \left(\| p \xi\|^{2} + \|p \zeta \|^{2} \right) \\
&< 2^{8} \|p \zeta - \zeta p\|^{2} \\
&\leq \frac{2^{9}}{\sin^{2}\theta}\left( \|p \eta - \eta p\|^{2} + \|p e^{\varphi}_{N' \cap M}(\eta) - e^{\varphi}_{N' \cap M}(\eta) p\|^{2} \right) \\
&\leq \frac{2^{9}}{\sin^{2}\theta}\left( \|p \eta - \eta p\|^{2} + \|p \xi - \xi p\|^{2} \right) \\
&\leq \frac{2^{10}}{\sin^{2}\theta} \|p \eta - \eta p\|^{2}.
\end{align*}
This finishes the proof of Claim \ref{claim4}.
\end{proof}
Using Claim \ref{claim4}, we prove the following generalization of \cite[Lemma 2.15]{Ha85} (see also \cite[Theorem 4.2, Step ${\rm VIII}$]{Po95}). Since our notion of angle is different from the one in \cite[Lemma 2.14]{Ha85}, we provide a detailed proof and then explain how to use the proof of \cite[Lemma 2.15]{Ha85}.
\begin{claim}\label{claim5}
Let $\delta > 0$, $\xi \in \mathord{\text{\rm L}}^2(N)^+$ any unit $M$-cyclic vector and $\eta \in \mathord{\text{\rm L}}^{2}(M)$ any unit vector such that $J\eta = \eta$ and $\eta \in \left( (N' \cap M) \xi\right)^{\perp}$. Then there exists a family of pairwise orthogonal projections $(e_i)_i$ in $N$ such that $\sum_{i \in I} e_i = 1$ and
\begin{equation}\label{eq:maximal}
2^{-18} \leq \left\| \eta - \sum_{i \in I} e_i\eta e_i \right\|^{2} \quad \text{ and } \quad \left\|\xi - \sum_{i \in I}e_i \xi e_i \right\|^{2} \leq \delta.
\end{equation}
\end{claim}
\begin{proof}[Proof of Claim \ref{claim5}]
Following the proof of \cite[Lemma 2.15]{Ha85}, we denote by $\mathcal F$ the nonempty inductive set of all families of nonzero pairwise orthogonal projections $(p_{i})_{i \in I}$ in $N$ that satisfy
\begin{align*}
\|\xi - p\xi p\|^{2} + \|\eta - p\eta p\|^{2} &\leq 2^{14} \left\|\eta - p\eta p - \sum_{i \in I} p_{i}\eta p_{i} \right\|^{2} \\
\left\|\xi - p\xi p - \sum_{i \in I} p_{i}\xi p_{i} \right\|^{2} &\leq \delta \left\|\eta - p\eta p - \sum_{i \in I} p_{i}\eta p_{i} \right\|^{2}
\end{align*}
where $p = 1 - \sum_{i \in I} p_{i}$. Let $(q_{i})_{i \in I}$ be a maximal element in $\mathcal F$ and put $q - \sum_{i \in I} q_{i}$. We show that the family of pairwise orthogonal projections $((q_{i})_{i \in I}, q)$ satisfies the conclusion of Claim \ref{claim5}. We have to show that $\left\|\eta - q\eta q - \sum_{i \in I} q_{i}\eta q_{i} \right\|^{2} \geq 2^{-18}$.
Assume by contradiction that $\left\|\eta - q\eta q - \sum_{i \in I} q_{i}\eta q_{i} \right\|^{2} < 2^{-18}$. Since
\begin{align*}
\|\xi - q\xi q\|^{2} + \|\eta - q\eta q\|^{2} &\leq 2^{14} \left\|\eta - q\eta q - \sum_{i \in I} q_{i}\eta q_{i} \right\|^{2} \\
&< 2^{14} \cdot 2^{-18} = \frac{1}{16},
\end{align*}
we have
\begin{align}\label{eq:claim5-0}
\max \left\{\|\xi - q \xi q\|, \|\eta - q \eta q\| \right\} &\leq \frac14 \\ \nonumber
\min \left\{\|q \xi q\|, \|q \eta q\| \right\} &\geq \frac34.
\end{align}
Then $q \neq 0$. Denote by $\mathord{\text{\rm E}}_{qNq} : qMq \to qNq$ the faithful normal conditional expectation obtained by restricting $\mathord{\text{\rm E}}_{N}$ to $qMq$. We regard the standard form $(qNq, \mathord{\text{\rm L}}^2(qNq), J_{qNq}, \mathord{\text{\rm L}}^2(qNq)^+)$ as a substandard form of the standard form $(qMq, \mathord{\text{\rm L}}^{2}(qMq), J_{qMq}, \mathord{\text{\rm L}}^{2}(qMq)^{+})$ via the isometric embedding $$\mathord{\text{\rm L}}^{2}(qNq)^{+} \hookrightarrow \mathord{\text{\rm L}}^{2}(qMq)^{+} : \zeta \mapsto (\langle \, \cdot \, \zeta , \zeta\rangle \circ \mathord{\text{\rm E}}_{qNq})^{1/2}.$$
Observe that $\left( \mathord{\text{\rm L}}^2(qNq) \subset \mathord{\text{\rm L}}^2(qMq) \right) = qJqJ \left( \mathord{\text{\rm L}}^2(N) \subset \mathord{\text{\rm L}}^2(M)\right)$ and $J_{qMq} = J \, qJqJ$ where $J = J_{M}$.
Put $\xi_q = \frac{q \xi q}{\|q \xi q\|} \in \mathord{\text{\rm L}}^2(qNq)^+$ and $\eta_q = \frac{q \eta q}{\|q\eta q\|} \in \mathord{\text{\rm L}}^2(qMq)$. Observe that $\xi_q \in \mathord{\text{\rm L}}^2(qNq)^+$ is a $qMq$-cyclic vector in $\mathord{\text{\rm L}}^2(qMq)$. Put $\varphi_q = \langle \, \cdot \, \xi_q, \xi_q\rangle \in (qMq)_\ast$ and observe that $\varphi_{q} \circ \mathord{\text{\rm E}}_{qNq} = \varphi_{q}$. Since the inclusion $(qNq)' \cap qMq \subset qMq$ is globally invariant under $\sigma^{\varphi_{q}}$, we may regard the standard form $((qNq)' \cap qMq, \mathord{\text{\rm L}}^{2}((qNq)' \cap qMq), J_{(qNq)' \cap qMq}, \mathord{\text{\rm L}}^{2}((qNq)' \cap qMq)^{+})$ as a substandard form of the standard form $(qMq, \mathord{\text{\rm L}}^{2}(qMq), J_{qMq}, \mathord{\text{\rm L}}^{2}(qMq)^{+})$ via the isometric embedding $$\mathord{\text{\rm L}}^{2}((qNq)' \cap qMq) \hookrightarrow \mathord{\text{\rm L}}^{2}(qMq) : x \xi_q \mapsto x \xi_q.$$
Denote by $e^{\varphi_q}_{(qNq)' \cap qMq} : \mathord{\text{\rm L}}^{2}(qMq) \to \mathord{\text{\rm L}}^{2}((qNq)' \cap qMq)$ the corresponding Jones projection.
We show that the angle $\angle_{\overline{((qNq)' \cap qMq)\xi_{q}}}(\eta_{q}) = \arccos(\|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q)\|)$, which generalizes the angle $\theta = \arccos(\langle \eta_{q}, \xi_{q}\rangle)$ that appears in \cite[Lemma 2.15]{Ha85}, satisfies $$\cos(\angle_{\overline{((qNq)' \cap qMq)\xi_{q}}}(\eta_{q})) = \|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q)\| \leq \frac12.$$
Since $(qNq)' \cap qMq = (N' \cap M)q$ by \cite[Lemma 2.1]{Po81}, we may choose a sequence $(x_n)_n$ in $N' \cap M$ such that $e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) = \lim_n x_nq \, \xi_q$. Note however that $(x_{n})_{n}$ need not be uniformly bounded. Regarding $\xi_q \in \mathord{\text{\rm L}}^{2}(qNq)^{+} \subset \mathord{\text{\rm L}}^2(N)^+$, for every $n \in \mathbf{N}$, we have
\begin{align}\label{eq:claim5-1}
\|x_n q \, \xi_q\|^2 &= \frac{1}{\|q\xi q\|}\langle x_n \xi, x_n \xi_q\rangle \quad (\text{since } qx_n = x_n q)\\ \nonumber
&= \frac{1}{\|q\xi q\|} \langle x_n^*x_n \xi, e_N(\xi_q)\rangle \quad (\text{since } \xi_q \in \mathord{\text{\rm L}}^2(N))\\ \nonumber
&= \frac{1}{\|q\xi q\|}\langle e_N(x_n^*x_n \xi), \xi_q\rangle \\ \nonumber
&= \frac{1}{\|q\xi q\|}\langle \mathord{\text{\rm E}}_N(x_n^*x_n) \xi, \xi_q\rangle \\ \nonumber
&= \|x_n \xi \|^2 \quad (\text{since } \mathord{\text{\rm E}}_N(x_n^*x_n) = \varphi(x_n^*x_n)1).
\end{align}
As before, since the inclusion $N' \cap M \subset M$ is globally invariant under $\sigma^{\varphi}$, we may regard the standard form $(N' \cap M, \mathord{\text{\rm L}}^{2}(N' \cap M), J_{N' \cap M}, \mathord{\text{\rm L}}^{2}(N' \cap M)^{+})$ as a substandard form of the standard form $(M, \mathord{\text{\rm L}}^{2}(M), J, \mathord{\text{\rm L}}^{2}(M)^{+})$ via the isometric embedding $$\mathord{\text{\rm L}}^{2}(N' \cap M) \hookrightarrow \mathord{\text{\rm L}}^{2}(M) : x \xi_{\varphi} \mapsto x \xi_{\varphi}.$$
Denote by $e_{N' \cap M}^{\varphi} : \mathord{\text{\rm L}}^{2}(M) \to \mathord{\text{\rm L}}^{2}(N' \cap M)$ the corresponding Jones projection. Regarding $\eta_{q} \in \mathord{\text{\rm L}}^{2}(qMq) \subset \mathord{\text{\rm L}}^{2}(M)$, we can compare $\|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) \|$ and $\|e^{\varphi}_{N' \cap M}(\eta_q) \|$ using \eqref{eq:claim5-1}. Indeed, we have
\begin{align*}
\|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) \|^2 &= |\langle e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q), \eta_q \rangle| \\
&= \lim_n |\langle x_nq \, \xi_q, \eta_q \rangle| \\
&= \lim_n \frac{1}{\|q\xi q\|} |\langle x_n \xi, \eta_q \rangle| \quad (\text{since } qx_n = x_nq, \forall n \in \mathbf{N})\\
&= \lim_n \frac{1}{\|q\xi q\|} |\langle e_{N' \cap M}^\varphi(x_n \xi), \eta_q \rangle| \quad (\text{since } x_{n} \in N' \cap M, \forall n \in \mathbf{N})\\
&= \lim_n \frac{1}{\|q\xi q\|} |\langle x_n \xi, e_{N' \cap M}^\varphi(\eta_q) \rangle| \\
&\leq \frac{1}{\|q\xi q\|}\limsup_n \|x_n \xi\| \cdot \|e_{N' \cap M}^\varphi(\eta_q)\| \\
&\leq \frac{1}{\|q\xi q\|}\limsup_n \|x_nq \, \xi_q\| \cdot \|e_{N' \cap M}^\varphi(\eta_q)\| \quad (\text{using } \eqref{eq:claim5-1}) \\
&= \frac{1}{\|q\xi q\|} \|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) \| \cdot \|e_{N' \cap M}^\varphi(\eta_q)\|
\end{align*}
and thus we obtain
\begin{equation}\label{eq:claim5-2}
\|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) \| \leq \frac{1}{\|q\xi q\|} \|e_{N' \cap M}^\varphi(\eta_q)\|.
\end{equation}
Since by assumption we have $e^{\varphi}_{N' \cap M}(\eta) = 0$, combining \eqref{eq:claim5-0} and \eqref{eq:claim5-2}, we obtain
\begin{align}\label{eq:claim5-3}
\|e^{\varphi_q}_{(qNq)' \cap qMq}(\eta_q) \| &\leq \frac{1}{\|q\xi q\| \cdot \|q \eta q\|} \|e_{N' \cap M}^\varphi(q\eta q)\| \\ \nonumber
&= \frac{1}{\|q\xi q\| \cdot \|q \eta q\|} \|e_{N' \cap M}^\varphi(q\eta q - \eta)\| \\ \nonumber
&\leq \frac{1}{\|q\xi q\| \cdot \|q \eta q\|} \|q\eta q - \eta\| \\ \nonumber
&\leq \frac14 \left(\frac43\right) < \frac12.
\end{align}
Since $N$ is a type ${\rm III}$ factor, there exists an isometry $v \in N$ such that $vv^{*} = q$. Then $\operatorname{Ad}(v^{*}) : qMq \to M$ is a $\ast$-isomorphism such that $\operatorname{Ad}(v^{*})(qNq) = N$ and such that $\operatorname{Ad}(v^{*}) \circ \mathord{\text{\rm E}}_{qNq} \circ \operatorname{Ad}(v) = \mathord{\text{\rm E}}_{N}$. We can now use \eqref{eq:claim5-3} in combination with Claim \ref{claim4} applied to the inclusion $\left( qNq \subset qMq \right) \cong \left( N \subset M\right)$ and the vectors $\xi_{q} \in \mathord{\text{\rm L}}^{2}(qNq)^{+}$ and $\eta_{q} \in \mathord{\text{\rm L}}^{2}(qMq)$ with $\delta >0$, and apply the rest of the proof of \cite[Lemma 2.15, pages 124-127]{Ha85} to obtain a contradiction.
\end{proof}
Using Claim \ref{claim5}, we obtain the following straightforward generalization of \cite[Lemma 2.16]{Ha85}.
\begin{claim}\label{claim6}
Let $\delta > 0$, $\xi \in \mathord{\text{\rm L}}^{2}(N)^{+}$ any unit $M$-cyclic vector and $\eta \in \mathord{\text{\rm L}}^{2}(M)$ any unit vector such that $\eta \in \left( (N' \cap M) \xi\right)^{\perp}$. Then there exists a nonzero projection $p \in N$ such that
\begin{equation}\label{eq:macroscopic-p}
2^{-21} \leq \left\| p\eta - \eta p\right\|^{2} \quad \text{ and } \quad \left\|p\xi - \xi p \right\|^{2} \leq \delta.
\end{equation}
\end{claim}
We can now finish the proof of $(\rm ii) \Rightarrow (\rm i)$. Let $\varphi \in M_{\ast}$ be any faithful state such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$. Then the inclusions $N' \cap M \subset \mathord{\text{\rm B}}(N \subset M, \varphi) \subset M$ are globally invariant under $\sigma^{\varphi}$. Let $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$ be any element such that $\mathord{\text{\rm E}}^{\varphi}_{N' \cap M}(x) = 0$. We show that $x = 0$. Put $\xi = \xi_{\varphi} \in \mathord{\text{\rm L}}^{2}(N)^{+}$ and $\eta = x \xi \in \mathord{\text{\rm L}}^{2}(M)$. Observe that $\eta \in \left( (N' \cap M) \xi\right)^{\perp}$. For every $n \geq 1$, applying Claim \ref{claim6}, there exists a projection $p_{n} \in N$ such that
\begin{equation*}
2^{-21} \|\eta\|^{2} \leq \left\| p_{n}\eta - \eta p_{n}\right\|^{2} \quad \text{ and } \quad \left\|p_{n}\xi - \xi p_{n} \right\|^{2} \leq \frac1n.
\end{equation*}
Then we have
\begin{align*}
\liminf_{n} \|p_{n}x - xp_{n}\|_{\varphi} &= \liminf_{n} \|p_{n} x\xi - xp_{n}\xi\| \\
&\geq \liminf_{n} \left(\|p_{n} \eta - \eta p_{n}\| - \|x\|_{\infty} \cdot \|p_{n}\xi - \xi p_{n}\| \right) \\
&= \liminf_{n} \|p_{n} \eta - \eta p_{n}\| \\
& \geq 2^{-21/2} \|\eta\|.
\end{align*}
Since $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$, we have $\lim_n \|p_{n}x - xp_{n}\|_{\varphi} = 0$ and so $\|\eta\| = 0$. Therefore, we have $x = 0$. This finally shows that $\mathord{\text{\rm B}}(N \subset M, \varphi) = N' \cap M$ and finishes the proof of $(\rm ii) \Rightarrow (\rm i)$.
\end{proof}
\subsection{A generalization of Connes--Takesaki relative commutant theorem}
In the setting of discrete irreducible inclusions with expectation, we prove a generalization of Connes--Takesaki relative commutant theorem \cite[Chapter II, Theorem 5.1]{CT76} (see also \cite[Theorem 4.3]{Po95} when $N \subset M$ is a finite index inclusion of type ${\rm III_{1}}$ factors).
\begin{thm}\label{thm-relative-connes-takesaki}
Let $N \subset M$ be any discrete irreducible inclusion of factors with separable predual and with expectation $\mathord{\text{\rm E}}_N : M \to N$. Assume that $N$ is a type ${\rm III}_1$ factor. Let $\psi$ be any dominant weight on $N$ and extend it to a dominant weight on $M$ by using $\mathord{\text{\rm E}}_N$.
Then we have $(N_{\psi})' \cap M = (N_\psi)' \cap M_\psi$. If the inclusion $N \subset M$ has finite index, then we moreover have $(N_\psi)' \cap M=\mathbf{C} 1$.
\end{thm}
\begin{proof}
By using \cite[Theorem ${\rm XII}$.1.1]{Ta03}, we can identify the inclusions
$$\left( N \subset M \right) = \left ( N_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \subset M_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \right)$$
where $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$ is a trace-scaling action that leaves $N_{\psi} \subset M_{\psi}$ globally invariant. We denote by $(v_{\lambda})_{\lambda > 0}$ the canonical unitaries in $N$ that implement the trace-scaling action $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$.
Let us first prove that $(N_\psi)' \cap M=\mathbf{C} 1$ when $N \subset M$ has finite index. Observe that $\mathord{\text{\rm E}}_N$ sends $(N_\psi)' \cap M$ onto $(N_\psi)' \cap N$ and $(N_\psi)' \cap N=\mathbf{C} 1$ by Connes--Takesaki relative commutant theorem \cite[Chapter II, Theorem 5.1]{CT76}. Then the restriction of $\mathord{\text{\rm E}}_N$ to $(N_\psi)' \cap M$, which is still of finite index, is in fact a faithful normal state. This means that $(N_\psi)' \cap M$ is finite dimensional. In particular, this implies that the relative flow of weights is inner on $(N_\psi)' \cap M$. Since the relative flow of weights is also ergodic on $(N_\psi)' \cap M$, this forces $(N_\psi)' \cap M=\mathbf{C} 1$ (see \cite[Theorem XI.3.11]{Ta03}).
Now, we deal with the case when $N \subset M$ is an arbitrary discrete irreducible inclusion. We freely use the terminology and the notation of \cite[Section 3]{ILP96}. We let $\Xi \subset \mathrm{Sect}(N)$ to be the set of all irreducible sectors of $N$ which appear in the $N$-$N$-bimodule decomposition of $\mathord{\text{\rm L}}^2(M)$. Each $\xi \in \Xi$ can be represented by a unital normal endomorphism $\rho_{\xi} : N \to N$ such that the inclusion $\rho_{\xi}(N) \subset N$ is irreducible, with expectation and has finite index. We denote by $\mathord{\text{\rm E}}_{\xi} : N \to \rho_{\xi}(N)$ the unique faithful normal conditional expectation. By \cite[Lemma 2.12 (\rm ii)]{ILP96} and up to replacing each $\rho_{\xi}$ by $\rho_{\xi} \circ \operatorname{Ad}(u)$ where $u \in \mathcal U(N)$, we may assume that $\psi|_{N} \circ \mathord{\text{\rm E}}_{\xi} = \psi|_{N}$ and $\psi|_{N} \circ \rho_{\xi} = \psi|_{N}$ for every $\xi \in \Xi$. For every $\xi \in \Xi$, denote by
$$\mathcal H_{\xi} = \{T \in M \mid \rho_{\xi}(x)T = Tx, \forall x \in N\}$$
the set of intertwiners in $M$ between $\text{\rm id}_{N}$ and $\rho_{\xi}$. Since $N' \cap M = \mathbf{C} 1$, $\mathcal H_{\xi}$ is a finite dimensional Hilbert space with inner product given by $\langle V, W\rangle 1 = V^{*}W$ (see \cite[Theorem 3.3]{ILP96}). We let $n_{\xi} = \dim \mathcal H_{\xi}$ and we fix an orthogonal basis $(V(\xi)_{i})_{1 \leq i \leq n_{\xi}}$ of $\mathcal H_{\xi}$.
Denote by $\Lambda \subset \Xi$ the subset of sectors of the form $\xi = [\sigma_t^\psi]$ for some $t \in \mathbf{R}$. For every sector $\xi = [\sigma_{t}^{\psi}] \in \Lambda$, we may and will take $\rho_\xi=\sigma_t^\psi$ as a representing element. For every $\xi \in \Lambda$, $\mathcal{H}_\xi$ is one-dimensional and $V(\xi)_1$ is a unitary in $M_{\psi}$. Indeed, let $\xi = [\sigma_{t}^{\psi}] \in \Lambda$ and simply write $u = V(\xi)_1 \in \mathcal U(M)$ such that $\sigma_t^\psi(x) = uxu^*$ for every $x \in N$. Since $N ' \cap M = \mathbf{C} 1$, \cite[Th\'eor\`eme 1.5.5]{Co72} implies that $\mathord{\text{\rm E}}_N : M \to N$ is the unique faithful normal conditional expectation from $M$ onto $N$. Then we have $\operatorname{Ad}(u) \circ \mathord{\text{\rm E}}_N = \mathord{\text{\rm E}}_N \circ \operatorname{Ad}(u)$ and so $\psi = u \psi u^*$. This shows that $u \in M_\psi$.
Let $x \in (N_{\psi})' \cap M$ be any element. For every $\xi \in \Xi$ and every $i \in \{1, \dots, n_{\xi}\}$, denote by $x(\xi)_{i} = \mathord{\text{\rm E}}_{N}(V(\xi)_{i} x) \in N$ the corresponding `Fourier coefficient' of $x \in M$. Fix $\xi \in \Xi$ and $i \in \{1, \dots, n_{\xi}\}$ such that $x(\xi)_{i} \neq 0$. For every $a \in N_{\psi}$, since $a x = x a$, we have $\rho_{\xi}(a) x(\xi)_{i} = x(\xi)_{i} a$ and so $x(\xi)_{i}^{*}x(\xi)_{i} \in (N_{\psi})' \cap N$. Since $(N_{\psi})' \cap N = \mathbf{C} 1$, we have $x(\xi)_{i}^{*}x(\xi)_{i} \in \mathbf{C} 1$. We also have $x(\xi)_{i}x(\xi)_{i}^{*} \in \rho_\xi(N_{\psi})' \cap N$. Since $\rho_\xi(N) \subset N$ is an irreducible inclusion of type ${\rm III}_1$ factors with finite index, it follows from the first part, that $\rho_\xi(N_{\psi})' \cap N=(\rho_\xi(N)_{\psi})' \cap N=\mathbf{C} 1$. This means that we also have $x(\xi)_{i}x(\xi)_{i}^{*} \in \mathbf{C} 1$. This shows that $x(\xi)_{i}=\mu u$ for some $\mu > 0$ and some unitary $u \in N$. Then we obtain $\rho_\xi(a)u=ua$ for every $a \in N_\psi$. Put $\pi=\operatorname{Ad}(u^*) \circ \rho_\xi$ so that $\pi(a)=a$ for every $a \in N_\psi$. For every $a \in N_\psi$ and every $\lambda > 0$, since $\theta_\lambda(a)v_\lambda= v_\lambda a$, we also have $\theta_\lambda(a)\pi(v_\lambda)=\pi(v_\lambda)a$ and so $u_\lambda^*\pi(u_\lambda) \in (N_\psi)' \cap N=\mathbf{C} 1$. The map $\chi : \lambda \mapsto v_\lambda^*\pi(v_\lambda) \in \mathbf{T}$ is a character, that is, there exists $t \in \mathbf{R}$ such that $\chi(\lambda)= \lambda^{{\rm i}t}$ for every $\lambda > 0$. Since $\pi(a)=a$ for every $a \in N_\psi$ and since $\pi(v_\lambda)=\lambda^{{\rm i}t} \, v_\lambda$ for every $\lambda > 0$, we conclude that $\pi = \sigma_t^\psi$ and so $[\rho_\xi]=[\sigma_t^\psi]$. By assumption on the choice of $\rho_\xi$, this forces $\rho_\xi=\sigma_t^\psi$ and $u \in \mathbf{C} 1$. We have shown that if $x(\xi)_i \neq 0$, then we must have $\xi \in \Lambda$ and $x(\xi)_1 \in \mathbf{C}$. Observe that for every $s \in \mathbf{R}$, we have $\sigma_{s}^{\psi}(x) \in (N_{\psi})' \cap M$. Then for every $s \in \mathbf{R}$, every $\xi \in \Xi$ and every $i \in \{1, \dots, n_{\xi}\}$, we have
$$(\sigma_{s}^{\psi}(x))(\xi)_i = 0 = x(\xi)_i \quad \text{if} \quad \xi \notin \Lambda$$
and
$$(\sigma_{s}^{\psi}(x))(\xi)_1 = \mathord{\text{\rm E}}_{N}(V(\xi)_{1} \sigma_{s}^{\psi}(x)) = \mathord{\text{\rm E}}_{N}(\sigma_{s}^{\psi}(V(\xi)_{1} x)) =\sigma_{s}^{\psi}( \mathord{\text{\rm E}}_{N}(V(\xi)_{1} x)) = x(\xi)_1 \quad \text{if} \quad \xi \in \Lambda$$
since $V(\xi)_{1} \in M_{\psi}$ and $x(\xi)_{1} \in \C1$. Since the Fourier coefficients uniquely determine $x \in M$ (see \cite[p.\ 45]{ILP96}), we have $\sigma_{s}^{\psi}(x) = x$ for every $s \in \mathbf{R}$. This implies that $x \in M_{\psi}$ and so $x \in (N_{\psi})' \cap M_{\psi}$.
\end{proof}
We give below a precise description of the relative commutant $(N_{\psi})' \cap M_{\psi}$. We need to introduce some more terminology. Let $G$ be any discrete group, $N$ any von Neumann algebra, $\alpha : G \curvearrowright N$ any action and $c \in \mathord{\text{\rm Z}}^2(G,\mathbf{T})$ any scalar $2$-cocycle. The \emph{twisted crossed product} $M = N \rtimes_{\alpha, c} G$ is the von Neumann algebra generated by $N$ and a family of unitaries $(u_g)_{g \in G}$ that is characterized by the following properties:
\begin{itemize}
\item [$(\rm i)$] $u_g x u_g^*=\alpha_g(x)$ for all $x \in N$ and all $g \in G$.
\item [$(\rm ii)$] $u_{gh}=c(g,h) \, u_gu_h$ for all $g,h \in G$.
\item [$(\rm iii)$] There exists a faithful normal conditional expectation $\mathord{\text{\rm E}}_N : M \rightarrow N$ that satisfies $\mathord{\text{\rm E}}_N(u_g)=0$ for all $g \in G \setminus \{e\}$.
\end{itemize}
When $N=\mathbf{C} 1$, we obtain the \emph{twisted group von Neumann algebra} that we denote by $\mathord{\text{\rm L}}_c(G)$ (see e.g.\ \cite{Su79}). Observe that $\mathord{\text{\rm L}}_c(G)$ is a tracial von Neumann algebra with canonical faithful normal tracial state $\tau$ that satisfies $\tau(u_{g}) = 0$ for all $g \in G \setminus \{e\}$.
We also need the following technical result that is probably known to specialists. We nevertheless include a proof for the reader's convenience.
\begin{lem}\label{lem:location}
Let $M$ be any $\sigma$-finite von Neumann algebra and denote by $(M, \mathord{\text{\rm L}}^{2}(M), J, \mathord{\text{\rm L}}^{2}(M)^{+})$ its standard form. Let $A \subset M$ by any von Neumann subalgebra and $\xi \in \mathord{\text{\rm L}}^2(M)^{+}$ any unit cyclic separating vector such that the faithful state $\varphi = \langle \, \cdot \, \xi, \xi\rangle \in M_{\ast}$ is tracial on $A$. Denote by $\overline{A \xi}$ the closure of $A \xi$ in $\mathord{\text{\rm L}}^2(M)$.
Then for every element $x \in M$ such that $x\xi \in \overline{A \xi}$, we have $x \in A$.
\end{lem}
\begin{proof}
Let $x \in M$ be any element such that $x \xi \in \overline{A \xi}$. We show that $x \in A$. Take a sequence $(x_n)_n$ of elements in $A$ such that $\lim_n \|(x - x_n)\xi\| = 0$. Note that $(x_{n})_{n}$ need not be uniformly bounded. Since $\varphi |_{A}$ is tracial and since $x_n \in A$ for every $n \in \mathbf{N}$, we have
\begin{align*}
\|x_n^*\xi - x_m^*\xi\|^2 &= \|x_n^* - x_m^*\|_\varphi^2 \\
&= \varphi((x_n - x_m)(x_n - x_m)^*) \\
&= \varphi((x_n - x_m)^*(x_n - x_m)) \\
&= \|x_n - x_m\|_\varphi^2 \\
&= \|x_n\xi - x_m\xi\|^2
\end{align*}
for all $m, n \in \mathbf{N}$. In particular, the sequence $(x_n^*\xi)_n$ is Cauchy in $\mathord{\text{\rm L}}^2(M)$ and so is convergent in $\mathord{\text{\rm L}}^2(M)$. Let $\eta \in \mathord{\text{\rm L}}^2(M)$ be such that $\lim_n \|\eta - x_n^*\xi\| = 0$. Recall that the (possibly unbounded) operator $S_0 : M\xi \to \mathord{\text{\rm L}}^2(M) : x\xi \to x^* \xi$ is closable and denote by $S$ its closure. Since $(x_n \xi, S(x_n \xi)) = (x_n \xi, x_n^*\xi) \to (x\xi, \eta)$ as $n \to \infty$ and since the graph of $S$ is closed, it follows that $\eta = S(x\xi) = x^*\xi$. This shows that $\lim_n \|x^* - x_n^*\|_\varphi = \lim_n \|x^*\xi - x_n^*\xi\| = 0$ and so
$$\lim_n \|x - x_n\|_\varphi^\sharp = 0.$$
Then \cite[Lemma 2]{EW76} shows that $x \in A$.
\end{proof}
Under the same assumption as in Theorem \ref{thm-relative-connes-takesaki}, we obtain the following precise description of the relative commutant $(N_{\psi})' \cap M_{\psi}$.
\begin{thm}\label{thm-description}
Let $N \subset M$ be any discrete irreducible inclusion of factors with separable predual and with expectation $\mathord{\text{\rm E}}_N : M \to N$. Assume that $N$ is a type ${\rm III}_1$ factor. Let $\psi$ be any dominant weight on $N$ and extend it to a dominant weight on $M$ by using $\mathord{\text{\rm E}}_N$.
Denote by $G \subset \mathbf{R}$ the subgroup of all the elements $t \in \mathbf{R}$ for which there exists $u_{t} \in \mathcal U(M)$ such that $\sigma_{t}^{\psi}(x) = u_{t}xu_{t}^{*}$ for every $x \in N$. Then there exists a scalar $2$-cocycle $c \in \mathord{\text{\rm Z}}^{2}(G, \mathbf T)$ such that
$$N \vee ((N_{\psi})' \cap M_{\psi}) = N \rtimes_{\sigma^{\psi}, c} G \quad \text{and} \quad (N_{\psi})' \cap M_{\psi} = \mathord{\text{\rm L}}_{c}(G).$$
Moreover, $\mathord{\text{\rm E}}_{N}|_{(N_{\psi})' \cap M_{\psi}} = \tau 1$ where $\tau$ is the canonical trace on $\mathord{\text{\rm L}}_{c}(G)$ and the relative flow of weights on $(N_{\psi})' \cap M_{\psi}$ satisfies $\theta^\psi_\lambda(u_t)=\lambda^{{\rm i}t} \, u_t$ for all $\lambda > 0$ and all $t \in G$.
\end{thm}
\begin{proof}
Write $W = \bigvee \{u_{t} \in \mathcal U(M) \mid t \in G\} \subset M$. Write $Q = N \vee W \subset M$. Since $\sigma^\psi : G \to \mathord{\text{\rm Aut}}(M)$ is a group homomorphism and since $N' \cap M = \mathbf{C} 1$, the map $c : G \times G \to \mathbf T : (s, t) \mapsto c(s, t)$ defined by $u_{s + t} = c(s, t) \, u_s u_t$ is a scalar $2$-cocycle, that is $c \in \mathord{\text{\rm Z}}^{2}(G, \mathbf T)$, such that
$$N \vee W = N \rtimes_{\sigma^{\psi}, c} G \quad \text{and} \quad W = \mathord{\text{\rm L}}_{c}(G).$$
(Observe that we have $\mathord{\text{\rm E}}_{N}(u_{t}) = 0$ for every $t \in G \setminus \{0\}$ by \cite[Lemme 1.5.6]{Co72}). Since $Q \subset M$ is globally invariant under $\sigma^{\psi}$ and since $\psi|_{Q}$ is semifinite, \cite[Theorem]{Ta71} implies that there exists a $\psi$-preserving conditional expectation $\mathord{\text{\rm E}}_{Q} : M \to Q$. Since $N' \cap M = \C1$, \cite[Th\'eor\`eme 1.5.5]{Co72} implies that $\mathord{\text{\rm E}}_{N} \circ \mathord{\text{\rm E}}_{Q} = \mathord{\text{\rm E}}_{N}$.
We show that $W = (N_{\psi})' \cap M_{\psi}$. Following the notation of the proof of Theorem \ref{thm-relative-connes-takesaki}, we have $\Lambda = \{[\sigma_{t}^{\psi}] \mid t \in G\} \subset \Xi$. For every $\xi = [\sigma_{t}^{\psi}] \in \Lambda$, we may assume that $V(\xi)_{1} = u_{t}$. We already observed that for every $t \in G$, $u_{t} \in (N_{\psi})' \cap M_{\psi}$. Thus, we have $W \subset (N_{\psi})' \cap M_{\psi}$. Let now $x \in (N_{\psi})' \cap M_{\psi}$ be any element. We proved in Theorem \ref{thm-relative-connes-takesaki} that for every $\xi \notin \Lambda$ and every $i \in \{1, \dots, n_{\xi}\}$, we have $x(\xi)_{i} = 0$ and for every $\xi \in \Lambda$, we have $x(\xi)_{1} \in \mathbf{C}$. Observe that $\mathord{\text{\rm E}}_{Q}(x) \in (N_{\psi})' \cap M_{\psi}$. Then for every $\xi \in \Xi$ and every $i \in \{1, \dots, n_{\xi}\}$, we have
$$(\mathord{\text{\rm E}}_{Q}(x))(\xi)_i = 0 = x(\xi)_i \quad \text{if} \quad \xi \notin \Lambda$$
and
$$(\mathord{\text{\rm E}}_{Q}(x))(\xi)_1 = \mathord{\text{\rm E}}_{N}(V(\xi)_{1} \mathord{\text{\rm E}}_{Q}(x)) = \mathord{\text{\rm E}}_{N}(\mathord{\text{\rm E}}_{Q}(V(\xi)_{1} x)) = \mathord{\text{\rm E}}_{N}(V(\xi)_{1} x) = x(\xi)_1 \quad \text{if} \quad \xi \in \Lambda$$
since $V(\xi)_{1} \in W \subset Q$ and $\mathord{\text{\rm E}}_{N} \circ \mathord{\text{\rm E}}_{Q} = \mathord{\text{\rm E}}_{N}$. Since the Fourier coefficients uniquely determine $x \in M$ (see \cite[p.\ 45]{ILP96}), we have $\mathord{\text{\rm E}}_{Q}(x) = x$ and so $x \in Q$. Choose any faithful state $\varphi \in M_{\ast}$ such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$. Observe that $\xi_{\varphi} \in \mathord{\text{\rm L}}^{2}(N)^{+} \subset \mathord{\text{\rm L}}^{2}(Q)^{+}$. The result obtained in Theorem \ref{thm-relative-connes-takesaki} and \cite[p.\ 45]{ILP96} show that in the standard form $\mathord{\text{\rm L}}^{2}(Q)$, we have the following convergence
$$x \xi_{\varphi} = \sum_{\xi \in \Lambda} x(\xi)_{1} V(\xi)_{1}\xi_{\varphi} = \sum_{t \in G} x_{t} u_{t}\xi_{\varphi} \in \overline{\mathord{\text{\rm L}}_{c}(G)\xi_{\varphi}}$$
where $x_{t} 1 = x(\xi)_{1} \in \mathbf{C} 1$ for $\xi = [\sigma_{t}^{\psi}]$. Since $\varphi |_{\mathord{\text{\rm L}}_{c}(G)}$ is the canonical trace $\tau$, Lemma \ref{lem:location} shows that $x \in \mathord{\text{\rm L}}_{c}(G) = W$.
It remains to compute the relative flow of weights on $(N_{\psi})' \cap M_{\psi}$. It is plain to see that $\mathord{\text{\rm E}}_{N}|_{(N_{\psi})' \cap M_{\psi}} = \tau 1$. By using \cite[Theorem ${\rm XII}$.1.1]{Ta03}, we can identify the inclusions
$$\left( N \subset M \right) = \left ( N_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \subset M_{\psi} \rtimes_{\theta} \mathbf{R}^*_+ \right)$$
where $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$ is a trace-scaling action that leaves $N_{\psi} \subset M_{\psi}$ globally invariant. We denote by $(v_{\lambda})_{\lambda > 0}$ the canonical unitaries in $N$ that implement the trace-scaling action $\theta : \mathbf{R}^*_+ \curvearrowright M_{\psi}$. For every $\lambda > 0$ and every $t \in \mathbf{R}$, we have
$$u_{t} v_{\lambda} u_{t}^{*} = \sigma_{t}^{\psi}(v_{\lambda}) = \lambda^{-{\rm i}t} \, v_{\lambda}$$
and so $\theta^\psi_{\lambda}(u_{t}) = v_{\lambda} u_{t} v_{\lambda}^{*} = \lambda^{{\rm i}t} \, u_{t}$.
\end{proof}
We deduce the following interesting consequence from Theorem \ref{thm-description}.
\begin{cor}
Let $N \subset M$ be any discrete irreducible inclusion of factors with separable predual and with expectation $\mathord{\text{\rm E}}_N : M \to N$. Assume that $N$ is a type ${\rm III}_1$ factor. Let $\psi$ be any dominant weight on $N$ and extend it to a dominant weight on $M$ by using $\mathord{\text{\rm E}}_N$.
The following assertions are equivalent:
\begin{itemize}
\item [$(\rm i)$] $(N_\psi)' \cap M=\mathbf{C} 1$.
\item [$(\rm ii)$] Every intermediate subfactor $N \subset P \subset M$ is of type ${\rm III}_1$.
\end{itemize}
\end{cor}
\begin{proof}
$(\rm i) \Rightarrow (\rm ii)$ Let $N \subset P \subset M$ be any intermediate subfactor. Observe that $\psi |_{P}$ is a dominant weight on $P$ and $P_\psi$ is a factor since $\mathcal{Z}(P_\psi) \subset (N_\psi)' \cap M = \mathbf{C} 1$. Thus, $P$ is a type ${\rm III_{1}}$ factor.
$(\rm ii) \Rightarrow (\rm i)$ By contraposition, if $(N_\psi)' \cap M \neq \mathbf{C} 1$, then by Theorem \ref{thm-description}, there exists $T > 0$ and a unitary $u \in M$ such that
$$ \forall x \in N, \quad \sigma_T^\psi(x)=uxu^*.$$
Then $P = \langle N, u\rangle$ is an intermediate subfactor such that $P \cong N \rtimes_{\sigma^\psi_T} \mathbf{Z}$ (observe that we have $\mathord{\text{\rm E}}_{N}(u^{n}) = 0$ for every $n \in \mathbf{Z} \setminus \{0\}$ by \cite[Lemme 1.5.6]{Co72}). Then \cite[Lemma 1]{Co85} implies that $P$ is a type ${\rm III}_\lambda$ factor where $\lambda = \exp(-\frac{2\pi}{T})$.
\end{proof}
The next proposition provides examples for which the open question from the introduction has a positive solution.
\begin{prop}\label{prop:open-question}
Let $N$ be any type ${\rm III_{1}}$ factor with separable predual and with trivial bicentralizer. Let $\iota : G \hookrightarrow \mathbf{R}$ be any injective homomorphism, where $G$ is an abelian countable discrete group and $c \in \mathord{\text{\rm Z}}^{2}(G, \mathbf T)$ any scalar $2$-cocycle. Let $\psi$ be any dominant weight on $N$ and define $\alpha = \sigma^\psi \circ \iota : G \curvearrowright N$ and put $M = N \rtimes_{\alpha, c} G$. Extend $\psi$ to $M$ by using the canonical conditional expectation $\mathord{\text{\rm E}}_{N} : M \to N$. Let $\varphi \in N_{\ast}$ be any faithful state.
Then $N \subset M$ is a discrete irreducible inclusion and there exists an isomorphism $$\pi : \mathord{\text{\rm B}}(N \subset M, \varphi) \rightarrow (N_\psi)' \cap M$$
such that $\theta^\psi =\pi \circ \beta^\varphi\circ \pi^{-1}$ and $\mathord{\text{\rm E}}_N(\pi(x))=\varphi(\mathord{\text{\rm E}}_N(x))1$ for all $x \in \mathord{\text{\rm B}}(N \subset M, \varphi)$.
\end{prop}
\begin{proof}
We know that $N \subset M$ is a discrete irreducible inclusion. Write $P = N \rtimes_{\sigma^\varphi \circ \iota, c} G$. By Connes' Radon--Nikodym cocycle theorem \cite[Theorem 1.2.1]{Co72}, the actions $\sigma^\varphi \circ \iota : G \curvearrowright N$ and $\sigma^\psi \circ \iota : G \curvearrowright N$ are cocycle conjugate and so the inclusions $N \subset P$ and $N \subset M$ are isomorphic. We also denote by $\mathord{\text{\rm E}}_{N} : P \to N$ the unique faithful normal conditional expectation.
Since $\mathord{\text{\rm B}}(N, \varphi) = \C1$, a straightforward argument using the Fourier expansion shows that $\mathord{\text{\rm B}}(N \subset P, \varphi) = \mathord{\text{\rm L}}_{c}(G) \subset P$ and $\mathord{\text{\rm E}}_{N}|_{\mathord{\text{\rm B}}(N \subset P, \varphi)} = \tau 1$ where $\tau$ is the canonical trace on $\mathord{\text{\rm L}}_{c}(G)$. Moreover, for every $x \in \mathord{\text{\rm B}}(N \subset P, \varphi)$, every $\lambda > 0$ and every $(a_{n})_{n} \in \ell^{\infty}(\mathbf{N}, N)$ such that $\lim_{n} \|a_{n}\varphi - \lambda \varphi a_{n}\| = 0$, we have $a_{n}x - \beta^{\varphi}_{\lambda}(x)a_{n} \to 0$ $\ast$-strongly. Since $\sigma_{t}^{\varphi}(a_{n}) - \lambda^{-{\rm i}t} a_{n} \to 0$ $\ast$-strongly for every $t \in \mathbf{R}$, it follows that $u_{g} a_{n} u_{g}^{*} - \lambda^{-{\rm i}\iota(g)} a_{n} \to 0$ $\ast$-strongly for every $g \in G$. This implies that $\beta^{\varphi}_{\lambda}(u_{g}) = \lambda^{{\rm i}\iota(g)} u_{g}$ for every $g \in G$. Combining this with the conclusion of Theorem \ref{thm-description} and since $\mathord{\text{\rm B}}(N \subset P, \varphi) \cong \mathord{\text{\rm B}}(N \subset M, \varphi)$, we obtain the desired isomorphism $\pi$.
\end{proof}
\begin{rem}
Let $N$ be any type ${\rm III_{1}}$ factor with separable predual and with trivial bicentralizer. Put $G = \mathbf{Z}^{2}$ and let $\iota : G \hookrightarrow \mathbf{R}$ be any injective homomorphism. Choose a scalar $2$-cocycle $c \in \mathord{\text{\rm Z}}^{2}(G, \mathbf T)$ such that $\mathord{\text{\rm L}}_{c}(G) \cong R$ is the hyperfinite type ${\rm II_{1}}$ factor (realize $R = \mathord{\text{\rm L}}^{\infty}(\mathbf T) \rtimes \mathbf{Z}$ where the action $\mathbf{Z} \curvearrowright \mathbf T$ comes from an irrational rotation). Let $\psi$ be any dominant weight on $N$ and define $\alpha = \sigma^\psi \circ \iota : G \curvearrowright N$ and put $M = N \rtimes_{\alpha, c} G$.
By Theorem \ref{thm-description}, since $(N_{\psi})' \cap M_{\psi} = \mathord{\text{\rm L}}_{c}(G) \cong R$ is a factor, it follows that $M_{\psi}$ is a factor and so $M$ is a type ${\rm III_{1}}$ factor. Then $N \subset M$ is a discrete irreducible inclusion of type ${\rm III_{1}}$ factors. Proposition \ref{prop:open-question} implies that the relative bicentralizer flow (which coincides with the relative flow of weights) is ergodic. We also point out that $M$ has trivial bicentralizer. Indeed, let $\varphi \in M_{\ast}$ be any faithful state such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$. By Proposition \ref{prop:open-question}, we may identify $M = N \rtimes_{\sigma^{\varphi} \circ \iota, c} G$ and we have $\mathord{\text{\rm B}}(N \subset M, \varphi) = \mathord{\text{\rm L}}_{c}(G)$. Since $\mathord{\text{\rm L}}_{c}(G)$ is a factor and $\mathord{\text{\rm L}}_{c}(G) \subset M_{\varphi}$ and since $\mathord{\text{\rm B}}(M, \varphi) \subset \mathord{\text{\rm B}}(N \subset M, \varphi) = \mathord{\text{\rm L}}_{c}(G)$ and $\mathord{\text{\rm B}}(M, \varphi) \subset (M_{\varphi})' \cap M$, it follows that $\mathord{\text{\rm B}}(M, \varphi) = \mathbf{C} 1$.
\end{rem}
\subsection{Proof of Corollary \ref{cor-characterization}}
\begin{proof}[Proof of Corollary \ref{cor-characterization}]
$(\rm ii) \Rightarrow (i)$ This follows from Proposition \ref{prop-masa}.
$(\rm i) \Rightarrow (i\rm i)$ Fix a dominant weight $\psi$ on $N$. Since $\mathord{\text{\rm c}}(N)' \cap \mathord{\text{\rm c}}(M) = \mathbf{C} 1$, we have $(N_{\psi})' \cap M = \mathbf{C} 1$ by Theorem \ref{thm-relative-connes-takesaki}. Since $N$ is amenable, so is $N_{\psi}$ and thus the inclusion $N_{\psi} \subset M$ satisfies the weak relative Dixmier property. By Theorem \ref{thm-relative-bicentralizer}, there exists a faithful state $\varphi \in M_{\ast}$ such that $\varphi \circ \mathord{\text{\rm E}}_{N} = \varphi$ and $(N_{\varphi})' \cap M = \mathbf{C} 1$. Finally, by \cite[Theorem 3.2]{Po81}, there exists an abelian von Neumann subalgebra $A \subset N_{\varphi}$ (with expectation) that is maximal abelian in $M$.
\end{proof}
\begin{proof}[Proof of Application \ref{app-crossed-product}]
We canonically have $\mathord{\text{\rm c}}(N \rtimes \Gamma) = \mathord{\text{\rm c}}(N) \rtimes \Gamma$. Application \ref{app-crossed-product} is now a consequence of Corollary \ref{cor-characterization} and \cite[Proposition 5.4]{HS88}.
\end{proof}
\begin{proof}[Proof of Application \ref{app-minimal-action}]
This is a consequence of Corollary \ref{cor-characterization} and \cite[Corollary 5.14]{Iz01} and its proof where it is shown that $\mathord{\text{\rm c}}(M^{\mathbf G})' \cap \mathord{\text{\rm c}}(M) = \C1$.
\end{proof}
\section{Bicentralizers of tensor product factors}
We first prove a relation between the bicentralizer of a tensor product and the product of the bicentralizers.
\begin{prop} \label{tensor-formula}
Let $M$ and $N$ be any two $\sigma$-finite factors with faithful states $\varphi \in M_{\ast}$ and $\psi \in N_{\ast}$. Then we have
$$ \mathord{\text{\rm B}}(M \mathbin{\overline{\otimes}} N,\varphi \otimes \psi) \subset \mathord{\text{\rm B}}(M,\varphi) \mathbin{\overline{\otimes}} \mathord{\text{\rm B}}(N,\psi).$$
If $M$ is a type ${\rm III}_1$ factor, then for all $x \in \mathord{\text{\rm B}}(M \mathbin{\overline{\otimes}} N,\varphi \otimes \psi)$ and all $\lambda > 0$, we have
$$ \beta^{\varphi \otimes \psi}_\lambda(x)=(\beta_\lambda^\varphi \otimes \text{\rm id})(x).$$
\end{prop}
\begin{proof}
Let $a \in \mathord{\text{\rm B}}(M \mathbin{\overline{\otimes}} N,\varphi \otimes \psi)$ and write $a\xi_{\varphi \otimes \psi} = \sum_{i \in I} \xi_i \otimes \eta_i$ where $(\xi_i)_{i \in I}$ is an orthonormal basis of $\mathord{\text{\rm L}}^2(M)$. Take $(x_n)_n \in \mathord{\text{\rm AC}}(N, \psi)$. Since $a \in \mathord{\text{\rm B}}(M \mathbin{\overline{\otimes}} N,\varphi \otimes \psi)$, we have $(1 \otimes x_n) a\xi_{\varphi \otimes \psi} - a\xi_{\varphi \otimes \psi} (1 \otimes x_n) \rightarrow 0$. Since $(\xi_i)_{i \in I}$ is orthonormal, this easily implies that $x_n \eta_i - \eta_{i} x_{n} \rightarrow 0$ for all $i \in I$. This shows that $\eta_i \in \overline{\mathord{\text{\rm B}}(N,\psi)\xi_{\psi}}$ and so $a \in M \mathbin{\overline{\otimes}} \mathord{\text{\rm B}}(N,\psi)$. Similarly, by decomposing over an orthonormal basis of $\mathord{\text{\rm L}}^2(N)$, we obtain $a \in \mathord{\text{\rm B}}(M,\varphi) \mathbin{\overline{\otimes}} \mathord{\text{\rm B}}(N,\psi)$.
\end{proof}
Now, we prove Theorem \ref{tensor_products_trivial}. For this, we will need the following criterion which is extracted from \cite[Theorem 2.3]{Ha85}.
\begin{thm} \label{local_haagerup}
Let $M$ be any $\sigma$-finite type ${\rm III}_1$ factor. Assume that there exists $\kappa > 0$ such that for every $\delta > 0$, every faithful state $\varphi \in M_{\ast}$ and every $x \in M$ such that $x \xi_\varphi=\xi_\varphi x^*$, $\varphi(x)=0$ and $\|x\|_\varphi=1$, we can find $a \in M$ such that
$$ \| a \|_\varphi + \| ax\|_\varphi < \kappa \| xa-a x \|_\varphi$$
and
$$ \|a \xi_\varphi - \xi_\varphi a\| < \delta \| xa-a x \|_\varphi. $$
Then $M$ has trivial bicentralizer.
\end{thm}
\begin{proof}
Note that in the proof of \cite[Theorem 2.3 $(1) \Rightarrow (2)$]{Ha85}, Condition $(1)$ is only used to obtain the following claim \cite[Lemma 2.9]{Ha85}:
\begin{itemize}
\item There exists a constant $\kappa > 0$ such that for every $\delta > 0$, every cyclic separating vector $\xi \in \mathord{\text{\rm L}}^2(M)^+$ and every unit vector $\eta=\eta^* \in \mathord{\text{\rm L}}^2(M)$ that is orthogonal to $\xi$, there exists $a \in M$ such that
$$ \| a \xi \| + \| a\eta\| < \kappa \| a\eta-a \eta \|$$
and
$$ \|a \xi - \xi a\| < \delta \| a\eta-\eta a \|. $$
\end{itemize}
Once this claim is obtained, the proof of \cite[Theorem 2.3]{Ha85} can be continued without any other assumption on $M$. Moreover, a closer look at the proof of \cite[Lemma 2.9]{Ha85} shows that one only needs to prove this claim when $\eta$ is of the form $\eta=x\xi=\xi x^*$ for some $x \in M$. Finally, by writing $\xi=\xi_\varphi$ for some faithful state $\varphi \in M_*$ and using the fact that
$$ \| a \eta- \eta a \| = \| a \, x \xi_\varphi- x \xi_\varphi \, a \| \leq \| ax-xa \|_\varphi + \| x \|_\infty \cdot \| a \xi_\varphi - \xi_\varphi a \|$$
we see that this claim can be reformulated as in our statement.
\end{proof}
\begin{proof}[Proof of Theorem \ref{tensor_products_trivial}]
Let $N$ be any factor such that $M \mathbin{\overline{\otimes}} N$ has trivial bicentralizer. Put $P=R_\infty$. If $N$ is of type ${\rm III}_\lambda$ $(0 < \lambda < 1)$, put instead $P=R_\lambda$. If $N$ is semifinite, then $M \mathbin{\overline{\otimes}} pNp$ has trivial bicentralizer for any nonzero finite projection $p \in N$ and so we may assume that $N$ is finite. In that case, put instead $P=\C1$. We have to show that $M \mathbin{\overline{\otimes}} P$ has trivial bicentralizer. In fact, since $(M \mathbin{\overline{\otimes}} P) \mathbin{\overline{\otimes}} N$ has trivial bicentralizer by Proposition \ref{tensor-formula}, we may assume that $M \cong M \mathbin{\overline{\otimes}} P$.
We use the criterion of Theorem \ref{local_haagerup}. Let $\delta > 0$ and $\varphi \in M_{\ast}$ be any faithful state. Let $x \in M$ be such that $x\xi_\varphi=\xi_\varphi x^*$ with $\| x \|_\varphi=1$ and $\varphi(x)=0$. Let $\psi \in N_{\ast}$ be any faithful state (if $N$ is of type ${\rm III}_\lambda$ with $0 < \lambda < 1$, we let $\psi$ to be a periodic state and if $N$ is finite, we let $\psi$ to be the trace). Since $M \mathbin{\overline{\otimes}} N$ has trivial bicentralizer, we can find an element $b \in \mathord{\text{\rm Ball}}(M \mathbin{\overline{\otimes}} N)$ such that $\| b\xi_{\varphi \otimes \psi}-\xi_{\varphi \otimes \psi} b\| \leq \delta$ and $\| b(x \otimes 1)-(x \otimes 1)b \|_{\varphi \otimes \psi} > \frac{1}{2}$ (see \cite[Lemma 3.2]{Ha85}).
Use the spectral theorem to identify $\mathord{\text{\rm L}}^2(N)$ with $\mathord{\text{\rm L}}^2(T,\mu)$ for some probability space $(T,\mu)$ in a such a way that $\Delta_\psi$ is identified with a multiplication operator by a borel function $f : (T, \mu) \rightarrow \mathbf{R}^*_+$ (if $N$ is of type ${\rm III}_\lambda$ with $0 < \lambda < 1$, then $f$ takes its values in $\lambda^\mathbf{Z}$ and if $N$ is finite then $f$ is constant equal to $1$). Identify $\zeta=b \xi_{\varphi \otimes \psi} $ with a function $t \mapsto \zeta(t)$ in $\mathord{\text{\rm L}}^2(T,\mu, \mathord{\text{\rm L}}^2(M))$. Note that
$$ \| b\xi_{\varphi \otimes \psi} -\xi_{\varphi \otimes \psi} b\|^2=\| (1-\Delta_{\varphi \otimes \psi}^{1/2}) \zeta \|^2=\int_T \| (1-f(t)^{1/2} \Delta_\varphi^{1/2})\zeta(t) \|^2 \, \mathord{\text{\rm d}} \mu(t). $$
Therefore, $\zeta(t)$ is in the domain of $\Delta_\varphi^{1/2}$ for almost every $t \in T$ and we have
$$ \int_T \| (1-f(t)^{1/2} \Delta_\varphi^{1/2})\zeta(t) \|^2 \, \mathord{\text{\rm d}} \mu(t) \leq \delta^2.$$
We also have
$$ \| (x \otimes 1)b -b(x\otimes 1)\|_{\varphi \otimes \psi}^2=\| (x \otimes 1) \zeta - \zeta (x^* \otimes 1) \|^2=\int_T \| x \zeta(t)-\zeta(t) x^*\|^2 \, \mathord{\text{\rm d}} \mu(t). $$
Then we have
$$ \int_T \| x \zeta(t)-\zeta(t) x^*\|^2 \, \mathord{\text{\rm d}} \mu(t) > \frac{1}{4}.$$
Similarly, we also have
$$ \int_T (\| \zeta(t) \|^2 + \|\zeta(t)x^*\|^2) \, \mathord{\text{\rm d}} \mu(t) = \|b\|_{\varphi \otimes \psi}^2+\|b(x \otimes 1)\|_{\varphi \otimes \psi}^2 \leq 2. $$
Gathering all these inequalities, we obtain
$$ \int_T \left( \| \zeta(t) \|^2 + \|\zeta(t)x^*\|^2+\frac{1}{\delta^{2}}\| (1-f(t)^{1/2} \Delta_\varphi^{1/2})\zeta(t) \|^2 \right) \mathord{\text{\rm d}} \mu(t) < 12 \int_T \| x \zeta(t)-\zeta(t) x^*\|^2 \, \mathord{\text{\rm d}} \mu(t). $$
Therefore, there exists $t \in T$ such that $\zeta(t)$ is in the domain of $\Delta_\varphi^{1/2}$ and
$$ \| \zeta(t) \|^2 + \|\zeta(t)x^*\|^2+\frac{1}{\delta^{2}}\| (1-f(t)^{1/2} \Delta_\varphi^{1/2})\zeta(t) \|^2 < 12 \| x \zeta(t)-\zeta(t) x^*\|^2. $$
Recall that the graph of $\Delta_\varphi^{1/2}$ is the closure of $\{ (c\xi, \xi c) \mid c \in M \}$. Then we can find $c \in M$ such that
$$ \| c\xi \|^2 + \|c \xi x^*\|^2+\frac{1}{\delta^{2}}\| c\xi-f(t)^{1/2} \xi c \|^2 < 12 \| x c \xi - c\xi x^*\|^2. $$
Now, since $M \cong M \mathbin{\overline{\otimes}} P$, we can find a nonzero partial isometry $v \in M' \cap M^\omega$ such that $v \xi^\omega=f(t)^{-1/2} \xi^\omega v$ (note that $f(t) \in \lambda^{\mathbf{Z}}$ if $N$ is of type ${\rm III}_\lambda$ with $0 < \lambda < 1$ and $f(t)=1$ if $N$ is finite). Since $\mathord{\text{\rm E}}_M(v^*v) > 0$ is a nonzero scalar, then in $\mathord{\text{\rm L}}^2(M^\omega)$, we obtain
$$ \| vc\xi^\omega \|^2 + \|vc \xi^\omega x^*\|^2+\frac{1}{\delta^{2}}\| vc\xi^\omega-\xi^\omega vc \|^2 < 12 \| x vc \xi^\omega - vc\xi^\omega x^*\|^2. $$
By letting $a=v_nc$ for $n$ large enough where $v=(v_n)^\omega$, we obtain
$$ \| a\xi_\varphi \|^2 + \|a \xi_\varphi x^*\|^2+\frac{1}{\delta^{2}}\| a\xi_\varphi-\xi_\varphi a \|^2 < 12 \| x a\xi_\varphi - a\xi_\varphi x^*\|^2. $$
Since $x\xi_\varphi=\xi_\varphi x^*$, this implies that
$$ \| a\|_\varphi^2 + \| ax\|_\varphi^2 < 12 \| xa - ax\|_\varphi^2 $$
and
$$\| a\xi_\varphi-\xi_\varphi a \|^2 < 12\delta^2 \| xa - ax\|_\varphi^2 $$
Since $\delta > 0$ is arbitrary, this finishes the proof.
\end{proof}
\begin{proof}[Proof of Application \ref{UPF}]
It is enough to prove the theorem for $n=2$. Also, it is enough to show that we can find some $i \in \{1, \dots, m \}$ such that $M_i \preceq_M N_1$. Indeed, we can then use \cite[Proposition 6.3 $(\rm v)$]{HMV16} and conclude by induction over $m \geq 1$.
We first note that $M_i$ has trivial bicentralizer for all $i \in \{1, \dots, m \}$ by \cite[Theorem C]{HI15}. Proposition \ref{tensor-formula} implies that $M$ has trivial bicentralizer. Therefore $N_1 \mathbin{\overline{\otimes}} R_\infty$ and $N_2 \mathbin{\overline{\otimes}} R_\infty$ also have trivial bicentralizer by Theorem \ref{tensor_products_trivial}. Therefore, we can apply \cite[Lemma 5.2]{HI15} to the decomposition
$$\mathcal{M}= (R_\infty \mathbin{\overline{\otimes}} R_\infty) \mathbin{\overline{\otimes}} M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m=(N_1 \mathbin{\overline{\otimes}} R_\infty) \mathbin{\overline{\otimes}} (N_2 \mathbin{\overline{\otimes}} R_\infty).$$
We obtain some $i \in \{1, \dots, m \}$ such that $M_i \preceq_{\mathcal{M}} N_1 \mathbin{\overline{\otimes}} R_\infty$. But this implies that $M_i \preceq_M N_1$ by \cite[Lemma 6.1]{HMV16}.
\end{proof}
|
1,108,101,565,452 | arxiv | \section{Introduction}
It is well-known that the pure state space $P({\mathcal A})$ of a quantum spin chain ${\mathcal A}$
(UHF-algebra, see subsection \ref{setting})
is homogeneous under the action of the asymptotically inner automorphisms \cite{powers}, \cite{brat}, \cite{fkk}.
In fact, the homogeneity is proven for much larger class, i.e.,
for all the separable simple $C^{*}$-algebras \cite{kos}.
In this paper, we focus on the subset $SP({\mathcal A})$ of $P({\mathcal A})$
consisting of pure states satisfying the split property. (See Definition \ref{split}.)
One equivalent condition for a state $\omega\in P({\mathcal A})$ to satisfy the split property is that
$\omega$ is quasi-equivalent to $\omega\vert_{{\mathcal A}_{L}}\otimes \omega\vert_{{\mathcal A}_{R}}$.
(See Remark \ref{splitrem}.)
Here, $\omega\vert_{{\mathcal A}_{L}}$, $\omega\vert_{{\mathcal A}_{R}}$ are restrictions of $\omega$
onto the left/right half-infinite chains. (See subsection \ref{setting}.)
A product state on ${\mathcal A}={\mathcal A}_{L}\otimes {\mathcal A}_{R}$ has no entanglement between
${\mathcal A}_{L}$ and ${\mathcal A}_{R}$ by definition.
In this sense, a state with the split property has small entanglement between ${\mathcal A}_{L}$
and ${\mathcal A}_{R}$. Using the result of \cite{powers}, \cite{brat}, \cite{fkk},\cite{kos},
one can easily see that for any $\omega_{0},\omega_{1}\in SP({\mathcal A})$,
there exist asymptotically inner automorphisms $\Xi_{L}$, $\Xi_{R}$ on ${\mathcal A}_{L}$, ${\mathcal A}_{R}$
such that $\omega_{1}\vert_{{\mathcal A}_{L}}\sim_{q.e.}\omega_{0}\vert_{{\mathcal A}_{L}}\circ \Xi_{L}$
and $\omega_{1}\vert_{{\mathcal A}_{R}}\sim_{q.e.}\omega_{0}\vert_{{\mathcal A}_{R}}\circ \Xi_{R}$.
(Here $\sim_{q.e.}$ means quasi-equivalence.)
From this and the split property of $\omega_{0}$, $\omega_{1}$, we see that $\omega_{1}$ and
$\omega_{0}\circ\left ( \Xi_{L}\otimes \Xi_{R}\right )$ are quasi-equivalent.
The product of automorphisms $\Xi_{L}\otimes \Xi_{R}$ clearly does not
create/destroy any entanglement between ${\mathcal A}_{L}$ and ${\mathcal A}_{R}$.
Hence any $\omega_{0}\in SP({\mathcal A})$
can get ''close to'' any $\omega_{1}\in SP({\mathcal A})$ without changing the entanglement.
In this sense, we may regard $SP({\mathcal A})$ to be ''homogeneous''.
What we would like to show in this paper is that the situation changes when symmetry comes into the game. This corresponds to the notion of symmetry protected topological phases in physics \cite{ogata}.
Let $SPG({\mathcal A})$ be the set of all states in $ SP({\mathcal A})$ which are invariant under the onsite action $\tau$
of a finite group $G$. (See Definition \ref{split}.)
We now require that the automorphisms $\Xi_{L}$, $\Xi_{R}$ above to preserve the symmetry
i.e., $\Xi_{L}\circ \tau_{L}(g)= \tau_{L}(g)\circ \Xi_{L}$
and $\Xi_{R}\circ \tau_{R}(g)= \tau_{R}(g)\circ \Xi_{R}$ for all $g\in G$.
(See (\ref{tgg}) for the definition of $\tau_{L}$ and $\tau_{R}$.)
For any $\omega_{0},\omega_{1}\in SPG({\mathcal A})$, can we always find such
automorphisms giving $\omega_{1}\sim_{q.e.}\omega_{0}\circ\left ( \Xi_{L}\otimes \Xi_{R}\right )$?
We show that the answer is no in general.
The obstacle is given by the second cohomology class of the projective representation of $G$
associated to $\omega\in SPG({\mathcal A})$.
We show that this second cohomology class is the complete invariant of this classification.
\subsection{Setting}\label{setting}
We consider the setting in this subsection throughout this paper.
We use the basic notation in section \ref{notasec} freely.
We start by summarizing standard setup of quantum spin chains on the infinite chain \cite{BR1,BR2}.
Throughout this paper, we fix some $2\le d\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$.
We denote the algebra of $d\times d$ matrices by $\mathop{\mathrm{M}}\nolimits_{d}$.
For each subset $\Gamma$ of ${\mathbb Z}$,
we denote the set of all finite subsets in $\Gamma$ by ${\mathfrak S}_{\Gamma}$.
We use the notation $\Gamma_{R}=[0,\infty)\cap {\mathbb Z}$ and $\Gamma_{L}=(-\infty,-1]\cap {\mathbb Z}$.
For each $z\in{\mathbb Z}$, let ${\mathcal A}_{\{z\}}$ be an isomorphic copy of $\mathop{\mathrm{M}}\nolimits_{d}$, and for any finite subset $\Lambda\subset{\mathbb Z}$, we set ${\mathcal A}_{\Lambda} = \bigotimes_{z\in\Lambda}{\mathcal A}_{\{z\}}$.
For finite $\Lambda$, the algebra ${\mathcal A}_{\Lambda} $ can be regarded as the set of all bounded operators acting on
the Hilbert space $\bigotimes_{z\in\Lambda}{{\mathbb C}}^{d}$.
We use this identification freely.
If $\Lambda_1\subset\Lambda_2$, the algebra ${\mathcal A}_{\Lambda_1}$ is naturally embedded in ${\mathcal A}_{\Lambda_2}$ by tensoring its elements with the identity.
For an infinite subset $\Gamma\subset {\mathbb Z}$,
${\mathcal A}_{\Gamma}$
is given as the inductive limit of the algebras ${\mathcal A}_{\Lambda}$ with $\Lambda\in{\mathfrak S}_{\Gamma}$.
We call ${\mathcal A}_{\Gamma}$ the quantum spin system on $\Gamma$.
In particular, we use notation
${\mathcal A}:={\mathcal A}_{{\mathbb Z}}$, ${\mathcal A}_{R}:={\mathcal A}_{\Gamma_{R}}$ and
${\mathcal A}_{L}:={\mathcal A}_{\Gamma_{L}}$.
Occasionally, we call them quantum spin chain, right infinite chain, left infinite chain, respectively.
Note that each of ${\mathcal A}_{\Lambda}$, ${\mathcal A}_{\Gamma}$ can be regarded naturally as a subalgebra of
${\mathcal A}$.
We also set ${\mathcal A}_{\rm loc,\Gamma}=\bigcup_{\Lambda\in{\mathfrak S}_\Gamma}{\mathcal A}_{\Lambda}
$, for any $\Gamma\subset {\mathbb Z}$.
We denote the standard basis of $\mathbb{C}^{d}$ by $\{e_i\}_{i=1,\ldots,d}$, and
denote the standard matrix unit of $\mathop{\mathrm{M}}\nolimits_{d}$ by $\{E_{i,j}\mid i, j=1,\ldots,d\}$.
For each finite $\Lambda\subset {\mathbb Z}$, we denote
the tensor product $\bigotimes_{k\in\Lambda}E_{i_{k},j_{k}}$ of $E_{{i_{k},j_{k}}}$ along $k\in\Lambda$,
by $E_{I,J}^{(\Lambda)}$ with $I:=(i_{k})_{k\in\Lambda}$ and
$J:=(j_{k})_{k\in\Lambda}$.
We also use the notation
\begin{align}\label{sldef}
{\mathcal S}_{\Lambda}:=\left\{E_{I,J}^{(\Lambda)}\mid I,J\in \{1,\ldots, d\}^{\times \Lambda}
\right\}.
\end{align}
Furthermore, we set $e_{I}^{(\Lambda)}:=\bigotimes_{k\in\Lambda}e_{i_{k}}\in\bigotimes_{\Lambda}{\mathbb C}^{d}$
for $I:=(i_{k})_{k\in\Lambda}$.
Throughout this paper we fix a finite group $G$ and its unitary representation $U$ on ${\mathbb C}^{d}$
satisfying
\begin{align}\label{uffl}
U(g)\notin {\mathbb C}\mathbb I_{{{\mathbb C}^{d}}},\quad \text{if} \quad g\neq e.
\end{align}
We denote the identity of $G$ by $e$.
Let $\Gamma\subset {\mathbb Z}$ be a non-empty subset.
For each $g\in G$, there exists a unique automorphism $\tau_{\Gamma}$ on ${\mathcal A}_{\Gamma}$
such that
\begin{align}\label{tgg}
\tau_{\Gamma}(g)\left ( a\right )=\mathop{\mathrm{Ad}}\nolimits\left (\bigotimes_{I} U(g)\right )\left ( a\right ),\quad a\in{\mathcal A}_{I},\quad g\in G,
\end{align}
for any finite subset $I$ of $\Gamma$.
We call the group homomorphism $\tau_{\Gamma}: G\to \mathop{\mathrm{Aut}}\nolimits {\mathcal A}_{\Gamma}$,
the on-site action of $G$
on ${\mathcal A}_{\Gamma}$ given by $U$.
In particular, when $\Gamma={\mathbb Z}$, (resp. $\Gamma=\Gamma_{R}$, $\Gamma=\Gamma_{L}$),
we denote $\tau_{\Gamma}$ by $\tau$ (resp. $\tau_{R}$, $\tau_{L}$).
For $\Gamma\subset {\mathbb Z}$, we denote by ${\mathcal A}_{\Gamma}^{G}$ the fixed point subalgebra of ${\mathcal A}_{\Gamma}$
with respect to $\tau_{\Gamma}$.
For simplicity, also use the notation ${\mathcal A}_{L}^{G}:={\mathcal A}_{\Gamma_{L}}^{G}$
and ${\mathcal A}_{R}^{G}:={\mathcal A}_{\Gamma_{R}}^{G}$.
\subsection{Projective representations of $G$}\label{pp}
A map $\sigma : G\times G\to {\mathbb T}$ is called a $2$-cocycle of $G$ if
\begin{enumerate}
\item
$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$, for all $g,h,k\in G$,
\item
$\sigma(g,e)=\sigma(e,g)=1$ for all $g\in G$.
\end{enumerate}
Define the product of two $2$-cocycles by their point-wise product.
The set of all $2$-cocycles of $G$ then becomes
an abelian group.
The resulting group we denote by
$Z^{2}(G,{\mathbb T})$.
The identity of $Z^{2}(G,{\mathbb T})$ is given by $1_{Z^{2}(G,{\mathbb T})}(g,h):=1$,
for $g,h\in G$.
For an arbitrary function $b: G\to {\mathbb T}$ such that $b(e)=1$,
\begin{align}\label{bound}
\sigma_{b}(g,h)=b(gh)^{-1} b(g)b(h), \quad g,h\in G
\end{align}
defines a $2$-cocycle.
The set of all 2-cocycles of this type
forms
a normal subgroup
$B^{2}(G,{\mathbb T})$ of $Z^{2}(G,{\mathbb T})$.
The quotient group $H^{2}(G,{\mathbb T}):=Z^{2}(G,{\mathbb T})/ B^{2}(G,{\mathbb T})$
is called the second cohomology group of $G$.
For each $\sigma\in Z^{2}(G,{\mathbb T})$, we denote by $[\sigma]_{H^{2}(G,{\mathbb T})}$ the second cohomology class that $\sigma$ belongs to.
A projective unitary representation of $G$
is a triple $({\mathcal H}, V,\sigma)$ consisting of a Hilbert space ${\mathcal H}$,
a map $V: G \to {\mathcal U} ({\mathcal H})$
and a $2$-cocycle $\sigma$ of $G$
such that $V(g)V(h) = \sigma(g, h)V(gh)$ for all $g,h\in G$. Note that we get $V(e)=\mathbb I_{{\mathcal H}}$ from the latter condition.
We call $\sigma$, the $2$-cocycle of $G$ associated to $V$, and call
$[\sigma]_{H^{2}(G,{\mathbb T})}$ the second cohomology class of $G$
associated to $V$. We occasionally say $({\mathcal H},V)$ is a projective unitary representation with $2$-cocycle $\sigma$.
The character of a finite dimensional projective unitary representation $({\mathcal H},V,\sigma)$
is given by $\chi_{V}(g)=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal H}}V(g)$, for $g\in G$.
We say a projective unitary representation
$({\mathcal H}, V,\sigma)$ of $G$ is irreducible if ${\mathcal H}$ and $0$ are the only $V$-invariant subspaces of ${\mathcal H}$.
As $G$ is a finite group, for any irreducible projective unitary representation $({\mathcal H}, V,\sigma)$ of $G$, the Hilbert space
${\mathcal H}$ is finite dimensional.
Projective unitary representations $({\mathcal H}_{1}, V_{1},\sigma_{1})$ and
$({\mathcal H}_{2}, V_{2},\sigma_{2})$ are said to be unitarily equivalent if there is a unitary
$W:{\mathcal H}_{1}\to{\mathcal H}_{2}$ such that $WV_{1}(g)W^{*}= V_{2}(g)$, with $g\in G$.
Clearly if $({\mathcal H}_{1}, V_{1},\sigma_{1})$ and
$({\mathcal H}_{2}, V_{2},\sigma_{2})$ are unitarily equivalent, the $2$-cocycles $\sigma_{1}$ and $\sigma_{2}$ coincides.
Schur's Lemma holds: let $({\mathcal H}_{1}, V_{1},\sigma_{1})$ and
$({\mathcal H}_{2}, V_{2},\sigma_{2})$ be irreducible projective unitary representations of $G$, and
$W:{\mathcal H}_{1}\to{\mathcal H}_{2}$ be a linear map such that
$W V_{1}(g)=V_{2}(g) W$ for all $g\in G$.
Then either $V=0$ or
$({\mathcal H}_{1}, V_{1},\sigma_{1})$ and
$({\mathcal H}_{2}, V_{2},\sigma_{2})$ are unitarily equivalent.
The proof is the same as that of the genuine representations (see \cite{simon} Theorem II.4.2
for example.)
For $\sigma\in Z^{2}(G,{\mathbb T})$, we denote by ${\mathcal P}_{\sigma}$,
the set of all unitarily equivalence classes of irreducible projective representations
with $2$-cocycle $\sigma$.
Note that ${\mathcal P}_{1_{Z^{2}(G,{\mathbb T})}}$ is equal to $\hat G$,
the dual of $G$.
For each $\alpha\in {\mathcal P}_{\sigma}$,
we fix a representative $({\mathcal H}_{\alpha}, V_{\alpha},\sigma)$.
We denote the dimension of ${\mathcal H}_{\alpha}$
(which is finite) by $n_{\alpha}$ and
fix an orthonormal basis
$\{\psi_k^{(\alpha)}\}_{k=1}^{n_{\alpha} }$ of ${\mathcal H}_{\alpha}$.
We introduce the matrix unit $\{f_{k,j}^{(\alpha)}\mid k,j=1,\ldots, n_{\alpha}\}$
of ${\mathcal B}({\mathcal H}_{\alpha})$ given by
\begin{align}
f_{k,j}^{(\alpha)}\xi=\braket{\psi_j^{(\alpha)}}{\xi}\psi_k^{(\alpha)},\quad
\xi\in {\mathcal H}_{\alpha}.\quad k,j=1,\ldots, n_{\alpha}.
\end{align}
We will use the following vector later, in section \ref{homogeneity}
\begin{align}\label{oa}
\Omega_{\alpha}:=
\frac 1{\sqrt{n_{\alpha}}}
\sum_{k=1}^{n_{\alpha}}
\psi_k^{(\alpha)}\otimes \psi_k^{(\alpha)}\in {\mathcal H}_{\alpha}\otimes {\mathcal H}_{\alpha}.
\end{align}
For each $\alpha\in {\mathcal P}_{\sigma}$
and $k,j=1,\ldots, n_{\alpha}$, define a function $\left ( V_{\alpha}\right )_{k,j}$
on $G$ by
\begin{align}
\left ( V_{\alpha}\right )_{k,j}(g)
:=\braket{\psi_k^{(\alpha)}}{ V_{\alpha}(g){\psi_j^{(\alpha)}}},\quad
g\in G.
\end{align}
As in Theorem III.1.1 of \cite{simon}, from Schur's Lemma,
we obtain the orthogonality relation:
\begin{align}\label{orthog}
\frac{1}{|G|} \sum_{g\in G}
\left ( V_{\alpha}\right )_{k,j}(g)
\overline{\left ( V_{\beta}\right )_{t,s}(g)}
=\frac{\delta_{\alpha,\beta}\delta_{j,s}\delta_{k,t}}{ n_{\alpha}},
\end{align}
for all $\alpha,\beta\in {\mathcal P}_{\sigma}$
and $k,j,t,s=1,\ldots, n_{\alpha}$.
Here $|G|$ denotes the number of elements in $G$.
In particular, ${\mathcal P}_{\sigma}$ is a finite set.
We freely identify $\alpha$ and $V_{\alpha}$.
For example, $\alpha\otimes \beta'$, $\alpha\otimes V$ should be understood as
$V_{\alpha}\otimes V_{\beta'}$,
$V_{\alpha}\otimes V$
for $\alpha\in{\mathcal P}_{\sigma}$, $\beta'\in {\mathcal P}_{\sigma'}$, and a projective unitary representation $V$.
We repeatedly use the following fact.
\begin{lem}\label{pd}
For any projective unitary representation $({\mathcal H}, V,\sigma)$, there are
Hilbert spaces ${\mathcal K}_{\alpha}$ labeled by $\alpha\in{\mathcal P}_{\sigma}$
and a unitary $W: {\mathcal H}\to \bigoplus_{\alpha\in {\mathcal P}_{\sigma}}{\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}$
such that
\begin{align}\label{osos}
WV(g)W^{*}=\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}
V_{\alpha}(g)\otimes \mathbb I_{{\mathcal K}_{\alpha}},\quad g\in G.
\end{align}
Furthermore, the commutant $V(G)':=\{ X\in {\mathcal B}({\mathcal H})\mid [X, V(g)]=0\}$
of $V(G)$ is of the form
\begin{align}\label{com}
V(G)'=W^{*}\left (
\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}
\mathbb I_{{\mathcal H}_{\alpha}}\otimes {\mathcal B}\left ( {{\mathcal K}_{\alpha}}\right )\rmk W
\end{align}
\end{lem}
\begin{proof}
For any $V$-invariant subspace of ${\mathcal H}$, its orthogonal complement is $V$-invariant as well.
Therefore, from
Zorn's Lemma, we may decompose $({\mathcal H}, V,\sigma)$
as an orthogonal sum of irreducible projective unitary representations with $2$-cocycle $\sigma$.
This proves (\ref{osos}).
The second statement (\ref{com}) follows from the orthogonality relation (\ref{orthog}).
\end{proof}
\begin{notation}\label{ddcom}
When (\ref{osos}) holds, we say that
$V$ (or $({\mathcal H}, V,\sigma)$) has an irreducible decomposition given by
Hilbert spaces $\{{\mathcal K}_{\gamma}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
We say $V$ (or $({\mathcal H}, V,\sigma)$) contains all elements of ${\mathcal P}_{\sigma}$ if ${\mathcal K}_{\alpha}\neq \{0\}$ for
all $\alpha\in {\mathcal P}_{\sigma}$.
We say $V$ (or $({\mathcal H}, V,\sigma)$) contains all elements of ${\mathcal P}_{\sigma}$ with infinite multiplicity
if $\mathop{\mathrm{dim}}\nolimits {\mathcal K}_{\alpha}=\infty$ for
all $\alpha\in {\mathcal P}_{\sigma}$.
We hence force omit $W$ in (\ref{osos}) and identify ${\mathcal H}$ and $\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}{\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}$ freely.
The Hilbert space ${\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}$ can be naturally regarded as a closed subspace of ${\mathcal H}$. We use this identification freely and call ${\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}$ the $\alpha$-component of $V$ (or $({\mathcal H}, V,\sigma)$).
\end{notation}
\begin{notation}\label{bfvee}
Let $({\mathcal H}, V,\sigma)$ be
a projective unitary representation. Let $b:G\to {\mathbb T}$ be a map such that $b(e)=1$.
Setting $\sigma_{b}$ as in (\ref{bound}),
we obtain $\sigma\sigma_{b}\in Z^{2}(G,{\mathbb T})$.
We also set $\left ( b\cdot V\right ) (g):=b(g)V(g)$, for $g\in G$.
Then
$({\mathcal H}, b\cdot V,\sigma\sigma_b)$ is a projective representation.
\end{notation}
\subsection{The split property and projective representations}
Next let us introduce the split property.
\begin{defn}\label{split}
Let $\omega$ be a pure state on ${\mathcal A}$. Let $\omega_R$ be the restriction of
$\omega$ to ${\mathcal A}_R$, and $({\mathcal H}_{\omega_R},\pi_{\omega_R},\Omega_{\omega_R})$ be the GNS triple of $\omega_R$.
We say $\omega$ satisfies the split property with respect to ${\mathcal A}_L$ and ${\mathcal A}_R$,
if the von Neumann algebra $\pi_{\omega_R}({\mathcal A}_{R})''$ is a type I factor.
We denote by $SP({\mathcal A})$ the set of all pure states on ${\mathcal A}$
which satisfy the split property with respect to ${\mathcal A}_L$ and ${\mathcal A}_R$.
We also denote by $SPG({\mathcal A})$, the set of all states $\omega$ in $SP({\mathcal A})$, which are
$\tau$-invariant.
\end{defn}
Recall that a type I factor is $*$-isomorphic to $B({\mathcal K})$, the set of all bounded operators
on a Hilbert space ${\mathcal K}$.
See \cite{takesaki}.
\begin{rem}\label{splitrem}
Let $\omega$ be a pure state on ${\mathcal A}$.
Let $\omega_L$ be
the restriction of $\omega$ to ${\mathcal A}_L$.
Then $\omega$ satisfies the split property if and only if
$\omega_{L}\otimes \omega_{R}$ is quasi-equivalent to $\omega$. ( See \cite{Matsui2}. In Proposition 2.2 of \cite{Matsui2}, it is assumed that the state is translationally invariant because of the first equivalent condition (i). However, the proof for the equivalence between
(ii) and (iii) does not require translation invariance.)
Therefore, by the symmetric argument, if $({\mathcal H}_{\omega_L},\pi_{\omega_L},\Omega_{\omega_L})$ is the GNS triple of $\omega_L$,
the the split property of $\omega$ implies that $\pi_{\omega_L}({\mathcal A}_{L})''$ is also a type I factor.
\end{rem}
For each $\omega\in SPG({\mathcal A})$, we may associate a second cohomology class of $G$.
\begin{prop}\label{unieq}
Let $\omega\in SPG({\mathcal A})$ and ${\varsigma}=L,R$.
Then
there exists an irreducible $*$-representation $\rho_{\omega,{\varsigma}}$
of ${\mathcal A}_{{\varsigma}}$ on a Hilbert space
${\mathcal L}_{\omega,{\varsigma}}$
that is quasi-equivalent to the GNS representation
of $\omega\vert_{{\mathcal A}_{{\varsigma}}}$.
For each of such irreducible $*$-representation $({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}})$,
there is a projective unitary representation $u_{\omega,{\varsigma}}$ of $G$
on ${\mathcal L}_{\omega,{\varsigma}}$ such that
\begin{align}\label{uintro}
\rho_{\omega,{\varsigma}}\circ \tau_{\varsigma}(g)
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{\omega,{\varsigma}}(g)\right )\circ \rho_{\omega,{\varsigma}},
\end{align}
for all $g\in G$.
Furthermore,
if another triple $(\tilde {\mathcal L}_{\omega,{\varsigma}}, \tilde\rho_{\omega,{\varsigma}}, \tilde u_{\omega,{\varsigma}})$ satisfies the same conditions as $({\mathcal L}_{\omega,{\varsigma}},
\rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}})$ above, then
there is a unitary $W:{\mathcal L}_{\omega,\varsigma}\to \tilde {\mathcal L}_{\omega,\varsigma}$ and $c: G\to {\mathbb T}$
such that
\begin{align}\label{Wp}
\left ( \mathop{\mathrm{Ad}}\nolimits W\right )\circ \rho_{\omega,\varsigma} =
\tilde \rho_{\omega,\varsigma},
\end{align}
\begin{align}\label{cg}
c(g) \cdot \left ( \mathop{\mathrm{Ad}}\nolimits W\right )\left ( u_{\omega,\varsigma}(g)\right )=\tilde u_{\omega,\varsigma}(g),\quad g\in G.
\end{align}
In particular,
for $2$-cocycle $\sigma_{\omega,\varsigma}$, $\tilde \sigma_{\omega,\varsigma}$
associated to $u_{\omega,\varsigma}$, $ \tilde u_{\omega,\varsigma}$ respectively,
we have
$[\sigma_{\omega,\varsigma}]_{H^{2}(G,{\mathbb T})}=[\tilde \sigma_{\omega,\varsigma}]_{H^{2}(G,{\mathbb T})}$.
\end{prop}
\begin{proof}
Let $({\mathcal H}_{\omega_{\varsigma}},\pi_{\omega_{\varsigma}},\Omega_{\omega_{\varsigma}})$ be the GNS triple of $\omega\vert_{{\mathcal A}_{{\varsigma}}}$.
The existence of irreducible $*$-representation $({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}})$
quasi-equivalent to $\pi_{\omega_{\varsigma}}$
follows from the definition of the split property.
To see the existence of $ u_{\omega,\varsigma} $ satisfying (\ref{uintro})
for such $({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}})$,
let $\iota_{\omega,{\varsigma}}: \pi_{\omega_{\varsigma}}\left ( {\mathcal A}_{\varsigma}\right )''
\to {\mathcal B}({\mathcal L}_{\omega,{\varsigma}})$ be the $*$-isomorphism
such that
$\rho_{\omega,{\varsigma}}=\iota_{\omega,{\varsigma}}\circ \pi_{\omega_{\varsigma}}$.
By the $\tau_{\varsigma}$-invariance of $\omega\vert_{{\mathcal A}_{{\varsigma}}}$,
the action $\tau_{\varsigma}$ of $G$ can be extended to an action
$\hat\tau_{\varsigma}$ on $\pi_{\omega_{\varsigma}}\left ( {\mathcal A}_{\varsigma}\right )''$,
so that $\hat\tau_{\varsigma}(g)\circ\pi_{\omega_{\varsigma}}=\pi_{\omega_{\varsigma}}\circ \tau_{\varsigma}(g)$,
for $g\in G$.
By the Wigner Theorem, the $*$-automorphism $\iota_{\omega,{\varsigma}}\circ \hat\tau_{\varsigma}(g)\circ
\iota_{\omega,{\varsigma}}^{-1}$ on ${\mathcal B}({\mathcal L}_{\omega,{\varsigma}})$
is given by a unitary $ u_{\omega,\varsigma} (g)$
so that
\begin{align}
\iota_{\omega,{\varsigma}}\circ \hat\tau_{\varsigma}(g)\circ
\iota_{\omega,{\varsigma}}^{-1}
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{\omega,\varsigma} (g)
\right ),\quad g\in G.
\end{align}
As $\hat \tau_{\varsigma}$ is an action of $G$, $u_{\omega,\varsigma}$ is a projective unitary representation.
We obtain (\ref{uintro}) by
\begin{align}
\rho_{\omega,{\varsigma}}\circ \tau_{\varsigma}(g)
=\iota_{\omega,{\varsigma}}\circ \pi_{\omega_{\varsigma}}\circ \tau_{\varsigma}(g)
=\iota_{\omega,{\varsigma}}\circ \hat\tau_{\varsigma}(g)\circ\pi_{\omega_{\varsigma}}
=\iota_{\omega,{\varsigma}}\circ \hat\tau_{\varsigma}(g)\circ\iota_{\omega,{\varsigma}}^{-1}\circ
\iota_{\omega,{\varsigma}}\circ\pi_{\omega_{\varsigma}}
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{\omega,{\varsigma}}(g)\right )\circ \rho_{\omega,{\varsigma}}.
\end{align}
Suppose that $(\tilde {\mathcal L}_{\omega,{\varsigma}}, \tilde\rho_{\omega,{\varsigma}}, \tilde u_{\omega,{\varsigma}})$ satisfies the same conditions as $({\mathcal L}_{\omega,{\varsigma}},
\rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}})$.
Then by the Wigner Theorem, there exists a unitary $W:{\mathcal L}_{\omega,\varsigma}\to \tilde {\mathcal L}_{\omega,\varsigma}$
satisfying (\ref{Wp}).
Note that
\begin{align}
\mathop{\mathrm{Ad}}\nolimits\left ( \tilde u_{\omega,{\varsigma}}(g)\right )\circ \tilde \rho_{\omega,{\varsigma}}
=
\tilde \rho_{\omega,{\varsigma}}\circ \tau_{\varsigma}(g)
=\mathop{\mathrm{Ad}}\nolimits W\circ \rho_{\omega,\varsigma}
\circ \tau_{\varsigma}(g)
=\mathop{\mathrm{Ad}}\nolimits W\circ
\mathop{\mathrm{Ad}}\nolimits\left ( u_{\omega,{\varsigma}}(g)\right )
\circ \mathop{\mathrm{Ad}}\nolimits W^*\circ \tilde \rho_{\omega,{\varsigma}}.
\end{align}
This implies that $\tilde u_{\omega,{\varsigma}}(g)^*\mathop{\mathrm{Ad}}\nolimits W\left ( u_{\omega,{\varsigma}}(g) \right )$
belongs to ${\mathbb T}\mathbb I_{\tilde {\mathcal L}_{\omega,\varsigma}}$ proving (\ref{cg}).
\end{proof}
\begin{defn}\label{pao}Let $\omega\in SPG({\mathcal A})$ and
$({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}})$
be a triple satisfying the conditions in Proposition \ref{unieq}.
Let $\sigma_{\omega,{\varsigma}}$ be the $2$-cocycle associated to
$u_{\omega,{\varsigma}}$.
We call
$({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}},\sigma_{\omega,{\varsigma}})$
a quadruple associated to $(\omega\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$.
Furthermore, we denote the second cohomology class $[\sigma_{\omega,{\varsigma}}]_{H^{2}(G,{\mathbb T})}$
by $c_{\omega,\zeta}$, and call it the second cohomology class of $G$ associated to
$(\omega\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$.
\end{defn}
\begin{rem}\label{rem17}
For a quadruple $({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}},\sigma_{\omega,{\varsigma}})$
associated to $(\omega\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$
and any map $b:G\to{\mathbb T}$,
$({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, b\cdot u_{\omega,{\varsigma}},\sigma_{b}\sigma_{\omega,{\varsigma}})$
is also a quadruple associated to
$(\omega\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$.
See (\ref{bound}) and Notation \ref{bfvee}.
\end{rem}
\subsection{Main Theorem}
Let us introduce $\mathop{\mathrm{AInn}}\nolimits^{G} (A_{\varsigma})$.
\begin{defn}
Let $\varsigma=L,R$.
An automorphism $\Xi_{\varsigma}$ of ${\mathcal A}_{\varsigma}$ is asymptotically inner in ${\mathcal A}_{\varsigma}^{G}$
if there is a norm continuous path $w_{\varsigma}:[0,\infty)\to {\mathcal U}\left ({\mathcal A}_{\varsigma}^{G}\right )$
with $w_{\varsigma}(0)=\mathbb I_{{\mathcal A}_{\varsigma}}$
that
\begin{align}
\Xi_{\varsigma}(a)=\lim_{t\to\infty}\mathop{\mathrm{Ad}}\nolimits\left ( w_{\varsigma}(t)\right )(a),\quad
a\in {\mathcal A}_{\varsigma}.
\end{align}
We denote by $\mathop{\mathrm{AInn}}\nolimits^{G} (A_{\varsigma})$ the set of all automorphisms which are
asymptotically inner in ${\mathcal A}_{\varsigma}^{G}$.
\end{defn}
In this paper, we consider the classification problem of $SPG({\mathcal A})$
with respect to the following equivalence relation.
\begin{defn}
For $\omega_{0}$, $\omega_{1}\in SPG({\mathcal A})$, we
write $\omega_{0}\sim_{{\rm split},\tau}\omega_{1}$
if there exist automorphisms $\Xi_{L}\in \mathop{\mathrm{AInn}}\nolimits^{G}({\mathcal A}_{L})$
and $\Xi_{R}\in \mathop{\mathrm{AInn}}\nolimits^{G}({\mathcal A}_{R})$ such that
$\omega_{1}$ and $\omega_{0}\circ\left (\Xi_{L}\otimes \Xi_{R}\right )$
are quasi-equivalent.
\end{defn}
Now we are ready to state our main theorem.
\begin{thm}\label{main}
For $\omega_{0}$, $\omega_{1}\in SPG({\mathcal A})$,
$\omega_{0}\sim_{{\rm split},\tau}\omega_{1}$
if and only if
$c_{\omega_{1},R}=c_{\omega_{0},R}$.
\end{thm}
The ''only if'' part of the Theorem \ref{main} is easy to prove.
In order to prove ''if'' part of the Theorem, we note that
if $c_{\omega_{1},R}=c_{\omega_{0},R}$ holds,
$\omega_{0}$ and $\omega_{1}$
give covariant representations of a twisted $C^{*}$-dynamical systems
$\Sigma_{\Gamma_{R}}^{(\sigma_{R})}$, $\Sigma_{\Gamma_{L}}^{(\sigma_{L})}$
(see section \ref{crossedproduct}), where
$\sigma_{R}$, $\sigma_{L}$ are
$2$-cocycles of $G$ such that
$[\sigma_{R}]_{H^{2}(G,{\mathbb T})}=c_{\omega_{1},R}=c_{\omega_{0},R}$
and $[\sigma_{L}]_{H^{2}(G,{\mathbb T})}=c_{\omega_{1},L}=c_{\omega_{0},L}$.
(See Remark \ref{rem17} and Lemma \ref{clmcr}.)
One of the basic idea is to encode the information of these
$2$-cocycles $\sigma_{R}$, $\sigma_{L}$
into $C^{*}$-algebras we consider.
Namely, instead of considering ${\mathcal A}_{R}$, ${\mathcal A}_{L}$,we consider
the the twisted crossed products $C^{*}(\Sigma_{\Gamma_{R}}^{(\sigma_{R})})$,
$C^{*}(\Sigma_{\Gamma_{L}}^{(\sigma_{L})})$.
We recall the twisted crossed product $C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ of $\Sigma_{\Gamma}^{(\sigma)}$ in section \ref{crossedproduct}.
In section \ref{projsec}, we show that
for any $\omega\in SPG({\mathcal A})$, and $\varsigma=L,R$, $u_{\omega,{\varsigma}}$
contains all elements of ${\mathcal P}_{\sigma_{\omega,{\varsigma}}}$.
Therefore, for any fixed $\alpha_{\varsigma}\in {\mathcal P}_{\sigma_{{\varsigma}}}$, both of $u_{\omega_{0},{\varsigma}}$ and $u_{\omega_{1},{\varsigma}}$
contains $\alpha_{\varsigma}$.
This fact allows us to regard
the problem as the homogeneity problem of
$ {\mathcal B}({\mathcal H}_{\alpha_{\varsigma}})\otimes C^{*}(\Sigma_{\Gamma_{\varsigma}}^{(\sigma_{\varsigma})})$,
with symmetry
(section \ref{homogeneity}).
The proof of the homogeneity relies on the
machinery developed in
\cite{powers}, \cite{brat}, \cite{fkk},\cite{kos}.
However, for our problem, we would like to take the path of
unitaries in the fixed point algebras ${\mathcal A}_{R}^{G}$, ${\mathcal A}_{L}^{G}$.
This requires some additional argument using the irreducible decompositions of
$u_{\omega_{0},{\varsigma}}$, $u_{\omega_{1},{\varsigma}}$.
This is given in section \ref{homogeneity}.
\section{Irreducible components in $u_{\omega,\varsigma}$}\label{projsec}
In this section we show that
$u_{\omega,\varsigma}$ contains all elements in
${\mathcal P}_{\sigma_{\omega,\varsigma}}$ with infinite multiplicity.
As $G$ is a finite group, its dual $\hat G$ is a finite set and we denote the number of the elements in $\hat G$ by $|\hat G|$.
We use the following notation for any unitary/projective unitary representations $V_1$, $V_2$.
We write
$V_1\prec V_2$ if $V_1$ is unitarily equivalent to a sub-representation of $V_2$.
We also say $V_1$ is included in $V_2$ in this case.
Clearly, $\prec$ is a preorder.
We write $V_{1}\cong V_{2}$ if $V_1$ and $V_2$ are unitarily equivalent.
For a unitary representation (resp. projective unitary representation) of $G$,
we denote by $\bar V$ the complex conjugate representation (resp. projective representation) of $V$.
(See \cite{simon} section II.6.)
\begin{lem}\label{simple}
There is an $l_{0}\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$
such that for any $l\ge l_{0}$, the tensor product $U^{\otimes l}$
contains any irreducible representation of $G$ as its irreducible component.
\end{lem}
\begin{proof}
Note that the character $\chi_{U}(g)$ is the sum of the eigenvalues of a unitary $U(g)$ acting on
${\mathbb C}^{d}$.
Therefore, the maximal possible value of $|\chi_{U}(g)|$ is $d$, which is equal to
$\chi_{U}(e)$.
This value is attained only if $U(g)\in{\mathbb T}\mathbb I_{{\mathbb C}^{d}}$.
By the condition (\ref{uffl}), for $g\in G\setminus \{e\}$,
$|\chi_{U}(g)|$ is strictly less than $d$.
Now for any irreducible representation $({\mathbb C}^{m},V)$ of $G$,
for any $l\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$, we have
\begin{align}\label{nncc}
\sum_{g\in G}\overline{\chi_{V}(g)}\chi_{U^{\otimes l}}(g)
=d^{l}\cdot m \left (
1+\sum_{g\in G\setminus \{e\}}
\frac{\overline{\chi_{V}(g)}}{m}
\left ( \frac{\chi_{U}(g)}{ d}
\right )^{l}
\right ).
\end{align}
Note that
\begin{align}
\frac{\overline{\chi_{V}(g)}}{m}
\left ( \frac{\chi_{U}(g)}{ d}
\right )^{l}
\end{align}
for $g\in G\setminus \{e\}$
converges to $0$ because of $|\chi_{U}(g)|<d$.
Therefore, for $l$ large enough,
the left hand side of (\ref{nncc}) is non-zero.
In other word, for $l$ large enough, $V$ is an irreducible component of
$U^{\otimes l}$.
As $\hat G$ is a finite set, this proves the Lemma.
\end{proof}
From this we obtain the following.
\begin{lem}\label{abu}
There is an $l_{0}\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$
such that
$\alpha\prec \beta\otimes U^{\otimes l}$
holds for any
$\sigma\in Z^{2}(G,{\mathbb T})$,
$\alpha,\beta\in{\mathcal P}_{\sigma}$,
and
$l_{0}\le l\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$.
\end{lem}
\begin{proof}
Let $l_{0}$ be the number given in Lemma \ref{simple}.
For any $\sigma\in Z^{2}(G,{\mathbb T})$ and
$\alpha,\beta\in{\mathcal P}_{\sigma}$,
$\alpha\otimes \bar \beta$ is a genuine representation of $G$.
Let $V\in \hat G$ be an irreducible component of $\alpha\otimes \bar \beta$.
By Lemma \ref{simple},
this $V$ is realized as an irreducible component of $U^{\otimes l}$ for $l\ge l_{0}$.
Therefore, for $l\ge l_{0}$, we have
\begin{align}
\sum_{g\in G}\overline{\chi_{\alpha}(g)}\chi_{ \beta\otimes U^{\otimes l}}(g)=
\sum_{g\in G}\overline{\chi_{\alpha}(g)\chi_{\bar \beta}(g)}\chi_{U^{\otimes l}}(g)
\ge
\sum_{g\in G}\overline{\chi_{V}(g)}\chi_{U^{\otimes l}}(g)>0.
\end{align}
This means $\alpha\prec \beta\otimes U^{\otimes l}$.
\end{proof}
\begin{lem}\label{smb}
Let $\sigma\in Z^{2}(G,{\mathbb T})$ be a fixed $2$-cocycle.
For any $m\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$, there exists an $N_{m}^{(\sigma)}\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$ satisfying the following:
For
any projective unitary representation $({\mathcal H},u)$ of $G$ with $2$-cocycle
$\sigma$,
$\alpha\in{\mathcal P}_{\sigma}$, and $\mathbb{N}}\newcommand{\bb}{\mathbb{B}\ni N\ge N_{m}^{(\sigma)}$,
we have
\begin{align}
m\cdot \alpha\prec U^{\otimes N}\otimes u.
\end{align}
(Here $m\cdot \alpha$ denotes the $m$ direct sum of $\alpha$. )
\end{lem}
\begin{proof}
First let us consider the case that
${\mathcal P}_{\sigma}$ consists of a unique element $\alpha\in {\mathcal P}_{\sigma}$.
Then for any $N\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$, and any projective representation $({\mathcal H},u,\sigma)$,
the multiplicity of $\alpha$ in $U^{\otimes N}\otimes u$ is $\frac{d^{N}\cdot\mathop{\mathrm{dim}}\nolimits{{\mathcal H}}}{n_{\alpha}}$,
which is bigger or equal to a $({\mathcal H},u)$-independent value $\frac{d^{N}}{n_{\alpha}}$.
The claim of Lemma \ref{smb} follows from this immediately for this case.
Next let us consider the case that the number of elements $|{\mathcal P}_{\sigma}|$ in ${\mathcal P}_{\sigma}$,
is larger than $1$.
From Lemma \ref{abu}, choose $l_{0}\in \mathbb{N}}\newcommand{\bb}{\mathbb{B}$ so that
$\alpha\prec\beta\otimes U^{\otimes l}$
for all $l\ge l_{0}$ and $\alpha,\beta\in{\mathcal P}_{\sigma}$.
For any $m\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$, choose $M_{m}\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$ so that
$|{\mathcal P}_{\sigma}|^{M_{m}}>m$. Here we use the condition that $|{\mathcal P}_{\sigma}|>1$.
We set $N_{m}^{(\sigma)}:=l_{0}(M_{m}+1)$.
Let $({\mathcal H},u)$ be a projective unitary representation of $G$ with $2$-cocycle
$\sigma$,
$\alpha\in{\mathcal P}_{\sigma}$, and $\mathbb{N}}\newcommand{\bb}{\mathbb{B}\ni N\ge N_{m}^{(\sigma)}$.
We would like to show that $m\cdot \alpha\prec U^{\otimes N}\otimes u$.
By the choice of $N_{m}^{(\sigma)}$, $N$ can be decomposed as
$N=k_{1}+k_{2}+\cdots+k_{M_{m}}+k_{M_{m}+1}$ with some $l_{0}\le k_{j}\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$, $j=1,\ldots,M_{m}+1$.
For each $j=1,\ldots,M_{m}+1$ and $\beta,\gamma\in{\mathcal P}_{\sigma}$,
we denote the multiplicity of $\gamma$ in $U^{\otimes k_{j}}\otimes \beta$
by $n_{\beta,\gamma}^{(j)}$.
From the choice of $l_{0}$, we have $1\le n_{\beta,\gamma}^{(j)}$
for any $j=1,\ldots,M_{m}+1$ and $\beta,\gamma\in{\mathcal P}_{\sigma}$.
Fix some $\beta_{0}\in{\mathcal P}_{\sigma}$ such that $\beta_{0}\prec u$.
From this, we get
\begin{align}
&m\cdot \alpha\prec |{\mathcal P}_{\sigma}|^{M_{m}}\cdot \alpha
=
\bigoplus_{\gamma_{1},\gamma_{2},\ldots,\gamma_{ {M_{m}}}}\alpha
\prec
\bigoplus_{\gamma_{1},\gamma_{2},\ldots,\gamma_{ {M_{m}}}, \gamma_{ {M_{m}+1}}}
n_{\beta_0,\gamma_{1}}^{(1)}n_{\gamma_{1},\gamma_{2}}^{(2)}\cdots
n_{\gamma_{{M_{m}}},{\gamma_{ {M_{m}+1}}}}^{( {M_{m}+1})}\cdot \gamma_{ {M_{m}+1}}\nonumber\\
&
\prec U^{\otimes k_{M_{m}+1}}\otimes U^{\otimes k_{M_{m}}}\otimes
\cdots U^{\otimes k_{2}}\otimes U^{\otimes k_{1}}\otimes\beta_0
=
U^{\otimes N}\otimes \beta_0\prec
U^{\otimes N}\otimes u.
\end{align}
This completes the proof.
\end{proof}
Now we are ready to show the main statement of this section.
From the following Lemma, we see that for any $\omega\in SPG({\mathcal A})$,
$u_{\omega,\varsigma}$ contains all elements of ${\mathcal P}_{\sigma_{\omega,\varsigma}}$
with infinite multiplicity.
\begin{thm}\label{zbdip}
Let $\Gamma$ be an infinite subset of ${\mathbb Z}$.
Let $({\mathcal L},\rho, u,\sigma)$ be a quadruple such that
\begin{description}
\item[(i)]
$\rho$ is a $*$-representation of ${\mathcal A}_{\Gamma}$ on a Hilbert space
${\mathcal L}$,
\item[(ii)]
$u$ is a projective unitary representation of $G$ on ${\mathcal L}$
with a $2$-cocycle $\sigma$,
\item[(iii)]
for any $g\in G$, we have
\begin{align}
\rho\circ \tau_{\Gamma}(g)
=\mathop{\mathrm{Ad}}\nolimits\left ( u(g)\right )\circ \rho.
\end{align}
\end{description}
Then $u$ contains all elements of ${\mathcal P}_{\sigma}$ with infinite multiplicity.
\end{thm}
\begin{proof}
Fix any $\alpha\in{\mathcal P}_{\sigma}$ and $m\in\mathbb{N}}\newcommand{\bb}{\mathbb{B}$.
We would like to show that $m\cdot \alpha\prec u$.
Let $N_{m}^{(\sigma)}$ be the number given in Lemma \ref{smb} for this fixed $m$.
Let $\Lambda$ be a subset of $\Gamma$ such that $|\Lambda|=N_{m}^{(\sigma)}$.
We may factorize $({\mathcal L}, \rho, u)$
to $\Lambda$-part and $\Gamma\setminus \Lambda$-part as follows:
There exist a $*$-representation $(\tilde {\mathcal L},\tilde \rho)$ of ${\mathcal A}_{\Gamma\setminus \Lambda}$ and a projective unitary representation $\tilde u$ of $G$ on $\tilde{\mathcal L}$
with $2$-cocycle $\sigma$, implementing $\tau_{\Gamma\setminus \Lambda}$.
There exists a unitary $W:{\mathcal L}\to \left ( \bigotimes_{\Lambda}{\mathbb C}^{d}\right )\otimes \tilde{\mathcal L}$
such that
\begin{align}\label{prev}
W\rho(a)W^{*}=\left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{\Lambda}}\otimes \tilde\rho\right ) (a),\quad a\in{\mathcal A}_{\Gamma},
\end{align}
and
\begin{align}\label{up}
W u(g)W^{*}=\left ( \bigotimes_{\Lambda}U(g)\right )\otimes \tilde u(g),\quad g\in G.
\end{align}
More precisely, set
$I_{0}=(i_{k})_{k\in\Lambda}\in\{1,\ldots, d\}^{\times \Lambda}$,
with $i_{k}=1$ for all $k\in\Lambda$.
We define the Hilbert space
$\tilde{\mathcal L}$ by $\tilde{\mathcal L}:=\rho\left ( E_{I_{0},I_{0}}^{(\Lambda)}\right ){\mathcal L}$,
and the $*$-representation $\tilde\rho$ of ${\mathcal A}_{\Gamma\setminus \Lambda}$ on $\tilde{\mathcal L}$ by
\begin{align}
\tilde\rho\left ( a\right ) :=\rho\left ( E_{I_{0},I_{0}}^{(\Lambda)}\otimes a\right ),\quad a\in {\mathcal A}_{\Gamma\setminus\Lambda}.
\end{align}
The unitary $W:{\mathcal L}\to \left ( \bigotimes_{\Lambda}{\mathbb C}^{d}\right )\otimes \tilde{\mathcal L}$
is defined by
\begin{align}
W\xi:=\sum_{I\in\{1,\ldots, n\}^{\times \Lambda}}e_{I}^{(\Lambda)}\otimes
\rho\left ( E_{I_{0},I}^{(\Lambda)}\right )\xi,\quad \xi\in{\mathcal L}.
\end{align}
It is straight forward to check (\ref{prev}).
By a straight forward calculation using (\ref{prev}), we can check that
$\left ( \left ( \bigotimes_{\Lambda}U(g)\right )^{*}\otimes \mathbb I_{\tilde {\mathcal L}}\right ) Wu(g)W^{*}$ with $g\in G$
commute with any element of $\left ( \bigotimes_{\Lambda}\mathop{\mathrm{M}}\nolimits_{d}\right )\otimes {\mathbb C} \mathbb I_{\tilde{\mathcal L}}$.
Hence there exists a unitary $\tilde u(g)$ on $\tilde {\mathcal L}$ such that
$\left (\lmk \bigotimes_{\Lambda}U(g)\right ))^{*}\otimes\mathbb I_{\tilde {\mathcal L}}\right ) Wu(g)W^{*}=\mathbb I_{\bigotimes_{\Lambda}{\mathbb C}^{d}}\otimes
\tilde u(g)$. This gives (\ref{up}).
It is straight forward to check that $\tilde u$ is
a projective unitary representation of $G$
with $2$-cocycle $\sigma$ implementing $\tau_{\Gamma\setminus \Lambda}$.
From (\ref{up}) and Lemma \ref{smb}, we have
\begin{align}
m\cdot \alpha \prec U^{\otimes N_{m}^{(\sigma)}}\otimes\tilde u=
U^{\otimes \Lambda}\otimes\tilde u
\cong u.
\end{align}
This completes the proof.
\end{proof}
Recall Definition \ref{pao}.
We note that $c_{\omega,R}$ and $c_{\omega,L}$
are not independent.
\begin{lem}\label{clmcr}
For any $\omega\in SPG({\mathcal A})$,
we have $c_{\omega,R}=c_{\omega,L}^{-1}$.
\end{lem}
\begin{proof}
Let $({\mathcal H},\pi,\Omega)$ be the GNS triple of $\omega$.
As $\omega$ satisfies the split property,
there are
Hilbert spaces ${\mathcal H}_{L},{\mathcal H}_{R}$ and a unitary
$W:{\mathcal H}\to{\mathcal H}_{L}\otimes {\mathcal H}_{R}$
such that
\begin{align}\label{mmd}
W\pi\left ( {\mathcal A}_{R}\right )''W^{*}={\mathbb C}\mathbb I_{{\mathcal H}_{L}}\otimes {\mathcal B}({\mathcal H}_{R}),\quad
W\pi\left ( {\mathcal A}_{R}\right )'W^{*}={\mathcal B}({\mathcal H}_{L})\otimes{\mathbb C}\mathbb I_{{\mathcal H}_{R}}.
\end{align}
(See Theorem 1.31 V \cite{takesaki}.)
From (\ref{mmd}), $\pi\left ( {\mathcal A}_{L}\right )''\subset \pi\left ( {\mathcal A}_{R}\right )'$
and
$\pi\left ( {\mathcal A}_{L}\right )''\vee \pi\left ( {\mathcal A}_{R}\right )''={\mathcal B}({\mathcal H})$,
we obtain
\begin{align}
W\pi\left ( {\mathcal A}_{L}\right )''W^{*}={\mathcal B}({\mathcal H}_{L})\otimes{\mathbb C}\mathbb I_{{\mathcal H}_{R}}.
\end{align}
Hence we obtain irreducible representations $({\mathcal H}_{L},\pi_{L})$
and $({\mathcal H}_{R},\pi_{R})$ of ${\mathcal A}_{L}$, ${\mathcal A}_{R}$
such that
\begin{align}
&W \pi\left (
a\otimes b
\right ) W^{*}
=\pi_{L}(a)\otimes\pi_{R}(b)
,\quad a\in{\mathcal A}_{L},\quad b\in{\mathcal A}_{R}.
\end{align}
The triple $({\mathcal H}_{L}\otimes {\mathcal H}_{R}, \pi_{L}\otimes\pi_{R}, W\Omega)$
is a GNS triple of $\omega$.
Therefore, $\omega\vert_{{\mathcal A}_{R}}$ is
$\pi_{R}$-normal.
As $\pi_{R}({\mathcal A}_{R})''$ is a factor,
$\pi_{R}$ and the GNS representation of $\omega\vert_{{\mathcal A}_{R}}$
are quasi-equivalent.
Similarly, $\pi_{L}$ and the GNS representation of $\omega\vert_{{\mathcal A}_{L}}$
are quasi-equivalent.
By the $\tau$-invariance of $\omega$,
there is a unitary representation $V$ of $G$ on ${\mathcal H}_{L}\otimes {\mathcal H}_{R}$
given by
\begin{align}
V(g)\left ( \pi_{L}\otimes\pi_{R}\right ) (a)W\Omega
=\left ( \pi_{L}\otimes\pi_{R}\right ) \left ( \tau(g)\left ( a\right )\rmk W\Omega,\quad
g\in G,\quad a\in{\mathcal A}.
\end{align}
On the other hand, by Proposition \ref{unieq}, there are
projective unitary representations $u_{{L}}$, $u_{{R}}$ of $G$
on ${\mathcal H}_{L}$, ${\mathcal H}_{R}$ such that
\begin{align}
\pi_{L}\circ \tau_{L}(g)\left ( a\right )
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{L}(g)\right )\circ \pi_{L}\left ( a\right ),\quad
\pi_{R}\circ \tau_{R}(g)\left ( b\right )
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{R}(g)\right )\circ \pi_{R}\left ( b\right ),
\end{align}
for all $a\in{\mathcal A}_{L}$, $b\in{\mathcal A}_{R}$ and $g\in G$.
Note that
\begin{align}
\mathop{\mathrm{Ad}}\nolimits\left ( V(g)\right )\circ \left ( \pi_{L}\otimes\pi_{R}\right )(x)
=\left ( \pi_{L}\otimes\pi_{R}\right )\circ \tau(g)(x)
=\mathop{\mathrm{Ad}}\nolimits \left ( u_{L}(g)\otimes u_{R}(g)\right ) \circ \left ( \pi_{L}\otimes\pi_{R}\right )(x),
\end{align}
for all $x\in{\mathcal A}$.
As $\left ( \pi_{L}\otimes\pi_{R}\right )({\mathcal A})''={\mathcal B}({\mathcal H}_{L}\otimes {\mathcal H}_{R})$,
this means that there is a map $b:G\to{\mathbb T}$
such that
\begin{align}\label{uuv}
u_{L}(g)\otimes u_{R}(g)=b(g) V(g),\quad g\in G.
\end{align}
Let $\sigma_{L},\sigma_{R}\in{\mathbb Z}^{2}(G,{\mathbb T})$
be $2$-cocycles of $u_{L}$, $u_{R}$ respectively.
From (\ref{uuv}),
we obtain
\begin{align}
\sigma_{L}\sigma_{R}=\sigma_{b}
\end{align}
(Here $\sigma_{b}$ is defined by (\ref{bound}).)
This means
\begin{align}
c_{\omega,R}=[\sigma_{R}]_{H^{2}(G,{\mathbb T})}
=[\sigma_{L}^{-1}]_{H^{2}(G,{\mathbb T})}
=c_{\omega,L}^{-1}.
\end{align}
\end{proof}
\section{Twisted $C^{*}$-dynamical system}\label{crossedproduct}
In this section we briefly recall basic facts about twisted $C^{*}$-crossed product.
Throughout this section, let $\Gamma$ be an infinite subset of ${\mathbb Z}$, and
$\sigma\in Z^{2}(G,{\mathbb T})$.
The quadruple $(G, {\mathcal A}_{\Gamma}, \tau_{\Gamma},\sigma)$
is a twisted $C^{*}$-dynamical system which we denote by $\Sigma_{\Gamma}^{(\sigma)}$.
(This is a simple version of \cite{bedos}.)
A covariant representation of $\Sigma_{\Gamma}^{(\sigma)}$ is a triple
$({\mathcal H},\pi, u)$ where $\pi $ is a $*$-representation of the $C^{*}$-algebra ${\mathcal A}_{\Gamma}$
on a Hilbert space
${\mathcal H}$
and $u$ is a projective unitary representation of $G$
with $2$-cocycle $\sigma$ on ${\mathcal H}$
such that
\begin{align}
u(g) \pi(a) u(g)^{*}= \pi\left (\tau_{\Gamma}(g)(a)\right ),\quad
a\in {\mathcal A}_{\Gamma},\quad g\in G.
\end{align}
In this paper, we say the covariant representation $({\mathcal H},\pi, u)$ is irreducible if
$\pi $ is an irreducible representation of ${\mathcal A}_{\Gamma}$.
Note that for a quadruple$({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}},\sigma_{\omega,{\varsigma}})$ associated to $(\omega\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$
with $\omega\in SPG({\mathcal A})$
(Definition \ref{pao}), $({\mathcal L}_{\omega,{\varsigma}}, \rho_{\omega,{\varsigma}}, u_{\omega,{\varsigma}})$
is an irreducible covariant representation of
$\Sigma_{\Gamma_{\varsigma}}^{(\sigma_{\omega,{\varsigma}})}$.
Let $C(G,{\mathcal A}_{\Gamma})$ be
the linear space of ${\mathcal A}_{\Gamma}$-valued functions
on $G$.
We equip $C(G,{\mathcal A}_{\Gamma})$ with a product and $*$-operation as follows:
\begin{align}
&f_{1}*f_{2}(h)
:=\sum_{g\in G}\sigma(g,g^{-1}h)\cdot f_{1}(g)\cdot \tau_{\Gamma}(g)\left ( f_{2}(g^{-1}h)\right ),\quad h\in G,
\label{multi}\\
&f^{*}(h):=
\overline{\sigma(h^{-1}, h)}\tau_{\Gamma}\left ( h\right ) \left (
f(h^{-1})^{*}
\right ),\quad h\in G,
\end{align}
for $f_{1},f_{2},f\in C(G,{\mathcal A}_{\Gamma})$.
The linear space $C(G,{\mathcal A}_{\Gamma})$ which is a $*$-algebra
with these operations is denoted by $C(\Sigma_{\Gamma}^{(\sigma)})$.
We will omit the symbol $*$ for the multiplication (\ref{multi}).
For a covariant representation $({\mathcal H},\pi, u)$ of
$\Sigma_{\Gamma}^{(\sigma)}$, we may introduce a $*$-representation
$({\mathcal H},\pi\times u)$ of
$C(\Sigma_{\Gamma}^{(\sigma)})$ by
\begin{align}
\left ( \pi\times u\right ) (f)
:=\sum_{g\in G}\pi\left ( f(g)\right ) u(g),\quad f\in C(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
The full twisted crossed product of $\Sigma_{\Gamma}^{(\sigma)}$, denoted $C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
is the completion of $C(\Sigma_{\Gamma}^{(\sigma)})$ with respect to the norm
\begin{align}
\left \Vert f\right \Vert_{u}:=
\sup\left\{
\left \Vert\left ( \pi\times u\right ) (f)\right \Vert
\mid
(\pi,u) : \text{covariant representation}
\right\},\quad f\in C(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
From any representation $({\mathcal H},\pi)$ of ${\mathcal A}_{\Gamma}$,
we can define a covariant representation $({\mathcal H}\otimes l^{2}(G)\simeq l^{2}(G,{\mathcal H}), \tilde\pi, \tilde u_{\pi})$ of $\Sigma_{\Gamma}^{(\sigma)}$
by
\begin{align}
\left ( \tilde \pi(a) \xi\right )(g):=\pi\left ( \tau_{\Gamma}(g^{-1})(a)\right )\xi(g),\quad
a\in{\mathcal A}_{\Gamma},\quad \xi\in l^{2}(G,{\mathcal H}),\quad g\in G,
\end{align}
and $\tilde u_{\pi}=\mathbb I_{{\mathcal H}}\otimes u^{\sigma}_{r}$.
Here, $u^{\sigma}_{r}$ is a projective unitary representation
with $2$-cocycle $\sigma$ on
$l^{2}(G)$ defined by
\begin{align}
\left ( u^{\sigma}_{r}(g)\xi\right ) (h)
=\sigma(g, g^{-1}h)\xi(g^{-1}h),\quad g,h\in G,\quad \xi\in l^{2}(G).
\end{align}
Note that $\pi$ is faithful because ${\mathcal A}_{\Gamma}$ is simple. Therefore, the representation
$ \tilde\pi\times \tilde u_{\pi}$ of $C(\Sigma_{\Gamma}^{(\sigma)})$ given by
$({\mathcal H}\otimes l^{2}(G)\simeq l^{2}(G,{\mathcal H}), \tilde\pi, \tilde u_{\pi})$
is faithful.
We define a $C^{*}$-norm $\left \Vert\cdot\right \Vert_{r}$ on $C(\Sigma_{\Gamma}^{(\sigma)})$ by
\begin{align}
\left \Vert f\right \Vert_{r}:=
\left \Vert
\tilde\pi\times \tilde u_{\pi}(f)
\right \Vert_{{\mathcal B}({\mathcal H}\otimes l^{2}(G))},\quad f\in C(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
The completion $C_{r}^{*}(\Sigma_{\Gamma}^{(\sigma)})$
of $C(\Sigma_{\Gamma}^{(\sigma)})$ with respect to this norm is the reduced twisted crossed product of $\Sigma_{\Gamma}^{(\sigma)}$.
As we are considering a finite group $G$, we have
$C(\Sigma_{\Gamma}^{(\sigma)})=C_{r}^{*}(\Sigma_{\Gamma}^{(\sigma)})=C^{*}(\Sigma_{\Gamma}^{(\sigma)})$, and
$\left \Vert\cdot \right \Vert_{r}=\left \Vert\cdot\right \Vert_{u}$.
For each $a\in {\mathcal A}_{\Gamma}$, $\xi_{a}:G\ni g\mapsto \delta_{g,e}a\in {\mathcal A}_{\Gamma}$ defines an element of $C^{*}(\Sigma_{\Gamma}^{(\sigma)})$.
The map $\xi:{\mathcal A}_{\Gamma}\ni a\mapsto \xi_{a}\in C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
is a unital faithful $*$-homomorphism.
Note that $\xi_{\mathbb I_{{\mathcal A}_{\Gamma}}}$ is the identity of $C^{*}\left (\Sigma_{\Gamma}^{(\sigma)}\right )$.
Hence
the $C^{*}$-algebra ${\mathcal A}_{\Gamma}$ can be regarded as a subalgebra
of $C(\Sigma_{\Gamma}^{(\sigma)})=C_{r}^{*}(\Sigma_{\Gamma}^{(\sigma)})=C^{*}(\Sigma_{\Gamma}^{(\sigma)})$.
Therefore, we simply write $a$ to denote $\xi_{a}$.
From the condition (\ref{uffl}),
for any $g\in G$ with $g\neq e$, the automorphism $\tau_{\Gamma}(g)$ is properly outer.
Therefore, by the argument in \cite{elliott} Theorem 3.2, $C(\Sigma_{\Gamma}^{(\sigma)})=C_{r}^{*}(\Sigma_{\Gamma}^{(\sigma)})=C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ is simple.
As ${\mathcal A}_{\Gamma}$ is unital, we have unitaries $\lambda_{g}\in C^{*}(\Sigma_{\Gamma}^{(\sigma)})$, $g\in G$,
defined by $G\ni h\mapsto \delta_{g,h}\mathbb I_{{\mathcal A}_{\Gamma}}\in {\mathcal A}_{\Gamma}$ such that
\begin{align}\label{ldef}
&\lambda_{g}\lambda_{h}=\sigma(g,h)\lambda_{gh},\quad g,h\in G,\nonumber\\
&\lambda_{g}a \lambda_{g}^{*}=\tau_{\Gamma}(g)\left ( a\right ),\quad a\in {\mathcal A}_{\Gamma},\; g\in G.
\end{align}
Note that $\lambda_{e}=\xi_{\mathbb I_{{\mathcal A}_{\Gamma}}}$ is the identity of $C^{*}\left (\Sigma_{\Gamma}^{(\sigma)}\right )$.
We set
\begin{align}
C^{*}\left (\Sigma_{\Gamma}^{(\sigma)}\right )^{G}
:=\left\{
f\in C^{*}\left (\Sigma_{\Gamma}^{(\sigma)}\right )\mid
\mathop{\mathrm{Ad}}\nolimits\left (\lambda_{g}\right )(f)=f,\quad g\in G
\right\}.
\end{align}
Let $({\mathcal H},\pi, u)$ be an irreducible covariant representation
of
$\Sigma_{\Gamma}^{(\sigma)}$.
The projective unitary representation $u$ has an irreducible decomposition given by some Hilbert spaces $\{{\mathcal K}_{\gamma}\mid \gamma\in {\mathcal P}_{\sigma}\}$
(Lemma \ref{pd} and Notation \ref{ddcom}). Namely we have
\begin{align}\label{bvp}
u(g)=\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}
V_{\alpha}(g)\otimes \mathbb I_{{\mathcal K}_{\alpha}},\quad g\in G,\quad \text{and}\quad
u(G)'=
\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}
\mathbb I_{{\mathcal H}_{\alpha}}\otimes {\mathcal B}\left ( {{\mathcal K}_{\alpha}}\right ).
\end{align}
Note that
\begin{align}\label{rg}
\left ( \pi\times u\right )(\lambda_{g})=u(g), \quad g\in G .
\end{align}
From this we have
\begin{align}
\left ( \pi\times u\right )
\left ( C^{*}\left (\Sigma_{\Gamma}^{(\sigma)}\right )^{G}\right )
\subset u(G)'=
\bigoplus_{\alpha\in {\mathcal P}_{\sigma}}
{\mathbb C}\mathbb I_{{\mathcal H}_{\alpha}}\otimes {\mathcal B}\left ( {{\mathcal K}_{\alpha}}\right ).
\end{align}
The following proposition is the immediate consequence of Theorem \ref{zbdip}.
\begin{prop}\label{zbdi}
Let $({\mathcal H},\pi,u)$ be an irreducible covariant representation of
$\Sigma_{\Gamma}^{(\sigma)}$.
Then $u$ contains
all elements of ${\mathcal P}_{\sigma}$ with infinite multiplicity.
\end{prop}
\section{Homogeneity}\label{homogeneity}
Throughout this section we fix $\Gamma=\Gamma_{L},\Gamma_{R}$, $\sigma\in Z^{2}(G,{\mathbb T})$, and
$\alpha\in {\mathcal P}_{\sigma}$.
We use the following notation.
\begin{notation}\label{nagai}Let $({\mathcal H},\pi,u)$ be an irreducible covariant representation of
$\Sigma_{\Gamma}^{(\sigma)}$ with
an irreducible decomposition of $u$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma}\mid \gamma \in {\mathcal P}_{\sigma}\}$. We use the symbol $\hat\pi$ to denote the irreducible representation
\begin{align}\label{phat}
\hat\pi=\mathop{\mathrm{id}}\nolimits_{{\mathcal B}({\mathcal H}_{\alpha})}\otimes \left (\pi\times u\right )
\end{align}
of
${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ on ${\mathcal H}_{\alpha}\otimes {\mathcal H}$.
For a unit vector $\xi\in {\mathcal K}_{\alpha}$,
we may define a state $\hat\varphi_{\xi}$ on ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
by
\begin{align}\label{nekosan}
\hat\varphi_{\xi}(x):=
\braket{\tilde\xi}
{\hat\pi
\left ( x\right ) \tilde\xi},\quad x\in {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
Here, $\tilde \xi$ is an element of $ {\mathcal H}_{\alpha}\otimes {\mathcal H}$,
\begin{align}\label{kaeru}
\tilde\xi:=\Omega_{\alpha}\otimes \xi\in {\mathcal H}_{\alpha}\otimes {\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}
\hookrightarrow
{\mathcal H}_{\alpha}\otimes {\mathcal H},
\end{align}
regarding $ {\mathcal H}_{\alpha}\otimes {\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha}$ as a subspace of $ {\mathcal H}_{\alpha}\otimes {\mathcal H}$. (See Notation \ref{ddcom}.)
Recall that $\Omega_{\alpha}$ is defined in (\ref{oa}).
We call this $\hat\varphi_{\xi}$ a state on ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
given by $({\mathcal H},\pi,u,\xi)$.
By the irreducibility of $\pi$, $\hat\pi$ is irreducible and $\hat \varphi_{\xi}$ is a pure state on ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$.
Note that $({\mathcal H}_{\alpha}\otimes {\mathcal H},\hat\pi,\tilde \xi)$
is a GNS triple of $\hat\varphi_{\xi}$.
\end{notation}
The goal of this section is to prove the following Proposition.
\begin{prop}\label{lem6}
Let $({\mathcal H}_{i},\pi_{i},u_{i})$ with $i=0,1$ be
irreducible covariant representations of $\Sigma_{\Gamma}^{(\sigma)}$
with irreducible decomposition of $u_{i}$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma,i}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
Let
$\xi_{i}\in{\mathcal K}_{\alpha,i}$ be unit vectors in ${\mathcal K}_{\alpha,i}$ for $i=0,1$.
(Recall Proposition \ref{zbdi} for existence of such vectors.)
Let $\hat\varphi_{{\xi}_{i}}$ be a state on ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
given by $({\mathcal H}_{i},\pi_{i},u_{i},\xi_{i})$, for each $i=0,1$.
Let
$\varphi_{i}$ be the restriction of $\hat\varphi_{{\xi}_{i}}$
onto ${\mathcal A}_{\Gamma}$.
Then there exists
a norm-continuous path $w:[0,\infty)\to {\mathcal U}({\mathcal A}_{\Gamma}^{G})$
with $w(0)=\mathbb I$,
such that
\begin{enumerate}
\item
for each $a\in{\mathcal A}_{\Gamma}$, the limit
\begin{align}
\lim_{t\to\infty}\mathop{\mathrm{Ad}}\nolimits\left ( w(t)\right )(a)=:\Xi_{\Gamma}(a)
\end{align}
exists and defines
an automorphism $\Xi_{\Gamma}$ on ${\mathcal A}_{\Gamma}$, and
\item the automorphism $\Xi_{\Gamma}$ in {\it 1.}
satisfies $\varphi_{1}=\varphi_{0}\circ\Xi_{\Gamma}$.
\end{enumerate}
\end{prop}
\begin{rem}
Basically, what we would like to do is to connect some $\pi_{0}$-normal state
$\varphi_{0}$ and some $\pi_{1}$-normal state
$\varphi_{1}$ via some $\Xi_{\Gamma}\in \mathop{\mathrm{AInn}}\nolimits^{G} (A_{\Gamma})$.
Without symmetry, $\varphi_{0}$ and
$\varphi_{1}$ can be taken to be pure states.
When the symmetry comes into the game, to guarantee that $\Xi_{\Gamma}$
commutes with $\tau_{\Gamma}(g)$,
we would like to assume that $\varphi_{0}$ and $\varphi_{1}$
are $\tau_{\Gamma}$-invariant.
If $\sigma$ is trivial, there is a $u_{i}$-invariant non-zero vector that we may
find such pure $\tau_{\Gamma}$-invariant states $\varphi_{0}$ and
$\varphi_{1}$.
But if the cohomology class of $\sigma$ is not trivial,
$u_{i}$ does not have a non-zero invariant vector.
However, there is still a rank $n_{\alpha}$ $u_{i}$-invariant
density matrix.
That is the reason why we consider ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$.
Note that the density matrix of $\varphi_{i}$
is a rank $n_{\alpha}$ operator which commutes with $u_{i}$.
\end{rem}
For the proof of Proposition \ref{lem6}, we use the machinery used in \cite{fkk} and \cite{kos}.
(See Appendix \ref{kosfkk}.)
However, as we would like to have a path in the fixed point algebra ${\mathcal A}_{\Gamma}^{G}$,
we need additional arguments.
For that purpose, the following Lemma plays an important role.
\begin{lem}\label{lem11iii}
Let $\Gamma_{0}$ be an infinite subset of ${\mathbb Z}$.
Let $({\mathcal H},\pi,u)$ be an irreducible covariant representation of
$\Sigma_{{\Gamma_0}}^{(\sigma)}$ with
an irreducible decomposition of $u$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
Then there exist irreducible $*$-representations $({\mathcal K}_{\gamma},\pi_{\gamma} )$, $\gamma\in {\mathcal P}_{\sigma}$
of ${\mathcal A}_{{\Gamma_0}}^{G}$
such that
\begin{align}\label{eq53}
\pi(a)=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma}(a),\quad
a\in {\mathcal A}^{G}_{{\Gamma_0}}.
\end{align}
Furthermore,
we have
\begin{align}\label{ttg}
\pi\left ({\mathcal A}^{G}_{{\Gamma_0}}\right )''=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}}\mathbb I_{{\mathcal H}_{\gamma}}\otimes
B({\mathcal K}_{\gamma}).
\end{align}
\end{lem}
\begin{notation}\label{agrep}
We call $\left\{({\mathcal K}_{\gamma}, \pi_{\gamma})\mid\gamma\in {\mathcal P}_{\sigma}\right\}$,
the family of representations of ${\mathcal A}^{G}_{{\Gamma_0}}$ associated to $({\mathcal H},\pi,u)$.
\end{notation}
\begin{proof}
For any $a\in{\mathcal A}_{{\Gamma_0}}^{G}$, we have $\pi(a)\in u(G)'$.
Therefore, from Lemma \ref{pd},
each $\pi(a)$ with $a\in{\mathcal A}_{{\Gamma_0}}^{G}$
has a form
\begin{align}
\pi(a)=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma}(a),
\end{align}
with uniquely defined $\pi_{\gamma}(a)\in{\mathcal B}({\mathcal K}_{\gamma})$,
for each $\gamma\in{\mathcal P}_{\sigma}$.
As $\pi$ is a $*$-representation, for each $\gamma\in {\mathcal P}_{\sigma}$, the map
$\pi_{\gamma}:{\mathcal A}^{G}_{{\Gamma_0}}\ni a\mapsto \pi_{\gamma}(a)\in {\mathcal B}({\mathcal K}_{\gamma})$ is
a $*$-representation and we have
\begin{align}
\pi\left ({\mathcal A}^{G}_{{\Gamma_0}}\right )''\subset\bigoplus_{\gamma\in {\mathcal P}_{\sigma}}\mathbb I_{{\mathcal H}_{\gamma}}\otimes
B({\mathcal K}_{\gamma}).
\end{align}
We claim that each $\pi_{\gamma}$ is an irreducible representation of ${\mathcal A}^{G}_{{\Gamma_0}}$, and
(\ref{ttg}) holds.
To see this, note that for any $x\in {\mathcal B}({\mathcal K}_{\gamma})$,
there exists a bounded net $\{a_{\lambda}\}_{{\lambda}}\in{\mathcal A}$ such that
$\pi(a_{\lambda})$ converges to $\mathbb I_{{\mathcal H}_{\gamma}}\otimes x\in
{\mathcal B}({\mathcal H}_{\gamma})\otimes {\mathcal B}({\mathcal K}_{\gamma})\subset {\mathcal B}({\mathcal H})$
in the $\sigma$-strong topology,
by the irreducibility of $\pi$ and the Kaplansky density theorem.
For this $\{a_{\lambda}\}_{{\lambda}}$,
we have
\begin{align}\label{access}
\frac 1{|G|} \sum_{g\in G} u_{g} \pi(a_{\lambda}) u_{g}^{*}
=\pi\left (
\frac 1{|G|} \sum_{g\in G}\tau_{{\Gamma_0}}(g)\left ( a_{\lambda}\right )
\right )
=
\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma}\left (
\frac 1{|G|} \sum_{g\in G}\tau_{{\Gamma_0}}(g)\left ( a_{\lambda}\right )\rmk
\in \pi\left ({\mathcal A}^{G}_{{\Gamma_0}}\right )''
\end{align}
because $\frac 1{|G|} \sum_{g\in G}\tau_{{\Gamma_0}}(g)\left ( a_{\lambda}\right )$
is $\tau_{{\Gamma_0}}$-invariant.
Since the left hand side of (\ref{access}) converges to
$\mathbb I_{{\mathcal H}_{\gamma}}\otimes x$
in the $\sigma$-strong topology,
we conclude that
$\mathbb I_{{\mathcal H}_{\gamma}}\otimes x\in \pi\left ({\mathcal A}^{G}_{{\Gamma_0}}\right )''$.
Hence (\ref{ttg}) holds. Looking at the $\gamma$-component of (\ref{access}),
we see that $\pi_{\gamma}\left ({\mathcal A}^{G}_{{\Gamma_0}}\right )''={\mathcal B}({\mathcal K}_{\gamma})$.
Hence $\pi_{\gamma}$ is irreducible.
This completes the proof.
\end{proof}
For each Lemma below, we use the machinery used in \cite{fkk} and \cite{kos}.
We remark arguments required to get a path inside of ${\mathcal A}_{\Gamma}^{G}$.
\begin{lem}\label{lem2}
Let $({\mathcal H}_{i},\pi_{i},u_{i})$ with $i=0,1$ be irreducible covariant representations of
$\Sigma_{\Gamma}^{(\sigma)}$ with
irreducible decomposition of $u_{i}$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma,i}\mid \gamma \in {\mathcal P}_{\sigma}\}$, $i=0,1$.
Let $\xi_{i}\in{\mathcal K}_{\alpha,i}$, $i=0,1$ be unit vectors.
Let $\hat\varphi_{{\xi}_{i}}$ be a state on ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
given by $({\mathcal H}_{i},\pi_{i},u_{i},\xi_{i})$, for each $i=0,1$.
(Recall Notation \ref{nagai}.)
Then for any ${\mathcal F}\Subset {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ and any $\varepsilon>0$,
there exists a self-adjoint $h\in {\mathcal A}_{\Gamma}^{G}$
such that
\begin{align}\label{lem2main}
\left \vert
\hat\varphi_{{\xi}_{0}}\left ( x\right )-\hat\varphi_{{\xi}_{1}}\circ \mathop{\mathrm{Ad}}\nolimits(e^{ih})(x)
\right \vert<\varepsilon,\quad x\in{\mathcal F}.
\end{align}
\end{lem}
\begin{proof}
First we prepare some notations.
We denote by $\tilde\xi_{i}$, $
\hat\pi_{i}$
the vector and the representation $\tilde\xi$, $
\hat\pi$
defined in Notation \ref{nagai}
with $({\mathcal H},\pi,u,\xi)$ replaced by $({\mathcal H}_{i},\pi_{i},u_{i},\xi_{i})$.
(See (\ref{kaeru}) and (\ref{phat}).)
The triple $( {\mathcal H}_{\alpha}\otimes {\mathcal H}_{i},\hat\pi_{i}, \tilde\xi_{i})$ is a GNS-triple
of $\hat\varphi_{{\xi}_{i}}$.
As $ {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ is simple, kernel of
$\hat\pi_{i}$ is zero for each $i=0,1$.
For each $k,j=1,\ldots, n_{\alpha}$,
we define an element
\begin{align}
Q_{k,j}^{(\alpha)}:=\frac{n_{\alpha}}{|G|}\sum_{g\in G}
\overline
{\braket{\psi_{k}^{(\alpha)}}{V_{\alpha}(g)\psi_{j}^{(\alpha)}}}
\lambda_{g}
\in C^{*}(\Sigma_{\Gamma}^{(\sigma)}),
\end{align}
with $\lambda_{g}\in C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
introduced in section \ref{crossedproduct} (\ref{ldef}).
We also set
\begin{align}
R^{(\alpha)}:=\frac1{n_{\alpha}}\sum_{k,j=1}^{n_{\alpha}} \ketbra{\psi_{k}^{(\alpha)}}{\psi_{j}^{(\alpha)}}\otimes Q_{k,j}^{(\alpha)}
\in{\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
We claim
\begin{align}\label{puq}
\left (\pi_{i}\times u_{i}\right )\left ( Q_{k,j}^{(\alpha)}\right )
=\ketbra{\psi_{k}^{(\alpha)}}{\psi_{j}^{(\alpha)}}\otimes\mathbb I_{{\mathcal K}_{\alpha,i}}
\in{\mathcal B}\left ( {\mathcal H}_{\alpha}\otimes {\mathcal K}_{\alpha,i}\right )
\subset {\mathcal B}\left ( {\mathcal H}_{i}\right ),
\end{align}
for each $k,j=1,\ldots, n_{\alpha}$ and $i=0,1$.
From this we have
\begin{align}\label{66}
\hat\pi_{i}\left ( R^{(\alpha)}\right )
=\ketbra{\Omega_{\alpha}}{\Omega_{\alpha}}\otimes \mathbb I_{{\mathcal K}_{\alpha,i}}
=:P^{(\alpha,i)},\quad i=0,1.
\end{align}
From (\ref{66}), we obtain
\begin{align}\label{or1}
\hat\varphi_{{\xi}_{i}}(R^{(\alpha)})=1,\quad i=0,1.
\end{align}
To see (\ref{puq}), recall the orthogonality relation (\ref{orthog}),
the irreducible decomposition of $u_{i}$ given by $\{{\mathcal K}_{\gamma,i}\mid \gamma \in {\mathcal P}_{\sigma}\}$ and
that $(\pi_{i}\times u_{i})(\lambda_{g})=u_{i}(g)$ (\ref{rg}).
Then we have
\begin{align}
&\left (\pi_{i}\times u_{i}\right )\left ( Q_{k,j}^{(\alpha)}\right )
=\frac{n_{\alpha}}{|G|}\sum_{g\in G}
\overline
{\braket{\psi_{k}^{(\alpha)}}{V_{\alpha}(g)\psi_{j}^{(\alpha)}}}
u_{i}(g)\nonumber\\
&=\frac{n_{\alpha}}{|G|}\sum_{g\in G}
\overline
{\braket{\psi_{k}^{(\alpha)}}{V_{\alpha}(g)\psi_{j}^{(\alpha)}}}
\left (
\bigoplus_{\gamma\in {\mathcal P}_{\sigma}}
V_{\gamma}(g)\otimes \mathbb I_{{\mathcal K}_{\gamma,i}}
\right )
=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}}\left (
\delta_{\alpha,\gamma}
\ketbra{\psi_{k}^{(\alpha)}}{\psi_{j}^{(\alpha)}}\otimes\mathbb I_{{\mathcal K}_{\alpha,i}}\right ).
\end{align}
We now start the proof of Lemma.
We fix an arbitrary ${\mathcal F}\Subset {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ and $\varepsilon>0$.
We then choose $0<\tilde\varepsilon$ small enough so that
\begin{align}\label{ted}
\tilde\varepsilon< \min\{1,\frac\varepsilon 2\},\quad
\text{and}\quad
4\max_{a\in{\mathcal F}}\left \Vert a\right \Vert
\tilde\varepsilon^{\frac 12}<\frac\varepsilon 2.
\end{align}
We also set
\begin{align}\label{frset}
\tilde{\mathcal F}:=
{\mathcal F}\cup
\left\{
R^{(\alpha)}
\right\}
\Subset B({\mathcal H}^{(\alpha)})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)}).
\end{align}
Applying Lemma \ref{kos1312} to this $\tilde\varepsilon$ and $\tilde{\mathcal F}$,
and pure states $\hat\varphi_{{\xi}_{i}}$, $i=0,1$ of a simple unital $C^{*}$-algebra $ B({\mathcal H}^{(\alpha)})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$,
we obtain
an $f\in \left ( B({\mathcal H}^{(\alpha)})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )_{+,1}$
and a unit vector $\zeta\in {\mathcal H}^{(\alpha)}\otimes {\mathcal H}_{1}$ such that
\begin{align}\label{zeta}
\hat\pi_{1}(f)\zeta=\zeta,\quad
\left \Vert
f\left ( a-\hat\varphi_{{\xi}_{0}}(a)\mathbb I \right ) f
\right \Vert<\tilde\varepsilon,\quad \text{for all}\quad a\in \tilde{\mathcal F}.
\end{align}
For $P^{(\alpha,1)}$ in (\ref{66}) and the $\zeta$ in (\ref{zeta}), we have
\begin{align}
&\left \Vert
\left ( \mathbb I-P^{(\alpha,1)}\right )\zeta
\right \Vert^{2}
=\braket{\zeta}{\hat\pi_{1}(f)\left ( \mathbb I- \hat\pi_{1}(R^{(\alpha)})\right ) \hat\pi_{1}(f)\zeta }
=\braket{\zeta}{\hat\pi_{1}(f)\left ( \hat\varphi_{{\xi}_{0}}(R^{(\alpha)})\mathbb I- \hat\pi_{1}(R^{(\alpha)})\right ) \hat\pi_{1}(f)\zeta }\nonumber\\
&\le \left \Vert
f\left ( \hat\varphi_{{\xi}_{0}}(R^{(\alpha)})\mathbb I- R^{(\alpha)}\right ) f
\right \Vert
<\tilde \varepsilon.
\end{align}
Here we used (\ref{or1}), for the second equality.
For the inequality we used (\ref{zeta}) and $R^{(\alpha)}\in\tilde {\mathcal F}$ (\ref{frset}).
Therefore, $P^{(\alpha,1)}\zeta$ is not zero, and we may define a unit vector
\begin{align}
\tilde\zeta:=
\frac 1{\left \Vert P^{(\alpha,1)}\zeta\right \Vert}P^{(\alpha,1)}\zeta\in {\mathcal H}_{\alpha}\otimes {\mathcal H}_{1}.
\end{align}
Furthermore, it
satisfies
\begin{align}
\left \Vert \zeta-\tilde\zeta\right \Vert\le
2\tilde\varepsilon^{\frac12}.
\end{align}
From this and two properties in (\ref{zeta})
for any $a\in {\mathcal F}$, we have
\begin{align}\label{est6}
&\left \vert
\hat\varphi_{{\xi}_{0}}(a)-\braket{\tilde\zeta}{ \hat\pi_{1}(a) \tilde\zeta}
\right \vert
\le
\left \vert
\braket{\hat\pi_{1}(f)\zeta}{\left ( \hat\varphi_{{\xi}_{0}}(a)- \hat\pi_{1}(a)\right ) \hat\pi_{1}(f)\zeta }
\right \vert
+\left \vert
\braket{\zeta}{\hat\pi_{1}(a) \zeta }
-\braket{\tilde\zeta}{\hat\pi_{1}(a)\tilde\zeta }
\right \vert\nonumber\\
&\le
\left \Vert
f\left ( a-\hat\varphi_{{\xi}_{0}}(a)\mathbb I \right ) f
\right \Vert
+2\max_{a\in{\mathcal F}}\left \Vert a\right \Vert \left \Vert \zeta-\tilde\zeta\right \Vert
<\tilde\varepsilon+2\max_{a\in{\mathcal F}}\left \Vert a\right \Vert 2\tilde\varepsilon^{\frac12}<\varepsilon,
\end{align}
by the choice of $\tilde\varepsilon$ (\ref{ted}).
Since $ P^{(\alpha,1)}\tilde\zeta= \tilde\zeta$,
there exists a unit vector $\eta\in{\mathcal K}_{\alpha,1}$ such that
$\tilde\zeta=\Omega_{\alpha}\otimes \eta$.
By Lemma \ref{lem11iii},
for each $\gamma\in{\mathcal P}_{\sigma}$, there exists an irreducible representation $ \pi_{\gamma,1}$
of ${\mathcal A}_{\Gamma}^{G}$ on ${\mathcal K}_{\gamma,1}$
such that
\begin{align}
\pi_{1}(a)=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma,1}(a),\quad
a\in {\mathcal A}^{G}_{\Gamma}.
\end{align}
Applying the Kadison transitivity theorem for unit vectors $\xi_{1}, \eta\in {\mathcal K}_{\alpha,1}$ and an irreducible representation $({\mathcal K}_{\alpha,1}, \pi_{\alpha,1})$ of ${\mathcal A}_{\Gamma}^{G}$,
we obtain a self-adjoint $h\in {\mathcal A}^{G}_{\Gamma}$ such that
$\pi_{\alpha,1}(e^{-ih})\xi_{1}=\eta$.
With this $h$, we can write $\tilde\zeta$ as
\begin{align}
\tilde\zeta=\hat\pi_{1}\left (
\mathbb I_{B({\mathcal H}_{\alpha})}\otimes e^{-ih}
\right )\tilde\xi_{1}.
\end{align}
Hence we obtain
\begin{align}
\hat\varphi_{{\xi}_{1}}\circ \mathop{\mathrm{Ad}}\nolimits\left ( e^{ih}\right )
=\braket{\tilde\zeta}{\hat\pi_{1}\left ( \cdot \right ) \tilde\zeta}.
\end{align}
Combining this with (\ref{est6}), we see that (\ref{lem2main}) holds.
\end{proof}
\begin{rem}
The main difference of the proof of Lemma \ref{lem2} from \cite{kos}, \cite{fkk}
is that in order to find $h$ in ${\mathcal A}_{\Gamma}^{G}$, we add
$R^{(\alpha)}$ to ${\mathcal F}$.
This allows us to replace $\zeta$ with $\tilde\zeta=\Omega_{\alpha}\otimes \eta$.
From this combined with Lemma \ref{lem11iii}, the problem is reduced to the Kadison transitivity
for the irreducible $({\mathcal K}_{\alpha,1},\pi_{\alpha,1}({\mathcal A}_{\Gamma}^{G}))$.
Note that $R^{(\alpha)}$ belongs to $B({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
but not in ${\mathcal A}_{\Gamma}$.
By extending the $C^{*}$-algebra we consider,
we are allowed to have the projection $P^{(\alpha,i)}$ (\ref{66})
corresponding to the irreducible component of $u_{i}$ in the $C^{*}$-algebra.
\end{rem}
\begin{notation}\label{gca}
For $\Lambda\Subset \Gamma$,
we introduce a finite subset of $B({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
given by
\begin{align}
{\mathcal G}_{\Lambda}:=
\left\{
\frac 1{\sqrt{|G|}} \ketbra{\psi_{j}^{(\alpha)}}{\psi_{1}^{(\alpha)}}\otimes
\lambda_{g} E_{I,I_{0}}^{(\Lambda)}\quad
\mid \quad
j=1,\ldots, n_{\alpha},\; I\in \{1,\ldots,d\}^{\times \Lambda},\;\;
g\in G
\right\}.
\end{align}
Here, we set $I_{0}:=(i_{k})_{k\in\Lambda}\in \{1,\ldots,d\}^{\times \Lambda}$, with $i_{k}=1$ for all $k\in\Lambda$.
\end{notation}
\begin{notation}\label{cond1}
We say an irreducible covariant representation $({\mathcal H},\pi,u)$ of $\Sigma_{\Gamma}^{(\sigma)}$
and unit vectors $\xi,\eta\in {\mathcal H}_{\alpha}\otimes{\mathcal H}$ satisfy
{\it Condition 1} for a pair $\delta>0$, $\Lambda\Subset \Gamma$,
if the representation $\hat\pi:=\mathop{\mathrm{id}}\nolimits_{{\mathcal B}({\mathcal H}_{\alpha})}\otimes \left ( \pi\times u\right )$
of $B({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ satisfies the following:
\begin{enumerate}
\item
For any $x,y\in {\mathcal G}_{\Lambda}$,
$\hat\pi(x)^{*}\xi$ and $\hat\pi(y)^{*}\eta$ are orthogonal.
\item For any $x,y\in {\mathcal G}_{\Lambda}$,
\begin{align}
\left \vert
\braket{\xi}{\hat\pi(x y^{*})\xi}
-\braket{\eta}{\hat\pi(x y^{*})\eta}
\right \vert<\delta.
\end{align}
\end{enumerate}
\end{notation}
Let $\delta_{2,a}$ be the function given in Lemma \ref{lem6kos}.
\begin{lem}\label{lem4}
For any $\varepsilon>0$ and
$\Lambda\Subset \Gamma$,
there exists a $\delta_{1}(\varepsilon, \Lambda)>0$
satisfying the following:
For any irreducible covariant representation $({\mathcal H},\pi,u)$ of $\Sigma_{\Gamma}^{(\sigma)}$
and unit vectors $\xi,\eta\in {\mathcal H}_{\alpha}\otimes{\mathcal H}$ satisfying
{\it Condition 1} for a pair $\delta_{1}(\varepsilon, \Lambda)>0$, $\Lambda\Subset \Gamma$,
there exists a positive element $h$ of $({\mathcal A}_{\Gamma\setminus\Lambda}^{G})_{1}$
such that
\begin{align}
\left \Vert
e^{i\pi\hat\pi (h)}\xi-\eta
\right \Vert<
\frac{1}{4\sqrt 2}\delta_{2,a}\left ( \frac\varepsilon 8\right ).
\end{align}
\end{lem}
\begin{proof}
Recall Lemma \ref{lem28}. We set
\begin{align}
\delta_{1}(\varepsilon,\Lambda):=\delta_{3,a}\left (\varepsilon, n_{\alpha} d^{|\Lambda|} |G|\right ),
\end{align}
with $\delta_{3,a}(\cdot,\cdot)$ in Lemma \ref{lem28}.
We prove that this $\delta_{1}$ satisfies the condition above.
Let us consider an arbitrary irreducible covariant representation $({\mathcal H},\pi,u)$ of $\Sigma_{\Gamma}^{(\sigma)}$
and unit vectors $\xi,\eta\in {\mathcal H}_{\alpha}\otimes{\mathcal H}$ satisfying
{\it Condition 1} for a pair $\delta_{1}(\varepsilon,\Lambda)>0$, $\Lambda\Subset \Gamma$.
We again use the notation $\hat\pi$
(\ref{phat}) for this $\pi$.
We apply Lemma \ref{lem28}, to an infinite dimensional Hilbert space
${\mathcal H}_{\alpha}\otimes {\mathcal H}$, a unital
$C^{*}$-algebra
$\hat\pi\left ( {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )$
acting irreducibly on ${\mathcal H}_{\alpha}\otimes {\mathcal H}$,
a finite subset $\hat\pi({\mathcal G}_{\Lambda})$
of $\hat\pi\left ( {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )$, and
unit vectors $\xi,\eta\in {\mathcal H}_{\alpha}\otimes{\mathcal H}$.
Note that $\sum_{x\in\hat\pi({\mathcal G}_{\Lambda}) }xx^{*}=\mathbb I$
by the definition of ${\mathcal G}_{\Lambda}$.
From {\it Condition 1}, $\xi,\eta$ satisfy the required conditions in Lemma \ref{lem28}.
By Lemma \ref{lem28} ,
there exists a positive $\tilde h\in \left ({\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )_{+,1}$
such that
\begin{align}\label{pmm}
\left \Vert
\hat\pi\left ( \bar h\right )\left (\xi+\eta\right )
\right \Vert
<\frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi},\quad \text{and}\nonumber\\
\left \Vert\left (\mathbb I-\hat \pi (\bar h)\right )
\left (\xi-\eta\right )
\right \Vert
<\frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi},
\end{align}
for
\begin{align}\label{hbardef}
\bar h:=\sum_{x\in {\mathcal G}_{\Lambda}} x\tilde hx^{*}.
\end{align}
Here the function $\delta_{2,a}$ is given in Theorem \ref{lem6kos}.
By this definition of $\bar h$, we see that
\begin{align}
\bar h\in \left ({{\mathcal B}({\mathcal H}_{\alpha})}\otimes {\mathcal A}_{\Lambda}\right )'\cap
\left\{
\lambda_{g}\mid g\in G
\right\}'\cap
\left ( {\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )_{+,1}.
\end{align}
We would like to replace $\bar h$ in (\ref{pmm}) to some positive element
$h \in ({\mathcal A}_{\Gamma\setminus\Lambda}^{G})_{+,1}$.
In order to do so, we factorize $({\mathcal H},\pi,u)$ to $\Lambda$-part and $\Gamma\setminus \Lambda$-part:
As in the proof of Theorem \ref{zbdip},
there exists an irreducible covariant representation $(\tilde {\mathcal H},\tilde \pi,\tilde u)$ of
$\Sigma_{\Gamma\setminus \Lambda}^{(\sigma)}$
and a unitary $W:{\mathcal H}\to \left ( \bigotimes_{\Lambda}{\mathbb C}^{d}\right )\otimes \tilde{\mathcal H}$
such that
\begin{align}\label{prev1}
W\pi(a)W^{*}=\left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{\Lambda}}\otimes \tilde\pi\right ) (a),\quad a\in{\mathcal A}_{\Gamma},
\end{align}
and
\begin{align}\label{up1}
W u(g)W^{*}=\left ( \bigotimes_{\Lambda}U(g)\right )\otimes \tilde u(g),\quad g\in G.
\end{align}
By Lemma \ref{pd}, $\tilde u$ has an irreducible
decomposition of given by
a set of Hilbert spaces
$\{{\mathcal K}_{\gamma}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
By Lemma \ref{lem11iii} and Lemma \ref{pd} we have
\begin{align}\label{utd}
\tilde \pi\left ({\mathcal A}^{G}_{\Gamma\setminus \Lambda}\right )''=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}}\mathbb I_{{\mathcal H}_{\gamma}}\otimes
B({\mathcal K}_{\gamma})=\tilde u(G)'.
\end{align}
Recall (\ref{pmm}).
Choose $\delta>0$ so that
\begin{align}\label{delt}
\left \Vert
\hat\pi\left ( \bar h\right )\left (\xi+\eta\right )
\right \Vert+\delta
<\frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi},\quad \text{and}\nonumber\\
\left \Vert\left (\mathbb I-\hat \pi (\bar h)\right )
\left (\xi-\eta\right )
\right \Vert+\delta
<\frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi}.
\end{align}
As $\bar h$ is in $\left ({{\mathcal B}({\mathcal H}_{\alpha})}\otimes {\mathcal A}_{\Lambda}\right )'$,
from (\ref{prev1}), we see that
there exists a positive $y\in {\mathcal B}(\tilde{\mathcal H})_{1}$ such that
\begin{align}
\left ( \mathbb I_{{\mathcal H}_{\alpha}}\otimes W\right )
\hat\pi(\bar h) \left ( \mathbb I_{{\mathcal H}_{\alpha}}\otimes W^{*}\right )
=\mathbb I_{{\mathcal H}_{\alpha}}\otimes\mathbb I_{ \bigotimes_{\Lambda}{\mathbb C}^{d}}\otimes y.
\end{align}
Furthermore, as $\bar h$ is in $\left\{
\lambda_{g}\mid g\in G
\right\}'$, from (\ref{up1}),
$y$ belongs to $\tilde u(G)'=\tilde\pi ({\mathcal A}^{G}_{\Gamma\setminus \Lambda})''$
by (\ref{utd}).
By the Kaplansky density theorem, there exists a positive $ h\in \left ({\mathcal A}^{G}_{\Gamma\setminus \Lambda}\right )_{+,1}$
such that
\begin{align}
\left \Vert
\left (
\hat\pi(h)-\hat\pi(\bar h)
\right )\left (
\xi\pm\eta
\right )
\right \Vert
=
\left \Vert
\left (
\mathbb I_{{\mathcal H}_{\alpha}}\otimes\mathbb I_{ \bigotimes_{\Lambda}{\mathbb C}^{d}}\otimes
\left ( \tilde\pi (h)-y\right )
\right )
\left (
\mathbb I_{{\mathcal H}_{\alpha}}\otimes W
\right )
\left (
\xi\pm\eta
\right )
\right \Vert<\delta.
\end{align}
This $h$ satisfies
\begin{align}
&\left \Vert
\hat\pi(h)\left (
\xi+\eta
\right )
\right \Vert
\le
\left \Vert
\left ( \hat\pi(h)-\hat\pi(\bar h)\right )\left (
\xi+\eta
\right )
\right \Vert+\left \Vert
\hat\pi(\bar h)\left (
\xi+\eta
\right )
\right \Vert
< \frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi},\quad \text{and}\nonumber\\
&\left \Vert
\left ( \mathbb I-\hat\pi(h)\right )\left (
\xi-\eta
\right )
\right \Vert
\le
\left \Vert
\left ( \hat\pi(h)-\hat\pi(\bar h)\right )\left (
\xi-\eta
\right )
\right \Vert+\left \Vert
\left ( \mathbb I-\hat\pi(\bar h)\right )\left (
\xi-\eta
\right )
\right \Vert
< \frac 1{4\sqrt2} \delta_{2,a}\left ( \frac \varepsilon 8\right ) e^{-\pi},
\end{align}
from the choice of $\delta$, (\ref{delt}).
We then obtain the required property of $h$:
\begin{align}
&\left \Vert
e^{i\pi\hat\pi (h)}\xi-\eta
\right \Vert
\le\left \Vert
\frac 12 \left (
e^{i\pi\hat\pi(h)}(\xi+\eta)-(\xi+\eta)\right )\right \Vert
+\left \Vert \frac 12 \left ( e^{i\pi\hat\pi(h)}(\xi-\eta)+(\xi-\eta)
\right )
\right \Vert\nonumber\\
&\le
\frac{e^{\pi}}2\left \Vert \hat\pi(h)\left (\xi+\eta\right )\right \Vert
+\frac{e^{\pi}}2\left \Vert \left ( \mathbb I -\hat\pi(h)\right )\left (\xi-\eta\right )\right \Vert
<
\frac{1}{4\sqrt 2}\delta_{2,a}\left ( \frac\varepsilon 8\right ).
\end{align}
\end{proof}
\begin{rem}
Note that an average over $G$ is contained in (\ref{hbardef}).
Because of this, we could take $\bar h$ to be $\mathop{\mathrm{Ad}}\nolimits\lambda_{g}$-invariant.
This is possible because $\lambda_{g}$
is included in the $C^{*}$-algebra we consider, i.e., in ${\mathcal B}({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$.
The main difference of Lemma \ref{lem4} compared to \cite{kos} is replacing
$\bar h$ with $h\in {\mathcal A}_{\Gamma\setminus \Lambda}^{G}$.
To carry it out, the decomposition (\ref{utd})
given from Lemma \ref{pd} Lemma \ref{lem11iii}
is used. This decomposition reduces the problem to the Kaplansky density Theorem for
$\hat\pi ({\mathcal A}_{\Gamma\setminus\Lambda}^{G})$.
\end{rem}
\begin{notation}\label{lef}
For any $\varepsilon>0$ and a finite set
${\mathcal F}\Subset {\mathcal A}$, there exists a $\Lambda(\varepsilon,{\mathcal F})\Subset\Gamma$
such that
\begin{align}\label{ab}
\inf\left\{\left \Vert a-b\right \Vert\mid b\in {\mathcal A}_{\Lambda(\varepsilon,{\mathcal F})}\right\}<\frac\varepsilon {16},\quad
\text{for all}\quad a\in{\mathcal F}.
\end{align}
For each $\varepsilon>0$ and
${\mathcal F}\Subset {\mathcal A}$, we fix such $\Lambda(\varepsilon,{\mathcal F})$.
If ${\mathcal F}$ is included in ${\mathcal A}_{\Lambda}$ for some $\Lambda\Subset \Gamma$,
we choose $\Lambda(\varepsilon,{\mathcal F})$ so that $\Lambda(\varepsilon,{\mathcal F})\subset \Lambda$.
For any $\varepsilon>0$ and
${\mathcal F}\Subset {\mathcal A}$,
set
\begin{align}
\delta_{2}(\varepsilon,{\mathcal F}):=\frac 12 \delta_{1}\left ( \frac \varepsilon 4,
\Lambda(\varepsilon,{\mathcal F})\right ).
\end{align}
Here we used
the function $\delta_{1}$ introduced in Lemma \ref{lem4}.
\end{notation}
\begin{lem}\label{lem5}
Let $\varepsilon>0$, and
${\mathcal F}\Subset ({\mathcal A}_{\Gamma})_{1}$.
Let $({\mathcal H},\pi,u)$ be
an irreducible covariant representation of $\Sigma_{\Gamma}^{(\sigma)}$
with an irreducible decomposition of $u$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
Let
$\xi,\eta$ be unit vectors in ${\mathcal K}_{\alpha}$.
Suppose that unit vectors
\begin{align}\label{txedef}
\tilde\xi:=\Omega_{\alpha}\otimes \xi,\quad
\tilde\eta:=\Omega_{\alpha}\otimes\eta\in {\mathcal H}_{\alpha}\otimes {\mathcal H}
\end{align}
satisfy
\begin{align}\label{dare}
\left \vert
\braket{\tilde \eta}{\hat \pi(xy^{*}) \tilde\eta}
-\braket{\tilde \xi}{\hat \pi(xy^{*}) \tilde \xi}
\right \vert
<\delta_{2}(\varepsilon,{\mathcal F}),\quad
\text{for all}\quad x,y\in {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}.
\end{align}
(Recall Notation \ref{nagai} and Notation \ref{gca}.)
Then there exists a norm-continuous path of unitaries
$v:[0,1]\to{\mathcal U}({\mathcal A}_\Gamma^{G})$ such that $v(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$,
\begin{align}\label{tetx}
\tilde\eta= \left (\mathbb I_{{\mathcal H}_{\alpha}}\otimes \pi(v(1))\right )\tilde\xi,
\end{align}
and
\begin{align}\label{adv}
\sup_{t\in[0,1]}\left \Vert
\mathop{\mathrm{Ad}}\nolimits v(t)(a)-a
\right \Vert<\varepsilon,\quad \text{for all}\quad a\in {\mathcal F}.
\end{align}
\end{lem}
\begin{proof}
We denote by ${\mathcal N}$, the finite dimensional subspace
spanned by $\{\hat \pi(xy^{*})\tilde\xi, \hat \pi(xy^{*})\tilde\eta\mid x,y\in {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}\}
$.
Then there exists a unit vector $\zeta$ in ${\mathcal N}^{\perp}$, the orthogonal complement of ${\mathcal N}$,
such that
\begin{align}\label{otcy}
\left \vert
\braket{\zeta}{\hat \pi(xy^{*}) \zeta}
-\braket{\tilde \xi}{\hat \pi(xy^{*}) \tilde \xi}
\right \vert<\delta_{2}(\varepsilon,{\mathcal F})\le\delta_{1}\left ( \frac \varepsilon 4,
\Lambda(\varepsilon,{\mathcal F})\right ),\quad x,y\in {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}.
\end{align}
To see this, note that the intersection of the set of all compact operators on
${\mathcal H}_{\alpha}\otimes {\mathcal H}$ and $\hat\pi\left ( \caB(\caH_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )$ is $0$ because
$\caB(\caH_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$ is simple.
Applying Glimm's Lemma
(Theorem \ref{glimm})
to $\delta_{2}(\varepsilon,{\mathcal F})>0$, a pure state $\braket{\tilde\xi}{\cdot \tilde\xi}$ on
$\hat\pi\left ( \caB(\caH_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})\right )$,
a finite dimensional subspace ${\mathcal N}$ of ${\mathcal H}_{\alpha}\otimes {\mathcal H}$
and
a finite subset
$\hat\pi\left ( {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}{\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}^{*}\right )$,
we obtain $\zeta$ above.
Combining (\ref{otcy}) with (\ref{dare})we also get
\begin{align}
&\left \vert
\braket{\zeta}{\hat \pi(xy^{*}) \zeta}
-\braket{\tilde \eta}{\hat \pi(xy^{*}) \tilde \eta}
\right \vert\le
\left \vert
\braket{\zeta}{\hat \pi(xy^{*}) \zeta}
-\braket{\tilde \xi}{\hat \pi(xy^{*}) \tilde \xi}
\right \vert
+\left \vert
\braket{\tilde \eta}{\hat \pi(xy^{*}) \tilde\eta}
-\braket{\tilde \xi}{\hat \pi(xy^{*}) \tilde \xi}
\right \vert
\nonumber\\
&<2\delta_{2}(\varepsilon,{\mathcal F})=\delta_{1}\left ( \frac \varepsilon 4,
\Lambda(\varepsilon,{\mathcal F})\right ),\quad x,y\in {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}.
\end{align}
Hence
$({\mathcal H},\pi,u)$
and unit vectors $\tilde\xi,\zeta$ (resp. $\tilde\eta, \zeta$) satisfy
{\it Condition 1.} (Notation \ref{cond1}) for a pair $\delta_{1}\left ( \frac \varepsilon 4,
\Lambda(\varepsilon,{\mathcal F})\right )>0$, $\Lambda(\varepsilon,{\mathcal F})\Subset \Gamma$.
Therefore, from Lemma \ref{lem4},
there exist positive elements $h_{1},h_{2}$ in
$({\mathcal A}_{\Gamma\setminus\Lambda(\varepsilon,{\mathcal F})}^{G})_{1}$
such that
\begin{align}\label{tiyu}
\left \Vert
e^{i\pi\hat\pi (h_{1})}\tilde\xi-\zeta
\right \Vert<
\frac{1}{4\sqrt 2}\delta_{2,a}\left ( \frac\varepsilon {32}\right ),\quad\text{and}\quad
\left \Vert
e^{i\pi\hat\pi (h_{2})}\tilde\eta-\zeta
\right \Vert<
\frac{1}{4\sqrt 2}\delta_{2,a}\left ( \frac\varepsilon {32}\right ).
\end{align}
Here $\delta_{2,a}$ is given in Theorem \ref{lem6kos}.
By the definition of $\tilde \xi$ (\ref{txedef}) and the decomposition
\begin{align}\label{ptpt}
\pi(a)=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma}(a),\quad
a\in {\mathcal A}^{G}_{\Gamma}
\end{align}
((\ref{eq53}) of Lemma \ref{lem11iii}),
with irreducible $*$-representations $({\mathcal K}_{\gamma}, \pi_{\gamma})$
of ${\mathcal A}_{\Gamma}^{G}$,
we have $e^{i\pi\hat\pi (h_{1})}\tilde\xi
=\Omega_{\alpha}\otimes e^{i\pi \pi_{\alpha}(h_{1)}}\xi$.
Similarly, we have
$e^{i\pi\hat\pi (h_{2})}\tilde\eta= \Omega_{\alpha}\otimes e^{i\pi \pi_{\alpha}(h_{2)}}\eta$.
Combining this with (\ref{tiyu}), we see that the unit vectors $e^{i\pi \pi_{\alpha}(h_{1)}}\xi, e^{i\pi \pi_{\alpha}(h_{2)}}\eta$ in ${\mathcal K}_{\alpha}$
satisfies
\begin{align}
\left \Vert e^{i\pi \pi_{\alpha}(h_{1})}\xi-e^{i\pi \pi_{\alpha}(h_{2})}\eta\right \Vert<\frac{1}{2\sqrt 2}\delta_{2,a}\left ( \frac\varepsilon {32}\right ).
\end{align}
Then from Lemma \ref{lem9kos}, there exists a unitary $v_{0}$ on ${\mathcal K}_{\alpha}$ such that
\begin{align}
v_{0}e^{i\pi \pi_{\alpha}(h_{1})}\xi=e^{i\pi \pi_{\alpha}(h_{2})}\eta,\quad\text{and}\quad
\left \Vert
v_{0}-\mathbb I_{{\mathcal K}_{\alpha}}
\right \Vert<\frac{1}{2}\delta_{2,a}\left ( \frac\varepsilon {32}\right )<\delta_{2,a}\left ( \frac\varepsilon {32}\right ).
\end{align}
From this and the fact that $\pi_{\alpha}$ is an irreducible
representation of ${\mathcal A}^{G}_{\Gamma}$,
applying Theorem \ref{lem6kos}, we obtain a self-adjoint
$k\in {\mathcal A}^{G}_{\Gamma}$ such that
\begin{align}\label{piyo}
e^{i\pi_{\alpha}(k)}e^{i\pi \pi_{\alpha}(h_{1})}\xi=e^{i\pi \pi_{\alpha}(h_{2})}\eta,\quad\text{and}\quad
\left \Vert k\right \Vert\le \delta_{1,a}\left (\frac{\varepsilon}{32}\right ).
\end{align}
Here the function $\delta_{1,a}$ is given in Notation \ref{exp}.
Now we define a continuous path of unitaries
$v:[0,1]\to{\mathcal U}({\mathcal A}_\Gamma^{G})$.
Set
\begin{align}
v_{1}(t):=e^{it\pi h_{1}}\in
{\mathcal U}\left ({\mathcal A}^{G}_{\Gamma\setminus \Lambda(\varepsilon,{\mathcal F})}\right ),\quad
v_{2}(t):=e^{itk}\in{\mathcal U}\left ({\mathcal A}^{G}_{\Gamma}\right ),\quad
v_{3}(t):=e^{-it\pi h_{2}}\in
{\mathcal U}\left ({\mathcal A}^{G}_{\Gamma\setminus \Lambda(\varepsilon,{\mathcal F})}\right )
\end{align}
for each $t\in[0,1]$.
For $i=1,3$, as $v_{i}$ takes value in ${\mathcal U}\left ({\mathcal A}^{G}_{\Gamma\setminus \Lambda(\varepsilon,{\mathcal F})}\right )$, $v_{i}(t)$ commutes with elements in $ {\mathcal A}_{\Lambda(\varepsilon,{\mathcal F})}$. From this and the fact that
the distance between ${\mathcal F}$ and $ {\mathcal A}_{\Lambda(\varepsilon,{\mathcal F})}$ is
less than $\frac{\varepsilon}{16}$ (Notation \ref{lef} (\ref{ab})),
we get $\left \Vert \mathop{\mathrm{Ad}}\nolimits v_{i}(t) (a)-a\right \Vert<\frac\varepsilon 8$, for all $a\in{\mathcal F}$,
$t\in[0,1]$, and $i=1,3$.
For $i=2$, from $\left \Vert k\right \Vert\le\delta_{1,a}\left (\frac{\varepsilon}{32}\right )$, recalling the definition of
$\delta_{1,a}$ in Notation \ref{exp},
we obtain $\left \Vert \mathop{\mathrm{Ad}}\nolimits v_{2}(t)(a)-a\right \Vert\le 2\left \Vert v_{2}(t)-\mathbb I \right \Vert\le\frac \varepsilon{16}$, for all $a\in{\mathcal F}\subset\left ( {\mathcal A}_{\Gamma}\right )_{1}$ and $t\in[0,1]$.
We define $v:[0,1]\to{\mathcal U}\left ( {\mathcal A}^{G}_{\Gamma}\right )$ by
\begin{align}
v(t):=\left\{
\begin{gathered}
v_{1}(3t),\quad \quad t\in\left[0, \frac 13\right],\\
v_{2}\left ( 3\left ( t-\frac 13\right )\rmk v_{1}(1),\quad\quad t\in\left[\frac 13,\frac 23\right],\\
v_{3}\left ( 3\left ( t-\frac 23\right )\rmk v_{2}(1) v_{1}(1),\quad\quad t\in\left[\frac 23,1\right].
\end{gathered}
\right.
\end{align}
Clearly $v(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$ and $v$ is norm-continuous, and it takes values in
${\mathcal U}\left ( {\mathcal A}^{G}_{\Gamma}\right )$.
From the above estimates on
$\left \Vert \mathop{\mathrm{Ad}}\nolimits v_{i}(t)(a)-a\right \Vert$ for $a\in{\mathcal F}$ and $i=1,2,3$, we also get (\ref{adv}).
Furthermore, we have
\begin{align}
\left (\mathbb I_{{\mathcal H}_{\alpha}}\otimes \pi(v(1))\right )\tilde\xi
=\Omega_{\alpha}\otimes \pi_{\alpha}\left ( v(1)\right )\xi
=\Omega_{\alpha}\otimes \pi_{\alpha}\left ( e^{-i\pi h_{2}}e^{ik}e^{i\pi h_{1}}\right )\xi
=\Omega_{\alpha}\otimes\eta=\tilde\eta.
\end{align}
Here, for the first equality, we used the fact that $v(1) $ is in in ${\mathcal A}^{G}_{\Gamma}$ and
(\ref{ptpt}).
The third equality is from (\ref{piyo}).
\end{proof}
\begin{rem}
By Lemma \ref{lem4}, we can take $h_{1}, h_{2}$ in
the fixed point algebra ${\mathcal A}^{G}_{\Gamma\setminus \Lambda_{(\varepsilon,{\mathcal F})}}$.
With the special form of $\tilde\xi,\tilde\eta$, in (\ref{txedef}),
the problem is reduced to the Kadison transitivity theorem
for $({\mathcal K}_{\alpha},\pi_{\alpha}({\mathcal A}_{\Gamma}^{G}))$.
The irreducibility of $\pi_{\alpha}$ is used there.
From this we may obtain $k$ interpolating
$e^{i\pi\hat\pi (h_{1})}\tilde\xi$ and $e^{i\pi\hat\pi (h_{2})}\tilde\eta$,
from ${\mathcal A}^{G}_{\Gamma}$.
\end{rem}
\begin{lem}\label{lem8}
For any $\varepsilon>0$ and ${\mathcal F}\Subset\left ( {\mathcal A}_{\Gamma}\right )_{1}$, the following holds:
Let $({\mathcal H}_{i},\pi_{i},u_{i})$ with $i=0,1$ be
irreducible covariant representations of $\Sigma_{\Gamma}^{(\sigma)}$
with irreducible decomposition of $u_{i}$ given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma,i}\mid \gamma \in {\mathcal P}_{\sigma}\}$.
Let
$\xi_{i}\in{\mathcal K}_{\alpha,i}$ be a unit vector in ${\mathcal K}_{\alpha,i}$ for $i=0,1$.
Suppose that the representation $\hat\pi_{i}:=\mathop{\mathrm{id}}\nolimits_{{\mathcal H}_{\alpha}}\otimes \left ( \pi_{i}\times u_{i}\right )$, $i=0,1$
of $B({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$
and unit vectors $\tilde\xi_{i}:=\Omega_{\alpha}\otimes \xi_{i}$
in ${\mathcal H}_{\alpha}\otimes {\mathcal H}_{i}$, $i=0,1$ satisfy
\begin{align}\label{toto}
\left \vert
\braket{\tilde \xi_{0}}{\hat\pi_{0}(x y^{*})\tilde\xi_{0}}-\braket{\tilde \xi_{1}}{\hat\pi_{1}(x y^{*})\tilde\xi_{1}}
\right \vert<\frac 12\delta_{2}\left ( \varepsilon,{\mathcal F}\right ),\quad
\text{for all }\quad x,y\in {\mathcal G}_{\Lambda(\varepsilon,{\mathcal F})}.
\end{align}
(Recall Notation \ref{lef} for $\delta_{2}$.)
Then for any $\varepsilon'>0$ and ${\mathcal F}'\Subset B({\mathcal H}_{\alpha})\otimes C^{*}(\Sigma_{\Gamma}^{(\sigma)})$,
there exists a norm-continuous path
$v:[0,1]\to {\mathcal U}({\mathcal A}_{\Gamma}^{G})$ with $v(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$ such that
\begin{align}\label{zen}
\left \vert
\braket{\tilde \xi_{0}}{\hat\pi_{0}(a)\tilde\xi_{0}}
-\braket{\tilde \xi_{1}}{\left ( \hat\pi_{1}\circ \mathop{\mathrm{Ad}}\nolimits\left ( v(1)\right )\rmk(a)\tilde\xi_{1}}
\right \vert<\varepsilon',\quad
\text{for all }\quad a\in{\mathcal F}',
\end{align}
and
\begin{align}
\left \Vert
\mathop{\mathrm{Ad}}\nolimits v(t)(y)-y
\right \Vert<\varepsilon,\quad
\text{for all }\quad y\in {\mathcal F},\quad
\text{and }\quad t\in[0,1].
\end{align}
\end{lem}
\begin{proof}
From Lemma \ref{lem2}, there exists a self-adjoint $h\in {\mathcal A}_{\Gamma}^{G}$
such that
\begin{align}\label{lem2mains}
\left \vert
\braket{\tilde \xi_{0}}{\hat\pi_{0}(a)\tilde\xi_{0}}-\braket{\tilde \xi_{1}}{\hat\pi_{1}\circ\mathop{\mathrm{Ad}}\nolimits \left ( e^{ih}\right )(a)\tilde\xi_{1}}
\right \vert<\min\left\{\varepsilon', \frac 12 \delta_{2}\left ( \varepsilon,{\mathcal F}\right )\right\}\quad \text{for all}\quad a\in {\mathcal F}'\cup {\mathcal G}_{\Lambda\left ( \varepsilon,{\mathcal F}\right )}\left ({\mathcal G}_{\Lambda\left ( \varepsilon,{\mathcal F}\right )}\right )^{*}.
\end{align}
From this and (\ref{toto}), we have
\begin{align}\label{108y}
\left \vert
\braket{\tilde \xi_{1}}{\hat\pi_{1}\circ\mathop{\mathrm{Ad}}\nolimits \left ( e^{ih}\right )(xy^{*})\tilde\xi_{1}}
-\braket{\tilde \xi_{1}}{\hat\pi_{1}(xy^{*})\tilde\xi_{1}}
\right \vert<
\delta_{2}\left ( \varepsilon,{\mathcal F}\right )\quad
\text{for all}\quad x,y\in {\mathcal G}_{\Lambda\left ( \varepsilon,{\mathcal F}\right )}.
\end{align}
Recall from Lemma \ref{lem11iii} that
\begin{align}\label{ptpt1}
\pi_{1}(a)=\bigoplus_{\gamma\in {\mathcal P}_{\sigma}} \mathbb I_{{\mathcal H}_{\gamma}}\otimes \pi_{\gamma,1}(a),\quad
a\in {\mathcal A}^{G}_{\Gamma}
\end{align}
with irreducible $*$-representations $({\mathcal K}_{\gamma,1}, \pi_{\gamma,1})$
of ${\mathcal A}_{\Gamma}^{G}$.
From this and $h\in{\mathcal A}_{\Gamma}^{G}$, we see that
$\hat\pi_{1}\left ( e^{-ih}\right )\tilde\xi_{1}=\Omega_{\alpha}\otimes \pi_{\alpha, 1}(e^{-ih})\xi_{1}$
By (\ref{108y}),
$({\mathcal H}_{1},\pi_{1},u_{1})$,
$\xi_{1}$, $\pi_{\alpha, 1}(e^{-ih})\xi_{1}$
satisfies the required condition in Lemma \ref{lem5}.
Applying Lemma \ref{lem5} for $({\mathcal H}_{1},\pi_{1},u_{1})$ and $ \xi_{1}$,
$\pi_{\alpha,1}(e^{-ih})\xi_{1}$,
we obtain a norm-continuous path of unitaries
$v:[0,1]\to{\mathcal U}({\mathcal A}_\Gamma^{G})$ such that $v(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$,
\begin{align}\label{tetx}
\hat\pi_{1}\left ( e^{-ith} \right ) \tilde\xi_{1}=\hat\pi_{1} \left ( v(1)^{*}\right )\tilde\xi_{1},
\end{align}
and
\begin{align}\label{adv1}
\sup_{t\in[0,1]}\left \Vert
\mathop{\mathrm{Ad}}\nolimits v(t)(a)-a
\right \Vert<\varepsilon,\quad \text{for all}\quad a\in {\mathcal F}.
\end{align}
From (\ref{lem2mains}) and (\ref{tetx}), we obtain (\ref{zen}).
\end{proof}
\begin{rem}
As in \cite{kos}, we replace $e^{ith}$ with $v(t)$ which satisfy (\ref{adv1}).
We may do so with $v(t)$ in ${\mathcal A}_{\Gamma}^{G}$ because of Lemma \ref{lem5}.
\end{rem}
After these preparation, the proof of Proposition \ref{lem6}
is the same as proof of Theorem 2.1 of \cite{kos}.
We give it here for the reader's convenience.
\begin{proofof}[Proposition \ref{lem6}]
We fix an increasing sequence $\Lambda_{n}$, $n=0,1,2,\ldots$
of non-empty finite subsets of $\Gamma$ such that $\Lambda_{n}\nearrow \Gamma$.
For each $i=0,1$, we use the notation $\hat\pi_{i}$, $\tilde\xi_{i}$, $\hat\varphi_{{\xi}_{i}}$ given in
Notation \ref{nagai}, replacing $({\mathcal H},\pi,u)$ and $\xi\in{\mathcal K}_{\alpha}$
with $({\mathcal H}_{i},\pi_{i},u_{i})$ and $\xi_{i}\in{\mathcal K}_{\alpha,i}$.
Let $({\mathcal K}_{\alpha,i}, \pi_{\alpha,i})$, $i=0,1$ be the irreducible $*$-representation of ${\mathcal A}_{\Gamma}^{G}$ obtained in Lemma \ref{lem11iii} (\ref{eq53})
with $({\mathcal H},\pi,u)$, $\Gamma_{0}$ replaced by $({\mathcal H}_{i},\pi_{i},u_{i})$, $\Gamma$.
Set ${\mathcal F}_{0}:={\mathcal S}_{\Lambda_{0}}$. (Recall (\ref{sldef}).)
Fix $\varepsilon>0$ or set $\varepsilon=1$.
Set ${\mathcal G}_{0}:=
{\mathcal G}_{\Lambda\left ({\varepsilon},{\mathcal F}_{0}\right )}{\mathcal G}_{\Lambda\left ({\varepsilon},{\mathcal F}_0\right )}^{*}$.
From Lemma \ref{lem2}, there exists a self-adjoint $h_{0}\in{\mathcal A}_\Gamma^{G}$.
such that
\begin{align}\label{kaze}
\left \vert
\hat\varphi_{{\xi}_{0}}\circ \mathop{\mathrm{Ad}}\nolimits\left ( e^{ih_{0}}\right )\left ( a\right )
-\hat\varphi_{{\xi}_{1}}(a)
\right \vert<\min\left\{\frac12 \delta_{{2}}\left (
{\varepsilon},{\mathcal F}_{0}
\right ), {\varepsilon}\right\},\quad
a\in {\mathcal G}_{0}\cup{\mathcal F}_{0}.
\end{align}
(Recall Notation \ref{lef} for $\delta_{2}$.)
We define $v_{0}:=[0,1]\to {\mathcal U}({\mathcal A}_\Gamma^{G})$ by
\begin{align}\label{v0def}
v_{0}(t)=e^{ith_{0}},\quad t\in[0,1].
\end{align}
We consider the following proposition $[P_{n}]$ for
each $n\in \mathbb{N}}\newcommand{\bb}{\mathbb{B}$:
\begin{quote} {[$P_{n}$]}
There exist norm-continuous paths $v_{k}:[0,1]\to {\mathcal U}({\mathcal A}_\Gamma^{G})$, $k=0,\ldots, 2n$
with $v_{k}(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$
satisfying the following:
Set
\begin{align}\label{fe}
{\mathcal F}_{2j}:= \left\{
x,
\mathop{\mathrm{Ad}}\nolimits\left (
v_{2j-1}(1)^{*} v_{2j-3}(1)^{*}\cdots v_{3}(1)^{* }v_{1}(1)^{*}
\right )(x)\mid x\in {\mathcal S}_{\Lambda_{2j}}
\right\}, \quad j=1,\ldots, n,
\end{align}
and
\begin{align}\label{fo}
{\mathcal F}_{2j-1}:= \left\{
x,
\mathop{\mathrm{Ad}}\nolimits\left (
v_{2j-2}(1)^{*} v_{2j-4}(1)^{*}\cdots v_{4}(1)^{* }v_{2}(1)^{*}v_{0}(1)^{*}
\right )(x)\mid x\in {\mathcal S}_{\Lambda_{2j-1}}
\right\}, \quad j=1,\ldots, n.
\end{align}
(Recall (\ref{sldef}).)
We also denote the finite subset ${\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2^{k}},{\mathcal F}_{k}\right )}{\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2^{k}},{\mathcal F}_{k}\right )}^{*}$
by ${\mathcal G}_{k}$, for each $k=0,1,\ldots,2n$.
Then
the following three
inequalities hold.
\begin{enumerate}
\item
For all $a\in {\mathcal G}_{2n}\cup{\mathcal F}_{2n}$,
\begin{align}\label{kono}
\left \vert
\hat\varphi_{{\xi}_{0}}\circ \mathop{\mathrm{Ad}}\nolimits\left (
v_{0}(1)v_{2}(1)\cdots v_{2n}(1)
\right )(a)
-\hat\varphi_{{\xi}_{1}}\circ \mathop{\mathrm{Ad}}\nolimits\left (
v_{1}(1)v_{3}(1)\cdots v_{2n-1}(1)
\right )(a)
\right \vert
<\min\left\{\frac12 \delta_{{2}}\left (
\frac{\varepsilon}{2^{2n}},{\mathcal F}_{2n}
\right ), \frac{\varepsilon}{2^{2n}}\right\}
\end{align}
\item
For all $a\in {\mathcal G}_{2n-1}\cup{\mathcal F}_{2n-1}$,
\begin{align}\label{hito}
\left \vert
\hat\varphi_{{\xi}_{0}}\circ \mathop{\mathrm{Ad}}\nolimits\left (
v_{0}(1)v_{2}(1)\cdots v_{2n-2}(1)
\right )(a)
-\hat\varphi_{{\xi}_{1}}\circ \mathop{\mathrm{Ad}}\nolimits\left (
v_{1}(1)v_{3}(1)\cdots v_{2n-1}(1)
\right )(a)
\right \vert
<\min\left\{\frac12 \delta_{{2}}\left (
\frac{\varepsilon}{2^{2n-1}},{\mathcal F}_{2n-1}
\right ), \frac{\varepsilon}{2^{2n-1}}\right\}
\end{align}
\item For all $t\in[0,1]$, $k=1,2,\ldots,2n$ with $k\le 2n$ and $x\in {\mathcal F}_{k-1}$,
we have
\begin{align}\label{vapd}
\left \Vert
\mathop{\mathrm{Ad}}\nolimits v_{k}(t)(x)-x
\right \Vert<\frac\varepsilon{2^{k-1}}.
\end{align}
\end{enumerate}
\end{quote}
Let us check that [$P_{1}$] with $v_{0}$ given in (\ref{v0def}) holds.
Set ${\mathcal F}_{1}$ as in (\ref{fo}) with $j=1$ and this $v_{0}$.
Set ${\mathcal G}_{1}:={\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2},{\mathcal F}_{1}\right )}{\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2},{\mathcal F}_{1}\right )}^{*}$.
From (\ref{kaze}), applying
Lemma \ref{lem8} for vectors $\xi_{1}$ and $\pi_{\alpha,0}(v_{0}(1)^{*})\xi_{0}$,
there exists a norm-continuous path
$v_{1}:[0,1]\to {\mathcal U}\left ({\mathcal A}_{\Gamma}^{G}\right )$ with $v_{1}(0)=\mathbb I$
such that
\begin{align}\label{p1}
\left \vert
\hat\varphi_{{\xi}_{0}}\circ\mathop{\mathrm{Ad}}\nolimits \left ( v_{0}(1)\right )\left ( a \right )
-\hat\varphi_{{\xi}_{1}}\circ\mathop{\mathrm{Ad}}\nolimits \left ( v_{1}(1)\right )\left ( a \right )
\right \vert
<\min\left\{\frac12 \delta_{{2}}\left (
\frac{\varepsilon}{2},{\mathcal F}_{1}\right ),
\frac{\varepsilon}{2}\right\}
,\quad
\text{for all }\quad a\in {\mathcal G}_{1}\cup{\mathcal F}_{1}
\end{align}
and
\begin{align}
\left \Vert
\mathop{\mathrm{Ad}}\nolimits v_{1}(t)(y)-y
\right \Vert<\varepsilon,\quad
\text{for all }\quad y\in {\mathcal F}_{0},\quad
\text{and }\quad t\in[0,1].
\end{align}
Set ${\mathcal F}_{2}$ as in (\ref{fe}) with $j=1$ for this $v_{1}$.
And set ${\mathcal G}_{2}:={\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2^{2}},{\mathcal F}_{2}\right )}{\mathcal G}_{\Lambda\left (\frac{\varepsilon}{2^{2}},{\mathcal F}_{2}\right )}^{*}$.
From (\ref{p1}), applying Lemma \ref{lem8} again to
vectors $\pi_{\alpha,0}\left ( v_{0}(1)^{*}\right )\xi_{0}$ and
$\pi_{\alpha,1}\left ( v_{1}(1)^{*}\right )\xi_{1}$, we obtain
a norm-continuous path
$v_{2}:[0,1]\to {\mathcal U}\left ({\mathcal A}_{\Gamma}^{G}\right )$ with $v_{2}(0)=\mathbb I$
such that
\begin{align}
\left \vert
\hat\varphi_{{\xi}_{0}}\circ\mathop{\mathrm{Ad}}\nolimits \left ( v_{0}(1)v_{2}(1)\right )\left ( a \right )
-\hat\varphi_{{\xi}_{1}}\circ\mathop{\mathrm{Ad}}\nolimits \left ( v_{1}(1)\right )\left ( a \right )
\right \vert
<\min\left\{\frac12 \delta_{{2}}\left (
\frac{\varepsilon}{2^{2}},{\mathcal F}_{2}\right ),
\frac{\varepsilon}{2^{2}}\right\}
,\quad
\text{for all }\quad a\in {\mathcal G}_{2}\cup{\mathcal F}_{2}
\end{align}
and
\begin{align}
\left \Vert
\mathop{\mathrm{Ad}}\nolimits v_{2}(t)(y)-y
\right \Vert<\frac \varepsilon 2,\quad
\text{for all }\quad y\in {\mathcal F}_{1},\quad
\text{and }\quad t\in[0,1].
\end{align}
Hence we have proven [$P_{1}$] with $v_{0}$ given in (\ref{v0def}).
The proof that [$P_{n}$] implies [$P_{n+1}$] with the same $v_{0},v_{1},\ldots, v_{2n}$
as in [$P_{n}$]
can be carried out in the same way, by the repeated use of Lemma \ref{lem8}.
Hence we obtain a sequence $\{v_{n}\}_{n=0}^{\infty}$ of norm-continuous paths
$v_{n}:[0,1]\to {\mathcal U}({\mathcal A}_{\Gamma}^{G})$ with $v_{n}(0)=\mathbb I_{{\mathcal A}_{\Gamma}}$
satisfying (\ref{kono}) (\ref{hito}) (\ref{vapd}).
We define norm continuous paths $y,z:[0,\infty)\to {\mathcal U}({\mathcal A}_{\Gamma}^{G})$
by
\begin{align}
&y(t):=v_{1}(t) v_{3}(t)\cdots v_{2j-1}(1)v_{2j+1}(t-[t]),\quad j\le t<j+1,\quad
j=0,1,2,\ldots,\nonumber\\
&z(t):=v_{0}(t) v_{2}(t)\cdots v_{2j-2}(1)v_{2j}(t-[t]),\quad j\le t<j+1\quad
j=0,1,2,\ldots.
\end{align}
Here $[t]$ denotes the largest integer less than or equal to $t$.
Then as in section 2 of \cite{kos},
for any $a\in{\mathcal A}_{\rm loc,\Gamma}$,
the limit
\begin{align}
\gamma_{0}(a):=\lim_{t\to\infty}\mathop{\mathrm{Ad}}\nolimits\left ( z(t)\right )(a),\quad
\gamma_{1}(a):=\lim_{t\to\infty}\mathop{\mathrm{Ad}}\nolimits\left ( y(t)\right )(a)
\end{align}
exist because of (\ref{vapd}) and the fact that ${\mathcal S}_{\Lambda_{n}}\subset {\mathcal F}_{n}$.
These limit define endomorphisms $\gamma_{0},\gamma_{1}$ on ${\mathcal A}_{\Gamma}$.
Furthermore, because of (\ref{vapd}) and the fact that
$\mathop{\mathrm{Ad}}\nolimits\left ( v_{n-1}(1)^{*}v_{n-3}(1)^{*}\cdots\right )\left ( {\mathcal S}_{\Lambda_{n}}\right )\subset {\mathcal F}_{n}$,
by the definition (\ref{fe}) and (\ref{fo}), for any $x\in{\mathcal A}_{\rm loc,\Gamma}$,
the limit
\begin{align}
&\lim_{j\to\infty} \mathop{\mathrm{Ad}}\nolimits\left ( v_{2j}(1)^{* } v_{2j-2}(1)^{*}\cdots v_{0}(1)^{*}\right )(x)
=:a_{x} \\
&\lim_{j\to\infty} \mathop{\mathrm{Ad}}\nolimits\left ( v_{2j-1}(1)^{* } v_{2j-3}(1)^{*}\cdots v_{1}(1)^{*}\right )(x)
=:b_{x}
\end{align}
exist. For these limits, we have
$\gamma_{0}(a_{x})=x$, and $\gamma_{1}(b_{x})=x$, for all $x\in{\mathcal A}_{\rm loc,\Gamma}$.
Therefore, $\gamma_{0}$ and $ \gamma_{1}$ are automorphisms.
By {\it 1.}, {\it 2.} of [$P_{n}$], we also have
\begin{align}\label{l1l2}
\varphi_{0}\circ\gamma_{0}=
\left.\hat\varphi_{{\xi}_{0}}\right\vert_{{\mathcal A}_{\Gamma}}\circ\gamma_{0}
=\left.\hat\varphi_{{\xi}_{1}}\right\vert_{{\mathcal A}_{\Gamma}}\circ\gamma_{1}
=\varphi_{1}\circ\gamma_{1}.
\end{align}
Let $\Xi_{\Gamma}$ be an automorphism given by $\Xi_{\Gamma}:=\gamma_{0}\circ\gamma_{1}^{-1}$ on ${\mathcal A}$.
Define a norm-continuous path $w:[0,\infty)\to {\mathcal U}({\mathcal A}_{\Gamma}^{G})$
by
\begin{align}
w(t):=z(t)y(t)^{*},\quad t\in [0,\infty).
\end{align}
We have
\begin{align}
\Xi_{\Gamma}(x)=\gamma_{0}\circ\gamma_{1}^{-1}(x)
=\lim_{t\to\infty} \mathop{\mathrm{Ad}}\nolimits(w(t))(x),\quad x\in{\mathcal A}_{\Gamma},
\end{align}
and $w(0)=\mathbb I$.
From (\ref{l1l2}), we have $\varphi_{0}\circ\Xi_{\Gamma}=\varphi_{1}$.
This completes the proof.
\end{proofof}
\section{Proof of the Main Theorem}
Now we are ready to prove Theorem \ref{main}.
Let $\omega_{0}$ and $\omega_{1}$ be elements of $SPG({\mathcal A})$.
\begin{proofof}["if'' part of Theorem \ref{main}]
Suppose that $c_{\omega_{0},R}=c_{\omega_{1},R}$.
From Lemma \ref{clmcr}, we have $c_{\omega_{0},L}=c_{\omega_{1},L}$.
For each $\zeta=L,R$ and $i=0,1$, let
$({\mathcal L}_{\omega_{i},{\varsigma}}, \rho_{\omega_{i},{\varsigma}}, u_{\omega_{i},{\varsigma}},\sigma_{\omega_{i},{\varsigma}})$
be a quadruple associated to $(\omega_{i}\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$.
By Remark \ref{rem17},
we may assume that $\sigma_{R}:=\sigma_{\omega_{0},{R}}=\sigma_{\omega_{1},{R}}$
and $\sigma_{L}:=\sigma_{\omega_{0},{L}}=\sigma_{\omega_{1},{L}}$.
For each $\zeta=L,R$ and $i=0,1$,
the triple $({\mathcal L}_{\omega_{i},{\varsigma}}, \rho_{\omega_{i},{\varsigma}}, u_{\omega_{i},{\varsigma}})$
is an irreducible covariant representations of
the twisted $C^{*}$-dynamical system $\Sigma_{\Gamma_{\varsigma}}^{(\sigma_{\varsigma})}$.
By Lemma \ref{pd}, $u_{\omega_{i},{\varsigma}}$ has an
irreducible decomposition given by a set of Hilbert spaces
$\{{\mathcal K}_{\gamma,i,\varsigma}\mid \gamma \in {\mathcal P}_{\sigma_{\varsigma}}\}$.
For each $\varsigma=L,R$, fix some $\alpha_{\varsigma}\in {\mathcal P}_{\sigma_{\varsigma}}$.
The spaces ${\mathcal K}_{\alpha_{\varsigma},i,\varsigma}$ $i=0,1$ are non-zero
because of Proposition \ref{zbdi}.
Fix unit vectors $\xi_{i,\varsigma}\in {\mathcal K}_{\alpha_{\varsigma},i,\varsigma}$ for each $\varsigma=L,R$ and $i=0,1$.
For each $\varsigma=L,R$ and $i=0,1$,
let $\hat\varphi_{\xi_{i,\varsigma}}$ be a state on ${\mathcal B}({\mathcal H}_{\alpha_{\varsigma}})\otimes C^{*}(\Sigma_{\varsigma}^{{(\sigma_{\varsigma})}})$
given by $({\mathcal L}_{\omega_{i},{\varsigma}}, \rho_{\omega_{i},{\varsigma}}, u_{\omega_{i},{\varsigma}},\sigma_{\omega_{i},{\varsigma}})$
(defined in Notation \ref{nagai} (\ref{nekosan}) with
${\mathcal H},\pi,u,\xi$ replaced by ${\mathcal L}_{\omega_{i},{\varsigma}}, \rho_{\omega_{i},{\varsigma}}, u_{\omega_{i},{\varsigma}},\xi_{i,\varsigma}$).
Let
$\varphi_{i,\varsigma}$ be the restriction of $\hat\varphi_{\xi_{i,\varsigma}}$
onto ${\mathcal A}_{\Gamma_{\varsigma}}$.
By the definition, $\varphi_{0,\varsigma}$, $\varphi_{1,\varsigma}$ are
quasi-equivalent to $\omega_{0}\vert_{{\mathcal A}_{\varsigma}}$, $\omega_{1}\vert_{{\mathcal A}_{\varsigma}}$,
respectively.
By Proposition \ref{lem6}, there exist $\Xi_{\varsigma}\in \mathop{\mathrm{AInn}}\nolimits^{G}({\mathcal A}_{\varsigma})$
such that $\varphi_{1,\varsigma}=\varphi_{0,\varsigma }\circ \Xi_{\varsigma}$,
$\varsigma=L,R$.
Recall that $\omega_{0}$, $\omega_{1}$ are quasi-equivalent to
$\omega_{0}\vert_{{\mathcal A}_{L}}\otimes\omega_{0}\vert_{{\mathcal A}_{R}}$ and
$\omega_{1}\vert_{{\mathcal A}_{L}}\otimes\omega_{1}\vert_{{\mathcal A}_{R}}$ respectively from the
split property. (Remark \ref{splitrem}.)
Hence we obtain
\begin{align}
&\omega_{1}\sim_{\rm q.e.}\omega_{1}\vert_{{\mathcal A}_{L}}\otimes\omega_{1}\vert_{{\mathcal A}_{R}}
\sim_{\rm q.e.}
\varphi_{1,L}\otimes\varphi_{1,R}
=
\left ( \varphi_{0,L}\circ \Xi_{L}\right )\otimes\left (\varphi_{0,R}\circ \Xi_{R}\right )
=\left (
\varphi_{0,L}\otimes\varphi_{0,R}\right )
\circ \left ( \Xi_{L}\otimes \Xi_{R}\right )\nonumber\\
&\sim_{\rm q.e.}
\left (
\omega_{0}\vert_{{\mathcal A}_{L}}\otimes\omega_{0}\vert_{{\mathcal A}_{R}}
\right )
\circ \left ( \Xi_{L}\otimes \Xi_{R}\right )
\sim_{\rm q.e.}
\omega_{0}
\circ \left ( \Xi_{L}\otimes \Xi_{R}\right ).
\end{align}
This completes the proof.
\end{proofof}
\begin{proofof}["only if'' part of Theorem \ref{main}]
Suppose
that $\omega_{0}\sim_{{\rm split},\tau}\omega_{1}$.
Then
there exist automorphisms $\Xi_{L}\in \mathop{\mathrm{AInn}}\nolimits^{G}({\mathcal A}_{L})$
and $\Xi_{R}\in \mathop{\mathrm{AInn}}\nolimits^{G}({\mathcal A}_{R})$ such that
$\omega_{1}$ and $\omega_{0}\circ\left (\Xi_{L}\otimes \Xi_{R}\right )$
are quasi-equivalent.
From the split property, we have
$\omega_{1}\sim_{\rm q.e.}\omega_{1}\vert_{{\mathcal A}_{L}}\otimes\omega_{1}\vert_{{\mathcal A}_{R}}$ and $\omega_{0}\circ\left (\Xi_{L}\otimes \Xi_{R}\right )\sim_{\rm q.e.}
\left (
\omega_{0}\vert_{{\mathcal A}_{L}}\circ\Xi_{L}
\right )\otimes
\left (
\omega_{0}\vert_{{\mathcal A}_{R}}\circ\Xi_{R}
\right )$.
Combining these, we see that
$\omega_{1}\vert_{{\mathcal A}_{R}}$ and $\omega_{0}\vert_{{\mathcal A}_{R}}\circ\Xi_{R}
$
are quasi-equivalent.
For each $\zeta=L,R$ and $i=0,1$, let
$({\mathcal L}_{\omega_{i},{\varsigma}}, \rho_{\omega_{i},{\varsigma}}, u_{\omega_{i},{\varsigma}},\sigma_{\omega_{i},{\varsigma}})$
be a quadruple associated to $(\omega_{i}\vert_{{\mathcal A}_{{\varsigma}}},\tau_{\varsigma})$.
From $\omega_{1}\vert_{{\mathcal A}_{R}}\sim_{\rm q.e.}\omega_{0}\vert_{{\mathcal A}_{R}}\circ\Xi_{R}$,
$\rho_{\omega_0,{R}}\circ\Xi_{R}$ is an irreducible $*$-representation of
${\mathcal A}_{R}$ on ${\mathcal L}_{\omega_{0},{R}}$, which is quasi-equivalent
to the GNS representation of $\omega_{1}\vert_{{\mathcal A}_{R}}$.
Furthermore, the projective unitary representation
$u_{\omega_{0},{R}}$ of $G$ with
$2$-cocycle $\sigma_{\omega_{0},{R}}$
satisfies
\begin{align}
\rho_{\omega_{0},{R}}\circ\Xi_{R}\circ \tau_{R}(g)\left ( a\right )
=\rho_{\omega_{0},{R}}\circ \tau_{R}(g) \circ\Xi_{R}\left ( a\right )
=\mathop{\mathrm{Ad}}\nolimits\left ( u_{\omega_{0},{R}}(g)\right )\circ \rho_{\omega_{0},{R}}\circ\Xi_{R}\left ( a\right )
,\quad a\in{\mathcal A}_{R},\quad g\in G.
\end{align}
From this,
$({\mathcal L}_{\omega_{0},{R}}, \rho_{\omega_{0},{R}}\circ\Xi_{R},
u_{\omega_{0},{R}},\sigma_{\omega_{0},{R}})$
is a quadruple associated to $(\omega_{1}\vert_{{\mathcal A}_{{R}}},\tau_{R})$.
Hence we obtain $c_{\omega_{1},R}=c_{\omega_{0},R}$.
This proves the claim.
\end{proofof}
{\bf Acknowledgment.}\\
{
This work was supported by JSPS KAKENHI Grant Number 16K05171 and 19K03534.
}
|
1,108,101,565,453 | arxiv | \section{Introduction}
\label{sec:introduction}
Web pages are constantly increasing in complexity and size. The HTTP Archive reports that the global average web page size has surpassed 1MB in April 2012 \cite{httparchive}. By the start of July 2013, visiting one of the top 1000 sites incurs, on average, 1246kB of web page resources over 100 separate requests \cite{httparchive}.
Such growth has been fueled by the emergence of advanced web-based services (Web 2.0, SaaS cloud services, etc.), enhanced client device capabilities (JavaScript browser runtimes \cite{Severance2012javascript}, display), and increased downlink speeds \cite{akamai2012q4soti}.
This growing complexity, however, can dramatically slow down page retrieval. Unfortunately, this has negative consequences \cite{skadberg2004visitors},
and very real ones in the case of commercial websites. It has been found that most users cannot tolerate in excess of 2 seconds of page load delay \cite{nah2004study}, and that increments of just 100ms on shopping websites can decrease sales by 1\% \cite{Kohavi2007experiments}. The converse is similarly true:
decreasing delay can have a powerful enhancing effect, with Google claiming to have increased ad revenue by 20\% through cutting 500ms from load times. To mitigate page load times, various extensions to HTTP have been proposed.
However, in practice, little progress has been made, with many web servers, proxies and browsers being slow to adopt these new tweaks (e.g.\ pipelining \cite{rfc2616}).
In light of these observations, some have proposed developing a new web protocol. Such efforts include Microsoft's Speed+Mobility \cite{SM-draft} and HTML5 Websockets; most prominent, however, is Google's SPDY \cite{SPDY-draft}. This has already begun to see deployment by prominent organisations such as Google, Twitter, Akamai and Facebook, whilst also being adopted as the base for HTTP/2.0 by the HTTPbis Working Group. Despite this, we still possess a limited understaning of its behaviour, overheads and performance: does it offer a fundamental improvement or just further tweaking?
In an attempt to answer the above question, a small number of early stage studies have explored the topic.
They offer a range of results, with some claiming significant gains and (curiously) others claiming rather negative results. This report seeks to resolve these issues, by analysing the circumstances under which SPDY can improve page load times, and the ones where the opposite is true.
To achieve this, we perform a large-scale evaluation of SPDY using an emulated testbed.
Based on NPN negotiation handshakes, we found out that the number of top 10,000 Alexa sites adopting SPDY was 208 on October 14\textsuperscript{th} 2012, rising to 271 on April 23\textsuperscript{rd} 2013.
Using some of these websites, we execute a large number of probes to measure the performance of SPDY in the wild. Confirming our suspicions, we find highly variable results between different websites and samples: SPDY has the potential to both benefit and damage page load times. Motivated by this, we perform a large body of controlled experiments in our local testbed to understand the reasons behind these performance variations. We identify the website types and network characteristics that SPDY thrives under, as well as how these benefits vary based on provider-side infrastructural decisions. To our knowledge, this is the first effort to offer such insights.
The rest of the report is organized as follows. Section \ref{sec:background} provides background and highlights related work. Section \ref{sec:meth} describes our measurement toolkit and environments. Section \ref{sec:results:live} presents the results of comparing SPDY to HTTPS on live websites. We then use an emulated network testbed in order to dissect the factors affecting SPDY performance, namely network characteristics (section \ref{sec:results:network}) and infrastructure setup (section \ref{sec:results:infra}).
Section \ref{sec:Conclusion} concludes and discusses future work.
\section{Background and Related Work}
\label{sec:background}
\subsection{SPDY} \label{sec:background:spdy}
SPDY is an application-layer web protocol that reuses HTTP's semantics \cite{rfc2616}. As such, it retains all features including cookies, ETags and Content-Encoding negotiations. SPDY only replaces the manner in which data is written to the network. The purpose of this is to reduce page load time. It does this by introducing the following mechanisms:
\begin{itemize}
\item \emph{Multiplexing}: A framing layer multiplexes streams over a single connection, removing the need to establish separate TCP connections for transferring different page resources.
\item \emph{Compression}: All header data is compressed to reduce the overheads of multiple related requests.
\item \emph{Universal encryption}: SPDY is negotiated over SSL/TLS and thus operates exclusively over a secure channel in order to address the increasing amounts of traffic sent over insecure paths (e.g.\ public WiFi).
\item \emph{Server Push/Hint}: Servers could proactively \textit{push} resources to clients (e.g.\ scripts and images that will be required). Alternatively, SPDY can send \textit{hints} advising clients to pre-fetch content.
\item \emph{Content prioritization}: A client can specify the preferred order in which resources should be transferred.
\end{itemize}
SPDY consists of two components. The first provides framing of data, thereby allowing things like compression and multiplexing. The framing layer works on top of secure (SSL/TLS) persistent TCP connections that are kept alive as long as the corresponding web pages are open. Clients and servers exchange \emph{control} and \emph{data} frames, both of which contain an 8 bytes header. Control frames are used for carrying connection management signals and configuration options, while data frames carry HTTP requests and responses. The second component maps HTTP communication into SPDY data frames. Multiple logical HTTP streams can be multiplexed using interleaved data frames over a single TCP connection.
\subsection{Related Studies}
There have been a small number of preliminary studies looking at the performance of SPDY.
The first was presented in a Google white paper \cite{SPDYWhitepaper}. By running 25 of the top 100 websites over simulated home networks, this publication showed significant performance benefits over both HTTP (27-60\%) and HTTPS (39-55\%). Whilst this does seem impressive, there are somewhat conflicting accounts of SPDY's performance in studies provided by Akamai \cite{NotAsSPDY} and Microsoft \cite{Padhye2012TR}. Tests performed by Akamai showed only a marginal benefit over HTTPS, alongside a decrease in performance when compared to HTTP. It was found that SPDY, on average, was only about 4.5\% faster than HTTPS, and about 3.4\% slower than HTTP. Microsoft offered slightly more positive results, but still did not attain the high levels previously reported by Google.
An Internet Draft by White \emph{et al.} \cite{whiteIEFT} also found a mix of results, highlighting that SPDY's performance would likely depend on a number of factors, e.g. number of servers, TCP configurations, network characteristics (specifically, latency and loss).
The authors found an average improvement of 29\% over HTTPS. They also reported that such benefits are not necessarily universal (i.e. they vary for different webpages). Nevertheless, the Internet Draft does not offer insight into such findings but rather focuses on coupling SPDY with an increase in the TCP initial congestion window.
It is difficult to derive a direct conclusion from the above studies as they offer a wide range of rather conflicting results. Some claim SPDY outperforms HTTP, whereas others claim the opposite. Consequently, the only clear conclusion is the SPDY has the potential to have highly variable performance. As of yet, these studies do not elucidate this observation, i.e.\ detailing the reasons behind such variations.
\section{Measurement Methodology}
\label{sec:meth}
\subsection{Measurement Toolkit}
\label{sec:meth:toolkit}
SPDY is designed to mitigate page load time for end users. We therefore focus on client-side measurements, for which we have built a toolkit based on the Chromium browser.
This seems a logical choice considering that both SPDY and Chromium were developed by Google. Chromium also offers sophisticated logging features that allow us to extract statistics via automated scripting. We use Chromium\,25 (running over Ubuntu Desktop 12.04.2) via the Chrome-HAR-capturer \cite{chrome-har-capturer} package, which interacts with Chromium through its remote debugging API. To ensure authenticity, we maintained all of Chromium's default settings, apart from disabling DNS pre-fetching in order to include DNS lookup time in all measurements.
When invoked, our measurement toolkit instructs Chromium to fetch a particular webpage. Once this is completed, the toolkit extracts detailed logs in the form of HTTP Archives (HAR) and Wireshark network traces. It then processes them to calculate metrics of interest. Traditionally, page load time has been measured by the Document Object Model (DOM) being fully loaded.
However, this is not suitable for our purposes, as it also captures browser processing time that is strictly HTML-related, e.g.\ arbitrating the style hierarchy. Instead, we wish to only measure the time spent performing network interactions. Thus, we use an alternate metric which we term the \textit{Time on Wire} (ToW). This is calculated using Wireshark network traces as the period between the first request and last response packets, giving us precise timestamps for the page transmission delay.
Using this toolkit, we employ Chromium in a non-obtrusive manner to retrieve a number of webpages in a range of different environments. The collated measurements allow us to explore the performance of SPDY. The rest of this section details the environments we utilised the measurement toolkit in.
\subsection{Measurement Setups}
\label{sec:meth:setups}
The measurements are separated into two groups, both using the above toolkit. First, we perform \emph{live tests}, probing real-world deployments (e.g.\ YouTube) to calculate the performance advantages of organisations currently using SPDY. Second, we expand on these results using \emph{emulated network tests}, creating our own controlled SPDY deployment in a local testbed. The latter allows us to deep-dive into SPDY's performance in a deterministic fashion by varying and monitoring the impact of various key factors (namely network conditions and website infrastructure setup). In both cases, a large number of samples are taken to ensure statistical significance. Overall, we have collected over 70,000 probes (12,000 live and 58,000 controlled).
\subsubsection{Live Tests}
\label{sec:meth:setups:live}
First, we perform live experiments using web sites that have already deployed SPDY in their real infrastructures. To discover these, we have implemented a crawler to probe the top 10k Alexa websites\footnote{All Alexa ranks henceforth are of April 23\textsuperscript{rd} 2013.}, recording their individual protocol support. We then select the top 8 Alexa websites that implement SPDY. We choose only the highest Alexa ranked website from every distinct online presence and disregard similar sites (facebook.com (\#1) but not fbcdn.net (\#202); google.com (\#2) but not google.co.in (\#12), google.com.hk (\#22), etc.; and so on). The list is shown in Table \ref{tab:LiveSPDYWebsites}.
\begin{table}[t!]
\centering
\caption{Live SPDY-enabled Websites}
\label{tab:LiveSPDYWebsites}
\begin{tabular}{l|rrrccc}
& \multicolumn{3}{c}{Resources} & SPDY & & Av.RTT \\ \cline{2-4}
Site & Count & Av. Size (kB) & Domains & Version & IW & (ms) \\ \hline
Facebook & 20 & 12.56 & 4 & 2 & 7 & 92 \\
Google & 7 & 41.29 & 2 & 3 & 7 & 8 \\
YouTube & 50 & 10.63 & 4 & 3 & 7 & 8 \\
Blogspot & 31 & 5.03 & 6 & 3 & 7 & 17 \\
Twitter & 7 & 46.40 & 3 & 3 & 10 & 158 \\
WordPress & 13 & 7.92 & 4 & 2 & 10 & 91 \\
imgur & 133 & 11.78 & 58 & 2 & 10 & 8 \\
youm7 & 270 & 11.07 & 54 & 2 & 10 & 150 \\
\end{tabular}
\end{table}
The selected websites provide a range of resource sizes, counts, and domains.
In terms of their respective delivery infrastructures, we note that all appear to use CDNs with the exception of WordPress. We confirm this using \texttt{whois} as well as other means (e.g.\ trying to directly access a CloudFlare IP address with a browser yields an error message \emph{from} CloudFlare).
It seems, unsurprisingly, that the employed CDN dictates the supported SPDY version and the TCP Initial Window (IW).\footnote{Note that increasing the IW size is another closely related component in Google's ``Make the Web Faster'' project. We determined IW by sending self-crafted TCP packets to the servers. Our program carried out a successful handshake followed by sending a HTTP GET for a large resource (typically a static image). We then noted the number of ensuing packets (which we never acknowledged) as the server's IW.} We therefore posit that our results for these sites are representative of the performance for other customers of the same respective CDN.
Using our measurement toolkit, we periodically probe each website from the Lancaster University campus using HTTP, HTTPS and SPDY. In this report, we focus on HTTPS as a baseline comparison as, like SPDY, it encrypts its data. However, where possible, we also include HTTP, considering that many websites have no interest in securing their connections. In both cases, when HTTP and HTTPS are used, we avoid bias by forcing Chromium to pursue the Next Protocol Negotiation (NPN) handshake nonetheless as SPDY does.
The probes are carried out for each website in an alternating sequence of protocols with 2 seconds between each run. For SPDY, we select the highest non-experimental version that the server supports (listed in Table\,\ref{tab:LiveSPDYWebsites}). Tests were carried out from different sites: Lancaster, Dublin, and Tokyo. We only discuss the Lancaster set as the other results provide very similar outcomes. The Lancaster tests ran on weekdays between 12pm and 5pm BST between 20/5/2013 and 23/5/2013. In total, we performed 1.06 million GET requests, with 500 samples taken for each website and protocol combination.
\subsubsection{Emulated Network Tests}
\label{sec:meth:setups:emu-net}
The above live experiments provide useful context to the current state of affairs, but are somewhat limited in what they can tell us. Although the comparison they provide is fair, it would be difficult to definitely and neutrally ascertain SPDY's performance as it is subject to variations in the network, and tightly bound to the particular deployment under test: its characteristics (e.g.\ web server, SPDY module/proxy) and its status (e.g.\ server load). To address these issues, we extend our tests by creating our own SPDY deployment in an emulated testbed interconnected via a LAN. This allows us to control the various network parameters to understand how they impact performance.
Our testbed consists of a client and server setup. The client runs our measurement toolkit and is connected via 100Mbps Ethernet. We then emulate various network conditions: the Linux \texttt{tc} utility is used to throttle bandwidth by shaping traffic with Hierarchy Token Bucket queuing \cite{htb}, and NetEm \cite{hemminger2005netem} is used (at the server) to specify a deterministic round trip time (RTT) and packet loss ratio (PLR).
The server runs Ubuntu Server 12.04 with the Apache 2.2.22 web server, supporting both HTTP and HTTPS. We use Apache's mod\_spdy 0.9.3.3-386 module which implements spdy/3. This is the most advanced SPDY implementation available, provided by Google's own SPDY project \cite{mod-spdy}. Using this server, we clone a set of the SPDY websites discovered in the wild; these are each intended to be representative of a broader class of top Alexa websites and are as follows:
\begin{itemize}
\item \emph{Twitter}: This is a simple page with only a few (7) resources (average size 46.4KB). This is comparable to other top Alexa websites such as Google and Blogspot and their regional versions, Wikipedia, and Soso (Chinese search engine).
\item \emph{YouTube}: This is a relatively complicated page, with a fair number (50) of resources (average size 10.63KB).
Examples of similar websites include Amazon's regional websites, AOL, Alibaba (Chinese e-commerce portal), About.com, and DailyMotion.
\item \emph{imgur}: This is a complicated page, with a large number (133) of images and flash (average size 11.78KB). Similar websites include QQ (Chinese messaging website), TaoBao (China's equivalent to eBay), The New York Times, CNN, and MSN.
\end{itemize}
The described setup enables us to have a single server-side SPDY implementation and a single SPDY-capable web browser, which rules out software discrepancies (note that we also later use multiple servers). This method also allows us to control the network characteristics in order to experiment with SPDY under different network conditions. Moreover, server and connection load are also controlled. Using this testbed, we apply the same methodology as in the live experiments, generating repeated page requests for the chosen webpages using HTTPS and SPDY.
The exact details of the parameters investigated will be presented in Sections \ref{sec:results:network} and \ref{sec:results:infra}.
\section{Live Results}
\label{sec:results:live}
We begin by performing experiments with existing deployments of SPDY. The aim here is not an exhaustive study but, rather, to form a general idea of the benefits being gained by some of those who have so far adopted SPDY. To achieve this, each website in Table \ref{tab:LiveSPDYWebsites}
is probed 500 times to calculate its ToW. Figure \ref{fig:plotLiveTop8-ToW} displays the cumulative distribution functions (CDFs) of these measurements, and Table \ref{tab:LiveSPDYgain} summarises them. Note that the HTTP results
are not included for websites that redirect such requests to HTTPS.
\begin{figure*}[t!]
\centering
\subfloat[Facebook]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-Facebook-cdf.pdf} }
\subfloat[Google]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-Google-cdf.pdf} }
\subfloat[YouTube]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-YouTube-cdf.pdf} }
\subfloat[Blogspot]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-Blogspot-cdf.pdf} }
\\
\subfloat[Twitter]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-Twitter-cdf.pdf} }
\subfloat[WordPress]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-Wordpress-cdf.pdf} }
\subfloat[imgur]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-imgur-cdf.pdf} }
\subfloat[youm7]{ \includegraphics[width=0.24\textwidth]{plotLive-ToW-youm7-cdf.pdf} }
\caption{ToW of Live SPDY-enabled Websites}
\label{fig:plotLiveTop8-ToW}
\end{figure*}
\vspace{0cm}
\begin{table}[h!]
\centering
\caption{Gain in ToW for Live SPDY-enabled Websites}
\label{tab:LiveSPDYgain}
\begin{tabular}{l|r}
& Average Gain in ToW \\
Site & (SPDY over HTTPS) \\ \hline
Facebook & 7.0\% \\
Google & -20.2\% \\
YouTube & 4.7\% \\
Blogspot & -6.0\% \\
Twitter & 10.6\% \\
WordPress & -15.1\% \\
imgur & 0.8\% \\
youm7 & 9.7\% \\
\end{tabular}
\end{table}
Confirming our analysis of past studies, the results are \emph{not} conclusive. We find no clear winner among the three protocols. Instead, we observe large performance variations between different websites, as well as between different samples for the same website. We find that notable improvements are, indeed, gained in some cases. On average, ToW is reduced by 7\% for Facebook, 4.7\% for YouTube, and 9.7\% for youm7. The biggest winner is the Twitter front-page, with an average ToW reduction of 10.6\%. This, however, is not a universal observation. In other cases, improvements are far more modest; for example, imgur only achieves a meager improvement of 0.8\%. Moreover, we find websites that suffer from their use of SPDY; an average ToW increase of
6.0\% for Blogspot and 15.1\% for Wordpress. Ironically, the biggest sufferer is Google with a 20.2\%
increase in ToW for their search homepage. There is certainly no one-size-fits-all operation with SPDY, as all websites alternate between SPDY and HTTP optimality.
These experiments therefore raise some interesting (yet serious) questions. From a research perspective, one might ask why these notable variations occur? From an administrator's perspective, the next logical question would then be if SPDY would benefit their deployment? The remainder of this report now explores these questions using emulated experiments. Whereas the live experiments limit our control to the client-side, emulated experiments allow us to dissect all aspects to understand the causes of such variations.
\section{Effect of Network Performance}
\label{sec:results:network}
To understand the reasons behind the variations in performance witnessed in the wild, we now perform controlled experiments in our testbed. We aim to examine the effect of different network conditions on the performance gain of SPDY over HTTPS. We mirror three of the above representative websites (Twitter, YouTube, imgur) and measure their ToW on a single client, single server testbed under a variety of network characteristics regarding delay, bandwidth, and loss.
\subsection{Delay}
\label{sec:results:network:rtt}
First, we inspect the impact that round trip time (RTT) has on SPDY's performance. In a real environment, this varies a lot between different requests due to client locations and path characteristics \cite{Kaune2009modelling,Elkhatib11}. To remove any variance, we fix bandwidth (BW) at 1Mbps and Packet Loss Ratio (PLR) at 0\%, whilst changing the RTT between the client and server in a range from 10ms to 490ms. Following this, we perform 20 requests for each website using each configuration with both SPDY and HTTPS. The results are presented in Figure \ref{fig:emu-delay-bw1} as the average percentage improvement in ToW of SPDY over HTTPS.
\begin{figure}[htb!]
\centering
\includegraphics[width=80mm]{emu-delay-bw1}
\caption{Effect of Round Trip Time (BW=1Mbps, PLR=0\%).}
\label{fig:emu-delay-bw1}
\end{figure}
In contrast to the live experiments, we see that SPDY \emph{always} achieves better performance than HTTPS in this setup. With low RTTs, these benefits are marginal: requests with RTTs below 150ms achieve under 5\% improvement on average. These benefits, however, increase dramatically as the RTT goes up. In the best case (490ms RTT for YouTube), SPDY beats HTTPS by 21.26\%.
The results effectively highlight the key benefit of SPDY: stream multiplexing. As RTT goes up, it becomes increasingly expensive for HTTPS to establish separate connections for each resource. \emph{Each} HTTPS connection costs one round trip on TCP handshaking and a further two on negotiating SSL setup. SPDY does this only once (per server) and hence reduces such large waste by multiplexing streams over a single
connection. By inspecting the HAR logs, we find that SPDY saves between 66\% and 94\% of SSL setup time, creating significant gains in high delay settings.
There is also a notable variation between the different webpages under test. In the cases of Twitter and YouTube, SPDY's ability to multiplex is well exploited by retrieving Twitter's 7 resources and YouTube's 50 resources in parallel. YouTube is by far the greatest beneficiary from SPDY with an average improvement of 13.81\% over HTTPS, whilst Twitter comes second with 6.87\%. The benefits for Twitter are less pronounced because there are fewer streams that can be multiplexed, therefore reducing the benefits of SPDY over HTTPS's maximum of 6 parallel TCP connections (note that this limit of six is hard coded, based on the amendment \cite{httpbisTicket131} to the limit set by RFC 2616 \cite{rfc2616}).
Perhaps more interesting, though, is the fairly steady behaviour exhibited by imgur across all delay values. At first, one would imagine imgur to benefit greatly from SPDY due to its ability to multiplex imgur's large number of resources (133). However, performance is very subdued: its overall average is 1.32\%. To understand this, we inspect the HAR logs to see what is occuring `under the bonnet'. Figure \ref{fig:plot-HARtimes-bw2-rtt490ratios} depicts a breakdown of the HTTPS and SPDY retrieval times for Twitter and imgur at RTT=490ms and BW=2Mbps. We choose this particular subset of our experiments as it provides network conditions where both websites achieve equal SPDY-induced improvement ($\approx$15\%), and hence provides a fair comparison. The figure shows the fraction of time spent in the five key stages of page retrieval: connect, send, wait, receive and SSL. We notice that the make-up of these retrievals is remarkably different. As expected, HTTPS spends a lot of time in the connect and SSL phases, establishing TCP and SSL connections (respectively). This increases for imgur, which has 19 times as many resources as Twitter. On the other hand, SPDY greatly reduces the connect and SSL stages but spends a huge proportion of time in wait. This phase begins when the browser issues a request for a resource, and ends when an intial response is received back. The receive phase is time spent receiving the response data until it is loaded into the browser's memory. In the case of SPDY, wait includes not just the network latency between the client and server but also the time requests are blocked until multiplexed onto the wire. For imgur, SPDY cuts connect time by 94\% but inflates wait time by more than 9 times. As emulated RTT was the same for both protocols, it appears that this inflation in wait time is an unfortunate product of SPDY's multiplexing, but we are unable to exactly ascertain \emph{why} multiplexing is creating this much delay.
In other words, SPDY's savings in establishing new connections is compromised with multiplexing overhead for highly complex webpages served over a single connection.
\begin{figure}[htb!]
\centering
\includegraphics[width=80mm]{HAR-key} \\
\includegraphics[width=80mm]{plot-HARtimes-bw2-rtt490ratios}
\caption{Breakdown of HAR Times for Twitter and imgur at RTT=250. SPDY spends 49\% in Wait for Twitter, and 97\% for imgur due to its high resource count.}
\label{fig:plot-HARtimes-bw2-rtt490ratios}
\end{figure}
\vspace{0cm}
\subsection{Bandwidth}
\label{sec:results:network:bw}
Next, we inspect the impact that client bandwidth has on performance. Once again, this parameter spans a wide range of values across the globe \cite{Dischinger2007broadband,Sundaresan2011broadband,akamai2012q4soti}.
This time we fix RTT at 150ms and PLR at 0.0\%, while setting client bandwidth to values between 64Kbps and 8Mbps. A first in first out tail-drop queue of 256 packets length is used to emulate commodity routers commonly used as residential and public gateways \cite{Li2007measuring,Gettys2011Bufferbloat}. The results are presented in Figure \ref{fig:emu-bw-rtt150}.
\begin{figure}[htb!]
\centering
\includegraphics[width=80mm]{emu-bw-rtt150}
\caption{Effect of Bandwidth (RTT=150ms, PLR=0\%).}
\label{fig:emu-bw-rtt150}
\end{figure}
\vspace{0cm}
This graph reveals a very different story to that of delay. Confirming the findings of the live experiments, we see that SPDY \emph{does} have the potential to lower performance, and significantly so. This occurs with a clear trend that favours lower capacities. At 64Kbps, on average, clients witness a 5.75\% improvement over HTTPS, compared to a 22.24\% decrease at 8Mbps. Initial impressions suggest that bandwidth variations have a larger detrimental impact on SPDY's performance.
We now have two dimensions of impact --- RTT and bandwidth --- where SPDY prefers high delay, low bandwidth ($<$1Mbps) environments. As previously discussed, the reason behind SPDY's sensitivity to RTT is relatively easy to measure by inspecting the HAR logs. However, its relationship with bandwidth is rather more complicated.
To understand this, we turn our attention to the network traces. We find that the separation between RTT and bandwidth is not particularly distinct. This is because HTTPS tends to operate in a somewhat network-unfriendly manner, creating queueing delays where bandwidth is low. The bursty use of HTTPS' parallel connections creates congestion at the gateway queues,
causing upto 3\% PLR and inflating RTT by upto 570\%\footnote{We also experimented with different gateway queue sizes. Generally, increasing queue size caused longer delays and more loss: upto a 920\% RTT increase and 5\% PLR with a 512 packets queue size, but only 296\% maximum RTT inflation and 1\% PLR with a queue of 64 packets.}. In contrast, SPDY causes negligible packet loss at the gateway.
The network friendly behaviour of SPDY is particularly interesting as Google has recently argued for the use of a larger IW for TCP \cite{rfc6928}. The aim of this is to reduce round trips and speed up delivery --- an idea which has been criticised for potentially causing congestion. One question here is whether or not this is a strategy that is specifically designed to operate in conjunction with SPDY. To explore this, we run further tests using IW=$\left\{3,7,10,16\right\}$ and bandwidth fixed at 1Mbps (all other parameters as above). For HTTPS, it appears that the critics are right: RTT and loss increase greatly with larger IWs. In contrast, SPDY achieves much higher gains when increasing the IW without these negative side effects. It therefore seems that Google have a well integrated approach in their ``Make the Web Faster'' project.
Interestingly, we observe that the key reason that this increase in RTT and loss adversely affects HTTPS is that it slows down the connection establishment phase, creating a similar situation to that presented earlier in Figure\,\ref{fig:emu-delay-bw1}. Obviously, this congestion also severely damages window ramping over the HTTPS connections. We can tangibly observe this by inspecting the client's TCP window size, which scales far faster with SPDY than any one of the parallel HTTPS connections; this alone leads to an average of $\approx$10\% more throughput than that of HTTPS.
While this explains SPDY's superior performance at low bandwidths, it does not explain its poor performance as capacities increase. As soon as bandwidth becomes sufficient to avoid the increased congestion caused by HTTPS, the benefits of SPDY begin to diminish. This is particularly the case for websites with fewer resources, like Twitter.
To understand this, we breakdown the operations performed by SPDY and HTTPS. Figure \ref{fig:plot-HARtimesBars-youtube} presents the results for YouTube as an example. Again, the two protocols have very different constitutions. HTTPS spends a large proportion of its time in the connect phase, setting up TCP and SSL. In contrast, SPDY spends the bulk of its time in the wait phase. Deep inspection reveals streams blocking until the connection is free to transmit.
In line with our previous findings, this highlights that SPDY does not always do an effective job of multiplexing. Whereas, previously, this was caused by the complexity of the webpage, here it appears that high capacity transmission is also a challenge.
Thus, as bandwidth increases, HTTPS can amortise the costs of TCP and SSL setup by exploiting the higher raw throughput afforded by opening parallel TCP sockets. We also observe that this situation occurs particularly when dealing with larger resources (e.g.\ images in Twitter), as window size can be scaled up before each connection ends in HTTPS. In contrast, SPDY appears to struggle to fill TCP's pipe as the server waits for new requests for each object from the client. Indeed, the Wireshark traces show TCP throughput reductions between 2\% and 10\% in the case of SPDY due to this problem compared to that of the parallel HTTPS connections. It would therefore seem that SPDY's default use of a single TCP connection might be unwise in circumstances of high bandwidth.
\begin{figure}[tbh!]
\centering
\includegraphics[width=80mm]{HAR-key} \\
\includegraphics[width=90mm, clip=true, trim=0 0 0 0.75cm]{plot-HARtimesBars-youtube}
\caption{HAR Times for different Bandwidth values (YouTube, RTT=150ms, PLR=0\%).}
\label{fig:plot-HARtimesBars-youtube}
\end{figure}
\vspace{0cm}
\subsection{Packet Loss Ratio}
\label{sec:results:network:loss}
Finally, we inspect the impact of packet loss on SPDY's performance. We fix RTT at 150ms and BW at 1Mbps, varying packet loss using the Linux kernel firewall with a stochastic proportional packet processing rule between 0 and 3\%\footnote{Packet loss in US mobile networks is reported to be as low as $\approx$ 0.2\% \cite{chen2012characterizing} and as high as 1.9\% \cite{heikkinen2012comparison}, but is considerably higher in other countries; e.g.\ 2-3\% in many European countries and $\geq$ 3\% in China, Russia and several South American states \cite{heikkinen2012comparison}.
It is also quite high for WiFi \cite{Sheth2007loss}. We therefore consider 0--3\% to be an appropriate parameter range.}. Figure \ref{fig:emu-loss-bw1-rtt150} presents the results.
\begin{figure}[tbh!]
\centering
\includegraphics[width=80mm]{emu-loss-bw1-rtt150}
\caption{Effect of Packet Loss (RTT=150ms, BW=1Mbps).}
\label{fig:emu-loss-bw1-rtt150}
\end{figure}
\vspace{0cm}
Immediately, we see that SPDY is far more adversely affected by packet loss than HTTPS is. This has been anticipated in other work \cite{Thomas2012spdying} but never before tested. It is also contrary to what has been reported in the SPDY white paper \cite{SPDYWhitepaper}, which states that SPDY is better able to deal with loss. The authors suggest because SPDY sends fewer packets, the negative effect of TCP backoff is mitigated. We find that SPDY does, indeed, send fewer packets (upto 49\% less due to TCP connection reuse). However, SPDY's multiplexed connections persist far longer compared to HTTPS. Thus, a lost packet in a SPDY connection has a more profound setback on the long term TCP throughput than it would in any of HTTPS' ephemeral connections, the vast majority of which do not last beyond the TCP slow start phase \cite{Sun2011cdns}. Furthermore, packet loss in SPDY affects all following requests and responses that are multiplexed over the same TCP connection. In contrast, a packet loss in one of the parallel HTTPS connections would not affect the other connections, neither concurrent nor subsequent (assuming HTTP pipelining is not used, which is commonly the case). In essence, HTTPS `spreads the risk' across multiple TCP connections. On average, we found that SPDY's throughput is affected by packet loss up to 7 times more than HTTPS (all experiments were performed using the default Linux CUBIC congestion avoidance algorithm).
It is also important to note that the probability of packet loss is higher in SPDY. According to \cite{Flach2013latency}, the probability of experiencing loss increases in proportion to the position of the packet in a burst chain. Hence, the chance of experiencing a packet tail drop is much higher for longer lived connections such as SPDY's.
Thus, not only does SPDY react badly to packet loss, the chance of it experiencing loss is also higher. This is effectively highlighted in Figure \ref{fig:emu-loss-bw1-rtt150}; imgur, which has the longest transfer time (by far), exhibits extremely poor performance under packet loss.
Finally, these results indicate that SPDY may not perform that well in mobile settings, one of its key target environments \cite{SPDY-perf-mobile}. Whilst both SPDY's high delay and low bandwidth support is desirable in this environment, the benefits can be undone by relatively low levels of packet loss (e.g.\ 0.5\%).
\section{Effect of Infrastructural Decisions}
\label{sec:results:infra}
The previous section has investigated the performance of SPDY under different network conditions
between a single client and server. However, our original crawling of the Alexa Top 10k highlighted a tendency for providers to implement a practice known as \emph{domain sharding}. This is the process of distributing page resources across multiple domains (servers), allowing browsers to open more parallel connections to download page resources.
Figure \ref{fig:PageResourceDomainCount} presents a CDF of the number of shards we discovered. We find that apart from front-less websites (such as media.tumblr.com and akamaihd.net), all websites employ some degree of domain sharding.
Here, we choose to deep dive into the practice of domain sharding to understand the implications of this infrastructural design choice on SPDY.
\begin{figure}[htb!]
\centering
\includegraphics[width=80mm]{plotDomainCount-cdf}
\caption{Number of Alexa Websites Resource Domains}
\label{fig:PageResourceDomainCount}
\end{figure}
\subsection{Number of Shards}
\label{sec:results:infra:shards}
To inspect the impact of sharding, we recreate the earlier experimental setup but mirror the webpages across multiple servers, as occurs in real setups. We consider 7 shards, i.e.\ servers, an appropiate upper limit here as our measurements find that 70 of the top 100 Alexa websites have 7 or fewer shards. Each shard is configured as in Section \ref{sec:meth:setups:emu-net}. We then distribute the webpage resources across these servers. We perform retrievals
using configurations between 1 and 7 shards, after adapting the HTML to reference shards in a round robin fashion. The client is configured with 1Mbps bandwidth, 150ms RTT and 0\% PLR. Figure \ref{fig:plot-shards7} presents the results of 100 runs at each configuration.
\begin{figure}[htb!]
\centering
\includegraphics[width=80mm]{plot-shards7}
\caption{Effect of the Number of Shards}
\label{fig:plot-shards7}
\end{figure}
\vspace{0cm}
We first note that sharding distinctly decreases SPDY's gain for YouTube and imgur. As the number of shards increases, so does the maximum number of parallel HTTPS connections. SPDY, too, is forced into creating multiple parallel TCP connections (one to each server). Hence, \emph{both} protocols are allowed to capitalise on increased parallelism. However, the benefits achieved by HTTPS outweigh those of SPDY as the former gains 6 new TCP connections per shard, a large performance boost that, in essence, offers SPDY-like multiplexing. This, therefore, reduces the overall improvement offered by SPDY. Another ramification of sharding evident from the examples of YouTube and imgur, is that as SPDY opens more connections, it multiplexes fewer streams per connection. This diminishes the returns of multiplexing which is SPDY's main competitive advantage over HTTPS. This suggests, based on our findings in Section \ref{sec:results:network:bw}, that increasing the servers' IW would give SPDY an advantage and the potential to tip the balance in its favour.
The case of Twitter provides a different insight. Here, fairly steady results are achieved across all sharding levels. SPDY gains marginal improvements over HTTPS by reducing the number of round trips, which is dictated by the number of resources in a page. For such a page with only 7 resources, SPDY saves between one and two round trips at 1 shard (depending on whether all resources were requested together or at different times as the page is rendered). With more shards, the number of round trips that SPDY potentially saves is reduced to only one, if any, due to its reduced ability to multiplex. Whereas, in the case of HTTPS, more shards means fewer resources (and hence fewer parallel connections) per shard. This has the effect of gradually decreasing HTTPS' parallelism as the number of shards increase, hence allowing SPDY to continue to retain an edge.
In summary, we deduce that SPDY loses its performance gains as a website is sharded more. However, these negative results are not ubiquitous and vary remarkably depending on the number of page resources. This raises a few questions about SPDY deployment. Are the benefits enough for designers and admins to restructure their websites to reduce sharding? What about third party resources that cannot be consolidated, e.g.\ ads and social media widgets? Can SPDY be redesigned to multiplex across domains? Is proxy deployment \cite{Thomas2012spdying} rewarding and feasible as a temporary solution? The success of SPDY (and thereupon HTTP/2.0) is likely to be dependent on the answers to precisely these questions.
\subsection{Number of Multiplexed Streams}
\label{sec:results:infra:mux}
So far, we have seen that sharding can create a significant challenge to SPDY's performance by forcing it into HTTP-like behaviour and by that limiting its ability to perform stream multiplexing. To further inspect this, we now directly study the impact of this multiplexing by artificially changing the maximum number of streams allowed per connection. This allows us to control exactly the degree of multiplexing afforded by SPDY. We vary this value from 1 to 100 (which is the default in Apache, as recommended by the SPDY draft \cite{SPDY-draft}) whilst mirroring the three websites on a single server. We perform these retrievals for a variety of RTTs. We choose to vary RTT because of the discovery that many of the bandwidth impacts are actually products of the inflated RTTs caused by queuing. Bandwidth is fixed at 1Mbps and PLR at 0\%.
In Figure \ref{fig:plot-heatmap-mux}, the average improvement in ToW (over HTTPS) for each multiplexing degree is displayed as a trend line, whilst the ToW reduction over different RTT values is shown as a heatmap to elicit more generalisable results. In all cases, SPDY's multiplexing has the potential to improve the ToW. For YouTube and imgur, we see a direct relationship between these benefits and the level of multiplexing afforded by SPDY. Here, these benefits plateau at 10 streams for YouTube and at 30 for imgur. In contrast, the results for Twitter
remain relatively steady for all levels of multiplexing.
\begin{figure}[tb!]
\centering
\subfloat[Twitter]{ \includegraphics[width=80mm]{plot-heatmap-mux-tw} } \\
\subfloat[YouTube]{ \includegraphics[width=80mm]{plot-heatmap-mux-yt} } \\
\subfloat[imgur] { \includegraphics[width=80mm]{plot-heatmap-mux-imgur} } \\
\caption{Effect of the Number of Multiplexed Streams per SPDY Connection over Varying RTTs.}
\label{fig:plot-heatmap-mux}
\end{figure}
To explore the different results for each page, we inspect the nature of their resources, as well as SPDY's recorded behaviour when accessing them. We confirm that these results are a product of the complexity of the webpages in terms of their resources. Twitter benefits little from increasing the multiplexing degree, as it only possesses 7 resources, i.e. no further benefits can be achieved when multiplexing beyond this level. The inverse case is found with YouTube (50 resources) and imgur (133 resources), which clearly can exploit multiplexing levels beyond 7 streams. Preventing this from happening has dire ramifications: when allowing SPDY to multiplex fewer than 6 streams for YouTube and imgur, it performs worse then HTTPS. This therefore confirms the negative impact that sharding will have on SPDY's deployment, where multiplexing capabilities could be severely undermined. We found these observations to be true for even more complicated websites, e.g.\ the New York Times website (148 resources).
To better understand the relationship between performance and page complexity, we perform regression analysis to look at the multiplexing level ($m$) required to outperform HTTPS for a website with a given number of resources ($r$). This is done for all websites under test in addition to three other websites we experimented with. We find that $m \approx r/4$, with a very strong fit ($R^2=0.98537$, $p$-value$=8.0684\e{-5}$). This is not a robust model and is not intended to be so; it effectively highlights the impact that page type will have on SPDY's performance. Another interesting point here is that intuition would perhaps lead towards a $r/6$ relationship, due to the maximum number of parallel HTTPS connections. Instead, $m$ is found to be of greater value. We are not able to pinpoint the reasons behind this, but it could be attributed to SPDY's multiplexing overheads diagnosed in section \ref{sec:results:network}.
\section{Conclusions \& Future Work}
\label{sec:Conclusion}
SPDY provides a low-cost upgrade of HTTP, aiming to reduce page load times leading to improved user experience. To do this, it introduces a variety of new features, including stream multiplexing and header compression. Currently, the behaviour and performance of SPDY are quite poorly understood, exacerbated by the often conflicting results reported by various early stage studies. Our own live experiments confirmed these observations, highlighting SPDY's ability to both decrease and increase page load times.
We therefore turned our efforts to identifying the conditions under which SPDY thrives. We found that SPDY offers maximum improvement when operating in challenged environments, i.e. low bandwidth and high delay. We concluded that stream multiplexing is at the heart of SPDY's performance. This feature allows it to minimise the number of round trips required to fetch resources. It also facilitates more disciplined congestion control, which allows SPDY to outshine HTTP on low bandwidth links and promotes further network enhancements such as increasing TCP's IW. On the other hand, SPDY's multiplexed connections last much longer than HTTP's, which makes SPDY more susceptible to loss and the subsequent issues with TCP backoff.
We then investigated the impact of infrastructural decisions on SPDY's performance, namely the prevalent practice of domain sharding. We observed that SPDY's benefits are reduced in sharded environments where SPDY is prevented from maximising on multiplexing. We predict this could have palpable implications on website design and deployment strategies. Finally, we observed throughout our experiments that page type has huge influence on SPDY's performance: SPDY favours pages with more and larger resources, as opposed to pages with a very large number of small resources which induces perceptible multiplexing overheads.
So far, we have investigated only a subset of SPDY's overall parameter space and, thus, our future work intends to focus on expanding these experiments. This includes alternate network configurations, but also extends to inspecting other SPDY features (e.g. Server Push and Hint). A particularly important aspect of our future work is to formulate a better understanding of SPDY's behaviour in relation to the different page characteristics, which we have discovered to have a profound impact on performance. Finally, this work should feed into the wider discussion regarding HTTP/2.0, and the future of the web.
\section{Acknowledgments}
This work was partly supported by the NERC EVOp (NE/I002200/1) and EPSRC IU-ATC (EP/J016748/1) projects. We thank Dr. Rajiv Ramdhany for access to testbed resources, and Prof. Gordon S. Blair for his valuable feedback.
\bibliographystyle{abbrv}
{\balance |
1,108,101,565,454 | arxiv | \section{Introduction}
\label{sec:instroduction}
Social media users are constantly creating content that connects them to others, but are users aware of the emotional influence that social media has on their moods or lives? While consuming and sharing information online is advantageous to connecting people, it may pose a risk to ignore the emotions that we (consciously or unconsciously) spread out to hundreds or thousands of people with a single post.
Over the past few years, emotional contagion through social media has been of great interest. Kramer et al.~\cite{kramer2014experimental} argue, ``Emotional states can be transferred to others via emotional contagion, leading people to experience the same emotions \textit{without their awareness}. Emotional contagion is well established in laboratory experiments, with people transferring positive and negative emotions to others.'' In addition, several studies, e.g., \cite{kramer2014experimental, fan2014anger, coviello2014detecting, ferrara2015measuring}, have found evidence of affect contagion in homophilic atmospheres, where people tend to connect with others who are socially similar to themselves. Furthermore, Tromholt~\cite{tromholt2016facebook} and Hunt et al.~\cite{hunt2018no} show that the content that we consume on platforms like Twitter or Facebook affects not only the emotions that we express on these platforms but also our general wellbeing{\color{black}, for example, through consuming toxic content or experiencing cyberbullying attacks~\cite{wulczyn2017ex, schmidt2017survey, rosa2019automatic}.}
\begin{figure}
\centering
\includegraphics[width=.62\linewidth]{basecard.pdf}
\caption{T-Moodifier options. Framed statistics are only present in T1.}
\label{fig:basecard}
\vspace{-1em}
\end{figure}
However, all of these insightful studies fail to address social media users' emotional awareness, {\color{black}i.e., users' ability to identify and differentiate emotions~\cite{boden2011you, huang2013distinguishing}}, of the content that they consume and share. {\color{black} On the one hand, we can leverage emotional awareness to help users improve the regulation of their emotions on social media~\cite{erisman2010preliminary, hill2012mindfulness} by assisting them with emotional differentiation~\cite{hill2012mindfulness, barrett2001knowing}. On the other hand, it has been shown that users tend to change their behavior on social media: They aim to deliver a positive impression of themselves that they want others to perceive \cite{ahn2013show, zhao2008identity, doi:10.1111/j.1468-2958.2007.00312.x}, self-censor their posts if they cannot filter their audience~\cite{sleeper2013post}, and try not to involve themselves in conflicts that may give others an overall bad impression~\cite{rainie2012tone}. In this study, we couple these users' intentions with a new tool that aims to help increase emotional awareness. We aim to understand to what extent Twitter users are aware of or reflect on the impact caused by the emotional content that they consume and create. In addition, we examine how users' sharing patterns change after they use this new tool.}
We introduce, build, deploy, and evaluate Tweet Moodifier (T-Moodifier), a Google Chrome extension that enables Twitter users to explore the emotional sentiment of posts in their News Feed (see figure \ref{fig:basecard}). The extension is powered by a machine learning algorithm that classifies Tweets into three different sentiment categories: \textit{positive} posts, which tend to use happy or surprising language; \textit{negative} posts, which tend to use sad, angry, or disgusting language; and posts without strong emotional language, which are classified as \textit{neutral}. Using T-Moodifier, participants have access to three new views (which can be regarded as different perspectives {\color{black}that can help users with emotional differentiation~\cite{barrett2001knowing}}) of their Twitter feed: (1) highlight with colors the emotional valence of each post in their News Feed, (2) filter out all negative and neutral content to keep only positive posts, and (3) filter out all positive and neutral content to keep only negative posts. Users also have the opportunity to click an emoji (positive, neutral, or negative) below each Tweet if they wish to indicate a different label for the automatically-labeled valence. Our hypotheses are that by making users aware of the emotions that surround them online, in this case through Tweets, users could:
\begin{enumerate}
\item Change their emotional sharing patterns.
\item Reflect more on, and consequently be more aware of, what they post before doing so.
\item Feel better or become mentally healthier.
\end{enumerate}
From the studies mentioned above \cite{coviello2014detecting,ferrara2015measuring}, one may conclude that showing positive content to people would make them happier. However, some researchers in the mental health community \cite{hunt2018no,primack2017social} propose that consuming only positive emotions can lead people to experience a negative feeling of isolation. Since it is not our primary goal to measure emotional contagion, T-Moodifier does not suggest that participants should spend more time in one T-Moodifier view than in another. {\color{black}Nevertheless, to prevent users from forgetting that they have the negative emotions view activated, if a user spends more than 15 minutes there, T-Moodifier shows a pop-up that blinks until the user restores his or her original feed (this pop-up was tested in a 10-person pilot without complaints, so we decided to keep it).} To the best of our knowledge, T-Moodifier is one of the first attempts to try to enhance and understand social media users' awareness of how Twitter affects them.
Results presented in this paper are based on 55 users who used T-Moodifier for at least 7 days each, for a total of 21 days studied per user (from two weeks before installing T-Moodifier to one week thereafter). Our findings suggest that the use of T-Moodifier increases users' emotional awareness. In addition, participants tend to neutralize and reflect more on their content when they have access to emotional statistics. This behavioral change suggests potential benefits of creating real-time mechanisms that increase social media users' awareness. Finally, participants who completed both pre- and post-experience surveys asserted more accurately and confidently the primary emotions they shared and perceived on Twitter.
The rest of this paper is organized as follows: In section~\ref{sec:protocol}, we describe the study protocol. In section~\ref{sec:classifier}, we explain the classifier in charge of giving emotional labels to users' Tweets. In section~\ref{sef:ppolicy}, we disclose our privacy policy. Finally, in sections~\ref{sec:results},~\ref{sec:conclusion}, and~\ref{sec:discussion} we present our results, conclusions, and discussion plus future work, respectively.
\section{Study protocol}
\label{sec:protocol}
First of all, we want to highlight that participants have full control over the T-Moodifier experience: a) They choose to use the plugin (or not), b) They have can stop using it at any time, and c) They are informed (and asked to explicitly accept the Privacy Policy; more details in section \ref{sef:ppolicy}) about which kinds of data will be collected on their usage and sharing patterns. Furthermore, we only use publicly available Twitter data.
Twitter users are invited to use T-Moodifier through its website\footnote{\url{https://tweetmoodifier.media.mit.edu}}, {\color{black}which, for this study, was sporadically promoted by the MIT Media Lab's social media accounts for two weeks\footnote{ $\sim$ 450K Twitter followers.}}. Upon installing the extension (participants are able to uninstall it at any time), they see a description of the tool's purpose and are prompted to answer a pre-experience survey and continue only if they accept our privacy policy and usage conditions.
Next, participants are randomly assigned to one of two different treatment groups, who are also compared to a control group based on public data. Participants are shown a set of options (see figure \ref{fig:basecard}) below their profile picture in the Twitter homepage, that allows them to:
\begin{itemize}
\item Treatment 1 (\texttt{T1}): see valences of Tweets in their News Feed, filter the emotions of the Tweets that they consume, and see personal statistics revealing the valence of what they tweeted.
\item Treatment 2 (\texttt{T2}): same as \texttt{T1} but NOT seeing any statistics about Tweets they generated.
\item Control: A random sample of ``friends,'' defined by Twitter as people our participants follow, who do not install the plugin. {\color{black}This group could include influencer/popular accounts whenever participants followed them.}
\end{itemize}
{\color{black}We define the control group assuming that it might reflect a homophilic atmosphere for participants. Then, if participants have been in synchrony with their homophilic group until T-Moodifier is introduced, and only participants change their behavior during treatment, we can say that T-Moodifier can account for these behavioral changes and not their homophilic atmosphere.}
When someone activates any of the T-Moodifier views, sentiment classification is obtained using the Tweets emotion classifier described in section \ref{sec:classifier}. In addition, T-Moodifier activates a visual mark as a reminder of the view they have enabled; T-Moodifier displays green borders in positive Tweets, red borders in negative Tweets, while neutral Tweets remain without a colored mark. Finally, after a seven-day experience, participants are taken to a post-survey. They are reminded of access to mental health resources {\color{black}(also available in our Privacy Policy \ref{sef:ppolicy})} each time they activate a T-Moodifier view {\color{black}(see Appendix for visual details)}.
While participants were told that T-Moodifier aimed to help them better understand which emotions they consume online, we also analyzed how they behaved while using T-Moodifier. We hypothesized that, since participants in \texttt{T1} had access to personal statistics about their publicly observable behavior on Twitter, they would be most likely to show a change in behavior regarding the emotional distribution of the Tweets they create. This study was developed under the Massachusetts Institute of Technology IRB Protocol \#1810563376.
\section{Emotions classifier}
\label{sec:classifier}
Aiming to deliver an efficient application, based on predictions of an affect sentiment classifier that is well known, {\color{black}relatively easy to interpret compared to most recent techniques}, and trained for Tweets (see Giachanou and Crestani's work for a survey~\cite{giachanou2016like}), we used the model by Go et al.~\cite{go2009twitter}. Their model uses Twitter APIs to generate its training and validation datasets. The model analyzes Tweets' text, but first, authors strip out all emojis in the Tweets, to later try to predict these emojis as emotions (positive, neutral, or negative) labels as a way of distant supervision. There are several emojis that can be classified as positive (e.g. :), :-), and :D), and negative (e.g. :(, :'(, and :@). The full list of emojis can be found in Go. et al.'s paper~\cite{go2009twitter}. Thus, their model uses self-labeled ``non-verbal'' emojis as its ground truth for sentiment labeling. We used this pre-trained classifier through its available API at \url{http://help.sentiment140.com/api}.
As with most machine learning systems, this classifier can make mistakes. Its authors reported accuracy score levels above 80\%. Unintended ``accidents'' can occur and be potentially harmful for people~\cite{amodei2016concrete}. Hence, we address this issue by allowing participants to rectify the emotional classification of what they observe (Tweets are relabeled only for them and not for all T-Moodifier users). {\color{black}As we collect users' re-labeled data, we could retrain the base model to provide more accurate or personalized emotion labels. However, the results reported here are under the assumption that by rectifying mislabeled Tweets, users can reflect on more accurate valences of their Tweets; hence, we do not retrain the base model.}
\section{Privacy Policy}
\label{sef:ppolicy}
{\color{black}We acknowledge that this line of research requires critical thinking about how to exploit and interpret users' data~\cite{panger2016reassessing}.} For Twitter experiments, some users (private users) may have decided to make their Tweets only visible to their Twitter followers. Hence, even though Twitter's Privacy Policy would allow us, T-Moodifier does not read or evaluate emotions of protected Tweets when analyzing participants' News Feeds. {\color{black}We make our privacy policy fully available to the community and users at \url{https://tweetmoodifier.media.mit.edu/privacy-policy}.}
\section{Results}
\label{sec:results}
We present results for the hypotheses stated in Section~\ref{sec:instroduction}. This analysis is based on 55 users; 20 female and 33 male participants reported their gender; \texttt{T1}: 24, \texttt{T2}: 28, and protected: 3{\color{black}, who are only included in the survey analysis}. They all used T-Moodifier for at least 7 days days, yielding 385 of T-Moodifier use. For the control group, we sample up to 100 friends of each participant, reaching 5089 unique users. {Data are collected using Twitter's Python API (TweePy\footnote{\url{http://tweepy.org}}). \color{black}A pre-experience survey also captured the following information:}
\begin{itemize}
\item Eighty-two percent of users were aged 18 to 34 years.
\item Seventy-five percent of participants
reported that they used Twitter at least once a day, whereas
62\% of all participants stated that they used it several times a day.
\item What option best describes what you use Twitter for?: The most popular answer (with a frequency of 51\%) among 10 options was that participants used Twitter to keep up with or share the news in general.
\end{itemize}
{\color{black}We use the t-test to compare between the treatments and control group and the paired t-test to compare behavioral changes and pre- versus post-survey responses. We use $p = 0.05$ as the cutoff for significance.}
\subsection{Sharing patterns}
We analyzed the proportion of positive, negative, and neutral Tweets posted by each member of \texttt{T1} and \texttt{T2}; {\color{black}this analysis also includes what users wrote in Tweets that retweeted others}. We compared how the average distribution, for public users, changed from before introducing T-Moodifier to one week thereafter. Table \ref{table:sharing-patterns-control-group} shows that the control group distribution of the emotions they share remains steady from two weeks prior to T-Moodifier through the week of treatment, with no statistically significant changes.
The number of Tweets pre/post experience did not change significantly, with an average of 11 and 15 Tweets per user for treatments and the control group, respectively. We define,
\begin{itemize}
\item $W_{-2}$: the second week before treatments.
\item $W_{-1}$: the week before treatments.
\item $W_{0}$: the week of treatments.
\item \texttt{T}: All T-Moodifier participants.
\vspace{-.25em}
\end{itemize}
\begin{table}[h!]
\centering
\caption{Emotional content of \textbf{control group's} Tweets in \%.}
\newcommand\items{3}
\arrayrulecolor{white}
\begin{tabular}{l*{\items}{|E}|}
\multicolumn{1}{l}{} &
\multicolumn{1}{c}{$W_{-2}$} &
\multicolumn{1}{c}{$W_{-1}$} &
\multicolumn{1}{c}{$W_{0}$} \\ \hhline{~*\items{|-}|}\hhline{~*\items{|-}|}
Positive & 29.3 & 30.3 & 28.2 \\ \hhline{~*\items{|-}|}
Neutral & 64.9 & 64.1 & 65.9 \\ \hhline{~*\items{|-}|}
Negative & 5.8 & 5.7 & 5.9 \\ \hhline{~*\items{|-}|}
\end{tabular}
\label{table:sharing-patterns-control-group}
\vspace{-1.5em}
\end{table}
\begin{table}[h!]
\centering
\caption{Emotional content of \textbf{T-Moodifier users'} Tweets in \%.}
\newcommand\items{4}
\arrayrulecolor{white}
\begin{tabular}{l*{\items}{|E}|}
\multicolumn{1}{l}{} &
\multicolumn{1}{c}{$W_{-2}$, \texttt{T}} &
\multicolumn{1}{c}{$W_{-1}$, \texttt{T}} &
\multicolumn{1}{c}{$W_{-1}$, \texttt{T1}} &
\multicolumn{1}{c}{$W_{0}$, \texttt{T1}} \\
Positive & 26.8 & 28.7 & 34.7 & 16.9 \\ \hhline{~*\items{|-}|}
Neutral & 68.2 & 66.3 & 62.8 & 79.7 \\ \hhline{~*\items{|-}|}
Negative & 5.0 & 5.0 & 2.5 & 3.4 \\ \hhline{~*\items{|-}|}
\end{tabular}
\label{table:sharing-patterns}
\vspace{-1.5em}
\end{table}
\begin{table}[h!]
\hspace{1.85em}
\arrayrulecolor{white}
\begin{tabular}{p{3.9cm}*{2}{|E}|}
\multicolumn{1}{l}{} &
\multicolumn{1}{c}{$W_{-1}$, \texttt{T2}} &
\multicolumn{1}{c}{$W_{0}$, \texttt{T2}} \\
Positive & 27.8 & 31.9 \\ \hhline{~*2{|-}|}
Neutral & 69.4 & 64.3 \\ \hhline{~*2{|-}|}
Negative & 2.8 & 3.8 \\
\end{tabular}
\end{table}
As seen in Table \ref{table:sharing-patterns}, the hypothesis that participants in \texttt{T1} would change their fraction of positive, neutral, and negative Tweets while using T-Moodifier ($W_{0}$, \texttt{T1}) was confirmed. In particular, they produced mainly neutral
content. This Table also illustrates that prior to introducing T-Moodifier ($W_{-1}$ and $W_{-2}$), participants (\texttt{T}) behaved similarly to the control group.
On the one hand, from Tables \ref{table:sharing-patterns} and \ref{table:stats} we observe significant differences for T-Moodifier users under Treatment 1 (\texttt{T1}), and since there is no evidence to claim that \texttt{T1} participants behave differently from their friends prior to using T-Moodifier, we can use their friends as a control group. Furthermore, we see that while \texttt{T1} participants use T-Moodifier they behave significantly different from themselves prior to T-Moodifier. They also behave significantly different from both the control and \texttt{T2} groups during the treatment week. On the other hand, we found that participants in \texttt{T2} did not significantly change their behavior as hypothesized. The only difference between treatments is that participants in \texttt{T1} had access to personal statistics about their posts' valences.
\vspace{-.5em}
\begin{table}[h!]
\centering
\caption{P-values for differences between groups in tables \ref{table:sharing-patterns-control-group} and \ref{table:sharing-patterns}}
\begin{tabular}{|llcc}
\multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{c}{Positive} & \multicolumn{1}{c}{Neutral}\\ \hline\hline
$W_{-1}$ & $W_{-1}$, \texttt{T1} & $0.4122$ & $0.8078$\\
$W_{0}$ & $W_{0}$, \texttt{T1} & $\textbf{0.0370}$ & $\textbf{0.0115}$ \\ \hline
$W_{-1}$, \texttt{T1} & $W_{0}$, \texttt{T1} & $\textbf{0.0305}$ & $\textbf{0.0351} $ \\
$W_{0}$, \texttt{T2} & $W_{0}$, \texttt{T1} & $\textbf{0.0459}$ & $\textbf{0.0482}$ \\\hline
\end{tabular}
\label{table:stats}
\end{table}
Hence, the results suggest that T-Moodifier, especially with personal feedback about the valence of what a user is posting, can potentially affect the affective valence of what that user tends to share on social media. {\color{black}These results resonate with Grosser's~\cite{grosser2014metrics} argument about Facebook users being driven by social metrics.}
Note that shifting toward a highly neutral distribution is not necessarily an expected change. Social media are powerful platforms for discussing social injustice issues that may make people experience negative emotions. Introducing T-Moodifier is not an attempt to calm down those essential human needs. Feeling negative or sharing negative information is sometimes very healthy, as is feeling positive and sharing positive information.
Table \ref{table:stats} shows the results of testing for (and confirming) the statistical significance of the hypothesized difference between $W_{0}$ and ($W_{0}$, \texttt{T1}). We also see confirmed a significant difference between ($W_{-1}$, \texttt{T1}) and ($W_{0}$, \texttt{T1}) as well as between ($W_{0}$, \texttt{T2}) and ($W_{0}$, \texttt{T1}). We present a summary of the standard deviations for these interesting sharing patterns changes in Table \ref{table:c-intervals}.
\vspace{-.75em}
\begin{table}[h!]
\centering
\caption{Standard deviations in \% of results in Tables \ref{table:sharing-patterns-control-group} and \ref{table:sharing-patterns}}
\begin{tabular}{lccccc}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{$W_{-1}$} &
\multicolumn{1}{c}{$W_{-1}$, \texttt{T1}} &
\multicolumn{1}{c}{$W_{0}$} &
\multicolumn{1}{c}{$W_{0}$, \texttt{T1}} &
\multicolumn{1}{c}{$W_{0}$, \texttt{T2}} \\
Positive & 26.2 & 30.7 & 26.5 & 19.4 & 31.5 \\
Neutral & 26.1 & 30.8 & 26.7 & 21.3 & 31.1 \\
Negative & 10.3 & 6.6 & 11.4 & 5.7 & 1.1 \\
\end{tabular}
\label{table:c-intervals}
\end{table}
Finally, we looked at the participants' engagement with T-Moodifier features. Participants used T-Moodifier views 25.4\% of the times they used Twitter (77\% of the time they preferred to highlight all three valences; 12\% only positive valences, and 11\% only negative valences). While using the T-Moodifier views, participants displayed 37.5 Tweets on a daily average (std: 57.5). Regarding mislabeling, 35\% of users relabeled 3\% of the Tweets displayed. We did not find tendencies toward specific mislabeled emotions.
\subsection{Self-reported awareness}
\label{sec:self-reported-aw}
How was awareness of mood assessed and was use of T-Moodifier associated with any changes in self-reported awareness of how Twitter impacts mood? We presented the participants with a pre- and post-questionnaire, each time asking them the following questions, whose responses are discrete numbers from 1 to 100 according to the stated limits (e.g. ``never - always,'' ``insignificant - enormous'').
\begin{enumerate}
\item[(1)] Twitter influences my mood (never - always).
\item[(2)] My connections on Twitter influence the emotions that I experience (never - always).
\item[(3)] Twitter has (insignificant - enormous) influence on the mood of others.
\item[(4)] I influence the emotions that others experience on Twitter (never - always).
\item[(5)] I am aware of (or reflect on) the emotions expressed in my Tweets before I post them (never - always).
\item[(6)] My confidence in the response to the question: ``Most of my Tweets are emotionally: negative or neutral or positive'' is (extremely weak - extremely strong).
\item[(7)] My confidence in the response to the question: ``Most of my friends' Tweets are emotionally: negative or neutral or positive'' is (extremely weak - extremely strong).
\end{enumerate}
\vspace{-.5em}
\begin{table}[h!]
\centering
\arrayrulecolor{black}
\caption{Pre-survey --- Twitter emotional influence questions.}
\begin{tabular}{cccccccc}
& \multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} & \multicolumn{1}{c}{(3)} & \multicolumn{1}{c}{(4)} & \multicolumn{1}{c}{(5)} & \multicolumn{1}{c}{(6)} & \multicolumn{1}{c}{(7)} \\ \hline
\textbf{Pre}, \texttt{T} & & & & & & & \\%\hline
Mean & 44.5 & 46.3 & 52.9 & \textbf{21.6} & 61.2 & 58.7 & 50.2 \\
Std & 28.1 & 24.1 & 28.5 & 22.1 & 30.9 & 31.6 & 24.3 \\\hline\hline
\multicolumn{7}{l}{Note that row (\textbf{Pre}, \texttt{T}) are results for all 55 participant.}
\end{tabular}
\label{table:surveys-1}
\end{table}
\subsubsection*{\textbf{Pre-survey results}, \texttt{T}}
In Table \ref{table:surveys-1}, from Questions (1), (2), and (3) we see that T-Moodifier users recognize that Twitter can influence emotions to some extent but only occasionally. However, (see Question (4)) they strongly believe that their Tweets or interactions on Twitter \textit{almost never} influence the emotions of others.
From Question (5), we see that the participants declare that they are somewhat aware of the emotions of what they share. However, they declare in Question (6) and (7) that they are not so sure about the emotional content they spread through, or consume from, social media. These last three questions show that users may recognize that what they consume or create has emotional content, and they may reflect on it somewhat, but they are not very confident as to whether the overall tone of what they are sending and receiving is more negative or more positive.
We also asked them to point to the emoji that best describes how scrolling through Twitter makes them feel; the results are shown in Figure~\ref{fig:emojis}. The large bias toward neutral might be associated with the fact that 51\% of the users claimed that they use Twitter mostly to keep up with or share the news in general (which could be thought as informative content) or it could also be associated with recognizing that they have a lot of both positive and negative emotions to things they see on Twitter, but they were forced to select only one emoji.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\linewidth]{emojis55p0.png}
\caption{Pre-questionnaire responses to: ``The emoji that best describes how scrolling through Twitter makes you feel is \dots'''.}
\label{fig:emojis}
\end{figure}
\subsubsection*{\textbf{Post-survey results, *}} In this section, we present results regarding only those participants who completed both pre-questionnaire (\textbf{Pre}, *) and post-questionnaire (\textbf{Post}, *) responses. They constituted 28 people (\texttt{T1}: 16, \texttt{T2}: 12).
\vspace{-.5em}
\begin{table}[h!]
\centering
\arrayrulecolor{black}
\caption{Pre/Post-survey, * --- Twitter emotional influence questions.}
\begin{tabular}{cccccccc}
& \multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} & \multicolumn{1}{c}{(3)} & \multicolumn{1}{c}{(4)} & \multicolumn{1}{c}{(5)} & \multicolumn{1}{c}{(6)} & \multicolumn{1}{c}{(7)} \\ \hline
\textbf{Pre}, * & & & & & & & \\%\hline
Mean & \textbf{50.0} & \textbf{50.8} & \textbf{56.4} & \textbf{28.3} & 52.3 & \textbf{52.7} & \textbf{46.0} \\%\vspace{1em}
Std & 27.1 & 28.2 & 26.8 & 23.5 & 29.5 & 30.7 & 23.1 \\\hline
\textbf{Post}, * & & & & & & & \\%\hline
Mean & \textbf{60.7} & \textbf{60.6} & \textbf{69.4} & \textbf{40.9} & 63.5 & \textbf{70.8} & \textbf{58.5} \\
Std & 24.3 & 24.5 & 14.3 & 16.9 & 25.5 & 23.0 & 20.1 \\\hline
P-value, * & 0.025 & 0.017 & 0.047 & 0.047 & 0.292 & 0.005 & 0.029 \\\hline\hline
\multicolumn{8}{l}{* participants who completed pre- and post-survey.}
\end{tabular}
\label{table:surveys-2}
\end{table}
We analyzed post-survey responses for \texttt{T1} and \texttt{T2} altogether. We expected an increase in T-Moodifier users' perception of emotional contagion (Questions (1)-(4)) while using T-Moodifier. As shown in Table \ref{table:surveys-2}, {\color{black}this perception changed significantly in the expected direction for all four questions}.
{\color{black}
Regarding Questions (1) and (2), after the T-Moodifier experience, users could recognize significantly more strongly that Twitter and their connections influenced them, suggesting that they gained awareness thanks to T-Moodifier. As for changes in Question (3)}, they might indicate that users perceived more clearly what their friends shared and, hence, believed that what was happening on Twitter influenced their friends' reactions. In addition, changes in Question (4) might reflect that users updated their belief regarding their power to influence others.
{\color{black}Even though Question (5) did not change significantly, it moved in the expected direction towards increasing users' emotional reflection before posting content.}
On the subject of the users' confidence regarding the emotions they shared and consumed on Twitter, we observed the following in the responses to Questions (6) and (7): the awareness of T-Moodifier users increased and they could assert more confidently the main emotion they shared and perceived on Twitter. {\color{black} For the emotions asked for Question (6), users changed from 41\% in accuracy before the experience to 50\% in accuracy afterwards; for the emotions asked for Question (7), they changed from 42\% to 58\%.
We conclude that these changes in awareness captured by the surveys, coupled with the observed behavioral changes, reflect T-Moodifier's potential to make users reflect more on what they see and share, increasing their emotional awareness.}\\
\subsubsection*{\textbf{Open-ended comments and design implications}}
Though we did not tell the participants that we would measure (publicly visible) changes in their behavior, in the open-ended comments, some of them acknowledged that they did reflect on how they reacted while using T-Moodifier. For example, one \texttt{T2} participant (who did not see their own statistics), stated ``[T-Moodifier] made me realize how I react to my feed's composition and people's positive or negative news.''
{\color{black}Furthermore, 75\% of those who returned the post-survey stated that T-Moodifier helped them better understand the emotions they consumed on Twitter. Most of them explicitly pointed to the tool's design as the means for that. They said that ``the filtration system [...] was a huge advantage,'' as well as ``the ability to toggle between positive and negative emotions.'' They also pointed to the labels and colored borders as a way of differentiating and ``giving more attention to emotions,'' or by giving them ``a reason to hope they [Tweets] would be positive before I even read them.'' These are examples of how T-Moodifier provoked emotional reflection, recognition, and awareness among social media users.}
\subsubsection*{\textbf{Why did people decide to use T-Moodifier?}}
Our pre-survey efforts included asking participants to comment on why they were installing T-Moodifier. One-third of them indicated that they decided to use T-Moodifier for various personal reasons. Another third of the participants indicated that they started using T-Moodifier because they were curious about the tool and how to use it. The final third stated that they were very interested in understanding and exploring their News Feed and how it affected them in greater depth.
Some people used the words ``unconsciously'' (ex: ``[I am] interested to see what I've followed unconsciously'') and ``subconsciously'' (ex: ``I want to understand what material I am subconsciously ingesting''). This indicated that they might have known they were unconsciously perceiving emotions on social media, but they did not have a tool, like T-Moodifier, to make those emotions explicit.
\subsection{Self-reported mental health}
{\color{black}
Previous works suggest that emotional awareness and recognition are crucial to wellbeing~\cite{schutte2011emotional, hill2012mindfulness, boden2015investigation} and can help to exploit emotions that improve one's mental health~\cite{lieberman2011subjective}.
We designed T-Moodifier to help with these and hope that, under prolonged use, it could help improve Twitter users' emotional awareness and, consequently, their mental wellbeing. To approach this assessment, we applied the Eight-Item Personal Health Questionnaire (PHQ-8)~\cite{kroenke2009phq} as a preliminary way to understand whether tools like T-Moodifier could have an impact on users' mental health. Participants who completed both pre- and post-questionnaire responses scored around 5.6 in both cases on average, with 7/55 (12.7\%) of them scoring at least 10 (which is considered a sign of depressive disorders); there were no significant differences between the two treatments. We can draw only limited conclusions based on the 28 returned post-surveys; however, we can say that people in this target group are willing to use applications that are aimed at helping them.
We do not have statistically significant evidence to support/refute our hypothesis regarding T-Moodifier helping users' wellbeing. For future studies, we plan to use the Warwick-Edinburgh Mental Wellbeing Scale~\cite{tennant2007warwick}, a more positively worded and less clinical-centric test, for assessment; this might also prevent concerns about inducing negative emotions instead of preventing them due to the application of a depression-centric test.}
\subsection{Users' inquiries and feedback about Tweet Moodifier}
Prior to running the experiment reported here, we ran a pilot version {\color{black}to get feedback on the T-Moodifier user experience}. Some participants in the pilot contacted us to ask for more details regarding the use of private information being collected, to know how T-Moodifier classifies Tweets' affective valences, and to report mislabeled Tweets. Consequently, we updated T-Moodifier and its website to provide all that information and implementation details from the beginning of the reported T-Moodifier experience. These questions show that transparency is key to reaching users and having them participate in this type of studies. As stated by Amodei et al~\cite{amodei2016concrete}, transparency is an emerging topic of concern among tech users.
\section{Conclusions}
\label{sec:conclusion}
We presented a new tool, Tweet Moodifier (T-Moodifier), a user-consented Chrome extension that aims to help users better understand the emotional valence of what they consume and share online when using Twitter. By allowing users to filter and make explicit the emotional content in their News Feed, T-Moodifier supports users in reflecting more confidently on the positive and negative nature of what emotions they consume and share. While the survey results in this work are preliminary (because of the small sample size completing both pre- and post-questionnaires), the behavioral results showed a significant association between receiving personalized statistical feedback on Tweet valence and an increase in the percentage of neutral Tweets sent. This result suggests that the use of T-Moodifier may help enhance Twitter users emotional awareness and may also influence their Twitter behavior.
\section{Discussion and future work}
\label{sec:discussion}
The reported results have several limitations that should be addressed in the future to obtain stronger conclusions and to explore other aspects of Twitter users' emotional awareness that T-Moodifier currently does not capture.
While participation yielded 385 days of T-Moodifier use, a major limitation is that the current sample of returned post-surveys is only 28, and the total days of use per person is only 7. Thus, the conclusions based on self-report are not as strong as those based on behavior, and the behavior is relatively short-term. Our next step will entail shedding light on what the long-term effects of T-Moodifier on a bigger audience might be. This could allow us to generalize our conclusions and increase our understanding of how social media affects us and how aware we are of that. Also, with having a bigger audience, we could differentiate between clusters of users, such as strongly negative-/positive-sharing users, or even focus the study on groups with depressive disorders or social media dependency. Understanding diverse users' awareness might be a significant step toward influencing them positively.
Another major limitation is that this study reduced the emotions of Tweets to positive and negative valences. This reduction introduces noise mainly to negative emotions, where we can find hate speech and compassionate expressions of grief under the same \textit{negative} label. It has been shown~\cite{fan2014anger} that different emotions produce different reactions in social media users. Hence, in further versions, T-Moodifier should be able to break down both positive and negative valences into more specific emotions. Also, we want to explore more sophisticated models to improve the emotions classifier accuracy.
While T-Moodifier could potentially reach all Twitter users, an important limitation is that its current version is only available for Google Chrome in a desktop version for English Tweets. It is shown that most users access social media using mobile devices~\cite{han2017we}, and that people's sharing patterns are different across devices. Therefore, it is highly probable that T-Moodifier will not capture all representative patterns. We would also like to understand the awareness of those users who are not willing to use applications like T-Moodifier.
Furthermore, by using T-Moodifier's current capabilities, we could try to understand what would be a healthy emotional diet/regimen for people in social media. So far, we explored participants organic behavior once they are aware of the emotional content that they perceive. We plan on extending this to give users the option to balance how much (percentage-wise) of each emotion they want to receive \cite{hunt2018no,primack2017social}.
{\color{black}As for T-Moodifier's design, we observed that the display of explicit feedback (i.e., personal statistics) revealing the valence of what users posted impacted them significantly. Also, users acknowledged that making emotions salient via colored marks and emojis in each Tweet helped them to differentiate and recognize emotions throughout their exposure to Twitter better. For a future version, we plan on giving a more positive tone to the availability of mental health resources by framing them as wellbeing resources, which is what T-Moodifier aims to be.}
Overall, we can see that T-Moodifier appears to be able to increase users' emotional awareness. However, it is too soon to say whether T-Moodifier causes a positive, negative or neutral effect in its users. We believe that prolonged use of tools that subtly elicit user emotional awareness could reduce the negative consequences of spending time in social media and help users take better control over their affective well-being.
\section*{Acknowledgment}
\footnotesize{}
The authors would like to thank the Network Computing Systems at the MIT Media Lab for providing the hardware required for deploying Tweet Moodifier. This research project was made possible thanks to the continuous guidance of the Committee on the Use of Humans as Experimental Subjects.
|
1,108,101,565,455 | arxiv | \section{Introduction}
\label{sec:introduction}
Hydrodynamics is an effective description of out-of-equilibrium systems in which the mean free path of particles is much shorter than any macroscopic time or length scale of the system~\cite{Kovtun:2012rj}. The equations of motion are the conservation laws of the energy-momentum tensor and charged currents, and they are supplemented by the so-called constitutive relations, i.e. expressions of these quantities in terms of fluid variables and organized in a derivative expansion. Quantum anomalies are associated with very robust mathematical properties of gauge fields at the non perturbative level~\cite{Zumino:1983ew,AlvarezGaume:1983}. In the presence of anomalies the currents are no longer conserved, and this has important effects in the hydrodynamic description of relativistic fluids. In addition to the ideal hydrodynamical contributions, there are extra terms in the constitutive relations which lead to dissipative and anomalous effects, i.e. for the charged currents~$\langle J^\mu \rangle = n u^\mu + \langle \delta J^\mu \rangle_{\textrm{\scriptsize diss \& anom}}$. In the presence of external electromagnetic fields or vortices in the fluid, parity is broken and some tensor structures appear in the constitutive relations associated to time reversal transport. This is the case of the {\it chiral magnetic effect} (CME), which is responsible for the generation of an electric current induced by an external magnetic field~\cite{Fukushima:2008xe}, and the {\it chiral vortical effect} (CVE) in which the electric current is induced by a vortex~\cite{Son:2009tf}, i.e. $\langle \delta J^\mu\rangle_{\textrm{\scriptsize anom}} = \sigma_B {\mathcal B}^\mu + \sigma_V \omega^\mu$. The corresponding susceptibilities, $\sigma_B$ and $\sigma_V$, are related to non-dissipative phenomena as they do not contribute to entropy production. These coefficients have been computed in a wide variety of methods, including kinetic theory~\cite{Stephanov:2012ki,Huang:2017tsq}, Kubo formulae~\cite{Kharzeev:2009pj,Landsteiner:2011cp,Landsteiner:2012kd} and fluid/gravity correspondence~\cite{Bhattacharyya:2008jc,Erdmenger:2008rm,Landsteiner:2011iq,Megias:2013joa}.
It has been recently proposed a new formalism to obtain the non-dissipative part of the anomalous constitutive relations based on the existence of an equilibrium partition function in a generic stationary background~\cite{Banerjee:2012iz,Jensen:2013vta}. There are several strategies to compute the effective action, including the solution of the anomaly equations~\cite{Manes:2018mth}, as well as differential geometry methods~\cite{Jensen:2013kka,Manes:2018llx,Manes:2019cqm}. Some applications of these techniques to the physics of anomalous fluids in thermal equilibrium have been presented in e.g. Refs.~\cite{Jensen:2012kj,Haehl:2013hoa}. In this work we will use this formalism to study the anomalous contributions to the constitutive relations in non-abelian theories. We also extend this analysis to study the hydrodynamics in presence of spontaneous symmetry breaking. In this case the equilibrium partition function is computed from the Wess-Zumino-Witten (WZW) functional, thus describing the low-energy interaction of Nambu-Goldstone (NG) bosons with external gauge fields. We apply these results to the analysis of chiral nuclear matter fluids in the presence of baryon, isospin and axial chemical potential. Our results show the existence of the chiral electric effect (CEE) first predicted in~\cite{Neiman:2011mj}, and confirm its non-dissipative nature.
\section{Equilibrium partition function formalism and hydrodynamics}
\label{sec:equilibrium_partition_function}
We will present in this section the main ingredients of the equilibrium partition function formalism relevant to compute the anomalous contributions to the constitutive relations~\cite{Banerjee:2012iz,Jensen:2013kka,Bhattacharyya:2013lha,Megias:2014mba}.
\subsection{Equilibrium partition function}
\label{subsec:equilibrium_partition_function}
Let us consider a relativistic invariant quantum field theory with a time independent ${\textrm {U}}(1)$ gauge connection on the manifold~\footnote{For simplicity, we will be restricted to abelian theories in this section, but we will generalize this analysis to the non-abelian case in Sec.~\ref{sec:non_abelian}.}
\begin{eqnarray}
ds^2 &=& G_{\mu\nu} dx^\mu dx^\nu = - e^{2\sigma(\vec{x})}(dt + a_i(\vec{x}) dx^i)^2 + g_{ij}(\vec{x}) dx^i dx^j \,, \label{eq:metric} \\
{\cal A} &=& {\cal A}_0(\vec{x}) dx^0 + {\cal A}_i(\vec{x}) dx^i \,.
\end{eqnarray}
The partition function of the system is
\begin{equation}
Z = {\textrm {Tr}} \, e^{-\frac{H-\mu_0 Q}{T_0}} \,,
\end{equation}
where $H$ is the Hamiltonian, $Q$ is the charge associated to the gauge connection, while $T_0$ and $\mu_0$ are the temperature and chemical potential at equilibrium. The dependence of $Z$ on the fields $\{\sigma, g_{ij}, a_i, \cal A_\mu\}$ should be consistent with: i) three-dimensional diffeomorphism invariance; ii) Kaluza-Klein (KK) invariance, i.e. $t \to t + \phi(\vec{x}) \,, \; \vec{x} \to \vec{x}$; and iii) ${\textrm {U}}(1)$ time-independent gauge invariance (up to an anomaly). From the partition function of the system, one can compute the energy-momentum tensor and ${\textrm {U}}(1)$ charged current by performing the appropriate $t$-independent variations, i.e.
\begin{equation}
\delta \log Z = \frac{1}{T_0}\int d^3x \sqrt{g_3} \, e^{\sigma} \left( -\frac{1}{2} T_{\mu\nu} \delta g^{\mu\nu} + J^\mu \delta {\cal A}_\mu \right) \,,
\end{equation}
where $g_3 = \det(g_{ij})$. The KK invariance of the partition function demands that $\log Z$ depends on the gauge fields through the following invariant combinations
\begin{equation}
A_0 = \cA_0 \,, \qquad A_i = \cA_i - a_i \cA_0 \,. \label{eq:A_KK}
\end{equation}
For a general dependence $\log Z = {\mathcal W}(e^\sigma, A_0, a_i, A_i, g^{ij},T_0,\mu_0)$, one gets the consistent currents and energy-momentum tensor~\cite{Banerjee:2012iz}
\begin{eqnarray}
&&\hspace{-2cm} \langle J_0 \rangle_{\textrm{\scriptsize cons}} = -\frac{T_0 e^{\sigma}}{\sqrt{g_3}}\frac{\delta \mathcal{W}}{\delta A_0} \,, \quad \langle J^i \rangle_{\textrm{\scriptsize cons}} = \frac{T_0 e^{-\sigma}}{\sqrt{g_3}}\frac{\delta \mathcal{W}}{\delta A_i} \,, \quad \langle T^{ij} \rangle = -\frac{2 T_0 e^{-\sigma}}{\sqrt{g_3}} g^{ik} g^{jl} \frac{\delta \mathcal{W}}{\delta g^{kl}} \,, \label{eq:Jcr} \\
&&\hspace{-2cm} \langle T_{00} \rangle =-\frac{T_0 e^{\sigma}}{\sqrt{g_3}}\frac{\delta \cW}{\delta \sigma} \,, \quad\hspace{0.4cm} \langle T_0^{\;i} \rangle = \frac{T_0 e^{-\sigma}}{\sqrt{g_3}}\left(\frac{\delta \mathcal{W}}{\delta a_i} - A_0 \frac{\delta \cW}{\delta A_i}\right) \,, \label{eq:Tcr}
\end{eqnarray}
so that ${\mathcal W}$ is a generating functional for the hydrodynamic constitutive relations.
\subsection{Derivative expansion}
\label{subsec:derivative_expansion}
Let us study the properties of the partition function in a derivative expansion. The most general equilibrium partition function up to zeroth order in derivatives is~\cite{Banerjee:2012iz}
\begin{equation}
\log Z = {\cal W}_{(0)} = \frac{1}{T_0} \int d^3x \sqrt{g_3} \, e^\sigma P(e^{-\sigma} T_0, e^{-\sigma} A_0) \,,
\end{equation}
where $P$ is an arbitrary function of two variables. Then the constitutive relations can be written as
\begin{eqnarray}
&& \langle J^0 \rangle = e^{-\sigma} \partial_b P \,, \qquad \langle J^i \rangle = 0 \,, \label{eq:crJ} \\
&&\hspace{-0.1cm} \langle T^{ij} \rangle = P g^{ij} \,, \qquad\hspace{0.3cm} \langle T_{00}\rangle = e^{2\sigma}(P - a \partial_a P - b \partial_b P) \,, \qquad \langle T_0^{\;i} \rangle = 0 \,, \label{eq:crT}
\end{eqnarray}
where we have used the notation $a \equiv e^{-\sigma} T_0$ and $b \equiv e^{-\sigma} A_0$. By comparison with the hydrodynamic constitutive relations of a perfect fluid (PF)
\begin{equation}
\langle J^\mu \rangle_{{\scriptsize\textrm{PF}}} = n u^\mu \,, \qquad \langle T^{\mu\nu} \rangle_{{\scriptsize\textrm{PF}}} = (\varepsilon + {\cal P}) u^\mu u^\nu + {\cal P} g^{\mu\nu} \,,
\end{equation}
where $\varepsilon$ is the energy density, ${\cal P}$ the pressure, $n$ the charge density and $u^\mu$ the local fluid velocity, one gets
\begin{equation}
\hspace{-1cm} u^\mu = e^{-\sigma}(1, 0, \dots, 0) \,, \quad {\cal P} = P \,, \quad \varepsilon = -P + a \partial_a P + b \partial_b P \,, \quad n = \partial_b P \,. \label{eq:umu_P}
\end{equation}
This implies that $\varepsilon$, ${\cal P}$ and $n$ are not independent functions, but they are determined in terms of a single {\it master} function, $P(a,b)$, which is the pressure. In addition, we can identify the local value of the temperature and chemical potential with $a$ and $b$ respectively.
Let us discuss now the properties of the equilibrium partition function at first order in the derivative expansion. The most general expression compatible with the symmetries mentioned above is~\cite{Banerjee:2012iz,Megias:2014mba}:
\begin{equation}
\hspace{-2cm} \cW_{(1)} = \int d^3x \sqrt{g_3} \left[ \alpha_1(\sigma,A_0) \epsilon^{ijk}A_i A_{jk} + \alpha_2(\sigma,A_0) \epsilon^{ijk} A_i f_{jk} \!+\! \alpha_3(\sigma,A_0) \epsilon^{ijk} a_i f_{jk} \right] \,, \label{eq:W1}
\end{equation}
where $A_{ij} = \partial_i A_j - \partial_j A_i$ and $f_{ij} = \partial_i a_j - \partial_j a_i$. The coefficients $\alpha_i(\sigma,A_0)$ depend on the particular theory considered, and they can be determined for instance by inserting Eq.~(\ref{eq:W1}) into Eqs.~(\ref{eq:crJ})-(\ref{eq:crT}), and comparing the result with the constitutive relations for that theory. In an ideal gas of Dirac fermions one finds
\begin{equation}
\hspace{-1.5cm} \alpha_1(\sigma,A_0) = - \frac{C}{6} \frac{A_0}{T_0} \,, \quad \alpha_2(\sigma,A_0) = -\frac{1}{2}\left( \frac{C}{6} \frac{A_0^2}{T_0^2} - C_2 \right) \,, \quad \alpha_3(\sigma,A_0) = 0 \,,
\end{equation}
where the coefficients $C = 1/(4\pi^2)$ and $C_2 = 1/24$ are related to the axial anomaly~\cite{Son:2009tf,Erdmenger:2008rm} and mixed gauge-gravitational anomaly~\cite{Landsteiner:2011cp}, respectively. These coefficients induce some contributions to the chiral magnetic and vortical conductivities, which read
\begin{equation}
\sigma_B = C \mu \,, \qquad \sigma_V = \frac{1}{2} C \mu^2 + C_2 T^2 \mu \,,
\end{equation}
where $T = e^{-\sigma} T_0$ and $\mu = e^{-\sigma} A_0$. In the following we will use this formalism that relates the partition function with the constitutive relations of the theory. $\cW$ will be obtained by solving the anomaly equations.
\section{Non-abelian anomalies}
\label{sec:non_abelian}
In this section we will provide a short introduction to chiral anomalies, and study the partition function for non-abelian theories.
\subsection{The chiral anomaly}
\label{subsec:chiral_anomaly}
Let us consider the theory of a chiral fermion coupled to an external gauge field $\cA_\mu \equiv \cA_\mu^a t_a$ described by the Lagrangian
\begin{equation}
{\mathcal L}_{{\textrm{\scriptsize YM}}} = i \overline\psi \gamma^\mu (\partial_\mu - i t_a \cA_\mu^a) \psi \,,
\end{equation}
where $t_a = t_a^\dagger$ are the Hermitian generators of the Lie algebra. To study gauge anomalies it is convenient to work with the effective action functional obtained by integrating out the fermion field
\begin{equation}
e^{i\Gamma[\cA]} \equiv \int {\mathcal D}\overline\psi {\mathcal D}\psi \, e^{i S_{\textrm{\tiny YM}}[\cA,\psi,\overline\psi]} \,.
\end{equation}
Under a general shift ${\mathcal A}_\mu^a \to {\mathcal A}_\mu^a + \delta {\mathcal A}_\mu^a$, the variation of $\Gamma[\cA]$ can be expressed as
\begin{equation}
\delta \Gamma[{\mathcal A}] = \int d^4 x \, \delta {\mathcal A}_\mu^a(x) \, J_{a\, {\textrm{\scriptsize cons}}}^\mu(x) \,, \label{eq:deltaGamma}
\end{equation}
where $J_{a\, {\textrm{\scriptsize cons}}}^\mu(x)$ is the consistent current. The axial anomaly is given by the failure of the effective action to be invariant under axial gauge transformations
\begin{equation}
{\mathcal A}_\mu \longrightarrow g^{-1} {\mathcal A}_\mu g - i g^{-1} \partial_\mu g \,, \qquad g(x) = \exp\left( - i \Lambda_a^{{\textrm{\scriptsize A}}}(x) t_a \right) \,.
\end{equation}
Under such a transformation
\begin{equation}
\delta_{{\textrm{\scriptsize gauge}}} \Gamma[{\mathcal A}]= - \int d^4 x \, \Lambda_a^{{\textrm{\scriptsize A}}}(x) \, G_a[{\mathcal A}(x)] \,,
\end{equation}
where $G_a[{\mathcal A}(x)]$ is the consistent anomaly. Particularizing Eq.~(\ref{eq:deltaGamma}) to $\delta {\mathcal A}_\mu^a = (D_\mu\Lambda^{{\textrm{\scriptsize A}}})^a$, one finds the (non)-conservation law for the consistent current
\begin{equation}
D_\mu J_{a \, {\textrm{\scriptsize cons}}}^\mu(x) = G_a[{\mathcal A}(x)] \,.
\end{equation}
The consequences of the anomaly to the hydrodynamics of fluids will be analyzed in the rest of the manuscript.
\subsection{The Bardeen form of the anomaly}
\label{subsec:Bardeen_form}
In the following we will consider a non-abelian theory with symmetry group ${\textrm {U}}(N_f) \times {\textrm {U}}(N_f)$, described by the Lagrangian
\begin{equation}
{\mathcal L}_{{\textrm{\scriptsize YM}}} = i \overline\psi_{\textrm{\scriptsize L}} \gamma^\mu (\partial_\mu - i t_a \cA_{{\textrm{\scriptsize L}}\,\mu}^a) \psi_{\textrm{\scriptsize L}} + i \overline\psi_{\textrm{\scriptsize R}} \gamma^\mu (\partial_\mu - i t_a \cA_{{\textrm{\scriptsize R}}\,\mu}^a) \psi_{\textrm{\scriptsize R}} \,.
\end{equation}
The Bardeen form of the anomaly in this theory is~\cite{Bardeen:1969md}
\begin{eqnarray}
G_a[\mathcal{V}, \mathcal{A}] &=& \frac{i N_c}{16 \pi^2} \epsilon^{\mu \nu \alpha \beta} {\textrm {Tr}} \Bigl\{ t_a \bigl[ {{\mathcal V}_{\mu \nu} {\mathcal V}_{\alpha \beta} + \frac{1}{3} {\mathcal A}_{\mu \nu} {\mathcal A}_{\alpha \beta}} - {\frac{32}{3} {\mathcal A}_\mu {\mathcal A}_\nu {\mathcal A}_\alpha {\mathcal A}_\beta } \\
&& \quad + {\frac{8}{3} i ({\mathcal A}_\mu {\mathcal A}_\nu {\mathcal V}_{\alpha \beta} +
{\mathcal A}_\mu {\mathcal V}_{\alpha \beta} {\mathcal A}_\nu + {\mathcal V}_{\alpha \beta} {\mathcal A}_\mu {\mathcal A}_\nu) \bigr]} \Bigr\} \,,
\end{eqnarray}
where
\begin{eqnarray}
{\mathcal V}_{\mu \nu} &=& \partial_\mu {\mathcal V}_\nu - \partial_\nu {\mathcal V}_\mu - i[{\mathcal V}_\mu, {\mathcal V}_\nu] - i[{\mathcal A}_\mu, {\mathcal A}_\nu] \,, \\
{\mathcal A}_{\mu \nu} &=& \partial_\mu {\mathcal A}_\nu - \partial_\nu {\mathcal A}_\mu - i[{\mathcal V}_\mu, {\mathcal A}_\nu] - i[{\mathcal A}_\mu, {\mathcal V}_\nu] \,,
\end{eqnarray}
are the field strengths for the vector and axial gauge fields, and $N_c$ is the number of colors~\footnote{We define the vector and axial gauge fields $({\mathcal V},{\mathcal A})$ in terms of $(\cA_{\textrm{\tiny L}},\cA_{\textrm{\tiny R}})$ by $\cA_{\textrm{\tiny L}} \equiv {\mathcal V} - {\mathcal A}$ and $\cA_{\textrm{\tiny R}} \equiv {\mathcal V} + {\mathcal A}$. Consequently, the corresponding vector and axial components of the currents are related to their left- and right-handed components as $J_{\textrm{\tiny R}}^\mu = \frac{1}{2}\left(J_{\textrm{\tiny V}}^\mu + J_{\textrm{\tiny A}}^\mu\right)$ and $J_{\textrm{\tiny L}}^\mu = \frac{1}{2}\left(J_{\textrm{\tiny V}}^\mu - J_{\textrm{\tiny A}}^\mu\right)$.}. $G_a$ includes triangle, square and pentagon one-loop diagram contributions. The anomaly arises from the breaking of gauge invariance under axial gauge transformations of the effective action~$\Gamma_0[{\mathcal V},{\mathcal A}]$, so that the action should satisfy
\begin{equation}
\mathscr{Y}_a(x) \Gamma_0[{\mathcal V}, {\mathcal A}] = 0 \,, \qquad \mathscr{X}_a(x) \Gamma_0[{\mathcal V}, {\mathcal A}] = G_a[{\mathcal V}, {\mathcal A}] \,, \label{eq:eq_anomaly}
\end{equation}
where $\mathscr{Y}_a(x)$ and $\mathscr{X}_a(x)$ are the local generators of vector and axial gauge transformations, respectively. The computation of $\Gamma_0[{\mathcal V}, {\mathcal A}]$ can be performed by solving Eq.~(\ref{eq:eq_anomaly}), leading to
\begin{eqnarray}
\hspace{-1.5cm} \Gamma_0[V, A, G] &=& -\frac{N_c}{32 \pi^2} \int dt \, d^3 x \sqrt{g_3} \,
\epsilon^{i j k} \, {\textrm {Tr}} \Biggl\{ \frac{32}{3} i \, V_0 A_i A_j A_k \nonumber \\
\hspace{-1.5cm}&& \quad + \frac{4}{3} (A_0 A_i + A_i A_0) A_{j k} + 4 (V_0 A_i + A_i V_0) V_{j k} \nonumber \\
\hspace{-1.5cm}&& \quad + \frac{8}{3} \bigl(A_0^2 + 3 V_0^2 \bigr) A_i \partial_j a_k \Biggr\} + C_2 T_0^2 \int dt \, d^3 x \sqrt{g_3} \, \epsilon^{i j k} {\textrm {Tr}} A_i \, \partial_j a_k \,, \label{eq:Gamma_0}
\end{eqnarray}
where $V_\mu$ and $A_\mu$ are KK invariant fields. $\Gamma_0[{\mathcal V},{\mathcal A}]$ can be determined also from differential geometry methods, cf. Refs.~\cite{Jensen:2013rga,Manes:2018llx,Manes:2019fyw,Manes:2020zdd} for details. As it is mentioned in Sec.~\ref{subsec:derivative_expansion}, the coefficient $C_2$ is related to the mixed gauge-gravitational anomaly. While in principle it would be possible to compute this contribution by taking into account the Riemann tensor effects, in the following we will neglect it as this analysis goes beyond the scope of the present work.
\section{Covariant currents and constitutive relations without Nambu-Goldstone bosons}
\label{sec:cov_currents}
In this section we will study the constitutive relations with unbroken chiral symmetry. We will particularize the result for a background with two light quark flavors.
\subsection{Covariant currents}
\label{subsec:cov_currents}
The charged currents obtained from the functional derivatives of the effective action are consistent currents, cf. Eq.~(\ref{eq:Jcr}). However, only covariant currents can enter in the constitutive relations, and these cannot be obtained directly from the functional derivative of an effective action, but they are defined by adding to the consistent currents the Bardeen-Zumino (BZ) polynomials~\cite{Bardeen:1984pm}, i.e.
\begin{equation}
J^\mu_{\textrm{\scriptsize cov}} = J^\mu_{\textrm{\scriptsize cons}} + J^\mu_{{\mathrm{BZ}}} \,, \label{eq:Jcov_Jcons_JBZ}
\end{equation}
with~\cite{Manes:2018llx,Manes:2019fyw}
\begin{eqnarray}
J_{{\textrm{\scriptsize V}} \, {\mathrm{BZ}}}^\mu &=& - \frac{N_c}{8 \pi^2} \epsilon^{\mu \nu \alpha \beta}
{\textrm {Tr}} \Bigr\{ t_a \bigl(\mathcal{A}_\nu \mathcal{V}_{\alpha \beta} + \mathcal{V}_{\nu \alpha} \mathcal{A}_\beta + \frac{8}{3} i \, \mathcal{A}_\nu \mathcal{A}_\alpha \mathcal{A}_\beta \bigr)\Bigr\} \,, \label{eq:JBZ_V} \\
J_{{\textrm{\scriptsize A}} \, {\mathrm{BZ}}}^\mu &=& - \frac{N_c}{24 \pi^2} \epsilon^{\mu \nu \alpha \beta}
{\textrm {Tr}} \Bigr\{ t_a \left(\mathcal{A}_\nu \mathcal{A}_{\alpha \beta} +
\mathcal{A}_{\nu \alpha} \mathcal{A}_\beta \right) \Bigr\} \,. \label{eq:JBZ_A}
\end{eqnarray}
Then the covariant currents and energy-momentum tensor in equilibrium can be obtained from ${\mathcal W}_0 = i\Gamma_0$ by using Eqs.~(\ref{eq:Jcr}), (\ref{eq:Tcr}) and (\ref{eq:Jcov_Jcons_JBZ}). The result is~\cite{Manes:2019fyw}
\begin{eqnarray}
\langle J^i_{a\, {\textrm{\scriptsize V}}} \rangle_{{\textrm{\scriptsize cov}}} &=&
\frac{N_c}{8 \pi^2} e^{-\sigma} \epsilon^{i j k} {\textrm {Tr}} \Bigl\{ t_a \bigl[
(A_0 V_{j k} + V_{j k} A_0) +
(V_0 A_{j k} + A_{j k} V_0) \nonumber \\
&&\quad + 2 (A_0 V_0 +V_0 A_0) \partial_j a_k \bigr] \Bigr\}, \label{eq:Ji_cov}\\
\langle J^i_{a \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c}{8 \pi^2} e^{-\sigma} \epsilon^{i j k} {\textrm {Tr}} \Bigl\{ t_a \bigl[ (A_0 A_{j k} + A_{j k} A_0) + (V_0 V_{j k} + V_{j k} V_0) \nonumber \\
&&\quad + 2 (A_0^2 +V_0^2) \partial_j a_k \bigr] \Bigr\} \,, \label{eq:Ji5_cov} \\
\langle T_0\,{}^i \rangle &=& -\frac{N_c}{8\pi^2} e^{-\sigma} \epsilon^{i j k} \, {\textrm {Tr}} \Bigl\{ \bigl(A_0^2 + V_0^2 \bigr) A_{j k} + (V_0 A_0 + V_0 A_0) V_{j k} \nonumber \\
&&\quad + \biggl( \frac{2}{3}A_0 ^3 + 2 A_0 V_0^2 \biggr) \partial_j a_k \Bigr\} \,, \label{eq:T0i_cov}
\end{eqnarray}
and vanishing values for the time components of the currents $\langle J_{0 \, a \, {\textrm{\scriptsize V}}} \rangle_{{\textrm{\scriptsize cov}}} = \langle J_{0 \, a \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} = 0$, and for the other components of the energy-momentum tensor $\langle T_{00} \rangle = \langle T^{ij} \rangle = 0$.
While these expressions are valid for a non-abelian theory with symmetry group ${\textrm {U}}(N_f) \times {\textrm {U}}(N_f)$, we will particularize them for a specific background. In presence of non-abelian charges, the maximal number of chemical potentials to be consistently introduced corresponds to the dimension of the Cartan subalgebra. Let us consider the following background for $N_f = 2$
\begin{equation}
V_\mu(\vec{x}) = V_{\mu \, 0}(\vec{x}) t_0 + \, V_{\mu \, 3}(\vec{x}) t_3 \,, \qquad A_0 = A_{0 \, 0} \, t_0 \,, \qquad A_i = 0 \,, \label{eq:background}
\end{equation}
where $t_0 = \frac{1}{2} 1_{2\times 2}$ and $t_3 = \frac{1}{2} \sigma_3$, while $\sigma_i$ are the Pauli matrices~\footnote{Notice that $A_i=0$ does not imply a vanishing spatial component of the gauge field, as this turns out to be $\mathcal{A}_i = a_i A_{0 \, 0} \, t_0$ by Eq.~(\ref{eq:A_KK}).}. In the following we will consider that $A_{0 \, 0}$ is constant. In addition, we can define the equilibrium velocity field, as well as equilibrium baryonic, isospin and axial chemical potentials, as
\begin{equation}
u_\mu = -e^\sigma (1, a_i) \,, \quad \mu_0 = e^{-\sigma} \, V_{0 \, 0} \,, \quad \mu_3 = e^{-\sigma} \, V_{0 \, 3} \,, \quad \mu_5 = e^{-\sigma} \, A_{0 \, 0} \,,
\end{equation}
respectively, where $\mu_5$ controls the chiral imbalance of the system~\cite{Gatto:2011wc,Andrianov:2013qta}. The coupling to the external gauge fields comes through the magnetic components of the non-abelian vector field strength, while the explicit dependence on $u_\mu$ is codified in terms of the vorticity vector. These quantities are defined by
\begin{equation}
{\mathcal B}_a^\mu = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta} u_\nu \mathcal{V}_{\alpha\beta \, a} \,, \qquad \omega^\mu = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta} u_\nu \partial_\alpha u_\beta \,.
\end{equation}
Then, the constitutive relations can be computed by using Eqs.~(\ref{eq:Ji_cov})-(\ref{eq:T0i_cov}), and the result expressed in Lorentz covariant form is
\begin{eqnarray}
\langle J_{0 \, {\textrm{\scriptsize V}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c }{8 \pi^2} \mu_5 {\mathcal B}_0^\mu , \qquad \langle J_{3\, {\textrm{\scriptsize V}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} = \frac{ N_c }{8 \pi^2} \mu_5 {\mathcal B}_3^\mu \,, \\
\langle J_{0\, {\textrm{\scriptsize A}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c}{8 \pi^2} \Big( \mu_0 {\mathcal B}_0^\mu + \mu_3 {\mathcal B}_3^\mu + (\mu_0^2 + \mu_3^2 - \mu_5^2) \omega^\mu \Big) \, , \\
\langle J_{3\, {\textrm{\scriptsize A}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c}{8 \pi^2} \Big( \mu_3 {\mathcal B}_0^\mu + \mu_0 {\mathcal B}_3^\mu + 2 \mu_0 \mu_3 \omega^\mu \Big) \,,
\end{eqnarray}
for the currents, and
\begin{equation}
\hspace{-2.2cm} \langle T^{\mu\nu} \rangle = u^\mu q^\nu + u^\nu q^\mu \quad \textrm{with} \quad q^\mu = \frac{N_c}{8 \pi^2} \mu_5 \bigg[ \mu_0 {\mathcal B}_0^\mu + \mu_3 {\mathcal B}_3^\mu + \left(\mu_0^2 + \mu_3^2 - \frac{1}{3}\mu_5^2\right) \omega^\mu \bigg] \,,
\end{equation}
for the energy-momentum tensor.
\subsection{Electromagnetic, baryon and isospin currents}
\label{subsec:em_bar_iso_currents}
Let $\psi$ be a flavor doublet of Dirac spinors made out of up and down quarks, i.e. $\psi = \left( \begin{array}{c}
u \\
d
\end{array} \right)$. Electromagnetism is identified with the ${\textrm {U}}(1)_{\textrm{\scriptsize V}}$ subgroup defined by the charge matrix for two light flavors with electric charges $+\frac{2}{3}e$ and $-\frac{1}{3}e$, i.e.
\begin{equation}
Q = \left(
\begin{array}{cc}
\frac{2}{3} & 0 \\
0 & -\frac{1}{3}
\end{array} \right) = \frac{1}{3}t_0 + t_3 \,. \label{eq:Q}
\end{equation}
Then, using that $J_{a\, V \, {\textrm{\scriptsize cons}}}^\mu = \overline\psi \gamma^\mu t_a \psi$, we can distinguish between the electromagnetic, baryonic and isospin currents, defined as
\begin{eqnarray}
J^\mu_{{\textrm{\scriptsize em}} \, {\textrm{\scriptsize cons}}} &=& e \overline\psi \gamma^\mu Q \psi = \frac{e}{3} J_{0\, {\textrm{\scriptsize V}} \, {\textrm{\scriptsize cons}}}^\mu + e J_{3\, {\textrm{\scriptsize V}} \, {\textrm{\scriptsize cons}}}^\mu \,, \nonumber \\
J^\mu_{{\textrm{\scriptsize bar}} \, {\textrm{\scriptsize cons}}} &=& \frac{2}{3} J_{0\, {\textrm{\scriptsize V}} \, {\textrm{\scriptsize cons}}}^\mu \,, \nonumber \\
J^\mu_{{\textrm{\scriptsize iso}} \, {\textrm{\scriptsize cons}}} &=& J_{3 \, {\textrm{\scriptsize V}} \, {\textrm{\scriptsize cons}}}^\mu \,,
\end{eqnarray}
respectively. These currents are not independent, but they fulfill the Gell-Mann-Nishijima (GMN) relation $J^\mu_{{\textrm{\scriptsize em}} \, {\textrm{\scriptsize cons}}} = \frac{e}{2} J^\mu_{{\textrm{\scriptsize bar}} \, {\textrm{\scriptsize cons}}} + e J^\mu_{{\textrm{\scriptsize iso}} \, {\textrm{\scriptsize cons}}}$. The same pattern is followed by the BZ terms of each current, and then the same relations are satisfied by the corresponding covariant currents. If we denote the physical magnetic field by ${\mathcal B}^\mu = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta} u_\nu {\mathscr{V}}_{\alpha\beta}$ where the physical potential is ${\mathscr{V}}_\mu$ and its KK invariant form is ${\mathbb{V}}_\mu$, i.e. ${\mathbb{V}}_0 = {\mathscr{V}}_0$ and ${\mathbb{V}}_i = {\mathscr{V}}_i - a_i {\mathscr{V}}_0$, then after making the replacements $V_{\mu\,0} = \frac{e}{3} {\mathbb{V}}_\mu$ and $V_{\mu\,3} = e {\mathbb{V}}_\mu$, one finds
\begin{equation}
\langle J^\mu_{{\textrm{\scriptsize em}}} \rangle_{{\textrm{\scriptsize cov}}} = \frac{5 e^2 N_c}{36 \pi^2} \mu_5 {\mathcal B}^\mu \,,
\end{equation}
where we have used that ${\mathcal B}_3^\mu = 3 {\mathcal B}_0^\mu = e {\mathcal B}^\mu$. This expression gives the transport coefficient associated with the CME. The absence of a CVE in the vector currents is a direct consequence of having considered the flavor group ${\textrm {U}}(2)_{\textrm{\scriptsize V}} \times {\textrm {U}}(2)_{\textrm{\scriptsize A}}$. This state of affairs contrasts with the ${\textrm {U}}(1)_{\textrm{\scriptsize V}} \times {\textrm {U}}(1)_{\textrm{\scriptsize A}}$ case, studied in~\cite{Landsteiner:2012kd,Jensen:2013vta}, where the cancellation leading to a vanishing value for the chiral vortical conductivity does not take place.
\section{Effective action in presence of spontaneous symmetry breaking: the Wess-Zumino-Witten partition function}
\label{sec:SBB}
In this section we will consider the physical situation in which the symmetry is spontaneously broken, either total or partially. A consequence is the appearance of NG bosons that can couple to external gauge fields and contribute to the anomaly. The WZW partition function describes the effects of the anomaly when the symmetry is spontaneously broken, and accounts for the anomaly-induced interactions between the external gauge fields ${\mathcal A}$ and the NG bosons~$\xi^a$~\cite{Kaiser:2000ck,Son:2007ny,Fukushima:2012fg,Brauner:2017mui}. The WZW action admits a simple expression in terms of the anomalous functional in absence of symmetry breaking $\Gamma_0$ studied in Sec.~\ref{sec:non_abelian}~\cite{Wess:1971yu,Witten:1983tw,Manes:1984gk,Chu:1996fr}
\begin{equation}
\Gamma^{{\textrm{\scriptsize WZW}}}[\mathcal A, \xi] = \Gamma_0[\mathcal A] - \Gamma_0[\mathcal A_{g}] \,, \label{eq:WZW}
\end{equation}
where $\mathcal A_{g} = g^{-1} \mathcal A g + g^{-1} dg$ is the gauge transformed field with group element $g \equiv \exp(-i \xi^a t_a)$. For applications to hadronic fluids, we are interested in the case $U(2)_{\textrm{\scriptsize L}} \times U(2)_{\textrm{\scriptsize R}} \to U(2)_{\textrm{\scriptsize V}}$, where the symmetry is broken down to the diagonal subgroup of vector gauge transformations. Then, one can make the replacements $\mathcal A \to (\mathcal A_{\textrm{\scriptsize L}}, \mathcal A_{\textrm{\scriptsize R}})$ and $g \to (U,{\mathbb I})$, where ${\mathbb I}$ is the identity element and
\begin{equation}
\hspace{-1.5cm} U(\xi) = \exp \Bigl( i X(\xi) \Bigr) \quad \textrm{with} \quad X(\xi) := 2\sum_{a=1}^3 \xi_a t_a = \frac{\sqrt{2}}{f_\pi} \left(
\begin{array}{cc}
\frac{1}{\sqrt{2}} \pi^0 & \pi^+ \\
\pi^- & -\frac{1}{\sqrt{2}} \pi^0
\end{array} \right)\,,
\end{equation}
includes three NG bosons $\{ \pi^0, \pi^\pm \}$ from the broken ${\textrm {SU}}(2)_{\textrm{\scriptsize A}}$ symmetry, while $f_\pi \approx 92 \, {\textrm{MeV}}$ is the pion decay constant~\footnote{The fourth NG boson $\xi_0$ is absent, as the ${\textrm {U}}(1)_{\textrm{\tiny A}}$ symmetry is violated by non-perturbative effects.}. Then the action at the lowest order in derivatives can be written as
\begin{equation}
\mathcal W_{(0)} = \frac{1}{T_0} \int d^3x \sqrt{g_3} \, e^\sigma \left[ P(T,\mu_0,\mu_3) + \mathcal L\right] \,, \label{eq:W0}
\end{equation}
where $P$ is the pressure in absence of NG bosons already introduced in Sec.~\ref{subsec:derivative_expansion}, and the Lagrangian contains the dependence on the pions
\begin{equation}
\mathcal{L} = \frac{f_\pi^2}{4} G^{\mu\nu} {\textrm {Tr}} \left\{ D_\mu U (D_\nu U)^\dagger \right\} \,.
\end{equation}
At first order in derivatives, the correction to the partition function is given by the WZW action evaluated in the background of Eq.~(\ref{eq:background}). The WZW action can be computed following the prescription of Eq.~(\ref{eq:WZW}), and the result is
\begin{eqnarray}
\hspace{-2.4cm} \mathcal W_{(1)}^{{\textrm{\scriptsize WZW}}} &=& \frac{N_c}{8 \pi^2 T_0} \int d^3 x \sqrt{g_3} \, \epsilon^{i j k} \Bigg[
- \frac{1}{2} V_{0 \, 0} V_{i \, 3} \partial_j {\textrm {Tr}} \bigl\{ (R_k + L_k) Q \bigr\} + \frac{i}{6} \, V_{0 \, 0} {\textrm {Tr}} \bigl\{ L_i L_j L_k \bigr\} \nonumber \\
\hspace{-2.4cm}&+& \frac{1}{2} \left( V_{0 \, 0} \, \partial_i V_{j \, 3} + V_{0 \, 3} \, \partial_i V_{j \, 0} + \frac{1}{2} V_{0 \, 0} V_{0 \, 3} \, f_{ij} \right) {\textrm {Tr}} \bigl\{ (R_k + L_k) Q \bigr\} \label{eq:W_WZW} \\
\hspace{-2.4cm}&+& \frac{1}{6} A_{0 \, 0} \left( \partial_i V_{j \, 3} + \frac{1}{2} V_{0 \, 3} f_{ij} \right) \left( {\textrm {Tr}} \bigl\{ (R_k -L_k) Q \bigr\} - 2 V_{k \, 3} {\textrm {Tr}} \bigl\{Q (Q - U^{-1} Q U) \bigl\} \right) \Bigg] \,, \nonumber
\end{eqnarray}
where we have introduced the notation
\begin{equation}
L_j = i \partial_j U \, U^{-1} \quad \textrm{and} \quad R_j = i U^{-1} \partial_j U \,.
\end{equation}
Since $U$ takes values on ${\textrm {SU}}(2)$, the generator $t_3$ can be interchanged with the charge matrix $Q$ inside the traces.
\section{Constitutive relations of the two-flavor hadronic fluid}
\label{subsec:constitutive_relations}
Physical quantities in hydrodynamics admit the decomposition in terms of their PF contributions and corrections containing higher terms in derivatives, i.e.
\begin{equation}
J^\mu = J^\mu_{{\scriptsize\textrm{PF}}} + \delta J^\mu \,, \qquad T^{\mu\nu} = T^{\mu\nu}_{{\scriptsize\textrm{PF}}} + \pi^{\mu\nu} \,.
\end{equation}
These corrections can be either dissipative or anomalous. Different definitions of the same physical variable may vary by gradient-dependent terms $(\delta T, \delta \mu_a, \delta u^\mu, \dots)$. This leads to high order terms ambiguities that should be compensated by the PF constitutive relations
\begin{equation}
J^\mu_{{\scriptsize\textrm{PF}}}(T_0 + \delta T, \mu_{a\, 0} + \delta \mu_a, \dots) \,, \qquad T^{\mu\nu}_{{\scriptsize\textrm{PF}}}(T_0 + \delta T, \mu_{a\, 0} + \delta \mu_a, \dots) \,,
\end{equation}
as the form of the currents and energy-momentum tensor cannot be changed by the ambiguities. The particular frame to fix these ambiguities is chosen in the following by requiring that one-derivative corrections to PF quantities vanish, so that contributions at this order come only from the terms $\delta J^\mu$ and $\pi^{\mu\nu}$. Using this frame, it was found that for systems with spontaneous symmetry breaking the energy-momentum tensor receives no corrections, while the corrections to the charged currents admit a decomposition in terms of their longitudinal and transverse components~\cite{Manes:2018llx,Manes:2019fyw}, i.e.
\begin{equation}
\pi^{\mu\nu} = 0 \,, \qquad \delta J^\mu = - (u_\nu \delta J^\nu) u^\mu + P^\mu{}_\nu \delta J^\nu \,,
\end{equation}
where $P^{\mu\nu} = G^{\mu\nu} + u^\mu u^\nu$ is the transverse projector to the local fluid velocity.
\subsection{Constitutive relations at the lowest order}
\label{subsec:CR_0}
The constitutive relations at the lowest order can be obtained by taking the corresponding functional derivatives on the effective action of Eq.~(\ref{eq:W0}), cf. Eqs.~(\ref{eq:Jcr}) and (\ref{eq:Tcr}). This leads to the result
\begin{eqnarray}
\hspace{-2.3cm} \langle J_{\mu\, 0} \rangle_{{\scriptsize\textrm{PF}}} &=& n_0 \, u_\mu \,, \\
\hspace{-2.3cm} \langle J_{\mu\, 3} \rangle_{{\scriptsize\textrm{PF}}} &=& n_3 \, u_\mu + i \frac{f_\pi^2}{4} {\textrm {Tr}} \left\{ [Q,U] \partial_\mu U^\dagger + [Q,U^\dagger] \partial_\mu U \right\} + \frac{f_\pi^2}{2} V_{\mu \, 3} {\textrm {Tr}} \left\{ [Q,U] [Q,U^\dagger] \right\} \,, \\
\hspace{-2.3cm} \langle T^{\mu\nu} \rangle_{{\scriptsize\textrm{PF}}} &=& (\varepsilon + P) u^\mu u^\nu + P G^{\mu\nu} + { \frac{f_\pi^2}{4} G^{\mu\alpha} G^{\nu\beta} {\textrm {Tr}}\left\{ D_{\alpha} U ( D_{\beta} U)^\dagger + D_{\beta} U ( D_{\alpha} U)^\dagger \right\} } \,,
\end{eqnarray}
where the number densities are defined by $n_a = \partial P/\partial \mu_a$ $(a = 0,3)$. These contributions to the constitutive relations have been expressed in a covariant form by writing them in terms of the metric $G_{\mu\nu}$ and the four velocity $u^\mu$, cf. Eqs.~(\ref{eq:metric}) and (\ref{eq:umu_P}). Notice that since the BZ terms contain one derivative of the gauge fields, there is no distinction between consistent and covariant currents at leading order in the derivative expansion.
\subsection{Corrections to the leading order constitutive relations}
\label{subsec:CR_1}
All dependence on the NG bosons matrix $U$ in the constitutive relations comes in terms of the following covariant expressions
\begin{eqnarray}
\mathcal{H} &=& {\textrm {Tr}} \bigl\{ \left( U^{-1} Q U - Q\right) Q \bigr\} \,, \qquad {\mathcal I}_\mu \equiv {\textrm {Tr}} \bigl\{ (R_\mu + L_\mu) Q \bigr\} \,, \label{eq:HI} \\
{\mathcal T}_\mu &\equiv& {\textrm {Tr}} \bigl\{ (R_\mu - L_\mu) Q \bigr\} + 2 \mathcal{V}_{\mu \, 3} {\textrm {Tr}} \bigl\{ \left( U^{-1} Q U - Q \right) Q \bigr\} \,. \label{eq:Tmu}
\end{eqnarray}
Then, the currents can be decomposed into their longitudinal and transverse components, and the result can be written as linear combinations of the following five pseudo-scalar quantities
\begin{eqnarray}
\hspace{-2cm} \mathbb S_{1(a)} &\equiv& {\mathcal I}_\mu \, {\mathcal B}_a^\mu \,, \qquad \mathbb S_2 \equiv {\mathcal I}_\mu \, \omega^\mu \,, \qquad \mathbb S_3 \equiv \epsilon^{\mu\nu\alpha\beta} u_\mu \left[ \mathcal{V}_{\nu \, 3} \, \partial_\alpha {\mathcal I}_\beta - \frac{i}{3} {\textrm {Tr}}\{ L_\nu L_\alpha L_\beta \} \right] \,, \\
\hspace{-2cm} \mathbb S_{4 (a)} &\equiv& {\mathcal T}_\mu \, {\mathcal B}_a^\mu \,, \qquad \mathbb S_5 \equiv {\mathcal T}_\mu \, \omega^\mu \,,
\end{eqnarray}
and the four transverse pseudo-vectors
\begin{eqnarray}
P_{1(a)}^\mu &=& \frac{1}{T} \epsilon^{\mu\nu\alpha\beta} u_\nu {\mathcal I}_\alpha \, \mathcal E_{\beta(a)} \,, \qquad P_2^\mu = \epsilon^{\mu\nu\alpha\beta} u_\nu \partial_\alpha {\mathcal I}_\beta \,, \\
P_{3(a)}^\mu &=& \frac{1}{T}\epsilon^{\mu\nu\alpha\beta} u_\nu {\mathcal T}_\alpha \, \mathcal E_{\beta(a)} \,, \qquad P_{4}^\mu = \epsilon^{\mu\nu\alpha\beta} u_\nu \partial_\alpha {\mathcal T}_\beta \,,
\end{eqnarray}
where $\mathcal E_{\mu(a)} = \mathcal V_{\mu\nu\, a} \, u^\nu = T \partial_\mu\left( \mu_a/T \right)$ is the electric field for $a= 0, 3$. Finally, the constitutive relations at first order in derivatives read
\begin{eqnarray}
\hspace{-1cm} u_\mu \langle \delta J_{0\, {\textrm{\scriptsize V}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& -\frac{N_c}{16\pi^2} \left[ \mathbb S_{1(3)} + \mathbb S_{3} \right] \,, \\
\hspace{-1cm} P^\mu{}_\nu \langle \delta J^\nu_{0\,{\textrm{\scriptsize V}}} \rangle_{{\textrm{\scriptsize cov}}} &=& - \frac{N_c}{16\pi^2} \left[ T \mathbb P_{1(3)}^\mu - \mu_3 \mathbb P_2^\mu - 2 \mu_5 \mathcal B_0^\mu \right] \,,\\
\hspace{-1cm} u_\mu \langle\delta J_{3\, {\textrm{\scriptsize V}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& - \frac{N_c}{48\pi^2} \left[ 3 \mathbb S_{1(0)} + 2 \mu_5 \mathbb S_{5} \right] \,, \\
\hspace{-1cm} P^\mu{}_\nu \langle \delta J^\nu_{3\, {\textrm{\scriptsize V}}} \rangle_{{\textrm{\scriptsize cov}}} &=& - \frac{N_c}{48\pi^2} \left[ 3 T \mathbb P_{1(0)}^\mu + \mu_5 \mathbb P_{4}^\mu + 4 \mu_3 \mu_5 \mathcal H \omega^\mu - 2 \mu_5 (\mathcal H + 3) \mathcal B_3^\mu \right] \,,
\end{eqnarray}
for vector currents, and
\begin{eqnarray}
u_\mu \langle \delta J_{0\, {\textrm{\scriptsize A}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& - \frac{N_c}{48\pi^2} \mathbb S_{4(3)} \,, \\
P^\mu{}_\nu \langle \delta J^\nu_{0 \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} &=& -\frac{N_c}{48\pi^2} \left[ T \mathbb P_{3(3)}^\mu + 2 \mu_3 {\mathcal H} \mathcal B_3^\mu + 4\mu_5^2 \omega^\mu \right] \,, \\
u_\mu \langle \delta J_{3\, {\textrm{\scriptsize A}}}^\mu \rangle_{{\textrm{\scriptsize cov}}} &=& -\frac{N_c}{48\pi^2} \left[ 3 \mathbb S_{4(0)} - 2 \mu_5 \mathbb S_2 \right] \,, \\
P^\mu{}_\nu \langle \delta J^\nu_{3 \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} &=& -\frac{N_c}{48\pi^2} \left[ 3 T P_{3(0)}^\mu + 6 \mu_3 {\mathcal H} {\mathcal B}_0^\mu - \mu_5 \mathbb P_2^\mu \right] \,,
\end{eqnarray}
for axial-vector currents. The BZ contributions, which correspond to the CME and CVE, are those terms proportional to the magnetic field ${\mathcal B}_a^\mu$ and vorticity vector $\omega^\mu$ without gothic fonts prefactors, and they only appear in the transverse components of the currents. Notice as well that the covariant currents are given in terms of the KK-variant gauge fields $({\mathcal V}_{\mu \, a }, {\mathcal A}_{\mu\, a})$ without any explicit reference to the KK gauge field~$a_i$. Finally, the terms proportional to $\mathbb P_{1(a)}^\mu$ and $\mathbb P_{3(a)}^\mu$ in the constitutive relations are the ones associated to the CEE, i.e. charge transport normal to the direction of the electric field~\cite{Neiman:2011mj}. To compare our results with other analyses in the literature we expand the covariant expressions of Eqs.~(\ref{eq:HI}) and (\ref{eq:Tmu}) in powers of the pion fields. Then, using the definitions of the currents and electromagnetic field given in Sec.~\ref{subsec:em_bar_iso_currents}, we can express the electromagnetic current in terms of the pion fields, electric and magnetic fields as
\begin{equation}
\hspace{-1.7cm} \langle \delta J^i_{{\textrm{\scriptsize em}}} \rangle_{{\textrm{\scriptsize cov}}} = \frac{e^2 N_c}{12\pi^2} \left[ \frac{1}{f_\pi} \epsilon^{ijk} \partial_j \pi^0 \mathcal E_k + \frac{2}{f_\pi^2} \mu \mu_5 \pi^+ \pi^- \omega^i + \frac{5}{3} \mu_5 {\mathcal B}^i \right] + \cdots \,,
\end{equation}
where the physical electric field $\mathcal E_i$ and the electric charge chemical potential $\mu$ are defined as
\begin{equation}
\mathcal E_i = e^{-\sigma} \partial_i {\mathscr{V}}_0 = T \partial_i\left( \frac{\mu}{T} \right) \,, \qquad \mu \equiv e^{-\sigma} {\mathscr{V}}_0 \,,
\end{equation}
while the relation with the baryonic and isospin chemical potentials is~$\mu_0 = \frac{1}{3} \mu_3 = \frac{e}{3} \mu$. Similar expressions can be written for the baryon and isospin currents, leading to
\begin{eqnarray}
\hspace{-1.7cm} \langle \delta J^i_{{\textrm{\scriptsize bar}}} \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{e N_c}{12\pi^2} \left[ \frac{1}{f_\pi} \epsilon^{ijk} \partial_j \pi^0 \mathcal E_k + \frac{1}{3} \mu_5 \mathcal B^i \right] + \cdots \,, \label{eq:Jibar} \\
\hspace{-1.7cm} \langle \delta J^i_{{\textrm{\scriptsize iso}}} \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{e N_c}{24\pi^2} \left[ \frac{1}{f_\pi} \epsilon^{ijk} \partial_j \pi^0 \mathcal E_k + \frac{4}{f_\pi^2} \mu \mu_5 \pi^+ \pi^- \omega^i + 3 \mu_5 {\mathcal B}^i \right] + \cdots \,. \label{eq:Jiiso}
\end{eqnarray}
All three currents are invariant under the gauge transformations of electromagnetism. Notice also that the terms proportional to the vorticity vector come always multiplied by $\mu_5$, which means that in the absence of chiral imbalance $(\mu_5 = 0)$ there are no contributions depending of the vorticity. Finally, by considering similar steps we can obtain some of the explicit contributions to the axial-vector covariant currents. The result is
\begin{eqnarray}
\hspace{-1.9cm} \langle \delta J^i_{0 \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c}{24 \pi^2 f_\pi^2} \left[ i e \epsilon^{ijk} (\pi^- \partial_j \pi^+ - \pi^+ \partial_j \pi^- ) \mathcal E_k + 2 e \mu \pi^+ \pi^- \mathcal B^i - 2\mu_5^2 \omega^i \right] + \cdots \,, \label{eq:Ji0A} \\
\hspace{-1.9cm} \langle \delta J^i_{3 \, {\textrm{\scriptsize A}}} \rangle_{{\textrm{\scriptsize cov}}} &=& \frac{N_c}{24 \pi^2 f_\pi^2} \left[ i e \epsilon^{ijk} (\pi^- \partial_j \pi^+ - \pi^+ \partial_j \pi^- ) \mathcal E_k - 2 e \mu \pi^+ \pi^- \mathcal B^i \right] + \cdots \,. \label{eq:Ji3A}
\end{eqnarray}
These currents contain chiral separation effect terms of electric, magnetic, and vortical type. Let us mention that written in terms of the KK-invariant magnetic field, $\mathbb B^\mu = \frac{1}{2} \epsilon^{\mu\nu\alpha\beta} u_\nu {\mathbb{V}}_{\alpha\beta} = \mathcal B^\mu + \frac{6}{e} \mu_0 \omega^\mu$, we find vorticity dependent terms mediated by the charged pion fields, that survive in the case $\mu_5 = 0$. The emergence of the CEE, i.e. terms proportional to $\epsilon^{ijk} \mathcal E_k$, is manifest in all the currents of Eqs.~(\ref{eq:Jibar})-(\ref{eq:Ji3A}). Despite the ongoing discussion in the literature concerning the non-dissipative character of the CEE (see e.g. Ref.~\cite{Chapman:2013qpa}), our derivation shows that the CEE is intrinsically non-dissipative, so that it does not lead to entropy production.
As a remark, let us mention that in this work we have computed the covariant currents by performing functional derivatives of the effective action and adding the corresponding BZ currents. However, there is a direct procedure to obtain the covariant currents in presence of spontaneous symmetry breaking by using a direct relation with the BZ currents, thus bypassing the need to use the WZW action. Basically, the covariant currents are given by $J^\mu_{{\textrm{\scriptsize cov}}}(\mathcal A,g) = g J^\mu_{{\mathrm{BZ}}}(\mathcal A_g) g^{-1}$~\cite{Jensen:2013kka,Manes:2018llx}, and one just have to make the replacements $\mathcal A \to (\mathcal A_{\textrm{\scriptsize L}}, \mathcal A_{\textrm{\scriptsize R}})$ and $g \to (U,{\mathbb I})$ in this relation to obtain
\begin{equation}
\hspace{-2cm} J^\mu_{{\textrm{\scriptsize L}}\, {\textrm{\scriptsize cov}}}(\mathcal A_{\textrm{\scriptsize L}},\mathcal A_{\textrm{\scriptsize R}},U) = U J^\mu_{{\textrm{\scriptsize L}} \, {\mathrm{BZ}}}(\mathcal A_{\textrm{\scriptsize L}}^U, \mathcal A_{\textrm{\scriptsize R}}) U^{-1} \,, \quad J^\mu_{{\textrm{\scriptsize R}}\, {\textrm{\scriptsize cov}}}(\mathcal A_{\textrm{\scriptsize L}},\mathcal A_{\textrm{\scriptsize R}},U) = J^\mu_{{\textrm{\scriptsize R}} \, {\mathrm{BZ}}}(\mathcal A_{\textrm{\scriptsize L}}^U, \mathcal A_R) \,, \label{eq:JcovJBZ}
\end{equation}
where the BZ currents are given by Eqs.~(\ref{eq:JBZ_V}) and (\ref{eq:JBZ_A}). Notice that there is no BZ contribution to the energy-momentum tensor, $T^{\mu\nu}_{\mathrm{BZ}} = 0$, and this is consistent with the fact that the anomalous energy-momentum tensor must vanish in a system with spontaneously broken symmetry, $\langle \pi^{\mu\nu}\rangle = 0$. This procedure has been studied in detail in Ref.~\cite{Manes:2019fyw}.
\section{Conclusions}
\label{sec:conclusions}
In this work we have studied non-dissipative transport effects of relativistic fluids up to first order in the hydrodynamic expansion in presence of non-abelian anomalies. The computation has been performed by using the equilibrium partition function formalism, which is suitable for the study of non-dissipative effects like the CME and CVE. We have extended the analysis to the case of spontaneous symmetry breaking, leading to relevant information about the hydrodynamics of NG bosons interacting with external electromagnetic fields and in presence of vortices in the fluid. After particularization of the results for two flavors, we have been able to provide explicit expressions for the constitutive relations of the covariant currents in a chiral hadronic fluid, and found that the presence of NG bosons induces the CME and CVE both in the vector and axial currents. The vorticity terms are controlled by the chemical potential governing chiral imbalance, $\mu_5$, and disappear in the limit $\mu_5 \to 0$. Our calculation also predicts the emergence of a CEE whose corresponding transport coefficient is explicitly evaluated. Our findings are in agreement with previous results in the literature in the presence of chiral imbalance.
Gravitational and/or mixed gauge-gravitational anomalies~\cite{Landsteiner:2011cp,Jensen:2012kj} can also be incorporated into the description by considering appropriate curvature terms within the differential geometry methods, a study that is indeed of interest given their recently discovered experimental signatures~\cite{Gooth:2017mbd}. Finally, let us stress that the techniques presented in this work can be extended to a wide variety of physical situations, ranging from superfluids~\cite{Lin:2011aa,Hoyos:2014nua} to condensed matter systems affected by triangle anomalies~\cite{Basar:2013iaa,Landsteiner:2013sja}.
\ack
The work of J.L.M. and M.V. has been supported by Spanish Science
Ministry grant PGC2018-094626-B-C21 (MCIU/AEI/FEDER, EU) and Basque
Government grant IT979-16. The research of E.M. is supported by
Spanish Science Ministry grant FIS2017-85053-C2-1-P, by FEDER/Junta de
Andaluc\'{\i}a-Consejer\'{\i}a de Econom\'{\i}a y Conocimiento
2014-2020 Operational Programme grant A-FQM-178-UGR18, by Junta de
Andaluc\'{\i}a grant FQM-225, and by Consejer\'{\i}a de Conocimiento,
Investigaci\'on y Universidad of Junta de Andaluc\'{\i}a and European
Regional Development Fund (ERDF) grant SOMM17/6105/UGR. The research
of E.M. is also supported by the Ram\'on y Cajal Program of the
Spanish Science Ministry grant RYC-2016-20678. M.A.V.-M. acknowledges
the financial support from the Spanish Science Ministry through
research grant PGC2018-094626-B-C22 (MCIU/AEI/FEDER, EU), as well as
from Basque Government grant IT979-16.
\section*{References}
|
1,108,101,565,456 | arxiv | \section{Schwinger Dyson Equations}
\label{DS}
In principle one can improve on the approximations for the fluctuation
field integral if the action $\E{\Action} $ is known.
In any case, more accurate methods are applicable
in the first step, since the starting action $S^0$
is known by assumption.
If $\E{\Action}$ is only known
approximately, there can be problems because the correction terms might
depend on details of the action which are neglected when a localization
approximation is made. This is the crucial point to discuss. It will turn out
that inclusion of two loop corrections requires that also the localization
approximation is improved to the next order. One needs to consider
third derivatives of the
normal ordered action evaluated at nearly constant fields.
Here we discuss the evaluation of the fluctuation field integral by
solution of Schwinger Dyson equations.
Let $ \Gamma $ be a positive semidefinite operator on ${\cal H} $. Then
the normalized Gaussian measure $d\mu_{\Gamma }$ on ${\cal H} $ is defined. Its
characteristic function is
\begin{equation}
\int d\mu_{\Gamma }(\phi )\exp i\scalar{q}{\phi} = \exp\left( -\frac 12
\scalar{q}{\Gamma q} \right) . \label{qGauss}
\end{equation}
If $A[\phi ]$ is a function of $\phi $ which is integrable with respect
to the Gaussian measure $ d\mu_{\Gamma }$,
it can be written in normal ordered form, as we know.
\begin{eqnarray}
A[\phi ] &=& :B [\phi ]:\\
B[\phi ] &=& \int d\mu_{\Gamma }(\xi ) A[\xi + \phi ]
\end{eqnarray}
The normal ordering prescription depends on $\Gamma $.
If $\Gamma $ is strictly positive and bounded away from zero,
then $B [\phi ]$ is an entire analytic
function of $\phi $ and can therefore be expanded in a power series. This
yields the expansion of $A$ in normal products of $\phi $. It exists
whether or not $A$ is continuous or differentiable. This is important
when one wishes to deal with singular cases such as a
discrete Gaussian model, where the action depends on the integer part of
a real lattice field.
Here we are interested in cases where $\Gamma $ has zero eigenvalues. The
1-di\-men\-sion\-al Dirac measure $ \delta (x)dx $ is the prototype of such a
measure; it is not a pathological case. In this degenerate case $\Gamma $
projects on a subspace ${\cal H}_C$ of ${\cal H} $ and $B[\phi + \xi ]$ is an entire analytic
function of $\xi \in {\cal H}_C$, assuming the
restriction $\Gamma_C$ of $\Gamma $ to
${\cal H}_C$ is bounded away from zero.
The normal ordering commutes with shifts of the field
and with derivatives with respect to $\phi $.
We need the following general integration by parts formula
\begin{equation}
\int d\mu_{\Gamma }(\phi ) A[\phi ] e^{-V[\phi]} =
B[\Gamma \frac {\delta}{\delta \xi } ]
\int d\mu_{\Gamma }(\phi ) e^{-V[\phi + \xi ]}\Einsch{\xi = 0}.
\label{PDiff}
\end{equation}
This generalizes the well known special case \cite{GlimmJaffe}
\begin{equation}
\int d\mu_{\Gamma }(\phi )\phi (z) e^{-V[\phi ]} =
\int_w \Gamma (z,w) \frac {\delta}{\delta \xi (w) }
\int d\mu_{\Gamma }(\phi ) e^{-V[\phi + \xi]}\Einsch{\xi = 0}
\end{equation}
and can be readily proven with the help of eq.~(\ref{qGauss}), as follows.
{\sc Proof of the general integration by parts formula:}
We use the Taylor formula in compact form
\begin{equation}
e^{a \frac {\delta }{\delta \phi } }F(\phi ) = F(\phi + a).
\end{equation}
Since an arbitrary functional $B[\phi]$ admits a functional Fourier transform,
it suffices to prove the formula (\ref{PDiff}) for the special case
$$
B[\phi] = e^{i\scalar{q}{\phi}}
$$
We compute
\begin{equation}
\begin{split}
& \int d\mu_{\Gamma}(\phi)
e^{i\scalar{q}{\Gamma\frac\delta{\delta\phi}}} A[\phi] \\
& = N_{\Gamma} \int D\phi A[\phi]
e^{-i\scalar{q}{\Gamma\frac\delta{\delta\phi}}}
e^{-\frac 12\scalar{\phi}{\Gamma^{-1}\phi}} \\
& = N_{\Gamma} \int D\phi A[\phi]
e^{-\frac 12\scalar{(\phi - iq\Gamma )}
{\Gamma^{-1}(\phi -i\Gamma q )}} \\
& \int d\mu_{\Gamma}(\phi) A[\phi]
e^{i\scalar{q}{\phi}+\frac 12\scalar{q}{\Gamma q}}\\
& \int d\mu_{\Gamma}(\phi) A[\phi]
:e^{i\scalar{q}{\phi}}: \qquad\qquad \text{{\sc q.e.d.}} \\
\end{split}
\end{equation}
The general integration by parts formula has many uses.
Suppose we define expectation values of an interacting
theory
\begin{eqnarray}
\mean{A} &=& Z^{-1}\int d\mu_{\Gamma }(\phi ) A[\phi ] e^{-V[\phi]}, \\
Z &=&\int d\mu_{\Gamma }(\phi ) e^{-V[\phi]},
\end{eqnarray}
and the generating function $G[\xi]$ of amputated Green
functions of the interacting theory
\begin{equation}
e^{G[\xi ]} = \int d\mu_{\Gamma }(\phi ) e^{-V[\phi+\xi]}.
\end{equation}
We preferred to introduce an unnormalized generating function,
so that $G[0]=\ln Z$. The normalized generating function is
$$
K[\xi]= G[\xi]-G[0].
$$
The integration by parts formula tells us that the expectation values equal
\begin{equation}
\mean{A} = e^{-G[0]}B[\Gamma \frac {\delta}{\delta \xi } ]
e^{G[\xi]}\Einsch{\xi= 0}.
\end{equation}
The Schwinger Dyson equation for the generating function $G $ is another
application of the integration by parts formula. It is obtained by
differentiating the definition of $e^G $ with respect to $\xi $ and
converting the resulting downstairs factor $ \frac {\delta}{\delta \xi }V $
into a differential operator. Suppose
$$
V[\phi] = :W[\phi]:.
$$
Then
\begin{eqnarray}
-\frac {\delta}{\delta \xi (z) }G[\xi] &=& e^{-G[\xi]}
W_{;z }[\xi + \Gamma \frac {\delta}{\delta \zeta } ]
e^{G[\xi+\zeta ]}\Einsch{\zeta=0}\
\label{SD} \\
W_{;z }&\equiv&\frac {\delta}{\delta \xi (z) }W.\nonumber
\end{eqnarray}
If $\Gamma $ has zero modes,
this is only valid for the directional derivative in ${\cal H}_C$ directions, i.e.
when multiplied from the left with $\Gamma $. Additive constants in $G$
drop out.
Let
\begin{eqnarray}
\E{\VPot} [\Psi , \zeta ] &=& \E{\Action} [\Psi + \zeta ] -
\frac 12 \scalar{\zeta}{\E{\FlucProp}_C[\Psi ]^{-1} \zeta},
\label{VDef} \\
e^{\E{\GAction}[\Psi , \xi ]}&=&
\int d\mu_{\Gamma }(\zeta )
e^{-{\E{\VPot} [\Psi ,\zeta + \xi ]}}
\ \ \ \mbox{for }\ \xi \in {\cal H}^i_{{\cal{C}}} ,
\label{GDef} \\
\Gamma &=& \E{\FlucProp} [\E{\Psi} [\Phi]].
\end{eqnarray}
We chose to work with an unnormalized generating function for amputated
Greens functions again; the normalized generating function is
$$
\E{\KAction} [\Psi , \xi ]= \E{\GAction}[\Psi , \xi ]-\E{\GAction}[\Psi , 0 ]
$$
$\E{\FlucProp}_C[\Psi ]^{-1}$ is the inverse of the restriction of
$\E{\FlucProp} [\Psi ] $ to its range ${\cal H}^i_{{\cal{C}}}$.
Dropping an additive constant, the exact effective action is
\begin{eqnarray}
\Z{\Action} [\Phi ] &=&
-\E{\GAction}[\Psi , 0] -
\frac 12 tr \ln \E{\FlucProp} [\Psi ], \label{GEffAction}\\
\Psi &=& \E{\Psi} [\Phi ].
\end{eqnarray}
The second term compensates for the transition from the unnormalized Gaussian
measure in definition (\ref{defEffective}) to a normalized one. It involves a
trace over ${\cal H}^i_{{\cal{C}}}$, cp. section \ref{GaussSection}.
The Schwinger Dyson equation (\ref{SD}) carries over literally;
$V,W,G$ all
depend parametrically on $\Psi $.
The single power of $\zeta $ which arises from differentiating
the second term in eq. (\ref{VDef}) converts into a differential operator.
As a result one obtains
\begin{eqnarray}
\E{\Action} [\Psi + \xi ] &=&
:\E{\NAction} [\Psi , \xi]:\\
\xi (z) &=& e^{-\E{\GAction} [\Psi , \xi ]}
\int_{w }\E{\FlucProp} (z ,w )\E{\NAction}_{;w }[\Psi ,
\xi + \E{\FlucProp} \frac {\delta}{\delta \zeta }] e^{\E{\GAction} [\Psi ,\xi+\zeta ]}
\Einsch{\zeta=0},\label{SDGeneral}\\
\E{\NAction}_{;w } &=& \frac {\delta}{\delta \xi (w )}\E{\NAction} . \label{SD1}
\end{eqnarray}
The above Schwinger Dyson equations
are true for any choice of the background field $\Psi $ and for any
choice of the propagator $\E{\FlucProp} $. A great simplification results if
they are chosen judiciously.
It is customary to call the amputated Green functions {\em vertices}. $n$-point
vertices are obtained as $n$-th $\xi$-derivatives of $\E{\KAction}$ at $\xi=0$. They
depend parametrically on the background field. In this terminology
the first and second derivative $\E{\KAction}_;$ and $\E{\KAction}_{;;}$
of $\E{\KAction}$ with respect to $\xi $ are the one- and two point vertices.
Only the directional derivatives in directions in ${\cal H}^i_{{\cal{C}}}$ are defined. We
require
\begin{eqnarray}
\zeta_1 \E{\KAction}_;[\Psi ,\xi = 0] &=& 0, \label{C1}\\
\zeta_1 \E{\KAction}_{;;}[\Psi ,\xi = 0]\zeta_2 &=& 0.\label{C2}
\end{eqnarray}
at $\Psi = \E{\Psi} [\Phi ]$ and for arbitrary
$\zeta_1, \zeta_2 \in {\cal H}^i_{{\cal{C}}}$. These conditions are shown
graphically in figure \ref{BildE}.
In the figures, the vertices obtained by differentiating $\E{\KAction}$
are cross hatched.
The Schwinger Dyson equations for the n-point functions
are obtained by differentiating
eq. (\ref{SDGeneral}) with respect to
$\xi$ at $\xi=0$. The resulting equations for one, two and three point
functions are shown in graphical form in figure \ref{BildF}.
\BildE{Condition on one- and two-point vertices in the Schwinger Dyson
approach}
\BildF{Schwinger Dyson equations for the one-,
two- and three-point vertices.
The first two equations serve to determine the background field and the
fluctuation field propagator.}
\BildG{Schwinger Dyson Gap equation in 1-loop approximation}
In principle the vertices can be computed from the
Schwinger Dyson equations
by iteration. As a consequence of conditions (\ref{C1},\ref{C2})
the equations for the one- and two point functions substitute for
the earlier equations for the
background field $\Psi $ and fluctuation propagator $\E{\FlucProp} $.
In zeroth approximation, the vertices with three and more legs are equal
to the corresponding normal ordered amplitudes (``bare vertices'').
The iteration proceeds by inserting the previous approximation for the
vertices on the right hand side of the
equations for the three-point functions, and similarly in
higher ones.
Because we have an infrared cutoff $\LatSpace_{i}^{-1}$, we can expect that
the iteration converges fast.
The effective action can be computed from eq. (\ref{GEffAction}). We
determine its $\Phi$-derivative by differentiating the definition
(\ref{GDef}) of $\E{\GAction}[\Psi , 0]$.
Using the change of covariance lemma (\ref{ChangeCov})
again, we obtain
\begin{eqnarray}
\frac{\delta}{\delta \Psi (z )} \E{\GAction}[\Psi ,0]
&=& e^{-\E{\GAction}[\Psi , 0]}\int d\mu_{\Gamma }(\zeta )\nonumber \\
& &
\left( \frac 12 \scalar{\frac{\delta}{\delta \zeta}} {\Gamma_{,z }
\frac{\delta}{\delta \zeta}} - \E{\VPot}_{,z}[\Psi , \zeta ]\right)
e^{-\E{\VPot}[\Psi , \zeta ]} \label{DG}
\end{eqnarray}
where $\Gamma = \E{\FlucProp}[\Psi ]$.
We restrict attention to
$\Phi$-derivatives.
From the definition of $\E{\VPot}$ we get
\begin{eqnarray}
\frac{\delta}{\delta \Phi (x )} \E{\VPot}[\E{\Psi}[\Phi ],\zeta ]
&=& \int \Psi _{,x} (z ) \left( \E{\Action}_{,z }+\frac 12
\scalar{\zeta} {\Gamma_C^{-1}\Gamma_{,z }\Gamma_C^{-1} \zeta}
\right) \\
&=& \int \Psi _{,x} (z ) \left(: \E{\NAction}_{;z }:+\frac 12
:\scalar{\zeta} {\Gamma_C^{-1}\Gamma_{,z }\Gamma_C^{-1} \zeta}:
+ \frac 12 \text{tr} \Gamma_{,z }\Gamma_C^{-1} \right) \nonumber
\end{eqnarray}
This is inserted into eq. (\ref{DG}). We convert again the downstairs factors
of $\zeta $ into differential operators, using the integration by parts
formula. One term cancels against the second term in eq. (\ref{GEffAction})
and we obtain the final result
\begin{eqnarray}
\frac {\delta}{\delta \Phi (x )}
\Z{\Action} [\Phi] &=&
\int_{z} \Psi _{,x }(z) \E{\QAction}_;{}[\Psi ,0 ](z)
\label{SDActionFirstDerivative} \\
\E{\QAction}_;{}[\Psi , \xi ](z)&\equiv &
\E{\NAction}_{;z }[\Psi , \xi +
\Gamma \frac {\delta}{\delta \zeta }]
e^{\E{\KAction}[\Psi ,\xi + \zeta ]}
\Einsch{\zeta=0}
\label{SDAction1}
\end{eqnarray}
The result is shown in graphical notation in figure \ref{BildJ}.
The derivative of the effective action can be computed when the
$\E{\KAction}$-vertices have been determined from the Schwinger Dyson equation.
We consider next the equation for the background field.
Comparing the above definition of $\E{\QAction}_;{}$ with the Schwinger Dyson equation
(eq. (\ref{SDGeneral}) at $\xi = 0$) for the 1-point function we see that
the equation for the background field takes the form
\begin{equation}
\zeta \E{\QAction}_;{}[\Psi ] =0 \ \ \mbox{for all } \ \zeta \in {\cal H}^i_{{\cal{C}}}
\end{equation}
The Schwinger Dyson equation for the 2-point function
implies
\begin{eqnarray}
\E{\FlucProp} \E{\QAction}_{;;}{} \E{\FlucProp} = \E{\FlucProp}.
\end{eqnarray}
One deduces (in the by now familiar way) the formula
for the derivative of the background field,
\begin{equation}
\E{\Psi}_{,x }(z ) = \Adj{ \tilde \E{\AvOp}}(z , x )
- \int_{w, u} \E{\FlucPropVar}(z , w)
\E{\QAction}_;{}{}_{\,,u}(w) \Adj{ \tilde \E{\AvOp}}(u , x ).
\label{backDerQ}
\end{equation}
$\E{\FlucPropVar}$ is defined as inverse of the mixed second derivative
$\E{\QAction}_{;}{}_{\,,}{}$ on ${\cal H}^i_{{\cal{C}}}$. Also we have
\begin{equation}
\E{\FlucProp}_{,z } = -\E{\FlucProp} \E{\QAction}_{;;}{}{}_{\,,z} \E{\FlucProp}.
\label{Q}
\end{equation}
One may finally
differentiate (\ref{SDActionFirstDerivative})
once more to get an equation for the second derivative of
the effective action. The term involving the second derivative of the
background field vanishes again because the constraints imply that
$\E{\Psi}_{,x y }\in {\cal H}^i_{{\cal{C}}}$ (see (\ref{MFAFieldPPinHCE}). As a result
\begin{multline}
\frac{\delta}{\delta \Phi (x )\delta \Phi (y )}
\Z{\Action} [\Phi ] = \\
\int_{zw} \Psi _{,x }(z)\nonumber \biggl(
\E{\NAction}_{;z\,,w } [\Psi ,
\Gamma \frac{\delta}{\delta \zeta } ] +
\int_u
\E{\NAction}_{;z u } [\Psi ,
\Gamma \frac{\delta}{\delta \zeta } ]
\left(\Gamma_{,w} \frac{\delta}{\delta \zeta }
\right)(u) \\
+ \E{\NAction}_{;z } [\Psi ,
\Gamma \frac{\delta}{\delta \zeta } ]
\E{\KAction}_{,w}[\Psi , \zeta ]
\biggr)
e^{\E{\KAction}[\Psi , \zeta ]}\Einsch{\zeta = 0}
\ \Psi _{,y }(w)
\label{SDAction2}
\end{multline}
One can insert (\ref{mixedReplacement})
and eq. (\ref{Q}) for the first term in order to eliminate a
reference to the first $\xi-$derivative of $\E{\NAction} $.
The equations for the first and second derivative of the effective action
are shown in Figures \ref{BildJ} and \ref{BildK}.
Again, there are no 1-particle reducible contributions.
\BildJ{Recursion relation for
${\E{\Action}}{}'$ in the Schwinger Dyson approach.}
\BildK{Recursion relation for
${\E{\Action}}{}''$ in the Schwinger Dyson approach. The prime denotes a
derivation with respect to the background field.}
From these equations one can see what are the leading corrections to the
approximations which were considered in the previous section. To
make the comparison we take the zeroth loop order of the Schwinger Dyson
equations. After inserting b) into c) in figure \ref{BildF} in zeroth order
we get the simplified vertices shown in figure \ref{BildO}. Using these in
the recursion relation for the derivatives of the effective action and
keeping only 1-loop graphs leads to
the familiar result of the previous section (figures \ref{BildN} and
\ref{BildH}).
\BildO{Schwinger Dyson equations of the first three vertices in zeroth order}
\subsection{Two loop corrections}
When we make appropriate
localization approximations, the Schwinger Dyson equations should
become integro-differential equations for matrix functions of a single
variable. We see, however, that the equations involve normal ordered
amplitudes with more than two high frequency legs.
The Schwinger Dyson equations for the vertices and the formulae for the
derivatives of the effective action require a separate discussion.
We see that the two loop terms in the recursion relation for the second
derivative of the effective action involves normal ordered amplitudes with
three high-frequency (``hard'') legs.
They are not known
if we only know the second derivative of the normal ordered action
evaluated at nearly constant field.
Therefore we need
consider {\em third} derivatives of normal ordered actions
evaluated at nearly constant fields. This introduces quantities
$ \E{\WPotloc}\left( z ,w , u |\eta \right) $ . They have exponential
decay with the tree distance of $z , w , u$ with decay length not
bigger than one lattice spacing.
We will also need recursion relations for these 3-point quantities. They can be
obtained from eq.~(\ref{SDAction2}) by differentiating once more. We will not
write the result explicitly, the graphs involved are similar to those in the
Schwinger Dyson equation (figure \ref{BildF}) for the 3-point vertices,
except that the external legs are soft ones. It appears
that this equation involves a normal ordered amplitude with {\em four}
hard legs (in the last graph), and so we seem to be in trouble again.
But this is actually not really so. Because the external legs are soft,
there are actually only {\em two} independent hard relative momenta
in this graph as in the remaining graphs.
The arguments of two hard legs which join to different
soft external legs may therefore be identified, and the
knowledge of up to third derivatives of the normal ordered
action suffices to determine this vertex accurately enough.
The strategy for the Schwinger Dyson equations is different. There are
vertices with still more hard legs. We regard the determination of the
fluctuation propagator and of the (cross hatched) vertices as one task,
whose input is the effective action $\E{\Action}$. We propose to make localization
approximations for the normal ordered actions $\E{\NAction} $, but not for
$\E{\Action}$. Therefore $\E{\Action}$ may be regarded as accurately known
for arbitrary fields from the previous application of the recursion formula.
Therefore one may take derivatives of any order to create hard legs.
One needs to insert the resulting formulae into the Schwinger Dyson
equations. As a result, one has to evaluate graphs involving propagators at
two different length scales. This complication arose already in the
Feynman Bogoliubov approximation in section \ref{FBapproximation}.
The practical evaluation of all the
two loop graphs requires a serious effort in
high performance computing, unless one is willing to approximate the
rather complicated exact formulae for the bare propagators and vertices.
We are not prepared to elaborate on this.
In this paper we only discuss the actual evaluation of 1-loop graphs.
\section{Formulas for saddle point approximation}
\subsection{Form of $\Gamma$}
\label{GammaFormAp}
We are looking for the propagator which corresponds to the quadratic term
$\frac 12\scalar{\E{\FlucField}}{{\E{\Action}}{}''[\E{\Psi}[\Phi]]\E{\FlucField}}$ in
(\ref{MFAintegral}). The fluctuation fields $\E{\FlucField}$ obey the
constraint $\E{\AvOp}\E{\FlucField}=0$.
Let\footnote{${\E{\Action}}{}''$ has in general no zero modes. If there are some
zero modes (e.g. for $\phi=0$) then they can be eliminated
with the help of projection operators before building the inverse
${{\E{\Action}}{}''}^{-1}$. This is well known (see \cite{GawKup}).}
\begin{equation}
\E{\FlucProp} = {{\E{\Action}}{}''}^{-1} -
{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}(\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}})^{-1}
\E{\AvOp}{{\E{\Action}}{}''}^{-1}\,.
\label{GammaForm}
\end{equation}
It obeys the constraint $\E{\AvOp}\E{\FlucProp}=0$.
Therefore $\E{\FlucProp}:\E{\cal H}\rightarrow{\cal H}^i_{{\cal{C}}}$. We now want to show that
\begin{equation*}
(\E{\FlucProp})^{-1} = \ActionPP
\end{equation*}
This means that
$$
\E{\FlucProp}{\E{\Action}}{}''\Einsch{{\cal H}^i_{{\cal{C}}}} = 1} %{\E{\id}_{{\cal H}^i_{{\cal{C}}}}
= {\E{\Action}}{}''\E{\FlucProp}\Einsch{{\cal H}^i_{{\cal{C}}}}\,.
$$
The first equation is equivalent to
$$
\E{\FlucProp}{\E{\Action}}{}''\E{\FlucField} = \E{\FlucField}
\qquad \forall\ \E{\FlucField}\in{\cal H}^i_{{\cal{C}}}\,.
$$
Applying ${\E{\Action}}{}''\E{\FlucField}$ to (\ref{GammaForm}) from the right
confirms this. \\
The second equation is true because it is the adjoint of the first.
\subsection{Proof of the relation for the background field}
\label{proofOfLemma}
Differentiating the defining equation (\ref{MFAFieldEq1}) for the
background field by the chain rule, we obtain
\begin{equation}
{\E{\Action}}{}'' [\E{\Psi}] {\E{\Psi}}{}' = \Adj{\E{\AvOp}} \E\lambda{}^{\prime}
\label{varfluc}
\end{equation}
for some $\E\lambda{}^{\prime} \in \Z{\cal H}$. Multiplying (\ref{varfluc})
with ${{\E{\Action}}{}''}^{-1}$
\begin{equation}
{\E{\Psi}}{}' = {{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}
\E\lambda{}^\prime \label{psiprime}
\end{equation}
and $\E{\AvOp}$ gives
\begin{equation*}
\E{\AvOp}{\E{\Psi}}{}' = \E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}
\E\lambda{}^\prime
\end{equation*}
By virtue of $\E{\AvOp}{\E{\Psi}}{}'=1} %{\Z{\id}$
it follows
\begin{equation*}
1} %{\Z{\id} = (\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}) \E\lambda{}^\prime
\end{equation*}
Therefore
\begin{equation}
(\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}})^{-1} = \E\lambda{}^\prime
\label{lambdaprime}
\end{equation}
We have now one possible form of ${\E{\Psi}}{}'$:
\begin{equation}
{\E{\Psi}}{}' = {{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}
(\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}})^{-1}
\label{psiprimeVariant}
\end{equation}
Multiplying $\E{\FlucProp}$ in (\ref{GammaForm})
with ${{\E{\Action}}{}''}$ from right and using (\ref{lambdaprime}) gives
\begin{align*}
\E{\FlucProp}{\E{\Action}}{}'' &= 1} %{\E{\id} -
{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}(\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}})^{-1}
\E{\AvOp} \\
&= 1} %{\E{\id} -
{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}\E\lambda{}^\prime \E{\AvOp}
\end{align*}
Inserting (\ref{psiprime}) leads to
\begin{equation}
\E{\FlucProp} {\E{\Action}}{}'' = 1 - {\E{\Psi}}{}' \E{\AvOp}.
\label{lemmaEquation}
\end{equation}
By multiplying $\Adj{\E{\AvOp}}$ from the right we get the result
\begin{equation}
{\E{\Psi}}{}' = (1-\E{\FlucProp} {\E{\Action}}{}'') \Adj{\widetilde{\E{\AvOp}}}.
\end{equation}
\subsection{Derivative of $\E{\FlucProp}$}
\label{FlucPropDer}
Multiplying (\ref{lemmaEquation}) with $\E{\FlucProp}$ from the right and using
$\E{\AvOp}\E{\FlucProp}=0$ gives
$$
\E{\FlucProp}{\E{\Action}}{}''\E{\FlucProp} = \E{\FlucProp}.
$$
Differentiation leads to
$$
\E{\FlucProp}{}' = \E{\FlucProp}{}'{\E{\Action}}{}''\E{\FlucProp}
+ \E{\FlucProp}{\E{\Action}}{}''' \E{\FlucProp}
+ \E{\FlucProp}{\E{\Action}}{}''\E{\FlucProp}{}'\,.
$$
${\E{\Action}}{}''\E{\FlucProp}$ and $\E{\FlucProp}{\E{\Action}}{}''$ can be replaced now by
(\ref{lemmaEquation}). Using $\E{\AvOp}\E{\FlucProp}{}'=0$ leads to the result
$$
\E{\FlucProp}{}' = - \E{\FlucProp}{\E{\Action}}{}''' \E{\FlucProp}
$$
or expressed with $\E{\WPot}$
\begin{equation}
\E{\FlucProp}_{,z} = \E{\FlucProp}\E{\WPot}_{,z}\E{\FlucProp}\,.
\label{GammaDerivative}
\end{equation}
\subsection{Determination of the background field}
An equation for $\E{\Psi}_{,x }[\Phi ] (z)$ was already derived in
\ref{proofOfLemma}. It involves quantities which are obtained together with the
propagator. Expressed in terms of $\E{\WPot}$ it looks
\begin{equation*}
{\E{\Psi}}{}' = (1+\E{\FlucProp} \E{\WPot} ) \Adj{\widetilde{\E{\AvOp}}}.
\end{equation*}
To determine the recursion relation of $\E{\WPot}$ we need also the second derivative.
Using the formula for the derivative of the fluctuation propagator
(\ref{GammaDerivative}) we find
\begin{equation}
\E{\Psi}_{,xy}[\Phi](z) =
\int\limits_{w,z_1,z_2\in\E{\Lat}}
\E{\FlucProp}(z,z_1)\, \E{\WPot}_{,w}(z_1,z_2)\,
\E{\Psi}_{,y}[\Phi](w)\, \E{\Psi}_{,x}[\Phi](z_2).
\label{secondDerMFA}
\end{equation}
\subsection{Recursion relation for $\E{\WPot}$}
\label{recursionRelation}
To get $\Z{\WPot}$ out of (\ref{actionPrel}) we need the second derivative of
$\Z{\Action}$.
Consider first
\begin{equation*}
\begin{split}
\dZx {x }
\text{tr} \ln \E{\FlucProp} [\E{\Psi} [\Phi ]] = &
\,\text{tr} \,\E{\FlucProp} [\E{\Psi} [\Phi ]] \,
\dZx {x } \E{\WPot} [\E{\Psi} [\Phi ]] \\
=& \int_{z} \text{tr} \Bigl(\E{\FlucProp} [\E{\Psi} [\Phi ]] \,
\E{\WPot}_{,z} [\E{\Psi} [\Phi ]]\Bigr)
\E{\Psi}_{,x }[\Phi](z ) \\
\end{split}
\end{equation*}
where we have used (\ref{GammaDerivative})
and $\E{\WPot}_{,z}[\E{\Psi}]$ or $\E{\Psi}_{,x }[\Phi]$
denote the functional derivatives
$$
\frac{\delta}{\delta\E{\Psi}(z)}\E{\WPot}[\E{\Psi}]\,,
\qquad\qquad\dZx {x} \E{\Psi}[\Phi]\,.
$$
We differentiate once more to obtain the second derivative and
omit the functional argument $\E{\Psi}[\Phi]$:
\begin{equation}
\begin{split}
& \dZxy {x} {y} \text{tr} \ln \E{\FlucProp} [\E{\Psi} [\Phi] ] \\[8pt]
= & \, \int\limits_{z,w\in\E{\Lat}} \!\!\!\! \text{tr} \Bigl(
\E{\FlucProp} \,\E{\WPot}_{,z}
\E{\FlucProp} \,\E{\WPot}_{,w}
+ \E{\FlucProp} \E{\WPot}_{,zw} \Bigr)
\E{\Psi}_{,x} [\Phi] (z)
\E{\Psi}_{,y} [\Phi] (w)
\\[5pt]
& + \int\limits_{z\in\E{\Lat}} \text{tr} \Bigl( \E{\FlucProp} \E{\WPot}_{,z} \Bigr)
\E{\Psi}_{,xy} [\Phi] (z)
\\[3pt]
= & \int\limits_{z,w,z_i\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,z} (z_2,z_3)
\E{\FlucProp} (z_3,z_4)
\E{\WPot}_{,w} (z_4,z_1)
\\[-8pt]
& \qquad\qquad\qquad
\E{\Psi}_{,x} [\Phi] (z) \E{\Psi}_{,y} [\Phi] (w)
\\[8pt]
& + \int\limits_{z,w,z_1,z_2\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,zw}(z_2,z_1)
\E{\Psi}_{,x }[\Phi ](z)
\E{\Psi}_{,y }[\Phi ](w) \\
& + \int\limits_{z,z_1,z_2\in\E{\Lat}}
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,z} (z_2,z_1)
\E{\Psi}_{,xy}[\Phi](z). \\
\end{split}
\label{WrecLast}
\end{equation}
Now we consider $\E{\Action}[\E{\Psi}[\Phi]]$.
By the chain rule
\begin{equation}
\dZxy {x}{y} \E{\Action} [\E{\Psi} [\Phi ]] =
\,\Adj {\E{\Psi}_{,x } [\Phi ]} \,
{\E{\Action}}{}'' [\E{\Psi} ] \,\E{\Psi}_{,y } [\Phi ]
+ {\E{\Action}}{}' [\E{\Psi} ] \,\E{\Psi}_{,x y } [\Phi ].
\label{Wsaddle}
\end{equation}
By the saddle point condition,
${\E{\Action}}{}' [\E{\Psi}] = \Adj {\E{\AvOp}}\E\lambda $.
Therefore the last term looks in coordinates
$$
\int\limits_{z\in\E{\Lat}} \dZxy {x}{y} \E{\Psi} [\Phi](z)
\int\limits_{x_1\in\Z{\Lat}}\Adj{\E{\AvOp}}(z,x_1)\E\lambda(x_1)\,.
$$
Inserting (\ref{secondDerMFA}) shows that this term vanishes because of
$\E{\FlucProp}\Adj{\E{\AvOp}}=0$.
Therefore
\begin{equation*}
\Z{\WPot}[\Phi](x,y) = \,\Adj {\E{\Psi}_{,x } [\Phi ]}\,
\E{\WPot}[\E{\Psi}[\Phi]] \,\E{\Psi}_{,y } [\Phi ]
+ \frac 12\dZxy {x} {y} \text{tr} \ln \E{\FlucProp} [\E{\Psi} [\Phi]]\,.
\end{equation*}
Using (\ref{WrecLast}) and (\ref{secondDerMFA}) leads to the result
(\ref{recursionW}).
\section{Simplification of mixed derivatives}
\label{derRelSimple}
In the case of Feynman Bogoliubov approximation we defined
the propagator $\E{\FlucProp} $ for arbitrary fields $\phi$ as
inverse of ${\E{\Action}}{}''$ on ${\cal H}^i_{{\cal{C}}}$ and derived some formulas in a general
manner. Afterwards we used special fields namely the background field
which depends on the block-spin. Now we regard it only
as a function of the background field.
It is convenient to extend this definition to arbitrary
fields by specifying that it is independent of the
high frequency components of the field,
\begin{equation}
\E{\FlucProp} [\E{\Psi} + \E{\FlucField} ] = \E{\FlucProp} [\E{\Psi}]
\label{sumDepP}
\end{equation}
for $\E{\FlucField} \in {\cal H}^i_{{\cal{C}}}$.
It is important that the formula for the derivative of the propagator
\begin{eqnarray}
\E{\FlucProp}_{,z}[\E{\Psi} ] \equiv
\frac {\delta}{\delta \E{\Psi} (z )}\E{\FlucProp} [\E{\Psi} ]
&=& - \E{\FlucProp} {\NActionE_{;;}}{}_{\,, z } \E{\FlucProp} \label{FlucPropDerN} \\
{\NActionE_{;;}}{}_{\,, z }
&=& \frac {\delta}{\delta \E{\Psi} (z )}{\NActionE_{;;}} \label{GammaDerN}
\end{eqnarray}
is then only valid in the restricted sense that it can be used
for evaluating $ \Phi$-derivatives of the propagator.
\sticky{
The equation is derived from eq.~(\ref{gap})
in the same way as in appendix \ref{FlucPropDer}.
}
With this specification of the propagator, the definition (\ref{pregap0})
of the normal ordered amplitude implies that
\begin{equation}
T [\E{\Psi} + \E{\FlucField} , \xi - \E{\FlucField} ] =
T [\E{\Psi} , \xi ] \label{sumDep}
\end{equation}
for $\E{\FlucField} \in {\cal H}^i_{{\cal{C}}}$. In other words,
$T [\E{\Psi} , \xi ]$ depends actually only on
$\E{\Psi} + \xi $ so long as $\xi \in {\cal H}^i_{{\cal{C}}} $.
We find it convenient, however, to extend the definition (\ref{defNO})
of $T [\E{\Psi} , \xi ]$ to arbitrary $\xi $.
Eq.~(\ref{sumDep}) remains valid, of course, but
$T [\E{\Psi},\xi]$ is no longer only a function of the
sum of its arguments.
When evaluating $\Phi $ derivatives, eq. (\ref{FlucPropDerN}) can be used
to express the derivative $\E{\FlucProp}_{,z }$ of the propagator.
On the other hand, $\E{\Psi} $-derivatives in ${\cal H}^i_{{\cal{C}}}$ directions
are the same as $\xi$-derivatives; the last term does not contribute in this
case because of eq. (\ref{sumDepP}).
Also the equation (\ref{backDerN}) for the derivative of the background
field simplifies, i.e. $\E{\FlucPropVar}=\E{\FlucProp}$:
\begin{equation}
\E{\Psi}_{,x }(z ) = \Adj{ \tilde \E{\AvOp}}(z , x )
- \int_{w, u} \E{\FlucProp}(z , w)
\E{\NAction}_{;w\,,u} \Adj{ \tilde \E{\AvOp}}
(u , x ), \label{backDerNsimple}
\end{equation}
It fulfills also (\ref{saddleConditionDerivative}) and the constraint
(\ref{constr}). One needs the identity
\begin{equation}
\int_{z ,w}\E{\FlucProp}(*,z )\PsiderivativeE{z}
\E{\NAction}_{;w } \E{\FlucProp} (w , *) = \E{\FlucProp} (*,*).
\end{equation}
$\E{\Psi} $- and $\xi$-derivatives of $\E{\NAction}$ agree in ${\cal H}^i_{{\cal{C}}}$-directions,
and the range of $\E{\FlucProp}$ is ${\cal H}^i_{{\cal{C}}}$. Therefore we may convert
the $\E{\Psi}$-derivative into a $\xi$-derivative. The assertion follows
now from the gap equation (\ref{gap}).
\section{Momentum space representation}
\label{GammaLattice}
Let $\E{\Lat}$ be a $D$-dimenional position space lattice with
lattice spacing $\LatSpace_{i}$ and length $\LatLength_{i}=\LatExt_{i}\LatSpace_{i}$.
The corresponding dual lattice $\E{\DualLat}$ has a lattice spacing
$\DualLatSpace_{i}=2\pi/\LatLength_{i}$ and length $\DualLatLength_{i}=2\pi/\LatSpace_{i}$.
The block factor between two successive lattices shall be denoted by
$s=\LatSpace_{i+1}/\LatSpace_{i}$.
It is convenient to introduce an additonal momentum space lattice
$\DualLat^{i,i+1}$ with spacing $\DualLatSpace_{i,i+1}=\DualLatLength_{i}$ and length
$\DualLatLength_{i,i+1}=s\DualLatLength_{i}$.
Then any $p\in\E{\DualLat}$ can be decomposed into $p=q+l$ with
$q \in \Z{\DualLat}$ and $l \in \DualLat^{i,i+1}$.
The definitions
\begin{alignat}{3}
\int \limits_{p} &:= \frac{\DualLatSpace_{i}^D}{(2\pi)^D}
\sum\limits_{p \in \E{\DualLat}}
&\quad,\quad
\int \limits_{q} &:= \frac{\DualLatSpace_{i+1}^D}{(2\pi)^D}
\sum\limits_{q \in \Z{\DualLat}}
& \quad,\quad
\int \limits_{l} & := \sum \limits_{l \in \DualLat^{i,i+1}}
\end{alignat}
imply the split
\begin{equation}
\int \limits_{p} = \int \limits_{q} \int \limits_{l}.
\end{equation}
We now want to derive the momentum space representation (\ref{flucmorep}) of
the fluctuation propagator.
First note that any linear operator $L:\E{\cal H}\rightarrow\E{\cal H}$ with
$L(z,z')=L(z-z')$ can
be written as
\begin{equation}
L(z-z') = \int \limits_{p} L(p) e^{ip(z-z')}.
\label{fourierrep}
\end{equation}
Examples are $\E{\WPotloc}(z,z'|\overline{\FieldE})$ in the saddle point approximation
and the laplacian $\E\Delta(z,z')$. In the latter case one gets
\begin{alignat}{2}
\E\Delta(p) & = - \hat{p}^2 &
\qquad,\qquad \hat{p}_{\mu}
&= \frac{2}{\LatSpace_{i}}\sin\left(\frac{p_{\mu}\LatSpace_{i}}{2}\right).
\end{alignat}
The representation (\ref{fourierrep}) also holds for the
averaging operator $\E{\AvOp}(x,z)$ since $x \in \Z{\cal H} \subset \E{\cal H}$
with
\begin{equation}
\E{\AvOp}(p) =
\prod \limits_{\mu=1}^{D}
\left(\frac{\sin(p_\mu s \LatSpace_{i}/2)}{s\sin(p_\mu \LatSpace_{i}/2)}
e^{+i p_\mu (s-1) \LatSpace_{i}/2}
\right)
\end{equation}
The fluctuation propagator $\E{\FlucProp}$ on the other hand
is only invariant under translations
$\E{\FlucProp}(z,z')= \E{\FlucProp}(z-x,z'-x)$
with $x \in \Z{\Lat}$ and therefore has the more complicated Fourier
representation
\begin{equation}
\E{\FlucProp}(z,z') =\int\limits_{l,q,l'}
\E{\FlucProp}(l,q,l') e^{i(q+l)z-i(q+l')z'}
\end{equation}
with
\begin{eqnarray}
\E{\FlucProp}(l,q,l')&=&
\E{v}(q+l)\delta_{l,l'} \nonumber\\
& & - \E{v}(q+l)\Adj\E{\AvOp}(q+l) \E{u}^{-1}(q)
\E{\AvOp}(q+l') \E{v}(q+l') \\
\E{u}(q) &=& \int \limits_{l} \E{\AvOp}(q+l)\E{v}(q+l)\Adj\E{\AvOp}(q+l)
\\
\E{v}(p) & :=& \E{\WPotloc}(\overline{\FieldE})^{-1}(p)
\end{eqnarray}
\section{Models whose fields assume discrete values}
Here we wish to explain how normal ordering can help to deal with models whose
fields assume a discrete set of values. Only the main idea will be presented;
detailed numerical studies remain to be done.
Consider as an example the discrete Gaussian Model in two or three dimensions.
It lives on a lattice $\Lat^0$. The field assumes integer
multiples of $2\pi $ as its values.
The starting action is
\begin{equation}
\Action^0 (n) = \frac 1 {2\beta} \int_{z} [ \nabla_{\mu }n (z )]^2 ,
\ \ \ n(z ) \in 2\pi {\bf Z}
\label{discreteGauss}
\end{equation}
In two dimensions, this model is related by a duality transformation to the
plane rotator model (with Villain action \cite{Villain}) which has
a Kosterlitz Thouless phase transition.
In three dimensions it is the dual
transform of a U(1)-lattice gauge theory. The renormalization group flow of
this model is well understood and lead to a rigorous proof that
the three dimensional U(1)-lattice gauge theory shows linear confinement for
arbitrary values of the coupling constant \cite{Confine}.
The overall factor in the action is written as
$1/\beta$ because the duality transformation
interchanges high and low temperatures.
The basic idea is to regard the discrete field as a function of a continuous
field $\phi $. One regards the action as a function of this continuous
field. One normal orders it in a self-consistent way. This furnishes at the
same time a split into a free action which is quadratic in $\phi $
and an interaction. In favorable cases, a good approximation to the
self-consistent split can be guessed a priori.
The normal ordered action is an entire function of the
field and can be expanded in powers of the field components which one wishes
to integrate out.
Let $N(\xi ) $ be the periodic function of $\xi \in {\bf R}$ with period
$2\pi \beta^{-1/2}$
which is defined by
$$
N(\xi ) = \xi \ \ \mbox{for } -\pi \beta^{-1/2}<\xi < \pi \beta^{-1/2}.
$$
We note that $N(\xi )$ is small for all values of $\xi $ if $\beta $ is large.
Now we may write
$$ n(z ) = \beta^{1/2}\left[ \phi (z) - N(\phi (z)) \right]$$
We may substitute an integration over $\phi (z )$ for
the sum over $n(z)$. The action takes the form
\begin{eqnarray}
\Action^0 (\phi )&=&
\frac 1 2 \int_{z} [ \nabla_{\mu }\phi (z )]^2 +
\VPot^0_1(\phi )+\VPot^0_2(\phi ) ,\\
\VPot^0_1 (\phi ) &=& \int_{z}
N(\phi (z))\Delta \phi (z) \\
\VPot^0_2 (\phi ) &=& \frac 1 2 \int_{z}
[\nabla_{\mu }N(\phi (z)]^2 .
\end{eqnarray}
Given the propagator $\Gamma $ with respect to which we want to normal order, the normal ordered form of the action is determined by Gaussian integration
(\ref{defNO}). In the case at hand, the result can be computed by Fourier
transformation. (There are also more direct methods which exploit the fact
that the Lagrangian depends only on the field at two lattice points
\cite{MackPordt}).
The periodic function $N(\varphi )$ of $\varphi \in {\bf R} $ admits a Fourier expansion
\begin{eqnarray}
N(\varphi )&=&\sum_{m \in {\bf Z}\setminus\{0\}} N_m e^{i m\beta^{1/2}
\varphi } , \\
N_m &=& \frac i{m}\beta^{-1/2} (-1)^m .
\end{eqnarray}
We wish to normal order in the fluctuation field
$\zeta $ which is to be integrated
out in one RG-step. We decompose as usual
$ \phi = \Psi + \zeta $,
$\Psi = \Psi ^0 [\Phi ]$.
Now we can write the three terms in the action in normal ordered form.
The quadratic term is trivial and remains quadratic. We use
partial integration and the formula (\ref{qGauss}) for the
characteristic function of a Gaussian measure to compute the normal ordered
form of $V^0_1$.
\begin{equation}
\begin{split}
& \int d\mu_{\Gamma }(\xi ) \VPot^0_1 (\phi + \xi ) =\\
& \int d\mu_{\Gamma }(\xi ) \int_{z} \left(
\Delta \phi (z)
N((\phi + \xi)(z))
+ \int_{w} \Delta \Gamma (w , z ) \frac {\delta}{\delta \xi (z)}
N(( \phi + \xi )(z))\right)\\
& = \sum_{m>0}2 i N_m
\int_{z} e^{-\beta m^2\Gamma (z , z)/2}\\
&
\left(
\Delta \phi (z )
\sin (m \beta^{1/2}\phi (z )) + \beta^{1/2}m \int_w
\Delta \Gamma (w , z ) \cos (m \beta^{1/2}\phi (z ))
\right) \\
& \phi = \Psi + \zeta .\\
\end{split}
\label{V1}
\end{equation}
The second term $V^0_2$ can be treated in a similar fashion. First the
lattice derivatives must be written as differences of two terms.
After that, the Fourier expansions are inserted and the Gaussian integration
can be performed as before. We will not give the somewhat complicated result
in full, but only an approximation to
the resulting effective action which is valid when $\beta $
is large. Although it is not quantitatively correct when $\beta $ is not large,
it will suffice to discuss qualitative features for orientation.
The $V_2^0$-term is small compared to the $V_1^0$-term in this limit.
The crucial aspect of the model is the breaking of the symmetry under
field translations
$$ \phi (z)\mapsto \phi(z) + 2\pi\beta^{-1/2} a. $$
The free action is symmetric under translations with $a \in {\bf R}$, but
the interaction breaks the symmetry down to translations with $a \in {\bf Z}$.
When $\beta $ is large, the expressions
$\exp \left(- \beta m^2 \Gamma(z,z)/2 \right)$ are exponentially
small for $m\not=0$, and the leading symmetry breaking terms are those with
$m=\pm 1$. We neglect to write the others, and also the small
symmetry breaking corrections to the generalized kinetic term .
The effective action $S^1$
after one step is given by eq. (\ref{Normalaction}). It depends
on the normal ordered form of $\Action^0$
at $\zeta = 0$, on the fluctuation propagator and on
the background field as we know. It takes the following form in $d$ dimensions
\begin{eqnarray}
S^1 (\Phi ) &=& \int_{z }
\left( \frac 12 [ \nabla \Psi (z ) ]^2 + \lambda (z )
\cos ( \beta^{1/2} \Psi (z )) - \frac 12\ln \Gamma (z ,z)
+ ... \right) \\
{\lambda }(z ) &=&
2e^{ - \beta \Gamma(z, z)/2} \int_w\Delta\Gamma(w,z).\\
\Psi &=& \Psi ^0[\Phi] \\
\Gamma &=& \Gamma^0 [\Psi ]. \label{effGauss}
\end{eqnarray}
For a translation invariant propagator $\Gamma $, $\lambda (z )$ would be
constant. The propagator depends on the dimension $d$.
Below
we will briefly discuss some qualitative implications on the basis of this
approximate expression,
neglecting a possible $\Psi $-dependence of $\Gamma $.
\footnote{The standard free $\Psi $-independent fluctuation propagator
of Kupiainen and Gawedzki \cite{GawKup}
gives a good a priori guess for the self-consistent split in this
particular model.}
Quantitative calculations could start from the exact
normal ordered expression for $\Action^0$.
The exact 1-loop effective action after one step is still given by
eq. (\ref{Normalaction}), which can be differentiated. A localization approximation
is made after the first RG-step, and the calculation proceeds in the usual way
from there on.
The fluctuation propagator depends on an infrared cutoff
$M=a_1^{-1} $. For orientation, we may imagine that the cutoff is
lowered by as much as we please in one step; we discuss what happens if
$M\mapsto 0$.
In $d=3$ dimensions, the propagator $\Gamma $ has a limit
when $M\mapsto 0$. Therefore, also ${\lambda }$ has a limit. However, to
judge the RG-flow, we need to rescale to the unit lattice. Therefore we need
to consider the dimensionless quantity $a_1^{d}{\lambda }$. It goes to
infinity. Therefore the cosine shaped potential wells become infinitely
high in the infrared limit, and there will always be spontaneous breaking
of the ${\bf Z}$-symmetry with a nonvanishing surface tension.
There is also a curvature at the minimum which produces a
mass. In the dual picture, the surface tension becomes the string tension
of the $U(1)$-lattice gauge theory.
In $d=2$ dimensions, $\Gamma $ diverges logarithmically when $M\mapsto 0$.
The asymptotic behavior of $a_1^{d}{\lambda }$ as $M\mapsto 0 $ depends then
on $\beta $. There will be a critical value $\beta_c$.
Its precise value depends on details which were neglected in (\ref{effGauss}).
For $\beta < \beta_c$, $a_1^2 {\lambda }$ tends to infinity, and for $\beta
>\beta_c $ it tends to zero. This corresponds to the two phases
of the plane rotator which are
separated by the Kosterless-Thouless phase transition. The
${\bf Z}$-symmetry is either spontaneously broken or enhanced to ${\bf R}$.
This phenomenon of symmetry enhancement was discovered long ago by Fr\"ohlich
and Spencer \cite{FS}.
It might also be of interest to consider the dual transform of the
4-dimensional $U(1)$-lattice gauge theory which is a
${\bf Z}$-gauge theory. In principle it could be treated with our method.
Discussions on how to deal with gauge theories are found in
\cite{balabanGauge,balabanJaffe,KMP}.
\section{Renormalization group flow on the lattice}
\label{introSection}
Suppose we start from the Euclidean action $\Action^0[\Field^0]$ of a
field theory
which lives on some lattice $\Lat^0 $ or on the continuum. It depends on a
field $\Field^0 $ on $\Lat^0$. We wish to compute a sequence of effective
actions $\E{\Action}[\E{\phi}]$ which live on lattices
$\E{\Lat}$ of increasing lattice spacing $\LatSpace_{i} $, i=0,1,2,... .
In principle the sequence of actions is defined once we fix a block-spin
definition. The block-spin definition determines a field
$\Phi=\Z{\phi}$
on the coarser lattice $\Z{\Lat}$ as some kind of average
\begin{equation*}
\Phi = \E{\AvOp} \phi
\end{equation*}
of the field $\phi=\E{\phi}$ on the lattice $\E{\Lat}$. The actions are
defined recursively by
\begin{equation}
\E Z[\Phi] = e^{-\Z{\Action}[\Phi]}
= \int D\phi \,\delta (\E{\AvOp} \phi - \Phi)
e^{-\E{\Action}[\phi]}. \label{defEffective}
\end{equation}
For given $\Phi$, $\E Z[\Phi]$ is the partition function of an
{\em auxiliary theory} in which only those variables are
integrated out which are to be interpreted later as
high frequency modes of the field.
We will also need certain expectation
values of this auxiliary theory.
If $A=A [\phi ]$ is an observable, we set
\begin{equation}
\meanE{A} = \E Z[\Phi ]^{-1}
\int D\phi A [\phi ] \delta (\E{\AvOp} \phi - \Phi)
e^{-\E{\Action}[\phi]}. \label{defExpVal}
\end{equation}
These expectation values depend on $\Phi$, but we neglect to indicate this
dependence explicitly.
In the calculations, two auxiliary $\Phi$-dependent quantities
will play
an important role, the {\em background field} $\E{\Psi} [\Phi ]$ and
the {\em fluctuation propagator} $\E{\FlucProp} [\E{\Psi} ]$.
The fluctuation propagator is
considered to depend on the block-spin $\Phi$ through the background field
$\E{\Psi} = \E{\Psi} [\Phi ]$.
Both auxiliary quantities are defined as expectation values of the
auxiliary theory.
\begin{eqnarray}
\E{\Psi}(z) &\equiv & \E{\Psi} [\Phi ](z ) =
\meanE{\phi (z )}, \label{MFAFieldDef} \\
\E{\FlucProp} [\E{\Psi} ](z , w ) &=&
\meanE{ \phi (z )\phi (w )}
- \meanE{\phi (z )}\meanE{\phi (w )}. \label{FlucPropDefIntro}
\end{eqnarray}
These definitions differ from those of Gawedzki and Kupiainen, cp.
\cite{GawKup,MackCargese}. In their work, expectation values of a free theory
were used. The present approach is more in the spirit of Balaban's work on
gauge theories, where expansions of the full action around its minimum
were used for reasons of gauge invariance \cite{balabanGauge}.
The validity of the method requires that the auxiliary theory has an
infrared cutoff of order $a_{i+1}^{-1}$. Therefore the fluctuation
propagator must decay exponentially with decay length no larger than about
$a_{i+1}$. When this condition is violated, it signals a bad choice
of the block-spin definition.
We are not able to calculate the functional integrals
(\ref{defEffective},\ref{defExpVal})
exactly. Therefore approximations are necessary.
The calculation is done in a self-consistent way. Both
the background field and the fluctuation propagator are used in the
calculation. Conditions will be imposed on them which ensure that
equations (\ref{MFAFieldDef}) and (\ref{FlucPropDefIntro}) are
fulfilled within the accuracy of the approximation.
We discuss in subsection \ref{localApproximation} how a truncation of
the effective action
to a manageable form can be justified. No expansion in powers of fields
is involved.
It follows from the definition
(\ref{FlucPropDefIntro}) that the fluctuation propagator satisfies the
constraints
\begin{equation}
\E{\AvOp} \E{\FlucProp} [\E{\Psi}] = 0 = \E{\FlucProp} [\E{\Psi} ] \Adj{\E{\AvOp}}.
\label{GammaConstraint0}
\end{equation}
The simplest choice of block-spin for a theory of scalar fields is as follows.
One identifies the sites $x$ of the coarser lattice $\Z{\Lat}$ with disjoint
hypercubes in the lattice $\E{\Lat}$, and one chooses the block-spin as
the average of the original field on these hypercubes
\begin{equation*}
\Phi(x ) = \int_{z }\E{\AvOp} (x ,z )\phi(z) =
\av{z \in x} \phi (z ) .
\end{equation*}
One can try to improve on the locality properties of the
effective actions by modifying the block-spin procedure \cite{Hasenfratz}.
We may imagine starting from a lattice of finite volume.
After a finite number of steps we arrive at a lattice $\Lat^N$ which
consists of only a single site. The field $\Phi $
on this lattice is some
average of an average ... of an average of the original field $\Field^0$.
Let us interpret it as magnetization. The action
$\Action^N[\Phi]$ will give us the constraint effective potential
-- i.e. the free energy --
as a function of the magnetization.
We will make the following assumptions on the choice of block-spin in this
paper.
We assume that $\E{\AvOp} $ is a linear map, and that $\E{\AvOp} \Adj{\E{\AvOp}} $
is invertible. (For gauge theories, linearity will have to be given up.)
We assume also that $\Adj{\E{\AvOp}} $ interpolates constant fields to constant
fields, and we impose the following normalization condition
\begin{equation*}
\int_{x\in \Z{\Lat}} \E{\AvOp} {\phi} (x ) = \int_{z\in \E{\Lat}}
\phi (z )
\end{equation*}
for constant fields. For the above mentioned choice of block-spin,
the equation is an identity for arbitrary fields $\phi $.
Let ${\cal H}^i$ be the Hilbert space of square summable functions on $\E{\Lat}$,
and let ${\cal H}^i_{{\cal{C}} } $
be the subspace of functions $\E{\FlucField} $
with vanishing block average $\E{\AvOp} \E{\FlucField} $ = 0.
Invertibility of $\E{\AvOp} \Adj{\E{\AvOp}} $ ensures the existence of the
orthogonal decomposition
$$
\E{\cal H} = \Adj {\E{\widetilde{\AvOp}}} \Z{{\cal H}} \oplus {\cal H}^i_{{\cal{C}}},
$$
where $\E{\widetilde{\AvOp}}=(\E{\AvOp}\Adj{\E{\AvOp}})^{-1}\E{\AvOp}$.
This is true because every $\phi \in \E{{\cal H} }$ can be decomposed as
$$
\phi = \Adj{\E{\widetilde{\AvOp}}}\Phi+\E{\FlucField}
$$
with $\Phi=\E{\AvOp}\phi\in\Z{\cal H}$ and
$\E{\FlucField}=\phi-\Adj{\E{\widetilde{\AvOp}}}\Phi$.
The first summand is obviously in $\Adj {\E{\widetilde{\AvOp}}} \Z{{\cal H}}$ and
the second is in ${\cal H}^i_{{\cal{C}}}$ because it vanishes when we apply $\E{\AvOp}$:
$
\E{\AvOp} \E{\FlucField} = \E{\AvOp} \phi - \E{\AvOp} \Adj {\E{\widetilde{\AvOp}}} \Phi
= \Phi - \Phi = 0.
$
Also the orthogonal decomposition is true
because the scalar product vanishes:
$
(\E{\FlucField},\Adj {\E{\widetilde{\AvOp}}} \Phi) =
((\E{\AvOp}\Adj{\E{\AvOp}})^{-1}\E{\AvOp}\E{\FlucField},\Phi)
= 0.
$
{\em Notation:} We will use letters $z , w ...$ for sites in
$\E{\Lat} $ and $x , y ... $ for sites in $\Z{\Lat}$.
\section{Improved saddle point method with normal ordering}
\label{FBapproximation}
\subsection{Advanced method with normal ordering}
The localization approximation becomes more and more accurate the larger
the scaling factor (ratio of lattice spacings) $s=\LatSpace_{i+1}/\LatSpace_{i} $.
On the other hand, the saddle point approximation becomes exact in the limit
$s\mapsto 1 $ (when the nature of the cutoff permits such a limit). It becomes
less accurate with increasing scaling factor because the phase space for
high frequency modes increases. This is one reason why we choose to consider
a fairly sophisticated improvement of the saddle point method.
The self consistent improvement of the saddle point
integration consists of two steps.
In the first step, the old action $\E{\Action} [\phi ]=
\E{\Action} [\E{\Psi} + \E{\FlucField} ]$ is self-consistently normal ordered in
the fluctuation field $\E{\FlucField}$; at the same time the background field
$\E{\Psi}=\E{\Psi}[\Phi]$ is determined. Symbolically
\begin{equation}
\E{\Action} [\phi ] = :\E{\NAction} [\E{\Psi} , \E{\FlucField} ]:
\label{NOSymb}
\end{equation}
The precise definition and properties of the normal ordered form $\E{\NAction} $
of the action $\E{\Action}$ will be described presently.
In the second step, the fluctuation field integration is performed, neglecting
normal products of higher order than second in the fluctuation field.
The integral is again Gaussian, and the result reads
\begin{equation}
\Z{\Action} [\Phi ] = \E{\NAction} [\E{\Psi} , 0 ]
-\frac 12 \text{tr} \ln \E{\FlucProp} [\E{\Psi} ] , \label{Normalaction}
\end{equation}
The trace is again over the space ${\cal H}^i_{{\cal{C}}}$
of functions $\E{\FlucField} $ on
$\E{\Lat} $ with vanishing block average, $\E{\AvOp} \E{\FlucField} =0$.
The method of self-consistent normal ordering
is very old - see \cite{Ruehl}; sometimes it is called
the Feynman Bogoliubov method. Therefore we will refer to our method here
as Feynman Bogoliubov approximation. The definition and properties of the
normal ordered action $\E{\NAction} $ are as follows.
Let $d\mu_{\Gamma }$ be the normalized Gaussian measure (free field measure)
with covariance (propagator) $\Gamma $.
Given the
fluctuation propagator $\E{\FlucProp} $,
the normal ordered amplitude $\E{\NAction} $ is defined as
\begin{eqnarray}
\E{\NAction} [\E{\Psi} , \E{\FlucField} ] &=&
\int d\mu_{\E{\FlucProp} [\E{\Psi}] }(\xi )
S [\E{\Psi} + \E{\FlucField} + \xi ]
\label{defNO}\\
&=&
\exp \left( \frac 12 \scalar{\frac{\delta}{\delta \xi}}
{\E{\FlucProp} [\E{\Psi} ] \frac{\delta}{\delta \xi}}
\right)
S [\E{\Psi} + \E{\FlucField} + \xi ]\Einsch{\xi = 0}\label{pregap0}
\end{eqnarray}
The second formula is valid when $\E{\Action}$ is differentiable. The first
formula is more general, it can be used to define a normal product expansion
also for discontinuous actions $\E{\Action}$ (cp. the section \ref{DS}).
We use the following notation for functionals with two arguments:
$$
\frac{\delta }{ \delta \E{\Psi} (z)} A[\E{\Psi},\E{\FlucField}]
= A_{,z}[\E{\Psi},\E{\FlucField}]\,,
\qquad \frac{\delta }{ \delta \E{\FlucField} (z)} A[\E{\Psi},\E{\FlucField}]
= A_{;z}[\E{\Psi},\E{\FlucField}]\,.
$$
The notation is a little sloppy. Because $\E{\AvOp} \E{\FlucField} =0$, the
second expression only makes sense after smearing with a test function $f$
which obeys $\E{\AvOp} f = 0$, viz.
$$ \int_z f(z) \frac{\delta }{ \delta \E{\FlucField} (z)}
A[\E{\Psi},\E{\FlucField}] = \frac {d}{d\tau }
A[\E{\Psi},\E{\FlucField} + \tau f]\Einsch{\tau =0}.
$$
The background field $\E{\Psi} = \E{\Psi} [\Phi ] $ will be determined
by the condition that the expansion of the action $\E{\NAction} $
in powers of the fluctuation field has no linear
term:
\begin{equation}
\E{\NAction} [\E{\Psi} , \E{\FlucField} ]
= T [\E{\Psi} , 0 ]
+ \frac 12 \scalar{\E{\FlucField}}
{{\NActionE_{;;}} [\E{\Psi} , 0 ]\E{\FlucField}}
+ ... \qquad \mbox{ for} \ \E{\AvOp} \E{\FlucField} = 0. \label{pregap1}
\end{equation}
That is the saddle point condition
\begin{equation}
\int_{z}\E{\FlucField} (z) \E{\NAction}_{;z} [\E{\Psi} ] = 0
\ \ \mbox{if} \ \E{\AvOp} \E{\FlucField} = 0 \label{saddleN}
\end{equation}
which is equivalent to
\begin{eqnarray}
{\NActionE_;} [\E{\Psi}] &=& \Adj {\E{\AvOp} }\E\lambda , \label{MFAFieldEq1N} \\
\E{\AvOp} \E{\Psi} &=& \Phi \ . \label{MFAFieldEq2N}
\end{eqnarray}
$\E\lambda[\Phi] (x ) $ are Lagrange multipliers, $x\in \Z{\Lat} $.
Now we specify the fluctuation propagator $\E{\FlucProp}$ as pseudo inverse
of ${\NActionE_{;;}}$ so that
\begin{eqnarray}
\E{\FlucProp} [\E{\Psi} ]{\NActionE_{;;}} [\E{\Psi} ]
\E{\FlucProp} [\E{\Psi} ]&=& \E{\FlucProp} [\E{\Psi} ].\label{gap}\\
\E{\AvOp} \E{\FlucProp} [\E{\Psi} ] = \E{\FlucProp} [\E{\Psi} ]\Adj{\E{\AvOp}} &=&0.
\end{eqnarray}
As in the saddle point approximation the fluctuation propagator satisfies
the constraints (\ref{GammaConstraint0}), is positive definite on
${\cal H}^i_{{\cal{C}}}$, and equal to the inverse of ${\NActionE_{;;}}$ on ${\cal H}^i_{{\cal{C}}}$
But ${\NActionE_{;;}} $ depends itself on the propagator $\E{\FlucProp}$. Therefore
(\ref{gap}) leads to a gap equation for $\E{\FlucProp}$.
One sees that the background field $\E{\Psi} $ is the minimum of the
normal ordered form $\E{\NAction} $ of the action $\E{\Action} $ on the surface with
prescribed block-spin $\Phi = \E{\AvOp} \phi $, and
$\E{\FlucProp} $ is the inverse of the Hessian ${\NActionE_{;;}} $ at
this minimum; it depends on $\E{\Psi} $. (The Hessian is a quadratic form
on the tangent space ${\cal H}^i_{{\cal{C}}} $ to the surface.)
The determination of the fluctuation field propagator $\E{\FlucProp} $ involves
solving a gap equation. The gap equation is obtained by differentiating
eq. (\ref{pregap0}) twice at $\E{\FlucField} = 0$.
\begin{eqnarray}
{\NActionE_{;;}} &=&
\expdiff {\E{\Action}}{}'' [\E{\Psi}+\xi]\Einsch{\xi=0} \label{gapeqn}\\
\E{\FlucProp} [\E{\Psi} ] &=& {\NActionE_{;;}} [\E{\Psi} , 0 ]^{-1} \ \mbox{on }
{\cal H}^i_{{\cal{C}}} .
\end{eqnarray}
The gap equation is shown in graphical form in figure \ref{BildI}.
Figure \ref{BildAb} explains the graphical notation.
Further derivatives can be applied and we
obtain a generalization of eq.(\ref{gapeqn}) as
shown in figure \ref{BildB}.
\BildAb{Graphical notation for normal ordered amplitudes}
\BildB{Definition of the normal ordered amplitudes. Further legs
generated by $\Psi $-derivatives will be marked by primes}
\BildI{Gap equation for the fluctuation propagator $\E{\FlucProp}$ in
Feynman-Bogoliubov approximation}
This completes the discussion of the improved saddle point approximation.
Eq.(\ref{Normalaction}) is still a formidable functional recursion formula.
We need to make it practical by our second approximation of localization.
The localization approximation has to be made for the second derivative
${\NActionE_{;;}} $ and not for ${\E{\Action}}{}''$.
In addition to the formula for the second derivative ${\E{\Action}}{}''$
there is a formula which expresses
the first derivative $ {\E{\Action}}{}'$ of the ordinary action in terms of the
first derivative ${\NActionE_;} $ of the previous normal ordered action; this
furnishes the potential up to an additive constant.
We have a similar formula for the derivative of the background field:
\begin{equation}
\E{\Psi}_{,x }(z ) =
(1 - \E{\FlucPropVar} {\NActionE{}_{\mathrm{;\,,}}} )\Adj{ \tilde \E{\AvOp}}(z,x).
\end{equation}
where $\E{\FlucPropVar}$ is defined as inverse of ${\NActionE{}_{\mathrm{;\,,}}}$ on ${\cal H}^i_{{\cal{C}}}$.
In contrast to $\E{\FlucProp}$ it is defined as inverse of the mixed
derivative ${\NActionE{}_{\mathrm{;\,,}}}$ which involves derivatives with respect to
the fluctuation field and the background field.
{\em Remark:} It can be shown with some effort that the two propagators are
actually equal, i.e. $\Gamma_B [\E{\Psi} ] = \E{\FlucProp} [\Psi ]$
when $\E{\Psi} $ is the background field. See appendix \ref{derRelSimple}.
\subsection{Gaussian integration}
We wish to evaluate the integral (\ref{defEffective}) for $\Z{\Action} $, using the
quadratic approximation (\ref{NOSymb},\ref{pregap1}) for $\E{\Action} $, viz.
$$
\E{\Action} [\E{\Psi} + \E{\FlucField} ]
= \E{\NAction} [\E{\Psi} , 0 ]
+ \frac 12 :\scalar{\E{\FlucField}} {{\NActionE_{;;}}[\E{\Psi},0]\E{\FlucField}}:
$$
We undo the normal ordering again. Dropping arguments $\xi=0$ we obtain
\begin{equation}
\E{\Action} [\E{\Psi} + \E{\FlucField} ]
= \E{\NAction} [\E{\Psi} ]
+ \frac 12 \scalar{\E{\FlucField}} {{\NActionE_{;;}} [\E{\Psi}]\E{\FlucField}}
- \frac 12 \text{tr} {\NActionE_{;;}} [\E{\Psi} ]
\E{\FlucProp} [\E{\Psi} ] . \label{newActionTemp}
\end{equation}
The trace is over ${\cal H}^i_{{\cal{C}}}$. By arguments similar to appendix \ref{GammaFormAp}
the fluctuation propagator has the explicit form
\begin{equation}
\E{\FlucProp} = {{\NActionE_{;;}}}^{-1} -
{{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}}(\E{\AvOp}{{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}})^{-1}
\E{\AvOp}{{\NActionE_{;;}}}^{-1}\,.
\label{GammaExplicit}
\end{equation}
As in the saddle point approximation
there is an alternative formula for the fluctuation propagator
\begin{equation}
\E{\FlucProp}[\E{\Psi} [\Phi ] ] = \lim_{\kappa \rightarrow \infty }\left(
{\NActionE_{;;}} [\E{\Psi} [\Phi ]] + \kappa \Adj{\E{\AvOp}}\E{\AvOp}
\right)^{-1}\,.
\end{equation}
We argue that the last term in (\ref{newActionTemp}) is a constant
independent of $\E{\Psi} $. Multiplying ${\NActionE_{;;}}$ from the right of
(\ref{GammaExplicit}) gives
\begin{equation}
\E{\FlucProp}{\NActionE_{;;}} = 1 -
{{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}}(\E{\AvOp}{{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}})^{-1}
\E{\AvOp} \,.
\label{GammaMalT}
\end{equation}
We take the trace of (\ref{GammaMalT})
\begin{equation*}
\begin{split}
\text{tr}(\E{\FlucProp}{\NActionE_{;;}}) &=\ \text{tr} 1
- \text{tr}\Bigl({{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}}
(\E{\AvOp}{{\NActionE_{;;}}}^{-1}\Adj{\E{\AvOp}})^{-1} \E{\AvOp}\Bigr) \\
&=\ \text{tr}\ \delta_{z}(0) - \text{tr}\ \delta_{x}(0) \\
&=\ \sum\limits_{z} 1 - \sum\limits_{x} 1 = \text{const.}
\qquad\qquad \text{q.e.d.} \\
\end{split}
\end{equation*}
Now we can again evaluate the functional integral (\ref{defEffective}) in Gaussian
approximation with the result
\begin{equation}
\Z{\Action} [\Phi ] = \E{\NAction} [\E{\Psi}[\Phi] , 0 ]
-\frac 12 \text{tr} \ln \E{\FlucProp} [\E{\Psi}[\Phi] ] .
\label{Normalaction1}
\end{equation}
\subsection{Relations between derivatives of the normal ordered action}
\label{derRel}
We derive a relation between the
$\E{\Psi}$- and $\xi $-derivative by using the following well known change
of covariance lemma for Gaussian measures \cite{GlimmJaffe}.
If the covariance $\Gamma_s$ depends parametrically on a variable $s$,
then
\begin{equation}
\frac {d}{ds} \int d\mu_{\Gamma_s }(\phi ) A[\phi ] =
\frac 12 \int d\mu_{\Gamma_s }(\phi )
\scalar{\frac{\delta}{\delta \phi }}
{\dot{\Gamma}_s \frac{\delta}{\delta\phi}A[\phi]}\,,
\label{ChangeCov}
\end{equation}
where $\dot{\Gamma}_s = d\Gamma_s /ds $.
We denote $\E{\Psi}$-derivatives by commas ``,'' and $\xi $-derivatives
by semicolons ``;'' as before.
Applying the change of covariance lemma to the definition (\ref{defNO})
of the normal ordered amplitude we obtain the relation
\begin{equation}
\E{\NAction}_{,z }[\E{\Psi} ] = \E{\NAction}_{;z }[\E{\Psi} ] + \frac 12
\text{tr} \Gamma{}_{, z }[\E{\Psi} ] {\NActionE_{;;}} [\E{\Psi} ].
\label{;to,}
\end{equation}
After another $\xi$-derivative we get
\begin{equation}
\E{\NAction}_{,z\,;w }[\E{\Psi} ] = \E{\NAction}_{;zw }[\E{\Psi} ]
+ \frac 12
\text{tr} \Gamma{}_{, z }[\E{\Psi} ] {\NActionE_{;;}}{}_{;w } [\E{\Psi} ].
\label{mixedReplacement}
\end{equation}
\subsection{Recursion relation for the first and second derivatives of the
action}
We differentiate the recursion relation (\ref{Normalaction1})
to obtain a formula
for the first derivative of the action $\Z{\Action}$. The derivative of the
$\text{tr} \ln $ is computed in the same way as before. The result reads
\begin{equation}
\Z{\Action}_{,x} = \int_{z} \E{\Psi}_{,x }(z ) \left(
\E{\NAction}_{,z} [\E{\Psi} ] -
\frac 12 \text{tr} \E{\FlucProp}_{,z} [\E{\Psi} ]
{\NActionE_{;;}}\right) .
\end{equation}
We re-express the $\E{\Psi}$-derivative in the first term using
eq.~(\ref{;to,}). A cancelation occurs and we
obtain the
final form of the recursion relation for the first derivative of the action.
\begin{equation}
\Z{\Action}_{,x} = \int_{z} \E{\Psi}_{,x }(z )
\E{\NAction}_{;z } [\E{\Psi}].
\label{actionP}
\end{equation}
Next we turn to the second derivative. We differentiate eq.~(\ref{actionP})
once more. Since $ \E{\NAction}_{;z }$ is the $\xi$-derivative
of the normal
ordered action we can use eq.~(\ref{;to,}) again to evaluate the derivative
of $\E{\NAction}_{;z}$ with the help of
\begin{equation}
\E{\FlucProp}_{,z}[\E{\Psi} ]
= - \E{\FlucProp}[\E{\Psi} ] {\NActionE_{;;}}{}_{\,, z }[\E{\Psi} ]
\E{\FlucProp}[\E{\Psi} ].
\end{equation}
Differentiating the factor $\E{\Psi}_{,x }(z ) $
produces a term
\begin{equation}
\int_{z}\E{\Psi}_{,x y }(z )T_{;z} \label{1PI}.
\end{equation}
Differentiating the constraint
(\ref{MFAFieldEq2N}) twice shows that $\E{\Psi}_{,xy}\in {\cal H}^i_{{\cal{C}}}$ because
\begin{equation}
\int\limits_{z} \E{\AvOp}(x',z) \E{\Psi}_{,x}(z)
= \delta(x'-x)\,, \label{constr}
\end{equation}
implies
\begin{equation}
\int\limits_{z} \E{\AvOp}(x',z) \E{\Psi}_{,xy}(z) = 0\,.
\label{MFAFieldPPinHCE}
\end{equation}
The saddle point condition (\ref{saddleN}) implies therefore that expression
(\ref{1PI}) vanishes. As a result we obtain the recursion formula for
the second derivative of the action
\begin{eqnarray}
\Z{\Action}_{,x y} &=&
\int_{z,w} \E{\Psi}_{,x }(z )\E{\NAction}_{;z\,,w}
\E{\Psi}_{,y}(w) \\
&=&
\int_{z,w} \E{\Psi}_{,x }(z )
\left(
\E{\NAction}_{;z w} - \frac 12 \, \text{tr}
(\E{\FlucProp} {\NActionE_{;;}}{}_{\,;z } \E{\FlucProp} {\NActionE_{;;}}{}_{\,,w})
\right) \E{\Psi}_{,y}(w)
\label{actionPP}
\end{eqnarray}
We neglected to indicate arguments $\E{\Psi} $ of propagators and normal
ordered action. We see that there is no 1-particle reducible contribution.
A subtle feature of this formula is the appearance of two different
vertices in the last term; one involves a $\E{\Psi}$-derivative and
the other a $\xi$-derivative to create the external leg. This feature is
familiar from Schwinger Dyson equations, cp. section \ref{DS}
below. Neglecting
the difference would only neglect a two loop correction, though.
The recursion relations in Feynman Bogoliubov approximation are shown in
figures \ref{BildN} and \ref{BildH}.
They involve propagators which are to be determined as
solutions of the gap equation (\ref{gapeqn}), shown graphically in figure
\ref{BildI}.
\BildN{Recursion relation for ${\Z{\Action}}{}' $ in the Feynman Bogoliubov
approximation}
\BildH{Recursion relations for the ${\Z{\Action}}{}''$ in the Feynman Bogoliubov
approximation. The second formula is obtained from the first with the help
of eq.~(\ref{mixedReplacement}).}
\subsection{Derivative of the background field}
The recursion relations involve the derivative $\E{\Psi}_{,x}$ of the
background field. Differentiating the saddle point condition
(\ref{saddleN}) we obtain
\begin{equation}
\int_{z,w}\E{\FlucField} (w)\E{\Psi}_{,x }(z )
\E{\NAction}_{;w,z} [\E{\Psi} ] = 0 \label{saddleConditionDerivative}
\end{equation}
We claim that this equation and the constraint (\ref{constr}) are solved by
\begin{equation}
\E{\Psi}_{,x }(z ) = \Adj{ \tilde \E{\AvOp}}(z , x )
- \int_{w, u} \E{\FlucPropVar}(z , w)
\E{\NAction}_{;w\,,u} \Adj{ \tilde \E{\AvOp}}
(u , x ), \label{backDerN}
\end{equation}
where $\Adj {\tilde \E{\AvOp}} =\Adj{\E{\AvOp}} (\E{\AvOp} \Adj{\E{\AvOp} })^{-1}$
as before and $\E{\FlucPropVar}$ is defined as the inverse of ${\NActionE{}_{\mathrm{;\,,}}}$
on ${\cal H}^i_{{\cal{C}}}$. The proof proceeds as in the saddle point section.
The remark at the end of the last subsection leads to a simplification
of (\ref{backDerN})
because $\E{\FlucProp} $ can be substituted for $\E{\FlucPropVar}$.
See appendix \ref{derRelSimple}.
\subsection{Discussion of the localization approximation}
There are new aspects of the localization approximation which need to be
discussed.
The consideration of this section assumed
that the starting amplitude $\E{\Action}$ is accurately known.
Because of this, it is appropriate to regard the recursion relations as
recursion relations for the normal ordered amplitudes $\E{\NAction}$,
and to make the
localization approximation for these amplitudes,
{\em not} for $\E{\Action}$. The localization approximation for
$\E{\NAction}$ is justified by
appealing to the same arguments as before:
In computing $\Z{\Action}$ we only need ${\NActionE_{;;}} $ for smooth fields.
Thus
we may regard one step in the recursion relations as composed of three parts
\begin{enumerate}
\item Localization approximation on $\E{\NAction} $ for smooth fields .
\item Computation of $\Z{\Action} $ from $\E{\NAction} $ for smooth fields
\item computation of
$\Z{\NAction} $ from $\Z{\Action} $
\end{enumerate}
When proceeding in this way, the result for $\Z{\Action}$ must
be substituted into the (gap) equations which determine the normal ordering.
In general there
result graphs which contain propagators from two successive length scales
$i$ and $i+1$. This complication does not arise when only 1-loop graphs are
retained.
There is one further technical complication. In contrast with
${\E{\Action}}{}''$,
${\NActionE_{;;}}$ is not necessarily exactly invariant
under arbitrary translations by
lattice vectors in $ \E{\Lat} $ but only under block
lattice translations.
Therefore it it is not easy to
calculate its pseudoinverse (the fluctuation field propagator) exactly.
One can proceed as follows, however.
Because ${\NActionE_{;;}}$ and $\E{\FlucProp}$ are determined by a gap equation
one can start with some guess for ${\NActionE_{;;}}$, e.g. with
the pseudoinverse of ${\E{\Action}}{}''$ which was used in the saddle point
approximation.
Then one iterates the gap equation.
The gap equations are here considered for reasons of accuracy
and not because there
are infrared problems. Quite on the contrary, the propagators must have
exponential decay. Therefore the iteration can be expected to converge very
fast.
\section{Numerical results}
\subsection{Getting started}
We want to calculate the constraint effective potential on the lattice
numerically.
We consider an example.
For starting action we take the standard $\varphi^4$-theory in two
dimensions,
\begin{equation}
\Action^0(\Field^0) = \,-\!\!\!\int\limits_{z,w\in\Lat^0}
\frac 12\Field^0(z)\Delta(z,w)\Field^0(w)
\ +\!\! \int\limits_{z\in\Lat^0}\Bigl( \frac 12 m_0^2 \Field^0(z)^2
+ \frac {\lambda_0} {4!} \Field^0(z)^4 \Bigr) .
\label{startAction}
\end{equation}
To calculate $\overline{W}{}^1[\Phi]$ and $\overline{V}{}^1[\Phi]$ we need
$\WPotloc^0[\Field^0]$, $\WPotloc^{0\prime}[\Field^0]$, $\WPotloc^{0\prime \prime}[\Field^0]$,
$\VPotloc^{0\prime}[\Field^0]$ and $\FlucProp^0[\Field^0]$ for constant fields
$\Field^0=\overline{\FieldE}$.
We write $\WPot^0$ (see also (\ref{startingAction})) in a symmetric form:
\begin{eqnarray}
\WPot^0(z_1,z_2|\Field^0) &=& \Delta(z_1,z_2)
-(m_0^2 +\frac{\lambda_0}2 \Field^0(z_1)^2)\delta(z_1-z_2) \\
&=& \Delta(z_1,z_2)
-\Bigl(m_0^2 +\frac{\lambda_0}4 (\Field^0(z_1)^2+\Field^0(z_2)^2)\Bigr)
\delta(z_1-z_2)
\end{eqnarray}
Therefore we have for $\WPotloc^0$
\begin{equation}
\WPotloc^0 (z_1,z_2|\xi) = \Delta(z_1,z_2)
-\Bigl(m_0^2 +\frac{\lambda_0}2 \xi^2 \Bigr) \delta(z_1-z_2)
\end{equation}
which is not an approximation because equation (\ref{Wloc}) is {\em exact} in
this case. Now it follows for the constant field $\overline{\FieldE}$
\begin{eqnarray}
\WPotloc^0 (z_1,z_2|\overline{\FieldE}) &=& \Delta(z_1,z_2)
-\Bigl(m_0^2 +\frac{\lambda_0}2 \overline{\FieldE}^2 \Bigr) \delta(z_1-z_2) \\
\WPotloc^{0\prime}(z_1,z_2|\overline{\FieldE}) &=& -\lambda \overline{\FieldE}\delta(z_1-z_2) \\
\WPotloc^{0\prime \prime}(z_1,z_2|\overline{\FieldE}) &=& -\lambda \delta(z_1-z_2)
\end{eqnarray}
Similarly, the potential
\begin{equation}
\VPotloc^{0\prime}(\overline{\FieldE}) \, = \, \VPot^0_{,z}[\Field^0]
\Einsch{\Field^0=\overline{\FieldE}}
\, = \, m_0^2\overline{\FieldE} +\frac\lambda{3!} \overline{\FieldE}^3
\end{equation}
\subsection{Relations in momentum space}
We have fixed the starting point. Next we want to calculate the recursion relations for
$\E{\WPotloc}$ and $\E{\VPotloc}$. To reduce the amount of work we switch to momentum space.
Because we insert constant fields we have translation symmetry. Therefore the
Fourier
transformation looks quite simple. For the notation see appendix \ref{GammaLattice}.
\begin{eqnarray}
\E{\WPotloc}(z_1,z_2|\overline{\FieldE}) &=& \int\limits_{p} \E{\WPotloc}(p|\overline{\FieldE})
e^{ip(z_1-z_2)} \\
\E{\Psi}_{,x}(z|\overline{\FieldE}) &=& \int\limits_{p} \E{\Psi}(p|\overline{\FieldE})
e^{ip(z-x)} \\
\E{\FlucProp}(z_1,z_2|\overline{\FieldE}) &=& \int\limits_{l,q,l'}
\E{\FlucProp}(l,q,l'|\overline{\FieldE}) e^{i((q+l)z_1-(q+l')z_2)}
\label{flucmorep}
\end{eqnarray}
and similar for the derivatives, which are now simple derivatives with respect to $\overline{\FieldE}$.
$\E{\FlucProp}$ has only translation symmetry on the block lattice. Therefore it is not
diagonal in momentum space (see appendix \ref{GammaLattice}).
Transforming the recursion relations into momentum space leads to
\begin{multline}
\Z{\WPotloc}(q|\overline{\FieldE}) = \int\limits_{l}
\Adj{{\E{\Psi}}{}'}(q+l|\overline{\FieldE})
\E{\WPotloc}(q+l|\overline{\FieldE}){\E{\Psi}}{}'(q+l|\overline{\FieldE}) \\
\begin{split}
& + \frac18\int\limits_{q',l_1,\ldots,l_4}
\E{\FlucProp}(l_1,q',l_2|\overline{\FieldE})
\E{\FlucProp}(l_3,q+q',l_4|\overline{\FieldE})
\Adj{{\E{\Psi}}{}'}(q-l_2+l_3|\overline{\FieldE})
{\E{\Psi}}{}'(q-l_1+l_4|\overline{\FieldE}) \\
& \quad
\Bigl\{ \E{\WPotloc}{}'(q+q'+l_3|\overline{\FieldE})
+\E{\WPotloc}{}'(q'+l_2|\overline{\FieldE}) \Bigr\}
\Bigl\{ \E{\WPotloc}{}'(q'+l_1|\overline{\FieldE})
+\E{\WPotloc}{}'(q+q'+l_4|\overline{\FieldE}) \Bigr\} \\
& + \frac14\int\limits_{q',l_1,l_2,l_3}
\E{\FlucProp}(l_1,q',l_2|\overline{\FieldE})
\Adj{{\E{\Psi}}{}'}(q-l_2+l_3|\overline{\FieldE})
{\E{\Psi}}{}'(q-l_1+l_3|\overline{\FieldE}) \\[-8pt]
& \qquad\qquad\qquad\quad
\Bigl\{ \E{\WPotloc}{}''(q'+l_1|\overline{\FieldE})
+\E{\WPotloc}{}''(q'+l_2|\overline{\FieldE}) \Bigr\} \\
& + \frac18\int\limits_{q',l_1,\ldots,l_4}
\E{\FlucProp}(l_1,q',l_2|\overline{\FieldE})
\E{\FlucProp}(-l_1+l_2,0,l_4|\overline{\FieldE})
\Adj{{\E{\Psi}}{}'}(q+l_3|\overline{\FieldE})
{\E{\Psi}}{}'(q+l_3+l_4|\overline{\FieldE}) \\
& \quad\qquad\qquad\quad
\Bigl\{ \E{\WPotloc}{}'(q'+l_1|\overline{\FieldE})
+\E{\WPotloc}{}'(q'+l_2|\overline{\FieldE}) \Bigr\}
\Bigl\{ \Adj{\E{\WPotloc}{}'}(q+l_3|\overline{\FieldE})
+\E{\WPotloc}{}'(l_4|\overline{\FieldE}) \Bigr\} \\
\end{split}
\end{multline}
and
\begin{multline}
\label{VPrecursionMom}
{{\Z{\VPotloc}}{}'}(\overline{\FieldE}) = {\E{\VPotloc}}{}'(\overline{\FieldE})
- \frac14 \int\limits_{q,l,l'}
\E{\FlucProp}(l,q,l'|\overline{\FieldE}) {\E{\Psi}}{}'(-l+l'|\overline{\FieldE}) \\[-6pt]
\Bigl\{\E{\WPotloc}{}'(q+l|\overline{\FieldE}) + \E{\WPotloc}{}'(q+l'|\overline{\FieldE})\Bigr\}\,.
\end{multline}
We need also the formula (\ref{psiDerivative}) in momentum space:
\begin{equation}
{\E{\Psi}}{}'(q+l|\overline{\FieldE}) = \Adj{\E{\AvOp}}(q+l) + \int\limits_{l'}
\E{\FlucProp}(l,q,l'|\overline{\FieldE})\E{\WPotloc}(q+l'|\overline{\FieldE})
\Adj{\E{\AvOp}}(q+l')
\label{psiDerivativeMomentum}
\end{equation}
\subsection{Algorithm}
We are now able to calculate all
quantities needed. The following algorithm gives an overview
of the procedure:
\begin{itemize}
\item[0.] $i=0:$ Given the starting action fix the values of
$\WPotloc^0(p|\overline{\FieldE}_j)$ and
$\VPotloc^{0\prime}(\overline{\FieldE}_j)$ for a number of constant field
values $\overline{\FieldE}_1,\ldots,\overline{\FieldE}_n$
and for all momenta $p$.
\item[1.] Calculate $\E{\WPotloc}{}'(p|\overline{\FieldE}_j)$ and $\E{\WPotloc}{}''(p|\overline{\FieldE}_j)$
by numerical differentiation.
\item[2.] Determine $\E{\FlucProp}(l,q,l'|\overline{\FieldE}_j)$
with the help of the explicit formula (\ref{gammaform}).
\item[3.] Calculate ${\E{\Psi}}{}'(q+l|\overline{\FieldE}_j)$ by (\ref{psiDerivativeMomentum}).
\item[4.] Insert all into recursion relations for $\Z{\WPotloc}(q|\overline{\FieldE}_j)$ and
${{\Z{\VPotloc}}{}'}(\overline{\FieldE}_j)$.
\item[5.] Increment $i$ and goto 1 until $i=N$.
\end{itemize}
We want to see numerically how this procedure works to produce the correct -- derivative of the --
effective
potential. Therefore we wrote three programs to compare various methods. A Monte-Carlo
program gives us reference values. A combined heatbath and metropolis
algorithm is used.
The heatbath is used with the kinetic term of the action to produce a Gaussian distributed
random value for the fluctuation field value. The following metropolis decides on the basis of the whole action if this value
is accepted.
As an alternative we calculated the effective potential perturbatively, using
the Gawedzki-Kupianen formalism \cite{GawKup}.
The action $\E{\Action}[\phi]$ is split into a kinetic term
$\frac 12\scalar{\phi}{(\E{\Propagator})^{-1}\phi}$ and a potential $\E{\VPot}[\phi]$.
$\E{\Propagator}$ is the massless
free propagator.
Again the notion of fluctuation field $\E{\FlucField}$ with $\E{\AvOp}\E{\FlucField}=0$
is used
and we have a background field $\E{\Psi}$:
\begin{equation}
\phi=\E{\Psi}+\E{\FlucField}\,.
\label{fieldSplit}
\end{equation}
The background field is an
interpolation of the block-spin field:
$$
\E{\Psi}=\E{\InOp}\Phi\,.
$$
The interpolation operator $\E{\InOp}$ is defined as
\begin{equation}
\E{\InOp} = \E{\Propagator}\Adj{\E{\AvOp}}(\Z{\Propagator})^{-1}
\label{InOperator}
\end{equation}
here $\Z{\Propagator}$ is the free propagator for the block-spin field:
\begin{equation}
\Z{\Propagator} = \E{\AvOp}\E{\Propagator}\Adj{\E{\AvOp}}
\label{Propa}
\end{equation}
Inserting the split of the field $\phi$ (\ref{fieldSplit}) into (\ref{defRG})
we get for the fluctuation integral
\begin{equation}
e^{-\Z{\Action}[\Phi]}
= e^{-\frac 12\scalar{\Phi}{(\Z{\Propagator})^{-1}\Phi} -\Z{\VPot}[\Phi]}
= \int_{{\cal H}^i_{{\cal{C}}}} D\E{\FlucField}
e^{-\frac 12\scalar{\E{\FlucField}}{\E{\FlucProp}^{-1}\E{\FlucField}}-\E{\VPot}[\E{\Psi}+\E{\FlucField}]}
\end{equation}
where the fluctuation propagator is defined as\footnote{Inserting (\ref{InOperator})
and (\ref{Propa}) one sees the same formula for $\E{\FlucProp}$
as in (\ref{gammaform}). For free fields $\E{\Propagator}$ and ${{\E{\Action}}{}''}^{-1}$ are identical.}
$$
\E{\FlucProp} = \E{\Propagator} - \E{\InOp}\Z{\Propagator}\Adj{\E{\InOp}}\,.
$$
The fluctuation integral can be rewritten as
\begin{equation}
e^{-\Z{\VPot}[\Phi]} = e^{\frac 12\scalar{\frac{\delta}{\delta\E{\Psi}}}
{\E{\FlucProp}\frac{\delta}{\delta\E{\Psi}}}}
e^{-\E{\VPot}[\E{\Psi}]}\,.
\end{equation}
This can be expanded. All graphs in one loop
order and up to three fluctuation propagators are taken into account.
This leads to the graphs in figure \ref{graphPert}.
\GraphPert{Perturbative expansion of the effective potential. External
lines represent $\E{\InOp}\Phi$ and internal lines $\E{\FlucProp}$.
2-point vertices have the weight $m_i^2$, 4-point vertices have the
weight $\lambda_i$ and 6-point vertices have the weight $\gamma_i$.}
The scale factor (block length) is the same as for the saddle point
approximation and we used a polynomial parameterization of the potentials:
$$
\E{\VPot}[\phi] = \int\limits_{z\in\E{\Lat}}\Bigl(
\frac 12{m_i}^2\phi(z)^2 + \frac{\lambda_i}{4!}\phi(z)^4
+ \frac{\gamma_i}{6!}\phi(z)^6\Bigr) \,.
$$
\subsection{Result}
As a result we get the following picture:
\begin{itemize}
\item For small field values the perturbative calculation is comparable
in accuracy to our method.
\item For larger field values the perturbative calculation goes rapidly wrong,
while the saddle point approximation remains accurate and has
the correct asymptotics (see figure \ref{overview}).
\item For the values of coupling constants which we consider,
larger blocks are better than small blocks (see figure \ref{largeBlock}).
This confirms our expectation from the discussion of the localization
approximation.
\end{itemize}
\newlength{\pswidth}
\setlength{\pswidth}{0.7\textwidth}
\newcounter{Angle}
\setcounter{Angle}{270}
\begin{figure}
\begin{center}
\epsfig{file=Graph1.ps,width=\pswidth,angle=\theAngle}
\caption{The derivative of the constraint effective potential
is plotted against the block-spin $\Phi$.
The bottom line is the original potential. The next upper line
is the saddle point approximation and the top line
the perturbative calculation.
The open squares represent the Monte Carlo
results. This was calculated on a $16\times16$
lattice with blocking factor 2, $m_0^2=-0.8$
and $\lambda_0=0.4$.
Note that the minimum of the potential is where $V'=0$ (see figure
\ref{mexican}.)
}
\label{overview}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{file=Graph2.ps,width=\pswidth,angle=\theAngle}
\caption{The derivative of the constraint effective potential
is plotted against the block-spin $\Phi$.
The bottom line is the original potential. \sticky{The next upper line
consists actually of two lines. The saddle point approximation
and the perturbative calculation (both with blocking factor 2) coincide
for small block-spin field values.}
The next upper line is the saddle point approximation with blocking
factor 2.
The top line is the saddle point approximation with blocking factor 4.
It is better than with blocking factor 2.
The open squares represent the Monte Carlo
results. This was calculated on a $16\times16$ for $m_0^2=-0.4$
and $\lambda_0=0.4$.}
\label{largeBlock}
\end{center}
\end{figure}
\noindent In figure \ref{flowOfPotential} the flow of the potential is shown.
\begin{figure}
\begin{center}
\epsfig{file=Graph3.ps,width=\pswidth,angle=\theAngle}
\caption{The flow of the derivative of the constraint effective potential
is plotted against the block-spin $\Phi$
in the vicinity of the minimum of the potential.
The lines represent ${\E{\VPotloc}}{}'$ for $i=0,1,2,3,4$ from the bottom
and the open squares the Monte Carlo
results. This was calculated on a $16\times16$
lattice with blocking factor 2, $m_0^2=-0.4$
and $\lambda_0=0.4$. The extension of the lattice is getting smaller
after every blocking step by factor 2. On the other side the
lattice spacing grows by factor 2, so that the lattice
volume stays constant. No rescaling to the unit lattice is performed.}
\label{flowOfPotential}
\end{center}
\end{figure}
\section{Relation to other approaches}
\subsection{Flow equation for one particle irreducible average actions}
In this paper we are chiefly interested in blocking to discrete ``flows''
from lattice
to lattice because we wish to investigate the accuracy of variants of the
method by comparison with Monte Carlo simulations. But the method itself is
not limited to this situation. We could consider actions $S^K$ which
depend on a continuously variable
cutoff $K$ in place of the discrete length scales $\LatSpace_{i}$, and we may
leave $\kappa$ finite, or take the limit $\kappa \mapsto \infty$ at the end.
The flow will depend on a substitute for the projector
\begin{equation}
R_K= \kappa
\Adj{\E{\AvOp}}\E{\AvOp} .
\end{equation}
Differentiating the recursion relation
in saddle point approximation, eq.~(\ref{action})
with respect to $t= \ln K$, we get
from the formula (\ref{GammaWithKappa}) for the fluctuation propagator
\begin{equation}
\frac {\partial}{\partial t}S^K
= \frac 12 tr (S^{K\prime \prime} + R_K)^{-1}
\frac {\partial}{\partial t}R_K . \label{Wet}
\end{equation}
This is Wetterich's flow equation for average actions \cite{Wetterich}.
The initial condition is also the same. We conclude that
Wetterich's effective actions can be interpreted
as Wilson effective actions, considered as functions of the background
field.
This is a remarkable
result because Wetterich's average actions are not by definition equal to
a Wilson effective action. They are one-particle irreducible while
a computation of a Wilsonian effective action in perturbation theory contains
one particle reducible pieces. A one particle reducible piece
occurs also in Polchinski's version of a
flow equation \cite{Polchinski}.
However, there are no one-particle reducible pieces
in the recursion relations when the Wilson effective action
is considered as a function of the background field.
For given values of the block-spin, the background field depends not only on
the block-spin but also on the action.
When the background field is determined accurately enough (e.g. in Feynman
Bogoliubov approximation) no one particle reducible graph appears.
\subsection{Perfect actions}
Let us also clarify the relation of our approach with the work of
Hasenfratz and Niedermayer \cite{Hasenfratz} on perfect actions. They also use
a background field $\Psi $. Let $\sigma : \E{\Lat} \mapsto \Z{\Lat} $ be
the map which scales the coordinates of every lattice point by a factor
$s=\LatSpace_{i+1} /\LatSpace_{i} $. This induces a map of fields
$\sigma^{\ast }: \Z{{\cal H}}\mapsto \E{{\cal H} }$ defined by
$\sigma^{\ast }\Phi (z ) = s^l \Phi( \sigma z )$, where $l$ is
determined by the dimension of the field.
When seeking a perfect action in tree approximation, the $tr \ln \E{\FlucProp}$
in the recursion relation is neglected, and the fixed point condition which
defines a perfect action in tree approximation reads
\begin{equation}
\E{\Action} (\E{\Psi} [\Phi ] )
=
\E{\Action} (\sigma^{\ast}\Phi ) .
\end{equation}
\subsection{Legendre transforms of higher order}
It was pointed out to us by Yu. Pismak \cite{Pismak}
that our method invites the
application of the formalism of Legendre transform of second order
\cite{Haymaker},
and of higher order when higher order corrections are considered.
This yields expansions in skeleton graphs similar to those familiar
from renormalization theory \cite{BD,MT}
with two-point functions and vertices which are
determined as solutions of the Schwinger Dyson equations.
\section{Theories with fermions}
In principle the considerations of this paper apply also to theories with
fermions, if one knows how to block them. The $\delta$-function in
the defining recursion relation (\ref{defRG}) for the actions
has only a symbolic meaning in this case, but the split of the integration over Fermi fields
into integration over high and low frequencies is nevertheless possible in
the same way as for Bose fields. But because of the different formula for
Grassmannian Gaussian integrals, the second term in the recursion relation
(\ref{action}) changes sign. In the formulae for normal ordered actions and
for derivatives of actions, this is taken into account by the familiar rule
\begin{center}
{\em a factor (-1) for every closed loop }
\end{center}
It is known how to block from fermions with the right number of flavors
in the continuum to
Kogut Susskind lattice fermions, and on from there.
How to do this in a way which preserves locality was first discovered by
G. Mai and is reviewed in \cite{KalkEtAl}; Bietenholz and Wiese found a
similar scheme in their studies of perfect actions \cite{BW}.
In fermionic theories the improvement of the saddle point method
by self-consistent normal ordering can be important, see the next section.
\section{How locality can fail}
\label{locSection}
Things can go badly wrong when locality of the effective action
fails due to a bad choice of block-spin.
It is therefore essential to monitor whether the method can be expected
to give reliable results by monitoring the range
\begin{equation} average
\frac{\int_{w } | (z -w )W(z , w |\Psi )|}
{\int_{w } | W(z , w |\Psi )|}
\end{equation}
of the interaction
for each value of the constant field which enters into the computation of the
desired quantity.
Failures of locality can occur for several reasons which require different
remedies.
One sees from the recursion relation for the derivatives of the action that
good locality properties can only be expected if the fluctuation
propagator $\E{\FlucProp} (z ,w |\Psi )$ decays with distance
$ (z -w) $ with decay length no more than one block lattice spacing .
For a $\phi^4$-action
\begin{equation}
\Gamma = \lim_{\kappa \mapsto \infty } \left(
-\Delta + m^2 + \frac 12 \lambda \phi (z)^2 +
\kappa \Adj{\E{\AvOp}}\E{\AvOp} \right)^{-1} .
\end{equation}
Following Balaban \cite{balaban}, one can estimate the decay properties by
the lowest eigenvalue of the operator in brackets with $\Delta $ replaced
by the Laplacian $\Delta^N$
with Neumann boundary conditions on the block boundary;
basically this eigenvalue should be strictly positive and of order at least
one in units where $\LatSpace_{i+1} = 1 $.
We are interested chiefly in (nearly) constant
$\phi $; in this case one can also find the decay by Fourier analysis
cp. ref. \cite{GawKup}.
The $ \kappa \Adj{\E{\AvOp}}\E{\AvOp} $-term effectively eliminates
the zero mode of $\Delta^N$ (constants) and the next eigenvalue of
$\Delta^N$ has the desired magnitude. Therefore things go wrong when
$ m^2 + \frac 12 \lambda \phi (z)^2 $ becomes too strongly negative. This can
happen when $\phi \approx 0 $ and $m^2 $ is too strongly negative -
i.e. near the top of a Mexican hat which is too high, cp.
figure \ref{mexican}.
\MexicanHat{Potential and its derivative. The middle region is unstable
and unphysical. It is separated from the stable region by a metastable
region. Our method cannot compute the constraint effective potential in
the unstable region because we assumed that translation invariance is
not spontaneously broken}
When $m^2$ is negative enough, something still more terrible happens.
Numerical work revealed that the minimal action for constant block-spin zero
is reached for background fields which have periodic domain walls of
alternating slope. In other words, the
auxiliary theory of section \ref{introSection} shows
spontaneous breaking of translation symmetry by block lattice vectors.
At intermediate RG-steps one could try to remedy this by choosing
block-spins of fixed length - i.e.
integrating out the modulus of the field - to obtain nonlinear
$\sigma$-model type effective actions.
But anyway we cannot calculate the dependence of the constraint
effective potential on the magnetization in the unstable region
(figure \ref{mexican}) because the assumption of no spontaneous
breaking of translational symmetry is physically wrong there.
Nonlocalities can appear in another more interesting way which is relevant for
studies of a dynamical Higgs mechanism, in particular {\em superconductivity.}
The remedy in this case is the introduction of a composite Bose field as
a block-spin. In superconductivity it represents Cooper pairs. This
mechanism was studied in some detail by Grabowski \cite{Grabowski}.
Nonlocalities of
3-point functions will also
induce nonlocalities of ${\E{\Action}}{}''$. The formal solution
of the Schwinger Dyson equation for the 3-point function
involves (among others) the famous
chain-of-bubble diagrams. The fluctuation propagators in these diagrams
have exponential decay. Nevertheless the sum of these diagrams can fail
to have the desired locality properties due to a pole in momentum space
below the UV-cutoff $\LatSpace_{i+1}^{-1}$ of the new action. This occurs in
fermionic theories when the coupling gets strong enough. In the
2-dimensional Gross Neveu model and in models of superconductivity,
one is always driven into this domain by the renormalization group flow.
When this happens, one is forced to introduce a composite block-spin
\cite{Grabowski}.
\section{Motivation for the improvement of the saddle point method}
In a $\phi^4$-theory the simple saddle point method works quite well.
But this is due to the simple form of the kinetic term in this theory.
Consider instead an action for an $n$-component field $\phi $
in two dimensions of the
form
\begin{equation}
S [\phi ] = \int \left(\frac{2n}f
(1 + \phi^2)^{-2}(\nabla \phi )^2 + n \delta (0) \ln (1 + \phi^2)
\right)
\end{equation}
where $f$ is a coupling constant. In the continuum, $\delta (0) $ is
quadratically divergent, on the lattice $\delta (0)=a^{-2}$.
This action comes from the $O(n+1)$-symmetric nonlinear $\sigma$-model
in stereographic coordinates; the term proportional $\delta (0) $ comes
from the measure on the sphere.
But let us forget where the action came from.
Suppose we wish to block from the continuum to the lattice. For simplicity,
consider first what happens if we apply a simple saddle point treatment to
the whole theory.
Expanding the action up to terms quadratic in the field, we get
\begin{equation}
S [\phi ] \approx \int \left(
\frac{2n}f (\nabla \phi )^2 + n\delta (0)\phi^2
\right) + ...
\end{equation}
Evidently we met with disaster. There is a quadratically divergent coefficient
in a quadratic action.
The same problem appears if we do the high frequency integral to
compute an effective lattice action from the continuum action.
If we normal order first before expanding in the field, the situation
is different. Let us imagine a finite volume and normal order with respect to
the massless free propagator $v_{Cb} $
without its zero mode. Normal ordering the
leading term $f(\nabla \phi )^2\phi^2 $ produces a quadratic term
$- f \Delta v_{Cb}(0) \phi^2 = f\delta (0) \phi^2 $
which cancels against the quadratically divergent term from the measure.
The disease of the simple saddle point method comes from its lack of
covariance under field reparameterization. This can lead to catastrophic
violations of Ward identities in theories with symmetries.
\footnote{It is known
that Ward identities guarantee a correct cancellation of divergences in the
nonlinear $\sigma$-model \cite{sigmaModel}.}
Since we shall want to apply our method to such theories in the future, it
was essential to go beyond a simple saddle point approximation.
Another advantage of the normal ordered version of our method (section
\ref{FBapproximation}) consists in the fact that it enlarges the class of
theories which can be dealt with to theories in which the field assumes a
{\em discrete} set of values, such as the discrete Gaussian model, Ising
and Potts models etc. This is true because any action can be expanded
in normal ordered products of the fields.
\section{The saddle point approximation}
\label{GaussSection}
\subsection{Parameterization of the action}
\label{introductionSaddle}
We wish to compute the action $\Z{\Action}[\Phi]$
which depends on a field $\Phi $ on the lattice
$\Z{\Lat}$ from the action $\E{\Action}[\phi] $ on the finer lattice $\E{\Lat}$.
We will make some
approximations to perform the calculation. We present a parameterization
of the actions which will be preserved by the approximate renormalization
group flow.
The action $\Z{\Action}[\Phi]$ will depend on the field $\Phi$ through
a functional $\E{\Psi}[\Phi]$.
The field $\E{\Psi} $
lives on the lattice $\E{\Lat} $. It is called the background field.
The recursion formula will have the form
\begin{eqnarray}
\Z{\Action}[\Phi] &=&
\E{\Action}[\E{\Psi}] -\frac 12 \text{tr} \ln \E{\FlucProp} [\E{\Psi} ] , \label{action}\\
\E{\Psi} &=& \E{\Psi} [\Phi ] .
\end{eqnarray}
Approximations will be made such that
\begin{enumerate}
\item
The parameterization of the action preserves its form
\item
The fluctuation propagator, the effective action and its first two
field derivatives, and
the background field $\E{\Psi}[\Phi]$ and its derivative
${\E{\Psi}}{}'[\Phi]$
with respect to the field $\Phi $ will enter into the recursion
relations, but they
need only be calculated for constant fields. This
computational problem is fit for a PC.
\end{enumerate}
We will use a dual notation for derivatives with respect to fields
\begin{equation*}
{\E{\Action}}{}'[\phi](z) \equiv
\E{\Action}_{,z }[\phi] = \dEx {z} \E{\Action} [\phi] \ ,
\end{equation*}
and similarly for second derivatives $\E{\Action}_{, z w} $.
The background field $\E{\Psi} $
will be determined as a functional of $\Phi $ by
the saddle point condition which involves the previous action
\begin{equation*}
\E{\Action} [\E{\Psi} ] = min \ \ \mbox{subject\ to}\ \
\E{\AvOp} \E{\Psi} = \Phi \ .
\end{equation*}
The fluctuation field propagator $\E{\FlucProp} $ lives on lattice
$\E{\Lat} $. It is a selfadjoint integral operator with kernel
$$ \E{\FlucProp} [\phi] (z , w) \ , \ \ z , w \in \E{\Lat} $$
and is defined as pseudoinverse of ${\E{\Action}}{}''[\phi]$ for all $\phi$
in the sense that
\begin{equation}
\E{\FlucProp} [\phi ]{\E{\Action}}{}'' [\phi ]
\E{\FlucProp} [\phi ]= \E{\FlucProp} [\phi ]. \label{FlucPropQinv}
\end{equation}
In the case where the field $\phi$ is equal to the background field
$\E{\Psi}[\Phi]$,
the fluctuation field propagator is positive definite on ${\cal H}^i_{{\cal{C}}}$. It
satisfies the constraints (\ref{GammaConstraint0}), viz.
\begin{equation}
\E{\AvOp} \E{\FlucProp} [\E{\Psi}] = 0 = \E{\FlucProp} [\E{\Psi} ] \Adj{\E{\AvOp}}.
\label{GammaConstraint}
\end{equation}
Therefore the space $ \E{\cal H} $ is mapped into $ {\cal H}^i_{{\cal{C}}}$ by $\E{\FlucProp} $,
and $\E{\FlucProp} $ vanishes on $\Adj {\E{\widetilde{\AvOp}}}\Z{\cal H} $.
The trace $\text{tr} $ which appears in the action (\ref{action}) is
to be understood as a trace over ${\cal H}^i_{{\cal{C}}}$.
We will explain later on how the background field and the fluctuation
propagator can be evaluated by a self
consistent approximation which involves neglect of higher order terms in
gradients of fields.
It is the crux of any real space renormalization group method to find
truncated forms of the effective actions which can be parameterized in a
manageable form.
Often this is done in an ad hoc fashion which throws away some pieces which
are irrelevant in a {\em perturbative} sense - higher powers of fields,
for instance. This is not really justified because
existing irrelevant terms get suppressed in the
next RG-step. But new irrelevant terms are created by the marginal and
relevant ones at the same time.
As a consequence, there is a kind of equilibrium, so that
(along the renormalized trajectory) the
irrelevant terms are determined by the relevant and marginal ones. They are
not necessarily very small and they influence the flow of the marginal and
relevant coupling constants. Throwing them away after each renormalization
group step introduces therefore a systematic error which accumulates.
In principle there is a better way. One could determine the irrelevant pieces
as a function of the marginal and relevant ones by solving a fixed point
equation. But until now this is practical only in simplified (hierarchical)
models \cite{Pordt}.
The motivation for our method of truncation will be described below. We will
argue that our approximation becomes more and more accurate the larger
the scaling factor (ratio of lattice spacings) $s=\LatSpace_{i+1}/\LatSpace_{i} $.
We will present
numerical evidence which confirms this.
On the other hand, the saddle point approximation becomes exact in the limit
$s\mapsto 1 $ (when the nature of the cutoff permits such a limit). It becomes
less accurate with increasing scaling factor because the phase space for
high frequency modes increases.
The truncation consists of a local approximation to the second derivative
${\E{\Action}}{}''=-W $ of the action. For more precise notation, set
\begin{eqnarray}
-\E{\Action}_{,z w}[\phi ] &=& \E{\WPot} [\phi](z , w) \\
\left( \E{\WPot} [\phi ] f \right)(z ) &=& \int_{w }
W [\phi] ( z , w) f (w )
\end{eqnarray}
We will approximate $\E{\WPot}$ by a function $\E{\WPotloc}(z , w |\xi ) $
which
depends only on the value $\xi $ of the field at one site of the lattice
$\E{\Lat} $. If $f\in \E{\cal H} $ is a function
on $\E{\Lat} $ then we approximate
\begin{equation}
\left( \E{\WPot} [\phi ]f\right) (z ) = \frac 12
\int_{w} \left[ \E{\WPotloc}\left( z ,w |\phi (z )\right) +
\E{\WPotloc}\left( z ,w |\phi (w )\right) \right] f (w ) \ .
\label{approximationIntro}
\end{equation}
$\E{\WPotloc} $ can be determined from the knowledge of $\E{\WPot} $
for {\em constant} fields $\phi $. Symmetrization is performed to maintain
hermiticity of $W$.
To understand the meaning of the approximation, suppose that
$\E{\Action} $ is the action of a $\phi^4$-theory. Then
\begin{equation}
\left( {\E{\Action}}{}'' f \right) (z ) = \left[ -\Delta + m^2
+ \frac 12 \lambda \phi (z)^2
\right] f(z). \label{startingAction}
\end{equation}
We see that our approximation is exact in this case because the field
dependent part of ${\E{\Action}}{}'' $ only involves the field
$\phi $ at a single point $z$.
In particular, the approximation is exact for free field theories. Since
the saddle point approximation is also exact for free fields,
our method is exact for free field theory.
A general action can be decomposed into a potential and a term
whose first derivative with respect to the field
vanishes for constant fields. We call it a generalized kinetic term.
\begin{equation}
\E{\Action} [\phi ]= \mbox{generalized kinetic term} + \E{\VPot}[\phi] \ .
\end{equation}
A field independent contribution to $\E{\VPot} $ is of no interest. Therefore it
suffices to know the derivative ${\E{\VPot}}{}' $. It is uniquely determined
if we know the derivative ${\E{\Action}}{}' $ of the action for {\em constant} fields.
Our method consists in deriving recursion relations for
$\E{\VPot} $ and $\E{\WPot}$ i.e. for the first and second derivatives
${\E{\Action}}{}' $ and $\E{\WPot} = -{\E{\Action}}{}'' $
of the action evaluated at {\em constant} fields.
Moreover we wish to obtain the effective actions as a function of the
block-spin and not only as a function of the background field. Therefore
we will need information on the block-spin dependence of the
background field.
It turns out that the use of the recursion formulae
requires knowledge of the derivative of the background field with respect to the
block-spin.
There is a formula for this:
\begin{equation}
\E{\Psi}_{,x }[\Phi](z ) \equiv
\dZx{x} \E{\Psi}[\Phi] =
(1 - \E{\FlucProp} {\E{\Action}}{}'' )\Adj{ \tilde \E{\AvOp}}(z,x).
\end{equation}
For a block-spin $\overline{\FieldZ}$ which lives on a lattice made of a single site,
translation invariance implies that
there exists a saddle point $\E{\Psi} = \E{\Psi} [\overline{\FieldZ} ]$
of the action which is also constant. Because of the
constraint $\E{\AvOp} \E{\Psi} = \Phi $ and our normalization conventions
it follows that
\begin{equation}
\E{\Psi} [\overline{\FieldZ} ] = \Adj{\E{\widetilde{\AvOp}}} \overline{\FieldZ} \ .
\label{Fieldloc}
\end{equation}
So we know the background field in this case, assuming the saddle point is the
minimum.
If it is not, this signals a breakdown of stability.
We will come back to such possibilities. They are familiar from
the Maxwell construction in thermodynamics.
The argument can be extended to show that to any block-spin
$\overline{\FieldZ}$ which is constant on $\Z{\Lat}$ there exists a saddle point
(\ref{Fieldloc}) which is constant, assuming that translational invariance
is not spontaneously broken. If
the invariance under translations by block lattice vectors is not
spontaneously broken in
the auxiliary theory with expectation values (\ref{defExpVal}), then the
appropriate saddle point $\Psi $ will be invariant under block lattice
translations. Therefore it is equal to the constraint minimum of the
action on a single block with periodic boundary conditions. Now we can appeal
to translation invariance of the action under translations on the fine lattice
$\E{\Lat}$ again to conclude that there exists a constant saddle point
unless translational symmetry is spontaneously broken.
It may happen that it is spontaneously broken or the
saddle point is not a minimum. Again, this signals a breakdown of stability.
In view of its interpretation as a propagator for the high frequency modes
of the field which are integrated out in a renormalization group step,
the kernel $\E{\FlucProp}[\phi] (z,w)$ of the fluctuation field
propagator
must decay exponentially with distance $|z -w |$ with decay length
at most one block lattice spacing $\LatSpace_{i+1} $ (see \cite{balaban}).
This is a condition which limits
the range of applicability of the present method. It could be monitored during
the computation of the renormalization group flow. When it is violated,
this will typically entail a violation of the locality properties
of the next action $\Z{\Action}$ as well. Locality requires that
$\E{\Action}_{, z w} $ decays exponentially with the distance between
$z$ and $w $ with decay length no larger than one lattice spacing
$\LatSpace_{i} $.
When this is violated
it is typically a sign that the choice of the block-spin definition
is not appropriate. This can happen for several reasons, see section
\ref{locSection}.
\subsection{The detailed procedure of saddle point approximation}
Let $\E{\Action}$ be a generic action.
We now want to evaluate the effective action $\Z{\Action}$
\begin{equation}
e^{-\Z{\Action}[\Phi]}
= \int D\phi \delta (\E{\AvOp} \phi - \Phi)
e^{-\E{\Action}[\phi]}
\label{defRG}
\end{equation}
in a saddle point approximation.
For this purpose we split the field
\begin{equation*}
\phi (z )= \E{\Psi}[\Phi](z) + \E{\FlucField} (z) \ \ \
(z \in \E{\Lat}).
\end{equation*}
into the background field $\E{\Psi}[\Phi]$ and a
fluctuation field $\E{\FlucField}$. The background field has also the meaning
of a mean field, because in the Gaussian approximation that we will use
for the action $\E{\Action}$ eq.(\ref{MFAFieldDef}) holds
The mean field $\E{\Psi}$ is determined as a functional of the block-spin
field $\Phi$ by
a saddle point condition which involves the previous action
\begin{equation*}
\E{\Action} [\E{\Psi} ] = min \ \ \mbox{subject\ to}\ \
\E{\AvOp} \E{\Psi} = \Phi \ .
\end{equation*}
The result is a nonlinear equation for $\E{\Psi}[\Phi]$
\begin{eqnarray}
{\E{\Action}}{}' [\E{\Psi}] &=& \Adj {\E{\AvOp} }\E\lambda , \label{MFAFieldEq1} \\
\E{\AvOp} \E{\Psi} &=& \Phi \ . \label{MFAFieldEq2}
\end{eqnarray}
$\E\lambda [\Phi](x) $ are Lagrange multipliers, $x\in \Z{\Lat} $.
Expanding the action around the mean field to second order in $\E{\FlucField}$ yields
\begin{equation}
\E{\Action}[\E{\Psi}+\E{\FlucField}] = \E{\Action} [\E{\Psi}] +
\int_{z } \E{\Action}_{,z}[\E{\Psi}]\E{\FlucField}(z)
+ \frac 12
\int_{z,w} \E{\Action}_{,z w}[\E{\Psi}]\E{\FlucField}(z)
\E{\FlucField} (w ).
\label{expand2Order}
\end{equation}
The linear term in the expansion vanishes
$$
({\E{\Action}}{}',\E{\FlucField})=(\Adj{\E{\AvOp}}\E\lambda,\E{\FlucField})
=(\E\lambda,\E{\AvOp}\E{\FlucField})=0,
$$
since due to (\ref{MFAFieldEq2}) $\E{\AvOp}\E{\FlucField}=0$.
Therefore we get
\begin{align}
e^{-\Z{\Action}[\Phi]} &\approx \int_{\E{\cal H}}
D\E{\FlucField} \delta ({\cal{C}}\E{\FlucField})
e^{-\E{\Action}[\E{\Psi}[\Phi]]
-\frac 12\scalar{\E{\FlucField}}
{{\E{\Action}}{}''[\E{\Psi}[\Phi]]\E{\FlucField}}
}
\label{MFAintegral}\\
&=e^{-\E{\Action}[\E{\Psi}[\Phi]]} \int_{{\cal H}^i_{{\cal{C}}}} D\E{\FlucField}
e^{-\frac 12\scalar{\E{\FlucField}}
{\E{\FlucProp}[\E{\Psi}[\Phi]]^{-1}\E{\FlucField}}
}
\label{GammaInvInt}
\end{align}
Note that ${\E{\Action}}{}''$ is a linear operator on $\E{\cal H}$.
In appendix \ref{GammaFormAp}
we show that $\E{\FlucProp}$ can be written as
\begin{equation}
\E{\FlucProp} = {{\E{\Action}}{}''}^{-1} -
{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}}(\E{\AvOp}{{\E{\Action}}{}''}^{-1}\Adj{\E{\AvOp}})^{-1}
\E{\AvOp}{{\E{\Action}}{}''}^{-1}\,.
\label{gammaform}
\end{equation}
Because of $\E{\AvOp}\E{\FlucProp}=0=\E{\FlucProp}\Adj{\E{\AvOp}}$ the fluctuation
propagator vanishes on $\Adj {\E{\AvOp}}\Z{\cal H}$ and is a linear operator
$\E{\cal H}\rightarrow{\cal H}^i_{{\cal{C}}}$. Therefore we can use the quantity
$(\E{\FlucProp})^{-1}={\E{\Action}}{}''\Einsch{{\cal H}^i_{{\cal{C}}}}$ in (\ref{GammaInvInt}).
The extension of $\E{\FlucProp}$ to all of $\E{\cal H}$ is a pseudoinverse of
${\E{\Action}}{}'' $ as described by (\ref{FlucPropQinv}).
The remaining fluctuation integral
is then Gaussian with covariance $\E{\FlucProp}[\E{\Psi}[\Phi]]$ and may
easily be performed. The result is
\begin{equation}
\Z{\Action} [\Phi ] = \E{\Action} [\E{\Psi} [\Phi ]]
-\frac 12 \text{tr} \ln \E{\FlucProp} [\E{\Psi} [\Phi ] ] , \label{actionPrel}
\end{equation}
The trace $\text{tr} $ is to be understood as a trace over ${\cal H}^i_{{\cal{C}}}$.
If one uses in (\ref{MFAintegral})
$$
\delta(\phi) = \lim\limits_{\kappa\rightarrow\infty}
e^{-\frac\kappa2 \int_z\phi(z)^2}\,,
$$
one obtains the following alternative expression for the
fluctuation propagator.
\begin{equation}
\E{\FlucProp}[\E{\Psi} [\Phi ] ] = \lim_{\kappa \rightarrow \infty }\left(
{\E{\Action}}{}'' [\E{\Psi} [\Phi ]] + \kappa \Adj{\E{\AvOp}}\E{\AvOp}
\right)^{-1}\,.
\label{GammaWithKappa}
\end{equation}
\subsection{Recursion relations}
By iterating the saddle point approximation we obtain
$\D{\Action}$ from $\Z{\Action}$:
$$
\D{\Action} [\varphi ] = \Z{\Action} [\Z{\Psi} [\varphi ]]
-\frac 12 \text{tr} \ln \Z{\FlucProp} [\Z{\Psi} [\varphi ] ].
$$
Now we need $\Z{\FlucProp}$. It is defined in terms of ${\Z{\Action}}{}''$
by an equation similar to (\ref{gammaform}). Therefore we try to
get a recursion relation for
\begin{equation*}
\E{\WPot}[\phi] = -{\E{\Action}}{}''[\phi]\,.
\end{equation*}
In appendix \ref{recursionRelation} the following
recursion relation will be deduced by differentiating eq.(\ref{action}).
(We neglect to write functional
$\E{\Psi}[\Phi]$-dependencies on the right hand side.)
\begin{multline}
\Z{\WPot} [\Phi](x,y) = \\
\begin{split}
& \int_{z,w\in\E{\Lat}}
\E{\WPot}(z,w)
\E{\Psi}_{,{y}}(z)
\E{\Psi}_{,{x}}(w) \\
& + \ \frac 12\int\limits_{z,w,z_i\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,z} (z_2,z_3)
\E{\FlucProp} (z_3,z_4)
\E{\WPot}_{,w} (z_4,z_1)
\E{\Psi}_{,x} (z) \E{\Psi}_{,y} (w)
\\[8pt]
& + \ \frac 12\int\limits_{z,w,z_1,z_2\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,zw}(z_2,z_1)
\E{\Psi}_{,x }(z)
\E{\Psi}_{,y }(w) \\
& + \ \frac 12\int\limits_{z,z_1,z_2\in\E{\Lat}}
\E{\FlucProp} (z_1,z_2)
\E{\WPot}_{,z} (z_2,z_1)
\E{\FlucProp}(z,z_3) \E{\WPot}_{,w}(z_3,z_4)
\E{\Psi}_{,y}(w) \E{\Psi}_{,x}(z_4)
. \\
\end{split}
\label{recursionW}
\end{multline}
This functional relation is an exact consequence of eq.(\ref{action}),
no localization approximation has been made yet.
We see that we need an equation for the derivative ${\E{\Psi}}{}' $
of the background field. In appendix \ref{proofOfLemma} it will be shown that
\begin{equation}
{\E{\Psi}}{}' = (1-\E{\FlucProp} {\E{\Action}}{}'') \Adj{\widetilde{\E{\AvOp}}}.
\label{psiDerivative}
\end{equation}
The recursion relation is shown in graphical form in figure
\ref{BildC}. The graphical notation is described in figure \ref{BildAa}.
We see that the recursion relation for ${\E{\Action}}{}''$ contains a 1-particle
reducible term. This is an artifact of the simple saddle point
approximation which will disappear in the improved version of section
\ref{FBapproximation}.
\BildAa{Definition of graphical notation for the derivatives of the
action and of the background field}
\BildC{Recursion relation for $\E{\WPot}=-{\E{\Action}}{}''$ in saddle point
approximation}
As described in section \ref{introductionSaddle} the action
$\E{\Action}$ can be split into a potential term
$\E{\VPot}[\phi]$ and
a generalized kinetic term whose first field derivative vanishes for
constant fields; the potential term is determined by the derivative
${\E{\Action}}{}'$ evaluated at constant field.
We give
the recursion relation for the first derivative ${\E{\Action}}{}'$ in a general form.
Using (see appendix \ref{FlucPropDer})
\begin{equation}
\E{\FlucProp}_{,z} = \E{\FlucProp}\E{\WPot}_{,z}\E{\FlucProp}\,.
\end{equation}
one gets by differentiation of (\ref{actionPrel})
\begin{equation}
\Z{\Action}_{,x}[\Phi] = \int\limits_{z} \E{\Psi}_{,x}[\Phi](z)
\Bigl\{ \E{\Action}_{,z}[\E{\Psi}[\Phi]]
- \frac 12\text{tr} \E{\FlucProp} [\E{\Psi}[\Phi]]
\E{\WPot}_{,z}[\E{\Psi}[\Phi]] \Bigr\}\,.
\label{SPrecursion}
\end{equation}
The graphical illustration is shown in figure \ref{BildD}.
\BildD{Recursion relation for $-{\Z{\Action}}{}'$ in saddle point approximation}
For constant block-spin field the derivative of the
generalized kinetic term vanishes.
Therefore the recursion relation for the potential term has
the same form as (\ref{SPrecursion}):
\begin{equation}
\Z{\VPot}_{,x}[\Phi] = \int\limits_{z} \E{\Psi}_{,x}[\Phi](z)
\Bigl\{ \E{\VPot}_{,z}[\E{\Psi}[\Phi]]
- \frac 12\text{tr} \E{\FlucProp} [\E{\Psi}[\Phi]]
\E{\WPot}_{,z}[\E{\Psi}[\Phi]] \Bigr\}\,.
\label{VPrecursion}
\end{equation}
\subsection{Justification of the localization approximation;
Application to constant fields}
\label{localApproximation}
Let us summarize what we have achieved so far. Given the functionals
$\E{\VPot}$ and $\E{\WPot}$ we can compute the fluctuation propagator
$\E{\FlucProp}$ via equation (\ref{gammaform}) and the derivative of the
background field ${\E{\Psi}}{}'$ via equation (\ref{psiDerivative}).
This enables us to calculate
$\Z{\WPot}$ and $\Z{\VPot}$ by means of recursion relations (\ref{recursionW}) and
(\ref{VPrecursion}). At least this can be done in principle.
Because of the functional nature of these equations the calculation
is too complicated for numerical purposes. We will arrive at a
significant simplification by arguing that we may focus attention on constant
fields.
\sticky
{
For constant block-spin $\overline{\FieldZ}$ the mean field $\E{\Psi}[\overline{\FieldZ}]$
is constant too and has the same value. To see this let us make an ansatz.
We split the field as
\begin{equation}
\phi = \Adj{\E{\widetilde{\AvOp}}} \overline{\FieldZ} + \E{\FlucField}\,.
\end{equation}
It fulfills the constraint (\ref{MFAFieldEq2}) and the first term on the
right-hand side is constant for our choice of the average operator.
Inserting this into (\ref{MFAFielDef}) leads to
\begin{equation}
\E{\Psi} [\overline{\FieldZ} ] = \Adj{\E{\widetilde{\AvOp}}} \overline{\FieldZ} \ .
\label{Fieldloc}
\end{equation}
The expectation value of the fluctuation field is zero because the
auxiliary theory is even in the fluctuation field, therefore it
vanishes:
\begin{equation}
\mean{\E{\FlucField}} = \frac
{ \int_{{\cal H}^i_{{\cal{C}}}} D\E{\FlucField} \,\E{\FlucField}
e^{-\frac 12\scalar{\E{\FlucField}}
{\E{\FlucProp}[\E{\Psi}[\Phi]]^{-1}\E{\FlucField}} } }
{ \int_{{\cal H}^i_{{\cal{C}}}} D\E{\FlucField} e^{-\frac 12\scalar{\E{\FlucField}}
{\E{\FlucProp}[\E{\Psi}[\Phi]]^{-1}\E{\FlucField}} } }
= 0
\end{equation}
}
Fields $\Phi $ with very large derivatives have very small probability
because of the kinetic term in the action. We may therefore assume that
$\Phi $ does not have such violent behavior.
$\E{\Psi} [\Phi ](z )$ is an interpolation of the field $\Phi$ from
the coarser lattice $\Z{\Lat}$ to the lattice $\E{\Lat}$. The interpolation is
determined by a minimality condition on the action.
Because of the kinetic term
in the action, we expect that the result of the interpolation is
a smooth function on $\E{\Lat} $.
This means
that we may consider $\E{\Psi}[\Phi ]$ as nearly constant over
distances $\LatSpace_{i}$, assuming that the scaling factor
$s=\LatSpace_{i+1}/\LatSpace_{i}$ is big enough
By locality $\E{\Action}_{, z w }[\E{\Psi}[\Phi ] ] $
is very nearly zero if
$z $ is not within about one lattice spacing
of $w $, and it depends only
on the field within a neighbourhood of about one lattice spacing
$\LatSpace_{i} $ of $z $ and $w $. Therefore we may approximate
$\E{\WPot} [\E{\Psi}]$ by a hermitian operator $\E{\WPotloc}$
which involves only the value $\xi = \E{\Psi} (z )$ of the field
at one point (see also (\ref{approximationIntro})),
\begin{equation}
\E{\WPot}[\E{\Psi}](z,w) \approx
\frac 12\Bigl(
\E{\WPotloc}(z,w|\E{\Psi}(z))+\E{\WPotloc}(z,w|\E{\Psi}(w))
\Bigr),
\label{Wloc}
\end{equation}
This justifies the localization approximation for $\Z{\WPot}$.
$\E{\WPotloc}$ has locality properties similar to a Laplacian.
To determine $\Z{\WPot}[\Phi]$ we need $\E{\FlucProp}[\phi]$ for
$\phi=\E{\Psi} [\Phi]$.
$\E{\FlucProp} (x , y) $ depends on the field $\phi$ on a domain of
diameter about $\LatSpace_{i+1}$. Unless $\Phi $ is constant, the field
$\phi $ need not be nearly constant over such a big domain. But
$\E{\FlucProp}[\phi]$ is determined by
${\E{\Action}}{}''[\phi]=-\E{\WPot} $ and we can use the localization approximation
for this latter quantity in order to determine $\E{\FlucProp}$ for general field
$\Phi$ if we need it.
Looking now at the recursion relation for $\Z{\WPot}(x,y)$ we see that three
different quantities are involved:
\begin{itemize}
\item The fluctuation propagator has
an exponential decay of one block lattice spacing
$\LatSpace_{i+1}$ (see \cite{balaban} and figure \ref{gammaDecay}).
\item The operator $\E{\WPot}(z,w)$ and its derivatives are very
local by assumption and have therefore a range of one lattice spacing
$\LatSpace_{i}$.
\item The derivative of the background field ${\E{\Psi}}{}'$
is also determined by these quantities and has exponential decay
with decay length $\LatSpace_{i+1}$.
\end{itemize}
\GammaDecay{The decay of the fluctuation propagator.}
Combining these arguments we see
that $\Z{\WPot}[\Phi]$ has
similar locality properties on the scale of the new lattice spacing
$\LatSpace_{i+1}$ as $\E{\WPot}$ had on the scale $\LatSpace_{i}$.
Therefore we will be entitled to
make similar approximations on the next level as before.
Now we can perform functional derivatives of $\E{\WPot}$ for almost constant fields:
\begin{align}
\E{\WPot}_{,z}[\phi](z_1,z_2) = \frac 12\Bigl(&
\E{\WPotloc}{}'(z_1,z_2|\phi(z_1))\delta(z-z_1) \nonumber\\
& + \E{\WPotloc}{}'(z_1,z_2|\phi(z_2))\delta(z-z_2)\Bigr)
\label{firstDerivWloc} \\
\E{\WPot}_{,zw}[\phi](z_1,z_2) = \frac 12\Bigl(&
\E{\WPotloc}{}''(z_1,z_2|\phi(z_1))\delta(z-z_1)\delta(w-z_1)
\nonumber \\
& +\E{\WPotloc}{}''(z_1,z_2|\phi(z_2))\delta(z-z_2)\delta(w-z_2)
\Bigr)\,,
\label{secondDerivWloc}
\end{align}
where $'$ denotes the derivative of $\E{\WPotloc}(z_1,z_2|\xi)$ with respect to
the single variable $\xi$. These formulas are valid when smeared with test functions in variables $z $ and $w$ which are nearly constant over
distances $\LatSpace_{i}$.
Setting the block-spin field constant the fluctuation propagator becomes a
function of the field value.
By virtue of (\ref{psiDerivative}) the derivative of the
background field can be written
as a simple derivative:
\begin{equation}
\E{\Psi}_{,x}[\,\overline{\FieldZ}\,](z) \equiv {\E{\Psi}}{}'(z,x|\overline{\FieldZ})
= \int\limits_{z_1}\Bigl(\delta(z,z_1)+\int\limits_{w}
\E{\FlucProp}(z,w|\overline{\FieldZ}) \E{\WPotloc}(w,z_1|\overline{\FieldZ})\Bigr)
\Adj{\widetilde{\E{\AvOp}}}(z_1,x)\,.
\label{PsiDerivativeCoord}
\end{equation}
For constant block-spin fields, inserting (\ref{firstDerivWloc}) and
(\ref{secondDerivWloc}) into the recursion relation (\ref{recursionW}) for
$W$ gives therefore a recursion relation for $\E{\WPotloc}$:
\begin{multline}
\Z{\WPotloc} (x,y|\overline{\FieldZ}) =
\int_{z,w\in\E{\Lat}}
\E{\WPotloc}(z,w|\overline{\FieldZ})
{\E{\Psi}}{}' (z,y|\overline{\FieldZ})
{\E{\Psi}}{}' (w,x|\overline{\FieldZ}) \\
\begin{split}
& + \ \frac18\int\limits_{z_i\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2|\overline{\FieldZ})
\E{\WPotloc}{}' (z_2,z_3|\overline{\FieldZ})
\E{\FlucProp} (z_3,z_4|\overline{\FieldZ})
\E{\WPotloc}{}' (z_4,z_1|\overline{\FieldZ}) \\[-8pt]
& \qquad\qquad\quad
\Bigl\{ {\E{\Psi}}{}'(z_2,x|\overline{\FieldZ}) {\E{\Psi}}{}'(z_4,y|\overline{\FieldZ})
+ {\E{\Psi}}{}'(z_2,x|\overline{\FieldZ}) {\E{\Psi}}{}'(z_1,y|\overline{\FieldZ}) \\
& \qquad\qquad\qquad
+ {\E{\Psi}}{}'(z_3,x|\overline{\FieldZ}) {\E{\Psi}}{}'(z_4,y|\overline{\FieldZ})
+ {\E{\Psi}}{}'(z_3,x|\overline{\FieldZ}) {\E{\Psi}}{}'(z_1,y|\overline{\FieldZ})
\Bigr\} \\
& + \ \frac14\int\limits_{z_1,z_2\in\E{\Lat} }
\E{\FlucProp} (z_1,z_2|\overline{\FieldZ})
\E{\WPotloc}{}''(z_2,z_1|\overline{\FieldZ}) \\[-8pt]
&\qquad\qquad\qquad
\Bigl\{ {\E{\Psi}}{}' (z_1,x|\overline{\FieldZ}) {\E{\Psi}}{}' (z_1,y|\overline{\FieldZ})
+ {\E{\Psi}}{}' (z_2,x|\overline{\FieldZ}) {\E{\Psi}}{}' (z_2,y|\overline{\FieldZ})
\Bigr\} \\
& + \ \frac18\int\limits_{z_1,z_2\in\E{\Lat}}
\E{\FlucProp} (z_1,z_2|\overline{\FieldZ})
\E{\WPotloc}{}'(z_2,z_1|\overline{\FieldZ})
\E{\WPotloc}{}'(z_3,z_4|\overline{\FieldZ}) {\E{\Psi}}{}' (z_4,x|\overline{\FieldZ}) \\[-8pt]
&\qquad\qquad\qquad
\Bigl\{ \E{\FlucProp}(z_2,z_3|\overline{\FieldZ}){\E{\Psi}}{}' (z_3,y|\overline{\FieldZ})
+ \E{\FlucProp}(z_2,z_3|\overline{\FieldZ}){\E{\Psi}}{}' (z_4,y|\overline{\FieldZ}) \\
&\qquad\qquad\qquad\quad
+ \E{\FlucProp}(z_1,z_3|\overline{\FieldZ}){\E{\Psi}}{}' (z_3,y|\overline{\FieldZ})
+ \E{\FlucProp}(z_1,z_3|\overline{\FieldZ}){\E{\Psi}}{}' (z_4,y|\overline{\FieldZ})
\Bigr\} . \\
\end{split}
\label{recursionWloc}
\end{multline}
For constant fields we have translation symmetry. Therefore the derivative of the
potential must be independent of the coordinate:
\begin{equation}
\E{\VPot}_{,z}[\overline{\FieldE}] = {\E{\VPotloc}}{}'(\overline{\FieldE}) = \text{const.}
\end{equation}
To obtain the recursion relation of the effective potential we insert the
local approximation for $\E{\WPot}$:
\begin{multline}
{{\Z{\VPotloc}}{}'}(\overline{\FieldZ}) = \int\limits_{z} \E{\Psi}_{,x}(z|\overline{\FieldZ})
{\E{\VPotloc}}{}'(\overline{\FieldZ}) \\
- \frac14 \int\limits_{z_1,z_2} \E{\FlucProp} (z_1,z_2|\overline{\FieldZ})
\E{\WPotloc}{}'(z_2,z_1|\overline{\FieldZ})
\Bigl(\E{\Psi}_{,x}(z_1|\overline{\FieldZ})
+ \E{\Psi}_{,x}(z_2|\overline{\FieldZ})\Bigr)\,.
\label{VPrecursionLocalPre}
\end{multline}
We simplify this. By assumption of section \ref{introSection}
$$
\int_{x\in\Z{\Lat}}\E{\AvOp}(x,z) = 1 \,.
$$
Differentiating the constraint (\ref{MFAFieldEq2})
\begin{equation}
\int\limits_{z} \E{\AvOp}(x',z) \E{\Psi}_{,x}(z) = \delta(x'-x)\,,
\end{equation}
and integrating over $x'$
we see that
$$
\int_{z\in\E{\Lat}} \E{\Psi}_{,x}[\Phi](z) = 1 \,.
$$
This leads to the simpler recursion relation
\begin{multline}
{{\Z{\VPotloc}}{}'}(\overline{\FieldZ}) = {\E{\VPotloc}}{}'(\overline{\FieldZ}) \\
- \frac14 \int\limits_{z_1,z_2} \E{\FlucProp} (z_1,z_2|\overline{\FieldZ})
\E{\WPotloc}{}'(z_2,z_1|\overline{\FieldZ})
\Bigl(\E{\Psi}_{,x}(z_1|\overline{\FieldZ})
+ \E{\Psi}_{,x}(z_2|\overline{\FieldZ})\Bigr)\,.
\label{VPrecursionLocal}
\end{multline}
One can use this to determine the constraint effective potential, i.e.
the potential on
the last lattice $\Lambda ^N$ which consists of one point only.
|
1,108,101,565,457 | arxiv | \section{Introduction}
\begin{wrapfigure}[14]{o}{0.25\textwidth}
\vspace{-3em}
\begin{center}
\includegraphics[scale=1]{qt-example}
\end{center}
%
\caption{\label{qt-example}Quasi-thres. graph with thick skeleton, grey root and dashed transitive closure.}
\end{wrapfigure}
Quasi-Threshold graphs, also known as \emph{trivially perfect} graphs, are defined as the $P_4$- and $C_4$-free graphs, i.e., the graphs that do not contain a path or cycle of length~4 as node-induced subgraph~\cite{ycc-q-96}.
They can also be characterized as the transitive closure of rooted forests \cite{w-antcg-65}, as illustrated in Figure~\ref{qt-example}.
These forests can be seen as skeletons of quasi-threshold graphs.
Further a constructive characterization exists: Quasi-threshold graphs are the graphs that are closed under disjoint union and the addition of isolated nodes and nodes connected to every existing node~\cite{ycc-q-96}.
Linear time quasi-threshold recognition algorithms were proposed in~\cite{ycc-q-96} and in~\cite{c-albfs-08}.
Both construct a skeleton if the graph is a quasi-threshold graph.
Further, \cite{c-albfs-08} also finds a $C_4$ or $P_4$ if the graph is no quasi-threshold graph.
Nastos and Gao \cite{ng-f-13} observed that components of quasi-threshold graphs have many features in common with the informally defined notion of communities in social networks.
They propose to find a quasi-threshold graph that is close to a given graph in terms of edge edit distance in order to detect the communities of that graph.
Motivated by their insights we study the quasi-threshold graph editing problem in this paper.
Given a graph $G=(V,E)$ we want to find a quasi-threshold graph $G'=(V,E')$ which is closest to $G$, i.e., we want to minimize the number $k$ of edges in the symmetric difference of $E$ and $E'$.
Figure~\ref{qt-edit-example} illustrates %
\begin{wrapfigure}[15]{o}{0.35\textwidth}
\begin{center}
\includegraphics[scale=1]{edit-example}
\end{center}
\caption{\label{qt-edit-example}Edit example with solid input edges, dashed inserted edges, a crossed deleted edge, a thick skeleton with grey root.}%
\end{wrapfigure}
an edit example.
Unfortunately, the quasi-threshold graph editing problem is NP-hard~\cite{ng-f-13}.
However, the problem is fixed parameter tractable (FPT) in $k$ as it is defined using forbidden subgraphs~\cite{c-f-96}.
A basic bounded search tree algorithm which tries every of the 6 possible edits of a forbidden subgraph has a running time in $O(6^k \cdot (|V|+|E|))$.
In \cite{dp-apktp-14} a polynomial kernel of size $O(k^7)$ was introduced.
Unfortunately, our experiments show that real-world social networks have a prohibitively large amount of edits.
We prove lower bounds on real-world graphs for $k$ on the scale of $10^4$ and $10^5$.
A purely FPT-based algorithm with parameter $k$ can thus not scale in practice.
The only heuristic we are aware of was introduced by Nastos and Gao~\cite{ng-f-13} but it examines all $\Theta(|V|^2)$ possible edits in each greedy editing step and thus needs time $\Omega(k\cdot|V|^2)$.
Even though this running time is polynomial it is still prohibitive for large graphs.
In this paper we fill this gap by introducing Quasi-Threshold Mover (QTM), the first scalable quasi-threshold editing algorithm.
The final aim of our research is to determine whether quasi-threshold editing is a useful community detection algorithm.
Designing an algorithm able of solving the quasi-threshold editing problem on large real-world graphs is a first step in this direction.
\subsection{Our Contribution}
Our main contribution is Quasi-Threshold Mover (QTM), a scalable quasi-threshold editing algorithm.
We provide an extensive experimental evaluation on generated as well as a variety of real-world graphs.
We further propose a simplified certifying quasi-threshold recognition algorithm.
QTM works in two phases: An initial skeleton forest is constructed by a variant of our recognition algorithm, and then refined by moving one node at a time to reduce the number of edits required.
The running time of the first phase is dominated by the time needed to count the number of triangles per edge.
The best current triangle counting algorithms run in $O(|E|\alpha(G))$ \cite{cn-asla-85,ob-tlabd-14} time, where $\alpha(G)$ is the arboricity.
These algorithms are efficient and scalable in practice on the considered graphs.
One round of the second phase needs $O(|V|+|E|\log \Delta)$ time, where $\Delta$ is the maximum degree.
We show that four rounds are enough to achieve good results.
\subsection{Preliminaries}
We consider simple, undirected graphs $G = (V, E)$ with $n = |V|$ nodes and $m = |E|$ edges.
For $v \in V$ let $N(v)$ be the adjacent nodes of $v$.
Let $d(v) := |N(v)|$ for $v \in V$ be the degree of $v$ and $\Delta$ the maximum degree in $G$.
Whenever we consider a skeleton forest, we denote by $p(u)$ the parent of a node $u$.
\section{Lower Bounds}
A lot of previous research has focused on FPT-based algorithms.
To show that no purely FPT-based algorithm parameterized in the number of edits can solve the problem we compute lower bounds on the number of edits required for real-world graphs.
The lower bounds used by us are far from tight.
However, the bounds are large enough to show that any algorithm with a running time superpolynomial in $k$ can not scale.
To edit a graph we must destroy all forbidden subgraphs $H$.
For quasi-threshold editing $H$ is either a $P_4$ or a $C_4$.
This leads to the following basic algorithm: Find forbidden subgraph $H$, increase the lower bound, remove all nodes of $H$, repeat.
This is correct as at least one edit incident to $H$ is necessary.
If multiple edits are needed then accounting only for one is a lower bound.
We can optimize this algorithm by observing that not all nodes of $H$ have to be removed.
If $H$ is a $P_4$ with the structure $A-B-C-D$ it is enough to remove the two central nodes $B$ and $C$.
If $H$ is a $C_4$ with nodes $A$, $B$, $C$, and $D$ then it is enough to remove two adjacent nodes.
Denote by $B$ and $C$ the removed nodes.
This optimization is correct if at least one edit incident to $B$ or $C$ is needed.
Regardless of whether $H$ is a $P_4$ or a $C_4$ the only edit not incident to $B$ or $C$ is inserting or deleting $\{A, D\}$.
However, this edit only transforms a $P_4$ into a $C_4$ or vice versa.
A subsequent edit incident to $B$ or $C$ is thus necessary.
$H$ can be found using the recognition algorithm.
However, the resulting running time of $O(k(n+m))$ does not scale to the large graphs.
In the appendix we describe a running time optimization to accelerate computations.
\section{Linear Recognition and Initial Editing}
\label{sec:linear_recognition}
The first linear time recognition algorithm for quasi-threshold graphs was proposed in \cite{ycc-q-96}.
In \cite{c-albfs-08}, a linear time certifying recognition algorithm based on lexicographic breadth first search was presented.
However, as the authors note, sorted node partitions and linked lists are needed, which result in large constants behind the big-O.
We simplify their algorithm to only require arrays but still provide negative and positive certificates.
Further we only need to sort the nodes once to iterate over them by decreasing degree.
Our algorithm constructs the forest skeleton of a graph $G$.
If it succeeds $G$ is a quasi threshold graph and outputs for each node $v$ a parent node $p(v)$.
If it fails it outputs a forbidden subgraph $H$.
To simplify our algorithm we start by adding a super node $r$ to $G$ that is connected to every node and obtain $G'$.
$G$ is a quasi threshold graph if and only if $G'$ is one.
As $G'$ is connected its skeleton is a tree.
A core observation is that higher nodes in the tree must have higher degrees, i.e., $d(v)\le d(p(v))$.
We therefore know that $r$ must be the root of the tree.
Initially we set $p(u)=r$ for every node $u$.
We process all remaining nodes ordered decreasingly by degree.
Once a node is processed its position in the tree is fixed.
Denote by $u$ the node that should be processed next.
We iterate over all non-processed neighbors $v$ of $u$ and check whether $p(u)=p(v)$ holds and afterwards set $p(v)$ to $u$.
If $p(u)=p(v)$ never fails then $G$ is a quasi-threshold graph as for every node $x$ (except $r$) we have that by construction that the neighborhood of $x$ is a subset of the one of $p(x)$.
If $p(u)\neq p(v)$ holds at some point then a forbidden subgraph $H$ exists.
Either $p(u)$ or $p(v)$ was processed first.
Assume without lose of generality that it was $p(v)$.
We know that no edge $(v, p(u))$ can exist because otherwise $p(u)$ would have assigned itself as parent of $v$ when it was processed.
Further we know that $p(u)$'s degree can not be smaller than $u$'s degree as $p(u)$ was processed before $u$.
As $v$ is a neighbor of $u$ we know that another node $x$ must exist that is a neighbor of $p(u)$ but not of $u$, i.e., $(u, x)$ does not exist.
The subgraph $H$ induced by the 4-chain $v-u-p(u)-x$ is thus a $P_4$ or $C_4$ depending on whether the edge $(v, x)$ exists.
We have that $u\neq r$ as $u$ is processed by the algorithm and $v\neq r$ as its degree is at most $d(u)$.
Further $p(u)\neq r$ as $p(v)$ was processed before $p(u)$ and $x\neq r$ as $r$ is a neighbor of $u$.
$H$ therefore does not use $r$ and is contained in $G$.
\paragraph{From Recognition to Editing.}
\label{sec:linear_editing}
We modify the recognition algorithm to construct a skeleton for arbitrary graphs.
This skeleton induces a quasi threshold graph $Q$.
We want to minimize $Q$'s distance to $G$.
Note that all edits are performed implicitly, we do not actually modify the input graph for efficiency reasons.
The only difference between our recognition and our editing algorithm is what happens when we process a node $u$ that has a non-processed neighbor $v$ with $p(u)\neq p(v)$.
The recognition algorithm constructs a forbidden subgraph $H$, while the editing algorithm tries to resolve the problem.
We have three options for resolving the problem: we ignore the edge $\{u, v\}$, we set $p(v)$ to $p(u)$, or we set $p(u)$ to $p(v)$.
The last option differs from the first two as it affects all neighbors of $u$.
The first two options are the decision if we want to make $v$ a child of $u$ even though $p(u) \neq p(v)$ or if we want to ignore this potential child.
We start by determining a preliminary set of children by deciding for each non-processed neighbor of~$u$ whether we want to keep or discard it.
These preliminary children elect a new parent by majority.
We set $p(u)$ to this new parent.
Changing~$u$'s parent can change which neighbors are kept.
We therefore reevaluate all the decisions and obtain a final set of children for which we set $u$ as parent.
Then the algorithm simply continues with the next node.
What remains to describe is when our algorithm keeps a potential child.
It does this using two edge measures: The number of triangles $t(e)$ in which an edge $e$ participates and a pseudo-$C_4$-$P_4$-counter $p_{\mathrm{c}}(e)$, which is the sum of the number of $C_4$ in which $e$ participates and the number of $P_4$ in which $e$ participates as central edge.
Computing $p_{\mathrm{c}}(x,y)$ is easy given the number of triangles and the degrees of $x$ and $y$ as $p_{\mathrm{c}}(\{x, y\}) = (d(x) - 1 - t(\{x, y\}))\cdot(d(y) - 1 - t(\{x, y\}))$ holds.
Having a high $p_{\mathrm{c}}(e)$ makes it likely that $e$ should be deleted.
We keep a potential child only if two conditions hold.
The first is based on triangles.
We know by construction that both $u$ and $v$ have many edges in $G$ towards their current ancestors.
Keeping $v$ is thus only useful if $u$ and $v$ share a large number of ancestors as otherwise the number of induced edits is too high.
Each common ancestor of $u$ and $v$ results in a triangle involving the edge $\{u,v\}$ in $Q$.
Many of these triangles should also be contained in $G$.
We therefore count the triangles of $\{u,v\}$ in $G$ and check whether there are at least as many triangles as $v$ has ancestors.
The other condition uses $p_{\mathrm{c}}(e)$.
The decision whether we keep $v$ is in essence the question of whether $\{u, v\}$ or $\{v, p(v)\}$ should be in $Q$.
We only keep $v$ if $p_{\mathrm{c}}(\{u, v\})$ is not higher than $p_{\mathrm{c}}(\{v, p(v)\})$.
The details of the algorithm can be found in the appendix.
The time complexity of this heuristic editing algorithm is dominated by the triangle counting algorithm as the rest is linear.
\section{The Quasi-Threshold Mover Algorithm}
\label{sec:local_moving}
\begin{figure}[tb]
\centering
\begin{subfigure}[b]{0.68\textwidth}%
\begin{algorithm}[H]
\ForEach{$v_m$-neighbor $u$}{
push $u$\;
}
\While{queue not empty}{
$u\leftarrow$ pop\;
determine $\mathrm{child}_{\mathrm{close}}(u)$ by DFS\;
$x\leftarrow \max$ over $\mathrm{score}_{\mathrm{max}}$ of reported $u$-children\;
$y\leftarrow \sum$ over $\mathrm{child}_{\mathrm{close}}$ of close $u$-children\;
\uIf{$u$ is $v_m$-neighbor}{
$\mathrm{score}_{\mathrm{max}}(u)\leftarrow \max\{x,y\} + 1$\;
}
\Else{
$\mathrm{score}_{\mathrm{max}}(u)\leftarrow \max\{x,y\} - 1$\;
}
\If{$\mathrm{child}_{\mathrm{close}}(u)>0$ or $\mathrm{score}_{\mathrm{max}}(u)>0$}{
report $u$ to $p(u)$\;
push $p(u)$\;
}
}
Best $v_m$-parent corresponds to $\mathrm{score}_{\mathrm{max}}(r)$\;
\end{algorithm}%
\caption{Pseudo-Code for moving $v_m$}%
\label{fig:moving-v-m-pseudo-code}
\end{subfigure}%
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
\includegraphics[width=2.2cm]{move.pdf}
\caption{ Moving $v_m$ example. }
\label{fig:move-v-m}
\end{subfigure}
\caption{In Figure \ref{fig:move-v-m} the drawn edges are in the skeleton. Crossed edges are removed while thick blue edges are inserted by moving $v_m$. $a$ is not adopted while $b$ is.}
\end{figure}
Our algorithm iteratively increases the quality of a skeleton $T$ using an algorithm based on local moving.
Local moving is a successful technique that is employed in many heuristic community detection algorithms \cite{bgll-f-08,gkw-edcgc-14,rn-m-11}.
As in most algorithm based on this principle, our algorithm works in rounds.
In each round it iterates over all nodes $v_m$ in random order and tries to move $v_m$.
In the context of community detection, a node is moved to a neighboring community such that a certain objective function is increased.
In our setting we want to minimize the number of edits needed to transform the input graph $G$ into the quasi-threshold graph $Q$ implicitly defined by $T$.
We need to define the set of allowed moves for $v_m$ in our setting.
Moving $v_m$ consists of moving $v_m$ to a different position within $T$ and is illustrated in Figure~\ref{fig:move-v-m}.
We need to chose a new parent $u$ for $v_m$.
The new parent of $v_m$'s old children is $v_m$'s old parent.
Besides choosing the new parent $u$ we select a set of children of $u$ that are \emph{adopted} by $v_m$, i.e., their new parent becomes $v_m$.
Among all allowed moves for $v_m$ we chose the move that reduces the number of edits as much as possible.
Doing this in sub-quadratic running time is difficult as $v_m$ might be moved anywhere in $G$.
By only considering the neighbors of $v_m$ in $G$ and a few more nodes per neighbor in a bottom-up scan in the skeleton, our algorithm has a running time in $O(n+m \log \Delta)$ per round.
While our algorithm is not guaranteed to be optimal as a whole we can prove that for each node $v_m$ we choose a move that reduces the number of edits as much as possible.
Our experiments show that given the result of the initialization heuristic our moving algorithm performs well in practice.
They further show that in practice four rounds are good enough which results in a near-linear total running time.
\paragraph{Basic Idea.}
Our algorithm starts by isolating $v_m$, i.e., removing all incident edges in $Q$.
It then finds a position at which $v_m$ should be inserted in $T$.
If $v_m$'s original position was optimal then it will find this position again.
For simplicity we will assume again that we add a virtual root $r$ that is connected to all nodes.
Isolating $v_m$ thus means that we move $v_m$ below the root $r$ and do not adopt any children.
Choosing $u$ as parent of $v_m$ requires $Q$ to contain edges from all ancestors of $u$ to $v_m$.
Further if $v_m$ adopts a child $w$ of $u$ then $Q$ must have an edge from every descendant of $w$ to $v_m$.
How good a move is depends on how many of these edges already exist in $G$ and how many edges incident to $v_m$ in $G$ are not covered.
To simplify notation we will refer to the nodes incident to $v_m$ in $G$ as \emph{$v_m$-neighbors}.
We start by identifying which children a node should adopt.
For this we define the \emph{child closeness} $\mathrm{child}_{\mathrm{close}}(u)$ of $u$ as the number of $v_m$-neighbors in the subtree of $u$ minus the non-$v_m$-neighbors.
A node $u$ is a \emph{close child} if $\mathrm{child}_{\mathrm{close}}(u)>0$.
If $v_m$ chooses a node $u$ as new parent then it should adopt all close children.
A node can only be a close child if it is a neighbor of $v_m$ or when it has a close child.
Our algorithm starts by computing all close children and their closeness using many short DFS searches in a bottom up fashion.
Knowing which nodes are good children we can identify which nodes are good parents for $v_m$.
A potential parent must have a close child or must be a neighbor of $v_m$.
Using the set of close children we can easily derive a set of parent candidates and an optimal selection of adopted children for every potential parent.
We need to determine the candidate with the fewest edits.
We do this in a bottom-up fashion.%
To implement the described moving algorithm we need to put $O(d_G(v_m))$ elements into a priority queue.
The running time is thus amortized $O(d_G(v_m) \log d_G(v_m))$ per move or $O(n+m \log \Delta)$ per round.
We start many small searches and analyze their running time complexity using tokens.
Initially only the $v_m$-neighbors have tokens.
A search consumes a token per step.
The details of the analysis are complex and are described in the appendix.
\paragraph{Close Children.}
To find all close children we attach to each node $u$ a DFS instance that explores the subtree of $u$.
Note that every DFS instance has a constant state size and thus the memory consumption is still linear.
$u$ is close if this DFS finds more $v_m$-neighbors than non-$v_m$-neighbors.
Unfortunately we can not fully run all these searches as this requires too much running time.
Therefore a DFS is aborted if it finds more non-$v_m$-neighbors than $v_m$-neighbors.
We exploit that close children are $v_m$-neighbors or have themselves close children.
Initially we fill a queue of potential close children with the neighbors of $v_m$ and when a new close child is found we add its parent to the queue.
Let $u$ denote the current node removed from the queue.
We run $u$'s DFS and if it explores the whole subtree then $u$ is a close child.
We need to take special care that every node is visited only by one DFS.
A DFS therefore looks at the states of the DFS of the nodes it visits.
If one of these other DFS has run then it uses their state information to skip the already explored part of the subtree.
To avoid that a DFS is run after its state was inspected we organize the queue as priority queue ordered by tree depth.
If the DFS of $u$ starts by first inspecting the wrong children then it can get stuck because it would see the $v_m$-neighbors too late.
The DFS must first visit the close children of $u$.
To assure that $u$ knows which children are close every close child must report itself to its parent when it is detected.
As all children have a greater depth they are detected before the DFS of their parent starts.
\paragraph{Potential Parents.}
Suppose we consider the subtree $T_u$ of $u$ and $w$ is a potential parent in $T_u$.
Consider the set of nodes $X_w$ given by the ancestors of $w$ and the descendants of all close children of $w$.
$X_w$ includes $w$ and its close children themselves.
Moving $v_m$ below $w$ requires us to insert an edge from $v_m$ to every non-$v_m$-neighbor in $X_w$.
We therefore want $X_w$ to maximize the number of $v_m$-neighbors minus the number of non-$v_m$-neighbors.
This value gives us a score for each potential parent in $T_u$.
We denote by $\mathrm{score}_{\mathrm{max}}(u)$ the maximum score over all potential parents in $T_u$.
Note that $\mathrm{score}_{\mathrm{max}}(u)$ is always at least -1 as we can move $v_m$ below $u$ and not adopt any children.
We determine in a bottom-up fashion all $\mathrm{score}_{\mathrm{max}}(u)$ that are greater than 0.
Whether $\mathrm{score}_{\mathrm{max}}(u)$ is -1 or 0 is irrelevant because isolating $v_m$ is never worse.
The final solution will be in $\mathrm{score}_{\mathrm{max}}(r)$ of the root $r$ as its ``subtree'' encompasses the whole graph.
$\mathrm{score}_{\mathrm{max}}(u)$ can be computed recursively.
If $u$ is a best parent then the value of $\mathrm{score}_{\mathrm{max}}(u)$ is the sum over the closenesses of all of $u$'s close children $\pm 1$.
If the subtree $T_w$ of a child $w$ of $u$ contains a best parent then $\mathrm{score}_{\mathrm{max}}(u)=\mathrm{score}_{\mathrm{max}}(w)\pm 1$.
The $\pm 1$ depends on whether $w$ is a $v_m$-neighbor.
Unfortunately not only potential parents $u$ have a $\mathrm{score}_{\mathrm{max}}(u)>0$.
However, we know that every node $u$ with $\mathrm{score}_{\mathrm{max}}(u)>0$ is a $v_m$-neighbor or has a child $w$ with $\mathrm{score}_{\mathrm{max}}(w)>0$.
We can therefore process all $\mathrm{score}_{\mathrm{max}}$ values in a similar bottom-up way using a tree-depth ordered priority queue as we used to compute $\mathrm{child}_{\mathrm{close}}$.
As both bottom-up procedures have the same structure we can interweave them as optimization and use only a single queue.
The algorithm is illustrated in Figure~\ref{fig:moving-v-m-pseudo-code} in pseudo-code form.
\section{Experimental Evaluation}
\label{sec:exp}
We evaluated the QTM algorithm on the small instances used by Nastos and Gao~\cite{ng-f-13}, on larger generated graphs and large real-world social networks and web graphs.
We measured both the number of edits needed and the required running time.
For each graph we also report the lower bound $b$ of necessary edits that we obtained using our lower bound algorithm.
We implemented the algorithms in C++ using NetworKit~\cite{ssm-nkait-14}.
All experiments were performed on an Intel Core i7-2600K CPU with 32GB RAM.
We ran all algorithms ten times with ten different random node id permutations.
\paragraph{Comparison with Nastos and Gao's Results.}
Nastos and Gao~\cite{ng-f-13} did not report any running times, we therefore re-implemented their algorithm.
Our implementation of their algorithm has a complexity of $O(m^2 + k \cdot n^2 \cdot m)$, the details can be found in the appendix.
Similar to their implementation we used a simple exact bounded search tree (BST) algorithm for the last 10 edits.
In Table~\ref{tab:resultsng} we report the minimum and average number of edits over ten runs.
Our implementation of their algorithm never needs more edits than they reported\footnote{Except on Karate, where they report 20 due to a typo. They also need 21 edits.}.
Often our implementation needs slightly less edits due to different tie-breaking rules.
For all but one graph QTM is at least as good as the algorithm of Nastos and Gao in terms of edits.
QTM needs only one more edit than Nastos and Gao for the grass\_web graph.
The QTM algorithm is much faster than their algorithm, it needs at most 2.5 milliseconds while the heuristic of Nastos and Gao needs up to 6 seconds without bounded search tree and almost 17 seconds with bounded search tree.
The number of iterations necessary is at most 5. As the last round only checks whether we are finished four iterations would be enough.
\begin{table}[htb]
\centering
\caption{Comparison of QTM and \cite{ng-f-13}. We report $n$ and $m$, the lower bound $b$, the number of edits (as minimum, mean and standard deviation), the mean and maximum of number of QTM iterations, and running times in ms.}
\label{tab:resultsng}
\begin{tabular}{lr@{~~}r@{~~}r@{~~}l@{~}|r@{~~}r@{~~}r@{~}|rr@{~}|r@{~~}r}
Name & $n$ & $m$ & $b$ & Algorithm & \multicolumn{3}{c|}{Edits} & \multicolumn{2}{c|}{Iterations} & \multicolumn{2}{c}{Time [ms]} \\
& & & & & min & mean & std & mean & max & mean & std \\
\midrule
\multirow{3}{*}{dolphins} & \multirow{3}{*}{62} & \multirow{3}{*}{159} & \multirow{3}{*}{24} & QTM & 72 & 74.1 & 1.1 & 2.7 & 4.0 & 0.6 & 0.1 \\
& & & & NG w/ BST & 73& 74.7 & 0.9 & - & - & 15\,594.0 & 2\,019.0 \\
& & & & NG w/o BST & 73& 74.8 & 0.8 & - & - & 301.3 & 4.0 \\[0.3em]
\multirow{3}{*}{football} & \multirow{3}{*}{115} & \multirow{3}{*}{613} & \multirow{3}{*}{52} & QTM & 251 & 254.3 & 2.7 & 3.5 & 4.0 & 2.5 & 0.4 \\
& & & & NG w/ BST & 255& 255.0 & 0.0 & - & - & 16\,623.3 & 3\,640.6 \\
& & & & NG w/o BST & 255& 255.0 & 0.0 & - & - & 6\,234.6 & 37.7 \\[0.3em]
\multirow{3}{*}{grass\_web} & \multirow{3}{*}{86} & \multirow{3}{*}{113} & \multirow{3}{*}{10} & QTM & 35 & 35.2 & 0.4 & 2.0 & 2.0 & 0.5 & 0.1 \\
& & & & NG w/ BST & 34& 34.6 & 0.5 & - & - & 13\,020.0 & 3\,909.8 \\
& & & & NG w/o BST & 38& 38.0 & 0.0 & - & - & 184.6 & 1.2 \\[0.3em]
\multirow{3}{*}{karate} & \multirow{3}{*}{34} & \multirow{3}{*}{78} & \multirow{3}{*}{8} & QTM & 21 & 21.2 & 0.4 & 2.0 & 2.0 & 0.4 & 0.1 \\
& & & & NG w/ BST & 21& 21.0 & 0.0 & - & - & 9\,676.6 & 607.4 \\
& & & & NG w/o BST & 21& 21.0 & 0.0 & - & - & 28.1 & 0.3 \\[0.3em]
\multirow{3}{*}{lesmis} & \multirow{3}{*}{77} & \multirow{3}{*}{254} & \multirow{3}{*}{13} & QTM & 60 & 60.5 & 0.5 & 3.3 & 5.0 & 1.4 & 0.3 \\
& & & & NG w/ BST & 60& 60.8 & 1.0 & - & - & 16\,919.1 & 3\,487.7 \\
& & & & NG w/o BST & 60& 77.1 & 32.4 & - & - & 625.0 & 226.4 \\
\end{tabular}
\end{table}
\paragraph{Large Graphs.}
\begin{table}[htbp]
\caption{Results for large real-world and generated graphs. Number of nodes $n$ and edges $m$, the lower bound $b$ and the number of edits are reported in thousands. Column ``I'' indicates whether we start with a trivial skeleton or not. $\bullet$ indicates an initial skeleton as described in Section~\ref{sec:linear_editing} and $\circ$ indicates a trivial skeleton. Edits and running time are reported for a maximum number of 0 (respectively 1 for a trivial initial skeleton), 4 and $\infty$ iterations. For the latter, the number of actually needed iterations is reported as ``It''. Edits, iterations and running time are the average over the ten runs.}
\label{tab:real-world}
\centering
\begin{tabular}{l@{~}|@{~}lr@{~~}r@{~~}l@{~}|@{~}r@{~~}r@{~~}r@{~}|r@{~}|@{~}r@{~~}r@{~~}r}
& Name & $n$ {[}K{]} & $b$ {[}K{]} & I & \multicolumn{3}{c|}{Edits {[}K{]}} & It & \multicolumn{3}{c}{Time {[}s{]}}\tabularnewline
& & $m$ {[}K{]} & & & 0/1 & 4 & $\infty$ & $\infty$ & 0/1 & 4 & $\infty$\tabularnewline
\midrule
\multirow{14}{*}{\begin{sideways}\hspace{-1em}
Social Networks
\end{sideways}} &
\multirow{2}{*}{Caltech} & 0.77 & \multirow{2}{*}{0.35} & $\bullet$ & 15.8 & 11.6 & 11.6 & 8.5 & 0.0 & 0.0 & 0.1\tabularnewline
& & 16.66 & & $\circ$ & 12.6 & 11.7 & 11.6 & 9.4 & 0.0 & 0.0 & 0.1\tabularnewline[0.3em]
& \multirow{2}{*}{amazon} & 335 & \multirow{2}{*}{99.4} & $\bullet$ & 495 & 392 & 392 & 7.2 & 0.3 & 5.5 & 9.3\tabularnewline
& & 926 & & $\circ$ & 433 & 403 & 403 & 8.9 & 1.3 & 4.9 & 10.7\tabularnewline[0.3em]
& \multirow{2}{*}{dblp} & 317 & \multirow{2}{*}{53.7} & $\bullet$ & 478 & 415 & 415 & 7.2 & 0.4 & 5.8 & 9.9\tabularnewline
& & 1\,050 & & $\circ$ & 444 & 424 & 423 & 9.0 & 1.4 & 5.2 & 11.5\tabularnewline[0.3em]
& \multirow{2}{*}{Penn} & 41.6 & \multirow{2}{*}{19.9} & $\bullet$ & 1\,499 & 1\,129 & 1\,127 & 14.4 & 0.6 & 4.2 & 13.5\tabularnewline
& & 1\,362 & & $\circ$ & 1\,174 & 1\,133 & 1\,129 & 16.2 & 1.0 & 3.7 & 14.4\tabularnewline[0.3em]
& \multirow{2}{*}{youtube} & 1\,135 & \multirow{2}{*}{139} & $\bullet$ & 2\,169 & 1\,961 & 1\,961 & 9.8 & 1.4 & 31.3 & 73.6\tabularnewline
& & 2\,988 & & $\circ$ & 2\,007 & 1\,983 & 1\,983 & 10.0 & 7.1 & 28.9 & 72.7\tabularnewline[0.3em]
& \multirow{2}{*}{lj} & 3\,998 & \multirow{2}{*}{1\,335} & $\bullet$ & 32\,451 & 25\,607 & 25\,577 & 18.8 & 23.5 & 241.9 & 1\,036.0\tabularnewline
& & 34\,681 & & $\circ$ & 26\,794 & 25\,803 & 25\,749 & 19.9 & 58.3 & 225.9 & 1\,101.3\tabularnewline[0.3em]
& \multirow{2}{*}{orkut} & 3\,072 & \multirow{2}{*}{1\,480} & $\bullet$ & 133\,086 & 103\,426 & 103\,278 & 24.2 & 115.2 & 866.4 & 4\,601.3\tabularnewline
& & 117\,185 & & $\circ$ & 106\,367 & 103\,786 & 103\,507 & 30.2 & 187.9 & 738.4 & 5\,538.5\tabularnewline
\midrule
\multirow{8}{*}{\begin{sideways}\hspace{-1em}
Web Graphs
\end{sideways}} &
\multirow{2}{*}{cnr-2000} & 326 & \multirow{2}{*}{48.7} & $\bullet$ & 1\,028 & 409 & 407 & 11.2 & 0.8 & 12.8 & 33.8\tabularnewline
& & 2\,739 & & $\circ$ & 502 & 410 & 409 & 10.7 & 3.2 & 11.8 & 30.8\tabularnewline[0.3em]
& \multirow{2}{*}{in-2004} & 1\,383 & \multirow{2}{*}{195} & $\bullet$ & 2\,700 & 1\,402 & 1\,401 & 11.0 & 7.9 & 72.4 & 182.3\tabularnewline
& & 13\,591 & & $\circ$ & 1\,909 & 1\,392 & 1\,389 & 13.5 & 16.6 & 65.0 & 217.6\tabularnewline[0.3em]
& \multirow{2}{*}{eu-2005} & 863 & \multirow{2}{*}{229} & $\bullet$ & 7\,613 & 3\,917 & 3\,906 & 13.7 & 6.9 & 90.7 & 287.7\tabularnewline
& & 16\,139 & & $\circ$ & 4\,690 & 3\,919 & 3\,910 & 14.5 & 22.6 & 85.6 & 303.5\tabularnewline[0.3em]
& \multirow{2}{*}{uk-2002} & 18\,520 & \multirow{2}{*}{2\,966} & $\bullet$ & 68\,969 & 31\,218 & 31\,178 & 19.1 & 200.6 & 1\,638.0 & 6\,875.5\tabularnewline
& & 261\,787 & & $\circ$ & 42\,193 & 31\,092 & 31\,042 & 22.3 & 399.8 & 1\,609.6 & 8\,651.8\tabularnewline
\midrule
\multirow{4}{*}{\begin{sideways}\hspace{-0.5em}Generated\end{sideways}} & Gen. & 100 & \multirow{2}{*}{42} & $\bullet$ & 200 & 158 & 158 & 4.6 & 0.2 & 3.5 & 4.1\tabularnewline
& 160K & 930 & & $\circ$ & 193 & 158 & 158 & 6.1 & 1.0 & 3.3 & 4.9\tabularnewline[0.3em]
& Gen. & 1\,000 & \multirow{2}{*}{0.391} & $\bullet$ & 1.161 & 0.395 & 0.395 & 3.0 & 3.3 & 43.8 & 43.8\tabularnewline
& 0.4K & 10\,649 & & $\circ$ & 182 & 5.52 & 5.52 & 6.1 & 15.9 & 52.9 & 78.8\tabularnewline
\end{tabular}
\end{table}
For the results in Table~\ref{tab:real-world} we used two Facebook graphs \cite{tmp-sf-12} and five SNAP graphs \cite{lk-snapd-14} as social networks and four web graphs from the 10th DIMACS Implementation Challenge \cite{bmsw-gpgcd-13,bcsv-ucasf-04,brsv-llpam-11,bv-twgfi-04}. We evaluate two variants of QTM. The first is the standard variant which starts with a non-trivial skeleton obtained by the heuristic described in Section~\ref{sec:linear_editing}. The second variant starts with a trivial skeleton where every node is a root. We chose these two variants to determine which part of our algorithm has which influence on the final result. For the standard variant we report the number of edits needed before any node is moved. With a trivial skeleton this number is meaningless and thus we report the number of edits after one round. All other measures are straightforward and are explained in the table's caption.
Even though for some of the graphs the mover needs more than 20 iterations to terminate, the results do not change significantly compared to the results after round 4.
In practice we can thus stop after 4 rounds without incurring a significant quality penalty.
It is interesting to see that for the social networks the initialization algorithm sometimes produces a skeleton that induces more than $m$ edits (e.g.\ in the case of the ``Penn'' graph) but still the results are always slightly better than with a trivial initial skeleton.
This is even true when we do not abort moving after 4 rounds.
For the web graphs, the non-trivial initial skeleton does not seem to be useful for some graphs.
It is not only that the initial number of edits is much higher than the finally needed number of edits, also the number of edits needed in the end is slightly higher than if a trivial initial skeleton was used.
This might be explained by the fact that we designed the initialization algorithm with social networks in mind.
Initial skeleton heuristics built specifically for web graphs could perform better.
While the QTM algorithm needs to edit between approximately 50 and 80\% of the edges of the social networks, the edits of the web graphs are only between 10 and 25\% of the edges.
This suggests that quasi-threshold graphs might be a good model for web graphs while for social networks they represent only a core of the graph that is hidden by a lot of noise.
Concerning the running time one can clearly see that QTM is scalable and suitable for large real-world networks.
As we cannot show for our real-world networks that the edit distance that we get is close to the optimum we generated graphs by generating quasi-threshold graphs and applying random edits to these graphs.
The details of the generation process are described in the appendix.
In Table~\ref{tab:real-world} we report the results of two of these graphs with $400$ and $160\,000$ random edits.
In both cases the number of edits the QTM algorithm finds is below or equal to the generated editing distance.
If we start with a trivial skeleton, the resulting edit distance is sometimes very high, as can be seen for the graph with 400 edits.
This shows that the initialization algorithm from Section~\ref{sec:linear_editing} is necessary to achieve good quality on graphs that need only few edits.
As it seems to be beneficial for most graphs and not very bad for the rest, we suggest to use the initialization algorithm for all graphs.
\begin{wrapfigure}[15]{o}{0.4\textwidth}
\includegraphics[width=\linewidth]{Caltech-Edited}
\caption{Edited Caltech network, edges colored by dormitories of endpoints.}
\label{fig:caltech}
\end{wrapfigure}
\paragraph{Case Study: Caltech.} The main application of our work is community detection.
While a thorough experimental evaluation of its usefulness in this context is future work we want to give a promising outlook.
Figure~\ref{fig:caltech} depicts the edited Caltech university Facebook network from~\cite{tmp-sf-12}.
Nodes are students and edges are Facebook-friendships.
The dormitories of most students are known.
We colored the graph according to this ground-truth.
The picture clearly shows that our algorithm succeeds at identifying most of this structure.
\section{Conclusion}
We have introduced Quasi-Threshold Mover (QTM), the first heuristic algorithm to solve the quasi-threshold editing problem in practice for large graphs.
As a side result we have presented a simple certifying linear-time algorithm for the quasi-threshold recognition problem.
A variant of our recognition algorithm is also used as initialization for the QTM algorithm.
In an extensive experimental study with large real world networks we have shown that it scales very well in practice.
We generated graphs by applying random edits to quasi-threshold graphs.
QTM succeeds on these random graphs and often even finds other quasi-threshold graphs that are closer to the edited graph than the original quasi-threshold graph.
A surprising result is that web graphs are much closer to quasi-threshold graphs than social networks, for which quasi-threshold graphs were introduced as community detection method.
A logical next step is a closer examination of the detected quasi-threshold graphs and the community structure they induce.
Further our QTM algorithm might be adapted for the more restricted problem of threshold editing which is NP-hard as well.\footnote{P{\aa}l Gr{\o}n{\aa}s Drange, personal communication (planned ESA submission)}
\paragraph{\textbf{Acknowledgment:}} We thank James Nastos for helpful discussions.
\printbibliography[segment=1]
\end{refsegment}
\newpage
\begin{refsegment}
\begin{appendix}
\section{Fast Computation of Lower Bounds}
As outlined in the main paper, the idea for computing lower bounds is to find a $C_4$ or $P_4$ and to remove two of the nodes, two neighboring nodes in the case of a $C_4$, the two central nodes in the case of the $P_4$, such that we destroy as few $C_4$ and $P_4$ as possible.
Finding a single $P_4$ or $C_4$ is possible in linear time using the certifying recognition algorithm.
The challenge when designing a fast algorithm for computing lower bounds is that the lower bound can be $n/2$ which results in a quadratic algorithm.
It is enough if we identify all central edges of a $P_4$ as we only want to remove the two nodes that are incident to that edge anyway.
For a $C_4$ it is also enough if we can identify any edge it is part of as we also just want to remove the two incident nodes.
Therefore it is enough if we can quickly find an edge that is part of a $C_4$ or a central edge of a $P_4$.
If we consider the neighbors of the two nodes $u$ and $v$ and want to find a $C_4$ or a $P_4$ where $\{u, v\}$ is the central edge, we only need to find two nodes $x \in N(u) \setminus \{v\}$ and $y \in N(v) \setminus \{u\}$ such that $x \notin N(v)$ and $y \notin N(u)$.
Common neighbors of $u$ and $v$ thus cannot be chosen for $x$ and $y$, however all other neighbors besides $u$ and $v$ can be chosen.
Therefore we know that such two nodes exist whenever $p_{\mathrm{c}}(\{u, v\}) = (d(u) - 1 - t(\{u, v\}))\cdot(d(v) - 1 - t(\{u, v\})) > 0$.
The algorithm we choose is based on this observation.
Initially, we count the triangles per edge for all edges.
Then we iterate over all nodes and for each node $u$ we choose a neighbor $v$ such that $p_{\mathrm{c}}(\{u, v\}) > 0$ and remove $u$ and $v$.
After removing $u$ and $v$ we update the triangle counters accordingly.
In order to destroy not too many $P_4$ and $C_4$, we sort the nodes initially be degree in ascending order.
We also choose the neighbor $v$ of $u$ such that the degree of $v$ is minimal.
Note that the initial iteration order does not necessarily reflect the degree order anymore after removing some of the nodes.
Given a graph structure that allows removing a node $u$ in amortized time $O(d(u))$ the whole algorithm can actually be implemented in time $O(\alpha(G)m)$ with $O(\alpha(G)m)$ memory consumption.
The running time $O(\alpha(G)m)$ comes from triangle listing \cite{cn-asla-85}.
The main idea is that we store for each edge the pairs of edges which form a triangle.
Whenever we delete a node, we check for each edge for all stored pairs if the two other edges still exist, and if yes, decrease their counter.
As we delete each edge only once this gives a total running time of $O(\alpha(G)m)$.
However in practice we found that the required amount of storage was too high for our compute servers.
Even if we have ``just'' 40 triangles per edge on average (for example the web graph ``eu-2005'' from the 10th DIMACS Implementation Challenge \cite{bmsw-gpgcd-13}) storing these triangles means that we need a lot more memory than for just storing $G$.
Therefore we used the trivial update algorithm that, for deleting the edge $\{u, v\}$, enumerates all triangles the edge is part of and updates the counters accordingly.
For deleting all edges this gives a $O(m \cdot \Delta)$ algorithm which only needs $O(m)$ memory.
In practice this was still fast enough for the graphs we considered.
In Table~\ref{tab:bounds} we report the lower bound and the running time of the lower bound calculation for the large real-world graphs we considered (refer to the experimental evaluation, Section~\ref{sec:exp} for details concerning the graphs).
The graphs are sorted by the number of edges $m$.
As in the experimental evaluation, we executed all experiments ten times with different random node id permutations.
Only for the largest graph, uk-2002, we used only one run with the original node ids for the lower bound calculation due to memory constraints.
In this table we report average and maximum bound and running time while in the experimental evaluation section we only reported the maximum.
However, as one can see, the average and the maximum do not differ significantly.
The running times clearly show that the running time does not only depend on $m$ but also on the degrees, i.e.\ graphs with a lower number of nodes but a comparable number of edges have a higher running time.
\begin{table}
\caption{Results for the lower bounds of the large real-world graphs we considered}
\label{tab:bounds}
\centering
\begin{tabular}{l@{~~}r@{~~}r@{~}|@{~}r@{~~}r@{~}|@{~}r@{~~}r}
Name & $n$ & $m$ & \multicolumn{2}{c|@{~}}{Lower Bound} & \multicolumn{2}{c}{Time [s]} \\
& & & mean & max & mean & max \\
\midrule
Caltech36 & 769 & 16\,656 & 349.5 & 350 & 0.0 & 0.0 \\
com-amazon & 334\,863 & 925\,872 & 99\,305.9 & 99\,413 & 0.6 & 0.6 \\
com-dblp & 317\,080 & 1\,049\,866 & 53\,656.9 & 53\,680 & 0.7 & 0.7 \\
Penn94 & 41\,554 & 1\,362\,229 & 19\,918.7 & 19\,920 & 1.8 & 1.8 \\
cnr-2000 & 325\,557 & 2\,738\,969 & 48\,500.0 & 48\,739 & 22.6 & 23.8 \\
com-youtube & 1\,134\,890 & 2\,987\,624 & 139\,006.5 & 139\,077 & 9.7 & 9.8 \\
in-2004 & 1\,382\,908 & 13\,591\,473 & 194\,849.9 & 195\,206 & 70 & 70.4 \\
eu-2005 & 862\,664 & 16\,138\,468 & 228\,457.1 & 228\,759 & 187.2 & 188.1 \\
com-lj & 3\,997\,962 & 34\,681\,189 & 1\,334\,663.3 & 1\,334\,770 & 65.8 & 66.3 \\
com-orkut & 3\,072\,441 & 117\,185\,083 & 1\,479\,977.2 & 1\,480\,007 & 394.2 & 395.4 \\
uk-2002 & 18\,520\,486 & 261\,787\,258 & & 2\,966\,359 & & 960.8 \\
\end{tabular}
\end{table}
\section{Details of the Initialization Algorithm}
In Algorithm~\ref{alg:full_init} we provide the full initialization heuristic as pseudo code.
Note that while for the parent calculation we use $\le$ for comparisons we use $<$ for the final selection of the neighbors to keep in order to not to wrongly assign too many neighbors to $u$.
\begin{algorithm}
\KwIn{$G = (V, E)$}
\KwOut{Parent assignment $p$ for each node}
Sort $V$ by degree in descending order using bucket sort\;
$p : V \rightarrow V \cup \{ \emptyset \}, u \mapsto \emptyset $\;
Count triangles $t(\{u, v\})$\;
\ForEach{$u \in V$}{
\tcp*[h]{Process node $u$}
$N \gets \{v \in N(u) \,|\, v $ not processed and $ p(u) = p(v)$ or ($p_{\mathrm{c}}(\{u, v\}) \le p_{\mathrm{c}}(\{v, p(v)\})$ and $\mathrm{depth}(v) \le t(\{u, v\}) + 1)\}$\;
$p_n \gets$ the most frequent value of $p(x)$ for $x \in N$\;
\If{$p_n \neq p(u)$}{
$p(u) \gets p_n$\;
$\mathrm{depth}(u) \gets 0$\;
$p_{\mathrm{c}}(\{u, p_n\}) \gets \infty$\;
}
\ForEach{$v \in N(u)$ that has not been processed}{
\If{$p(u) = p(v)$ or $(p_{\mathrm{c}}(\{u, v\}) < p_{\mathrm{c}}(\{v, p(v)\})$ and $\mathrm{depth}(v) < t(\{u, v\}) + 1)$}{
$p(v) \gets u$\;
$\mathrm{depth}(v) \gets \mathrm{depth}(v) + 1$\;
}
}
}
\caption{The Initialization Algorithm}
\label{alg:full_init}
\end{algorithm}
\section{The Quasi-Threshold Mover in Detail}
Here we want to describe the quasi-threshold mover algorithm in more detail.
Apart from giving more details how we actually implemented the algorithm in order to achieve the claimed running time we will also give proofs for its correctness and running time.
The QTM algorithm iteratively modifies the forest that defines a quasi-threshold graph.
In the following we assume again that our forest has a virtual root $r$ that is connected to all nodes in the original graph, i.e.\ we consider only the case of a tree.
For a single node $v_m$ the algorithm solves the following problem optimally:
Find a parent $u$ in the forest and a set of close children $C$ of that parent $u$ such that inserting $v_m$ as child of $u$ and moving $C$ to be children of $v_m$ minimizes the number of edits among all choices of $u$ and $C$.
One iteration of the algorithm consists of solving this problem for every node of the graph.
We will show later that for a single node $v_m$ this is possible in time $O(d(v_m) \log(d(v_m)))$ time amortized over an iteration, so the time for a whole iteration is in $O(n + m \log (\Delta))$.
For $t > 0$ iterations the total running time is $O(t \cdot (n + m \log (\Delta)))$.
The main idea why this works is that we do not need to consider all possible parents but only those parents which are adjacent to $v_m$ or which have a close child, i.e.\ a child of which more than half of the descendants are adjacent to $v_m$.
Otherwise the existing edges do not compensate for the missing edges and we could as well add $v_m$ as child of $r$, i.e.\ delete all edges in the original graph that are incident to $v_m$.
We will show how to determine these possible parents and close children by visiting only a constant number of nodes for each neighbor of $v_m$ that are determined by populating counters from the bottom to the top of the tree.
\begin{algorithm}[tb]
\caption{Core algorithm of QTM: finding a new parent and children to be adopted.}
\label{alg:local_moving}
\CommentSty{// Assumption: $v_m$ is not in the tree}\;
Insert neighbors of $v_m$ in the queue\; \nllabel{lm:pqinit}
\While{Queue is not empty}{
$u \gets$ pop node from queue\;
mark $u$ as touched\; \nllabel{lm:touched}
\lIf{$\mathrm{child}_{\mathrm{close}}(u) > \mathrm{score}_{\mathrm{max}}(u)$}{$\mathrm{score}_{\mathrm{max}}(u) \gets \mathrm{child}_{\mathrm{close}}(u)$}\nllabel{lm:gupdate}
\If{$u$ is marked as neighbor}{$\mathrm{child}_{\mathrm{close}}(u) \gets \mathrm{child}_{\mathrm{close}}(u) + 2$, $\mathrm{score}_{\mathrm{max}}(u) \gets \mathrm{score}_{\mathrm{max}}(u) + 2$\;} \nllabel{lm:neighborupdate}
$\mathrm{child}_{\mathrm{close}}(u) \gets \mathrm{child}_{\mathrm{close}}(u) - 1$,
$\mathrm{score}_{\mathrm{max}}(u) \gets \mathrm{score}_{\mathrm{max}}(u) - 1$\;\nllabel{lm:tokendec}
\If(\CommentSty{// Start a DFS from $u$}){$\mathrm{child}_{\mathrm{close}}(u) \ge 0$
and $u$ has children}{ \nllabel{lm:dfsstart}
$x \gets $ first child of $u$\;
\While{$x \neq u$}{\nllabel{lm:dfswhile}
\eIf{$x$ not touched or $\mathrm{child}_{\mathrm{close}}(x) < 0$}{
$\mathrm{child}_{\mathrm{close}}(u) \gets \mathrm{child}_{\mathrm{close}}(u) - 1$\;
$x \gets \mathrm{DFS}(x)$\;
\If{$\mathrm{child}_{\mathrm{close}}(u) <
0$}{\nllabel{lm:dfsend}
$\mathrm{DFS}(u) \gets x$\;
break\;
}
$x \gets$ next node in DFS
order after $x$ below $u$\;
}{
$x \gets$ next node in DFS order
after the subtree of $x$ below $u$\;\nllabel{lm:dfsskip}
}
}
}
\If(\CommentSty{// Propagate information to parent}){$u
\neq r$}{
\If{$\mathrm{child}_{\mathrm{close}}(u) > 0$}{
$\mathrm{child}_{\mathrm{close}}(p(u)) \gets \mathrm{child}_{\mathrm{close}}(p(u)) + \mathrm{child}_{\mathrm{close}}(u)$\;
Insert $p(u)$ in queue\;\nllabel{lm:einsertpq}
}
\If{$\mathrm{score}_{\mathrm{max}}(u) > \mathrm{score}_{\mathrm{max}}(p(u))$}{
$\mathrm{score}_{\mathrm{max}}(p(u)) \gets \mathrm{score}_{\mathrm{max}}(u)$\;
Insert $p(u)$ in queue\;
}
}
}
\end{algorithm}
In one iteration, the QTM algorithm simply iterates over all nodes in a random order.
For each node $v_m$ we search the optimal parent and children.
Algorithm~\ref{alg:local_moving} contains the pseudo code for the main part of this search.
In order to avoid complicated special cases we first remove $v_m$ from the tree\footnote{This is equivalent to isolating $v_m$ by inserting $v_m$ below the virtual root $r$.}
In the end we want to move $v_m$ back to its initial position if no better position was found.
If the initial position of $v_m$ was the best position, then the algorithm will find it again.
However, if there are multiple positions in the skeleton that induce the same number of edits, the algorithm will find any of these positions.
In order to make sure that the algorithm terminates even if we do not limit the number of iterations we store the initial position of $v_m$, i.e.\ its children and its parent.
We also count the number of edits that were necessary among its neighbors.
If no improvement was possible, we move the node back to this initial position in the end.
For a single node $v_m$ that shall possibly be moved we will process its neighbors and possibly $O(d(v_m))$ other nodes ordered by decreasing depth.
We maintain the list of these nodes in a priority queue that is initialized with the neighbors of $v_m$ and sorted by depth.
As we do not want to dynamically determine the depth of a node we calculate the depth initially and update it whenever we remove or insert a node in the forest.
A marker is set for all neighbors of $v_m$ in order to make it possible to determine in constant time if a node is adjacent to $v_m$.
When we process a node $u$ of the queue, we first determine if $u$ is the best parent in the subtree of $u$, then we possibly visit some nodes below $u$ using a special DFS in order to determine the child closeness of $u$ and possibly insert its parent into the queue.
We will later explain the details of the DFS.
We store the score of the best solution in the subtree of $u$ in $\mathrm{score}_{\mathrm{max}}(u)$ and the child closeness of $u$ in $\mathrm{child}_{\mathrm{close}}(u)$. In order to avoid special cases we initialize $\mathrm{score}_{\mathrm{max}}(u)$ with $-1$ and $\mathrm{child}_{\mathrm{close}}(u)$ with $0$.
Furthermore we store at each node $u$ the state of the DFS that has possibly been started at $u$.
In order to store the state we only store the last visited node.
We store this node in $\mathrm{DFS}(u)$.
We initialize $\mathrm{DFS}(u)$ with $u$.
At the end, we can find the number of edits that can be saved over isolating $v_m$ in $\mathrm{score}_{\mathrm{max}}(r)$ and we can also additionally track which parent lead to that score.
As already mentioned, we compare this to the number of edits at the old position of $v_m$ and move $v_m$ back to the old position if no improvement was possible.
If an improvement is possible, we insert $v_m$ below the parent that we identified as best parent $u$.
The missing part are the children that shall be moved from $u$ to $v_m$.
We can determine them by visiting all previously visited nodes (we can store them) and check for each visited node $c$ if it is a close child of $u$, i.e.\ if attaching it to $v_m$ would save edits which we have stored in $\mathrm{child}_{\mathrm{close}}(c)$.
\paragraph{Proof of Correctness}
In this section we want to give a formal proof why the local moving algorithm is correct, i.e.\ always selects the best parent and the best selection of children.
We do this by giving exact definitions of all used variables and proofing their correctness.
We begin with the child closeness $\mathrm{child}_{\mathrm{close}}(u)$ which is the number of edits that we can save if $v_m$ is attached below $u$:
\begin{proposition}
\label{prop:e_correct}
Either $\mathrm{child}_{\mathrm{close}}(u)$ is the number of neighbors of $v_m$ in the subtree of $u$ minus the
number of non-neighbors, or there are more non-neighbors than neighbors in the
subtree of $u$. In the latter case, if $u$ has been processed, then $\mathrm{child}_{\mathrm{close}}(u) = -1$.
More precisely if $u$ has been processed, $\mathrm{child}_{\mathrm{close}}(u)$ is the number of existing neighbors minus the number of missing neighbors of all nodes in DFS order between $u$ and $D(u)$ and additionally all subtrees of children $c$ with $\mathrm{child}_{\mathrm{close}}(c) > 0$ that are not in the DFS order between $u$ and $\mathrm{DFS}(u)$.
\end{proposition}
\begin{proof}
We will give the proof by structural induction.
As first step we want to establish that all nodes where $\mathrm{child}_{\mathrm{close}}(u) \ge 0$ are processed.
As $\mathrm{child}_{\mathrm{close}}(u) < 0$ if there are no neighbors of $v_m$ in the subtree of $u$ only neighbors of $v_m$ and their ancestors can have $\mathrm{child}_{\mathrm{close}}(u) \ge 0$.
All neighbors of $v_m$ are processed (line~\ref{lm:pqinit}).
For non-neighbors $u$, $\mathrm{child}_{\mathrm{close}}(u) \ge 0$ means that one of their children $c$ is close, i.e.\ has $\mathrm{child}_{\mathrm{close}}(c) > 0$.
As in this case $c$ inserts $v_m$ into the queue (line~\ref{lm:einsertpq}) also in this case $u$ will be processed.
As we process all nodes by descending depth (only parents, i.e.\ nodes of smaller depth, are inserted in the queue) we can assume that if we are at a node $u$, all descendants of $u$ that need to be processed have been processed and that when the algorithm terminates all nodes $u$ with $\mathrm{child}_{\mathrm{close}}(u) \ge 0$ have been processed.
In line~\ref{lm:neighborupdate} and \ref{lm:tokendec} $\mathrm{child}_{\mathrm{close}}(u)$ is updated such that the proposition is true if we consider only $u$ itself.
This means that the proposition is true for leafs which is also the initial step of our induction.
As the claim is true for all children of $u$ we can also assume that all children $c$ with $\mathrm{child}_{\mathrm{close}}(c) > 0$ already updated $\mathrm{child}_{\mathrm{close}}(u)$ accordingly, i.e.\ $\mathrm{child}_{\mathrm{close}}(u)$ already correctly considers $u$ and the values of all children with $\mathrm{child}_{\mathrm{close}}(c) > -1$.
If we have $\mathrm{child}_{\mathrm{close}}(u) = -1$ in line~\ref{lm:dfsstart} $\mathrm{child}_{\mathrm{close}}(u)$ must have been 0 initially as in the following it can only be decreased by 1 at maximum.
Therefore we are in the situation that $u$ is no neighbor of $v_m$ and $u$ has no close children, i.e.\ children with $\mathrm{child}_{\mathrm{close}}(c) > 0$.
In this situation this is already the final result as if this result was incorrect, i.e.\ $\mathrm{child}_{\mathrm{close}}(u) > -1$, then there must be at least as many neighbors as non-neighbors of $v_m$ among the nodes in the subtree of $u$.
As $u$ is no neighbor of $v_m$ the descendants must contain at least one more neighbor than there are non-neighbors among them and this must also be true for at least one of the children of $u$.
Therefore $\mathrm{child}_{\mathrm{close}}(c) > 0$ for this child which is contradiction to the situation that there are no children $c$ with $\mathrm{child}_{\mathrm{close}}(c) > 0$.
So now we only need to consider the case that $\mathrm{child}_{\mathrm{close}}(u) > -1$ in line~\ref{lm:dfsstart} which means that the algorithm will start a DFS.
As first step we want to have a closer look at the DFS that is executed in the algorithm.
If we say in the following that the DFS ``visits'' a node we mean that it is the value of $x$ in line~\ref{lm:dfswhile}.
On visiting certain nodes, we decrease $\mathrm{child}_{\mathrm{close}}(u)$.
If $\mathrm{child}_{\mathrm{close}}(u) < 0$, we stop the DFS and store the last visited node in $\mathrm{DFS}(u)$.
This means that at the end of a DFS either $\mathrm{child}_{\mathrm{close}}(u) \ge 0$ or $\mathrm{DFS}(u)$ points to the last visited node.
Whenever we visit a node, there are three possible cases:
\begin{enumerate}
\item The easiest case is that $c$ has been processed and $\mathrm{child}_{\mathrm{close}}(c) > -1$. In this
case the edits of $c$ are already considered by the parent of $c$ and we do
not need to deal with it. Furthermore, we know that $\mathrm{child}_{\mathrm{close}}(c)$ is the correct number
of neighbors minus non-neighbors of the whole subtree of $c$ so it is correct
that the algorithm skips these nodes in line~\ref{lm:dfsskip}.
\item If $c$ has not been processed yet, it is a not a neighbor of $v$
(otherwise it would have been processed). We decrease $\mathrm{child}_{\mathrm{close}}(u)$ which is correct as
this is a missing neighbor. Then we can continue the DFS. The same is true if
$c$ has been processed, $\mathrm{child}_{\mathrm{close}}(c) < 0$ but $\mathrm{DFS}(c) = c$, i.e.\ no DFS has been
executed.
\item If $\mathrm{child}_{\mathrm{close}}(c) < 0$ and $\mathrm{DFS}(c) \neq c$ we know from the induction
hypothesis that $\mathrm{child}_{\mathrm{close}}(c) = -1$ and that this is exactly the number of existing
minus the number of missing neighbors from $c$ up to $\mathrm{DFS}(c)$ in DFS order (including $\mathrm{DFS}(c)$)
plus the number of children $c'$ of $c$ with $\mathrm{child}_{\mathrm{close}}(c') > -1$ which we ignore anyway
in the DFS.
We decrease $\mathrm{child}_{\mathrm{close}}(u)$ which correctly considers the nodes between $c$ and $\mathrm{DFS}(c)$ in DFS order (both included).
Then we jump to $\mathrm{DFS}(c)$ and do not visit $\mathrm{DFS}(c)$ but the next node in DFS order which is obviously correct as $\mathrm{DFS}(c)$ has already been considered by decreasing $\mathrm{child}_{\mathrm{close}}(u)$.
\end{enumerate}
When the DFS ends, either we have now considered all edits of the descendants
of $u$ or the DFS ended with $\mathrm{child}_{\mathrm{close}}(u) < 0$ and we have stored the location of the
last visited node in $\mathrm{DFS}(u)$. In the latter case, all nodes up to this point
have been considered as we have outlined before. Therefore the claim is now
also true for $u$. \qed
\end{proof}
If we want to know for a potential parent $u$ how many edits we can save by moving some of its children to $v_m$ this is the sum of $\mathrm{child}_{\mathrm{close}}(c)$ for all close children of $u$, i.e.\ children with $\mathrm{child}_{\mathrm{close}}(c) > 0$.
This is the value that we store in $\mathrm{child}_{\mathrm{close}}(u)$ before $u$ is processed by setting $\mathrm{child}_{\mathrm{close}}(p(c))$ for all close children $c$ of $u$, i.e.\ children $c$ with $\mathrm{child}_{\mathrm{close}}(c) > 0$.
Obviously, this is positive if a node $u$ has at least one close child.
In order to not to need to evaluate all nodes as potential parents we make use of the following observation:
\begin{proposition}
\label{prop:possibleparents}
Only nodes with close children and neighbors of $v_m$ need to be considered as parents of $v_m$.
\end{proposition}
\begin{proof}
Assume otherwise: The best parent $u$ has no close children and is not a neighbor of $v_m$.
Then attaching children of $u$ to $v_m$ makes no sense as this would only increase the number of needed edits so we can assume that no children will be attached.
However then choosing $p(u)$ as parent of $v_m$ will save one edit as $u$ is no neighbor of $v_m$.
This is a contradiction to the assumption that $u$ is the best parent. \qed
\end{proof}
So far we have only evaluated edits below nodes and identified all possible parents which are also processed as we have established before.
The part that is still missing is the evaluation of the edits above a potential parent $u$.
\begin{theorem}
Consider the subtree $T_u$ of $u$.
Then for the subgraph of $T_u$, $\mathrm{score}_{\mathrm{max}}(u)$ stores the maximum number of edits from $v_m$ to nodes inside the subgraph of $T_u$ that can be saved by choosing the parent of $v_m$ in $T_u$ instead of isolating $v_m$ or $-1$ if no edits can be saved.
\end{theorem}
\begin{proof}
The proof is given by structural induction on the tree skeleton.
We start with the initial step which is a node $u$ that is a leaf of the tree.
As $T_u$ only consists of $u$, we have only one edge from $v_m$ to $u$ and therefore only two cases:
$u$ is a neighbor of $v_m$ or not.
In both cases, $\mathrm{score}_{\mathrm{max}}(u)$ is initialized with $-1$ as there are no children that could propagate any values.
In the second case, $u$ will not be processed but the result is already correct anyway: no edits can be saved by choosing $u$ as parent of $v_m$.
In the first case, as $\mathrm{child}_{\mathrm{close}}(u)$ is initialized to $0$, $\mathrm{child}_{\mathrm{close}}(u) > \mathrm{score}_{\mathrm{max}}(u)$ and therefore $\mathrm{score}_{\mathrm{max}}(u) \gets 0$ (line~\ref{lm:gupdate}).
We end with $\mathrm{score}_{\mathrm{max}}(u) = 1$ which is correct, we can save an edit over isolating $u$ as the edge $(u, v_m)$ does not need to be deleted when we chose $u$ as parent of $v_m$.
When we set $\mathrm{score}_{\mathrm{max}}(u)$ we can also store $u$ as best parent together with $\mathrm{score}_{\mathrm{max}}(u)$.
Now we can assume that the theorem holds for all children of $u$.
As not all nodes are processed, we need to explain why $u$ is processed at all if $\mathrm{score}_{\mathrm{max}}(u) > -1$ should hold.
There are two possibilities: Either it could make sense to use $u$ as parent or we could use a node below $u$ as parent.
In the first case by Proposition~\ref{prop:possibleparents} either $u$ is a neighbor of $v_m$ or $u$ has a close child which means that $u$ is processed.
Assume that in the second case $u$ was not processed but it should be $\mathrm{score}_{\mathrm{max}}(u) > -1$.
Further we can assume that $u \notin N(v_m)$ as otherwise $u$ was processed.
Let $x$ be the best parent in $T_u$ and let $c$ be the direct child of $u$ such that $x$ is in the subtree of $c$ $T_c$ (it is possible that $x = c$).
If it makes sense to use $x$ as parent of $v_m$ then by inserting $v_m$ below $x$ also the edge $\{u, v_m\}$ must be inserted.
This means that in the subtree of $c$ we can save one more edit as the edit $\{u, v_m\}$ is not necessary which means that $\mathrm{score}_{\mathrm{max}}(c) = \mathrm{score}_{\mathrm{max}}(u) + 1 > 0$.
This means that by induction $c$ must have been processed and $c$ must have propagated $\mathrm{score}_{\mathrm{max}}(c)$ to $u$ and also inserted $u$ in the queue which is a contradiction to the assumption that $u$ is not processed.
When $u$ is processed, we need to make the decision if $u$ is the best parent in $T_u$ or if we should choose the parent below $u$.
The edit of the edge $\{v_m, u\}$ is needed or not independent of the choice of the parent in $T_u$.
Therefore we do not need to reconsider any decisions that were made below $u$.
If we do not choose $u$, then we need to choose the best parent below $u$, i.e.\ the one of the subtree of the child $c$ with the highest value of $\mathrm{score}_{\mathrm{max}}(c)$.
This is also what the algorithm does by propagating $\mathrm{score}_{\mathrm{max}}(c)$ to the parent as maximum of $\mathrm{score}_{\mathrm{max}}(c)$ and $\mathrm{score}_{\mathrm{max}}(p(c))$.
Therefore $\mathrm{score}_{\mathrm{max}}(u)$ is initialized to the best solution below $u$.
If we want to determine how good $u$ is as parent, we need to look at the closeness of its children.
More specifically, we can save as many edits as the sum of $\mathrm{child}_{\mathrm{close}}(c)$ for all close children $c$ of $u$.
This is the value to which $\mathrm{child}_{\mathrm{close}}(u)$ is initialized by its close children, therefore we only need to compare $\mathrm{score}_{\mathrm{max}}(u)$ to $\mathrm{child}_{\mathrm{close}}(u)$.
Therefore it is correct to set $\mathrm{score}_{\mathrm{max}}(u)$ to $\mathrm{child}_{\mathrm{close}}(u)$ if $\mathrm{child}_{\mathrm{close}}(u)$ is larger than $\mathrm{score}_{\mathrm{max}}(u)$.
After this initial decision, we increase or decrease $\mathrm{score}_{\mathrm{max}}(u)$ by one depending on whether the edge $\{u, v_m\}$ exists or not, this is obviously correct.
\qed
\end{proof}
As $T_r$ is the whole graph, $\mathrm{score}_{\mathrm{max}}(r)$ determines the best solution of the whole graph.
Therefore the QTM algorithm optimally solves the problem of finding a new parent and a set of its children that shall be adopted.
\paragraph{Proof of the Running Time}
After showing the correctness of the algorithm, we will now show that the running time is indeed $O(m \log(\Delta))$ per iteration and amortized $O(d \log(d))$ per node.
During the whole algorithm we maintain a depth value for each node that specifies the depth in the forest at which the node is located.
Whenever we move a node, we update these depth values.
This involves decreasing the depth values of all descendants of the node in its original position and increasing the depth values of all descendants of the node at the new position.
Unfortunately it is not obvious that this is possible in the claimed running time as a node $v_m$ might have more than $O(d(v_m))$ descendants.
Note that a node is adjacent to all its descendants and ancestors in the edited graph.
This means that every ancestor or descendant that is not adjacent to the node causes an insert.
Therefore the node must be neighbor of at least half of the ancestors and children after a move operation as otherwise the less than half of the degree deletes are cheaper than the more than half of the degree inserts.
This means that updating the depth values at the destination is possible in $O(d)$ time.
For the update of the values in the original position we need a different, more complicated argument.
First of all we assume that initially the total number of edits never exceeds the number of edges as otherwise we could simply delete all edges and get less edits.
For amortizing the number of needed edits of nodes that have more descendants and ancestors than their degree we give each node tokens for all their neighbors in the edited graph.
As the number of edits is at most $m$ the number of initially distributed tokens is in $O(m)$.
Whenever we move a node $v_m$, it generates tokens for all its new neighbors and itself, i.e. in total at most $2 \cdot d(v_m)$ tokens.
Therefore a node has always a token for each of its ancestors and descendants and can use that token to account for updating the depth of its previous descendants.
In each round only $O(m)$ tokens are generated, therefore updating the depth values of a node is in amortized time $O(d)$ per node and $O(m)$ per iteration.
Using the same argument we can also account for the time that is needed for updating the pointers of each node to its parents and children and for counting the number of initially needed or saved edits.
What we have shown so far means that once we know the best destination we can move a node and update all depth values in time $O(d)$ amortized over an iteration where all nodes are moved.
The remaining claim is that we can determine the new parent and the new children in time $O(d \log(d))$ per node.
More precisely we will show that only $O(d)$ nodes are inserted in the queue and we need amortized constant time for processing a node.
A standard max-heap that needs $O(\log(n))$ time per operation can be used for the implementation of the queue.
All values that are stored per node need to be initialized for the whole iteration.
All nodes whose values are changed, which are exactly the nodes that have been in the queue at some moment, need to be stored so their values can be reset at the end of the processing of a node.
The basic idea of the main proof is that each neighbor of $v_m$ gets four tokens.
This is represented by the fact that we increase $\mathrm{child}_{\mathrm{close}}(u)$ and $\mathrm{score}_{\mathrm{max}}(u)$ by 2 for all neighbors $u$ of $v_m$.
When we process a node $u$, one token is consumed if this node is no neighbor of $v_m$, then the DFS consumes tokens of $\mathrm{child}_{\mathrm{close}}(u)$ and at the end the rest of the tokens are passed to the parent.
Note that all nodes that are processed have $\mathrm{child}_{\mathrm{close}}(u) > 0$ or $\mathrm{score}_{\mathrm{max}}(u) > 0$, either initially or after accounting for the fact that they are neighbors of $v_m$.
First of all let us only consider processed nodes $u$ with $\mathrm{child}_{\mathrm{close}}(u) > 0$ initially or after accounting for the fact that $u$ is a neighbor of $v_m$.
We consume one token for processing this node in line~\ref{lm:tokendec}.
This is for the whole processing of the node apart from the DFS where the accounting is more complicated.
Apart from the DFS only constant work is done per node, so consuming one taken is enough for that.
First of all note that for each visited node in the DFS only a constant amount of work is needed as traversing the tree, i.e.\ possibly traversing a node multiple times can be accounted to the first visit.
Obviously without keeping a stack this needs a tree structure where we can determine the next child $c'$ after a child $c$ of a node $u$ can be determined in constant time.
This can be implemented by storing in node $c$ the position of $c$ in the array (or list) of children in $p(c)$.
This also allows deleting entries in the children list in constant time (in an array deletion can be implemented as swap with the last child).
Whenever we visit a node that has not been touched yet or that has $\mathrm{child}_{\mathrm{close}}(x) < 0$, we consume one token of $\mathrm{child}_{\mathrm{close}}(u)$.
When this is not the case, i.e.\ $\mathrm{child}_{\mathrm{close}}(x) > -1$, the node has been processed already and we account our visiting of $x$ to the processing of $x$.
This is okay as we visit each node only once during a DFS:
After the DFS starting at $u$ has finished, either $\mathrm{child}_{\mathrm{close}}(u) > -1$ and an upcoming DFS will not descend into the subtree of $u$ anymore or we ended the DFS in line~\ref{lm:dfsend} and thus have set $\mathrm{DFS}(u)$ to the last visited node which means that when we visit $u$ in an upcoming DFS, this DFS will directly jump to $\mathrm{DFS}(u)$ after visiting $u$.
Note that by decreasing $\mathrm{child}_{\mathrm{close}}(u)$ to $-1$ we actually consume one more token than we had.
However for this we only need a constant amount of work which can be accounted for by the processing time of $u$.
Now we consider nodes $u$ that are processed with $\mathrm{score}_{\mathrm{max}}(u) > 0$ initially.
If we ignore line~\ref{lm:gupdate} everything seems to be simple: we consume one token and pass the rest to the parent (using the maximum instead of the sum) if a token is left.
However if we set $\mathrm{score}_{\mathrm{max}}(u)$ to $\mathrm{child}_{\mathrm{close}}(u)$ we are getting new tokens out of nowhere.
Fortunately it turns out we can explain that these tokens are also from the $\mathrm{score}_{\mathrm{max}}(c)$ of all children $c$ of $u$ but the sum instead of the maximum:
Note that $\mathrm{child}_{\mathrm{close}}(c) \le \mathrm{score}_{\mathrm{max}}(c)$ for any node $c$ that has been processed, i.e.\ after line~\ref{lm:gupdate} $\mathrm{child}_{\mathrm{close}}(c) \le \mathrm{score}_{\mathrm{max}}(c)$ holds, then in line~\ref{lm:neighborupdate} both are increased by 2 and after that only $\mathrm{child}_{\mathrm{close}}(c)$ is decreased.
As $\mathrm{child}_{\mathrm{close}}(u)$ is initially the sum of all positive $\mathrm{child}_{\mathrm{close}}(c)$ of the children $c$ of $u$, it follows that initially $\mathrm{child}_{\mathrm{close}}(u)$ is smaller or equal to the sum of all $\mathrm{score}_{\mathrm{max}}(c)$ of the children $c$ of $u$.
Therefore actually each child $c$ has passed a part of the tokens of $\mathrm{score}_{\mathrm{max}}(c)$ to $u$ in form of $\mathrm{child}_{\mathrm{close}}(u)$.
Therefore also line~\ref{lm:gupdate} does not create new tokens.
This means that in total we only process $O(d)$ nodes and do amortized constant work per node as we have claimed.
\section{Details of the Algorithm proposed by Nastos and Gao}
Nastos and Gao \cite{ng-f-13} describe that in their greedy algorithm they test each possible edge addition and deletion (i.e.\ all $O(n^2)$ possibilities) in order to choose the edit that results in the largest improvement, i.e.\ the highest decrease of the number of induced $P_4$ and $C_4$.
After executing this greedy heuristic they revert the last few edits and execute the bounded search tree algorithm.
If this results in a solution with fewer edits, they repeat this last step until no improvement is possible anymore.
We chose 10 for the number of edits that are reverted.
The main question for the implementation is thus how to select the next edit.
As far as we know it is an open problem if it is possible to determine the edit that destroys most $P_4$ and $C_4$ in time $o(n^2)$.
Therefore we concentrate on the obvious approach that was also implied by Nastos and Gao: execute each possibility and see how the number of $P_4$ and $C_4$ changes.
The main ingredient is thus a fast update algorithm for this counter.
As far as we are aware the fastest update algorithm for counting node-induced $P_4$ and $C_4$ subgraphs needs amortized time $O(h^2)$ for each update where $h$ is the $h$-index of the graph \cite{egst-e-12}.
While the worst-case bound of the $h$-index is $\sqrt{m}$ it has been shown that many real-world social networks have a much lower $h$-index \cite{es-tigia-09}.
However this algorithm requires constant-time edge existence checks and stores many counts for pairs and triples of edges (though only if they are non-zero).
We implemented a different algorithm which has the same worst-case complexity if we ignore the actual value of $h$: $O(m)$.
Furthermore this algorithm is much simpler to implement and while it needs $O(n)$ additional memory during updates only the counter itself needs to be stored between updates.
The initial counting is thus possible in time $O(m^2)$, therefore this results in an $O(m^2 + k \cdot n^2 \cdot m)$ algorithm.
Note that the time needed for the initial counting is dominated by the time needed for each edit.
The main idea of the algorithm is that we examine the neighborhood structure of the edge that shall be deleted or inserted.
Using markers we note which neighbors are common or exclusive to the two incident nodes.
We iterate once over each of these three groups of neighbors and over their neighbors which needs at most $O(m)$ time.
Based on the status of the markers of these neighbors of the neighbors we can count how many times certain structures occur on the neighborhood of the edge.
Using these counts we can determine how many $P_4$ and $C_4$ were destroyed and created by editing the edge.
Apart from applying the update algorithm $m$ times there is also a simpler $O(m^2)$ counting algorithm which we used for the initialization.
Here the idea is again that we determine the common and exclusive neighborhoods for each edge.
Then we only need to iterate over the exclusive neighbors of one of the two nodes and check for each of them how many of its neighbors are exclusive to the other node.
This gives us the number of $C_4$ that edge is part of.
The product of the sizes of the exclusive neighborhoods gives us the number of $P_4$ where the edge is the central edge plus the number of $C_4$ the edge is part of.
Combining both we can get the number of $P_4$ where the edge is the central edge.
While the sum of these values already gives the number of $P_4$, the sum of the $C_4$-counts still needs to be divided by 4.
Note that when the graph is a quasi-threshold graph, i.e.\ there are not $P_4$ and $C_4$, this needs only $O(m\cdot\Delta)$ time.
\section{Generated Graphs}
Each connected component of the quasi-threshold graph was generated as reachability graph of a rooted tree.
For generating a tree, $0$ is the root and each node $v \in \{1, \dots, n-1\}$ chooses a parent in $\{0, \dots, v-1\}$.
As shown by \cite{lksf-ccscn-10} many real-world networks including social networks exhibit a community size distribution that is similar to a power law distribution.
Therefore we chose a power law sequence with 10 as minimum, $0.2 \cdot n$ as maximum and $-1$ as exponent for the component sizes and generated trees of the respective sizes.
For $k$ edits we inserted $0.8 \cdot k$ new edges and deleted $0.2 \cdot k$ old edges of the quasi-threshold graph chosen uniformly at random.
Therefore after these modifications the maximum editing distance to the original graph is $k$.
We used a more insertions than deletions as preliminary experiments on real-world networks showed that during editing much more edges are deleted than inserted.
\begin{table}
\caption{Results for the generated graphs}
\label{tab:genresults}
\centering
\begin{tabular}{l@{~}r@{~~}r@{~~}r@{~}|@{~}r@{~~}r@{~~}r@{~}|@{~}r@{~}|@{~}r@{~~}r@{~~}r}
Rand & $n$ & $b$ & I & \multicolumn{3}{c|@{~}}{Edits} & It & \multicolumn{3}{c}{Time {[}s{]}}\tabularnewline
Ed. & $m$ & & & 0 & 4 & $\infty$ & $\infty$ & 0 & 4 & $\infty$\tabularnewline
\midrule
\multirow{2}{*}{20} & 100 & \multirow{2}{*}{16} & $\bullet$ & 34.4 & 20.0 & 20.0 & 2.4 & 0.0 & 0.0 & 0.0\tabularnewline
& 269 & & $\circ$ & 36.7 & 21.4 & 21.4 & 3.9 & 0.0 & 0.0 & 0.0\tabularnewline[0.3em]
\multirow{2}{*}{400} & 100 & \multirow{2}{*}{49} & $\bullet$ & 420.5 & 352.0 & 351.9 & 3.9 & 0.0 & 0.0 & 0.0\tabularnewline
& 497 & & $\circ$ & 372.0 & 363.5 & 363.5 & 3.9 & 0.0 & 0.0 & 0.0\tabularnewline[0.3em]
\multirow{2}{*}{20} & 1\,000 & \multirow{2}{*}{19} & $\bullet$ & 38.0 & 19.0 & 19.0 & 2.0 & 0.0 & 0.0 & 0.0\tabularnewline
& 4\,030 & & $\circ$ & 166.4 & 21.7 & 21.7 & 4.1 & 0.0 & 0.0 & 0.0\tabularnewline[0.3em]
\multirow{2}{*}{400} & 1\,000 & \multirow{2}{*}{225} & $\bullet$ & 585.3 & 391.2 & 391.2 & 3.4 & 0.0 & 0.0 & 0.0\tabularnewline
& 4\,258 & & $\circ$ & 594.4 & 393.6 & 393.6 & 4.4 & 0.0 & 0.0 & 0.0\tabularnewline[0.3em]
\multirow{2}{*}{8K} & 1\,000 & \multirow{2}{*}{494} & $\bullet$ & 8\,268 & 7\,219 & 7\,218.5 & 5.2 & 0.0 & 0.0 & 0.0\tabularnewline
& 8\,818 & & $\circ$ & 7\,647 & 7\,511 & 7\,490.6 & 8.3 & 0.0 & 0.0 & 0.0\tabularnewline[0.3em]
\multirow{2}{*}{20} & 10\,000 & \multirow{2}{*}{20} & $\bullet$ & 47.8 & 20.0 & 20.0 & 2.0 & 0.0 & 0.1 & 0.1\tabularnewline
& 66\,081 & & $\circ$ & 1\,669 & 69.8 & 69.8 & 4.6 & 0.1 & 0.2 & 0.2\tabularnewline[0.3em]
\multirow{2}{*}{400} & 10\,000 & \multirow{2}{*}{366} & $\bullet$ & 849.3 & 390.6 & 390.6 & 3.2 & 0.0 & 0.2 & 0.2\tabularnewline
& 66\,309 & & $\circ$ & 2\,143 & 440.7 & 440.7 & 4.8 & 0.1 & 0.2 & 0.2\tabularnewline[0.3em]
\multirow{2}{*}{8K} & 10\,000 & \multirow{2}{*}{3\,256} & $\bullet$ & 11\,626 & 7\,902 & 7\,902 & 3.9 & 0.0 & 0.2 & 0.2\tabularnewline
& 70\,869 & & $\circ$ & 10\,184 & 7\,912 & 7\,911 & 5.1 & 0.1 & 0.2 & 0.3\tabularnewline[0.3em]
\multirow{2}{*}{160K} & 10\,000 & \multirow{2}{*}{4\,985} & $\bullet$ & 157\,114 & 144\,885 & 144\,880 & 5.8 & 0.0 & 0.6 & 0.8\tabularnewline
& 162\,069 & & $\circ$ & 150\,227 & 148\,206 & 147\,892 & 9.7 & 0.1 & 0.5 & 1.2\tabularnewline[0.3em]
\multirow{2}{*}{20} & 100\,000 & \multirow{2}{*}{20} & $\bullet$ & 64.3 & 20.0 & 20.0 & 2.0 & 0.2 & 1.7 & 1.7\tabularnewline
& 833\,565 & & $\circ$ & 17\,785 & 529.9 & 529.6 & 5.3 & 0.8 & 2.8 & 3.7\tabularnewline[0.3em]
\multirow{2}{*}{400} & 100\,000 & \multirow{2}{*}{384} & $\bullet$ & 1\,047 & 391.3 & 391.3 & 3.2 & 0.2 & 2.6 & 2.6\tabularnewline
& 833\,793 & & $\circ$ & 18\,319 & 900.0 & 899.4 & 5.5 & 0.8 & 2.9 & 3.9\tabularnewline[0.3em]
\multirow{2}{*}{8K} & 100\,000 & \multirow{2}{*}{6\,519} & $\bullet$ & 18\,550 & 7\,889 & 7\,889 & 3.4 & 0.2 & 2.8 & 2.8\tabularnewline
& 838\,353 & & $\circ$ & 26\,144 & 8\,381 & 8\,381 & 5.4 & 0.8 & 2.8 & 3.8\tabularnewline[0.3em]
\multirow{2}{*}{160K} & 100\,000 & \multirow{2}{*}{42\,021} & $\bullet$ & 199\,558 & 158\,021 & 158\,021 & 4.6 & 0.2 & 3.5 & 4.1\tabularnewline
& 929\,553 & & $\circ$ & 193\,071 & 158\,031 & 158\,025 & 6.1 & 1.0 & 3.3 & 4.9\tabularnewline[0.3em]
\multirow{2}{*}{3.2M} & 100\,000 & \multirow{2}{*}{49\,913} & $\bullet$ & 2\,728\,804 & 2\,647\,566 & 2\,647\,564 & 5.8 & 1.1 & 12.5 & 16.8\tabularnewline
& 2\,753\,553 & & $\circ$ & 2\,655\,538 & 2\,654\,738 & 2\,654\,736 & 5.7 & 3.0 & 11.9 & 16.9\tabularnewline[0.3em]
\multirow{2}{*}{20} & 1\,000\,000 & \multirow{2}{*}{20} & $\bullet$ & 68.9 & 20.0 & 20.0 & 2.2 & 3.6 & 32.5 & 32.3\tabularnewline
& 10\,648\,647 & & $\circ$ & 181\,540 & 5\,116 & 5\,111 & 6.0 & 16.4 & 54.3 & 79.7\tabularnewline[0.3em]
\multirow{2}{*}{400} & 1\,000\,000 & \multirow{2}{*}{391} & $\bullet$ & 1\,161 & 395.1 & 395.1 & 3.0 & 3.3 & 43.8 & 43.8\tabularnewline
& 10\,648\,875 & & $\circ$ & 182\,248 & 5\,523 & 5\,518 & 6.1 & 15.9 & 52.9 & 78.8\tabularnewline[0.3em]
\multirow{2}{*}{8K} & 1\,000\,000 & \multirow{2}{*}{7\,447} & $\bullet$ & 25\,085 & 7\,912 & 7\,912 & 3.5 & 3.4 & 50.1 & 50.0\tabularnewline
& 10\,653\,435 & & $\circ$ & 189\,504 & 13\,006 & 13\,001 & 6.0 & 16.6 & 53.4 & 78.0\tabularnewline[0.3em]
\multirow{2}{*}{160K} & 1\,000\,000 & \multirow{2}{*}{112\,814} & $\bullet$ & 369\,501 & 158\,808 & 158\,808 & 4.1 & 3.5 & 57.5 & 58.8\tabularnewline
& 10\,744\,635 & & $\circ$ & 346\,462 & 163\,337 & 163\,330 & 6.4 & 17.4 & 54.8 & 84.5\tabularnewline[0.3em]
\multirow{2}{*}{3.2M} & 1\,000\,000 & \multirow{2}{*}{476\,562} & $\bullet$ & 3\,747\,793 & 3\,163\,277 & 3\,163\,273 & 5.8 & 4.7 & 71.8 & 101.2\tabularnewline
& 12\,568\,635 & & $\circ$ & 3\,820\,935 & 3\,164\,175 & 3\,163\,848 & 7.5 & 26.2 & 78.2 & 134.7\tabularnewline
\end{tabular}
\end{table}
In Table~\ref{tab:genresults} we show the results for all graphs that we have generated.
The first column shows the number of random edits we performed.
As already mentioned in the experimental evaluation for all generated graphs the QTM algorithm finds a quasi-threshold graph that is at least as close as the original one.
Omitting the initialization gives much worse results for low numbers of edits and slightly worse results for higher numbers of edits.
The lower bound is relatively close to the generated and found number of edits for low numbers of edits, for very high numbers of edits it is close to its theoretical maximum, $n/2$.
All in all this shows that the QTM algorithm finds edits that are reasonable but it depends on a good initial heuristic.
\printbibliography[segment=2]
\end{appendix}
\end{refsegment}
\end{document}
|
1,108,101,565,458 | arxiv | \section{Introduction}\label{Section Introduction}
After the electronification of delta-one trading, where high-frequency trading companies provide the vast majority of the liquidity on several thousands of assets, systematic options trading seems to be the next main challenge in quantitative trading. For assets listed in a central limit order book, as in the equity world, execution and market making are carried out using algorithms. However, for less mature markets such as a great proportion of fixed income securities, systematic market making activities are driven by request-for-quote (RFQ for short) systems: the client sends a request to obtain a buy or sell price, for a given quantity of a security, to one or several market makers, who propose prices based on their current positions. Given the prices, the client accepts or refuses one or several transactions. On OTC markets, such as the corporate bonds market, the proportion of the volume traded with electronic market makers is increasing. \\
For more than three decades, the optimal market making problem on cash markets has been the object of many academic studies. The two primary references are \cite{grossman1988liquidity, ho1981optimal}. In \cite{grossman1988liquidity}, the authors proposed a simple three-period economic model representing the interaction between market makers and market-takers and analyzed the equilibrium state. In \cite{ho1981optimal}, the authors studied the behavior of a market maker facing a stochastic demand and an inventory risk and obtained his optimal strategy using the stochastic optimal control theory. In the well-known paper of Avellaneda and Stoikov \cite{avellaneda2008high} inspired by this framework, they proposed a model applicable for market making on the order-driven market at the high-frequency. However, due to the continuous nature of the market maker's spreads, and the assumption that the underlying asset is a diffusion process, this model is more suited to quote-driven markets such as corporate bonds market. \\
By providing a rigorous analysis of the stochastic control problem of \cite{avellaneda2008high}, the authors of \cite{gueant2013dealing} show, in the case of a CARA utility function, that the market maker's problem boils down to a system of linear ordinary differential equations. A large part of the contribution to the market making literature comes from works of Cartea and Jaimungal, who enriched the initial model by introducing alpha signals, ambiguity aversion, competition with other agents, see, for example, \cite{cartea2017algorithmic, cartea2018enhancing, cartea2016incorporating, cartea2015algorithmic}. In these works, they consider a risk-adjusted expectation maximization. As shown in \cite{manziuk2019optimal}, the solution of such formulation can also be obtained through CARA utility maximization after a suitable intensity function transformation. More recently, multi-asset market making, still on linear markets, has been addressed through reinforcement learning techniques, see \cite{gueant2019deep}, and dimensionality reduction techniques, as in \cite{bergault2019size}. \\
Regardless of how rich is the part of academic literature considering linear markets, the part studying optimal market making on options is far less extensive. A reasonable market making model for options has to take into account a lot of stylized facts. First, option market makers trade simultaneously derivatives and the corresponding underlying, which implies the construction of more complex trading strategies taking into account, for example, the Delta-Vega hedging. Consequently, one needs to impose a factorial stochastic volatility model, possibly with jumps, on the underlying asset. Second, option market makers need to manage several thousands of positions, which lead to very high-dimensional problems that cannot be solved using classical numerical schemes. Even if machine learning techniques are used, involving, for example, deep reinforcement learning methods (see \cite{gueant2019deep, weinan2017deep}), the computation time can still be an obstacle. The market maker has to answer a request from a client in a given time, which can be insufficient to recalibrate the model if some parameter changes need to be applied (for example, the correlation structure). Finally, when dealing with short maturity options, the market maker has to manage the positions individually to avoid sudden high exposure due to the Gamma of a specific position. This specificity prevents the use of some dimensionality reduction techniques. \\
In the existing academic literature, options market making is addressed in \cite{baldacci2019algorithmic, el2015stochastic, stoikov2009option}. In \cite{stoikov2009option}, the authors consider three different settings for a market maker managing a single option and its underlying. The first setting is a complete market with continuous trading in the perfectly liquid underlying. The second is a complete market with an illiquid underlying where the market maker sets bid and ask quotes in the option and the stock. The third is an incomplete market with residual risks due to stochastic volatility and overnight jumps in the stock price. In \cite{el2015stochastic}, the authors consider a market maker in charge of a single option in a framework à la Avellaneda-Stoikov, where an underlying follows a one-factor stochastic volatility model, and the market maker is always Delta-hedged. They provide optimal bid and ask quotes for the option taking into account the risk of model misspecification. Finally, in \cite{baldacci2019algorithmic}, the authors consider a perfectly Delta-hedged market maker in charge of a book of options with long maturities, whose prices are driven by a stochastic volatility model. The only risk factor comes from the Brownian motion driving the volatility of the underlying. Using a first-order approximation of the Vega of the portfolio, they show that the problem of an options market maker boils down to a three-dimensional Hamilton-Jacobi-Bellman (HJB) equation, which can be solved using classical finite difference schemes. By linearizing the value function of the market maker around the Vegas at the initial time, they provide a way to relax the constant Vega assumption. However, the disadvantage of this approach is its time-consumption due to the necessity to simulate inventory trajectories. Moreover, the constant Vega assumption, making the control problem time-inconsistent, is only valid for a market maker in charge of long-dated options where possible jumps in the underlying do not influence the global risk position drastically. Finally, if one adds other Greeks such as Vanna and Vomma, the model becomes hardly tractable as the HJB equation is in dimension $5$. \\
In this article, our goal is to propose a market making algorithm that considers the three specificities mentioned above, more flexible and applicable in practice. To this end, we consider a market maker in charge of a book of options on different underlyings. The assets follow a one-factor stochastic volatility model with jumps, and the Brownian motions driving the underlying and the volatility of each asset are correlated. We first consider the case of a perfectly Delta-hedged market maker who manages his volatility Greeks, namely the Vega, the Vanna, and the Vomma, for all his positions. Inspired by \cite{evangelista2018new}, we approximate the jump-diffusion HJB equation corresponding to the optimization problem of the market maker with an elliptic Partial Differential Equation (PDE for short). Using an ansatz quadratic in the inventories, we approximate the value function by a system of non-linear PDEs, which can be easily solved via classical numerical methods for a small number of assets. For a number of underlyings above two, we recast the ansatz by adding a non-local term, enabling the use of the Deep Galerkin method as in \cite{hirsa2020unsupervised} to solve the system of PDEs rapidly due to its simple non-linearity.\\
The method presented in this paper has several advantages. First, contrary to \cite{el2015stochastic} and similarly to~\cite{baldacci2019algorithmic}, the market maker can design trading strategies on a high number of options. Contrary to \cite{baldacci2019algorithmic}, the market maker controls each position individually, which is particularly important for short-dated options that must be managed one by one. Moreover, it enables us to reproduce classic option market making behavior where one option is hedged with another. Second, we allow continuous updates of the Greeks (Delta, Vega, Vanna, Vomma) of each option, and the dependence of the intensities of orders arrival on the dynamics of the underlying and its stochastic volatility. This is a major improvement compared to \cite{baldacci2019algorithmic}, as the quotes of the market maker are adjusted dynamically with respect to the evolution of both an underlying and stochastic volatility, allowing the problem to be solved in a time-consistent way. Third, we can use a model for the underlying dynamics with an arbitrary number of factors without increasing the computation time. We show numerically how this algorithm outperforms the one in \cite{baldacci2019algorithmic} in terms of average PnL for a portfolio of options, where Vegas vary significantly. \\
The paper has the following structure: in Section \ref{sec_framework}, we present the framework of options market making and the corresponding optimization problem faced by the market maker. In Section \ref{Ansatz}, we show how to simplify the problem by approximating the value function. Finally, Section \ref{sec_numerics} is devoted to numerical experiments.
\section{Framework}\label{sec_framework}
\subsection{The option book}
We consider a filtered probability space $(\Omega,\mathcal{F},\mathbb{P})$ where all stochastic processes are defined, and a time horizon $T>0$. We consider $d>1$ stocks with the following one factor stochastic volatility dynamics with jumps:
\begin{align}\label{eq_stock_dynamics}
\left\{
\begin{array}{ll}
dS^i_{t} = b_{\mathbb{P}}^i (t,S_t^i) dt +\sigma^i(t,S_t^i, \nu_t^i)dW_{t}^{i,S} + \int_{\mathbb{R}} Z^i(dt,dz) \\
d\nu^i_{t}=a^i_{\mathbb{P}}(t,\nu^i_t)dt+v^i_{\mathbb{P}}(t,\nu_t^i)dW_{t}^{i,\nu},
\end{array}
\right.
\end{align}
where $(W_{t}^{i,S},W_{t}^{i,\nu})_{t\in \mathbb{R}^{+}}$ is a couple of Brownian motions with quadratic covariation given by the coefficients $\rho^{i}=\frac{d\langle W^{i,S},W^{i,\nu}\rangle}{dt} \in (-1,1)$, and $a^i_{\mathbb{P}},b^i_{\mathbb{P}},v^i_{\mathbb{P}},\sigma^i$ are such that the SDEs \eqref{eq_stock_dynamics} admit a unique strong solution\footnote{In particular, for the sake of readability, we assume that there is no correlation between the volatility process of an asset and the variations of another asset. This assumption can be directly relaxed.}. The processes $Z^i(dt,dz)$ are marked point processes independent from the Brownian motions, with intensity kernels $\kappa_t^i(dz)$. We also assume that there exists covariance matrices $\Sigma^S,\Sigma^{\nu} \in \mathcal{M}_d(\mathbb{R})$ which correspond to the correlation structure of the stocks and the stochastic volatility in the option book. There also exists a risk-neutral probability measure $\mathbb{Q}$ such that
\begin{align*
\left\{
\begin{array}{ll}
dS^i_{t} = \sigma^i(t,S_t^i, \nu_t^i)d\hat{W}_{t}^{i,S} + \int_{\mathbb{R}}Z^i(dt,dz) \\
d\nu^i_{t}=a^i_{\mathbb{Q}}(t,\nu^i_t)dt+v^i_{\mathbb{Q}}(t,\nu_t^i)d\hat{W}_{t}^{i,\nu},
\end{array}
\right.
\end{align*}
where $(\hat{W}_{t}^{i,S},\hat{W}_{t}^{i,\nu}), i\in \{1,\dots,d\}$ are $\mathbb{Q}-$Brownian motions.
\begin{remark}
As the reader will see in the following, by applying the ansatz detailed in Section \ref{Ansatz}, one can use a multi-factor stochastic volatility model for the underlying without increasing the complexity of the algorithm. For example, one can work with the well-known two-factor Bergomi model easily, see \cite{bergomi2008smile, bergomi2015stochastic}.
\end{remark}
On every underlying $i\in \{1,\dots,d\}$ we consider a set of $N^i$ European options $\mathcal{O}^{i,j}$ of maturity~$T^{i,j}$, for $j\in \{1,\dots,N^i\}$. In the above one-factor model, we know that for all $(i,j)\in \{1,\dots,d\}\times \{1,\dots,N^i\}$, and all $t\in [0,T^{i,j}]$ such that $T\!<\!\min_{i,j} T^{i,j}$, $\mathcal{O}_{t}^{i,j}=O^{i,j}(t,S_{t}^i,\nu^i_{t})$ where $O^{i,j}$ is a solution on $[0,T^{i,j})\times \mathbb{R}_{+}^2$ of the following partial differential equation under the probability $\mathbb{Q}$:
\begin{align*}
\begin{split}
0 = \,\, & \partial_{t}O^{i,j}(t,S^i,\nu^i)+a^i_{\mathbb Q}(t,\nu^i)\partial_{\nu^i}O^{i,j}(t,S^i,\nu^i) +\frac{1}{2}\big(\sigma^i (t,S^i,\nu^i)\big)^{2}\partial^{2}_{S^iS^i}O^{i,j}(t,S^i,\nu^i)\nonumber\\
& +\rho^{i,i} \nu^i_{\mathbb Q}(t,\nu^i)\sigma^i(t,S^i,\nu^i)\partial^{2}_{\nu^i S^i}O^{i,j}(t,S^i,\nu^i)+\frac{1}{2}\big(v_{\mathbb{Q}}^i(t,\nu^i)\big)^2\partial^{2}_{\nu^i\nu^i}O^{i,j}(t,S^i,\nu^i) \\
& + \int_{\mathbb{R}} \Big(O^{i,j}(t,S^i+\gamma^i(t,z),\nu^i)-O^{i,j}(t,S^i,\nu^i)\Big)\kappa^i(dz).
\end{split}
\end{align*}
As the time horizon $T$ is small compared to the maturity of the options (which can be from one day up to several years), the terminal condition of the PDEs does not have to be specified. In Section~\ref{sec_numerics}, numerical experiments are addressed using European call options but any other option with a path-independent payoff can be considered. We now define the market maker's problem.
\subsection{The market maker's problem on OTC markets}
We consider a market maker in charge of providing bid and ask quotes for the $\sum_{i\in \{1,\dots,d\}} N^i$ options over the period $[0,T]$ where $T<\min_{i,j} T^{i,j}$. The bid and ask prices on the option $j\in \{1,\dots,N^i\}$ of stock $i\in \{1,\dots,d\}$ are defined, for transaction size $z$, by
\begin{align*}
P_t^{i,j,b} = \mathcal{O}^{i,j}_t - \delta_t^{i,j,b}(z), \quad P_t^{i,j,a} = \mathcal{O}^{i,j}_t + \delta_t^{i,j,a}(z),
\end{align*}
where $\big(\delta_t^{i,j,b}(\cdot),\delta_t^{i,j,a}(\cdot)\big)\in \mathcal{A}$, where $\mathcal{A}$ is the set of uniformly bounded $\mathcal{F}$-predictable processes. They represent the spread on the bid or ask side of the option $\mathcal{O}^{i,j}$. The number or transactions on these options are defined by marked point processes $N^{i,j,b}(dt,dz),N^{i,j,a}(dt,dz)$, with almost surely no simultaneous jumps, whose respective intensity processes are given by
\begin{align*}
\Lambda_t^{i,j,b}(S,\nu,dz)=\lambda^{i,j,b}\big(S,\nu,\delta_t^{i,j,b}(z)\big)\mu^{i,j,b}(dz), \quad \Lambda_t^{i,j,a}(S,\nu,dz)=\lambda^{i,j,a}\big(S,\nu,\delta_t^{i,j,a}(z)\big)\mu^{i,j,a}(dz).
\end{align*}
The couples $(\mu^{i,j,b},\mu^{i,j,a})$ are probability measures on $\mathbb{R}_+^\star$ modeling the distribution of transaction sizes for the options. Note that, in our framework, the intensities are allowed to depend on both the underlying and the stochastic volatility of the assets. \\
The market maker manages his inventory process on each option, that is
\begin{align*
dq_t^{i,j} = \int_{\mathbb{R}_+^\star}z \big(N^{i,j,b}(dt,dz) - N^{i,j,a}(dt,dz)\big).
\end{align*}
For the sake of simplicity, we represent the vector of inventories as follows:
\begin{align*}
q^{\mathbf{T}} = \big(q^{1,1},\dots,q^{1,N^1},\dots,q^{d,N^d} \big) \in \mathcal{M}_{\sum_{l=1}^{d} N^l,1}(\mathbb{R}).
\end{align*}
Assuming perfect Delta-hedging\footnote{This assumption can be relaxed by assuming that the market maker acts on the stock market. This way, the mean-variance objective function will take into account the Delta of the portfolio.}, the $\Delta$ of the portfolio on the $i$-th asset, $i\in \{1,\dots,d\}$, is given by
\begin{align*}
\Delta_t^i = \sum_{j\in \{1,\dots,N^i\}}\partial_{S^i}O^{i,j}(t,S_t^i,\nu_t^i)q_t^{i,j},\quad \Delta_t = \sum_{i\in \{1,\dots,d\}} \Delta_t^i.
\end{align*}
The cash process of the market maker at time $t$ is defined by
\begin{align*}
dX_t &= \!\!\!\! \sum_{\substack{(i,j)\in \\ \{1,\dots,d\} \times \\ \{1,\dots,N^i\}}}\!\!\!\!\!\Big(\int_{\mathbb{R}_+^\star}\!\!\!\! z\big(\delta_t^{i,j,b}(z)N_t^{i,j,b}(dt,dz) + \delta_t^{i,j,a}(z)N_t^{i,j,a}(dt,dz)\big) - \mathcal{O}_t^{i,j}dq_t^{i,j}\Big) \!\!+\!\!\!\! \sum_{i\in \{1,\dots,d\}} \big(S_t^i d\Delta_t^i + d\langle\Delta^{i},S^i\rangle_t\big).
\end{align*}
We finally define the Mark-to-Market value of the portfolio of the market maker as
\begin{align*}
V_t = X_t - \sum_{i\in \{1,\dots,d\}} \Delta_t^i S_t^i + \sum_{(i,j)\in\{1,\dots,d\}\times\{1,\dots,N^i\}}q_t^{i,j}\mathcal{O}_t^{i,j}.
\end{align*}
For all $(i,j)\in\{1,\dots,d\}\times\{1,\dots,N^i\}$, the Vega, the Vomma and the Vanna of the option $\mathcal{O}_t^{i,j}$ are defined as
\begin{align*}
&\mathcal{V}_t^{i,j}= \partial_{\sqrt{\nu^i}}O^{i,j}(t,S_t^i,\nu_t^i) = 2\sqrt{\nu^i}\partial_{\nu^i}O^{i,j}(t,S_t^i,\nu_t^i), \\
&(\mathcal{VO})_t^{i,j}=\partial_{\sqrt{\nu^i}\sqrt{\nu^i}}O^{i,j}(t,S_t^i,\nu_t^i) = 4\nu^i \partial^2_{\nu^i \nu^i}O^{i,j}(t,S_t^i,\nu_t^i), \\
& (\mathcal{VA})_t^{i,j}= \partial_{S\sqrt{\nu^i}}O^{i,j}(t,S_t^i,\nu_t^i) = 2\sqrt{\nu^i}\partial_{S\nu^i}O^{i,j}(t,S_t^i,\nu_t^i).
\end{align*}
We also define the vectors $e^{i,j}\in \mathbb{R}^{\sum_{l=1}^{d} N^l}$ where $e_k^{i,j}=\mathbf{1}_{\{k=\sum_{l=1}^{i-1} N^l+j\}}$ and $(e^1,\dots,e^d)$ as the canonical basis of $\mathbb{R}^d$. If we denote by $\Gamma_t^i = \frac{v_{\mathbb{P}}^i(t,\nu_t^i)}{2\sqrt{\nu_t^i}}\sum_{j\in \{1,\dots,N^i\}} q_t^{i,j} \mathcal{V}_t^{i,j}$, we can write the market maker's problem as
\begin{align}\label{pb_MM}
\sup_{\delta \in \mathcal{A}}\mathbb{E}\Big[V_T - \frac{\gamma}{2}\sum_{(i,k)\in \{1,\dots,d\}^2}\int_0^T \Gamma_t^i \Gamma_t^k \Sigma^{\nu,i,k}dt \Big].
\end{align}
Here we penalize the portfolio's total Vega. Any other penalization could be used, as long as it is quadratic in $q$. For example, this includes more complicated penalties linked to another position to hedge, or some target for the Greeks. We define the Hamiltonians
\begin{align*}
H^{i,j,a}(S,\nu,p) = \sup_{\delta }\lambda^{i,j,a}(S,\nu,\delta) \big(\delta-p\big), \quad H^{i,j,b}(S,\nu,p) = \sup_{\delta }\lambda^{i,j,b}(S,\nu,\delta) \big(\delta-p\big),
\end{align*}
and the following processes $\mathcal{G}(t,S,\nu) \in \mathbb{R}^{\sum_{l=1}^{d} N^l}$ such that
\begin{align*}
&\mathcal{G}_{j}(t,S,\nu) = \mathcal{V}_t^{k_j,j-\left(\sum_{l=1}^{k_j-1} N^l\right)}\frac{a^{k_j}_{\mathbb{P}}(t,\nu^{k_j})-a^{k_j}_{\mathbb{Q}}(t,\nu^{k_j})}{2\sqrt{\nu^{k_j}}} \\
& \qquad \qquad + \rho^{k_j}(\mathcal{VA})_t^{k_j,j-\left(\sum_{l=1}^{k_j-1} N^l\right)}\frac{v^{k_j}_{\mathbb{P}}(t,\nu^{k_j})-v^{k_j}_{\mathbb{Q}}(t,\nu^{k_j})}{2\sqrt{\nu^{k_j}}}\sigma^{k_j}(t,S^{k_j},\nu^{k_j})\\
& \qquad \qquad + (\mathcal{VO})_t^{k_j,j-\left(\sum_{l=1}^{k_j-1} N^l\right)}\frac{\big(v^{k_j}_{\mathbb{P}}(t,\nu^{k_j})\big)^2-\big(v^{k_j}_{\mathbb{Q}}(t,\nu^{k_j})\big)^2}{4\nu^{k_j}},
\end{align*}
where $k_j=i$ if $j\in \{\sum_{l=1}^{i-1} N^l,\dots,\sum_{l=1}^{i} N^l\}$, for $i\in \{1,\dots,d\}$. We also define $\mathcal{R}(t,S,\nu)\! \in\! \mathcal{M}_{\sum_{l=1}^{d}\!\! N^l,d}(\mathbb{R)}$ such that
\begin{align*}
& \mathcal{R}_{j,i}(t,S,\nu) = \frac{v_{\mathbb{P}}^i(t,\nu^i)}{2\sqrt{\nu_t^i}}\mathcal{V}_t^{i,j-\left(\sum_{l=1}^{k_j-1} N^l\right)}, \quad \text{for } j\in \{\sum_{l=1}^{i-1} N^l,\dots,\sum_{l=1}^{i} N^l\},i\in \{1,\dots,d\},
\end{align*}
and $0$ otherwise. Finally, denote the diffusion part of the HJB equation as
\begin{align*}
\mathcal{L}(t,S,\nu,q,u) = & \sum_{i\in \{1,\dots,d\}} b^i_{\mathbb P}(t,S^i) \partial_{S^i}u(t,S,\nu,q) + \sum_{i\in \{1,\dots,d\}} a^i_{\mathbb P}(t,\nu^i) \partial_{\nu^i}u(t,S,\nu,q)\\
& + \frac{1}{2}\sum_{(i,k)\in \{1,\dots,d\}^2} \partial_{S^i S^k}u(t,S,\nu,q) \sigma^i(t,S^i,\nu^i)\sigma^j(t,S^k,\nu^j)\Sigma^{S,i,k} \\
& + \sum_{i\in \{1,\dots,d\}}\int_{\mathbb{R}}\kappa^i(dz)\Big(u\big(t,S + e^i\gamma^i(t,z),\nu,q\big)-u(t,S,\nu,q)\Big) \\
& + \frac{1}{2}\sum_{(i,k)\in \{1,\dots,d\}^2} \partial_{\nu^i \nu^j}u(t,S,\nu,q) v^i_{\mathbb{P}}(t,\nu^i)v^k_{\mathbb{P}}(t,\nu^k)\Sigma^{\nu,i,k} \\
& + \sum_{i\in \{1,\dots,d\}}\partial_{\nu^i S^i}u(t,S,\nu,q) \rho^{i} v_{\mathbb{P}}^i (t,\nu^i) \sigma^i(t,S^i,\nu^i).
\end{align*}
The HJB equation associated to \eqref{pb_MM} with compact notations is
\begin{align}\label{HJB_Compact}
\begin{split}
0 = & \,\, \partial_t u(t,S,\nu,q) + \mathcal{L}(t,S,\nu,q,u) + q^{\mathbf{T}}\mathcal{G}(t,S,\nu) - \frac{\gamma}{2} q^{\mathbf{T}} \mathcal{R}(t,S,\nu) \Sigma^{\nu} \mathcal{R}^{\mathbf{T}}(t,S,\nu)q \\
& + \sum_{\substack{(i,j)\in \\ \{1,\dots,d\} \times \\ \{1,\dots,N^i\}}} \int_{\mathbb{R}_+}z H^{i,j,b}\Big(S,\nu,\frac{u(t,S,\nu,q)-u(t,S,\nu,q+z e^{i,j})}{z}\Big)\mu^{i,j,b}(dz) \\
& + \sum_{\substack{(i,j)\in \\ \{1,\dots,d\} \times \\ \{1,\dots,N^i\}}} \int_{\mathbb{R}_+}z H^{i,j,a}\Big(S,\nu,\frac{u(t,S,\nu,q)-u(t,S,\nu,q-z e^{i,j})}{z}\Big)\mu^{i,j,a}(dz).
\end{split}
\end{align}
with terminal condition $u(T,S,\nu,q)=0$. The proof of existence and uniqueness of a viscosity solution to \eqref{HJB_Compact} associated to the control problem \eqref{pb_MM} relies on classic arguments of second order viscosity solutions with jumps, see for example \cite{barles2008second,bastien2020bid,bergault2019size}.
\section{Solving the market maker's problem with a system of non-linear PDEs}\label{Ansatz}
Equation \eqref{HJB_Compact} is intractable with classical numerical methods when dealing with several options on several underlyings. Notably, the method proposed in \cite{baldacci2019algorithmic} to overcome the constant Vega assumption requires Monte-Carlo simulations of high-dimensional inventory trajectories, which is very time-consuming. In this section, inspired by \cite{evangelista2018new}, we propose an approximation of the value function of the market maker, quadratic with respect to the vector of inventories to reduce the dimensionality of the problem. \\
A Taylor expansion at $0$ on the third variable with respect to $\epsilon$ gives
\begin{align*}
& H^{i,j,b}\Big(S,\nu,\frac{u(t,S,\nu,q)-u(t,S,\nu,q+\epsilon z e^{i,j})}{z}\Big) + H^{i,j,a}\Big(S,\nu,\frac{u(t,S,\nu,q)-u(t,S,\nu,q-\epsilon z e^{i,j})}{z}\Big) \\
& = H^{i,j,b}(S,\nu,0) + H^{i,j,a}(S,\nu,0) + \epsilon\big( H^{' i,j,a}(S,\nu,0) - H^{i,j,b}(S,\nu,0)\big) \partial_{q}u(t,S,\nu,q) \\
& +\frac{\epsilon^2}{2} \Big(H^{''i,j,a}(S,\nu,0)\big(\partial_q u(t,S,\nu,q)\big)^2 - zH^{'i,j,a}(S,\nu,0)\partial_{qq} u(t,S,\nu,q) \Big) \\
& + \frac{\epsilon^2}{2} \Big(H^{''i,j,b}(S,\nu,0)\big(\partial_q u(t,S,\nu,q)\big)^2 - zH^{'i,j,b}(S,\nu,0)\partial_{qq} u(t,S,\nu,q) \Big) + o (\epsilon^3),
\end{align*}
and by taking $\epsilon=1$, Equation \eqref{HJB_Compact} becomes
\begin{align}\label{HJB_ansatz}
0 = \,\, & \partial_t u(t,S,\nu,q) + \mathcal{L}(t,S,\nu,q,u) +q^{\mathbf{T}} \mathcal{G}(t,S,\nu) - \frac{\gamma}{2} q^{\mathbf{T}} \mathcal{R}(t,S,\nu) \Sigma^{\nu} \mathcal{R}^{\mathbf{T}}(t,S,\nu)q \nonumber\\
& + \sum_{\substack{(i,j)\in \\ \{1,\dots,d\} \times \\ \{1,\dots,N^i\}}} \int_{\mathbb{R}_+} \Bigg( H^{i,j,b}(S,\nu,0) - H^{i,j,b}(S,\nu,0) \partial_{q}u(t,S,\nu,q) \nonumber\\
& + \frac{1}{2} \Big(H^{''i,j,b}(S,\nu,0)\big(\partial_q u(t,S,\nu,q)\big)^2 - zH^{'i,j,b}(S,\nu,0)\partial_{qq} u(t,S,\nu,q) \Big)\Bigg)
\mu^{i,j,b}(dz) \\
& + \sum_{\substack{(i,j)\in \\ \{1,\dots,d\} \times \\ \{1,\dots,N^i\}}} \int_{\mathbb{R}_+}\Bigg (H^{i,j,a}(S,\nu,0) + H^{' i,j,a}(S,\nu,0) \partial_{q}u(t,S,\nu,q) \nonumber\\
& +\frac{1}{2} \Big(H^{''i,j,a}(S,\nu,0)\big(\partial_q u(t,S,\nu,q)\big)^2 - zH^{'i,j,a}(S,\nu,0)\partial_{qq} u(t,S,\nu,q) \Big) \Bigg)\mu^{i,j,a}(dz).\nonumber
\end{align}
In the following we will show how a simple ansatz, quadratic with respect to the vector of inventories, leads to significant simplifications. For the sake of the simplicity of the notation, assume that $H^{i,j,a}=H^{i,j,b}=H^{i,j}$ (extension to asymmetric intensities is straightforward). By setting
\begin{align*
u(t,S,\nu,q)= \theta^0(t,S,\nu) + q^{\mathbf{T}}\theta^1(t,S,\nu) - q^{\mathbf{T}}\theta^2(t,S,\nu)q,
\end{align*}
where $\theta^0 \in \mathbb{R}, \theta^1 \in \mathbb{R}^{\sum_{l=1}^{d} N^l}, \theta^2 \in \mathcal{M}_{\sum_{l=1}^{d} N^l}(\mathbb{R})$ are solutions of the following system of non-linear PDEs:
\begin{align}\label{system_pde}
\begin{cases}
0 = & \partial_t \theta^0(t,S,\nu) + \overline{\mathcal{L}}(t,\theta^0,\nu,S) + 2 \sum_{(i,j)\in \{1,\dots,d\}\times \{1,\dots,N^i\}} H^{i,j}(S,\nu,0) \\
& + \int_{\mathbb R_+}\Big( 2 z H^{'i,j}(S,\nu,0)\theta^2_{j,j}(t,S,\nu)
+ H^{'' i,j}(S,\nu,0)\big(\theta_j^1(t,S,\nu)\big)^2\Big)\mu^{i,j}(dz) \\
0 = & \partial_t \theta^1(t,S,\nu) +\overline{\mathcal{L}}(t,\theta^1,\nu,S) + \mathcal{G}(t,S,\nu)+ 4 \theta^2(t,S,\nu)\text{diag}\big(H^{''}(S,\nu,0)\big) \theta^1(t,S,\nu) \\
0 = & \partial_t \theta^2(t,S,\nu) + \overline{\mathcal{L}}(t,\theta^2 ,\nu,S) - \frac{\gamma}{2} \mathcal{R}(t,S,\nu) \Sigma^{\nu} \mathcal{R}^{\mathbf{T}}(t,S,\nu) \\
& {} + 4 \theta^2(t,S,\nu) \text{diag}\big(H^{''}(S,\nu,0)\big)\theta^2(t,S,\nu),
\end{cases}
\end{align}
where
\begin{align*}
\overline{\mathcal{L}}(t,\theta,\nu,S) = & \sum_{i\in \{1,\dots,N\}} b^i_{\mathbb P}(t,S^i) \partial_{S^i}\theta(t,S,\nu) + \sum_{i\in \{1,\dots,N\}} a^i_{\mathbb P}(t,\nu^i) \partial_{\nu^i}\theta(t,S,\nu) \\
& {}+ \!\frac{1}{2} \!\!\sum_{(i,k)\in \{1,\dots,d\}^2}\!\!\!\!\!\!\!\!\!\!\! \big(\! \partial_{S^i S^j}\theta(t,S,\nu) \sigma^i\!(t,S^i\!,\nu^i)\sigma^k\!(t,S^k\!\!,\nu^k)\Sigma^{S\!,i\!,k} \!\!+\! \partial_{\nu^i \nu^k}\theta(t,S\!,\nu) v^i_{\mathbb{P}}(t,\nu^i)v^k_{\mathbb{P}}(t,\nu^k)\Sigma^{\nu,i,k}\!\big) \\
&+ \!\!\!\sum_{i\in \{1,\dots,d\}}\!\!\bigg(\partial_{\nu^i S^i}\theta(t\!,S\!,\nu) \rho^{i} v_{\mathbb{P}}^i (t,\nu^i) \sigma^i(t\!,S^i\!,\nu^i)
\! + \!\!\int_{\mathbb{R}}\!\kappa^i(dz)\Big(\theta\big(t\!,S\! +\! e^i\gamma^i(t,z)\!,\nu\big)\!\!-\!\!\theta(t\!,S\!,\nu)\Big)\bigg) .
\end{align*}
and $\theta^0(T,S,\nu)=0,\theta^1(T,S,\nu)=\mathbf{0}_{\sum_{l=1}^{d} N^l,1},\theta^2(T,S,\nu)=\mathbf{0}_{\sum_{l=1}^{d} N^l}$. In system \eqref{HJB_ansatz}, one can note that the PDE with respect to $\theta^2$ is independent from the two others, which reduces the overall complexity. It can easily be solved for a small number of underlyings and a large number of options using finite difference schemes. Note that a higher order expansion does not yield a polynomial solution. However, it is possible to truncate the high degree terms to obtain a polynomial solution. This does not lead to a significant change of the value function or the controls if the penalty term is at most quadratic.\\
We now show some numerical applications of the methodology.
\section{Numerical results}\label{sec_numerics}
To perform a comparison with respect to the existing methods, we first recall the methodology of \cite{baldacci2019algorithmic}. In this article, the authors consider a market maker managing a book of options on a single underlying, and they suppose he is perfectly delta-hedged. We have the following set of market parameters:
\begin{itemize}
\item $d=1,N^1=N=20$: there are $20$ call options on a single underlying.
\item Stock price at time $t=0$: $S_0=100$\euro{}.
\item Instantaneous variance at time $t=0$: $\nu_0=0.04\text{ year}^{-1}$.
\item Heston model parameters: $b_{\mathbb{P}}(t,S)=\mu S$, $\sigma(t,S,\nu)=S\sqrt{\nu}$, $v_{\mathbb{P}}(t,\nu)=v_{\mathbb{Q}}(t,\nu)=\xi\sqrt{\nu}$, with $\xi=0.7\text{ year}^{-1}$.
\item $a_{\mathbb{P}}(t,\nu)=\kappa_{\mathbb{P}}(\theta_{\mathbb{P}}-\nu),a_{\mathbb{Q}}(t,\nu)=\kappa_{\mathbb{Q}}(\theta_{\mathbb{Q}}-\nu)$, with $\kappa_{\mathbb{P}}=\kappa_{\mathbb{Q}}=2\text{ year}^{-1}$, $\theta_{\mathbb{P}}=\theta_{\mathbb{Q}}=0.04\text{ year}^{-1}$.
\item $Z(dt,dz)= 0$: there is no jump in the dynamics of the underlying.
\item Spot-variance correlation: $\rho=-0.7$.
\end{itemize}
We consider the case of a market maker dealing with $20$ European call options written on that stock where the strike$\times$maturity couples are the elements $(K^j,T^j),j\in \{1,...,20\}$ of the set $\mathcal K\times \mathcal T$, where
\begin{align*}
\mathcal{K}=\{97,98,99,100\}, \quad \mathcal{T}=\{0.3 \text{ year} ,0.4 \text{ year}, 0.5 \text{ year}, 0.6 \text{ year}, 0.7 \text{ year}\}.
\end{align*}
These market parameters provide the implied volatility surface as in Figure \ref{impvolsurf}.
\vspace{-4mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.6\textwidth]{img/impvolsurf.pdf}
\caption{Implied volatility surface associated with the market parameters.}\label{impvolsurf}
\end{center}
\end{figure}
\vspace{-5mm}
We consider mainly in-the-money options with maturity ranging from $3$ to $6$ months so that, due to the influence of both Vanna and Vomma, the Vega of the portfolio changes noticeably and the prices of options are non negligible. \\
We define the following intensity functions:
\begin{align*}
\Lambda^{j,a}(S,\nu,\delta)=\Lambda^{j,b}(S,\nu,\delta)=\frac{\lambda^j}{1+\exp\big(\alpha+\frac{\beta}{\mathcal{V}_t^j}\delta\big)},
\end{align*}
for $j\in \{1,\dots,N\}$, where $\lambda^j= \frac{252\times 50}{1+0.7\times |S_0-K^j|}\text{ year}^{-1}$, $\alpha=-0.7$, and $\beta=10 \text{ year}^{\frac{1}{2}}$. The choice of $\lambda^j$ corresponds to $50$ requests per day for at-the-money options, and decreases to 13.2 for the most in-the-money options. The choice of $\alpha$ corresponds to a probability of $\frac{1}{1+e^{-0.7}}\approx 66\%$ to trade when the answered quote is the mid-price (i.e $\delta=0$). The choice of $\beta$ corresponds to a probability of $\frac{1}{1+e^{-0.8}}\approx 68\%$ to trade when the answered quote corresponds to an implied volatility $1\%$ better for the client and a probability of $\frac{1}{1+e^{-0.6}}\approx 64\%$ to trade when the answered quote corresponds to an implied volatility $1\%$ worse for the client. \\
We assume transactions of constant size with $z^j = \frac{5\times 10^5}{\mathcal{O}_0^j}$ contracts for option $j$, in other words, the measures $\mu^{j,b},\mu^{j,a}$ are Dirac masses at~$z^j$. This corresponds approximately to $500000$\euro{} per transaction. \\
We finally set $T=0.004$ year (i.e $1$ day), and a risk aversion parameter $\gamma= 2\dot 10^{-5}$\euro{}$^{-1}$. \\
The HJB equation using the constant Vega assumption of \cite{baldacci2019algorithmic} is
\begin{align*}
0 = \,\, &\partial_t u(t,\nu,\mathcal{V}^\pi) + a_{\mathbb{P}}(t,\nu)\partial_{\nu}u(t,\nu,\mathcal{V}^\pi) + \frac{1}{2}\nu \xi^2 \partial_{\nu \nu} u(t,\nu,\mathcal{V}^\pi) + \mathcal{V}^\pi \frac{a_{\mathbb{P}}(t,\nu)-a_{\mathbb{Q}}(t,\nu)}{2\sqrt{\nu}}-\frac{\gamma\xi^2}{8} (\mathcal{V}^\pi)^2 \\
& + \!\!\!\sum_{j\in \{1,\dots,N\}} \!\!\!z^j H^{j,b}\Big(\frac{u(t,\nu,\mathcal{V}^\pi)-u(t,\nu,\mathcal{V}^\pi+z^j \mathcal{V}^j)}{z^j} \Big) +\!\!\! \sum_{j\in \{1,\dots,N\}} \!\!\! z^j H^{j,a}\Big(\frac{u(t,\nu,\mathcal{V}^\pi)-u(t,\nu,\mathcal{V}^\pi-z^j \mathcal{V}^j)}{z^j} \Big),
\end{align*}
with terminal condition $u(T,\nu,\mathcal{V}^\pi)=0$, and
\begin{align*}
& \mathcal{V}_t^{\pi} = \sum_{j\in \{1,\dots,N\}} z^j \mathcal{V}^j q_t^{j}, \\
& H^{j,a/b}(p)=\sup_{\delta^{j,a/b}} \Lambda^{j,a/b}(\delta^{j,a/b})(\delta^{j,a/b}-p).
\end{align*}
In the case where Vega are not constant, we use the following ansatz:
\begin{align*}
u(t,S,\nu,q)= \theta^0(t,S,\nu) + q^{\mathbf{T}}\theta^1(t,S,\nu) + q^{\mathbf{T}}\theta^2(t,S,\nu)q,
\end{align*}
where $\theta^0 \in \mathbb{R},\theta^1\in \mathbb{R}^N,\theta^2\in \mathcal{M}_N(\mathbb{R})$. Define
\begin{align*}
\tilde{\mathcal{L}}(t,S,\nu,\theta) = a_{\mathbb{P}}(t,\nu)\partial_{\nu}\theta(t,S,\nu)+ \frac{1}{2}\nu \xi^2 \partial_{\nu \nu}\theta(t,S,\nu) + \frac{1}{2}\nu S^2 \partial_{SS}\theta(t,S,\nu) + \rho \nu S \xi \partial_{\nu S}\theta(t,S,\nu),
\end{align*}
and assume symmetry of intensity functions, that is $H^{j,b}=H^{j,a}=H^j$, we obtain the following system of coupled PDEs:
\begin{align}\label{sys_pde_matrix}
\begin{cases}
0 = & \partial_t \theta^0(t,S,\nu) + \tilde{\mathcal{L}}(t,S,\nu,\theta^0) + 2 \sum\limits_{j\in \{1,\dots,N\}}H^j(S,\nu,0) + 2\sum\limits_{j\in \{1,\dots,N\}}z^j H^{'j}(S,\nu,0)\theta^2_{j,j}(t,S,\nu) \\
& {}+ \sum\limits_{j\in \{1,\dots,N\}}H^{'' j }(S,\nu,0)\big(\theta_j^1(t,S,\nu)\big)^2 \\
0 = & \partial_t \theta^1(t,S,\nu) + \tilde{\mathcal{L}}(t,S,\nu,\theta^1) + \mathcal{V}_t \frac{a_{\mathbb{P}}(t,\nu)-a_{\mathbb{Q}}(t,\nu)}{2\sqrt{\nu}} + 4 \theta^2(t,S,\nu)\text{diag}\Big(H^{''}(S,\nu,0)\Big) \theta^1(t,S,\nu) \\
0 = & \partial_t \theta^2(t,S,\nu) + \tilde{\mathcal{L}}(t,S,\nu,\theta^2) - \frac{\gamma\xi^2}{8}\text{diag}\big(\mathcal{V}_t\big)\frac{\mathbf{1}\mathbf{1}^T}{N}\text{diag}\big(\mathcal{V}_t\big) \\
&{} + 4 \theta^2(t,S,\nu) \text{diag}\big(H^{''}(S,\nu,0)\big)\theta^2(t,S,\nu),
\end{cases}
\end{align}
where
\begin{align*}
& \mathcal{V}_t = \big(\partial_{\sqrt{\nu}}O^1(t,S,\nu),\dots, \partial_{\sqrt{\nu}}O^N(t,S,\nu)\big)^{\mathbf{T}}, \qquad \mathbf{1}= (1,\dots,1)^{\mathbf{T}} \in \mathbb{R}^N.
\end{align*}
We first show in Figures \ref{vf_0_1} and \ref{vf_0_19} some plots of the value function obtained by solving \eqref{sys_pde_matrix}.
\begin{figure}[H]
\begin{center}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vf_0_1_nu002.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vf_0_1_nu004.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\vspace{-5mm}
\includegraphics[width=1\textwidth]{img/vf_0_1_nu006.pdf}
\end{subfigure} \\
\vspace{-3mm}
\caption{Value function for different inventories in $(97,0.3)$ and $(98,0.3)$ options, inventories in other options assumed to be equal 0, for different values of $\nu$.}
\label{vf_0_1}
\end{center}
\end{figure}
\vspace{-6mm}
\begin{figure}[H]
\begin{center}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vf_0_19_nu002.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vf_0_19_nu004.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\vspace{-5mm}
\includegraphics[width=1\textwidth]{img/vf_0_19_nu006.pdf}
\end{subfigure}
\vspace{-3mm}
\caption{Value function for different inventories in $(97,0.3)$ and $(100,0.7)$ options, inventories in other options assumed to be equal 0, for different values of $\nu$.}
\label{vf_0_19}
\end{center}
\end{figure}
The value function often has higher values on the diagonals. The market maker can compensate a long position in an option with a short position in another one. The values are noticeably lower for higher values of the volatility. \\
We present in Figure \ref{askdenu} the evolution of the optimal ask quotes with respect to the stochastic volatility for the spot $S=100$.
\begin{figure}[H]
\begin{center}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdenu_03.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdenu_04.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdenu_05.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdenu_06.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdenu_07.pdf}
\end{subfigure}
\caption{Optimal ask quotes with respect to $\nu$ for different options maturities.}
\label{askdenu}
\end{center}
\end{figure}
We observe the usual increasing behavior of the optimal quotes with respect to both maturity and volatility of the underlying. \\
In Figure \ref{askdespot}, we plot the evolution of the optimal ask quotes with respect to the underlying asset for the volatility $\nu=0.04$.
\begin{figure}[H]
\begin{center}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdespot_03.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdespot_04.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdespot_05.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdespot_06.pdf}
\end{subfigure} \\
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/askdespot_07.pdf}
\end{subfigure}
\caption{Optimal ask quotes with respect to $S$ for different options maturities.}
\label{askdespot}
\end{center}
\end{figure}
\vspace{-5mm}
The behavior of the optimal quotes with respect to the strike depends on the expiry. We can see that the quotes are of the U-shaped nature, the quotes are decreasing in the spot price until some point depending on the strike and the expiry, and then become increasing. The inflection point decreases with the strike decreasing, and conversely for the expiry date. This way we can see that, for example, the quote for the option $(K,T)=(97,0.7)$ is monotonously increasing in the spot price for the considered grid, which is fairly representative of the possible prices during one day. Conversely, for the option $(K,T)=(100,0.3)$ the quote is decreasing for almost all values of the grid.\\
In Figure \ref{pnlinaday}, we show the average PnL per request of the trader during the day over 1000 simulations, using the constant Greek approximation of \cite{baldacci2019algorithmic} and our algorithm. \\
At the beginning of the trading day, both methods yield a similar PnL per request. Notice that the PnL per request for the method with constant Greek approximation is slightly higher. Indeed the parameters at the beginning of the day correspond to the calibration parameters, and our algorithm is more conservative as it takes into account the risk that the underlying price could change. However, after roughly a tenth of the trading day the method with constant Greek approximations starts to underperform our algorithm. This underperformance increases along the day as the constant Vega approximation becomes less accurate. On the contrary, with our method the PnL per request remains constant: there is no need for recalibration.
\vspace{-3mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.6\textwidth]{img/pnlinaday.jpg}
\vspace{-3mm}
\caption{Average PnL per request over the trading day using constant and non-constant Greek approximations.}\label{pnlinaday}
\end{center}
\end{figure}
\vspace{-7mm}
In Figure \ref{vegas_example}, we show one of 1000 simulation examples of the trajectories for the Vega of each option. We see that Vegas for this set of options are changing considerably during the day.
\vspace{-0mm}
\begin{figure}[H]
\begin{center}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vegas_example1.pdf}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\includegraphics[width=1\textwidth]{img/vegas_example2.pdf}
\end{subfigure}
\vspace{-3mm}
\caption{Example of Vega trajectories.}\label{vegas_example}
\end{center}
\end{figure}
\vspace{-7mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.6\textwidth]{img/cdfpnls.png}
\vspace{-2mm}
\caption{Cumulative distribution functions of the PnL over the trading day using both methods.}\label{cdfpnls}
\end{center}
\end{figure}
Finally, we present in Figure \ref{cdfpnls} the cumulative distribution function of the PnL of the trader using the constant Greek approximation of \cite{baldacci2019algorithmic} and our algorithm. We observe that the tail distribution of the PnL using our non-constant Greek approximation is higher compared to the method in \cite{baldacci2019algorithmic}.
\begingroup
\setcounter{section}{0}
\renewcommand\thesection{Appendix \Alph{section}}
\section{The market maker's problem for large number of underlyings}
In this appendix, we present the system of low-dimensional PDEs analogous to \eqref{system_pde} for more complex cases such as the market making problem on several underlyings or the case where a number of different options' parameters is large (over one hundred). \\
We can rewrite the system of $(\sum_{i\in\{1,...,d\}}N^i)^2$ equations \eqref{system_pde} on $\theta^2$ as a set of $d^2$ equations by adding the strike and the maturity to the state variables. The same can be applied for the $\theta^1$ equation. This way we obtain a smaller set of equations, though having more dimensions and some non-local terms.\\
Let $\mathbb{O}^i = \big\{(T^{i,j},K^{i,j}), j\in\{1,...N^i\}\big\}$ be the set of parameters of options on the underlying $i\in\{1,...,d\}$ and let us define $\hat\theta^1_i:[0,T]\times\mathbb{R}\times\mathbb{R}_+\times\mathbb{O}^i \to \mathbb{R}$ such that, for all $j\in\{1,...N^i\}$,
\begin{align*}
\hat\theta^1_i(t,S,\nu,(T^{i,j},K^{i,j})) = \theta^1_{\sum_{l=1}^{i-1} N^l+j}(t,S,\nu).
\end{align*}
Similarly for $i_1,i_2\in\{1,...,d\}$, define $\hat\theta^2_{i_1,i_2}:[0,T]\times\mathbb{R}\times\mathbb{R}_+\times\mathbb{O}^{i_1}\times\mathbb{O}^{i_2}\to \mathbb{R}$ such that, for any $j\in\{1,...N^{i_1}\}$ and $l\in\{1,...N^{i_2}\}$,
\begin{align*}
\hat\theta^2_{i_1,i_2}(t,S,\nu,(T^{i_1,j},K^{i_1,j}), (T^{i_2,l},K^{i_2,l})) = \theta^2_{\sum_{l=1}^{i_1-1} N^l+j,\sum_{l=1}^{i_2-1} N^l+l}(t,S,\nu).
\end{align*}
Then the system of non-linear PDEs \eqref{system_pde} can be rewritten as
\begin{align*}
\begin{cases}
0 =& \partial_t \theta^0(t,S,\nu) + \overline{\mathcal{L}}(t,S,\nu,\theta^0) + 2 \underset{i\in\{1,\dots,d\}}{\sum}\underset{(T,K)\in\mathbb{O}^i}{\sum}H^i(S,\nu,0)(T,K) \\
&{}+ 2\underset{i\in\{1,\dots,d\}}{\sum}\underset{(T,K)\in\mathbb{O}^i}{\sum}\int_{\mathbb R_+}z H^{'i}(S,\nu,0)(T,K)\hat\theta^2_{i,i}\big(t,S,\nu,(T,K),(T,K)\big)\mu^{i,(T,K)}(dz) \\
& + \underset{i\in\{1,\dots,d\}}{\sum}\underset{(T,K)\in\mathbb{O}^i}{\sum}H^{''i}(S,\nu,0)(T,K)\Big(\hat\theta^1_i\big(t,S,\nu,(T,K)\big)\Big)^2 \\
0 =& \partial_t \hat\theta^1_i(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1)) + \overline{\mathcal{L}}^1\big(t,S,\nu,\hat\theta^1_i,(\mathcal{T}^1,\mathcal{K}^1)\big) +\mathcal{G}_i(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1))\\
&{}+ 4 \underset{i_2\in\{1,\dots,d\}}{\sum}\underset{(T,K)\in\mathbb{O}^{i_2}}{\sum}\hat\theta^2_{i,i_2}(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1),(T,K))H^{''i_2}(S,\nu,0)(T,K) \hat\theta^1_{i_2}\big(t,S,\nu,(T,K)\big) \\
0 =& \partial_t \hat\theta^2_{i_1,i_2}\big(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1),(\mathcal{T}^2,\mathcal{K}^2)\big) + \overline{\mathcal{L}}^2\big(t,S,\nu,\hat\theta^2_{i_1,i_2} ,(\mathcal{T}^1,\mathcal{K}^1),(\mathcal{T}^2,\mathcal{K}^2)\big) \\
&{}-\frac{\gamma}{2} \mathcal{R}_{i_1}\big(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1)\big) \Sigma^{\nu,i_1,i_2} \mathcal{R}_{i_2}\big(t,S,\nu,(\mathcal{T}^2,\mathcal{K}^2)\big)\\
& {} + 4 \underset{i_3\in\{1,\dots,d\}}{\sum}\underset{(T,K)\in\mathbb{O}^{i_3}}{\sum}\hat\theta^2_{i_1,i_3}\big(t,S,\nu,(\mathcal{T}^1,\mathcal{K}^1),(T,K)\big) \hat H^{''i_3}(S,\nu,0)(T,K)\hat\theta^2_{i_3,i_2}\big(t,S,\nu,(T,K),(\mathcal{T}^2,\mathcal{K}^2)\big),
\end{cases}
\end{align*}
where $\big((\mathcal{T}^1,\mathcal{K}^1),(\mathcal{T}^2,\mathcal{K}^2)\big)\in \big(\prod_{i\in \{1,\dots,d\}} \mathbb{O}^i\big)^2$ and, for $j\in\{1,...N^{i_1}\}$, $l\in\{1,...N^{i_2}\}$,
\begin{align*}
&H^{i}(S,\nu,0)(T^{i,j},K^{i,j}) = H^{i,j}(S,\nu,0),\\ &\mathcal{G}_i(t,S,\nu,(T^{i,j},K^{i,j})) = \mathcal{G}(t,S,\nu)_{\sum_{l=1}^{i-1} N^l+j},\\
&\mathcal{R}_i(t,S,\nu,(T^{i,j},K^{i,j})) = \mathcal{R}(t,S,\nu)_{\sum_{l=1}^{i-1} N^l+j,i},\\
&\mu^{i,(T^{i,j},K^{i,j})}=\mu^{i,j},\\
&\overline{\mathcal{L}}^1\big(t,S,\nu,\hat\theta^1_i,(T^{i,j},K^{i,j})\big) = \overline{\mathcal{L}}(t,S,\nu,\theta^1)_{\sum_{l=1}^{i-1} N^l+j},\\ &\overline{\mathcal{L}}^2\big(t,S,\nu,\hat\theta^2_{i_1,i_2},(T^{i_1,j},K^{i_1,j}),(T^{i_2,l},K^{i_2,l})\big) = \overline{\mathcal{L}}(t,S,\nu,\theta^2)_{\sum_{l=1}^{i_1-1} N^l+j,\sum_{l=1}^{i_2-1} N^l+l}.
\end{align*}
In particular, if $d=1$, $\hat\theta^1$ and $\hat\theta^2$ are solutions of non-local PDEs in dimensions 5 and 7 respectively. The observed regularity of the solution with respect to the strike and expiry implies that the high-dimensional PDEs can be solved, for example, by a non-local variant of the Deep Galerkin Method, see~\cite{hirsa2020unsupervised, Sirignano_2018}.
\bibliographystyle{abbrv}
|
1,108,101,565,459 | arxiv | \section{\large\textbf{Introduction}}\label{I}
\textit{Counting invariants}, also called \textit{coloring invariants} or
\textit{coloring-counting invariants}, are a type of integer-valued
invariant of knots or other knotted objects (links, braids, tangles, spatial
graphs, surface-links etc.). They are defined by attaching elements of some
algebraic structure, envisioned as ``colors'', to portions of diagrams
according to rules, typically stated in the form of algebraic axioms, which
ensure that the number of such colorings is unchanged by the relevant
diagrammatic moves. Underlying this simplistic combinatorial picture of
diagrams and colorings lurks a more sophisticated algebraic structure, a
set of morphisms from a categorical object associated to the knotted object
to a (generally finite) coloring object. Perhaps the simplest nontrivial
example is Fox tricoloring, where the simple rule of making all three colors
match or all three differ at each crossing secretly encodes group
homomorphisms from the fundamental group of the knot complement to the group
of integers modulo 3. Examples of coloring structures include groups, kei,
quandles, biquandles and many more.
An \textit{enhancement} of a counting invariant is a stronger invariant from
which the counting invariant can be recovered \cite{EN}. One strategy
which has proven successful for defining enhancements is to seek invariants
$\phi$ of colored knots; then for a given $\phi$, the multiset of $\phi$
values over the set of colorings of our knot is a new invariant of knots
whose cardinality is the original counting invariant but which
carries more information about the original knot. One of the first such
examples was the \textit{quandle cocycle invariant} introduced in \cite{CJKLS},
in which integer-valued invariants of quandle-colored knots known as
\textit{Boltzmann weights} are defined using a cohomology theory for quandles.
The multiset of such Boltzmann weights is then an invariant of the original
uncolored knot; it is stronger than the quandle counting invariant in
question since different multisets of Boltzmann weights can have the same
cardinality.
A \textit{quantum enhancement}, then, is a quantum invariant of $X$-colored
knots for some knot coloring structure $X$. In \cite{NR} these are
conceptualized as $X$-colored tangle functors, i.e. assignments of matrices
of appropriate sizes to the various $X$-colored basic tangles (positive and
negative crossings, maximum, minimum and vertical strand) which make up
tangles via sideways stacking interpreted as tensor product and vertical
stacking as matrix composition. In \cite{NOR} some examples are found via
structures known as \textit{biquandle brackets}, skein invariants modeled
after the Kauffman bracket but with coefficients which depend on biquandle
colorings at crossings. In \cite{NOSY} biquandle brackets are generalized
to include a virtual crossing as a type of smoothing. A special case of
biquandle brackets was described independently in \cite{A}. In \cite{IM} a
type of biquandle bracket whose skein relation includes a vertex
is considered. In \cite{NO} biquandle brackets are defined using \textit{trace
diagrams} in order to allow for recursive expansion as opposed to the
state-sum definition in \cite{NOR}.
This paper is organized as follows. In Section \ref{B} we survey some knot
coloring structures and look in detail at one such structure, biquandles. In
Section \ref{BB} we see the definition and examples of biquandle brackets
as an example of a quantum enhancement. In Section \ref{O} we summarize
a few other examples of quantum enhancements, and we end in Section \ref{Q}
with some questions for future research.
\section{\large\textbf{Biquandles and Other Coloring Structures}}\label{B}
A \textit{knot coloring structure} is a set $X$ whose elements we can think
of as colors or labels to be attached to portions of a knot or link
diagram, together with coloring rules chosen so that the number of valid
colorings of a knot diagram is not changed by Reidemeister moves and hence
defines an invariant. In this section we will look in detail at one such
structure, known as \textit{biquandles}, and then briefly consider some other
examples. For more about these topics, see \cite{EN}.
\begin{definition}
Let $X$ be a set. A \textit{biquandle structure} on $X$ is a pair of binary
operations $\, \underline{\triangleright}\, ,\, \overline{\triangleright}\, :X\times X\to X$ satisfying the following axioms:
\begin{itemize}
\item[(i)] For all $x\in X$, we have $x\, \underline{\triangleright}\, x=x\, \overline{\triangleright}\, x$,
\item[(ii)] The maps $S:X\times X\to X\times X$ and
$\alpha_x,\beta_x:X\to X$ for each $x\in X$ defined by
\[\alpha_x(y)=y\, \overline{\triangleright}\, x,\ \beta_x(y)=x\, \underline{\triangleright}\, y\
\mathrm{and} \ S(x,y)=(y\, \overline{\triangleright}\, x,x\, \underline{\triangleright}\, y)\]
are invertible, and
\item[(iii)] For all $x,y,z\in X$, we have the \textit{exchange laws}:
\[
\begin{array}{rcll}
(x\, \underline{\triangleright}\, y)\, \underline{\triangleright}\, (z\, \underline{\triangleright}\, y) & = & (x\, \underline{\triangleright}\, y)\, \underline{\triangleright}\, (z\, \overline{\triangleright}\, y) & (iii.i) \\
(x\, \underline{\triangleright}\, y)\, \overline{\triangleright}\, (z\, \underline{\triangleright}\, y) & = & (x\, \overline{\triangleright}\, y)\, \underline{\triangleright}\, (z\, \overline{\triangleright}\, y) & (iii.ii) \\
(x\, \overline{\triangleright}\, y)\, \overline{\triangleright}\, (z\, \overline{\triangleright}\, y) & = & (x\, \overline{\triangleright}\, y)\, \overline{\triangleright}\, (z\, \underline{\triangleright}\, y) & (iii.iii).
\end{array}
\]
\end{itemize}
\end{definition}
The biquandle axioms encode the Reidemeister moves using a coloring scheme
with elements of $X$ coloring semiarcs in an oriented link diagram with the
pictured coloring rules at crossings:
\[\includegraphics{bbs-1.pdf}\]
Then using the following generating set of oriented Reidemeister moves,
\[\includegraphics{bbs-2a.pdf}\]
\[\includegraphics{bbs-2b.pdf}\]
\[\includegraphics{bbs-2c.pdf}\]
the following theorem is then easily verified:
\begin{theorem}
Given an oriented link diagram $D$ with a coloring by a biquandle $X$,
for any diagram $D'$ obtained from $D$ by a single Reidemeister move, there
is a unique coloring of $D'$ by $X$ which agrees with the coloring on $D$
outside the neighborhood of the move.
\end{theorem}
Hence we obtain:
\begin{corollary}
The number of colorings of a knot or link diagram $D$ by a biquandle $X$
is an integer-valued invariant of the knot or link $K$ represented by
$D$, called the \textit{biquandle counting invariant} and denoted
by $\Phi_X^{\mathbb{Z}}(K)$.
\end{corollary}
\begin{example} (Alexander biquandles)
A straightforward example of a biquandle structure is to let $X$ be any
commutative ring with identity $R$ with choice of units $s,t$ and define
binary operations
\[\begin{array}{rcl}
x\, \underline{\triangleright}\, y& = & tx+(s-t)y \\
x\, \overline{\triangleright}\, y & = & sx.
\end{array}
\]
For instance, setting $X=\mathbb{Z}_5$ with $t=3$ and $s=2$, we
obtain biquandle operations
\[\begin{array}{rcl}
x\, \underline{\triangleright}\, y& = & 3x+(2-3)y=3x+4y \\
x\, \overline{\triangleright}\, y & = & 2x.
\end{array}
\]
To compute the biquandle counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for an
oriented knot or link $K$ represented by a diagram $D$, we can then
solve the system of the linear equations obtained from the crossings of $D$
using the coloring rule above. For example, the $(4,2)$-torus link has
system of coloring equations below.
\[\raisebox{-0.75in}{\includegraphics{bbs-3.pdf}} \quad
\begin{array}{rcl}
3x_1+4x_8 & = & x_2 \\
2x_8 & = & x_7 \\
3x_8+4x_1 & = & x_5 \\
2x_1 & = & x_4 \\
3x_3+4x_6 & = & x_4 \\
2x_6 & = & x_5 \\
3x_6+4x_3 & = & x_7 \\
2x_3 & = & x_2 \\
\end{array}
\]
Then row-reducing over $\mathbb{Z}_5$ we have
\[
\left[\begin{array}{rrrrrrrr}
3 & 4 & 0 & 0 & 0 & 0 & 0 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 4 & 2 \\
4 & 0 & 0 & 0 & 4 & 0 & 0 & 3 \\
2 & 0 & 0 & 4 & 0 & 0 & 0 & 0 \\
0 & 0 & 3 & 4 & 0 & 4 & 0 & 0 \\
0 & 0 & 0 & 0 & 4 & 2 & 0 & 0 \\
0 & 0 & 4 & 0 & 0 & 3 & 4 & 0 \\
0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 \\
\end{array}\right]
\sim
\left[\begin{array}{rrrrrrrr}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 3 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 4 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}\right]
\]
and the space of colorings is one-dimensional, so
$\Phi_X^{\mathbb{Z}}(K)=|\mathbb{Z}_5|=5$. This distinguishes
$K$ from the unlink of two components, which has
$\Phi_X^{\mathbb{Z}}(U_2)=|\mathbb{Z}_5|^2=25$ colorings by $X$.
\end{example}
A coloring of a diagram $D$ representing an oriented knot or link $K$ by
biquandle $X$ determines a unique homomorphism $f:\mathcal{B}(K)\to X$
from the \textit{fundamental biquandle of $K$}, $\mathcal{B}(K)$, to X.
Hence the set of colorings may be identified with the homset
$\mathrm{Hom}(\mathcal{B}(K),X)$. In particular, an $X$-labeled diagram
$D_f$ can be identified with an element of $\mathrm{Hom}(\mathcal{B}(K),X)$,
and we have
\[\Phi_X^{\mathbb{Z}}(K)=|\mathrm{Hom}(\mathcal{B}(K),X)|.\]
See \cite{EN} for more about the fundamental biquandle of an oriented
knot or link.
The key idea behind enhancements of counting invariants is the observation
that it's not just the number of colorings of a diagram which is invariant, but
the set of colored diagrams itself. More precisely, given a biquandle
$X$ and an oriented knot or link diagram $D$, the set of $X$-colorings
of $D$ is an invariant of $K$ in the sense that changing $D$ by a Reidemeister
move yields a set of colorings of $D'$ in one-to-one correspondence with
the set of colorings of $D$. Then any invariant $\phi$ of $X$-colored
oriented knot or link diagrams can give us a new invariant of the original
knot or link, namely the multiset of $\phi$-values over the set of colorings
of $D$. We call such an invariant an \textit{enhancement} of the counting
invariant.
\begin{example}
Perhaps the simplest enhancement is the \textit{image enhancement}, which
sets $\phi$ for a coloring of a diagram to be the size of the image
sub-biquandle of the coloring. For example, the trefoil knot has nine colorings
by the Alexander biquandle $X=\mathbb{Z}_3$ with $t=2$ and $s=1$. Three of these
colorings are monochromatic, while six are surjective colorings. Then the
counting invariant value $\Phi_X^{\mathbb{Z}}(3_1)=9$ is enhanced to the
multiset $\Phi_X^{\mathrm{Im},M}(3_1)=\{1,1,1,3,3,3,3,3,3\}$. For convenience,
we can convert the multiset to a polynomial by converting the multiplicities
to coefficients and the elements to powers of a formal variable $u$, so the
image enhanced invariant becomes $\Phi_X^{\mathrm{Im},M}(3_1)=3u+6u^3$. This
notation, adapted from \cite{CJKLS}, has the advantage that evaluation of the
polynomial at $u=1$ yields the original counting invariant. See \cite{EN}
for more about enhancements.
\end{example}
\begin{example}\label{ex:q}
The earliest example of an enhancement of the counting invariant is the family
of \textit{quandle 2-cocycle invariants}, introduced in \cite{CJKLS}. In this
type of enhancement, we consider biquandles $X$ with operation $x\, \overline{\triangleright}\, y=x$,
known as \textit{quandles}, and consider maps $\phi:X\times X\to A$ where
$A$ is an abelian group. For each $X$-coloring of an oriented knot or link
diagram $D$, we obtain a contribution $+\phi(x,y)$ at positive crossings and
$-\phi(x,y)$ at negative crossings as depicted:
\[\includegraphics{bbs-4.pdf}\]
The sum of such contributions over the all crossings in an $X$-colored
diagram is called the \textit{Boltzmann weight} of the colored diagram.
The conditions on $\phi$ which make the Boltzmann weight unchanged by
$X$-colored Reidemeister moves can be expressed in terms of a cohomology
theory for quandles: the Boltzmann weight is preserved by Reidemeister III
moves if $\phi$ is a \textit{rack 2-cocycle} and preserved by
Reidemeister I moves if $\phi$ evaluates to zero on \textit{degenerate}
cochains, which form a subcomplex; invariance under Reidemeister II moves
is automatic from the way the Boltzmann weights are defined. The quotient
of rack cohomology by the degenerate subcomplex yields \textit{quandle
cohomology}. In particular, 2-coboundaries always yield a Boltzmann weight
of zero, so cohomologous cocycles define the same enhancement.
See \cite{CJKLS,EN} for more.
\end{example}
Other examples of knot coloring structures include but are not limited to
the following:
\begin{itemize}
\item \textit{Groups}. Any finite group $G$ defines a counting invariant
consisting of the set of group homomorphisms from the \textit{knot group},
i.e., the fundamental group of the knot complement, to $G$.
\item \textit{Quandles}. As mentioned in Example \ref{ex:q}, quandles are
biquandles $X$ whose over-action operation is trivial, i.e. for all
$x,y\in X$ we have $x\, \overline{\triangleright}\, y=x$. Introduced in \cite{J}, the
\textit{fundamental quandle} of a knot determines both the knot group and the
peripheral structure, and hence determines the knot up to ambient
homeomorphism.
\item \textit{Biracks}. Biracks are biquandles for framed knots and links, with
the Reidemeister I move replaced with the framed version. To get invariants of
unframed knots and links using biracks, we observe that the lattice of framings
of a link is an invariant of the original link; then the lattice of, say, birack
colorings of the framings of an unframed link $L$ forms an invariant of $L$.
\end{itemize}
For each of these and other coloring structures, enhancements can
be defined, resulting in new invariants.
\section{\large\textbf{Biquandle Brackets}}\label{BB}
A \textit{biquandle bracket} is a skein invariant for biquandle-colored knots
and links. The definition was introduced in \cite{NOR} (and independently, a
special case was introduced in \cite{A}) and has only
started to be explored in other recent work such as \cite{NO,NOSY,IM}.
\begin{definition}
Let $X$ be a biquandle and $R$ a commutative ring with identity. A
\textit{biquandle bracket} $\beta$ over $X$ and $R$ is a pair of maps
$A,B:X\times X\to R^{\times}$ assigning units $A_{x,y}$ and $B_{x,y}$ to each
pair of elements of $X$ such that the following conditions hold:
\begin{itemize}
\item[(i)] For all $x\in X$, the elements $-A_{x,x}^2B_{x,x}^{-1}$ are all
equal, with their common value denoted by $w$,
\item[(ii)] For all $x,y\in X$, the elements
$-A_{x,y}B^{-1}_{x,y}-A_{x,y}^{-1}B_{x,y}$ are all equal, with their common
value denoted by $\delta$, and
\item[(iii)] For all $x,y,z\in X$ we have
\[\begin{array}{rcl}
A_{x,y}A_{y,z}A_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & = & A_{x,z}A_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}A_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
A_{x,y}B_{y,z}B_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & = & B_{x,z}B_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}A_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
B_{x,y}A_{y,z}B_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & = & B_{x,z}A_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}B_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
A_{x,y}A_{y,z}B_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & = &
A_{x,z}B_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}A_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
& & +A_{x,z}A_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}B_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
& & +\delta A_{x,z}B_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}B_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
& & +B_{x,z}B_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}B_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z} \\
B_{x,y}A_{y,z}A_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & & \\
+A_{x,y}B_{y,z}A_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & & \\
+\delta B_{x,y}B_{y,z}A_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y} & & \\
+B_{x,y}B_{y,z}B_{x\, \underline{\triangleright}\, y,z\, \overline{\triangleright}\, y}
& = & B_{x,z}A_{y\, \overline{\triangleright}\, x,z\, \overline{\triangleright}\, x}A_{x\, \underline{\triangleright}\, z,y\, \underline{\triangleright}\, z}. \\
\end{array}\]
\end{itemize}
\end{definition}
We can specify a biquandle bracket $\beta$ over a ring $R$ and finite biquandle
$X=\{x_1,\dots, x_n\}$ by giving an $n\times 2n$ block matrix with entries in
$R$ whose left block lists the $A_{x,y}$ values and whose right block lists
the $B_{x,y}$ values:
\[\beta=\left[\begin{array}{rrrr|rrrr}
A_{x_1,x_1} & A_{x_1,x_2}& \dots & A_{x_1,x_n} & B_{x_1,x_1} & B_{x_1,x_2}& \dots & B_{x_1,x_n} \\
A_{x_2,x_1} & A_{x_2,x_2}& \dots & A_{x_2,x_n} & B_{x_2,x_1} & B_{x_2,x_2}& \dots & B_{x_2,x_n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
A_{x_n,x_1} & A_{x_n,x_2}& \dots & A_{x_n,x_n} & B_{x_n,x_1} & B_{x_n,x_2}& \dots & B_{x_n,x_n} \\
\end{array}\right].\]
The biquandle bracket axioms are the conditions required for invariance of the
\textit{state-sum} value obtained by summing the products of smoothing
coefficients and powers of $\delta$ and $w$ for each Kauffman state
of an $X$-colored diagram under $X$-colored Reidemeister moves. More
precisely, we write skein relations
\[\includegraphics{bbs-5.pdf}\]
and assign $\delta$ to be the value of a component in a smoothed state,
$w$ the value of a positive kink and $w^{-1}$ the value of a negative kink.
\[\includegraphics{bbs-6.pdf}\]
More formally, we have:
\begin{definition}
Let $\beta$ be a biquandle bracket over a finite biquandle $X$ and commutative
ring with identity $R$ and let $D$ be an oriented knot or link diagram. Then
for each $X$-coloring $D_f$ of $D$, let $\beta(D_f)$ be the state-sum value
obtained by summing over the set of Kauffman states the products of smoothing
coefficients $\phi_{x,y}\in \{A_{x,y}^{\pm 1},B_{x,y}^{\pm 1}\}$ at each
crossing as determined by the colors and smoothing type times $\delta^kw^{n-p}$
where $k$ is the number of components in the state, $n$ is the number of
negative crossings and $p$ is the number of positive crossings.
That is, for each $X$-coloring $D_f$ of $D$, we have
\[\beta(D_f)=\sum_{\mathrm{Kauffman\ States}} \left(\left(\prod_{\mathrm{crossings}} \phi_{x,y}\right)\delta^kw^{n-p}\right).\]
Then the multiset of $\beta(D_f)$-values over the set of $X$-colorings of $D$ is
denoted
\[\Phi_X^{\beta,M}(D)=\{\beta(D_f)\ |\ D_f\in \mathrm{Hom}(\mathcal{B}(K),X)\}.\]
The same data may be expressed in ``polynomial'' form (scare quotes since the
exponents are not necessarily integers but elements of $R$) as
\[\Phi_X^{\beta}(D)=\sum_{D_f\in\mathrm{Hom}(\mathcal{B}(K),X)} u^{\beta(D_f)}.\]
\end{definition}
Hence we have the following theorem (see \cite{NOR}):
\begin{theorem}
Let $X$ be a finite biquandle, $R$ a commutative ring with identity and
$\beta$ a biquandle bracket over $X$and $R$. Then for any oriented knot or
link $K$ represented by a diagram $D$, the multiset $\Phi_X^{\beta,M}(D)$
and the polynomial $\Phi_X^{\beta}(D)$ are unchanged by Reidemeister moves
and hence are invariants of $K$.
\end{theorem}
\begin{example}
A biquandle bracket in which $A_{x,y}=B_{x,y}$ for all $x,y\in X$ defines
a biquandle 2-cocycle $\phi\in H^2_B(X)$. In this case the polynomial invariant
$\Phi_X^{\beta}(D)$ is the product of the biquandle $2$-cocycle enhancement
$\Phi_X^{\phi}(K)$ with the Kauffman bracket polynomial of $K$ evaluated at
$A=-1$. See \cite{NOR} for more details.
\end{example}
\begin{example}
A biquandle bracket $\beta$ over the biquandle of one element $X=\{x_1\}$
is a classical skein invariant. For example, the biquandle bracket
$\beta$ over $R=\mathbb{Z}[A^{\pm 1}]$ with $A_{x_1,x_1}=A$ and $B_{x_1,x_1}=A^{-1}$
(and hence $\delta=-A^2-A^{-2}$ and $w=-A^3$) is the Kauffman bracket polynomial.
\end{example}
Thus, biquandle brackets provide an explicit unification of classical skein
invariants and biquandle cocycle invariants. Even better though, there are
biquandle brackets which are neither classical skein invariants nor cocycle
invariants, but something new.
\begin{example}\label{ex:toy}
Let $R=\mathbb{Z}_7$ and $X=\mathbb{Z}_2=\{1,2\}$ with the biquandle operations
$x\, \underline{\triangleright}\, y=x\, \overline{\triangleright}\, y= x+1$ (note that we are using the symbol $2$ for the class of
zero in $\mathbb{Z}_2$ so that our row and column numberings can start with
1 instead of 0). Then via a computer search, one can check that
\[\beta=\left[\begin{array}{rr|rr}
1 & 5 & 3 & 1 \\
4 & 1 & 5 & 3
\end{array}\right]\]
defines a biquandle bracket, with $\delta=-1(3)^{-1}-1^{-1}3=-5-3=-1=6$ and
$w=-(1)^2(3)^{-1}=-5=2$. The skein relations at positive crossings are as shown:
\[\scalebox{0.85}{\includegraphics{bbs-7.pdf}}\]
Let us illustrate in detail the computation of $\Phi_X^{\beta,M}(L)$ where
$L$ is the oriented Hopf link with two positive crossings. There are four
$X$-colorings of the Hopf link and indeed of every classical link -- the
unenhanced counting invariant with this choice of coloring biquandle $X$
can detect component number of classical links and nothing else. However, the
biquandle bracket enhancement gives us more information.
\[\scalebox{0.9}{\includegraphics{bbs-9.pdf}}\]
For each coloring, we compute the state-sum value:
\[\scalebox{0.9}{\includegraphics{bbs-8.pdf}}\]
yields $[(1)(1)6^2+(1)(3)6+(3)(1)6+(3)(3)6^2]2^{-2} = (1+4+4+2)2=1$;
\[\scalebox{0.9}{\includegraphics{bbs-10.pdf}}\]
yields $[(5)(4)6^2+(1)(4)6+(5)(5)6+(1)(5)6^2]2^{-2} = (6+3+3+5)2=6$;
\[\scalebox{0.9}{\includegraphics{bbs-11.pdf}}\]
yields $[(4)(5)6^2+(5)(5)6+(4)(1)6+(5)(1)6^2]2^{-2} = (6+3+3+5)2=6$, and
\[\scalebox{0.9}{\includegraphics{bbs-12.pdf}}\]
yields $[(1)(1)6^2+(3)(1)6+(1)(3)6+(3)(3)6^2]2^{-2} = (1+4+4+2)2=1$. Then
the multiset form of the invariant is $\Phi_X^{\beta,M}(L)=\{1,1,6,6\}$,
or in polynomial form we have $\Phi_X^{\beta}(L)=2u+2u^6$. Since the unlink
of two components has invariant value $\Phi_X^{\beta,M}(U_2)=\{6,6,6,6\}$,
the enhanced invariant is stronger than the unenhanced counting invariant.
\end{example}
Example \ref{ex:toy} is merely a small toy example meant to illustrate the
computation of the invariant, of course. Biquandle brackets over larger
biquandles and larger (finite or infinite) rings have already proved their
utility as powerful knot and link invariants, with cocycle invariants at
one extreme (information concentrated in the coloring) and skein invariants
at the other (information concentrated in the skein relations). So far,
the known examples of biquandle brackets which are neither classical
skein invariants nor cocycle invariants have been largely found by
computer search; it is our hope that other examples can be found by
more subtle methods.
\section{\large\textbf{Other Quantum Enhancements}}\label{O}
Biquandle brackets are one example of a more general phenomenon known as
\textit{quantum enhancements}, broadly defined as quantum invariants
of $X$-colored knots or other knotted structures for an appropriate
coloring structure $X$. In this section we collect a few other recent
examples of quantum enhancements.
In \cite{NOSY}, the author together with coauthors
K. Oshiro, A. Shimizu and Y. Yaguchi defined
\textit{biquandle virtual brackets}, a generalization of biquandle
brackets which includes a virtual crossing as a type of smoothing, i.e.,
\[\includegraphics{bbs-13.pdf}\]
A biquandle bracket is then a biquandle virtual bracket in which the virtual
coefficients are all zero. This framework gives another way of recovering
the biquandle cocycle invariants, this time without the factor of the
Kauffman bracket evaluated at $-1$, by having classical smoothing
coefficients all zero. Examples of these invariants are shown to be able to
detect mirror image and orientation reversal. In particular, these are
examples of invariants of classical knots and links which are defined in
a way that fundamentally requires virtual knot theory; it is our hope that
these invariants can provide a reason for classical knot theorists to care
about virtual knot theory.
In \cite{NO}, the author together with coauthor N. Oyamaguchi addressed the
issue of how to compute biquandle brackets in a recursive term-by-term
expansion as opposed to the state-sum approach described in Section \ref{BB}.
Our method uses \textit{trace diagrams}, trivalent spatial graphs with
decorations carrying information about smoothings which enable maintaining
a biquandle coloring throughout the skein expansion.
\[\includegraphics{bbs-14.pdf}\]
These are equivalent to the original state-sum biquandle brackets but
can allow for faster hand computation as well as for allowing moves
and diagram reduction during the course of the expansion. Technical conditions
are identified for which trace moves (e.g., moving a strand over, under or
through a trace) are permitted depending on certain
algebraic conditions being satisfied by the bracket coefficients.
In \cite{IM}, another skein relation is used in the biquandle bracket setting,
superficially similar to the biquandle virtual brackets described above
but with the virtual smoothing replaced with a 4-valent vertex.
\[\includegraphics{bbs-15.pdf}\]
This family of quantum enhancements includes Manturov's parity bracket
invariant as special case, as well as the biquandle brackets defined in
Section \ref{BB}.
In \cite{NR}, the author together with coauthor V. Rivera (a high school
student at the time) introduced the notion of quantum enhancements in the form
of $X$-colored TQFTs or $X$-colored tangle functors for the case of involutory
biracks $X$. These are given by matrices
$X_{x,y}^{\pm 1},I,U$ and $N$ over a commutative ring with identity $R$
corresponding to the basic $X$-colored tangles
\[\includegraphics{bbs-16.pdf}\]
such that the tensor equations representing the
$X$-colored Reidemeister moves and planar isotopy moves
are satisfied, where we interpret vertical stacking as matrix product and
horizontal stacking as tensor (Kronecker) product. Computer searches
for solutions to these equations
proved inefficient even for small rings, so we considered $X$-colored
braid group representations as a first step. Indeed, biquandle brackets
have so far been the best method for producing examples of this
type of quantum enhancement. For example, the biquandle bracket in
example \ref{ex:toy} defines the following quantum enhancement:
\[X_{11}=X_{22}=\left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 0 & 3 & 0 \\
0 & 3 & 6 & 0 \\
0 & 0 & 0 & 1
\end{array}\right],\quad
X_{12}=\left[\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 2 & 0 \\
0 & 0 & 0 & 5
\end{array}\right],\quad
X_{21}=\left[\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 0 & 5 & 0 \\
0 & 5 & 3 & 0 \\
0 & 0 & 0 & 4
\end{array}\right],
\]
\[I=\left[\begin{array}{rr}
1 & 0 \\
0 & 1
\end{array}\right],\quad
N=\left[\begin{array}{rrrr}
0 & 1 & 4 & 0 \\
\end{array}\right],\quad
U=\left[\begin{array}{r}
0 \\
5 \\
1 \\
0 \\
\end{array}\right].
\]
To compute a quantum enhancement in this format, our $X$-colored oriented
diagrams $D_f$ are decomposed into matrix products of tensor products of
the five basic tangles which are then replaced with the appropriate matrices
and multiplied to obtain $1\times 1$ matrices, i.e., ring elements. These
are then multiplied by the appropriate writhe correction
factor $w^{n-p}$ and collected into a multiset. For example, the Hopf link
with pictured coloring decomposes as
\[\raisebox{-0.65in}{\includegraphics{bbs-17.pdf}} \quad (U\otimes U)(I\otimes X_{yx}\otimes I)(I\otimes X_{xy}\otimes I)(N\otimes N)\]
The enhanced invariant for the Hopf link in Example \ref{ex:toy}
is then the multiset
\[\left\{
\begin{array}{l}
(U\otimes U)(I\otimes X_{11}\otimes I)^2(N\otimes N)w^{-2}, \\
\ (U\otimes U)(I\otimes X_{22}\otimes I)^2(N\otimes N)w^{-2}, \\
\ (U\otimes U)(I\otimes X_{12}\otimes I)(I\otimes X_{21}\otimes I)(N\otimes N)w^{-2}, \\
\
(U\otimes U)(I\otimes X_{21}\otimes I)(I\otimes X_{12}\otimes I)(N\otimes N)w^{-2}
\end{array}\right\}
=\{3(2),3(2),4(2),4(2)\}
=\{1,1,6,6\}\]
as in Example \ref{ex:toy}.
\section{\large\textbf{Questions}}\label{Q}
We end this short survey with some questions and directions for future
research.
\begin{itemize}
\item As mentioned in \cite{NOR}, biquandle 2-cocycles define biquandle
brackets, and the operation of componentwise multiplication of the biquandle
bracket matrix of a 2-cocycle with a bracket representing a 2-coboundary
yields a biquandle bracket representing a cohomologous 2-cocycle. Weirdly,
this also works with biquandle brackets which do not represent 2-cocycles,
extending the equivalence relation of cohomology to the larger set of biquandle
brackets. What exactly is going on here?
\item
So far, biquandle brackets over finite biquandles have been found primarily by
computer search using finite coefficient rings. We would like to find
examples of biquandle brackets over larger finite biquandles and over larger
rings, especially infinite polynomial rings.
\item
The first approach for generalization, examples of which have been considered
in \cite{NOSY} and \cite{IM}, is to apply the biquandle bracket idea to
different skein relations. One may find that skein relations which do not
yield anything new in the uncolored case can provide new invariants in various
biquandle-colored cases.
\item
In addition to biquandle brackets, we would like to find other examples of
quantum enhancements. Possible avenues of approach include representations
of biquandle-colored braid groups, biquandle-colored Hecke algebras,
biquandle-colored TQFTs and many more.
\item
Like the Jones, Homflypt and Alexander polynomials, every biquandle bracket
should be susceptible to Khovanov-style categorification, providing another
infinite source of new knot and link invariants.
\item
Since biquandle brackets contain both classical skein invariants and
cocycle enhancements as special cases, we can ask which other known
(families of) knot and link invariants are also describable in this way or
recoverable from biquandle bracket invariants. For example, can every
Vassiliev invariant be obtained as a coefficient in some biquandle bracket
invariant over a polynomial ring, like the coefficients of the Jones polynomial?
\item
Finally, we can define quantum enhancements for coloring structures other than
biquandles and for knotted objects other than classical knots. The
possibilities are limitless!
\end{itemize}
|
1,108,101,565,460 | arxiv | \section{Introduction}
\label{section:intro}
The study of large underdense voids in the large-scale matter distribution of the Universe has become increasingly important in recent years, with the creation of a number of public catalogues of voids in galaxy survey data \citep{Pan:2012,Sutter:2012wh,Nadathur:2014a} and a wide variety of statistical analyses based on them.
Voids are interesting primarily because of the cosmological information they may contain. Various studies have suggested that they could be used to constrain the expansion history of the Universe and the equation of state of dark energy \citep[e.g.][]{Ryden:1995,Lee:2009,Bos:2012,Hamaus:2014b}, to test modified theories of gravity (\citealt*{Li:2012,Clampitt:2013,Cai:2015}; \citealt{Zivick:2014}), to calibrate measurements of galaxy bias (\citealt{Hamaus:2013}; \citealt*{Chan:2014}), to constrain initial conditions of structure formation \citep*{Kamionkowski:2009}, or to probe more exotic theories such as coupled dark energy \citep{Sutter:2014d}. The primary void observables used in such studies are their abundances and size distributions, the distortion of their shapes in redshift space \citep{Alcock:1979}, their dark matter density profiles, and the void-galaxy or void-void position correlations. Given the exciting potential applications, a rigorous comparison of theoretical predictions of these properties and those seen for voids in $N$-body simulations and galaxy surveys is very important.
However, this aim is complicated by the degree of ambiguity surrounding a very fundamental question: what exactly \emph{is} a `void'? From a theoretical perspective, there is a clear answer, provided by the spherical evolution model of \citet{Sheth:2003py}, who identify voids as those non-linear underdense regions which have evolved to reach \emph{shell crossing}. This identification is convenient, as voids can then be modelled analogously to collapsed overdense haloes using the excursion set formalism \citep{Press:1974,Bond:1991,Lacey:1993}, and therefore allows clear predictions to be made for void observables.
In practical terms, however, the definition of a void is not so clear. When dealing with either $N$-body simulations or galaxy survey data, an algorithmic approach is required to identify regions as voids, which is complicated by the fact that voids are naturally poorly sampled by observable tracers, making shot noise a serious issue. A number of different void finders have been proposed \citep[see][for a review of methods]{Colberg:2008}, which unfortunately do not always agree with each other. Watershed void finders (e.g., \citealt*{Platen:2007qk,Neyrinck:2008,Sousbie:2011a,Cautun:2013}) form an interesting class of algorithms. They use tessellation techniques \citep{Schaap:2000} to reconstruct the density field from discrete data points, and the watershed algorithm for creating a void hierarchy. They present a number of advantages for practical studies, as they are more resilient to shot noise in the density reconstruction \citep[though see also e.g.][for other interesting proposals]{Elyiv:2015,Shandarin:2014}, and do not make prior assumptions about void geometries. They are also the most commonly used. Watershed voids may therefore be considered a reasonable practical definition of a void.
However, watershed algorithms make no reference to shell crossing, which is the defining characteric of theoretical models. The obvious question is therefore how, or whether, these two void definitions are related to each other. The answer is important to the practical use of watershed voids in cosmology, as well as to the development of further theoretical predictions. A number of studies (e.g., \citealt{Sutter:2014b,Chan:2014,Pisani:2014,Chongchitnan:2015}; \citealt*{Achitouv:2015}) which apply the \citet{Sheth:2003py} formalism to describe watershed voids \emph{assume} that the two approaches describe the same or closely related objects. There is known to be a significant disagreement between model predictions for void abundances as a function of their size and results obtained for watershed voids in dark matter simulations. This can be partially resolved (at least at large void sizes) by the ad hoc assumption that shell crossing and void formation occur at less extreme densities than predicted by the spherical model, but as we will also show, such an approach lacks self-consistency. A more direct comparison of void properties with the model is therefore desirable.
At the same time, a number of properties of watershed voids remain imperfectly understood. A generic property of watershed void finders is that voids containing the deepest density minima should have the largest sizes. Yet the fits provided by \citet*{Hamaus:2014a} to describe density profiles about void centres appears to suggest the opposite behaviour \citep[though note that the applicability of this fitting form is not universally accepted, e.g.][]{Nadathur:2015a}. Perhaps related to this problem is the question of how to define the `centre' of a void --- which is also important for void correlation studies. The standard procedure assigns the centre to a weighted average of the positions of the tracers of mass within a void \citep[e.g.][]{Lavaux:2012,Sutter:2012wh,Nadathur:2014a,Sutter:2014b}, but it is not clear that this will correctly identify the region with the greatest \emph{absence} of mass. Another interesting question is how densities reconstructed from discrete tracer distributions relate to the true underlying density field. In studies of voids from simulations, the full simulation output is typically randomly down-sampled to provide a set of tracers, which can have a dramatic effect on the recovered void properties \citep{Sutter:2014b}.
Our goals in this paper are two-fold. We wish to understand the relationship between watershed voids and theoretical models. To do this, we move beyond the fitting of void number functions alone and identify other important characteristics of voids which can be used to test the assumption of shell crossing more broadly. We also want to empirically examine the properties of watershed voids in simulations in order to understand the working of the algorithm and clarify some of the issues above.
To do so we make use of the popular {\small ZOBOV} watershed algorithm \citep{Neyrinck:2008}. To enable comparison of our results with others in the literature, we will mostly use the options for {\small ZOBOV} implemented in the {\small VIDE} toolkit \citep{VIDE:2015}. We analyse voids identified using randomly subsampled dark matter particles as tracers, and relate them to true densities determined from the full resolution simulation output. We propose a new definition of the void centre, which is designed to better identify the true location of the underdensity within the void. The details of the $N$-body simulation, the watershed algorithm and the methods for identifying void centres and measuring density profiles are described in Section~\ref{section:numerical}.
Section~\ref{section:theory} provides a summary of the spherical model for void evolution, which we use to extract general identifying characteristics of shell-crossed voids for comparison with simulation results. This comparison is performed in Section~\ref{section:properties}, where we also outline the general properties of voids in our simulation. We show that larger voids do correspond to deeper density minima, as expected for the watershed algorithm. We examine the viability of the fitting formula of \citet{Hamaus:2014a} to describe the density profiles of voids, and show that the fits described in that paper do not provide a good quantitative or qualitative description of the variation of the average profile within the void population. In addition, subsampled tracers almost always overestimate the density contrast in voids. All of these results have practical implications for future studies that use watershed void finders. We compare these results from simulated voids to theory and argue that there is no evidence that watershed void finders in general, and {\small VIDE} and {\small ZOBOV} in particular, satisfy the primary defining criteria of the \cite{Sheth:2003py} model. This leads us to reassess the viability of describing watershed voids using existing theoretical techniques. We summarize and conclude in Section~\ref{section:conclusions}.
\section{Numerical methods}
\label{section:numerical}
\subsection{Simulation}
\label{subsec:MultiDark}
In this paper we use $N$-body simulation data from the MultiDark Run1 (MDR1) release of the MultiDark project \citep{Prada:2012}.\footnote{Publicly available from \url{www.cosmosim.org}.} MDR1 uses an Adaptive-Refinement-Tree (ART) code, based on adaptive mesh refinement, to simulate $2048^3$ dark matter particles within a cubic volume of $1\,(h^{-1}\mathrm{Gpc})^3$, in a $\Lambda$CDM cosmological model with parameters $(\Omega_m,\Omega_\Lambda,\Omega_b,h,n_s,\sigma_8)=(0.27,0.73,0.0469,0.7,0.95,0.82)$. The simulation has mass resolution $m_p = 8.721\times10^9\;h^{-1}M_\odot$ and force resolution $7\;h^{-1}$kpc. Initial conditions were set using the Zeldovich approximation at redshift $z=65$.
From the full particle output at redshift 0 we randomly subsample the dark matter particles down to a number density of $\overline{n}=3.2\times10^{-3}\;h^3$Mpc$^{-3}$, similar to that of typical galaxy samples \citep[e.g.][]{Zehavi:2011}. This corresponds to a mean nearest-neighbour separation of $\overline{n}^{-1/3}\sim7\;h^{-1}$Mpc. We refer to the resulting sample as the \emph{Main} sample, and use these particles as tracers for the void finding. In addition, we have used a control sample with a higher tracer density $2\times10^{-2}\;h^3$Mpc$^{-3}$, which we refer to as the \emph{Dense} sample. However, our primary conclusions regarding the properties of watershed voids do not depend strongly on the tracer number density. Therefore unless otherwise stated, all results presented in this paper refer to voids from the \emph{Main} sample.
Note that a random subsampling of tracers introduces shot noise but does not change the fundamental clustering properties of the dark matter field. Therefore despite having the same tracer number density, the properties of voids in such subsampled tracers and those in the galaxy distribution would not be expected to be (and are not) the same, since galaxies are biased tracers of the matter density. However for our purposes of understanding the general properties of watershed voids in this work, a random subsampling is sufficient. We consider the effects of galaxy bias separately in a companion paper \citep{Nadathur:2015c}.
Although our tracers are themselves dark matter particles, as the subsampling procedure increases shot noise we will distinguish between the tracer \emph{number} density, and the underlying dark matter density. The dark matter density in MDR1 is determined from the full resolution particle output of the simulation at redshift 0, by using a cloud-in-cell interpolation on a $1024^3$ grid, followed by smoothing with a Gaussian kernel with width equal to one grid cell. The sub-Mpc resolution of this grid is much smaller than the typical void size scales, so that this procedure in effect provides a continuous underlying dark matter density, of which the subsampled tracer particles are an approximately Poisson realization.
In the following, we will reserve the symbols $\rho$ and $\Delta$ for dark matter densities determined using this gridded smoothed density field, and use the symbols $n$ and $\Delta_n$ for the equivalent quantities determined from the tracer number densities.
\subsection{Void finding}
\label{subsec:voidfinding}
To identify voids in the dark matter particle distribution, we make use of the {\small ZOBOV} watershed void finder \citep{Neyrinck:2008}, with the options implemented in the {\small VIDE} toolkit \citep{VIDE:2015}. Although there are known issues with the application of {\small VIDE} to galaxy survey data with irregular survey volumes and masks \citep{Nadathur:2014a}, when dealing with a simulation cube with periodic boundary conditions these do not present a problem. However, note that in some cases, especially the definition of the void centre described below, we use our own modification of the {\small ZOBOV} algorithm.
{\small ZOBOV} works by reconstructing the density field based on a Voronoi tessellation of the simulation cube around the discrete distribution of tracer particles. The tessellation associates each particle with a Voronoi cell consisting of the region of space closer to it than to any other particle. The volume of the Voronoi cell $i$ relative to the mean volume is then used to estimate the local tracer number density $n_i$ at the particle location. This reconstruction, known as the Voronoi tessellation field estimator (VTFE), is naturally scale-adaptive and thus far more resilient against shot noise effects than naive counts-in-cells measurements.
After reconstructing the density field, the algorithm identifies all local minima of the reconstructed density field and determines the ``catchment basins" around each minimum, known as zones. Zones are then merged to form a nested hierarchy of voids according to the watershed principle \citep{Platen:2007qk}, such that the zone with the smallest minimum density $n_\mathrm{min}$ then acquires neighbouring higher-density zones as sub-voids, in increasing order of the minimum density on the watershed ridge separating them. For each void thus formed, we define an effective void radius $R_v$ as the radius of a sphere with equivalent volume $V$,
\begin{equation}
R_v = \left(\frac{3}{4\pi}V\right)^{1/3}\;.
\end{equation}
Even in the absence of any merging, deeper density minima typically correspond to zones of larger volume and thus larger $R_v$. However, the watershed merging procedure also ensures that voids with deepest density minima contain greater numbers of merged sub-voids and therefore have the largest sizes. This correlation of minimum density and void size is a common property of all watershed void-finders and is not unique to the {\small ZOBOV} algorithm.
To avoid excessive merging leading to essentially infinite void sizes, {\small VIDE} imposes a restriction preventing the merger of two zones unless the minimum link density along the watershed ridge separating them satisfies $n_\mathrm{link}<n_\mathrm{max}$, where $n_\mathrm{max}=0.2\overline{n}$. This condition applies only to the lowest density point on such a ridge, and does not prevent voids from containing regions of much higher densities. The value of 0.2 therefore has no theoretical motivation, and $n_\mathrm{max}$ should be considered an arbitrary free parameter. Indeed alternative values of $n_\mathrm{max}$ have been considered in other works \citep{Nadathur:2014a,Hotchkiss:2015a,Achitouv:2015,Nadathur:2015a}, and properties such as the abundance of root-level voids, the distribution of void sizes and void density profiles will all depend on the value chosen. A fuller discussion of the effects of this arbitrary choice is provided by \citet{Nadathur:2015c}. However, for ease of comparison with previous results we shall restrict ourselves in this paper to the default value hard-coded in {\small VIDE}.
Further selection cuts might be desirable at this stage, since {\small ZOBOV} simply reports all local density minima as potential voids, without regard to the value of the minimum density within them or any reference to shell crossing. {\small VIDE} provides an optional selection cut which purports to remove those voids which have a tracer number density within a defined central region greater than $0.2$ times the mean. However, this density is measured by naive number counting on a scale smaller than the mean inter-particle separation. Therefore, as pointed out by \citet{NH:2013b}, it is very badly affected by shot noise and the values determined by {\small VIDE} are almost completely uncorrelated with the true central density. In any case this cut only excludes a small fraction of final voids, so we do not apply it. We also do not apply the much more conservative cuts on $n_\mathrm{min}$ suggested by \citet{Nadathur:2014a}, \citet{Hotchkiss:2015a} and \citet{Nadathur:2015a}.
A selection cut based on the void radius has sometimes been advocated in the literature, to remove voids with $R_v<\overline{n}^{-1/3}$, which are claimed to be below the resolution limit. In fact the adaptive nature of the tessellation means that {\small ZOBOV} automatically excludes small voids below its resolution limit, as we show in Section \ref{section:properties}. Therefore no further cut on $R_v$ is necessary.
By applying these criteria we find in total 27 450 voids in the \emph{Main} tracer sample. Of these, 26 919 are root-level voids in the hierarchy, i.e. they are not subvoids of any parent voids and their volumes do not overlap each other.
\subsection{Void centres}
\label{subsec:centres}
Since voids obtained from the watershed algorithm have arbitrary shapes, different prescriptions may be used to define the location of the void `centre'. The most commonly used definition, which is also the definition implemented in {\small VIDE}, is the volume-weighted barycentre of the void member particles
\begin{equation}
\label{eq:barycentre}
\bm{X}^\rmn{bc}_v = \frac{1}{\sum_i V_i}\sum_i V_i\bm{x}_i\;,
\end{equation}
where $\bm{x}_i$ is the position of the $i$th particle, $V_i$ is the volume of it's corresponding Voronoi cell, and the sum runs over all member particles of void $v$.
In the low-density interior of a void, Voronoi cells in the tessellation are typically greatly elongated, and the particles contained within them lie far from their geometrical centres. This means that the position $\bm{x}_i$ corresponding to each cell is an imprecise measure of the location of the cell. In addition, watershed voids contain a great number of member particles -- the median number for our void sample is $82$, and many voids contain several hundreds -- the vast majority of which reside in the overdense walls and filaments on the outskirts of voids. A combination of these two factors means that although the barycentre position defined by equation~\ref{eq:barycentre} is roughly symmetrically located with respect to the overdense void walls, it is typically very far from the position of minimum density. This is because the barycentre definition is fundamentally based on the locations where tracers are present, rather than locations from where they are \emph{absent}. A consequence of this is that the location of the barycentre is very sensitive to the sub-void fraction and thus to the arbitrary condition controlling void merging described above.
For some purposes, it may be more logical to define the void centre to coincide with the location of the minimum density within it. This is particularly important when void centre locations are subsequently used to measure density-dependent effects, such as void lensing or ISW contributions. To achieve this, we adopt the following procedure. We identify the core particle of the void as the particle with the largest Voronoi cell (i.e., corresponding to the minimum tracer density $n_\mathrm{min}$), and examine the tessellation output to identify all Voronoi cells adjacent to it. From this set we select the lowest density neighbouring particle, and then, in order of increasing density, two other particles that are adjacent to both the core particle and the previous selections. This provides us with the four lowest density mutually adjacent Voronoi cells in the void; we now define the void centre to lie at the point of intersection of these four cells, which is also the circumcentre of the tetrahedron formed by the four tracer particles. This point represents the location within the void that is maximally distant from all tracers.
We shall refer to this alternative definition of the void centre as the \emph{circumcentre} and denote its location by $\bm{X}^\rmn{cc}_v$. In Section~\ref{section:properties} we show that both the number density of tracers and the underlying dark matter density are indeed significantly lower at the circumcentre than the barycentre. In Appendix \ref{appendixB} we also show that the circumcentre location is more resilient to shot noise effects arising due to subsampling.
\subsection{Density profile determination}
\label{subsec:profile_methods}
A fundamental quantity of interest is the average distribution of tracers and dark matter about the void centre, and the variation of this distribution with void properties. We study this behaviour by constructing stacked density profiles for subsets of voids satisfying different criteria. To do so we rescale distances within each qualifying void in units of the void radius $R_v$, and then estimate the average density in the stack in concentric spherical shells about the void centre.
Estimating the tracer number density in this way is complicated by shot noise effects, since the interiors of voids by definition contain very few tracer particles which can be used for number density measurements. \citet{Nadathur:2015a} showed that an unbiased estimate accounting for Poisson noise can be obtained using the volume-weighted estimator for the average number density in the $j$th radial shell,
\begin{equation}
\label{eq:Poissonestimator}
\overline{n}^j=\frac{\sum_{v=1}^{N_v}N_i^j+1}{\sum_{i=1}^{N_v}V_i^j}\,,
\end{equation}
where the $j$th shell has width $\Delta\tilde{r}$ in units of the rescaled radial distance $\tilde{r}$ for each void, $V_i^j$ is the true volume of the $j$th shell of the $i$th void and $N_i^j$ is the number of tracer particles contained within it, and the sum over $i$ runs over all voids included in the stack. Note that the $N_i^j$ in this formula includes all tracer particles within the shell, not just those that are identified as members of the void by the watershed algorithm. Under the assumption that the individual numbers $N_i^j$ are Poisson realizations of the true underlying density, the error in equation~\ref{eq:Poissonestimator} can then be estimated at any desired confidence level directly from the definition of the Poisson distribution. In this paper plotted errorbars indicate the 68\% confidence limits on $n$.
As the resolution of the dark matter density field is much finer than the typical void size, no such complications are required when estimating $\rho$ over the stack of voids. We simply sample the dark matter density at all grid points contained within the radial shell and calculate the mean and standard deviation of the values obtained. We present our results for the stacked dark matter densities in terms of the average total enclosed density within a radius $r$, $1+\overline{\Delta(r)}$, since this allows a more direct contact with the theory described in Section~\ref{section:theory}.
All density profiles are calculated out to three times the void radius, and are measured in radial bin steps of $0.1$ times the radius.
\section{Excursion set models of voids}
\label{section:theory}
Most existing theoretical descriptions of voids are derived from the framework presented by \citet{Sheth:2003py}. This in turn derives from the original excursion set approach of \citet{Press:1974,Epstein:1983,Bond:1991} and is based on the model of spherical evolution of mass shells \citep{Gunn:1972,Lilje:1991}. In this Section we briefly summarize such models in order to highlight the key areas of comparison with the results of watershed void finders.
In this picture the evolution of a spherical mass shell of radius $r$ is determined by the total enclosed density contrast within the radius of the shell at time $t$, $1+\Delta(r,t)$, where
\begin{equation}
\label{eq:Delta}
\Delta(r,t) = \frac{3}{r^3}\int_0^r\left[\frac{\rho(y,t)}{\overline\rho(t)}-1\right]y^2dy,
\end{equation}
and by the time evolution of the cosmological density parameter $\Omega(t)$. Underdense spherical regions contain a density deficit (i.e., $\Delta(r,t)<0$) which causes shells to expand outwards. This deficit is stronger for inner shells, which therefore expand faster than outer shells, and mass evacuated from the centre of the underdensity begins to pile up at its edges. For a steep enough starting density profile, at some point in the evolution inner shells catch up with shells which were initially further out from them, in an event known as \emph{shell crossing}. The moment of shell crossing marks a transition in the evolution of the underdensity, as it subsequently expands outwards self-similarly \citep*{Suto:1984,Fillmore:1984,Bertschinger:1985}.
Within the spherical model, it can be shown that shell crossing occurs when the average density enclosed within the void is
\begin{equation}
\label{eq:shellcrossing}
\rho_\rmn{enc}/\overline{\rho}=1+\Delta(r,t) \simeq0.2.
\end{equation}
This corresponds to a linearly extrapolated average density contrast of
\begin{equation}
\label{eq:linDelta}
\Delta_\rmn{lin}=\delta_\rmn{v}\simeq-2.71\;,
\end{equation}
at the epoch of shell crossing, with this value independent of radius $r$ and largely independent of the cosmological parameters governing the background evolution. This is analogous to the case of spherical collapse of clusters, which occurs above a linear overdensity threshold of $\Delta_\rmn{lin}=\delta_\rmn{c}\simeq1.69$.
Following \citet{Blumenthal:1992,Dubinski:1992}, \citet{Sheth:2003py} then identify the population of \emph{voids} with only those mature evolved underdensities that have reached the stage of shell crossing. If the initial Gaussian density fluctuation field is smoothed on a range of different smoothing scales $R$, this physical picture identifies fluctuations which exceed the density threshold $\delta_\rmn{v}$ on smoothing scale $R$ with potential voids of radius $R$ today. These fluctuations can be characterized by their depth in units of the rms fluctuation of the density field on scale $R$,\footnote{Note that the definition $\nu=\delta_\rmn{v}/\sigma_0$ is also commonly used. In this case equation~\ref{eq:SvdWdist} would need to be appropriately modified, as done by \citet{Chan:2014}.}
\begin{equation}
\label{eq:nu}
\nu\equiv\delta_\rmn{v}^2/\sigma_0^2(R)\;,
\end{equation}
where $\sigma_0(R)$ is one of the set of spectral moments
\begin{equation}
\label{eq:sigma}
\sigma_j^2(R)\equiv\int\frac{k^{2+2j}}{2\pi^2}W^2(kR)P(k)\rmn{d}k\;,
\end{equation}
with $P(k)$ the power spectrum of the unsmoothed density fluctuation field and $W(kR)$ the smoothing filter.
At this point, the abundance and size distribution of voids can be predicted by a number of models of varying degrees of sophistication. \citet{Sheth:2003py} use the excursion set approach \citep[e.g.,][]{Bond:1991,Sheth:1998} to account for fluctuations which cross the $\delta_\rmn{v}$ threshold on some small scale but are overdense with $\Delta_\rmn{lin}>\delta_\rmn{c}$ on some larger scale. Such underdensities would be crushed by the collapse of the surrounding cluster and so would not be visible as voids today (the \emph{void-in-cloud} effect). This amounts to a two-barrier problem. According to this model, assuming void number density is conserved on evolving from Lagrangian to Eulerian space, this number density can be expressed as a function of the Eulerian void radius $R_v$ (e.g., \citealt*{Jennings:2013}; \citealt{Chan:2014}) as
\begin{equation}
\label{eq:numdens}
\frac{\rmn{d}N}{\rmn{d}R_v} = \left(\frac{3}{4\pi R_\rmn{L}^3}\right) f(\nu)\frac{\rmn{d}\nu}{\rmn{d}R_\rmn{L}}\,,
\end{equation}
where
\begin{equation}
\label{eq:SvdWdist}
f(\nu)\simeq \sqrt{\frac{1}{2\pi\nu}}\exp\left(-\frac{\nu}{2}\right)\exp\left(-\frac{|\delta_\rmn{v}|}{\delta_\rmn{c}}\frac{\mathcal{D}^2}{4\nu}-2\frac{\mathcal{D}^4}{\nu^2}\right)\;,
\end{equation}
and
\begin{equation}
\label{eq:SvdWD}
\mathcal{D}\equiv \frac{|\delta_\rmn{v}|}{(\delta_\rmn{c}+|\delta_\rmn{v}|)}\;.
\end{equation}
Here the Lagrangian radius $R_\rmn{L}=0.58R_v$, a relationship determined by the shell crossing condition above.
However, it is well known that equation~\ref{eq:numdens} does not provide a good fit to the distribution of voids found by watershed algorithms, since it predicts a sharp cutoff in void sizes above $\sim5\;h^{-1}$Mpc, much smaller than observed for watershed voids. A number of studies \citep{Jennings:2013,Sutter:2014b,Chan:2014,Pisani:2015}) have attempted to improve fits by relaxing the shell crossing condition $\delta_\rmn{v}=-2.71$ and treating $\delta_\rmn{v}$ as a free parameter instead.\footnote{\citet{Jennings:2013} also propose an alternative adaptation of this model, but this cuts off the distribution at even smaller $R_v$, so would make the discrepancy worse.} This procedure is not justified by any specific theoretical model. However, if $\delta_\rmn{v}$ is allowed to vary, self-consistency requires that the relationship between Eulerian and Lagrangian radius be correspondingly modified to
\begin{equation}
\label{eq:LagrangianR}
R_\rmn{L} = \frac{R_v}{\left(1-\delta_\rmn{v}/c\right)^{c/3}}\,,
\end{equation}
where $c\simeq1.594$ \citep{Bernardeau:1994,Jennings:2013}. Despite this modification, equation~\ref{eq:numdens} with a variable $\delta_\rmn{v}$ still fails to describe the distribution of small voids, and the fit values of $\delta_\rmn{v}$ for large voids vary widely. \citet{Chan:2014} obtain $\delta_\rmn{v}\simeq-1$ and find little redshift dependence of this value, contrary to theoretical expectation --- however, they keep $R_\rmn{L}=0.58R_v$ fixed when varying $\delta_\rmn{v}$ instead of using equation~\ref{eq:LagrangianR}, so their model is not self-consistent. On the other hand, \cite{Sutter:2014b} find a range of different $\delta_\rmn{v}$ values for voids from different samples, ranging from $-0.26$ to $-0.5$. \cite{Pisani:2015} quote $\delta_\rmn{v}=-0.45$. It is hard to conceive of an explanation for why $\delta_\rmn{v}$ should lie so far from the theoretical prediction if the shell crossing model were true.
\begin{figure}
\includegraphics[width=85mm]{fitted_num_densities.pdf}
\caption{The differential number density of voids in simulation as a function of their size, for both simulation samples. Error bars are calculated assuming the void numbers in each bin are Poisson distributed. The dashed line shows the best fit of the \citet{Sheth:2003py} model to the $R_v>25\;h^{-1}$Mpc data, with $\delta_\rmn{v}=-0.40$. The solid line shows an exponential cutoff model which describes the same data better.}
\label{fig:numdens}
\end{figure}
In any case, insofar as allowing $\delta_\rmn{v}$ to vary allows a fit to the distribution of the largest voids, it does so simply by replicating an exponential cutoff in the distribution at large $R_v$. To demonstrate this, in Fig.~\ref{fig:numdens} we show the distribution of void sizes obtained from both our simulation samples, together with the prediction obtained from eqs.~\ref{eq:numdens} and \ref{eq:LagrangianR} with value $\delta_\rmn{v}=-0.40$, and a simple exponential curve $\propto\exp(-R_v^{0.60})$. These numerical values represent the best fits obtained from fitting to the $R_v>25\;h^{-1}$Mpc data from the higher resolution \emph{Dense} sample (note that the size distribution at large $R_v$ is itself resolution-dependent). The minimum radius cut is imposed since neither model can fit the data at all scales. The best-fit parameters are sensitive to the exact choice of this cut, but the relative quality of the fits does not change significantly. Despite having an extra parameter, the exponential cutoff model significantly outperforms the modified excursion set model on the basis of the Akaike Information Criterion \citep{Akaike:1974}.
An approximate exponential cutoff at large void radii is a rather generic feature of alternative descriptions of voids, and would also apply if for instance voids were modelled simply as minima of a Gaussian density field at a fixed smoothing scale (\citealt{BBKS}; \citealt*{Flender:2013}; \citealt{Nadathur:2014b}). We do not intend to attempt a fully-fledged alternative theoretical description of voids here; instead our point is that the ability or otherwise of equation~\ref{eq:numdens} with variable $\delta_\rmn{v}$ to fit the void distribution in a limited size range should not be taken as evidence that the excursion set model provides a good description of watershed voids.
However, other aspects of the excursion set model can also be directly tested. The crucial ingredient of the model is identification of underdensities as voids \emph{only if they have undergone shell crossing}. Various modifications of the model (e.g., \citealt*{Paranjape:2012a,Musso:2012,Paranjape:2012b}; \citealt{Jennings:2013,Achitouv:2015}) do not change this fundamental picture. Since watershed algorithms in general make no explicit reference to shell crossing when defining a `void', it is desirable to test this assumption much more directly than through the ad hoc fitting of the number function described above.
A key property of shell crossing is that it occurs at the same enclosed density contrast $\Delta$ for all voids, irrespective of their size. (This is also what justifies the use of a single $\delta_\rmn{v}$ in fitting void abundances.) Adding additional complexities to the spherical evolution model, such as the effect of a shear field, could relax the condition $\Delta=-0.8$ to some extent, and may introduce a small scatter in the values of $\Delta$ over the void population. Nevertheless, strong variation of the enclosed density with properties of watershed voids would be a clear sign that they do not correspond to similar shell-crossed objects.
A related property of shell-crossed voids is that if the enclosed density contrast is to be the same for voids of all sizes, equation~\ref{eq:nu} requires that larger voids must correspond to more extreme fluctuations of the parameter $\nu$. It can be shown that this in turn means that larger voids should on average correspond to shallower but broader initial density profiles $\delta(r)$, while smaller voids correspond to deeper and steeper profiles. That is, smaller shell-crossed voids should contain deeper density minima than large voids.\footnote{We thank Ravi Sheth for drawing our attention to this point.} The analogous situation for collapsing haloes is that the most massive haloes should be the least centrally concentrated, which is indeed the case \citep*[e.g.][]{NFW:1996,NFW:1997}. More generally, voids with the deepest density minima should have the steepest density profiles, and vice versa.
These qualitative properties provide clear tests of the assumption that watershed voids have undergone shell crossing. However, as we show in the next Section, neither of them hold true for the voids obtained using {\small VIDE} and {\small ZOBOV}, nor should one expect them to hold for other watershed void finders.
\section{Properties of watershed voids}
\label{section:properties}
\begin{figure*}
\hspace{4em}\includegraphics[width=130mm]{nmin_Rv_hist.pdf}
\caption{The distribution of the minimum tracer number densities within voids and void sizes in the \emph{Main} sample. There is a clear trend towards increasing void size as the minimum density decreases. The dotted lines show the contours enclosing $95\%$ and $99\%$ of all `voids' identified in a random uniform distribution of points with the same number density and in the same volume. The arrow indicates the value $R_v=\overline{n}^{-1/3}$, roughly the mean inter-particle separation, which has sometimes been suggested as minimum size cut. The dashed line shows the true minimum achievable void size resolution as a function $n_\rmn{min}$: most voids automatically lie well away from this limit.}
\label{fig:nmin_R}
\end{figure*}
\subsection{Sizes and densities}
\label{subsec:sizes}
Fig.~\ref{fig:nmin_R} shows the distribution of void sizes and minimum tracer number densities for all voids in our \emph{Main} tracer sample. It is immediately obvious that lower minimum number densities are correlated with larger void sizes, as we argued would always be the case for watershed void finders in general and {\small ZOBOV} in particular. The characteristic banana-shaped distribution is similar to that found by \citet{Nadathur:2015a,Nadathur:2015c}, indicating that this is a universal property of the void finder, independent of whether the tracer particles used are dark matter particles or galaxies in haloes. It is also noteworthy that neither {\small VIDE} nor {\small ZOBOV} impose any restriction on the allowed minimum densities, resulting in a range of $n_\rmn{min}$ values extending all the way up to the mean.
The minimum achievable void size resolution is dictated by the process of reconstructing the density field from the Voronoi tessellation described in Section~\ref{subsec:voidfinding}, which sets a natural cutoff of $R_{v,\,\rmn{min}}=\left(3/4\pi\right)^{1/3}r_{N}\left(n_\rmn{min}/\overline{n}\right)^{-1/3}$, where $r_{N}\equiv\overline{n}^{-1/3}\sim7\;h^{-1}$Mpc is roughly the mean inter-particle separation. This cutoff is shown by the dashed black line, and $r_N$ by the vertical arrow. In fact most voids naturally lie well away from this limit, simply because most zones contain several tracer particles. This means that the selection criterion $R_v>r_N$ advocated by some studies \citep[e.g.][]{Sutter:2012wh,Sutter:2014b} has no practical effect, whereas a tighter criterion $R_v>2r_N$ \citep{Hamaus:2014a} is unnecessarily conservative.
Also shown are contours showing the $95$ and $99$ per cent confidence limit contours for the distribution of spurious `voids' identified by the same algorithm in a random uniform distribution of points with the same volume and same number density as the \emph{Main} sample. There is clearly a considerable overlap between the two distributions, but care is required in its interpretation, as $P\left((n_\rmn{min},R_v)|\rmn{Poisson}\right)$ is not the same as $P\left(\rmn{Poisson}|(n_\rmn{min},R_v)\right)$. In the absence of information on the true dark matter content of such voids, conservative cuts to the void catalogue based on the properties of Poisson voids have previously been advocated \citep[e.g.][]{Neyrinck:2008,Nadathur:2014a, Hotchkiss:2015a}, but these may use available data sub-optimally. Since we have access to this information from the simulation, we analyse all voids without imposing such cuts a priori, and in fact we find that while Poisson contamination increases in the overlap region, statistically speaking voids of all $(n_\rmn{min},R_v)$ values on average correspond to true dark matter underdensities. Similarly, we find that low values of the density ratio $r$ \citep{Neyrinck:2008} also do not serve as a reliable indicator of Poisson contamination. This issue is discussed further in Appendix \ref{appendixB}.
\begin{figure}
\includegraphics[width=85mm]{DMdens_vs_Rv.pdf}
\caption{Binned average values of the dark matter density at the location of the void centre, as a function of the void size. The top and bottom panels show data for voids in the \emph{Dense} and \emph{Main} samples respectively. Circles (green) and squares (red) refer to the two alternative definitions of the void centre; the circumcentre is clearly a better locator of the true minimum density in the void.}
\label{fig:rho_R}
\end{figure}
\begin{figure*}
\hspace{4em}\includegraphics[width=125mm]{navg_Rv_hist.pdf}
\caption{The distribution of the average tracer number densities within voids and void sizes in the \emph{Main} sample. The dotted lines show the contours enclosing $95\%$ and $99\%$ of voids in random distributions as in Fig.~\ref{fig:nmin_R}, and the arrow indicates the approximate mean inter-particle separation.}
\label{fig:navg_R}
\end{figure*}
Another noteworthy aspect of Fig.~\ref{fig:nmin_R} is the apparent saturation of the minimum densities within voids with increasing $R_v$. This is a consequence of the finite tracer number density, and the saturation value is dependent on the mean density $\overline{n}$. Subsampling tracer particles lowers $\overline{n}$ and thus reduces the apparent tracer density contrast in voids. Conversely, \citet{Nadathur:2015c} show that at the same mean tracer density, more highly biased tracers result in much lower values of $n_\rmn{min}$ within voids.
Given the uncertainties associated with the tracer number density discussed below, the true dark matter density at void locations is perhaps a more informative quantity. To measure this we make use of the dark matter density field described in \ref{subsec:MultiDark} and simply measure its value $\rho$ in the grid cell corresponding to the position of the void centre.
Fig.~\ref{fig:rho_R} shows the binned average central densities as a function of the void radius $R_v$, for both \emph{Dense} and \emph{Main} voids, and for both definitions of the void centre described in Section~\ref{subsec:centres}. The error bars represent the $2\sigma$ uncertainty in the mean. As expected, in all cases the circumcentre definition is a superior indicator of the location of minimum density within the void. It is also clear that this minimum density decreases with increasing void size, contrary to the excursion set prediction. Curiously, for voids in the \emph{Dense} sample, the density at the barycentre first decreases and then increases with $R_v$. This is because in this case voids with large $R_v$ tend to be formed from the merger of several sub-voids. As such sub-voids necessarily correspond to shallower density minima they do not affect the location of the minimum density, but they do shift the location of the barycentre via equation~\ref{eq:barycentre}. In contrast, the location of the circumcentre is independent of the sub-void fraction and the choice of criteria to control void merging.
Fig.~\ref{fig:navg_R} shows the distribution of the \emph{average} tracer density $n_\rmn{avg}$ within the void, calculated from the number of void member particles and the void volume, and $R_v$. As pointed out by \citet{Nadathur:2014a,Achitouv:2015}, $n_\rmn{avg}$ is typically $\gtrsim1$ and much larger than $n_\rmn{min}$, simply because the watershed definition means that voids always extend to include high density regions on the separating ridges. This feature of {\small ZOBOV} and {\small VIDE} indirectly demonstrates that most tracers in identified voids reside in overdensities, which explains why the barycentre is a poor locator of the minimum underdensity, and also suggests that when $n_\rmn{avg}>1$ the void radius is a significant overestimate of the size of the true underdense region.
\begin{figure}
\includegraphics[width=85mm]{central_densities.pdf}
\caption{Binned average values of the true dark matter density at the location of the void centre, as a function of the minimum tracer number density within the void, for both definitions of the void centre. The top and bottom panels show the data for voids from the \emph{Dense} and \emph{Main} tracer samples respectively. A $45^\circ$ line is shown for reference in each case.}
\label{fig:rho_nmin}
\end{figure}
It is worth noting that a selection cut on the minimum void radius alone is a sub-optimal way of excluding voids that are on average overdense, since it would eliminate many with the lowest $n_\rmn{avg}$ values as well.
\subsection{Tracer density versus dark matter density}
\label{subsec:tracervDM}
The relationship between the tracer number density and the true underlying dark matter density in the simulation is also of interest. Even though the tracers in our case are down-sampled dark matter particles, we find that these two quantities are in general not the same. There is already an inherent shot noise in the number densities of dark matter particles arising from the fact that they constitute a discrete realization of the underlying continuous density field. Randomly down-sampling the dark matter particles enhances this shot noise, meaning that, particularly in void regions, tracer number densities tend to be larger than the true dark matter density. This problem is to a large extent mitigated by the self-adaptive nature of the Voronoi tessellation, but cannot be completely removed. A second consequence of reducing the total number of tracers is to increase the effective smoothing scale at which tracer number densities are measured. As mentioned above, for the VTFE density reconstruction this scale is $\sim r_N\sim7\,h^{-1}$Mpc, whereas the dark matter density field is smoothed at a scale of $<1\,h^{-1}$ Mpc. This change of smoothing scales acts in the opposite direction, tending to make dark matter underdensities appear shallower in $n_\rmn{min}$.
The relative strengths of these two effects are illustrated in Fig.~\ref{fig:rho_nmin}, which shows the relationship between the dark matter density $\rho$ at the position of the void centre and the minimum tracer number density within the void as determined from the tessellation, for both definitions of the void centre, and for voids from both the \emph{Main} and \emph{Dense} tracer samples. That $\rho$ exceeds $n_\rmn{min}$ at the barycentre for both samples is to be expected, since the barycentre typically lies quite far from the location of the tracer density minimum. But even at the circumcentre, which is guaranteed to lie in the region of minimum tracer number density, the tracer number density does not accurately reflect the dark matter density, although there is a clear linear relationship between $\rho$ and $n_\rmn{min}$. Particularly in the most underdense voids, shot noise enhancement dominates, causing $n_\rmn{min}/\overline{n}<\rho_\rmn{min}/\overline{\rho}$. However, for the shallowest voids the smoothing effect becomes more important, reversing the relationship. The same qualitative trends are seen for both the \emph{Dense} and \emph{Main} tracer samples, and more generally hold for any tracer population obtained from subsampling dark matter particles.
\subsection{Density profiles}
\label{subsec:profile_results}
\begin{figure*}
\begin{center}
\includegraphics[width=158mm]{n_RQ_profiles.pdf}
\caption{Stacked tracer number density profiles for voids of different sizes. Stacks are chosen to include equal numbers of voids in each. \emph{Left}: Profiles for void stacks centred on void barycentres. The solid lines show the best-fit forms of the fitting formula of \citet{Hamaus:2014a,Sutter:2014b} (eqs.~\ref{eq:HSW}, \ref{eq:HSWalpha} and \ref{eq:HSWbeta}), which generally provides a poor fit to the data. Due to discreteness artefacts at small $r$, data points shown with open symbols are excluded from the fitting procedure. \emph{Right}: Profiles for the same voids but with the stacks centred around the circumcentres. By construction the circumcentre more accurately locates the region of minimum tracer density within the void.}
\label{fig:n_profiles}
\end{center}
\end{figure*}
We now turn to the distribution of tracer particles and dark matter around void centres. Anticipating that the form of the density profiles will depend on the void size, we first examine the average profiles for stacks of voids within different ranges of $R_v$, chosen such that each stack contains an equal number of voids.
Fig.~\ref{fig:n_profiles} shows the resulting tracer number density profiles for stacks centred on the void barycentres and circumcentres in the left and right panel respectively. The barycentre stacks show a strong trend for decreasing central density as the void size increases, as is expected from Fig.~\ref{fig:nmin_R}. Voids are generally surrounded by overdense walls, which are much higher for small voids than for large ones. The general asymmetry of the circumcentre location with respect to particles in the void walls is also apparent in the fact that the stacked profiles about this location are less able to resolve the high densities in these walls. On the other hand, central densities are much lower for the circumcentre stacks. This is the essential tradeoff between the two centre definitions: the barycentre has a greater degree of symmetry with respect to the surrounding overdensities, whereas the circumcentre identifies the true location of the \emph{under}density.
A curious feature is apparent in the stacked barycentre profiles at smaller void radii: the tracer density does not show a minimum at the void barycentre, but instead at some distance \emph{away} from the centre. In fact for the smallest voids the average tracer density at the barycentre is indistinguishable from the mean. As argued in Section~\ref{subsec:centres}, the void barycentre is always displaced away from the location of the minimum tracer density within the void; in particular, for small voids the barycentre is often at or very close to the location of a tracer particle within the void. The tracer number density $n(r)$ is measured by naively counting the numbers of tracer particles within volumes on scales generally much smaller than the mean inter-particle separation. As a result, when the barycentre location is close to a tracer particle, high central values for $n(r)$ are obtained.
The converse effect can be seen by comparison with the right panel of Fig.~\ref{fig:n_profiles}, which shows the stacked density profiles for the same voids, but based around the void circumcentre. The circumcentre is also a special point, as it is by construction as far as possible from all tracers in the void. Unsurprisingly therefore, sufficiently small spheres around the circumcentre contain no particles at all and $n(r)\sim0$ for voids of all sizes, even though $n_\rmn{min}$ values are never so small and vary with void size. The Voronoi tessellation avoids this issue because of its self-adaptive resolution. It can be seen from Fig. \ref{fig:rho_nmin} that the Voronoi reconstructed $n_\rmn{min}$ is a much better predictor of the true dark matter density than number counts --- which is why it is preferable to reconstruct the density field from the tessellation in the first place. For this reason, such stacked number density profiles should not be relied upon for quantitative analysis without calibration.
For this purpose we instead make use of the full dark matter density field at high resolution. Profiles of the average enclosed dark matter density $1+\Delta(r)$ are shown in Fig.~\ref{fig:rho_R_profiles}, for the same void stacks as before. These confirm some properties of watershed voids which are of significance for the attempts to model them theoretically. First, as already seen in Figs.~\ref{fig:nmin_R} and \ref{fig:rho_R}, larger voids contain deeper density minima. Secondly, the enclosed density contrast within these voids is $\Delta(r)>-0.8$, for all void sizes and at all distances $r$. The condition for shell crossing to occur is thus not satisfied at any point within the average void. Finally, the central matter densities are much lower for circumcentre stacks than those centred on the barycentre, as expected from Fig.~\ref{fig:rho_R}.
\begin{figure*}
\begin{center}
\includegraphics[width=158mm]{rho_RQ_profiles.pdf}
\caption{Stacked profiles of the total enclosed dark matter density, $1+\Delta(r)$, within radius $r$ of the void centre. The stacks are the same as in Fig.~\ref{fig:n_profiles}. The left panel shows profiles for stacks centred around the void barycentres, and the right panel for stacks centred around the circumcentres.}
\label{fig:rho_R_profiles}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=160mm]{rho_nmin_profiles.pdf}
\caption{Stacked profiles of the total enclosed dark matter density, $1+\Delta(r)$, for voids within the same size range $15<R_v<20\;h^{-1}$Mpc, but with different minimum tracer densities. Profiles in the left panel are stacked about the barycentres, and in the right panel about the circumcentres.}
\label{fig:rho_nmin_profiles}
\end{center}
\end{figure*}
Our results in Fig.~\ref{fig:n_profiles} may also be compared to those of \citet{Hamaus:2014a}, who postulate a `universal' profile for voids based on the functional form
\begin{equation}
\label{eq:HSW}
\frac{n(r)}{\overline{n}}=1+\delta_c\left(\frac{1-\left(r/r_s\right)^\alpha}{1+\left(r/R_v\right)^\beta}\right)\,.
\end{equation}
This profile form has two free parameters, $\delta_c$ and $r_s$, with $\alpha(r_s)$ and $\beta(r_s)$ fixed by eqs. \ref{eq:HSWalpha} and \ref{eq:HSWbeta}. Note that although \citet{Hamaus:2014a} refer to this as the density profile $\rho(r)$, their fits to data are in fact based on measurements of the tracer \emph{number} density $n(r)$ --- as emphasized above, for sub-sampled dark matter tracers these two quantities are not the same. We determine the best-fit values of these two parameters by fitting to the $n(r)$ data for each void stack. To avoid the discreteness artefacts described above, in each stack we exclude data points with $\left(r/R_v\right)\overline{R_v}<\left(3/4\pi\right)^{1/3}r_N$ from the fitting procedure, where $\overline{R_v}$ is the mean void radius for the stack. The resulting fits are shown by the solid lines in the left-hand panel of Fig. \ref{fig:n_profiles}. It can be seen that this `universal' profile function generally provides a poor fit to the data, both within the void interior and in the overdense walls.
Similar results are obtained when fitting to the true dark matter density profiles $\rho(r)$. The behaviour of the best-fit parameters in equation~\ref{eq:HSW} as functions of the void size $R_v$ also significantly differs from that claimed by \citet{Hamaus:2014a} and \citet{Sutter:2014b}. In particular, these authors suggest that the central density contrast $\delta_c$ is an approximately linear, increasing function of $R_v$ for voids of all sizes and in all tracer populations. This behaviour is central to their claim that equation~\ref{eq:HSW} can provide a `universal' description of void density profiles in all tracer populations through a simple rescaling of void sizes. On the contrary, as shown in Fig.~\ref{fig:rho_R}, $\delta_c(R_v)$ is a strictly decreasing function for \emph{Main} sample voids, and is a non-linear, U-shaped function for \emph{Dense} sample voids. Further discussion of the fitting profile is provided in Appendix \ref{appendixA}, where we show that the behaviour of other parameter fits also disagrees with the claimed universality.
We conclude that the fits provided by \citet{Hamaus:2014a,Sutter:2014b} fail both qualitatively and quantitatively to describe the density profiles we observe. Equally, we find no evidence for the self-similarity of tracer density profiles seen by \citet{Nadathur:2015a}, but in this case differences in methodology, the use of dark matter particles instead of galaxies as tracers and the different selection criteria applied to voids may preclude direct comparison (see \citealt{Nadathur:2015c} for a fuller discussion).
So far, following earlier works \citep{Hamaus:2014a,Nadathur:2015a} we have only considered the variation in the mean profile with the size of the voids included in the stack, but it is clear that this cannot be the only important variable. In fact, as shown in Fig.~\ref{fig:rho_nmin_profiles}, voids of similar sizes but different minimum densities $n_\rmn{min}$ have very different density profiles. Voids with different $n_\rmn{min}$ clearly do not enclose the same density contrasts, and deeper density minima do not correspond to steeper density profiles. The enclosed density contrast $\Delta$ clearly varies widely over the void population, providing further evidence that the population of watershed voids does not satisfy the foundational assumption of the excursion set model.
It is also clear that a more complete description of the density profiles around voids is obtained by accounting for the extent of variation in both dimensions of the $\left(n_\rmn{min},R_v\right)$ plane. Fitting formulae such as those provided by \citet{Hamaus:2014a} or \citet{Nadathur:2015a}, which account only for variation with void radius, will in principle be unable to describe the full variety of watershed voids.
However, it is worth stressing that the distribution of highly biased galaxies trace dark matter underdensities rather differently than the randomly down-sampled dark matter particles we have used in this work \citep{Nadathur:2015c}, and it is the dark matter profiles of galaxy voids which are of greater practical interest in cosmology.
\section{Conclusions}
\label{section:conclusions}
Our aim in this paper was to provide an empirical investigation into the properties of watershed voids in order to better understand the operation of void finding algorithms such as {\small VIDE} and {\small ZOBOV} and the relation to theory. Several previous studies have focused on the distribution of void sizes alone, and have attempted to fit this using modifications of the spherical evolution model. Such an approach however misses the important relationship between void size and density: \emph{larger voids correspond to deeper density minima}. This is a fundamental feature of {\small ZOBOV} that holds irrespective of whether the tracers used for void identification are simulation dark matter particles, haloes or galaxies. It is also a more general property that should apply to \emph{any} watershed void finder.
The conclusion that follows from this relationship --- and which we also demonstrate directly through stacked density profiles around void centres --- is that watershed voids cannot correspond to a population of objects which all enclose the same density contrast, which is the principal starting assumption of theoretical descriptions deriving from the model of \citet{Sheth:2003py}. It has long been known that the void number function prediction of this model fails to match that of watershed voids by many orders of magnitude. It has sometimes been argued without proof \citep[e.g.][]{Sutter:2014b,Chan:2014} that the void formation threshold $\delta_\rmn{v}$ might differ from the shell crossing value in the spherical model due to the generally aspherical nature of watershed voids. This assumption has led several authors to treat $\delta_\rmn{v}$ as a free parameter but without altering the basic model.
However, given the range in enclosed density contrasts $\Delta$ over the watershed void population, a single value of $\delta_\rmn{v}$ for all voids does not seem tenable. Even more suggestive is the fact that for no subset of these voids does the average enclosed density contrast satisfy the criterion for shell crossing, $\Delta\simeq-0.8$, at \emph{any} radial distance from the centre, let alone at the void radius $R_v$. Nor do smaller voids correspond to deeper density minima as expected in the model.
The simplest interpretation of this evidence is that watershed voids simply do not correspond to objects that have undergone shell crossing. With hindsight this should not seem surprising --- {\small ZOBOV} uses only information on the local topology of the density field, and makes no reference to shell crossing. Furthermore, neither {\small VIDE} nor {\small ZOBOV} apply any meaningful conditions even on the minimum tracer density $n_\rmn{min}$ within voids, instead reporting \emph{all} local density minima. Attempts to explain how the shell crossing density criterion may be altered in such voids therefore seem to be misguided. A simpler starting proposition would be to give up the enforced assumption of shell crossing and to describe watershed voids simply as what they are: regions of density minima.
We should stress that breaking this link to theoretical models of shell-crossed voids does not necessarily make the results obtained from watershed void finders less useful for practical cosmological studies. For instance, these voids can still be used to identify large-scale underdense environments. Some of them (though not all) will also correspond to maxima of the gravitational potential, and so they can still be used for studies of lensing \citep{Melchior:2014} or the ISW effect \citep{Cai:2013ik,Hotchkiss:2015a,Planck:2015ISW}. Equally, we do not intend to claim that the \citet{Sheth:2003py} model does not correctly describe shell-crossed underdensities on much smaller scales \citep[although see][]{Falck:2015,Achitouv:2015}. Our statement is simply that this and related models do not match simulation or observational data because the word `void' has a different meaning in the two contexts.
Another interesting feature of our results is the relationship between the underdensity in voids measured using subsampled tracers and using the full resolution dark matter density. We show that values of $n$ and $\rho$ do not completely agree, and apparent tracer underdensities in deep voids are deeper than the true dark matter minima. The relationship between $n$ and $\rho$ depends on the mean sampling density of tracers; it will certainly also change if biased tracers are used. This does not affect the basic operation of {\small ZOBOV}, which only uses relative tracer densities to identify minima, but it argues against the use of absolute values of the central tracer density in applying selection cuts, as has sometimes been suggested \citep[e.g.][]{Sutter:2012wh,Jennings:2013,Sutter:2014b,Nadathur:2014a}. In other words, selecting a region which apparently satisfies the shell-crossing criteria in terms of the tracer number density does not ensure that it does so in the true matter density. This was already pointed out by \citet{Furlanetto:2006} for the case when the tracers are galaxies; our results show that it applies even if the tracers are a subset of dark matter particles in the simulation.
\section{Acknowledgements}
We thank Ravi Sheth for stimulating correspondence and Alexis Finoguenov for helpful discussions. SH acknowledges support from the Science and Technology Facilities Council [grant number ST/L000652/1].
The MultiDark Database used in this paper and the web application providing online access to it were constructed as part of the activities of the German Astrophysical Virtual Observatory as result of a collaboration between the Leibniz-Institute for Astrophysics Potsdam (AIP) and the Spanish MultiDark Consolider Project CSD2009-00064. The MultiDark simulations were run on the NASA's Pleiades supercomputer at the NASA Ames Research Center.
|
1,108,101,565,461 | arxiv | \section{Introduction}
One of the central problems in constructing precision tests of
a quantum field theory such as quantum chromodynamics is the elimination of
theoretical ambiguities such as the dependence on the renormalization
scale $\mu$ in perturbative expansions in the coupling $\alpha_s(\mu)$.
However, any prediction which
relates one physical quantity to another cannot depend on theoretical
conventions such as the choice of renormalization scheme or renormalization scale.
This is the principle underlying ``commensurate scale relations" (CSR) \cite{CSR},
which are general QCD predictions relating physical observables to each other.
For example, the ``generalized Crewther relation", which is discussed in more
detail below, provides a scheme-independent relation between the QCD
corrections to the Bjorken (or Gross Llewellyn-Smith) sum rule for deep inelastic
lepton-nucleon scattering, at a given momentum transfer $Q$, to the radiative
corrections to the annihilation cross section $\sigma_{e^+ e^- \to
\rm hadrons}(s)$, at a corresponding ``commensurate" energy scale $\sqrt s$.
\cite{CSR,BGKL} The specific relation between the physical scales $Q$
and $\sqrt s$ reflects the fact that the radiative corrections to each process have
distinct quark mass thresholds.
The generalized Crewther relation can be derived by calculating the
QCD radiative corrections to the deep inelastic sum rules and $R_{e^+ e^-}$ in a
convenient renormalization scheme such as the modified minimal subtraction
scheme $\overline{\rm MS}$. One then algebraically eliminates $\alpha_{\overline
{MS}}(\mu)$.
Finally, BLM scale-setting \cite{BLM} is used to eliminate the $\beta$-function
dependence of the coefficients. The form of the resulting relation between the
observables thus matches the result which would have been obtained had QCD
been a
conformal theory with zero $\beta$ function. The final result relating the
observables is independent of the choice of intermediate
$\overline{\rm MS}$ renormalization scheme.
In quantum electrodynamics, the running coupling $\alpha_{QED}(Q^2)$,
defined from
the Coulomb scattering of two heavy test charges at the momentum transfer
$t = -Q^2$, is
taken as the standard observable. Similarly, one can take the
momentum-dependent coupling
$\alpha_V(Q^2)$, defined from the potential scattering for heavy color
charges, as a
standard QCD observable. Commensurate scale relations between $\alpha_V$
and the QCD
radiative corrections to other observables have no scale or scheme
ambiguity, even in
multiple-scale problems such as multijet production. As is the case in
QED, the
momentum scale which appears as the argument of
$\alpha_V$ reflect the mean virtuality of the exchanged gluons.
Furthermore, we can
write a commensurate scale relation between $\alpha_V$ and an analytic
extension of
the
$\alpha_{\overline {MS}}$ coupling, thus transferring all of the
unambiguous scale-fixing
and analytic properties of the physical $\alpha_V$ scheme to the $\overline {MS}$
coupling.
Commensurate scale relations thus provide
fundamental and precise scheme-independent tests of QCD, predicting how
observables
track not only in relative normalization, but also in their commensurate scale
dependence.
\section{The Generalized Crewther Relation}
Any perturbatively calculable physical quantity can be used to define an
effective charge \cite{Grunberg,DharGupta,GuptaShirkovTarasov} by incorporating
the entire radiative correction into its definition. All such effective charges
$\alpha_A(Q)$ satisfy the Gell-Mann-Low renormalization group
equation. In the case of massless quarks, the first two terms in the
perturbative
expansion for the
$\beta$ function of each effective charge,
$\beta_0$ and $\beta_1$, are universal; different schemes or effective
charges only
differ through the third and higher coefficients. Any
effective charge can be used as a reference running coupling constant in
QCD to define
the renormalization
procedure. More generally, each effective charge or renormalization
scheme, including
$\overline{\rm MS}$, is a special case of the universal coupling function
$\alpha(Q, \beta_n)$.
For example, consider the Adler function \cite{Adler} for the $e^+ e^-$
annihilation cross section
\begin{equation} D(Q^2)=-12\pi^2 Q^2{d\over dQ^2}\Pi(Q^2),~
\Pi(Q^2) =-{Q^2\over 12\pi^2}\int_{4m_{\pi}^2}^{\infty}{R_{e^+ e^-}(s)ds\over
s(s+Q^2)}.
\end{equation}
The entire radiative correction to this function is defined as
the effective charge
$\alpha_D(Q^2)$ :
\begin{eqnarray}
D \left( Q^2/ \mu^2, \alpha_{\rm s}(\mu^2) \right) &=&
D \left (1, \alpha_{\rm s}(Q^2)\right) \label{3} \\
&\equiv&
3 \sum_f Q_f^2 \left[ 1+ {3\over 4} C_F{\alpha_D(Q^2) \over \pi}
\right]
+( \sum_f Q_f)^2C_{\rm L}(Q^2) \nonumber \\
&\equiv& 3 \sum_f Q_f^2 C_D(Q^2)+( \sum_f Q_f)^2C_{\rm L}(Q^2),
\nonumber
\end{eqnarray}
where $C_F={N_C^2-1\over 2 N_C}. $
The coefficient $C_{\rm L}(Q^2)$ appears at the third order in
perturbation theory and is related to the ``light-by-light scattering type"
diagrams. (Hereafter $\alpha_{\rm s}$ will denote the ${\overline{\rm MS}}$
scheme strong coupling constant.)
Similarly, we can define the entire radiative correction to the Bjorken sum rule
as the effective charge $\alpha_{g_1}(Q^2)$ where
$Q$ is the corresponding momentum transfer:
\begin{equation}
\int_0^1 d x \left[ g_1^{ep}(x,Q^2) - g_1^{en}(x,Q^2) \right]
\equiv {1\over 6} \left|g_A \over g_V \right|
C_{\rm Bj}(Q^2)
= {1\over 6} \left|g_A
\over g_V \right| \left[ 1- {3\over 4} C_F{\alpha_{g_1}(Q^2) \over
\pi} \right] .
\end{equation}
It is straightforward to algebraically relate $\alpha_{g_1}(Q^2)$ to
$\alpha_D(Q^2)$ using the known expressions to three loops in the
$\overline{\rm MS}$ scheme. If one chooses the renormalization scale to resum
all of the quark and gluon vacuum polarization corrections into
$\alpha_D(Q^2)$, then
the final result turns out to be remarkably simple \cite{BGKL}
$(\widehat\alpha = 3/4\, C_F\ \alpha/\pi):$
\begin{equation}
\widehat{\alpha}_{g_1}(Q)=\widehat{\alpha}_D( Q^*)-
\widehat{\alpha}_D^2( Q^*)+\widehat{\alpha}_D^3( Q^*) +
\cdots,
\end{equation}
where
\begin{eqnarray}
\ln \left({ {Q}^{*2} \over Q^2} \right) &=&
{7\over 2}-4\zeta(3)+\left(\frac{\alpha_D ( Q^*)}{4\pi}
\right)\Biggl[ \left(
{11\over 12}+{56\over 3} \zeta(3)-16{\zeta^2(3)}
\right) \beta_0\cr &&
+{26\over 9}C_{\rm A}
-{8\over 3}C_{\rm A}\zeta(3)
-{145\over 18} C_{\rm F}
-{184\over 3}C_{\rm F}\zeta(3)
+80C_{\rm F}\zeta(5)
\Biggr].
\label{EqLogScaleRatio}
\end{eqnarray}
where in QCD, $C_{\rm A}=N_C = 3$ and $C_{\rm F}=4/3$.
This relation shows how
the coefficient functions for these two different processes are
related to each other at their respective commensurate scales. We emphasize
that the $\overline{\rm MS}$ renormalization
scheme is used only for calculational convenience; it serves simply as an
intermediary between observables. The renormalization group
ensures
that the forms of the CSR relations in perturbative QCD are independent
of the choice of an intermediate renormalization scheme.
The Crewther relation was originally derived assuming that the theory is
conformally
invariant; {\it i.e.}, for zero $\beta$ function. In the physical case, where the
QCD coupling runs, all non-conformal effects are resummed into the
energy and momentum transfer scales of
the effective couplings $\alpha_R$ and $\alpha_{g1}$. The general
relation between these two effective charges for nonconformal
theory thus takes the form of a geometric series
\begin{equation}
1- \widehat \alpha_{g_1} =
\left[ 1+ \widehat \alpha_D( Q^*)\right]^{-1} \ .
\end{equation}
We have dropped the small light-by-light scattering contributions.
This is again a special advantage of relating observable to
observable.
The coefficients are independent of
color and are the same in Abelian, non-Abelian, and conformal gauge theory. The
non-Abelian structure of the theory is reflected in the expression for the scale
${Q}^{*}$.
Is experiment consistent with the generalized Crewther relation? Fits
\cite{MattinglyStevenson} to the experimental measurements of the
$R$-ratio above the thresholds for the production of $c\overline{c}$ bound
states
provide the empirical constraint:
$\alpha_{R}({\sqrt s} =5.0~{\rm GeV})/\pi \simeq 0.08\pm 0.03.$
The prediction for the effective coupling for
the deep inelastic sum rules at the commensurate momentum transfer $Q$
is then
$\alpha_{g_1}(Q=12.33\pm 1.20~{\rm GeV})/\pi
\simeq \alpha_{\rm GLS}(Q=12.33\pm 1.20~{\rm GeV})/\pi
\simeq 0.074\pm 0.026.$
Measurements of the Gross-Llewellyn Smith sum rule have so far only been
carried out
at relatively small values of $Q^2$ \cite{CCFRL1,CCFRL2};
however, one can use the results of the theoretical
extrapolation \cite{KS} of the experimental data presented in \cite{CCFRQ}:
$ \alpha_{\rm GLS}^{\rm extrapol}(Q=12.25~{\rm GeV})/\pi\simeq 0.093\pm 0.042.$
This range overlaps with the prediction from the generalized
Crewther relation. It is clearly important to have higher precision
measurements to
fully test this fundamental QCD prediction.
\section{General Form of Commensurate Scale Relations}
In general, commensurate scale
relations connecting the effective charges for observables $A$ and $B$ have the
form
\begin{equation}
\alpha_A(Q_A) =
\alpha_B(Q_B) \left(1 + r^{(1)}_{A/B} {\alpha_B(Q_B)\over \pi} + r^{(2)}_{A/B}
{\alpha_B(Q_B)\over
\pi}^2 + \cdots\right),
\label{eq:CSRg}
\end{equation}
where the coefficients $r^{{n}}_{A/B}$
are
identical to the coefficients obtained in a con\-formally invariant theory
with $\beta_B(\alpha_B) \equiv (d/d\ln Q^2) \alpha_B(Q^2) = 0$.
The ratio of the scales $Q_A/Q_B$ is thus fixed by the requirement that
the couplings sum all of the effects of the non-zero $\beta$ function. In
practice the
NLO and NNLO coefficients and relative scales can be identified from the flavor
dependence of the perturbative series; {\it i.e.}\ by shifting scales such that the
$N_F$-dependence associated with $\beta_0 = 11/3 C_A - 4/3 T_F N_F$ and
$\beta_1 =
-34/3 C_A^2 + {20\over 3} C_A T_F N_F + 4 C_F T_F N_F$
does not appear in
the coefficients. Here $C_A=N_C$, $C_F=(N^2_C-1)/2N_C$ and $T_F=1/2$.
The shift in scales which gives conformal coefficients in effect pre-sums
the large and strongly divergent terms in the PQCD series which grow as
$n! (\beta_0
\alpha_s)^n$, {\it i.e.}, the infrared renormalons associated with coupling-constant
renormalization. \cite{tHooft,Mueller,LuOneDim,BenekeBraun}
The
renormalization scales $Q^*$ in the BLM method are physical in the sense that
they reflect the mean virtuality of the gluon propagators. This
scale-fixing procedure is consistent with scale fixing in QED, in agreement
with
in the Abelian limit, $N_C \to 0$. \cite{BrodskyHuet}
\cite{BLM,LepageMackenzie,Neubert,BallBenekeBraun}
The ratio of scales
$\lambda_{A/B} = Q_A/Q_B$ guarantees that the
observables $A$ and $B$ pass through new quark thresholds at the same physical
scale. One can also show that the commensurate scales satisfy the
transitivity rule
$\lambda_{A/B} = \lambda_{A/C} \lambda_{C/B},$ which ensures that predictions
are independent of the choice of an intermediate renormalization scheme or
intermediate observable $C.$
\section{Commensurate Scale Relations and Fixed Points}
In general, we can write the relation between any two effective charges at
arbitrary
scales
$\mu_A$ and
$\mu_B$ as a correction to the corresponding relation obtained in a
conformally invariant
theory:
\begin{equation}
\alpha_A(\mu_A) = C_{AB}[\alpha_B(\mu_B)] +
\beta_B[\alpha_B(\mu_B)] F_{AB}[\alpha_B(\mu_B)]
\label{eq:ak}
\end{equation}
where
\begin{equation}
C_{AB}[\alpha_B] = \alpha_B + \sum_{n=1} C_{AB}^{(n)}\alpha^n_B
\label{eq:al}
\end{equation}
is the functional relation when $\beta_B[\alpha_B]=0$. In fact, if $\alpha_B$
approaches a fixed point $\bar\alpha_B$ where
$\beta_B[\bar\alpha_B]=0$,
then $\alpha_A$ tends to a fixed point given by
\begin{equation}
\alpha_A \to \bar\alpha_A = C_{AB}[\bar\alpha_B].
\label{eq:am}
\end{equation}
The commensurate scale relation for observables $A$ and $B$ has a similar
form, but
in this case the relative scales are fixed such that the non-conformal term
$F_{AB}$ is
zero.
Thus the commensurate scale relation $\alpha_A(Q_A) = C_{AB}[\alpha_B(Q_B)]$
at general commensurate scales is also the relation connecting the values
of the fixed
points for any two effective charges or schemes. Furthermore, as
$\beta\rightarrow 0$,
the ratio of commensurate scales $Q^2_A/Q^2_B$ becomes the ratio of
fixed point scales $\bar Q^2_A/\bar Q^2_B$ as one approaches
the fixed point regime.
\section{Implementation of $\alpha_V$ Scheme}
\unboldmath
Is there a preferred effective charge which we should use to characterize the
coupling strength in QCD? In QED, the running coupling $\alpha_{QED}(Q^2)$,
defined from
the potential between two infinitely heavy test charges, has traditionally
played that
role. In the case of QCD, the heavy-quark potential $V(Q^2)$ is defined as the
two-particle-irreducible scattering amplitude of test color charges; {\it i.e.} \ the
scattering of an infinitely heavy quark and antiquark at momentum transfer $t =
-Q^2.$ The relation $V(Q^2) = - 4 \pi C_F
\alpha_V(Q^2)/Q^2$ then defines
the effective charge $\alpha_V(Q).$ This coupling can provide a
physically based alternative to the usual ${\overline {MS}}$ scheme.
As in the corresponding case of Abelian QED, the scale $Q$ of the coupling
$\alpha_V(Q)$ is identified with the exchanged momentum. Thus there is never
any ambiguity in the interpretation of the scale. All vacuum polarization
corrections
due to fermion pairs are incorporated in $\alpha_V$ through the usual
vacuum polarization
kernels which depend on the physical mass thresholds. Of course, other
observables could be used to define the standard QCD coupling, such as the
effective
charge defined from heavy quark radiation. \cite{Uraltsev}
The relation of $\alpha_V(Q^2)$ to the conventional $\overline {MS}$
coupling is now known to NNLO, \cite{Peter}
but in the following only the NLO relation will be used. The commensurate
scale relation is given by \cite{BGMR}
\begin{eqnarray}
\label{eq:csrmsovf}
\alpha_{\overline{\mbox{\tiny MS}}}(Q)
& = & \alpha_V(Q^{*}) + \frac{2}{3}N_C{\alpha_V^2(Q^{*}) \over \pi}
\nonumber \\
& = & \alpha_V(Q^{*}) + 2{\alpha_V^2(Q^{*}) \over \pi}\ ,
\end{eqnarray}
which is valid for $Q^2 \gg m^2$. The coefficients in the
perturbation expansion have their conformal values, {\it i.e.}, the same coefficients
would occur even if the theory had been conformally invariant with $\beta=0$.
The commensurate scale is given by
\begin{eqnarray}
Q^* & = & Q\exp\left[\frac{5}{6}\right] \ .
\end{eqnarray}
The scale in the $\overline {MS}$ scheme is thus a factor
$\sim 0.4$ smaller than the physical scale. The coefficient $2 N_C/3$ in the NLO
coefficient is a feature of the non-Abelian couplings of QCD; the same
coefficient occurs even if the theory were conformally invariant with
$\beta_0=0.$
Using the above QCD results, we can transform any NLO prediction
given in $\overline{MS}$ scheme to a scale-fixed expansion in
$\alpha_V(Q)$.
We can also derive the connection between the $\overline{MS}$ and $\alpha_V$
schemes for Abelian perturbation theory using the limit $N_C \to 0$ with
$C_F\alpha_s$ and $N_F/C_F$ held fixed. \cite{BrodskyHuet}
The use of $\alpha_V$ and related physically defined effective charges such as
$\alpha_p$ (to NLO the effective charge defined from the (1,1) plaquette,
$\alpha_p$ is the same as $\alpha_V$) as expansion parameters has been
found to be valuable in lattice
gauge theory, greatly increasing the convergence of perturbative expansions
relative to
those using the bare lattice coupling. \cite{LepageMackenzie} Recent lattice
calculations of the
$\Upsilon$- spectrum \cite{Davies} have been used with BLM
scale-fixing to determine a NLO normalization
of the static heavy quark potential: $
\alpha_V^{(3)}(8.2 {\rm GeV}) = 0.196(3)$ where the effective number of light
flavors is
$n_f = 3$. The
corresponding modified minimal subtraction coupling evolved to the
$Z$ mass and five flavors is $ \alpha_{\overline{MS}}^{(5)}(M_Z) =
0.1174(24)$. Thus a high precision value for $\alpha_V(Q^2)$ at a specific
scale is
available from lattice gauge theory. Predictions for other QCD observables
can be
directly referenced to this value without the scale or scheme ambiguities,
thus greatly
increasing the precision of QCD tests.
One can also use $\alpha_V$ to characterize the coupling which appears in the hard
scattering contributions of exclusive process amplitudes at large momentum
transfer,
such as elastic hadronic form factors, the photon-to-pion transition form
factor at
large momentum transfer \cite{BLM,BJPR} and exclusive weak decays of heavy
hadrons.\cite{Henley} Each gluon
propagator with four-momentum $k^\mu$ in the hard-scattering quark-gluon
scattering amplitude $T_H$ can be associated with the coupling
$\alpha_V(k^2)$ since the
gluon exchange propagators closely resembles the interactions encoded in the
effective potential $V(Q^2)$. [In Abelian theory this is exact.]
Commensurate scale
relations can then be
established which connect the hard-scattering subprocess amplitudes which
control exclusive processes to other QCD observables.
We can anticipate that eventually
nonperturbative
methods such as lattice gauge theory or discretized light-cone quantization will
provide a complete form for the heavy quark potential in $QCD$. It
is reasonable to assume that $\alpha_V(Q)$ will not diverge at small space-like
momenta. One possibility is that $\alpha_V$ stays relatively constant
$\alpha_V(Q) \simeq 0.4$ at low momenta, consistent with fixed-point behavior.
There is, in fact, empirical evidence for freezing of the $\alpha_V$ coupling
from the observed systematic dimensional scaling behavior of exclusive
reactions. \cite{BJPR} If this is in fact the case, then the range of QCD
predictions can be extended to quite low momentum scales, a regime normally
avoided because of the apparent singular structure of perturbative
extrapolations.
There are a number of other advantages of the $V$-scheme:
\begin{enumerate}
\item
Perturbative expansions in $\alpha_V$ with the scale set by the momentum
transfer cannot
have any
$\beta$-function dependence in their coefficients since all running
coupling effects are
already summed into the definition of the potential. Since
coefficients involving
$\beta_0$ cannot occur in an expansions in $\alpha_V$, the divergent infrared
renormalon series of the form $\alpha^n_V\beta_0^n n!$ cannot occur. The
general convergence properties of the scale $Q^*$ as an expansion in $\alpha_V$
is not known. \cite{Mueller}
\item
The effective coupling $\alpha_V(Q^2)$ incorporates vacuum polarization
contributions with finite fermion masses. When continued to time-like
momenta, the coupling has the correct analytic dependence dictated by the
production thresholds in the $t$ channel. Since $\alpha_V$ incorporates quark
mass effects exactly, it avoids the problem of explicitly computing and
resumming
quark mass corrections.
\item
The $\alpha_V$ coupling is the natural expansion parameter for
processes involving non-relativistic momenta, such as heavy quark production at
threshold where the Coulomb interactions, which are enhanced at low relative
velocity $v$ as $\pi \alpha_V/v$, need to be
re-summed. \cite{Voloshin,Hoang,Fadin}
The effective Hamiltonian for nonrelativistic QCD is thus most naturally
written in
$\alpha_V$ scheme.
The threshold corrections
to heavy quark production in $e^+ e^-$ annihilation depend on
$\alpha_V$ at specific scales $Q^*$. Two distinct ranges of scales arise
as arguments of
$\alpha_V$ near threshold: the relative momentum of the quarks governing the
soft gluon exchange responsible for the Coulomb potential, and a high momentum
scale, induced by
hard gluon exchange, approximately equal to twice the quark mass for the
corrections.
\cite{Hoang} One thus can use threshold production to obtain a direct
determination of $\alpha_V$ even at low scales. The corresponding QED results
for $\tau$ pair production allow for a measurement of the magnetic moment of the
$\tau$ and could be tested at a future $\tau$-charm
factory. \cite{Voloshin,Hoang}
\end{enumerate}
We also note that
computations in different sectors of the Standard Model have been
traditionally carried out using different renormalization schemes.
However, in a grand
unified theory, the forces between all of the particles in the fundamental
representation should become universal above the grand unification scale.
Thus it is
natural to use $\alpha_V$ as the effective charge for all sectors of a grand
unified theory, rather than in
a convention-dependent coupling such as $\alpha_{\overline {MS}}$.
\section{The Analytic Extension of the $\bar{MS}$ Scheme}
The standard ${\overline {MS}}$ scheme
is not an analytic function of the renormalization scale at heavy quark thresholds;
in the running of the coupling the quarks are taken as massless, and
at each quark threshold the value of $N_F$ which appears in the $\beta$
function is
incremented. Thus Eq. (\ref{eq:csrmsovf}) is technically only valid far
above a heavy quark threshold. However, we can use this commensurate scale
relation to define
an extended
$\overline {MS}$ scheme which is continuous and analytic at any scale. The new
modified scheme inherits all of the good properties of the $\alpha_V$ scheme,
including its correct analytic properties as a function of the quark masses
and its
unambiguous scale fixing. \cite{BGMR}
Thus we define
\begin{equation}
\widetilde {\alpha}_{\overline{\mbox{\tiny MS}}}(Q)
= \alpha_V(Q^*) + \frac{2N_C}{3} {\alpha_V^2(Q^{**})\over\pi} +
\cdots ,
\label{alpmsbar2}
\end {equation}
for all scales $Q$. This equation not only provides an analytic
extension of the $\overline{MS}$ and similar schemes, but it also ties down the
renormalization scale to the physical masses of the quarks as they
enter into the vacuum polarization contributions to $\alpha_V$.
The modified scheme \amst\ provides an analytic interpolation of
conventional $\overline{MS}$ expressions by utilizing the mass dependence of the
physical \mbox{$\alpha_{V}$}\ scheme. In effect, quark thresholds are treated
analytically to all orders in $m^2/Q^2$; {\it i.e.}, the evolution of the analytically
extended coupling in the intermediate regions reflects the actual mass
dependence of a physical effective charge and the analytic properties of
particle production.
Just as in Abelian QED, the mass dependence of the effective potential
and the analytically extended scheme \amst\ reflects the analyticity of the
physical thresholds for particle production in the
crossed channel. Furthermore, the definiteness of the dependence in the quark
masses automatically constrains the renormalization scale. There
is thus no scale ambiguity in perturbative expansions in \mbox{$\alpha_{V}$}\ or \amst.
In leading order the effective number of flavors in the modified scheme
\amst\ is given to a very good approximation by the simple form \cite{BGMR}
\begin{equation}
\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}\left(\frac{m^2}{Q^2}\right)
\cong \left(1 + {5m^2 \over {Q^2\exp({5\over 3})}} \right)^{-1}
\cong \left( 1 + {m^2 \over Q^2} \right)^{-1}.
\end{equation}
Thus the contribution from one flavor is $\simeq 0.5$ when
the scale $Q$ equals the quark mass $m_i$. The standard procedure
of matching $\alpha_{\overline{\mbox{\tiny MS}}}(\mu)$ at the quark
masses serves as a zeroth-order approximation to the continuous $N_F$.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=4in
\epsfbox{nfsum.eps}
\end{center}
\caption[*]{The continuous
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}$ in the analytic
extension of the $\overline{\mbox{MS}}$ scheme as a
function of the physical scale $Q$. (For reference the
continuous $N_F$ is also compared with
the conventional procedure of taking $N_F$ to be a step-function at the
quark-mass thresholds.)}
\label{fig:nfsum}
\end{figure}
Adding all flavors together gives the total
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$
which is shown in Fig.~\ref{fig:nfsum}. For reference, the
continuous $N_F$ is also compared with
the conventional procedure of taking $N_F$ to be a step-function at the
quark-mass thresholds.
The figure shows clearly that there are hardly any plateaus at all
for the continuous
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$ in
between the quark masses.
Thus there is really no scale below 1 TeV where
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$
can be approximated by a constant; for all $Q$ below 1 TeV there is always
one quark
with mass $m_i$ such that $m_i^2 \ll Q^2$ or $Q^2 \gg m_i^2$ is not
true.
We also note that if one would use any other scale than the
BLM-scale for $\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$,
the result would be to increase the difference between the analytic
$N_F$ and the standard procedure of using the step-function at the
quark-mass thresholds.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=4in
\epsfbox{adiff.eps}
\end{center}
\caption[*]{The solid curve shows the relative difference between the
solutions to
the 1-loop renormalization group equation using continuous $N_F$,
$\widetilde{\alpha}_{\overline{\mbox{\tiny MS}}}(Q)$, and conventional discrete
theta-function thresholds, $\alpha_{\overline{\mbox{\tiny MS}}}(Q)$.
The dashed (dotted) curves shows the same quantity but using the scale $2Q$
($Q/2$)
in $\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}$. The solutions
have been
obtained numerically starting from the world average \cite{Burrows}\
$\alpha_{\overline{\mbox{\tiny MS}}}(M_Z) = 0.118$.}
\label{fig:adiff}
\end{figure}
Figure~\ref{fig:adiff} shows the relative difference between the two
different solutions of the 1-loop renormalization group equation,
{\it i.e.}\ $(\widetilde{\alpha}_{\overline{\mbox{\tiny MS}}}(Q)-
{\alpha}_{\overline{\mbox{\tiny MS}}}(Q) )/
\widetilde{\alpha}_{\overline{\mbox{\tiny MS}}}(Q)$.
The solutions have been obtained numerically starting from the
world average \cite{Burrows}
$\alpha_{\overline{\mbox{\tiny MS}}}(M_Z) = 0.118$.
The figure shows that
taking the quark masses into account in the running leads to
effects of the order of one percent, most especially
pronounced near thresholds.
To illustrate how to compute an
observable using the analytic extension of the $\overline{\mbox{MS}}$\ scheme and compare
with the standard treatment in
the $\overline{\mbox{MS}}$\ scheme we consider the
QCD corrections to the quark part of the non-singlet hadronic width of
the Z-boson, $\Gamma_{had,q}^{NS}$. Writing the QCD corrections in terms
of an effective charge we have
\begin{equation}
\Gamma_{had,q}^{NS}=\frac{G_FM_Z^3}{2\pi\sqrt{2}}
\sum_{q}\{(g_V^{q})^2+(g_A^{q})^2\}
\left[1+\frac{3}{4}C_F\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi}\right]
\end{equation}
where the effective charge $\alpha_{\Gamma,q}^{NS}(s)$ contains all
QCD corrections,
\begin{eqnarray}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(\mu)}{\pi}
\Bigg\{1+\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(\mu)}{\pi}
\nonumber \\ && \times
\left[\sum_{q=1}^{N_L}\left(-\frac{11}{12}+\frac{2}{3}\zeta_3
+ F\left(\frac{m_q^2}{s}\right)
-\frac{1}{3}\ln\left(\frac{\mu}{\sqrt{s}}\right)\right)
\right. \nonumber \\ && \left. \left.
+\sum_{Q=N_L+1}^{6}G\left(\frac{m_Q^2}{s}\right)\right] + \ldots \right\}
\end{eqnarray}
To calculate $\alpha_{\Gamma,q}^{NS}(s)$ in the
analytic extension of the $\overline{\mbox{MS}}$\ scheme one first applies
the BLM scale-setting procedure
in order to absorb all the massless effects of non-zero $N_F$ into the
running of the
coupling. This gives
\begin{eqnarray}
\label{eq:agms}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(Q^*)}{\pi}
\\ & & \times
\left\{1+\frac{\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}(Q^*)}{\pi}
\left[\sum_{q=1}^{N_L}F\left(\frac{m_q^2}{s}\right)
+\sum_{Q=N_L+1}^{6}G\left(\frac{m_Q^2}{s}\right)\right] + \ldots \right\}
\nonumber
\end{eqnarray}
where
\begin{equation}
Q^*=\exp\left[3\left(-\frac{11}{12}+\frac{2}{3}\zeta_3\right)\right]\sqrt{s}
=0.7076\sqrt{s}.
\end{equation}
Operationally, one then simply drops
all the mass dependent terms in the above expression and replaces the
fixed $N_F$ coupling $\alpha_{\overline{\mbox{\tiny MS}}}^{(N_L)}$
with the analytic \amst. (For an observable calculated with massless quarks
this step reduces to replacing the coupling.)
In this way both the massless $N_F$ contribution,
as well as the mass-dependent contributions from double bubble diagrams,
are absorbed into the coupling.
We are thus left with a very simple expression,
\begin{eqnarray}
\label{eq:aganalytic}
\frac{\alpha_{\Gamma,q}^{NS}(s)}{\pi} & = &
\frac{\amst(Q^*)}{\pi},
\end{eqnarray}
reflecting the fact that the QCD effects of quarks in the
perturbative coefficients, both massless and massive, should be absorbed
into the running of the coupling.
In order to compare the analytic extension of the $\overline{\mbox{MS}}$\ scheme
with the standard $\overline{\mbox{MS}}$\ result for
$\alpha_{\Gamma,q}^{NS}(s)$, we will apply
the BLM scale-setting procedure also for the standard $\overline{\mbox{MS}}$\ scheme.
This is to ensure that any differences are due to the different
ways of treating quark masses and not due to the scale choice.
In other words we want to compare Eqs.~(\ref{eq:agms})
and (\ref{eq:aganalytic}). As the normalization point we use
$\alpha_{\overline{\mbox{\tiny MS}}}^{(5)}(M_Z)=0.118$
which we evolve down to $Q^*=0.7076M_Z$ using leading order
massless evolution with $N_F=5$. This value is then used to calculate
$\alpha_{\Gamma,q}^{NS}(M_Z)=0.1243$ in the $\overline{\mbox{MS}}$\ scheme using
Eq.~(\ref{eq:agms}). Finally, Eq.~(\ref{eq:aganalytic}) gives the
normalization point for $\amst(Q^*)$.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=4in
\epsfbox{gdiff.eps}
\end{center}
\caption[*]{The relative difference between the calculation of
$\alpha_{\Gamma,q}^{NS}(s)$ in the analytic extension of the $\overline{\mbox{MS}}$\ scheme
and the standard treatment of masses in the $\overline{\mbox{MS}}$\ scheme. The discontinuities
are due to the mismatch between the $s/m^2$ and $m^2/s$ expansions of the
functions $F$ and $G$.}
\label{fig:gdiff}
\end{figure}
Figure~\ref{fig:gdiff} shows the relative difference between the two
expressions for $\alpha_{\Gamma,q}^{NS}(s)$ given by Eqs.~(\ref{eq:agms})
and (\ref{eq:aganalytic}) respectively. As can be seen from the figure the
relative difference is remarkably small, less than $0.2\%$ for scales above
1 GeV. Thus
the analytic extension of the $\overline{\mbox{MS}}$\ scheme takes the mass corrections
into account in a very simple way without having to include an infinite
series of higher dimension operators or doing complicated multi-loop
diagrams with explicit masses.
The form of $N_F(Q)$ at NNLO has recently been computed to two loop order
in QCD for the
$\alpha_V$ scheme. The application to the analytic extension of $\overline{\mbox{MS}}$\ scheme
will be
discussed in a forthcoming paper. \cite{Melles}
\section{Conclusion}
Commensurate scale relations have a number of attractive properties:
\begin{enumerate}
\item
The ratio of physical scales $Q_A/Q_B$ which appears in commensurate scale
relations
reflects the relative position of physical thresholds, {\it i.e.}\, quark
anti-quark pair
production.
\item
The functional dependence and perturbative expansion of the CSR are identical to
those of a conformal scale-invariant theory where $\beta_A(\alpha_A)=0$
and $\beta_B(\alpha_B)=0$.
\item
In the case of theories approaching fixed-point behavior
$\beta_A(\bar\alpha_A)=0$ and
$\beta_B(\bar\alpha_B)=0$, the commensurate scale relation relates both
the ratio of
fixed point couplings $\bar\alpha_A/\bar\alpha_B$, and the ratio of
scales as the fixed point is approached.
\item
Commensurate scale relations satisfy the Abelian correspondence principle
\cite{BrodskyHuet};
{\it i.e.}\ the non-Abelian gauge theory prediction reduces to Abelian theory for
$N_C \to 0$ at
fixed $ C_F\alpha_s$ and fixed $N_F/C_F$.
\item
The perturbative expansion of a commensurate scale relation has the same
form as a
conformal theory, and thus has no
$n!$ renormalon growth arising from the $\beta$-function.
It is an interesting conjecture whether the perturbative expansion relating
observables to observable are in fact free of all $n!$ growth. The
generalized Crewther relation, where the commensurate relation's perturbative
expansion forms a geometric series to all orders, has convergent behavior.
\end{enumerate}
Virtually any perturbative QCD prediction can be written in the form of a
commensurate
scale relation, thus eliminating any uncertainty due to renormalization
scheme or scale
dependence. Recently it has been shown \cite{BPT} how the commensurate scale
relation between the radiative corrections to $\tau$-lepton decay and
$R_{e^+e^-}(s)$
can be generalized and empirically tested for arbitrary $\tau$ mass and nearly
arbitrarily functional dependence of the $\tau$ weak decay matrix element.
An essential feature of the \mbox{$\alpha_{V}$}(Q) scheme is the absence of any
renormalization scale ambiguity, since $Q^2$ is, by definition, the square of
the physical momentum transfer. The \mbox{$\alpha_{V}$}\ scheme naturally takes into
account quark mass
thresholds, which is of particular phenomenological importance to QCD
applications in the mass region close to threshold.
As we have seen, commensurate scale relations provide
an analytic extension of the conventional $\overline{\mbox{MS}}$\ scheme in which many of
the advantages of the \mbox{$\alpha_{V}$}\ scheme are inherited by the \amst\ scheme,
but only minimal changes have to be made.
Given the commensurate scale relation connecting \amst\ to \mbox{$\alpha_{V}$}\, expansions in
\amst\ are effectively expansions in \mbox{$\alpha_{V}$}\ to the given order in perturbation
theory at a corresponding commensurate scale. Taking finite quark mass
effects into account analytically
in the running, rather than using a fixed flavor number $N_F$ between
thresholds, leads
to effects of the order of $1\%$ for the one-loop running coupling, with the
largest differences occurring near thresholds. These differences are important
for observables which are calculated neglecting quark masses, and could
turn out to be
significant when comparing low and high energy measurements of the strong
coupling.
Unlike the conventional \mbox{$\alpha_{\overline{\mbox{\tiny MS}}}$}\ scheme, the modified \amst\ scheme is
analytic at quark mass thresholds, and it thus provides a natural
expansion parameter for perturbative representations of observables.
In addition, the extension of the $\overline{\mbox{MS}}$\ scheme, including quark mass
effects analytically, reproduces the standard treatment of quark masses in the
$\overline{\mbox{MS}}$\ scheme to within a fraction of a percent. The standard treatment amounts
to either calculating multi-loop diagrams with explicit quark masses or adding
higher dimension operators to the effective Lagrangian. These corrections
can be viewed as compensating for the fact that the number of flavors in the
running is kept constant between mass thresholds. By utilizing the BLM scale
setting procedure, based on the massless $N_F$ contribution, the analytic
extension of the $\overline{\mbox{MS}}$\ scheme correctly absorbs both massless and mass dependent
quark contributions from QCD diagrams, such as the double bubble diagram, into
the running of the coupling. This gives the opportunity to convert
any calculation made in the $\overline{\mbox{MS}}$\ scheme with massless quarks into an
expression which includes quark mass corrections from QCD diagrams
by using the BLM scale and replacing \mbox{$\alpha_{\overline{\mbox{\tiny MS}}}$}\ with \amst.
Finally, we note the potential importance of utilizing the \mbox{$\alpha_{V}$}\
effective charge or the equivalent analytic \amst\ scheme in
supersymmetric and grand unified theories, particularly since the
unification of couplings and masses would be expected to occur in terms
of physical quantities rather than parameters defined by theoretical
convention.
\section*{Acknowledgments}
Much of this work is based on collaborations with Michael Melles, Mandeep
Gill, Hung Jung Lu, Andrei Kataev, and Gregory Gabadadze.
We also thank the organizers of RADCOR98, particularly Professor Joan
Sola of the Universitat Autonoma de Barcelona, for their outstanding arrangements
and hospitality.
|
1,108,101,565,462 | arxiv | \section{Introduction}
\label{sec:introduction}
As one of the brightest binary stars in the sky, Capella
($\alpha$\,Aurigae, HD\,34029, HR\,1708, HIP\,24608,
\ion{G8}{3}+\ion{G0}{3}, $P_{\rm orb} = 104$ days, $V = 0.07$) has
been studied for more than a century with a wide range of techniques
and at all observable wavelengths.\footnote{Capella has a wide common
proper motion companion that is itself a visual binary composed of M
dwarfs. The system is therefore a hierarchical quadruple. Revised
properties of the M dwarf pair are reported in the Appendix.} A
persistent source of frustration for several decades has been the
difficulty in determining accurate absolute masses for the components,
despite the wealth of astrometric and spectroscopic measurements
available. The history of this problem has been related by several
authors \citep[e.g.,][]{Batten:91, Barlow:93}, and most recently in
our earlier paper \citep[][hereafter T09]{Torres:09}. The challenge
associated with the masses has hindered efforts to pin down the
precise evolutionary state of the more massive primary star, which has
widely been considered to be a core helium-burning object, based
mostly on timescale arguments. Disappointingly, current stellar
evolution models have so far largely failed to confirm that notion, as
it has not been possible to achieve a satisfactory fit to the global
properties of both stars simultaneously at a single age when assuming
the bulk chemical composition the system is believed to have. The
secondary, on the other hand, is clearly on its way across the
Hertzprung gap.
The uncertainty in the masses was thought to have been solved in the
T09 analysis, which improved the formal precision by about a factor of
three compared to previous estimates, and documented efforts to
control systematic errors in the radial velocities that have likely
plagued the determination of the velocity amplitude of the
rapidly-rotating secondary star for decades, as described there.
Despite this, it was still not possible to establish the state of the
primary component unambiguously when enforcing a single age.
In the interim, \cite{Weber:11} have presented a new spectroscopic
study of Capella based on much higher-quality observational material
that leads to significantly larger masses for both stars than in our
2009 study, by several times the stated uncertainties. In particular,
the spread in mass between the stars increased from 1\% to about
3.5\%, which is an enormous difference for a pair of giants, and could
drastically change the assessment of their relative state of
evolution. In addition, we have now made a new determination of the
chemical composition of Capella that is appreciably different from the
abundance assumed in the earlier paper, and is a key ingredient for
the comparison with stellar evolution models. These two developments
motivate us to take a fresh look at the system in order to investigate
the impact of the new measurements. Furthermore, \cite{Weber:11} have
presented evidence that the orbit of Capella may be very slightly
eccentric, unexpectedly, whereas all previous studies including our
own assumed it is circular. It is of interest, therefore, to revisit
our 2009 study of tidal evolution in the system (orbit circularization
and rotational synchronization) with more sophisticated models than
used there, especially in light of the new masses.
We have organized the paper as follows. In Sect.~\ref{sec:RVs} we
describe the new spectroscopic observations of \cite{Weber:11} and
comment on the issue of systematics in the radial velocities, in
comparison with our previous results from 2009. A revised orbital fit
for Capella is the presented in Sect.~\ref{sec:orbit}, using also
astrometric and other measurements from T09. Sect.~\ref{sec:chemical}
reports a new detailed chemical analysis of both stars from
disentangled spectra, along with a determination of the atmospheric
parameters. The revised physical properties of the stars are collected
in Sect.~\ref{sec:properties}, and are compared against three
different sets of stellar evolution models in
Sect.~\ref{sec:stellarevolution}. Key chemical diagnostics available
for Capella are also compared with model predictions in this section.
Then in Sect.~\ref{sec:tidalevolution} we examine the evolution of
orbital and stellar properties subjected to the influence of tidal
mechanisms, as a test of that theory. Sect.~\ref{sec:discussion}
presents a discussion of the results and concluding remarks. Finally,
the Appendix provides an update on the orbital properties of the wide
common proper motion companion of Capella.
\section{Radial velocities}
\label{sec:RVs}
The numerous historical radial-velocity (RV) measurements of Capella
have been discussed at length in our T09 study, which highlighted how
challenging it has been to determine accurate values for the rapidly
rotating secondary star ($v \sin i \approx 35$~\kms), whereas those of
the sharp-lined primary ($v \sin i \approx 4$~\kms) have been quite
consistent over the last century. T09 presented 162 new RV
determinations for both components based on spectra obtained at the
Harvard-Smithsonian Center for Astrophysics (CfA) covering only a very
narrow wavelength range (45\,\AA). The RVs were measured using the
two-dimensional cross-correlation algorithm TODCOR \citep{Zucker:94},
with synthetic templates appropriate for each star. Because of the
limited wavelength coverage, those measurements are susceptible to
systematic errors resulting mostly from lines shifting in and out of
the spectral window as a function of orbital phase. Therefore, an
effort was made to control those biases by performing numerical
simulations to determine corrections to the velocities, which were at
the level of the final uncertainties in the individual measurements
for the secondary, and slightly larger for the primary. Final errors
in the RVs as measured from the scatter in the orbital fit were about
0.44~\kms\ for the primary and 0.89~\kms\ for the secondary. A sign
that systematic errors remained at some level in the CfA velocities
was evident in the residuals of the secondary star shown in Figure~2
of T09, in which a pattern can be seen as a function of orbital phase,
with a peak semi-amplitude of about twice the typical error. Possible
explanations for this, as discussed by T09, include the presence of
spots on the active secondary star, or template mismatch.\footnote{In
particular, due to limitations in the available library of synthetic
spectra they used, the macroturbulent velocity of the templates
($\zeta_{\rm RT} = 1.5$~\kms) was not quite as large as appropriate
for giant stars. This also resulted in an overestimation of the
rotational velocities of the components, as discussed by T09.} An
additional indication of possible biases was the fact that a small
offset ($0.267 \pm 0.079$~\kms) was found between the primary and
secondary velocities in the global orbital fit of T09 that could not
be accounted for by differences in the gravitational redshift between
the stars, and was ascribed to similar reasons as the secondary
residual pattern.
More recently \cite{Weber:11} have reported new RV measurements for
Capella based on a very large set of more than 400 spectra obtained
with the STELLA \'echelle spectrograph \citep{Strassmeier:10} on a
1.2\,m robotic telescope in Tenerife, Spain. These spectra are of far
superior quality than those of T09, both in terms of wavelength
coverage (two orders of magnitude larger) and signal-to-noise
ratios. \cite{Weber:11} derived velocities using a similar
two-dimensional cross-correlation approach as T09, and also performed
numerical simulations to assess and correct for systematic errors
caused by the measuring technique. The velocity scatter from their
orbital fit is 0.064~\kms\ for the primary and 0.297~\kms\ for the
secondary, seven and three times smaller than in T09, respectively.
The key difference in this data set compared to T09 is in the
resulting velocity semi-amplitude of the secondary star ($K_{\rm B} =
26.840 \pm 0.024$~\kms), which is more than 6$\sigma$ larger than
reported by T09 ($K_{\rm B} = 26.260 \pm 0.087$~\kms). This difference
alone leads to absolute masses for Capella that are 4\% larger than
in T09 for the primary, and 2\% for the secondary, a very significant
change that exceeds the formal mass errors by a factor of many. The
semi-amplitudes $K_{\rm A}$ of the primary star, on the other hand,
are in virtually perfect agreement (see below).
Despite the much improved random errors of \cite{Weber:11}, the
residuals of the secondary velocities from their spectroscopic orbital
model (see their Figure~2) still display a phase-dependent pattern
reminiscent of the one in T09, also with a semi-amplitude of roughly
twice the errors, but at a much lower level in absolute terms.
Moreover, they note that a small offset is seen again between the
primary and secondary velocities (0.059~\kms) that cannot be explained
by the gravitational redshift effect. This suggests that systematic
errors may still be lurking in these new measurements, for possibly
some of the same reasons as before. Nevertheless, any remaining
systematics are likely to be significantly less important than in T09,
as expected not only from the much higher quality of the spectroscopic
material, but suggested also by the significantly smaller magnitudes
of \emph{i}) the corrections for systematics applied by
\cite{Weber:11}, \emph{ii}) the formal uncertainties in the individual
RVs, or equivalently, the scatter from the spectroscopic orbital
solution, \emph{iii}) the amplitude of the residual pattern for the
secondary, and \emph{iv}) the unexplained residual primary/secondary
offset. In the next section we therefore incorporate these
measurements in a new orbital analysis of Capella.
Of the 438 RV measurements reported in Table~1 of \cite{Weber:11},
their final solution excluded 14 for the primary and 7 for the
secondary. We have done the same here as we found them to give
unusually large residuals, and we adopted also the measurement
uncertainties as published.
\section{Revised orbital fit}
\label{sec:orbit}
The global orbital solution in our T09 study of Capella combined all
usable astrometric observations in the literature with the CfA RVs for
both stars, as well as radial velocities for the primary star from
many of the historical data sets. The latter were carefully examined
to ensure that they imply $K_{\rm A}$ values consistent with those
from the CfA RVs in separate spectroscopic solutions using the same
fixed orbital period (see T09, Table~2). A similar solution of the
\cite{Weber:11} velocities shows that the primary semi-amplitude,
$K_{\rm A} = 29.959 \pm 0.005$~\kms, is essentially the same as that
from the CfA RVs, $K_{\rm A} = 29.96 \pm 0.04$~\kms. Therefore, for
our revised global solution below we have incorporated the
\cite{Weber:11} measurements for both stars, the CfA velocities for
the primary (but not the secondary), and the primary velocities from
the same historical data sets as used in T09. Although the
\cite{Weber:11} observations certainly dominate by weight, the older
measurements are still useful because they extend the baseline more
than a century, improving the orbital period.
The extensive astrometry available for Capella includes measurements
made by many authors with the technique of long-baseline
interferometry, beginning with the work of \cite{Merrill:22}, as well as
speckle interferometry, direct imaging, and the intermediate
astrometric measurements from \hip. These have all been described and
tabulated in our earlier study, and we refer the reader to that work
for details. So far as we are aware, no further astrometric
observations have been published for Capella except for those of
\cite{Huby:13}, which we do not use here, however, because of concerns
expressed by these authors about possible systematic errors
affecting their measurements.
Our global orbital fit follows closely that described by T09, and
includes the following parameters: orbital period ($P_{\rm orb}$),
relative angular semi-major axis ($a\arcsec$), inclination angle
($i_{\rm orb}$), eccentricity ($e$), longitude of periastron of the
more massive and cooler star ($\omega_{\rm A}$)\footnote{Note that
this is the fainter star at optical wavelengths (see T09), which we
refer to as star `A'.}, position angle of the ascending node for the
equinox J2000.0 ($\Omega_{\rm J2000}$), time of periastron passage
($T_{\rm peri}$), center-of-mass velocity ($\gamma$), the velocity
semi-amplitudes for each star ($K_{\rm A}$ and $K_{\rm B}$), the
angular semimajor axis of the photocenter ($a''_{\rm phot}$),
corrections to the \hip\ catalog values of the sky position of the
barycenter ($\Delta\alpha \cos\delta$, $\Delta\delta$) at the mean
catalog reference epoch of 1991.25, and corrections to the proper
motion components ($\Delta\mu_{\alpha} \cos\delta$,
$\Delta\mu_{\delta}$). To account for differences in the zero points
of the various RV data sets relative to the primary star measurements
of \cite{Weber:11}, we have also solved for 10 velocity offsets, one
for each set. An additional parameter, $f_\rho$, was included to
correct the scale of the angular separation measurements from two of
the astrometric data sets (see T09 for details).
\cite{Weber:11} discussed several adjustments made to their secondary
velocities to place them on the same system as their primary
velocities of Capella. These adjustments were intended to correct for
differences in the gravitational redshift of the two components, and
other shifts of unknown origin (see above). From their discussion it
is not entirely clear to us whether these constant shifts have been
applied to the velocities they reported, so our global solution
includes one additional offset, $\Delta_{\rm AB}$, to account for
possible residual effects. With this, the total number of adjustable
parameters in our fit is 27.
The solution in our T09 study assumed a circular orbit for Capella, as
have all previous analyses of the binary. We noted, however, that
there were hints of a non-zero eccentricity in the interferometric
measurements of \cite{Hummel:94}, though not in the CfA RVs or in any
of the other data sets. We ascribed this to the transformation that
\cite{Hummel:94} made between their original interferometric
visibilities ($V^2$) and the nightly relative positions in polar coordinates
that they published, given that their own solution using the original
visibilities indicated a circular orbit. As pointed out by
\cite{Weber:11}, however, a spectroscopic fit using their RVs also
indicates a statistically significant non-zero eccentricity, of very
nearly the same magnitude as we had seen, and even with a consistent
longitude of periastron. This suggests we may have been too quick to
dismiss the possibility of a non-circular orbit in T09, as unexpected
as this may be for a pair of giants (one being a clump star) in a period of 104 days \citep[see, e.g.,][]{Massarotti:08}. We
discuss this further in Sect.~\ref{sec:tidalevolution}. For our new
global solution we have chosen to allow the orbit to be eccentric, on
the assumption that the effect is real.
\begin{deluxetable}{lc}
\tabletypesize{\scriptsize}
\tablecolumns{2}
\tablecaption{Revised orbital solution for Capella.\label{tab:elements}}
\tablehead{
\colhead{~~~~~~~~~~~~~~~Parameter~~~~~~~~~~~~~~~} & \colhead{Value}}
\startdata
\noalign{\vskip -4pt}
\sidehead{Adjusted quantities} \\
\noalign{\vskip -9pt}
~~~~$P_{\rm orb}$ (days)\dotfill & 104.02128~$\pm$~0.00016\phn\phn \\
~~~~$a''$ (mas)\dotfill & 56.442~$\pm$~0.023\phn \\
~~~~$i_{\rm orb}$ (deg)\dotfill & 137.156~$\pm$~0.046\phn\phn \\
~~~~$e$\dotfill & 0.00089~$\pm$~0.00011 \\
~~~~$\omega_{\rm A}$ (deg)\dotfill & 342.6~$\pm$~9.0\phn\phn \\
~~~~$\Omega_{\rm J2000}$ (deg)\dotfill & 40.522~$\pm$~0.039\phn \\
~~~~$T_{\rm peri}$ (HJD$-$2,400,000)\dotfill & 48147.6~$\pm$~2.6\phm{2222} \\
~~~~$\gamma$ (\kms)\tablenotemark{a}\dotfill & $+29.9387$~$\pm$~0.0032\phn\phs \\
~~~~$K_{\rm A}$ (\kms)\dotfill & 25.9611~$\pm$~0.0044\phn \\
~~~~$K_{\rm B}$ (\kms)\dotfill & 26.860~$\pm$~0.017\phn \\
~~~~$a''_{\rm phot}$ (mas)\dotfill & 2.14~$\pm$~0.70 \\
~~~~$\Delta\alpha^*$ (mas)\dotfill & $-$0.53~$\pm$~0.81\phs \\
~~~~$\Delta\delta$ (mas)\dotfill & $-$0.37~$\pm$~0.57\phs \\
~~~~$\Delta\mu_{\alpha} \cos\delta$ (mas~yr$^{-1}$)\dotfill & $+$0.33~$\pm$~1.00\phs \\
~~~~$\Delta\mu_{\delta}$ (mas~yr$^{-1}$)\dotfill & $-$0.04~$\pm$~0.60\phs \\
~~~~$f_{\rho}$\tablenotemark{b}\dotfill & 1.0400~$\pm$~0.0032 \\
~~~~$\Delta_{\rm AB}$ for WS (\kms)\tablenotemark{c}\dotfill & $+$0.050~$\pm$~0.013\phs \\
~~~~$\Delta_{1\phn}$ $\langle$WS$-$C01$\rangle$ (\kms)\dotfill & $-$0.19~$\pm$~0.14\phs \\
~~~~$\Delta_{2\phn}$ $\langle$WS$-$N00$\rangle$ (\kms)\dotfill & $+$2.35~$\pm$~0.45\phs \\
~~~~$\Delta_{3\phn}$ $\langle$WS$-$G08$\rangle$ (\kms)\dotfill & $-$0.67~$\pm$~0.26\phs \\
~~~~$\Delta_{4\phn}$ $\langle$WS$-$S39$\rangle$ (\kms)\dotfill & $-$1.52~$\pm$~0.16\phs \\
~~~~$\Delta_{5\phn}$ $\langle$WS$-$S53$\rangle$ (\kms)\dotfill & $+$0.62~$\pm$~0.14\phs \\
~~~~$\Delta_{6\phn}$ $\langle$WS$-$B86$\rangle$ (\kms)\dotfill & $-$0.10~$\pm$~0.13\phs \\
~~~~$\Delta_{7\phn}$ $\langle$WS$-$S90$\rangle$ (\kms)\dotfill & $-$2.07~$\pm$~0.60\phs \\
~~~~$\Delta_{8\phn}$ $\langle$WS$-$B91$\rangle$ (\kms)\dotfill & $-$0.62~$\pm$~0.13\phs \\
~~~~$\Delta_{9\phn}$ $\langle$WS$-$B93$\rangle$ (\kms)\dotfill & $+$0.79~$\pm$~0.10\phs \\
~~~~$\Delta_{10}$ $\langle$WS$-$T09$\rangle$ (\kms)\dotfill & $+$0.289~$\pm$~0.035\phs \\
\sidehead{Derived quantities} \\
\noalign{\vskip -9pt}
~~~~$M_{\rm A}$ ($M_{\sun}$)\dotfill & 2.5687~$\pm$~0.0074 \\
~~~~$M_{\rm B}$ ($M_{\sun}$)\dotfill & 2.4828~$\pm$~0.0067 \\
~~~~$q\equiv M_{\rm B}/M_{\rm A}$\dotfill & 0.96653~$\pm$~0.00062 \\
~~~~$a$ ($10^6$ km)\dotfill & 111.11~$\pm$~0.10\phn\phn \\
~~~~$a$ ($R_{\sun}$)\dotfill & 159.72~$\pm$~0.15\phn\phn \\
~~~~$a$ (au)\dotfill & 0.74272~$\pm$~0.00069 \\
~~~~$\pi_{\rm orb}$ (mas)\dotfill & 75.994~$\pm$~0.089\phn \\
~~~~Distance (pc)\dotfill & 13.159~$\pm$~0.015\phn \\
~~~~$\mu_{\alpha} \cos\delta$ (mas~yr$^{-1}$)\dotfill & $+$75.85~$\pm$~1.00\phn\phs \\
~~~~$\mu_{\delta}$ (mas~yr$^{-1}$)\dotfill & $-$427.17~$\pm$~0.60\phn\phn\phs \\
~~~~$(\ell_{\rm B}/\ell_{\rm A})_{H_p}$\tablenotemark{d}\dotfill & 1.204~$\pm$~0.060
\enddata
\tablecomments{References for the RV offsets $\Delta_1$ to $\Delta_{10}$ are:
C01 = \cite{Campbell:01}; N00 = \cite{Newall:00}; G08 = \cite{Goos:08};
S39 = \cite{Struve:39}; S53 = \cite{Struve:53}; B86 = \cite{Beavers:86};
S90 = \cite{Shcherbakov:90}; B91 = \cite{Batten:91}; B93 = \cite{Barlow:93}; and
T09 = \cite{Torres:09}. The physical constants used in the analysis are those specified by \cite{Torresetal:10}.}
\tablenotetext{a}{On the reference frame of the RVs of \cite{Weber:11}.}
\tablenotetext{b}{Scale factor for the angular separation measurements by
\cite{Merrill:22} and \cite{Kulagin:70}.}
\tablenotetext{c}{Primary/secondary offset for the \cite{Weber:11} velocities (WS), in
the sense $\langle$primary \emph{minus} secondary$\rangle$.}
\tablenotetext{d}{Flux ratio between the secondary and primary in the \hip\ passband,
derived from the angular semimajor axis, the semimajor axis of the photocentric
orbit as measured by the satellite, and the velocity semi-amplitudes (see T09).}
\end{deluxetable}
The parameters of our fit are presented in Table~\ref{tab:elements},
along with other properties derived directly from the orbital
elements. With the exception of $K_{\rm B}$ and the quantities that
depend on it, the other results are rather similar to those in T09.
The eccentricity is essentially the same as derived by
\cite{Weber:11}.
\section{Spectroscopic analysis}
\label{sec:chemical}
Until recently the only detailed chemical analysis available for
Capella was that by \cite{McWilliam:90}, indicating a sub-solar
composition of ${\rm [Fe/H]} = -0.37 \pm 0.22$ on the scale of
\cite{Grevesse:84}, equivalent to ${\rm [Fe/H]} = -0.20 \pm 0.22$ on a
more modern scale in which the solar iron abundance is ${\rm A(Fe)} =
7.50$.\footnote{We use the standard abundance notation in which ${\rm
A(X)} = \log [n({\rm X})/n({\rm H})]+12$, where $n({\rm X})$ and
$n({\rm H})$ are the numbers of atoms per unit volume of element X
and of hydrogen.} This determination is presumably based on the
sharp lines of the primary star, but there is no indication that the
presence in the spectrum of the nearly equally bright secondary was
properly accounted for, and in addition, the analysis adopted an
incorrect primary temperature. A new metallicity determination was
reported by \cite{Fuhrmann:11} that gives a rather higher abundance of
${\rm [Fe/H]} = +0.05 \pm 0.08$, apparently on the scale of
\cite{Grevesse:96} in which the solar iron abundance is also ${\rm
A(Fe)} = 7.50$. This study is based on spectral synthesis applied to
eight \ion{Fe}{1} lines and one \ion{Fe}{2} line from a single
composite spectrum with some degree of line blending, using an
unspecified primary/secondary flux ratio.
Below we describe our new determination of the chemical composition
and atmospheric parameters of Capella based on the technique of
spectral disentangling \citep{Simon:94}, which bypasses the line
blending problems inherent in previous analyses that used composite
spectra.
\subsection{Disentangling}
\label{sec:disentangling}
For our spectroscopic analysis we made use of public archival spectra
taken in 2003 and 2004 with the ELODIE spectrograph \citep{Baranne:96}
on the 1.93\,m telescope at the Observatoire de Haute-Provence, in
France. The nominal resolving power of the instrument is $R = 42,000$,
and the 15 spectra used span the approximate wavelength range
4000--6800\,\AA, with signal-to-noise ratios ranging from about 130 to
560 per pixel at 5550\,\AA. We have disentangled these spectra using
the {\sc FDBinary} program of \cite{Ilijic:04}, in the same way as
described recently by \cite{Torres:14}. {\sc FDBinary} implements
spectral disentangling in the Fourier domain according to
\cite{Hadrava:95}. The signal-to-noise ratios of the resulting
disentangled spectra are approximately 510 for the cooler primary and
590 for the secondary. Renormalization of the disentangled spectra for
a proper abundance analysis \citep[see][]{Pavlovski:05, Lehmann:13}
requires knowledge of the relative flux contribution of each star at
each wavelength. Flux ratios for Capella have been measured
throughout the UV, optical, and near-infrared range, as reported by
T09. Figure~\ref{fig:fluxratio} shows the predicted flux ratio based
on PHOENIX model spectra from \cite{Husser:13} for parameters near
those of the components, along with the measurements tabulated by T09
as well as others from the recent study by \cite{Huby:13} over the
range 6112--8430\,\AA. The agreement is very good. For our purposes we
have used a smoothed version of this relation. Sample segments of the
disentangled spectra of the two components of Capella are presented in
Figure~\ref{fig:disentangled}.
\begin{figure}
\epsscale{1.15}
\plotone{fig01.eps}
\figcaption[]{Measured flux ratios for the components of Capella (cool
star relative to hot star, i.e., primary relative to
secondary). Values from T09 are indicated with dots and error boxes,
in which the horizontal length of each box indicates the wavelength
coverage. Other measurements from \cite{Huby:13} are shown by the
lighter gray squares in the range 0.611--0.843 $\mu$m. Overplotted is
the predicted flux ratio based on synthetic spectra by
\cite{Husser:13} scaled according to the radius ratio given by
T09.\label{fig:fluxratio}}
\end{figure}
\begin{figure}
\epsscale{1.15}
\plotone{fig02.eps}
\figcaption[]{Portions of the disentangled ELODIE spectra of
Capella. Secondary shifted vertically for
clarity.\label{fig:disentangled}}
\end{figure}
\subsection{Atmospheric parameters and abundance analysis}
\label{sec:abundances}
The general methodology for determining the atmospheric parameters and
abundances from the disentangled spectra follows the procedures
described by \cite{Torres:14}. The {\sc uclsyn} code
\citep{Smalley:01} was used to synthesize spectra, based on initial
temperatures and surface gravities from T09 and a built-in grid of LTE
model atmospheres by \cite{Castelli:03} with a scaled-solar mixture. Line broadening was modeled
adopting an initial composition matching the Sun, and microturbulent
velocities of $\xi_{\rm t} = 1.5$\,\kms\ for both components
(fine-tuned below). Least-squares fitting then yielded a
macroturbulent velocity for the primary of $\zeta_{\rm A} = 6.6 \pm
0.4$\,\kms, and a projected rotational velocity of $v_{\rm A} \sin i =
4.4 \pm 0.5$\,\kms. Given the rapid rotation of the secondary,
macroturbulence has a negligible effect on the line profiles and we
made no attempt to determine it from the ELODIE spectra. The
rotational velocity of the companion was measured to be $v_{\rm B}
\sin i = 34.5 \pm 0.7$\,\kms.
Elemental abundances were determined using spectral lines suitable for
giant stars over the wavelength range 4600--6750\,\AA. Line lists and
atomic data were taken from the work of \cite{Reddy:12},
\cite{Bocek:15}, and \cite{Lyubimkov:15}. Equivalent widths measured
within {\sc uclsyn} for the very numerous iron lines were used to set
the microturbulent velocities from the condition of a null correlation
between the abundance and the reduced equivalent widths. We derived
values of $\xi_{\rm t,A} = 1.48 \pm 0.08$\,\kms\ for the primary and
$\xi_{\rm t,B} = 1.55 \pm 0.11$\,\kms\ for the secondary. We also made
an estimate of the effective temperatures from the usual condition of
excitation equilibrium, iterating with the measurement of $\xi_{\rm
t}$, with the following results: $T_{\rm eff,A} = 4980 \pm 80$\,K
and $T_{\rm eff,B} = 5750 \pm 110$\,K. There is very good agreement
between these values and others reported by T09; we discuss them
further below. The surface gravities in our analysis were held fixed
at the estimates reported by T09, which are very close to our final
values described in the next section.
Detailed abundances were determined for 22 species in both stars, and
oxygen in the primary only. They are listed in
Table~\ref{tab:abundances}, which includes also the values relative to
the Sun on the scale of \cite{Asplund:09}. No adjustments have been
applied for NLTE effects. The uncertainties account for possible
errors in $T_{\rm eff}$ as reported above, and also include a
contribution from a representative error of 0.1\,\kms\ in $\xi_{\rm
t}$. The uncertainties in $\log g$ have a negligible impact.
The choice of the mixture adopted in the model atmospheres,
particularly the CNO composition, also has a minimal effect on our
abundance determinations \citep[see also][]{Morel:14}. We find no
dependence of the Fe abundance on wavelength, which is an indication
that our adopted wavelength-dependent flux ratios (Figure 1) are
accurate and do not introduce significant systematic errors in the
abundances. A further indication of the robustness of our
determinations is the fact that the abundances are quite similar for
the two components (except for species affected by evolution; see
below), as expected for a binary system.
The weighted average iron abundance of the two stars from \ion{Fe}{1}
is ${\rm [Fe/H]} = -0.04 \pm 0.06$, or very nearly solar, in contrast
with the sub-solar composition adopted by T09, and in better agreement
with the estimate by \cite{Fuhrmann:11}. We find no significant
enhancement of the $\alpha$ elements in Capella: ${\rm [\alpha/Fe]} =
-0.02 \pm 0.04$. A graphical representation of the abundance pattern
for the two components is seen in Figure~\ref{fig:abundances},
compared to the solar composition.
The lithium abundance has long been known to be very different for the
two components of this binary \citep{Wallerstein:64, Wallerstein:66} as
a result of chemical evolution in the primary. We used spectral lines
and atomic data from \cite{Lyubimkov:12} in the vicinity of the
\ion{Li}{1} $\lambda$6708 doublet to make new estimates for each star,
and obtained values of ${\rm A(Li)} = 1.08 \pm 0.11$ and ${\rm A(Li)}
= 3.28 \pm 0.13$ for the primary and secondary, respectively. These
are consistent with previous measurements. The equivalent widths we
determined are $21.4 \pm 0.7$\,m\AA\ and $297.4 \pm 8.0$\,m\AA.
\begin{deluxetable*}{ll c ccc c ccc c c}
\tablewidth{0pc}
\tablecaption{Abundances from our disentangled ELODIE spectra of Capella.\label{tab:abundances}}
\tablehead{
\colhead{} &
\colhead{} & &
\multicolumn{3}{c}{Primary} & &
\multicolumn{3}{c}{Secondary} & &
\colhead{} \\
\cline{4-6} \cline{8-10} \\ [-1ex]
\colhead{A} &
\colhead{X} & &
\colhead{Abundance} &
\colhead{[X/H]} &
\colhead{$N$} &&
\colhead{Abundance} &
\colhead{[X/H]} &
\colhead{$N$} & &
\colhead{$\log\epsilon_{\sun}$}
}
\startdata
3 & \ion{Li}{1} && $1.08 \pm 0.11$ & $-0.07 \pm 0.15$ & 3 && $3.28 \pm 0.13$ & $+2.30 \pm 0.16$ & 3 && $1.05 \pm 0.10$ \\
6 & \ion{C}{1} && $8.25 \pm 0.14$ & $-0.18 \pm 0.15$ & 4 && $8.28 \pm 0.11$ & $-0.15 \pm 0.12$ & 5 && $8.43 \pm 0.05$ \\
8 & \ion{O}{1} && $8.55 \pm 0.11$ & $-0.14 \pm 0.12$ & 1 && \nodata & \nodata & \nodata && $8.69 \pm 0.05$ \\
11 & \ion{Na}{1} && $6.13 \pm 0.09$ & $-0.11 \pm 0.10$ & 4 && $6.33 \pm 0.07$ & $+0.09 \pm 0.08$ & 4 && $6.24 \pm 0.04$ \\
12 & \ion{Mg}{1} && $7.60 \pm 0.09$ & $+0.00 \pm 0.10$ & 2 && $7.42 \pm 0.10$ & $-0.18 \pm 0.11$ & 5 && $7.60 \pm 0.04$ \\
14 & \ion{Si}{1} && $7.69 \pm 0.04$ & $+0.18 \pm 0.05$ & 10 && $7.59 \pm 0.07$ & $+0.08 \pm 0.08$ & 7 && $7.51 \pm 0.03$ \\
20 & \ion{Ca}{1} && $6.27 \pm 0.11$ & $-0.07 \pm 0.12$ & 8 && $6.38 \pm 0.11$ & $+0.04 \pm 0.08$ & 7 && $6.34 \pm 0.04$ \\
21 & \ion{Sc}{1} && $3.13 \pm 0.15$ & $-0.02 \pm 0.16$ & 5 && $3.16 \pm 0.10$ & $+0.01 \pm 0.11$ & 6 && $3.15 \pm 0.04$ \\
21 & \ion{Sc}{2} && $3.12 \pm 0.07$ & $-0.03 \pm 0.08$ & 8 && $3.10 \pm 0.08$ & $-0.05 \pm 0.09$ & 11 && $3.15 \pm 0.04$ \\
22 & \ion{Ti}{1} && $4.96 \pm 0.12$ & $+0.01 \pm 0.13$ & 15 && $5.02 \pm 0.09$ & $+0.07 \pm 0.10$ & 11 && $4.95 \pm 0.05$ \\
22 & \ion{Ti}{2} && $4.93 \pm 0.05$ & $-0.02 \pm 0.07$ & 3 && $4.91 \pm 0.09$ & $-0.04 \pm 0.10$ & 6 && $4.95 \pm 0.05$ \\
23 & \ion{V}{1} && $4.07 \pm 0.12$ & $+0.14 \pm 0.14$ & 14 && $4.10 \pm 0.07$ & $+0.17 \pm 0.11$ & 13 && $3.93 \pm 0.08$ \\
24 & \ion{Cr}{1} && $5.64 \pm 0.09$ & $+0.00 \pm 0.10$ & 9 && $5.67 \pm 0.07$ & $+0.03 \pm 0.08$ & 11 && $5.64 \pm 0.04$ \\
24 & \ion{Cr}{2} && $5.61 \pm 0.09$ & $-0.03 \pm 0.10$ & 7 && $5.57 \pm 0.07$ & $-0.07 \pm 0.08$ & 6 && $5.64 \pm 0.04$ \\
25 & \ion{Mn}{1} && $5.32 \pm 0.09$ & $-0.11 \pm 0.05$ & 8 && $5.31 \pm 0.08$ & $-0.12 \pm 0.09$ & 5 && $5.43 \pm 0.05$ \\
26 & \ion{Fe}{1} && $7.47 \pm 0.06$ & $-0.03 \pm 0.07$ & 42 && $7.44 \pm 0.08$ & $-0.06 \pm 0.09$ & 41 && $7.50 \pm 0.04$ \\
26 & \ion{Fe}{2} && $7.39 \pm 0.07$ & $-0.11 \pm 0.08$ & 8 && $7.38 \pm 0.06$ & $-0.12 \pm 0.07$ & 11 && $7.50 \pm 0.04$ \\
27 & \ion{Co}{1} && $4.87 \pm 0.08$ & $-0.12 \pm 0.10$ & 8 && $5.03 \pm 0.07$ & $+0.04 \pm 0.10$ & 5 && $4.99 \pm 0.07$ \\
28 & \ion{Ni}{1} && $6.20 \pm 0.04$ & $-0.02 \pm 0.06$ & 16 && $6.21 \pm 0.07$ & $-0.01 \pm 0.08$ & 17 && $6.22 \pm 0.04$ \\
39 & \ion{Y}{2} && $2.11 \pm 0.09$ & $-0.10 \pm 0.10$ & 4 && $2.23 \pm 0.05$ & $+0.02 \pm 0.07$ & 5 && $2.21 \pm 0.05$ \\
40 & \ion{Zr}{1} && $2.54 \pm 0.07$ & $-0.04 \pm 0.08$ & 5 && $2.25 \pm 0.12$ & $-0.33 \pm 0.13$ & 3 && $2.58 \pm 0.04$ \\
57 & \ion{La}{2} && $1.11 \pm 0.08$ & $+0.01 \pm 0.09$ & 5 && $1.23 \pm 0.05$ & $+0.13 \pm 0.06$ & 5 && $1.10 \pm 0.04$ \\
60 & \ion{Nd}{2} && $1.49 \pm 0.06$ & $+0.07 \pm 0.07$ & 8 && $1.52 \pm 0.05$ & $+0.10 \pm 0.06$ & 7 && $1.42 \pm 0.04$
\enddata
\tablecomments{Columns list the atomic number, the element and
ionization degree, the logarithm of the number abundance on the
usual scale in which A(H) = 12, the logarithmic abundance relative
to the Sun, and the number of spectral lines measured. The last
column gives the reference photospheric solar values from
\cite{Asplund:09}. }
\end{deluxetable*}
\begin{figure}
\epsscale{1.15}
\plotone{fig03.eps}
\figcaption[]{Photospheric abundance pattern measured for the Capella
components, compared to the standard solar composition of
\cite{Asplund:09} (gray shading). Abundances from different ions of
the same element have been averaged.\label{fig:abundances}}
\end{figure}
Additional chemical diagnostics for Capella have been reported by T09,
and include the $^{12}$C/$^{13}$C carbon isotope ratio for the primary
star \citep[$27 \pm 4$;][]{Tomkin:76} and the C/N abundance ratios for
both components ($0.57 \pm 0.06$ for the primary and $3.30 \pm 0.16$
for the secondary). We adopt those as published.
\section{Physical properties}
\label{sec:properties}
The revised masses of Capella that incorporate the new RVs of
\cite{Weber:11} have formal uncertainties of 0.3\%, and differ only
slightly from theirs through a combination of a marginally larger
$K_{\rm B}$ value in our analysis and a smaller orbital inclination
angle than they used. The new masses are 4.2\% and 1.6\% larger than
those listed by T09, which is a significant difference due almost
entirely to the change in the velocity amplitude of the secondary. The
radii, based on the individual angular diameters from T09 and the
revised orbital parallax, are approximately 0.9\% larger than before,
and have precisions of about 5\% and 4\% for the primary and
secondary, respectively. These are limited by the angular diameters,
as the uncertainty in the orbital parallax is only 0.11\%.
\begin{deluxetable}{lc@{~~~~}c}
\tablecolumns{3}
\tablewidth{0pc}
\tablecaption{Revised physical parameters of Capella.\label{tab:dimensions}}
\tablehead{
\colhead{~~~~~~~~Parameter~~~~~~~~} &
\colhead{Primary} &
\colhead{Secondary}
}
\startdata
Mass ($M_{\sun}$)\dotfill & 2.5687~$\pm$~0.0074 & 2.4828~$\pm$~0.0067 \\
$q\equiv M_{\rm B}/M_{\rm A}$\dotfill & \multicolumn{2}{c}{0.96653~$\pm$~0.00062} \\
$a$ ($10^6$ km)\dotfill & \multicolumn{2}{c}{111.11~$\pm$~0.10\phn\phn} \\
$a$ (au)\dotfill & \multicolumn{2}{c}{0.74272~$\pm$~0.00069} \\
$\pi_{\rm orb}$ (mas)\dotfill & \multicolumn{2}{c}{75.994~$\pm$~0.089\phn} \\
Distance (pc)\dotfill & \multicolumn{2}{c}{13.159~$\pm$~0.015\phn} \\
Radius ($R_{\sun}$)\dotfill & 11.98~$\pm$~0.57\phn & 8.83~$\pm$~0.33 \\
$\log g$ (cgs)\dotfill & 2.691~$\pm$~0.041 & 2.941~$\pm$~0.032 \\
$T_{\rm eff}$ (K)\dotfill & 4970~$\pm$~50\phn\phn & 5730~$\pm$~60\phn\phn \\
Luminosity ($L_{\sun}$)\tablenotemark{a}\dotfill & 78.7~$\pm$~4.2\phn & 72.7~$\pm$~3.6\phn \\
$BC_V$ (mag)\dotfill & $-$0.304~$\pm$~0.055\phs & $-$0.089~$\pm$~0.051\phs \\
$M_V$ (mag)\dotfill & 0.296~$\pm$~0.016 & 0.167~$\pm$~0.015 \\
$v \sin i$ (\kms)\tablenotemark{b}\dotfill & 4.1~$\pm$~0.4 & 35.0~$\pm$~0.5\phn \\
$P_{\rm rot}$ (days)\tablenotemark{c}\dotfill & 104~$\pm$~3\phn\phn & 8.5~$\pm$~0.2 \\
Age (Myr)\tablenotemark{d}\dotfill & \multicolumn{2}{c}{590--650} \\
${\rm [Fe/H]}$\dotfill & \multicolumn{2}{c}{$-$0.04~$\pm$~0.06\phs} \\
A(Li)\dotfill & 1.08~$\pm$~0.11 & 3.28~$\pm$~0.13 \\
$^{12}$C/$^{13}$C\,\tablenotemark{d}\dotfill & 27~$\pm$~4\phn & \nodata \\
C/N\,\tablenotemark{c}\dotfill & 0.57~$\pm$~0.06 & 3.30~$\pm$~0.16
\enddata
\tablenotetext{a}{Computed from $V$, $\pi_{\rm orb}$, and $BC_V$ from
\cite{Flower:96}, adopting $M_{\rm bol}^{\sun} = 4.732$ (see T09 and \citealt{Torres:10}).}
\tablenotetext{b}{Average of 5 measurements from the literature for the primary and 10 for
the secondary that account for macroturbulence, including our own
(see text).}
\tablenotetext{c}{Measured values adopted from T09.}
\tablenotetext{d}{Age range from the MESA and Granada models (see text).}
\tablenotetext{e}{Measurement by \cite{Tomkin:76}.}
\end{deluxetable}
T09 reported three independent estimates of the effective temperatures
for the two components. One is from a comparison of their
spectroscopic observations with synthetic spectra with solar
metallicity, giving $T_{\rm eff,A} = 4900 \pm 100$\,K and $T_{\rm
eff,B} = 5710 \pm 100$\,K. Another came from the use of the measured
angular diameters of the stars along with their apparent magnitudes,
the parallax, and bolometric corrections. The updated parallax in the
present work does not alter those estimates significantly; they are
$T_{\rm eff,A} = 4970 \pm 160$\,K and $T_{\rm eff,B} = 5690 \pm
130$\,K. A third determination by T09 was based on the measured color
indices for the stars, and the use of the color/temperature
calibrations of \cite{Ramirez:05}, which depend on metallicity. A
sub-solar composition ${\rm [m/H]} = -0.37 \pm 0.07$ had been assumed
by T09, whereas we now derive a value much closer to solar. Using our
determination of ${\rm [Fe/H]} = -0.04 \pm 0.06$
(Sect.~\ref{sec:abundances}), the revised photometric estimates become
4940\,K and 5680\,K, which are 30\,K and 70\,K higher than
before. Furthermore, a careful examination of the zero point of the
\cite{Ramirez:05} calibrations by \cite{Casagrande:10} suggests that
the scale of those relations is too cool by about 85\,K compared to
the best available absolute scale, at least in the temperature range
of the Capella components. We have therefore applied this offset,
obtaining corrected photometric estimates of $T_{\rm eff,A} = 5025 \pm
110$\,K and $T_{\rm eff,B} = 5765 \pm 120$\,K. The uncertainties
include a contribution of 100\,K added in quadrature to the
photometric and calibration errors, to be conservative. Finally, a
fourth temperature determination was reported in the previous section
from the disentangled ELODIE spectra, giving $T_{\rm eff,A} = 4980 \pm
80$\,K and $T_{\rm eff,B} = 5750 \pm 110$\,K. The weighted average of
the four values for each component is $T_{\rm eff,A} = 4970 \pm 50$\,K
and $T_{\rm eff,B} = 5730 \pm 60$\,K, in which the uncertainties
account not only for the individual weights but also for the scatter
of the measurements, and are believed to be realistic. These averages
are 50\,K hotter than in T09.
The available determinations of $v \sin i$ for both components were
summarized in our previous work (T09, Table~14). Our present
measurements from the ELODIE spectra are consistent with those of
others, as well as with the measurements reported by
\cite{Fuhrmann:11}, which are $v_{\rm A} \sin i = 3.5 \pm
0.8$\,\kms\ and $v_{\rm B}\sin i = 35.4 \pm 3.2$\,\kms. The weighted
averages of all independent determinations (5 for the primary, 10 for
the secondary) that have taken account of macroturbulence broadening,
especially for the primary component, are $v_{\rm A}\sin i = 4.1 \pm
0.4$\,\kms\ and $v_{\rm B}\sin i = 35.0 \pm 0.5$\,\kms, which we adopt
for the remainder of the paper.
The masses, radii, temperatures, and other derived properties are
summarized in Table~\ref{tab:dimensions}. Note that the bolometric
luminosities are independent of the temperatures and radii, and are
based on the apparent magnitudes, the orbital parallax, and bolometric
corrections from \cite{Flower:96}, as in T09. If we instead compute
them from $T_{\rm eff}$ and $R$, the results are consistent, but have
larger formal uncertainties: $L_{\rm A} = 78.7 \pm 8.1$\,$L_{\sun}$
and $L_{\rm B} = 75.4 \pm 6.4$\,$L_{\sun}$.
\section{Comparison with stellar evolution models}
\label{sec:stellarevolution}
Up until now the ability of stellar evolution models to match all of
the global properties of both components of Capella simultaneously at
a single age has not been entirely satisfactory, likely at least in
part because there are so many observational constraints
available. This has made it difficult to establish the evolutionary
status of the primary star unambiguously, although it has widely been
thought to be a core helium-burning (clump) star, based on timescale
arguments (see T09). The significantly different (and more precise)
masses obtained above, and evidence that the chemical composition is
rather different from that previously assumed, motivate us to revisit
the comparison with stellar evolution models here.
An initial test was made using the PARSEC isochrones of
\cite{Bressan:12}.\footnote{http://stev.oapd.inaf.it/cmd~.} These
models adopt the solar distribution of heavy elements from the
compilation by \cite{Grevesse:98}, with adjustments to some elements
following \cite{Caffau:11} such that the solar photospheric
metallicity is $Z_{\sun} = 0.01524$. On this scale the measured
abundance of Capella (Sect.~\ref{sec:abundances}) corresponds
approximately to $Z = 0.0133$. The helium abundance $Y$ follows an
adopted enrichment law with a slope $\Delta Y/\Delta Z = 1.78$, which
results in a value for Capella of $Y = 0.272$. Convection is treated
in the standard mixing length theory approximation. The calibration to
the Sun leads to a mixing length parameter of $\alpha_{\rm MLT} =
1.74$, which is held fixed in these models. Convective core
overshooting is also included, with an efficiency parameter of
$\Lambda_{\rm c} = 0.5$ for Capella, representing the mean free path
of convective bubbles across the border of the convective region,
expressed in units of the pressure scale height $H_{\rm p}$. This is
roughly equivalent to $\alpha_{\rm ov} = 0.25$ pressure scale heights
above the convective boundary in the more commonly used formulation of
this phenomenon (see below). Mass loss from stellar winds is not
considered in these models for the mass range of interest for Capella,
nor is rotation.
With the chemical composition and convective parameters fixed as
described above, we searched for the common age giving the best fit to
the masses, radii, temperatures, and luminosities of both stars using
the $\chi^2$ statistic
\begin{displaymath}
\chi^2 = \sum\left(\left[\frac{\Delta M}{\sigma_M}\right]^2 +
\left[\frac{\Delta T_{\rm eff}}{\sigma_{T_{\rm eff}}}\right]^2 +
\left[\frac{\Delta L}{\sigma_L}\right]^2 +
\left[\frac{\Delta R}{\sigma_R}\right]^2\right)
\end{displaymath}
as the figure of merit, where the sum is over both stars and the
$\Delta$ quantities represent the difference between the predicted and
measured properties. A reasonably good fit was obtained for an age of
622 Myr, matching the properties of the stars within about 1.4 times
their uncertainties, with the largest discrepancy being in the primary
temperature. We note, however, that the mass ratio of Capella is known
much more precisely than the individual masses, with $\sigma_q \approx
0.06$\% compared to mass errors $\sigma_M$ of 0.29\% and 0.27\%. The
masses in the above fit were allowed to vary independently, and as a
result the best-fit mass ratio differs from the measured value by
about 3$\sigma$. We therefore repeated the fit constraining $q$ to be
near its measured value by using a corresponding penalty term $[\Delta
q/\sigma_q]^2$ in $\chi^2$ instead of the secondary mass term. We
obtained a solution of similar quality (all properties reproduced
within 1.4$\sigma$) and nearly the same age of 625 Myr. This fit is
illustrated in Figure~\ref{fig:girardi}, and it places the primary
star in the core helium-burning phase (clump).
\begin{figure}
\epsscale{1.15}
\plotone{fig04.eps}
\figcaption[]{Comparison of the observed properties of Capella ($M$,
$R$, $T_{\rm eff}$, $L$, $q$) with a PARSEC isochrone from the model
series by \cite{Bressan:12} in the $L$--$T_{\rm eff}$ and
$R$--$T_{\rm eff}$ planes. Metallicity and age are as indicated in
the lower panel. The insets show enlargements around the primary
star, with a short line segment connecting the measured location in
each plane with the best-fit position along the isochrone.
Corresponding insets for the secondary are not shown as the match to
its properties is better.\label{fig:girardi}}
\end{figure}
Aside from the global structural properties of the stars considered
above, several chemical diagnostics including the lithium abundance,
the C/N ratios, and the isotopic carbon abundance ratio
$^{12}$C/$^{13}$C are available for Capella that are not normally
tabulated with published stellar evolution models, but that are
nevertheless interesting to compare with theoretical predictions. To
that end, we have performed an additional test against a second set of
models by \cite{Claret:04}, occasionally referred to below as the
Granada models.
These models adopt also the solar abundance distribution by
\cite{Grevesse:98}, with some adjustments such that the solar
metallicity becomes $Z_{\sun} = 0.0189$. The abundance of Capella then
corresponds approximately to $Z = 0.0172$, which we held fixed. The
enrichment law $\Delta Y/\Delta Z = 2.0$ typically used in these
models results in a helium abundance for Capella of $Y = 0.274$,
similar to that used above. The solar-calibrated value of the mixing
length parameter is $\alpha_{\rm MLT} = 1.68$, and convective core
overshooting $\alpha_{\rm ov}$ is parametrized such that the mean free
path above the convective boundary is $d_{\rm ov} = \alpha_{\rm ov}
H_{\rm p}$. Rotation was initially not included in our tests.
A grid of Granada evolutionary tracks was computed for the measured
masses and a range of convective parameters for each component of
Capella, as in principle there is no reason to expect stars in such
different evolutionary states to have the same convective properties.
We varied $\alpha_{\rm MLT}$ between values of 1.0 and 2.2 in steps of
0.1, and $\alpha_{\rm ov}$ over the range 0.15--0.40, with a step of
0.05. Mass loss was included in these calculations following the
prescription by \cite{Reimers:75}. Preliminary tests indicated a
minimal loss of mass for both stars up to the present age, but we
nevertheless incremented the initial values slightly by
0.005\,$M_{\sun}$ and 0.002\,$M_{\sun}$, respectively, so as to
reproduce the measured masses exactly at the best-fit age. An
excellent fit to the radii, temperatures, and luminosities of both
stars was found for mixing length parameters of 1.80 for the primary
and 1.50 for the secondary, and convective core overshooting
parameters of 0.35 and 0.30, respectively, with estimated
uncertainties in each of these of about 0.05. Deviations in $R$,
$T_{\rm eff}$, and $L$ from the measured values are all smaller than
0.4$\sigma$. The best-fit age we obtained, about 649 Myr, was
constrained to be the same for the two stars. We illustrate this
solution in Figures~\ref{fig:claret1} and \ref{fig:claret2} for the
$L$--$T_{\rm eff}$ and $R$--$T_{\rm eff}$ diagrams, respectively.
We point out that the age in this best-fit solution is driven entirely
by the properties of the secondary, specifically, by its effective
temperature. This is because that star is in such a rapid phase of
evolution that the temperature is predicted to change drastically (by
many times the observational uncertainty) in just 1~Myr.
Figures~\ref{fig:claret1} and \ref{fig:claret2} show, for example,
that between the ages of 648 and 649 Myr the temperature of the
secondary decreases by 1130\,K, cooling by a further 660\,K over the
next million years. The primary, on the other hand, stays at
essentially the same temperature over this time, and only changes its
radius and luminosity, but at a much slower pace (see top insets in
the figures). Consequently, it does not constrain the age nearly as
much. Because of the rapid evolution of the secondary, the formal
uncertainty in the age that comes from its temperature error is
negligible. A more meaningful uncertainty may be obtained by varying
its mass within allowed limits, which results in an age range of
approximately $\pm 5$~Myr. This does not include possible systematic
errors having to do with the physics in the models. As found above
from the PARSEC models, the primary star is seen to be located in the
clump, on the hot side of the giant loop (end of core helium-burning
phase), where the radius and luminosity are increasing with time.
A test with Granada models that include rotation for both components
gave a very similar fit, with an age of 655 Myr that is only
marginally older than before.
\begin{figure}[t]
\epsscale{1.15}
\plotone{fig05.eps}
\figcaption[]{Comparison of the observed properties of Capella (dots
and error boxes) with the Granada models by \cite{Claret:04} in the
$L$--$T_{\rm eff}$ diagram. The evolutionary tracks shown are for
the measured masses (incremented by 0.005\,$M_{\sun}$ for the
primary and 0.002\,$M_{\sun}$ for the secondary, to account for mass
loss; see text) and the measured metallicity ($Z = 0.0172$ for these
models). Reference ages are marked along the secondary track, and
the insets show enlargements around the position of each star, with
a short line segment connecting the observations to the predicted
positions.\label{fig:claret1}}
\end{figure}
\begin{figure}
\epsscale{1.15}
\plotone{fig06.eps}
\figcaption[]{Similar to Figure~\ref{fig:claret1} for the $R$--$T_{\rm
eff}$ plane.\label{fig:claret2}}
\end{figure}
The evolution of the surface chemistry in giants is the result of
mixing and is directly related to the depth of the convection zone,
which changes drastically as the stars approach the so-called first
dredge-up, during their initial ascent of the giant branch. Other
changes can occur later. The first dredge-up event is illustrated in
Figure~\ref{fig:dredgeup}, which shows predictions from the Granada
models for the abundance of lithium, the $^{12}$C/$^{13}$C carbon
isotope ratio, and the C/N ratio as a function of time. Also shown for
reference are the changes in the location of the bottom of the
convection zone for each star (lower panel). The measurements of
these key chemical diagnostics are represented with dots at the
best-fit age of 649 Myr. Generally there is good agreement between
theory and observation, except for the C/N ratios that deviate the
most.\footnote{The predicted \emph{difference} between the ratios,
however, is in better agreement with the measured difference (to
within 1.7$\sigma$).} We note, however, that the C/N measurements
rely on emission fluxes from spectral lines in the lower transition
layers between the stellar chromosphere and the corona (specifically,
\ion{C}{4}~$\lambda$1550.8 and \ion{N}{5}~$\lambda$1238.8; see T09),
so they may not strictly represent the abundances in the photosphere
\citep[despite some evidence that they do; see,
e.g.,][]{Bohm-Vitense:92}. The convection zone in the secondary
star is seen to have just begun deepening, and should reach maximum
depth approximately 7 Myr from now, according to these models.
\begin{figure}
\epsscale{1.15}
\plotone{fig07.eps}
\figcaption[]{Evolution of chemical diagnostics for Capella occurring
near the first dredge-up, as predicted by the models of
\cite{Claret:04}. Solid lines represent the primary, and dashed
lines the secondary. The best-fit age of about 649 Myr is marked by
the vertical dotted lines. The lower panel shows the depth of the
bottom of the convective zone as a function of time, in units of the
stellar radius. The dots represent the measurements, with error bars
that are barely visible on this scale. Measured and predicted values
are listed near each point, with the deviations listed in
parenthesis next to the predictions, in units of the observational
errors.\label{fig:dredgeup}}
\end{figure}
A final test was performed against stellar evolution tracks computed
using the Modules for Experiments in Stellar Astrophysics \citep[MESA,
revision 7385; see][]{Paxton:11,
Paxton:13}.\footnote{http://mesa.sourceforge.net/~.} These models
use the scaled solar abundances of \cite{Asplund:09}, according to
which $Z_{\sun} = 0.0134$. The measured composition of Capella
corresponds to a metal mass fraction of about $Z = 0.012$, and the
helium abundance follows from an adopted $\Delta Y/\Delta Z = 1.67$,
and is $Y = 0.270$. Mass loss is again computed according to the
\cite{Reimers:75} prescription, in this case with an efficiency
parameter (multiplicative scale factor) of $\eta_{\rm R} = 0.2$. For
this paper we used the `{\tt grey\_and\_kap}' surface boundary
condition \citep{Paxton:13}, with opacities and equation of state as
discussed extensively by \citet{Paxton:11, Paxton:13}.
Overshoot mixing across convective boundaries is treated slightly
differently than in the models considered previously. MESA uses the
exponential decay formalism of \citet{Herwig:97}, in which the product
of the free parameter $f_{\rm ov}$ and the local pressure scale height
provide the scale length over which the diffusion coefficient decays
from its value in the convective region. Although MESA allows for the
free parameter to take on different values depending on the nuclear
burning present in each convective zone, we have opted to use the same
$f_{\rm ov}$ value for H- and He-burning regions. For reference, with
the choices listed above, we obtained a solar-calibrated mixing length
parameter $\alpha_{\rm MLT} = 1.84$, and \citet{Herwig:97} suggest a
value of $f_{\rm ov} \simeq 0.02$ for overshoot mixing above the
H-burning core.
A grid of MESA evolutionary tracks with the specified composition was
computed for each star over broad ranges in $\alpha_{\rm MLT}$ (1.70
to 2.00, in steps of 0.05) and $f_{\rm ov}$ (0.00--0.04, in steps of
0.01). The models were evolved from the fully-convective pre-main
sequence to the end of core He burning. Due to the inclusion of mass
loss, we increased the initial masses of the stars, as we did before
for the Granada models, by 0.0048\,$M_{\sun}$ and 0.0027\,$M_{\sun}$
in this case such that the tracks reach the measured masses at their
respective present locations in the H-R diagram. An excellent
fit to the properties of both components was achieved at a common age
of 588.5~Myr, with all residuals being smaller than 1.2$\sigma$. The
optimal convective parameters were found to be $\alpha_{\rm MLT} =
1.85$ for the primary and $\alpha_{\rm MLT} = 1.80$ for the secondary,
with $f_{\rm ov} = 0.02$ for both stars. The matches in the
$L$--$T_{\rm eff}$ and $R$--$T_{\rm eff}$ diagrams are illustrated in
Figures~\ref{fig:mesa1} and \ref{fig:mesa2}. Once again the models
place the primary at the end of the core helium-burning phase. The age
is somewhat younger than obtained from the Granada models (a
$\sim$10\% difference), while the age found earlier from the PARSEC
models is intermediate between these two. These age differences
correlate strongly and have to do mostly with the $Z$ value used in
the calculations for each model. The differences in $Z$ at a fixed
(measured) [Fe/H] value are in turn a consequence of the different
heavy-element mixtures adopted for the Sun in each case.
\begin{figure}
\epsscale{1.15}
\plotone{fig08.eps}
\figcaption[]{Similar to Figure~\ref{fig:claret1} for the MESA
models. The measured metallicity corresponds to $Z = 0.0120$ in these
models.\label{fig:mesa1}}
\end{figure}
\begin{figure}
\epsscale{1.15}
\plotone{fig09.eps}
\figcaption[]{Similar to Figure~\ref{fig:mesa1} for the $R$--$T_{\rm
eff}$ plane.\label{fig:mesa2}}
\end{figure}
The predictions from the MESA models regarding the evolution of the
surface chemistry of the stars are similar to those from the Granada
models. In particular, the predicted $^{12}$C/$^{13}$C ratio for the
primary at the age of 588.5~Myr is 20.7, and the expected C/N ratios
for the primary and secondary are 0.77 and 3.98, respectively, both
being somewhat higher than measured.
\section{Tidal evolution}
\label{sec:tidalevolution}
The considerable amount of information available for Capella offers a
valuable opportunity to test our understanding of tidal theory in
binary stars, in ways those models have not often been challenged
before. Such calculations are capable of making detailed predictions
about the evolution of the size and shape of the orbit, as well as the
rotational properties of the individual components including their
spin rate, $\Omega_{\rm rot} = 2\pi/P_{\rm rot}$, and the spin-orbit
angle $\phi$ (angle between the spin axis and the total angular
momentum vector of the orbit, sometimes referred to as `obliquity').
In addition to the known orbital elements of Capella, estimates are
available also of the rotation periods of both stars, $P_{\rm rot}$
(see Table~\ref{tab:dimensions}, and T09), and of their projected
rotational velocities, $v \sin i$. Our earlier study of the binary
examined its tidal evolution considering the turbulent dissipation and
radiative damping mechanisms by \citet[and references
therein]{Zahn:92}, as well as the hydrodynamical mechanism of
\citet[and references therein]{Tassoul:97}. These theoretical
formulations involve a number of assumptions and simplifications
discussed by \cite{Zahn:77} and \cite{Hut:81}. In particular, the
equations are linearized around the equilibrium state, and are
strictly valid only for relatively small eccentricities and
near-synchronous rotation, as well as relatively small obliquities.
In this work we have chosen to use the more general equations of tidal
evolution by \cite{Hut:81}, which are valid for arbitrary
eccentricities and rotation rates, although they are still restricted
to relatively low mutual inclination angles $\phi$. We used a
fourth-order Runge-Kutta algorithm to integrate the six coupled
differential equations describing the time-dependent changes in the
orbital semimajor axis ($da/dt$), eccentricity ($de/dt$), angular
rotational velocities of both stars ($d\Omega_{\rm rot,A}/dt$,
$d\Omega_{\rm rot,B}/dt$), and their spin-orbit angles ($d\phi_{\rm
A}/dt$, $d\phi_{\rm B}/dt$). In what follows we normalize the
rotation rates to the mean orbital rate $\Omega_{\rm orb}$, for
convenience. The relevant stellar properties that also evolve with
time, such as the radius, were taken at each integration step directly
from the best-fit evolutionary tracks of \cite{Claret:04} discussed in
the previous section. The turbulent dissipation timescale for the
stellar phases with convective envelopes (later stages for Capella)
was adopted from \cite{Zahn:77}, whereas the timescale for earlier
phases with radiative envelopes follows \cite{Claret:97}. The initial
conditions, which are unknown, were set by trial and error to match
the measured values of the orbital period, the eccentricity, and the
spin rates at the current age as closely as possible.
The outcome of these calculations is illustrated in the top four
panels of Figure~\ref{fig:tidal}. Setting the initial values to $P_0
= 220$~days, $e_0 = 0.70$, $(\Omega_{\rm rot,A}/\Omega_{\rm orb})_0 =
320$, and $(\Omega_{\rm rot,B}/\Omega_{\rm orb})_0 = 260$ leads to
evolved properties that are very close to those observed at the
present age. In particular, the observed super-synchronous rotation
rate of the secondary ($\Omega_{\rm rot,B}/\Omega_{\rm orb} = P_{\rm
orb}/P_{\rm rot,B} \approx 12$) is well reproduced. Qualitatively
the largest difference is perhaps in the orbital eccentricity, which
theory predicts should strictly have fallen to zero some 35 Myr ago,
driven almost exclusively by the evolution of the primary star.
Quantitatively, however, the difference in $e$ between theory and
observation is small, as the measured value (if real) is only $e =
0.00089 \pm 0.00011$. While the agreement reached in these four
observed properties is not entirely unexpected because we have allowed
for four free parameters (the initial values), we note that the good
fit was only possible by increasing the nominal efficiency of the
tidal dissipation by more than an order of magnitude. Without this
increase, we find that the predicted rotational velocities of the
stars near the zero-age main-sequence are unreasonably low ($v \sin i
\sim 20$\,\kms) for early A-type stars, such as the Capella components
would have been. In order to yield more reasonable projected
rotational velocities in excess of 100\,\kms\ we had to increase the
efficiency of the tidal mechanisms by a factor of $\sim$40. A similar
shortcoming in the efficiency of theory was found earlier by
\cite{Claret:97}, in their analysis of tidal synchronization and
circularization of a sample of detached eclipsing binaries.
\begin{figure}
\epsscale{1.15}
\plotone{fig10.eps}
\figcaption[]{Predicted tidal evolution for Capella according to
\cite{Hut:81}, compared with the observations (filled circles; error
bars are smaller than the point size). (a) Eccentricity
evolution. The vertical dotted line in this and the other panels
marks the current age of the binary (649 Myr) according to the
models by \cite{Claret:04} described in the text. (b) Evolution of
the orbital period. (c) and (d) Evolution of the spin rate of the
primary and secondary, normalized for convenience in terms of the
orbital angular velocity. The dot-dashed line represents the
evolution of the pseudo-synchronous rotation rate, computed
following Eq.(42) of \cite{Hut:81}. (e) Evolution of the fractional
gyration radius of each star, for reference (solid line for primary,
dashed for secondary). (f) Evolution of the spin-orbit inclination
angle for each star (lines as in previous panel). (g) and (h)
Predicted projected rotational velocities of the two
stars.\label{fig:tidal}}
\end{figure}
Figure~\ref{fig:tidal}f displays the evolution of the spin-orbit
angles for the two stars near the present age, where the integrations
have been performed with arbitrary initial values of $\phi_0 = 0.4$
radians (about 23\arcdeg) for both stars, as we have no direct handle
on those angles. Tests with other (non-zero) values show
that the behavior is qualitatively always the same: the primary's spin
axis aligns itself with the orbital axis (i.e., $\phi$ reaches zero)
well before the current age, whereas the secondary remains formally
misaligned. The alignment of the primary happens at very nearly the
same time that its rotation becomes synchronous and that the orbit
circularizes. The spin-orbit angle of the secondary quickly shrinks to
zero shortly after the present age (4--5 Myr later), at which time it
also synchronizes its rotation with the mean orbital motion. In both
cases the changes are the result of significant structural adjustments
in the stars, such as a sharp increase in the radius of gyration,
$r_{\rm gyr}$, related to the moment of inertia through $I = M (r_{\rm
gyr} R)^2$ (see Figure~\ref{fig:tidal}e).
While we cannot measure the present-day values of $\phi$, it is
possible to gain indirect knowledge about these angles using our
spectroscopic estimates of $v \sin i$ along with the measurements of
$P_{\rm rot}$ and the radii of both stars. These quantities are
trivially related by
\begin{equation}
\label{eq:vsini}
v \sin i = \frac{2\pi R}{P_{\rm rot}} \sin i_{\rm rot}\,,
\end{equation}
where the inclination angle on the left-hand side (inaccessible to
direct observation) is strictly that of the stellar spin axis relative
to the line of sight ($i_{\rm rot}$), and also appears on the right.
We may thus solve for the $\sin i_{\rm rot}$ term on the right-hand
side. Figure~\ref{fig:sini} displays the distributions of $\sin
i_{\rm rot}$ derived from the propagation of all observational errors
in a Monte Carlo fashion. For reference we also show the sine of the
orbital inclination angle relative to the line of sight ($i_{\rm
orb}$), which is directly measurable from the astrometric
observations (Table~\ref{tab:dimensions}). The close agreement between
$\sin i_{\rm rot}$ and $\sin i_{\rm orb}$ for both stars is highly
suggestive that the spin axes may actually be parallel to the orbital
axis in space, and this in turn would imply $\phi = 0$. We cannot
rule out a difference in quadrants such that the spin axes are tilted
with respect to the axis of the orbit while still maintaining the same
sine value (e.g., $i_{\rm rot} = 180\arcdeg - i_{\rm orb}$), but such
a coincidence for both stars seems rather unlikely.\footnote{Barring
this type of situation, conversion of the distributions in
Figure~\ref{fig:sini} to inclination angles yields $i_{\rm rot,A} =
135.3^{+6.1}_{-6.8}$ deg and $i_{\rm rot,B} = 138.2^{+2.3}_{-2.4}$
deg, compared to the orbital value of $i_{\rm orb} = 137.156 \pm
0.046$ deg.} This indirect empirical evidence that the obliquity may
currently be zero for both stars (which would hardly happen by chance) appears to point to a discrepancy
with the prediction from tidal theory for the secondary component
(Figure~\ref{fig:tidal}f), whereas for the primary there is good
agreement.
\begin{figure}
\epsscale{1.15}
\plotone{fig11.eps}
\figcaption[]{The histograms represent the empirical distributions of
$\sin i_{\rm rot}$ derived from the measurement of $v \sin i$,
$P_{\rm rot}$, and the radius of each star as given in
Table~\ref{tab:dimensions}. The dashed lines mark the value of $\sin
i_{\rm orb}$ determined from our orbital fit. The good agreement
strongly suggests that the spin axes of both stars may in fact be
parallel to the orbital axis.\label{fig:sini}}
\end{figure}
It is possible that part of the reason for this difference lies in the
small-angle approximation implicit in the tidal differential equations
of \cite{Hut:81} for the angle $\phi$. However, we note also that the
secondary of Capella happens to be in such a rapid state of evolution
that theoretical predictions for the spin-orbit angle are extremely
sensitive to other details, including those related to structural
changes in the stars happening at this stage. Those changes may actually have
a larger impact than the dissipation processes themselves (though the latter must
also matter). In particular, changes in both the spin rate and the
spin-orbit angle are strongly driven in part by the sudden change in
the moment of inertia (or equivalently, the gyration radius)
illustrated in Figure~\ref{fig:tidal}e. An additional complication is
the large ad-hoc increase in the efficiency of the tidal mechanism
that was required to match other observations, as mentioned earlier.
Thus, a definitive assessment of the accuracy of tidal theory related
to its other approximations is difficult in this case.
An alternate way of comparing observations with the predictions from
tidal and evolution models combined is by examining the evolution of
$v \sin i$, which is a spectroscopically measured quantity. It is a
function of the stellar radius, the spin rate, and the inclination
angle of the rotation axis to the line of sight, $i_{\rm rot}$
(Eq.[\ref{eq:vsini}]). The latter angle is not directly predicted by
theory, but is related to other angles by
\begin{equation}
\label{eq:lambda}
\cos\phi = \cos i_{\rm orb} \cos i_{\rm rot} + \sin i_{\rm orb} \sin i_{\rm rot} \cos\lambda\,,
\end{equation}
in which the obliquity $\phi$ can be predicted, $i_{\rm orb}$ is
known, and $\lambda$ is the angle between the sky projections of the
spin axis and the orbital axis (`projected obliquity'). The angle
$\lambda$ changes with time and is directly measurable in eclipsing
systems by observing the Rossiter-McLaughlin effect. As Capella does
not eclipse we have no knowledge of this angle in this case, except
for the weak condition that it represents a lower limit to the
three-dimensional angle $\phi$ \citep[see, e.g.,][]{Fabrycky:09}.
Eq.[\ref{eq:lambda}] may be solved for the quantity $\sin i_{\rm rot}$
that we need in order to compute $v \sin i$ with eq.[\ref{eq:vsini}],
resulting in the quadratic equation
\begin{equation*}
A \sin^2 i_{\rm rot} + B \sin i_{\rm rot} + C = 0
\end{equation*}
where
\begin{align*}
A &= \cos^2 i_{\rm orb} + \sin^2 i_{\rm orb} \cos^2\lambda \\
B &= -2 \cos\phi\, \sin i_{\rm orb} \cos\lambda \\
C &= \cos^2\phi - \cos^2 i_{\rm orb}\,.
\end{align*}
For the primary star the prediction that $\phi = 0$ near the current
epoch means that $\lambda$ is also zero. Using this value in the
equations above to compute the expected evolution of the projected
rotational velocity leads to the trend shown in
Figure~\ref{fig:tidal}g, where we refer to $v \sin i$ more properly as
$v_{\rm A} \sin i_{\rm rot,A}$. Values at times when $\phi \neq 0$
will be somewhat less accurate because the projected obliquity
$\lambda$ may also be different from zero. The predicted value of
4.0~\kms\ at the current age is in excellent agreement with the
measurement ($4.1 \pm 0.4$~\kms). The corresponding evolution of
$v_{\rm B} \sin i_{\rm rot,B}$ is seen in Figure~\ref{fig:tidal}h. In
this case the predicted value at the current age does not match the
measurement, and the discrepancy is in part a reflection of the
evolution of $\phi$ that was discussed above, but also has to do with
the very rapid changes in the structure of the star ($R$, $r_{\rm
gyr}$) at the present time. Tests in which we changed $\lambda$
within reason yielded very similar results, and cannot explain the
difference.
Finally, it is of interest to verify that the components of Capella
have always been detached, as any significant mass transfer earlier in
their lives (e.g., through Roche lobe overflow in the primary) would
invalidate our comparison with stellar evolution models, which are
designed for normal (unperturbed) stars. The fraction of its Roche
lobe filled by the primary component depends on its size and also on
the size of the orbit, which changes due to tidal forces
(Figure~\ref{fig:tidal}b). The same Granada evolutionary track used
above indicates the star attained a maximum radius at the tip of the
giant branch of about 38\,$R_{\sun}$ at an age of 617~Myr (MESA models
predict a similar maximum size of 36\,$R_{\sun}$), and the filling
factor ($R/R_{\rm Roche}$) at the time was approximately 0.63. This
indicates that mass transfer through Roche-lobe overflow has not taken
place in Capella.
\section{Discussion and concluding remarks}
\label{sec:discussion}
For the first time current stellar evolution models are shown here to
provide a satisfactory fit to the observed global properties of both
components of Capella at a single age, for a chemical composition
equal to that measured. The comparison confirms the long-held but
largely unsubstantiated belief that the primary is a clump star, and
more precisely, it suggests Capella~A is near the end of its core
He-burning phase, as indicated consistently by three different sets of
models. The principal factors that have allowed the better match are
an improvement in the accuracy of the absolute masses, made possible
by the high-quality radial-velocity measurements of \cite{Weber:11},
and the first robust determination of the metallicity of Capella
derived here. Our detailed chemical analysis of the disentangled
spectra of the components yields essentially the same near-solar
composition for the two stars, which is rather different from the
sub-solar abundance the binary was previously assumed to have in our
earlier T09 study. All measured properties (masses, radii,
temperatures, and independently derived luminosities) are now
simultaneously in agreement with the models to within
0.4--1.4$\sigma$, depending on the model. This result strengthens our
confidence in evolutionary calculations for evolved stars.
Chemical indicators of evolution that differ greatly between the
components, such as the $^{12}$C/$^{13}$C ratio, the lithium
abundance, and the C/N ratio, are also broadly in agreement with
theoretical predictions, though with somewhat larger differences that
may be due to either shortcomings in the mixing prescriptions in the
models, or observational errors in these delicate measurements. Rarely
has it been possible to perform this type of test involving key
chemical diagnostics for a binary with two evolved components having
well-determined absolute dimensions ($M$, $R$, etc.), such as Capella.
On the other hand, the performance of tidal theory as described by the
tidal evolution equations of \cite{Hut:81} is not as good. As was
noticed previously by \cite{Claret:97}, we again find that the
efficiency of the tidal mechanisms involved seems to be too low by a
factor of $\sim$40, if predictions are to be consistent with the large
rotational velocities Capella A and B presumably had earlier in their
lives as A-type stars ($v \sin i > 100$\,\kms), and at the same time
the much lower velocities they now have as evolved stars. Application
of an ad-hoc adjustment of this magnitude brings agreement, although
theory still forecasts a significant misalignment between the spin
axis of the secondary and the orbital axis at the present age, whereas
empirical evidence based on the direct measurement of the star's
radius, rotation period, and $v \sin i$ seems to favor an obliquity
consistent with zero. Model predictions about the synchronous
rotation of the primary and its spin-orbit alignment do agree with
observational clues, as does the supersynchronous rotation of the
secondary. A small additional disagreement in that the orbit seems to
be very slightly eccentric even though it is expected to be perfectly
circular may or may not be significant, as the difference is small.
Finally, another important consequence of our revised chemical
composition for Capella concerns the relationship with the coronal
abundances, which have been measured by many authors as summarized by
T09. Coronal abundances in the Sun are known to depend on the first
ionization potential (FIP), in such a way that low-FIP elements (less
than 10 eV) are overabundant compared to those with higher FIP
\citep[see, e.g.,][]{Feldman:02}. T09 showed this to be the case for
Capella as well. Other stars display the opposite effect \citep[see,
e.g.,][]{Brinkman:01, Laming:04}. What is less clear from evidence
in other stars is whether the low-FIP elements are enhanced relative
to the photospheric abundance, or whether it is the high-FIP elements
that are depleted compared to the photosphere. T09 adopted a sub-solar
photospheric composition for Capella of ${\rm [m/H]} = -0.34 \pm
0.07$, based on the work of \cite{McWilliam:90}, and since this agreed
with the measured coronal abundances of the high-FIP elements (see
their Figure~18), they concluded it was the low-FIP elements that were
enhanced. Interestingly, our revised photospheric abundance of ${\rm
[Fe/H]} = -0.04 \pm 0.06$ leads to precisely the opposite
conclusion: now it is the low-FIP elements that agree with the
photosphere, and therefore the high-FIP elements are depleted. A
recent study by \cite{Peretz:15} of the coronae of stars with
super-solar photospheric abundances found the same effect shown by
Capella in $\alpha$~Cen A and B, which have somewhat similar
temperatures, although these are dwarfs so it is unclear how
significant this may be. In any case, Capella now represents a robust
point of reference for such studies in evolved and active stars.
\acknowledgements
We thank Brian Mason (U.S.\ Naval Observatory) for providing the
historical positional measurements of the Capella HL system, and the
anonymous referee for helpful comments. The research has made use of
the SIMBAD database, operated at CDS, Strasbourg, France, of NASA's
Astrophysics Data System Abstract Service, of the Washington Double
Star Catalog maintained at the U.S.\ Naval Observatory, and of data
products from the Two Micron All Sky Survey (2MASS), which is a joint
project of the University of Massachusetts and the Infrared Processing
and Analysis Center/California Institute of Technology, funded by NASA
and the NSF.
|
1,108,101,565,463 | arxiv | \section{Introduction}
Clusters serve as laboratories for investigating the dependence of
galaxy morphology on the density of the environment. \citet{dre80}
found that the fraction of early type galaxies increases with local
density of galaxies. He also found that around 80\% of galaxies in
nearby clusters are of the early type. In recent years, deep, high
resolution imaging by the \textit{Hubble Space Telescope (HST)} has
helped to extend the study of galaxy morphology as a function of the
environment to $z \sim 1$. The observed fraction of different
morphological types can be explained in terms of both 'nature' and
'nurture' scenarios. The former argues that a galaxy's morphological
type is determined by initial conditions at formation
\citep[e.g. ][]{egg62}, while the latter depends on the influence of the
environment and of secular evolution for determining the final
morphological type \citep[e.g. ][]{too77}. Numerical simulations play an
important role in investigating the relative importance of the two
scenarios under different physical conditions.
Following Dressler's pioneering work, many observational studies have
been done to measure and understand the morphology density relation
(MDR), and the dependence of morphological type on distance of the
galaxy from the cluster centre
\citep{dre97,got03,smi05,pos05,hol07,tre03,cap07}. \citet{smi05}
found that the early type fraction is constant in low density
environments over the last 10 Gyr, but there is significant evolution
in this fraction in higher density regions. According to
\citet{smi05}, this suggests that most of the ellipticals in clusters
formed at high redshift, and the increase in the fraction of early
type galaxies is because of the physical processes in dense regions
which transform disk galaxies with ongoing star formation to early
types. Dissipationless merging of cluster galaxies may also be
responsible for this increase.
There is increasing evidence to show that massive ellipticals formed
by dissipationless (dry) merger of two or more systems. Such
ellipicals must have formed at later times than their low luminosity
counterparts \citep{luc06}. \citet{dok05} analysed tidal debris of
elliptical galaxies and concluded that $\sim70\%$ of the bulge
dominated galaxies have experienced a merger. The analysis of nearby
bulge dominated galaxies has shown that the gas to stellar mass ratio
is very small and these mergers are mostly 'dry'. Using a
semi-analytic model for galaxy formation, \citet{naa06} found that
both the photometric and kinematic properties of massive elliptical
galaxies are in agreement with the scenario where massive elliptical
galaxies are produced by mergers of lower mass ellipticals. They
suggested that the merger of two spiral galaxies alone cannot
reproduce the observed properties, and that the large remnant mass ($>
6\times 10^{11} M_\odot$) implies that they must have undergone
elliptical - elliptical mergers. They found that this process is
independent of the environment and redshift, which means that dry
mergers can occur at low redshift as well. By analysing spirals from
cluster and group environments at intermediate redshift ($z\sim 0.5$),
\citet{mor07} found that the Tully-Fisher relation shows larger
scatter for cluster spirals than for those in the field. They also
found that the central surface mass density of spirals in clusters is
small beyond the cluster virial radius, and argued that these
observations provide evidence for merger/harrasment.
Mergers are not common in clusters, as the velocity dispersion of
virialized clusters is large. So if ellipticals in clusters have
formed by mergers, that most likely happened during the early stage of
cluster collapse \citep{roo82}. There is some observational evidence
to support this idea; \citet{dok99} showed that there is a large
fraction of ongoing mergers in a cluster at $z = 0.83$. The
observation of the unvirialized cluster $\textrm{RX J}0848+4453$ at
$z=1.27$ revealed many ongoing dissipationless mergers of galaxies
\citep{dok01}. These observations go against the view of monolithic
collapse where all the ellipticals formed at the same time, at very
high redshift.
In this letter we report on the evolution of galaxies in the core
region of clusters, measured using bulge-disk decomposition. We show
that in the case of brightest cluster galaxies, the fraction of
galaxies with $B/T>0.4$ and $n>2.5$ evolved significantly over the
redshift range 0.31 to 0.83. Throughout the paper we use the standard
concordance cosmology with $\Omega_\Lambda = 0.73$, $\Omega_m = 0.27$
and $H_0 = 71$ km s$^{-1}$ Mpc$^{-1}$.
\section{Cluster sample and Decomposition Technique}
We have exclusively used archival Hubble Space Telescope (HST) data in
this work. All of the observations used by us were obtained with
either the ACS or WFPC2 cameras on-board the HST. We constructed our
sample of clusters by an extensive literature survey of HST
observations of moderate redshift clusters. The clusters we study
span the redshift range 0.31 to 0.837. The clusters were selected such
that they each had at least 15 spectroscopically confirmed cluster
members listed in the literature
\citep{cou98,sma97,dre99,wil99,hal04,dem05}. We restricted our study
only to those clusters whose second brightest member galaxy has an
absolute magnitude between -25.0 and -27 in rest-frame B-band. Here
the magnitudes are corrected for the cosmological surface brightness
dimming. This criterion allows us to restrict ourselves to clusters of
roughly comparable luminosity. Further, we only included clusters
with imaging data around the rest-frame $B$ filter i.e. in filters
where the central wavelength of the filter corresponded to a rest
frame wavelength in the range 350-550 nm. This has the advantage that
the k-correction for transformation from $R$ or $I$ filters in the
observer frame, to the $B$ filter in the rest frame, has only a weak
dependence on galaxy spectral type in the redshift range $0.3 \lesssim
z \lesssim 0.8$ \citep{boh07}. Incompleteness of the spectroscopy is a
potential problem; since our data are drawn from a heterogeneous set
of observations, it is not possible to correct for incompleteness in a
completely consistent way. However, for the five clusters in our
sample which are part of the ESO Distant Clusters Survey (EDisCS),
\citet{hal04} have shown that incompleteness does not introduce a
significant bias. The level of incompleteness is also relatively small
because they (like us) restrict themselves to the brighter cluster
members. Nine clusters satisfied our selection conditions at the time
of commencing this study; the basic data for these clusters are given
in Table \ref{clustab}.
The images we downloaded from the HST data archive were processed in a
standard way using the On-the-Fly Reprocessing (OTFR) pipeline at
STScI. The processed images were dark and bias subtracted and
flat-fielded by the OTFR pipeline. We combined them using the
Multidrizzle package \citep{koe02} to flag and remove cosmic rays and
to correct for geometric distortion and produced a single coadded
image. For some clusters, multiple disjoint pointings have been used;
in such cases we obtained multiple multidrizzled images. The
correlated pixel noise introduced by the drizzle process, was
corrected for, using the prescription of \citet{cas00}.
We computed the center of each cluster as the centroid of the
brightest cluster galaxy. We then selected all galaxies with a
spectroscopic redshift confirming their cluster membership and located
within a 1 Mpc projected distance from the cluster center. Our study
is therefore restricted to galaxies lying (mostly) within the core
region of the clusters. Note that for three clusters -- AC 114,
CL 0303+17, and 3C 295 -- the HST imaging is not complete for the 1 Mpc
projected distance.
Galaxies are known to undergo luminosity evolution, to account for which we
adopted the following simple scheme: $M_V(z) = M_V(z=0) - 0.8 z$, where $z$ is the redshift of the object and
$M_V(z=0) = -19.5$ \citep{pos05}. The magnitude cutoff in the observed HST filter was then calculated using
\begin{equation}
m_{lim} = M_V(z) + DM - (M_V - M_{HST}) + k_{HST}
\end{equation}
where DM is the distance modulus, $M_V - M_{HST}$ is the rest-frame
color between $V$ and the observed HST filter and $k_{HST}$ is the
k-correction in the observed HST filter. The k-correction for
each cluster was computed using the elliptical SED provided by \citet{pog97}.
All magnitudes are in the Vega system.
With all the above constraints, we obtained a sample of 379 galaxies
in nine clusters. We used the GALFIT \citep{pen02} program for 2D
bulge disk decomposition of the galaxy images. To fit the galaxies,
we have developed an automated pipeline (Vinu {et al.} \ 2010, in
preparation) which completely automates the fitting procedure and
organizes the results into a database, supplemented by a number of
diagnostic plots to detect anomalies. The basic steps in performing
the 2D decomposition are: (1) make a cut-out image for each galaxy,
including neighbouring galaxies. (2) estimate starting values of
fitting parameters using the parameters measured by Sextractor (3)
make masks to exclude stars, faint galaxies and other artifacts and
(4) Run GALFIT. Bright neighboring galaxies were fitted simultaneously
with the target galaxy. We modeled each galaxy as a linear sum of a
S\'ersic (for the bulge) and an exponential (for the disk)
component. The centers, position angles, ellipticities and central
surface brightnesses of bulge and disk were fitted simultaneously. The
S\'ersic index was left unconstrained. The estimation of galaxy
parameters, particularly the S\'ersic index, may be systematically affected
if the sky or PSF are incorrectly estimated. To
test for the effect of using an incorrect PSF, we ran the decomposition for
each galaxy using two stellar PSFs, one constructed using the nearest
star and the other using the second nearest star and compared the
fitted values of $B/T$ and $n$ for every galaxy. To test for incorrect
sky, we ran the decomposition in two modes: one in which the sky was
left as a free parameter and one in which it was fixed to the local
sky value as determined by Sextractor. In both tests, $\sim80$\%
of galaxies showed random changes in the extracted parameters at $< 10$\% level,
with no obvious systematics. Our simple tests are consistent with the extensive simulations of \citet{hau07} which showed that GALFIT estimates parameters accurately for HST data,
even at relatively shallow depths. As an additional precaution, we
examined the fit results, by eye, for every galaxy. The fit diagnostic
plots were used to evaluate the quality of the fit. We used the
reduced $\chi^2$ given by GALFIT to identify galaxies with large
residuals. A large residual in the central part of the galaxy may be
caused by improper estimation of the point spread function
(PSF). Also, simultaneous fitting with neighbor galaxies needs special
care as the number of free parameters is significantly larger. For
galaxies that show a large residual, we took these caveats into
consideration and refit the galaxies until the residual became
small. We then assigned a quality factor for each galaxy based on the
magnitude of the residual and the deviation of fitted position angle
from the galaxy position angle estimated by eye. If a fit was below a
threshold quality, we excluded that galaxy from further analysis. We
found that most of the galaxies that failed the quality check, either
have peculiar morphology or strong spiral arms. We successfully fit
337 out of 379 galaxies in our sample. Unless otherwise stated, all
further discussion in this letter only applies to these 337 galaxies.
The cluster-wise breakup of galaxies with a good fit is given in Table
\ref{clustab}.
\begin{table*}
\centering
\begin{minipage}{150mm}
\caption{Summary data for sample clusters}
\label{clustab}
\begin{tabular}{@{}llllllllll@{}}
\hline
Cluster & RA & Dec & z & Camera & Filter & $m_{lim}$ & $N_{Tot}$ & $N_{Fit}$ & Reference\\
\hline
AC 114 & 22 58 48.4 & -34 48 60 & 0.31 & WFPC2 & F702W & 20.77 & 72 & 68 & \citet{cou98}\\
CL 0303+17 & 03 06 15.9 & +17 19 17 & 0.42 & WFPC2 & F702W & 21.64 & 28 & 26 & \citet{sma97}\\
3C 295 & 14 11 19.5 & +52 12 21 & 0.46 & WFPC2 & F702W & 21.93 & 37 & 32 & \citet{sma97}\\
CL 1232-1250 & 12 32 30.3 & -12 50 36 & 0.54 & ACS & F814W & 21.66 & 46 & 41 & \citet{whi05}\\
CL 1054-1146 & 10 54 24.4 & -11 46 19 & 0.697 & ACS & F814W & 22.44 & 30 & 25 & \citet{whi05}\\
CL 1040-1155 & 10 40 40.3 & -11 56 04 & 0.704 & ACS & F814W & 22.47 & 25 & 19 & \citet{whi05}\\
CL 1054-1245 & 10 54 43.5 & -12 45 51 & 0.75 & ACS & F814W & 22.68 & 29 & 28 & \citet{whi05}\\
CL 1216-1201 & 12 16 45.3 & -12 01 17 & 0.794 & ACS & F814W & 22.89 & 50 & 43 & \citet{whi05}\\
RX J0152.7-1357 & 01 52 27.4 & -13 55 01 & 0.837 & ACS & F775W & 23.50 & 62 & 55 & \citet{bla06}\\
\hline
\end{tabular}
$m_{lim}$ : Faint magnitude cutoff in the observed HST filter, $N_{Tot}$ : Total number of galaxies, $N_{Fit}$ : Number of galaxies with good fit
\end{minipage}
\end{table*}
\section{Results}
\subsection{Evolution of mean bulge-to-total luminosity ratio ($\langle B/T \rangle$) and S\'ersic index $n$}
In Table \ref{results} we list mean and median values of a few
parameters of interest for all the nine clusters. Note that the
bulge-to-total luminosity ratio is computed using the parameters of
the best fit model. We find the $\langle B/T \rangle$ of cluster
galaxies in the central 1 Mpc of the clusters changes from
0.59$^{+0.03}_{-0.02}$ at redshift z = 0.31 to 0.48 $\pm$ 0.03 at z =
0.837 (Figure \ref{avg-BT}). Errors were measured using the bootstrap
resampling method \citep{efr93}, in this and subsequent figures. The
increase in $\langle B/T \rangle$ ratio has been found qualitatively
(i.e. using visual morphological classification) by previous studies
\citep{dre97,fas00,smi05}; the present work obtains the result
quantitatively using bulge-disk decomposition. It must be noted that,
if our sample is somewhat biased towards luminous red galaxies at
high-z, and if such galaxies are early type even at those redshifts,
then our estimated $\langle B/T\rangle$ value represents an upper
limit at $z \sim 0.8$. In addition, incompleteness will also tend to
make our estimate of $\langle B/T\rangle$ too high because at fainter
magnitudes we would preferentially miss low $B/T$ late-type galaxies
\citep{des07}. This leads to our estimated $\langle B/T\rangle$ to be
the upper limit for these clusters. So the selection bias, if any,
will lead to an apparent weaker evolution. So our estimation of
evolution of $\langle B/T\rangle$ may be an underestimate.
We also find that the mean value of S\'ersic index decreases with
lookback time over this redshift range. The value changes from
4.13$^{+0.43}_{-0.39}$ to 2.74$^{+0.28}_{-0.30}$ from $z = 0.31$ to $z
= 0.84$. Figure \ref{avg-n} shows the evolution of S\'ersic index
against lookback time.
\begin{table}
\caption{Mean and median parameters obtained through bulge-disk decomposition}
\label{results}
\begin{tabular}{@{}lllll}
\hline
Cluster & $\langle B/T\rangle$ & $\langle n\rangle$ & \~{n} & $f_b$\\
\hline
AC 114 & $0.59^{+0.03}_{-0.02}$ & $4.13^{+0.43}_{-0.39}$ & $3.47^{+0.27}_{-0.30}$ & $0.55^{+0.03}_{-0.04}$\\
CL 0303 & $0.52^{+0.05}_{-0.05}$ & $4.24^{+1.02}_{-0.79}$ & $3.31^{+0.20}_{-0.83}$ & $0.46^{+0.05}_{-0.07}$\\
3C 295 & $0.52^{+0.04}_{-0.05}$ & $3.47^{+0.59}_{-0.56}$ & $3.17^{+0.72}_{-0.78}$ & $0.43^{+0.03}_{-0.04}$\\
CL 1232-1250 & $0.48^{+0.04}_{-0.04}$ & $4.15^{+0.56}_{-0.52}$ & $3.30^{+0.41}_{-0.52}$ & $0.47^{+0.04}_{-0.05}$\\
CL 1054-1146 & $0.45^{+0.05}_{-0.05}$ & $3.60^{+0.76}_{-0.65}$ & $2.41^{+1.02}_{-0.43}$ & $0.26^{+0.05}_{-0.06}$\\
CL 1040-1155 & $0.42^{+0.08}_{-0.07}$ & $3.50^{+0.68}_{-0.68}$ & $3.25^{+0.71}_{-0.56}$ & $0.28^{+0.08}_{-0.08}$\\
CL 1054-1245 & $0.47^{+0.05}_{-0.05}$ & $2.80^{+0.38}_{-0.41}$ & $2.55^{+0.56}_{-0.55}$ & $0.37^{+0.05}_{-0.06}$\\
CL 1216-1201 & $0.48^{+0.03}_{-0.04}$ & $3.25^{+0.54}_{-0.44}$ & $2.57^{+0.40}_{-0.51}$ & $0.32^{+0.04}_{-0.04}$\\
RX J0152.7-1357 & $0.48^{+0.03}_{-0.03}$ & $2.74^{+0.28}_{-0.30}$ & $2.49^{+0.43}_{-0.28}$ & $0.40^{+0.02}_{-0.02}$\\
\hline
\end{tabular}
\medskip
$\langle B/T\rangle$ : mean value of $B/T$ bulge-to-total luminosity ratio, $\langle n\rangle$ : mean value of Se\'rsic index, \~{n} : median value of Se\'rsic index, $f_b$ : fraction of bulge-like galaxies (see text for definition)
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig1.eps}
\caption{Evolution of mean value of the bulge-to-total luminosity ratio $B/T$}
\label{avg-BT}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig2.eps}
\caption{Evolution of mean value of S\'ersic index}
\label{avg-n}
\end{figure}
\subsection{Evolution of bulge-like galaxy fraction}
Since the $\langle B/T\rangle$ decreases significantly with lookback
time, one expects to see a simultaneous decrease in the fraction of
early-type galaxies with a bulge-like morphology. We classify a
galaxy as bulge-like if its $B/T \geq 0.4$ and $n > 2.5$. The second
condition is required to exclude galaxies with a very strong disk
which can, on occasion, be incorrectly modeled as a bulge with
$n\sim1$ (an exponential), with a correspondingly high $B/T$
luminosity ratio. With this definition, we are able to compute a
bulge-like galaxy fraction for each cluster. In figure \ref{evo-early}
we plot this fraction against lookback time. Note that the fraction
is normalized by the total number of galaxies in the cluster, not the
number of galaxies with a successful fit. The galaxies that are
poorly fit are dominated by galaxies of irregular morphology. We see
a near monotonic decrease with lookback time in the bulge dominated
fraction of galaxies. We find 40.0 $^{+2}_{-2}$ \% of galaxies at
redshift z = 0.837 are bulge-like. This increases to 55 $^{+3}_{-4}$
\% within $\sim$ 3.5 Gyr.
\subsection{Discussion}
In the last decade, several studies have reported evolution of the
morphological content of galaxy clusters by a visual study of galaxy
morphology \citep{sma97,cou98,fas00,des07}. Other studies focused on
the evolution of the morphology-density relation (MDR), where
morphology changes were studied as a function of redshift and galaxy
density \citep{dre97,tre03,pos05,smi05}.
In this work, we have taken the first approach; however, we have used
{\it quantitative} measures of galaxy morphology rather than a
qualitative morphological classification by eye. We decompose the
galaxy light into bulge and disk components and study the evolution of
morphology of galaxies in clusters. We find that the bulge component
of the {\it bright} galaxies in clusters is, on average, becoming
stronger as the Universe evolves. The fraction of bulge-like
galaxies, defined as having $B/T > 0.4$ and $n > 2.5$ also increases
from 40\% to 55\%.
There is also a significant increase in the mean value of the S\'ersic
index as the Universe ages.
Our results on the evolution of morphological fraction are consistent
with previous work on the subject \citep{dre97,pos05,smi05}. However,
\citet{des07}, using ACS observations of galaxy clusters with $0.5 < z
< 0.8$ found no evolution of morphological fraction. At first glance,
this seems to contradict our results. However, it must be noted that
in our study, we are only including bright, spectroscopically
confirmed cluster galaxies. If fainter galaxies are included, as was
done in the sample of \citet{des07}, the $\sim10$\% change we see in
the morphological fraction may easily be washed out. This explanation
also agrees with the results of \citet{hol07} who showed that there is
evolution in the early type fraction with redshift if a luminosity limited sample is used.
\citet{pos05,des07,pog09} found that the E+S0 fraction correlates with the velocity dispersion of the cluster. The fraction is large for clusters with high dispersion. This effect would be consistent with the evolution we see, provided velocity dispersion systematically decreases with redshift. We see no systematic dependence of velocity dispersion with redshift except that the two highest redshift clusters in our sample have high velocity dispersions ($> 1000 $ km/s). This high value, which implies higher $\langle B/T\rangle$ is consistent with the mild increase of $\langle B/T\rangle$ and the fraction of bulge dominated galaxies for the two high $z$ clusters. But the dominating effect seems to be morphological evolution.
The $\langle B/T\rangle$ of a cluster may increase either due to an increase in
the strength of the bulge component of galaxies or by the fading of
disk component. The increase in average bulge strength may be caused
by merging which tends to produce a elliptical like morphology
\citep{alb82,bar92,her92,bou05}. On the other hand, the fading
of the disk could be a by-product of the morphological transformation
of galaxies. In clusters, a variety of mechanisms such as galaxy
harassment \citep{moo96,moo98,moo99}, minor mergers and ram pressure
stripping \citep{gun72,aba99} may contribute to the disk fading.
The increase in the mean S\'ersic index $\langle n \rangle$ with cosmic time seems
to indicate that mergers play a role \citep{sca03}. Merger events are common at
intermediate redshift \citep{dre94}. The increase in the fraction of
galaxies with high $B/T$ and S\'ersic index is possible if galaxies
gradually evolve into a phase where the spheroidal component
increasingly dominates. The end point of such evolution is an elliptical
galaxy.
It has recently been suggested that the bright S0 population has likely
formed through monolithic collapse or major mergers
\citep{bar07,bar09}. Numerical simulations have also shown that
dissipative merger of two unequal mass disk galaxies
\citep{bek98,bek05} can produce lenticulars. Recent observations of
low redshift clusters suggest that a majority of the infall population
is merging or interacting \citep{mos06}. Coupled with the constraints
from \citet{car86}, it is becoming increasingly clear that formation
of spheroidal (mostly lenticular) galaxies through mergers is the
dominant mechanism behind systematic changes in galaxy morphology.
It must be noted that the trends we see are weak and statistical in
nature; they are only visible when averaged over a large number of
galaxies in a large number of clusters over a wide range of redshift.
The detailed physics operating in each cluster, doubtless modifies the
morphological evolution of galaxies in that cluster. Nevertheless, the
fact that we see trends indicates that they are real and strong enough
not to be drowned by the different physical conditions and processes
operating in individual clusters. Using the large database of bulge/disk
decomposition results we have obtained, we are attempting to
disentangle cluster specific effects from cosmological ones.
\begin{figure}
\centering
\includegraphics[scale=0.35]{fig3.eps}
\caption{Evolution of fraction of bulge-dominated galaxies}
\label{evo-early}
\end{figure}
\section*{Acknowledgments}
Vinu Vikram acknowledges financial support from the Council of Scientific and Industrial Research (CSIR). We thank the referee for insightful comments and suggestions that greatly improved this paper.
|
1,108,101,565,464 | arxiv | \section{Introduction}
In response to large-scale integration of intermittent sources of energy \cite{lund2015review} in power systems,
system operators seek novel approaches to balancing the load and generation.
The aggregation of distributed energy resources (DERs) and regulation of their aggregate power output has emerged as a viable balancing mechanism \citep[e.g.]{SG2010, Hiskens2011,pinson2014benefits} for many system operators.\footnote{
In Australia, for instance, it is estimated that there are 2.5 million rooftop solar installations with a total capacity of more than 10 GW and 73,000 home energy storage systems with a 1.1 GW capacity.}
For the transmission system operator, the load aggregator provides a single point of contact, which is responsible for the management of risks involved in the coordination of an ensemble of DERs and loads, and the legal and accounting aspects associated with modern grid codes and contracts governing the provision of ancillary services to the grid.
The load aggregator then coordinates an ensemble of participants, which can take many forms, including DERs \citep[e.g.]{SG2010a}, interruptible or deferrable loads \citep[e.g.]{Alagoz2013,mathieu2014arbitraging}, and eventually, vehicle-to-grid (V2G) services \citep[e.g.]{EV2018,lee2020adaptive}.
Load aggregation is sometimes also known as a virtual power plant \cite{el2010virtual,crisostomi2014plug,8013070}.
Independent of the form of the ensemble, load aggregation introduces a novel feedback loop \cite{6197252,li2020real} into regulation problems in power systems.
Recent regulation\footnote{In the European Union, Regulation (EU) 2019/943 of the European Parliament and of the Council of 5 June 2019 on the internal market for electricity sets ``fundamental principles [to] facilitate aggregation of distributed demand and supply'', while requiring ``non-discriminatory market access'' and ``fair rules''. Its implementation in individual member states varies and is still in progress in many member states since mid-2021.} mandates the use of load aggregation, and hence makes the closed-loop analysis of load aggregation very relevant.
A recent pioneering study of the closed-loop aspects of load aggregation by Li et al. \cite{li2020real} leaves three issues open:
how to go beyond a linearisation of the physics of the alternating-current (AC) model?
How to model the uncertainty inherent in response to the control signal provided by the load aggregator?
Finally, what criteria of a good-enough behaviour of such a closed-loop system to consider?
The first issue is particularly challenging: While the consideration of losses in the AC model may be necessary \cite{baker2021solutions} for the sake of the accuracy of the model, it introduces a wide range of challenging behaviours \citep{754100,rogers2012power} in the non-linear dynamics.
In this paper, we address all three issues in parallel. First, we suggest that good-enough behaviour involves ``predictability and fairness'' properties that are rarely considered in control theory and require new analytical tools.
These include:
\begin{itemize}
\item[(a)] whether each DER and load, on average, receives similar treatment in terms of load reductions, disconnections, or power quality,
\item[(b)] whether the prices or incentives depend on initial conditions, such as consumption at some point in the past,
\item[(c)] whether the prices, incentives, or limitations of service are stable quantities, preferably not sensitive to noise entering the system,
and predictable.
\end{itemize}
Building upon the recent work of \cite{ErgodicControlAutomatica}, we show that these criteria can be phrased in terms of properties of a specific stochastic model. In particular, these criteria are achieved whenever one can guarantee the existence of the {\em unique invariant measure} of a stochastic model of the closed-loop system. Thus, the design of feedback systems for deployment in the demand-response application should consider both the traditional notions of regulation and optimality and provide guarantees concerning the existence of the unique invariant measure.
Next, we show that many familiar control strategies, in straightforward situations, do not necessarily give rise to feedback systems that possess the unique stationary measure.
In theory, we use the model of \cite{ErgodicControlAutomatica} to show that
with any controller ${\mathcal C}$ that is linear marginally stable with a pole $s_1 =e^{q j \pi}$ on the unit circle, where $q$ is a rational number, the closed-loop system cannot have a unique invariant measure.
In practice, we illustrate this behaviour in simulations using Matpower \cite{zimmerman2010matpower}.
As our main result, we show that one can design controllers that allow for predictability and fairness from the point of view of a participant, such as an operator of DERs, even in the presence of losses in the AC model. These results are based on a long work history on incremental input-to-state stability, which we have extended to a stochastic setting.
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{diagram-power.pdf}
\caption{An illustration of our closed-loop model, following \cite{ErgodicControlAutomatica}.}
\label{system}
\end{figure}
\section{Related Work}
There is a rich history of work on dynamics and control in power systems, summarised in several excellent textbooks of \cite{kundur1994power,sauer1998power,machowski2011power,rogers2012power}, and \cite{bevrani2014robust}. Without aiming to present a comprehensive overview, one should like to point out that traditionally, load frequency control (LFC) is performed using controllers with poles on the unit circle, such as proportional-integral (PI) controllers \citep[e.g.]{4074993,32469}.
There are many alternative proposals, some of them excellent \cite[e.g.]{Ugrinovskii2005,Perfumo2012,zarate2010opf,li2016sqp,wang2018sdp,li2019sequential,nojavan2020voltage,nikkhah2020stochastic,9511198}, but even the most recent textbook of \cite[Chapter 4, ``Robust PI-Based Frequency Control'']{bevrani2014robust} suggests that ``complex state feedback or higher-order dynamic controllers [...] are impractical for [the] industry''.
Although the idea of demand response has been studied for several decades \citep[e.g.]{4112061}, only recently it has been shown to have commercial potential \citep{4275305,5608535,6558529, Biegel2014}. Notably, with a capacity as low as 20--70 kWh, load aggregators may break even \cite{Biegel2014} in the spot market in developed economies, although many grid codes still require load aggregators to aggregate a substantially more enormous amount of active power (e.g., 1 MW).
Correspondingly, there has been much interest in incorporating load aggregation in load frequency control \citep{5764847,6112198,6239581,6224203,6315670,6397579,6508915,6509996,6702462,6717056,6874591,6839081,dorfler2017gather,7004818,7065326,7944568}.
We refer to \cite{Dehghanpour2015} for a recent survey.
In a recent pioneering study, Li et al. \cite{li2020real} considered the closed-loop system design while quantifying the available flexibility from the aggregator to the system operator,
albeit without the non-linearity of the alternating-current model.
While some of the more recent approaches utilize model-predictive control \cite[e.g.]{7065326}, many of the early proposals extend the use of PI. In contrast, we utilise {\em iterated function systems} \citep{elton1987ergodic,BarnsleyDemkoEltonEtAl1988,barnsley1989recurrent}, a class of Markov processes, to design novel controllers following \cite{ErgodicControlAutomatica}.
\section{Notation, Preliminaries and Our Model}\label{sec:model}
Let us introduce some definitions and notation:
\begin{dfn}\label{df:class-K}
A function $\gamma : \mathbb R^{+}\to \mathbb R^{+}$ is is said to be of class $\mathcal K$ if it is continuous, increasing and $\gamma(0)=0$. It is of class $\mathcal K_{\infty}$ if, in addition, it is proper, i.e., unbounded.
\end{dfn}
\begin{dfn}\label{df:class-KL}
A continuous function $\beta: \mathbb R^{+}\times \mathbb R^{+}\to \mathbb R^{+}$ is said to be of class $\mathcal K \mathcal L$, if for all fixed $t$ the function $\beta(\cdot, t)$ is of class $\mathcal K$ and for all fixed $s$, the function $\beta(s, \cdot)$ is is non-increasing
and tends to zero as $t\to \infty$.
\end{dfn}
In keeping with \cite{ErgodicControlAutomatica}, our model is based on a feedback signal $\pi(k)\in \Pi \subseteq \mathbb{R}$ sent to $N$ agents.
Each agent $i$ has a state $x_i\in \mathbb{R}^{n_i}$ and an associated output $y_i \in\mathbb{D}_i$, where the latter is a finite set.
The non-de\-ter\-min\-is\-tic response of agent $i$ is within a finite set of actions:
$\mathbb{A}_i =\{
a_1,\ldots,a_{L_i} \}\subset \mathbb{R}^{n_i}$.
The finite set of possible resource demands of agent $i$ is
$\mathbb{D}_i$:
\begin{equation}
\label{eq:5}
\mathbb{D}_i := \{ d_{i,1},
d_{i,2}, \ldots, d_{i,m_i}\}.
\end{equation}
We assume there are $w_i \in \mathbb{N}$ state transition maps ${\mathcal W}_{ij}: \mathbb{R}^{n_i} \to \mathbb{R}^{n_i}$, $j=1,\ldots,w_i$
for agent $i$
and $h_i \in \mathbb{N}$ output maps ${\mathcal H}_{i\ell}: \mathbb{R}^{n_i} \to
\mathbb{D}_i$, $\ell= 1,\ldots,h_i$ for each agent $i$.
The evolution of the states and
the corresponding demands then satisfy:
\begin{align}\label{eq:general}
x_i(k+1) & \in \{ {\mathcal W}_{ij}(x_i(k)) \;\vert\; j = 1, \ldots, w_i\}, \\
y_i(k) & \in \{ {\mathcal H}_{i\ell}(x_i(k)) \;\vert\; \ell = 1, \ldots, h_i\},
\end{align}
where the choice of agent $i$'s response at time $k$ is governed by probability
functions $p_{ij} : \Pi \to [0,1]$, $j=1,\ldots,w_i$, respectively
$p'_{i\ell} : \Pi \to [0,1]$, $\ell=1,\ldots,h_i$. Specifically, for each agent $i$, we have
for all $k\in\mathbb{N}$ that
\begin{subequations} \label{eq:problaws}
\begin{align}\label{eq:problaw-1}
&\mathbb{P}\big( x_i(k+1) = {\mathcal W}_{ij}(x_i(k)) \big) = p_{ij}(\pi(k)),\\
&\mathbb{P}\big( y_i(k) = {\mathcal H}_{i\ell}(x_i(k)) \big) = p'_{i\ell}(\pi(k)).
\intertext{Additionally, for all $\pi \in \Pi$, $i=1,\ldots,N$ it holds that}
&\sum_{j=1}^{w_i} p_{ij}(\pi) = \sum_{\ell=1}^{h_i} p'_{i\ell}(\pi) = 1.
\end{align}
\end{subequations}
The final equality comes from the fact $p_{ij}$, $p'_{i\ell}$ are
probability functions. We assume that, conditioned on $\{ x_i(k) \}, \pi(k)$,
the random variables $\{ x_i(k+1) \mid i = 1,\ldots,N \}$ are stochastically independent. The outputs $y_i(k)$ each depend on $x_i(k)$ and the signal $\pi(k)$ only.
\label{sec:markov}
Let $\Sigma$ be a closed subset of $\mathbb{R}^n$ with the usual Borel
$\sigma$-algebra $\mathbb{B}(\Sigma)$. We call the elements of $\mathcal{B}(\Sigma)$
events. A Markov chain on $\Sigma$ is a sequence
of $\Sigma$-valued random vectors $\{ X(k)\}_{k\in\mathbb{N}}$ with the Markov property, which is the equality of a probability of an event conditioned on past events
and probability of the same event conditioned on the current state, \emph{i.e.}, we always have
\begin{align}
\mathbb{P} \left( X(k+1) \in \mathcal{G} \mid X(j)=x_{j}, \, j=0, 1, \dots, k \right) \nonumber \\
= \mathbb{P}\big(X(k+1)\in \mathcal{G} \mid X(k)=x_{k}\big), \nonumber
\end{align}
where $\mathcal{G}$ is an event
and $k \in \mathbb{N}$. We assume the Markov chain is time-homogeneous. The transition operator $\mathcal P$ of the Markov chain is defined
for $x \in \Sigma$, $\mathcal{G} \in \mathcal{B}\left(\Sigma\right)$ by
\begin{align}
P(x,\mathcal{G}) := \mathbb{P}(X(k+1) \in \mathcal{G} \; \vert \; X(k) = x).
\end{align}
If the initial condition $X(0)$ is distributed according to an initial distribution
$\lambda$, we denote by $\mathbb{P}_\lambda$ the probability measure induced on
the path space, i.e., space of sequences with values in $\Sigma$. Conditioned on an
initial distribution $\lambda$, the random variable $X(k)$ is distributed
according to the probability measure $\lambda_k$ which is determined inductively by $\lambda_0=\lambda$ and
\begin{equation}
\label{eq:measureiteration}
\lambda_{k+1}(\mathcal{G}) := \int_{\Sigma} \mathcal P(x, \mathcal{G}) \, \lambda_k(d x),
\end{equation}
for $\mathcal{G} \in \mathbb{B}$.
A probability measure $\mu$ is called
invariant with respect to the Markov process $\{ X(k) \}$ if it is a fixed
point for the iteration described by \eqref{eq:measureiteration},
i.e., if
\begin{align}\label{df: inv-meas}
\mathcal P \mu = \mu\quad
\end{align}
An invariant probability measure $\mu$ is called attractive, if for every probability measure $\nu$ the sequence $\{\lambda_k \}$ defined by \eqref{eq:measureiteration} with initial condition $\nu$ converges to $\mu$ in distribution.
In an iterated function systems with place dependent probabilities \citep{Elton1987,Barnsley1988(1),Barnsley1989}, we are given a set of maps $\{ f_j : \Sigma \to \Sigma\; \vert \; j \in \mathcal{J} \}$, where $\mathcal{J}$ is a (finite or countably infinite) index set. Associated to these maps, there are probability functions
\begin{align}
p_j: \Sigma \to [0,1]\text{ with } \sum_{j\in \mathcal{J}} p_j(x) = 1 \text{ for all } x\in \Sigma.
\label{df: prob-funct}
\end{align}
The state $X(k+1)$ at time $k+1$ is then given by $f_j(X(k))$ with probability $p_j(X(k))$,
where $X(k)$ is the state at time $k$. Sufficient conditions for the existence of a unique attractive
invariant measure can be given in terms of ``average contractivity'' \citep{elton1987ergodic,BarnsleyDemkoEltonEtAl1988,barnsley1989recurrent}.
Any discrete-time Markov chain can be generated by an \emph{iterated function system with probabilities} see Kifer\cite[Section 1.1]{Kifer2012} or \cite[Page 228]{Bhattacharya2009}, although such representation is not unique, see, Stenflo \cite{Stenflo1999}. Although not well known, iterated function systems (IFS) are a convenient and rich class of Markov processes. The class of stochastic systems arising from the dynamics of multi-agent interactions can be modelled and analysed using IFS in a particularly natural way. A wealth of results\citep{Elton1987,Barnsley1988(2),Barnsley1989,Barnsley2013,Stenflo2001(1),Szarek2003(1),Steinsaltz1999, Walkden2007,Barany2015,Diaconis1999,Iosifescu2009,Stenflo2012(s)} then applies.
\section{The Closed-Loop Model as Iterated Random Functions}
Let us now model the system of Figure \ref{system} by a Markov chain on a state space representing all the system components. In general, one could consider:
\begin{align}
\label{eq:nonlinear-agents-all}
\phantom{{\mathcal A}} &\left\{ \begin{array}{ccl}
x_i(k+1) &\in& \{ {\mathcal W}_{ij}(x_i(k)) \; \vert \; j=1,\ldots, w_i \} \\
y_i(k) &\in& \{ {\mathcal H}_{ij}(x_i(k)) \; \vert \; j=1,\ldots, h_i \}, \\
\end{array} \right. \\
& \hspace*{1.6cm} y(k) = \sum_{i=1}^N y_i(k),
\end{align}
\begin{equation}
\label{eq:nonlinear-filter}
{\mathcal F} ~:~ \left\{ \begin{array}{ccl}
x_f(k+1) &=& {\mathcal W}_{f}(x_f(k),y(k)) \\
\hat y(k) &=& {\mathcal H}_{f}(x_f(k),y(k)),
\end{array} \right.
\end{equation}
\begin{equation}
\label{eq:nonlinear-cont}
{\mathcal C} ~:~ \left\{ \begin{array}{ccl}
x_c(k+1) &=& {\mathcal W}_{c}(x_c(k),\hat y(k),r) \\
\pi(k) &=& {\mathcal H}_{c}(x_c(k),\hat y(k),r),
\end{array} \right.
\end{equation}
In addition, we have (e.g., Dini) continuous probability functions
$p_{ij},p'_{il}:\Pi \to [0,1]$ so that
the probabilistic laws \eqref{eq:problaws} are satisfied.
If we denote by $\mathbb{X}_i, i=1,\ldots,N, \mathbb{X}_C$ and $\mathbb{X}_F$ the state spaces of the
agents, the controller and the filter, then the system evolves on the
overall state space $\mathbb{X} := \prod_{i=1}^N \mathbb{X}_i \times \mathbb{X}_C \times \mathbb{X}_F$
according to the dynamics
\begin{equation}
\label{eq:6}
x(k+1) := \begin{pmatrix}
(x_i)_{i=1}^N \\
x_f \\
x_c
\end{pmatrix} (k+1) \in \{ F_m(x(k)) \,\vert\, m \in {\mathbb M}\}.
\end{equation}
where each of the maps $F_m$ is of the form
\begin{equation}
\label{eq:F_m-definition}
F_m(x(k)) \mathrel{\mathop:}= \begin{pmatrix}
( {\mathcal W}_{ij}(x_i (k)) )_{i=1}^N \\
{\mathcal W}_f(x_f(k), \sum_{i=1}^N {\mathcal H}_{i\ell}(x_i (k))) \\
{\mathcal W}_c(x_c(k), {\mathcal H}_f(x_f(k), \sum_{i=1}^N {\mathcal H}_{i\ell}(x_i (k))))
\end{pmatrix}
\end{equation}
and the maps $F_m$ are indexed by indices $m$ from the set
\begin{equation}
\label{eq:8}
\mathbb{M} \mathrel{\mathop:}= \prod_{i=1}^N \{ (i,1), \ldots, (i,w_i) \} \times \prod_{i=1}^N \{ (i,1), \ldots, (i,h_i) \}.
\end{equation}
By the independence assumption on the choice of the transition maps and
output maps for the agents, for each multi-index
$m=((1,j_1),\ldots,(N,j_N),(1,l_1),\ldots,(N,l_N))$ in this set, the
probability of choosing the corresponding map $F_m$ is given by
\begin{multline}
\label{eq:9}
\mathbb{P} \left( x(k+1)=F_m(x(k)) \right) = \\ \left(\prod_{i=1}^N
p_{ij_i}(\pi(k)) \right) \left( \prod_{i=1}^N p'_{il_i}(\pi(k))
\right) =: q_m(\pi(k)).
\end{multline}
We have obtained an iterated function system of the previous subsection \ref{sec:markov}.
\subsection{Our Objective}
We aim to achieve regulation, with probability $1$
\begin{equation}
\label{eq:1}
\sum_{i = 1}^N y_i(k) = y(k) \leq r
\end{equation}
to some reference $r>0$, given by the installed capacity.
Further, we aim for predictability, in the sense that for each agent $i$ there exists a
constant $\overline{r}_i$ such that
\begin{equation}
\label{eq:3}
\lim_{k\to \infty} \frac{1}{k+1} \sum_{j=0}^k y_i(j) = \overline{r}_i,
\end{equation}
where the limit is independent of initial conditions.
This can be guaranteed, when one has a unique invariant measure for the closed-loop model as iterated random functions.
The requirement of \textbf{fairness} is then for the limit $\overline{r}_i$ to coincide
for all $1 \le i \le N$.
\subsection{A Motivating Negative Result}\label{sec:comments-pi-negative}
The motivation for our work stems from the fact that controllers with integral action \citep{Franklin, Franklin_dig}, such as the Proportional-Integral (PI) controller, fail to provide unique ergodicity, and hence the fairness properties.
In many applications, controllers with integral action are widely adopted, including leading control systems \cite{barker2007speedtronic} in power generation.
A simple PI control can be implemented as:
\begin{equation}\label{pid1}
\pi(k) = \pi(k-1) + \kappa \big[ e(k) - \alpha e(k-1) \big],
\end{equation}
which means its transfer function from $e$ to $\pi$, in terms of the ${\mathcal Z}$ transform, is given by
\begin{equation}\label{pid2}
C(z) = \kappa\frac{1 - \alpha z^{-1}}{1 - z^{-1}}.
\end{equation}
Since this transfer function is not asymptotically stable, any associated realisation matrix will not be Schur.
Note that this is the case for any controller with any sort of integral
action, \emph{i.e.}, a pole at $z = 1$.
\begin{thm}[\cite{ErgodicControlAutomatica}]
\label{thm:pole}
Consider $N$ agents with states $x_i, i=1,\ldots,N$. Assume
that there is an upper bound $m$ on the different values the agents
can attain, \emph{i.e.}, for each $i$ we have $x_i \in \mathbb{A}_i =\{
a_1,\ldots,a_{m_i} \}\subset \mathbb{R}$
for a given set ${\mathbb{A}}_i$ and $1 \leq m_i \leq m$.
Consider the feedback system in Figure \ref{system}, where ${\mathcal
F}\, : \, y \mapsto \hat y$ is a finite-memory moving-average (FIR)
filter. Assume the controller ${\mathcal C}$ is a linear marginally
stable single-input single-output (SISO) system with a pole $s_1 =
e^{q j \pi}$ on the unit circle where $q$ is a rational number,
$j$ is the imaginary unit,
and $\pi$ is Archimedes' constant $3.1416$. In
addition, let the
probability functions $p_{il} : \mathbb{R} \to [0,1]$ be continuous for all
$i=1, \ldots, N, l=1,\ldots,m_i$, \emph{i.e.}, if
$\pi(k)$ is the output of ${\mathcal C}$ at time $k$, then $\mathbb{P}(x_i(k+1)=a_l)=p_{il}(\pi(k))$. Then
the following holds.
\begin{enumerate}
\item[(i)] The set ${\mathbb O}_{\mathcal F}$ of possible output values of the filter ${\mathcal F}$ is finite.
\item[(ii)] If the real additive group ${\mathcal E}$ generated by
$\{ r - \hat y \mid \hat y \in {\mathbb O}_{\mathcal F} \}$ with $r$ from \eqref{eq:1} is discrete, then
the closed-loop system cannot be uniquely ergodic.
\item[(iii)] For rational ${\mathbb A}_i
\subset \mathbb{Q}$ for all $i=1,\ldots,N$, $r
\in \mathbb{Q}$, and rational coefficients of the FIR filter ${\mathcal F}$, the closed-loop
system cannot be uniquely ergodic.
\end{enumerate}
\end{thm}
This suggests that it is perfectly possible
for the closed loop both to perform its regulation function well and, at the same time, to
destroy the ergodic properties of the closed loop.
We demonstrate this in a realistic load-aggregation scenario in Section \ref{sec:numeric}.
\section{The Main Result}\label{sec:nonlinear}
For a linear controller and filter, \cite{ErgodicControlAutomatica} have established conditions that assure the existence of a unique invariant measure for the closed-loop. Let us consider non-linear controllers and filters, as required in power systems, and seek unique ergodicity conditions.
Incremental stability is well-established concept to describe the asymptotic property of differences between any two solutions. One can utilise the concept of incremental input-to-state stability, which is defined as follows:
\begin{dfn}[Incremental ISS, \cite{angeli2002lyapunov}]
Let $\mathcal U
$ denote the set of all
input functions $u: \mathbb Z_{\ge k_0}\to \mathbb R^d
. Suppose, $F: \mathbb R^d \times \mathbb{R}^n\to \mathbb R^n$ is continuous, then the discrete-time non-linear dynamical system
\begin{align}
x(k+1) = F(x(k),u(k)),
\label{system-inciss}
\end{align}
is called (globally) \emph{incrementally input-to-state-stable} (incrementally ISS), if there exist $\beta\in \mathcal{KL}$ and $\gamma\in\mathcal K$ such that
for any pair of inputs $u_1, u_1\in\mathcal{U}$ and any pair of initial condition $\xi_1, \xi_2 \in \mathbb{R}^n$:
\begin{align*}
&\|x(k, \xi_1, u_1)-x(k, \xi_2, u_2)\|\\
&\le \beta(\|\xi_1 - \xi_2\|, k)+ \gamma(\|u_1-u_2\|_{\infty}),\quad \forall k\in \mathbb{N}.
\end{align*}
\end{dfn}
\begin{dfn}[Positively Invariant Set]
A set $\mathcal X\subseteq \mathbb R^n$ is called positively invariant under the system \eqref{system-inciss} if for any initial state $x(k_0)=\xi\in \mathcal X$ we have $\phi(k)\in \mathcal X \quad \forall k\ge k_0$.
\end{dfn}
\begin{dfn}[Uniform Exponential Incremental Stability] The system \eqref{system-inciss} is uniformly exponentially incrementally
stable in a positively invariant set $\mathcal X$ if there exists $\kappa \ge 1$ and $\lambda >1$ such that for any pair of initial states $\xi_1, \xi_2 \in \mathcal X$, for any pair of $u_1, u_2\in \mathcal U$, and for all $k\ge k_0$ the following holds
\begin{align}
&\|\phi (k,k_0,\xi_1, u_1)-\phi (k,k_0,\xi_2, u_2)\|\nonumber\\
&
\le \kappa \|\xi_1-\xi_2\| \lambda^{-(k-k_0)}\|u_1-u_2\|_{\infty}.
\end{align}
If $\mathcal X= \mathbb R^n$, then the system \eqref{system-inciss} is called uniformly globally exponentially incrementally stable.
\end{dfn}
With this notation, we can employ contraction arguments for iterated function systems \cite{BarnsleyDemkoEltonEtAl1988} to prove:
\begin{thm} \label{thm02Ext}
Consider the feedback system depicted in Figure \ref{system}.
Assume that each agent
$i \in \{1,\cdots,N\}$ has a state governed by the non-linear iterated
function system
\begin{align}
\label{eq:nonlinear-agents}
x_i(k+1) &= {\mathcal W}_{ij}(x_i(k)) \\
y_i(k) &= {\mathcal H}_{ij}(x_i(k)),
\end{align}
where:
\begin{itemize}
\item[(i)] we have globally Lipschitz-continuous and continuously differentiable functions ${\mathcal W}_{ij}$ and ${\mathcal H}_{ij}$ with
global Lipschitz constant $l_{ij}$, resp. $l'_{ij}$
\item[(ii)]
we have Dini continuous probability functions
$p_{ij},p'_{il}$ so that
the probabilistic laws \eqref{eq:problaws} are satisfied
\item[(iii)] there are scalars $\delta, \delta' > 0$ such that
$p_{ij}(\pi) \geq \delta > 0$,
$p'_{ij}(\pi) \geq \delta' > 0$ for all $\pi\in \Pi$ and all $(i,j)$
\item[(iv)] the composition $F_m$
\eqref{eq:F_m-definition} of agents' transition maps and the probability functions satisfy the contraction-on-average condition. Specifically, consider the space $\mathbb{X}_a := \prod_{i=1}^{N} \mathbb {X}_i$ of all agents and the maps
$F_{a,m}: \mathbb{X}_a \to \mathbb{X}_a$, $(x_i)_{i=1}^N\mapsto (\mathcal{W}_{im}(x_i))_{i=1}^N$ together with the probabilities $p_m:\Pi\to[0,1]$, $m\in \mathbb{M}$, form an average contraction system in the sense that there exists a constant $0<\tilde c<1$ such that for all $x,\hat{x} \in \mathbb{X}_a, x\neq\hat{x}, \pi\in\Pi$ we have
\begin{equation}
\label{eq:average-contraction}
\sum_{m}p_m(\pi) \frac{\|F_{a,m}(x)-F_{a,m}(\hat{x})\|}{\|x-\hat{x}\|} < \tilde c.
\end{equation}
\end{itemize}
Then, for every incrementally input-to-state stable controller $\mathcal{C}$ and every incrementally input-to-state stable filter $\mathcal{F}$ compatible with the feedback structure, the feedback loop has a unique, attractive
invariant measure. In particular, the system is uniquely ergodic.
\end{thm}
\begin{proof}
To prove the result, we have to show that the assumptions of Theorem~2.1 in \cite{BarnsleyDemkoEltonEtAl1988} are satisfied. To this end, we note that the maps $F_m$ satisfy, by (i), the necessary Lipschitz conditions. Further, the probability maps have the required regularity and positivity properties by (ii) and (iii). Thus, it only remains to show that the maps $F_m$ defined in \eqref{eq:F_m-definition} have the average contraction property. The existence of a unique attractive invariant measure then follows from \cite{BarnsleyDemkoEltonEtAl1988} and unique ergodicity follows from \cite{elton1987ergodic}.
First we note that the filter and the controller are incrementally stable systems that are in cascade. It is
shown in \cite[Theorem 4.7]{angeli2002lyapunov} that cascades of incrementally ISS systems are incrementally ISS. The proof is given for continuous-time systems but readily extends to the discrete-time case. It will therefore be sufficient to treat filter and controller together as one system with state $z$.
To show the desired contractivity property, consider two distinct points $x=(x_a,z)=((x_i),z), \hat{x}=(\hat{x}_a,\hat{z})=((\hat{x}_i),\hat{z})\in \mathbb{X}$. Let $L$ be an upper bound for the Lipschitz constants of the output functions $\mathcal{H}_{ij}$, $\mathcal{H}_f$. The assumptions on incremental ISS ensure that for each index $m$
\begin{multline*}
\|F_m(x) - F_m(\hat{x}) \| \leq \\ \max \{ \|F_{a,m}(x_a) - F_{a,m}(\hat{x}_a) \|, \\ \beta_f(\|z-\hat{z}\|,1) + \gamma_f(L\|x_a - \hat{x}_a\|) \}.
\end{multline*}
As the maps $F_{a,m}$ satisfy the average contractivity condition, it is an easy exercise to see that also the sets of iterates satisfy the contractivity condition. The result then can be obtained using Theorem 15 of \cite{tran2018convergence}, where it is shown that incrementally stable systems are contractions,
using the converse Lyapunov theorem of \cite{jiang2002converse}.
The proof is concluded by invoking Theorem~2.1 in \cite{BarnsleyDemkoEltonEtAl1988}.
\end{proof}
Subsequently, one may wish to design a controller that is \emph{a priori} incrementally input-to-state-stable, e.g. considering extending the work on ISS-stable model-predictive controller \cite{1185106} or model-predictive controllers \cite{zarate2010opf,li2016sqp,wang2018sdp,li2019sequential,nojavan2020voltage,nikkhah2020stochastic,9511198} that are constrained to be small-signal stable. Alternatively, one may ask whether other controllers guaranteeing fairness are available, without explicitly adding incrementally input-to-state-stable constraints.
The implementation depends on the setting under consideration.
\section{Numerical results}\label{sec:numeric}
Our numerical results are based on simulations utilising Matpower \cite{zimmerman2010matpower}, a standard open-source toolkit for power-systems analysis in version 7.1 running on Mathworks Matlab 2019b.
We use Matpower's power-flow routine to implement a non-linear filter, which models the losses in the alternating-current model.
The model consists of an ensemble of $N$ DERs or partially-controllable loads divided into two groups that differ in
their probabilistic response to the control signal provided by the load aggregator.
In particular, following the closed-loop of Figure \ref{system}:
The response of each system $\mathcal S_i$ to the control signal $\pi(k)$ provided by the load aggregator at time $k$ takes the form of
probability functions, which suggest the probability a DERs is committed at time $k$ as a function of the control signal $\pi(k)$.
In our simulations, the two functions satisfying \eqref{df: prob-funct} are:
\begin{align}\label{eq:prob-func}
g_{i1}(x_i (k+1)=1)&=0.02+\frac{0.95}{1+\exp(-\xi\pi(k)-x_{01})} \\
g_{i2}(x_i (k+1)=1)&=0.98-\frac{0.95}{1+\exp(-\xi\pi(k)-x_{02})}
\end{align}
where we have implemented both PI and its lag approximant for the control of an ensemble of DERs or partially-controllable loads.
The binary-valued vector:
\begin{align}\label{eq:cmmtd-gnrtr}
u(k)\in \{0,1\}^N
\end{align}
captures the output of the probability functions \eqref{eq:prob-func} in a particular realization.
The commitment of the DERs within the ensemble is provided to Matpower, which using a standard power-flow algorithm computes
the active power output $P(k)$ of the individual DERs:
\begin{align}\label{eq:output-gnrtr}
P(k)\in \mathbb{R}^N
\end{align}
which is aggregated into the total active power output $p(k) = \sum P(k)$ of the ensemble.
Then, a filter $\mathcal F$ is applied:
\begin{align}\label{eq:filtr}
\hat p(k)= \frac{p(k)+p(k-1)}{2} + \textrm{losses}(k).
\end{align}
Notice that here we both smooth the total active power output with a moving-average filter and accommodate the losses in the AC model.
The error
\begin{align}\label{eq:err}
e(k)=r-\hat p(k)
\end{align}
between the reference power output $r$ and the filtered value $\hat p(k)$ is then used as the input for the controller.
Output of the controller $\pi(k)$ is a function of error $e(k)$ and an inner state of the controller $x_c (k)$.
Signal of the controller is given by the PI or its lag approximant:
\begin{align}\label{eq:two-contrlr}
\pi_{\text{PI}}(k+1)&=\left[K_p e(k)+K_i \left(x_c (k)+e(k)\right)\right]\\
\pi_{\text{Lag}}(k+1)&=\left[K_p e(k)+K_i \left(0.99x_c (k)+e(k)\right)\right]
\end{align}
The procedure is demonstrated in the following pseudocode.
\begin{algorithm}[bht]
\SetAlgoLined
$P(0) = \textrm{power-flow}(u(0))$ for a given initial commitment $u(0)$ \\
\For{$i\gets0$ \KwTo $\infty$, i.e., each run of the simulation }{
\For{$i\gets1$ \KwTo $k_{\max}$, where the length $k_{\max}$ of the time horizon is given}{
$p(k) = \sum P(k-1)$, cf. \eqref{eq:output-gnrtr} \\
$\hat p(k) = {\mathcal F}(p(k))$, e.g. \eqref{eq:filtr} \\
$e(k) = r - \hat p(k)$, cf. \eqref{eq:err} \\
$\pi(k) = {\mathcal C}(e(k))$, cf. \eqref{eq:two-contrlr} \\
$u(k) = {\mathcal S}(\pi(k))$, cf. \eqref{eq:prob-func} and \eqref{eq:cmmtd-gnrtr} \\
$P(k) = \textrm{power-flow}(u(k))$, cf. \eqref{eq:output-gnrtr}
}
}
\caption{Pseudo-code capturing the closed loop.}
\end{algorithm}
\subsection{A Serial Test Case}
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{linear_03plus/linear_network_diagram.png}
\caption{A single-line diagram of the serial test case.}
\label{fig:distribution}
\end{figure}
First, let us consider a simple serial test case depicted in Figure \ref{fig:distribution}, where there are a number of buses connected in series, as they often are in distribution systems. In particular, the buses are:
\begin{itemize}
\item a slack bus, such as a distribution substation,
\item 5 buses with 12 DERs of 5 MW capacity connected to each bus,
with probability function $g_{i2}$ of \eqref{eq:prob-func} modelling the response of these first 60 DERs to the signal,
\item 5 buses with 12 DERs of 5 MW capacity connected to each bus, with probability function $g_{i1}$ of \eqref{eq:prob-func}
modelling the response of these 60 DERs to the signal,
\item a single load bus with a demand of 500 MW.
\end{itemize}
We aim to regulate the system to the reference signal $r$, which is $300$ MW plus losses at time $k=0$.
Initially, the first 60 ($= 5 \cdot 12$) generators are off and the second following 60 ($= 5 \cdot 12$)
generators are on.
Considering the ensemble is composed of 120 DERs, with a total capacity of 600 MW,
this should yield a load of less than 200 MW on the slack bus.
We repeated the simulations 300 times, considering the time horizon of 2000 time steps each. The results in
Figures \ref{fig:linear03main1}--\ref{fig:linear03main3} present the mean over the 300 runs with a solid line and the region of mean $\pm$ standard deviation as a shaded area.
Notice that we are able to regulate the aggregate power output (cf. Figure ~\ref{fig:linear03main1}). With the PI controller, however, the state of the controller, and consequently the signal and the power generated at individual buses over the time horizon, are determined by the initial state of the controller (cf. Figures \ref{fig:linear03main2}--\ref{fig:linear03main3}).
\begin{figure*}[h]
\centering
\includegraphics[width=0.45\textwidth]{linear_03plus/01_Ensemble_power.png}
\includegraphics[width=0.45\textwidth]{linear_03plus/07_System_losses.png}
\caption{Results of simulations on the serial test case regulated to 300 MW plus losses:
Aggregate power produced by the ensemble (left) and the corresponding losses in transmission (right) as functions of time for the two controllers and two initial states of each of the two controllers.}
\label{fig:linear03main1}
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=0.45\textwidth]{linear_03plus/04_pi.png}
\includegraphics[width=0.45\textwidth]{linear_03plus/05_x_c.png}
\caption{Results of simulations on the serial test case:
Control signal (left) and the state of the controllers (right) as functions of time, for the two controllers and two initial states of each of the two controllers.}
\label{fig:linear03main2}
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=0.45\textwidth]{linear_03plus/02_Gen_1_power.png}
\includegraphics[width=0.45\textwidth]{linear_03plus/03_Gen_2_power.png}
\caption{Results of simulations on the serial test case:
Powers at the first 60 DERs (left) utilising probability functions $g_{i2}$ of \eqref{eq:prob-func} and following 60 DERs (right) utilising probability functions $g_{i1}$ of \eqref{eq:prob-func}.}
\label{fig:linear03main3}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{case118_pf_01/118_07_pi_signal.png}
\caption{Results of simulations on IEEE 118-bus test case:
Control signal, as a function of time, for the two controllers and two initial states of each of the two controllers.}
\label{fig:sim2sig}
\end{figure}
\subsection{A Transmission-System Test Case}
Next, let us demonstrate the analogous results on the standard IEEE 118-bus test case with minor modifications. Notably, the ensemble is connected to two buses of the transmission system. At bus 10, 1 generator with a maximum active power output of 450 MW was replaced by 8 DERs with 110 MW of power each, out of which 4 DERs used probability function $g_{i1}$, where $x_{01}=660$.
The other four DERs used function $g_{i2}$, where $x_{02}=110$ and $\xi=100$.
At bus 25, 1 generator with maximum active power output of 220 MW was replaced by four generators with active power output of 110 MW, out of which
two generators used probability function $g_{i1}$, and the other two used function $g_{i2}$ with $r=660$.
As in the serial test case, we have repeated the simulations 500 times, considering the time horizon of 2000 time steps each.
Figure \ref{fig:sim2sig} yields analogical results to
Figure \ref{fig:linear03main2} in the serial test case, where a different initial state $x_c (0)$ of the controller yields a very different control signal produced by the PI controller. This is not the case for the lag approximant.
Consequently, some DERs may be activated much more often than others, all else being equal, solely as an artifact of the use of PI control.
Many further simulation results illustrating this behaviour are available in the Supplementary material on-line.
\section{Conclusions and Further Work}
We have introduced a notion of predictability of fairness in load aggregation and demand-response management, which relies on the concept of unique ergodicity \cite{9445023}, which in turn is based on the existence of a unique invariant measure for a stochastic model of a closed-loop model of the system.
Related notions of unique ergodicity have recently been used in social-sensing \cite{9445023}, and two-sided markets \cite{ghosh2021unique}; we envision it may have many further applications in power systems.
The notions of predictability and fairness, which we have introduced, may be violated, if controls with poles on the unit circle (e.g., PI) are applied to load aggregation.
In particular, the application of the PI controller destroys the ergodicity of the closed-loop model.
Considering the prevalence of PI controllers within load frequency control, this is a serious concern.
On the other hand, we have demonstrated that certain other controllers guarantee predictability and fairness.
An important direction for further study is to consider unique ergodicity of stability-constrained ACOPF \cite{zarate2010opf,li2016sqp,wang2018sdp,li2019sequential,nojavan2020voltage,nikkhah2020stochastic,9511198}, or related controllers based on stability-constrained model-predictive control (MPC). We believe there are a wealth of important results ahead.
\paragraph*{Acknowledgements}
Ramen and Bob were supported in part by Science Foundation Ireland grant 16/IA/4610.
Jakub has been supported by OP VVV project CZ.02.1.01/0.0/0.0/16\_019/0000765 ``Research Center for Informatics''.
\FloatBarrier
|
1,108,101,565,465 | arxiv | \section{Introduction}
High-energy electrons have a wide range of applications, ranging from fundamental research in particle physics to medical diagnostics and treatment. However, conventional acceleration methods require long acceleration lengths, which also acts to limit the maximum attainable energy to some tens of GeV.
One promising alternative technique is wakefield acceleration, first proposed by Tajima and Dawson\cite{lwfa-tajimadawson}. A short driver, either a laser pulse or charged particle beam, is used to excite a plasma wave. The resulting charge imbalance can lead to large electric fields -- much larger than can be supported by conventional media. A witness bunch within the wake may then be accelerated.
\edit{The majority of works to date have considered internal injection, due to the relative ease with which high-energy electrons can be achieved. The first experiments made use of self-injection, in which the plasma wave is driven beyond the wavebreaking limit, allowing plasma electrons to be trapped within the accelerating phase of the wake \cite{lwfa-modena-selfinjection}. Several schemes have since been proposed to allow better control of the injection process \cite{lwfa-malka-review}, with a view to improve the energy-spread and emittance of the accelerated electron beam. The use of an external source of electrons may also offer accelerated beams of higher quality \cite{lwfa-grebenyuk-external2014}.}
The AWAKE project\cite{pwfa-AWAKE-2016} will make use of a proton driver, potentially allowing electron acceleration to TeV energies\cite{pwfa-caldwell-protondriven,pwfa-caldwell-selfmodulation}, orders of magnitude higher than current state of the art. The long proton driver interacts with its own wake via the self-modulation instability, leading to the creation of a train of bunches\cite{pwfa-kumar-selfmodulation}, making efficient wakefield generation possible. \edit{In the proposed experiment, an externally injected electron bunch will be accelerated.} Alternatively, high energies could be achieved by making use of a staged laser-driven wakefield accelerator\cite{wakefield-steinke-staged}, with the accelerated electrons from each stage injected into the next. In both cases the dynamics of external electron injection will play an important role.
In this paper, we investigate the influence of injection position on the energy gain of electrons in a plasma wake. In Section~\ref{dependence} we show that this dependence is more complex than previously realised, with narrow filaments away from the centre of the wake in which high acceleration can be achieved. Section~\ref{channels} explains the underlying physics of these quasi-stable injection channels, resulting from electron dephasing with the wake. The stability of the effect and the applicability to experiments is discussed in Section~\ref{experiment}, and conclusions are drawn in Section~\ref{conclusion}.
\section{Acceleration dependence on injection position}\label{dependence}
In order to gauge the influence of injection position on energy gain, a parameter scan was carried out using a series of test particle simulations. A slice in the $x$--$y$ plane was sampled, intersecting with the centre of the wake. The results are shown in Fig.~\ref{fig:scan2d}.
The electromagnetic wakefields were generated using the fully three-dimensional quasistatic version of the PIC code VLPL \cite{pic-pukhov-quasistatic}. A section of the associated potential is shown in Fig.~\ref{fig:scan2d}a. \edit{A short, pre-modulated proton beam was used to give a wake with parameters relevant to the AWAKE experiment, with a driver energy of 400~GeV, and a plasma density of $7\times10^{14}$~cm$^{-3}$ (corresponding to $\lambda_p=2\pi c/\omega_p=1.26$~mm). For simplicity, injection into a low-amplitude wake, for which the plasma response remains linear, is first considered. A plasma modulation depth of 8.5\% is therefore used, giving a maximum accelerating field of 220~MV/m. Comparison to a larger accelerating field, as predicted for AWAKE, as well as a discussion of the use of a laser or electron driver, is made in Section~\ref{experiment}.}
\edit{The wake is assumed to remain static in the co-moving frame, $\xi=x-v_gt$, with $v_g$ the group velocity of the driver, which is valid for a slowly evolving drive beam. This is somewhat different to the proposed AWAKE experiment, in which the electron beam is injected into the wake during self-modulation, leading to a growing wake with a non-constant phase velocity\cite{pwfa-pukhov-selfmodulationphase,pwfa-schroeder-growthandphasevelocity}. The case treated here is instead applicable to injection into the developed wake\cite{pwfa-muggli-injection,pwfa-AWAKE-2014}.}
\edit{In the figure, the driver is located at some position in the $+\xi$ direction. Electrons were placed in the wake with an initial energy of 5~MeV, propagating parallel to the driver. Experimentally, this could be realised via injection by a co-propagating electron beam entering the plasma behind the drive beam. Although not considered here, it is worth noting that such injection is complicated by edge-effects upon entering the plasma\cite{pwfa-AWAKE-path,pwfa-lotov-selfmodulationtrapping,pwfa-AWAKE-2014}.}
Particles with different initial positions were propagated in the wakefield using a Boris push. Calculations show that radiation reaction is negligible in this regime. The energy attained after a driver propagation distance of 25~cm (corresponding to $\sim200\lambda_p$) is shown in Fig.~\ref{fig:scan2d}b. As expected, the largest acceleration occurs for particles which are injected into the wake where the fields are both focussing and accelerating. However, it can immediately be seen that there additionally exist narrow filaments away from the centre of the wake in which high energy can be achieved.
Figure~\ref{fig:scan2d}c shows the final transverse position of injected electrons. It can be seen that for fixed $\xi$, there exist regions in which electrons are alternately ejected in the positive and negative $y$ directions, corresponding to the particles executing a different number of half-oscillations in the wake before being ejected. This behaviour is strongly nonlinear, arising due to the nonlinearity of the focussing electric fields, and also due to the larger forward acceleration experienced by particles nearer the axis, which increases both their inertia and the time taken to dephase with the wake. At each transition there is a narrow band in which the final transverse displacement remains small, which correlate with the high-energy filaments.
\begin{figure}
\includegraphics{fancy_5eV.eps}
\caption{Influence of injection position on acceleration after 25~cm ($\sim200\lambda_p$). a) Wakefield potential, $-k_p\phi$, assumed to be static in this frame. Blue areas will act to trap electrons. The driver is located in the $+\xi$ direction. b) Final electron electron energy against initial position. c) Final transverse position against initial position.}
\label{fig:scan2d}
\end{figure}
\section{Quasi-stable injection channels}\label{channels}
In order to better understand the underlying physics of the filament structure, we consider the trajectories of three individual particles. Figure~\ref{fig:traj}a shows the final energy for varying initial transverse position $y_0$ for a fixed $\xi_0=0.5$ \comment{(equivalent to a slice taken through Fig.~\ref{fig:scan2d})}. We choose three particles with initial positions, marked on the plot, corresponding to a local maximum in the final energy ($y_0/\lambda_p=0.4963$) and two straddling points, taken at $\pm 0.02\lambda_p$. Figures ~\ref{fig:traj}b, c and d show the corresponding evolution of $\xi$, $y$ and the energy.
\begin{figure}
\includegraphics{evolution.eps}
\caption{Trajectories of individual particles in the wake. a) Slice showing final electron energy against initial transverse position for $\xi_0=0.5$. The three initial positions to be tracked are marked. b) Evolution of $\xi$ against time. c) Evolution of $y$ against time. d) Evolution of the energy against time.}
\label{fig:traj}
\end{figure}
The initial energy of the injected electrons ($5$~MeV) is much lower than that of the driver, and so the particles rapidly fall backwards in the co-moving frame, as seen in Fig~\ref{fig:traj}b. As they dephase, they pass through the focussing/accelerating phase of the wake and begin to gain energy, as seen in Fig.~\ref{fig:traj}d.
The particles are injected away from the centre of the wake, and so they oscillate in the $y$ plane, and as such experience a smaller average accelerating field than those injected close to the axis. As a result they do not gain enough energy to become trapped in the wake, and continue to dephase until they reach the defocussing/accelerating phase of the wake. The defocussing field then causes the electrons to be ejected from the wake in either the $+y$ or $-y$ direction. The direction in which particles are ejected depends on their position and momentum as they exit the focusing phase, which in turn depends on their initial position. On the threshold between ejection in the positive and negative directions, there exist particles which achieve a temporary equilibrium, allowing them to remain in the wake for a longer period, and in doing so gain more energy than their neighbours. For the particles considered here, the central particle remains in the wake more than $50\,T_p$ longer than its neighbours, where $T_p = 2\pi/\omega_p$. We refer to these trajectories as ``quasi-stable'' -- the particles are globally unstable, but are able to climb the potential gradient in the defocussing phase, gaining more energy as they do.
The effect is illustrated in Fig.~\ref{fig:isotraj}, which shows the particle trajectories superimposed on the wakefield potential. Particles oscillate in the trapping potential (blue), before dephasing, falling back to the defocussing phase (red). Depending on the angle with which the particles approach the defocussing potential, they will be deflected to one side or the other. The angle of the central particle, however, is such that it climbs the potential gradient in the defocussing phase, rather than immediately being deflected. The transverse velocity of the particle decreases as it approaches the centre of the wake, allowing it to stay in the accelerating field for significantly longer, leading to increased energies.
\begin{figure}
\centering
\includegraphics{itraj_thick.eps}
\caption{Trajectories and energy evolution for the three particles in Fig.~\ref{fig:traj}, superimposed on the wakefield potential. The potential is shown in the vertical direction, with blue areas acting to trap electrons. The colour of the electron trajectories corresponds to the energy.}
\label{fig:isotraj}
\end{figure}
These quasi-stable injection channels are a direct consequence of the wakefield structure. As the accelerating and focussing fields are a quarter-wavelength out of phase, particles can continue to be accelerated after leaving the focussing field.
\section{Stability and applicability to experiments}\label{experiment}
\begin{figure}
\includegraphics{fancy_compare4.eps}
\caption{Influence of injection position on electron energy after 25~cm ($\sim200\lambda_p$) for different momenta of the injected electrons (Figs.~a,b) and different wake amplitudes (Figs.~c,d). a) 5~MeV electrons injected from below at a 1$^\circ$ angle to the driver into a 8.5\% modulated wake. b) 16~MeV electrons injected parallel to the driver into a 8.5\% modulated wake. c) 5~MeV electrons injected parallel to the driver into a 15\% modulated wake. d) 5~MeV electrons injected parallel to the driver into a 40\% modulated wake.}
\label{fig:scan2d2}
\end{figure}
The emergence of these quasi-static injection channels appears to be remarkably robust. Although the wakefields used here are fully electromagnetic, the same structure can be observed using a simple analytical form for a purely electrostatic field. \edit{Figures \ref{fig:scan2d2}a,b show the influence of changing the initial momentum of the injected electrons. Other parameters are the same as in Fig.~\ref{fig:scan2d}. In Fig.~\ref{fig:scan2d2}a, electrons are injected from below at a $1^\circ$ angle, i.e. propgating in the $+\xi$, $+y$ direction. Although this breaks the symmetry in the $y$ plane, it can be seen that the injection channels remain. Reflection of the beam, as discussed in Ref.~\onlinecite{pwfa-lotov-optimalangle}, is not observed in this case. Reflection is expected when the angle relative to the driver is below the optimal angle, which for these parameters is $0.3^\circ$. Modelling the effect would therefore require a longer propagation distance than the 25~cm used here.}
\edit{The effect of increasing the energy of the injected electrons is shown in Figure~\ref{fig:scan2d2}b. An electron energy of 16~MeV is used, propagating parallel to the driver. These faster electrons take longer to dephase, and so execute more oscillations in the trapping potential before being ejected from the wake. As the quasi-stable channels occur on the threshold between different number of oscillations, this results in the channels being more closely spaced in $y_0$. Changing the velocity of the driver will have a similar effect -- if the driver velocity is decreased, the injected electrons will take longer to dephase. Conversely, a higher-velocity driver will cause dephasing to occur sooner, and so the injection channels will be more widely spaced. The main influence of the choice of driver, whether it be an ion, electron or laser beam, is the resulting phase velocity of the wake. The propagation distance over which the driver can be considered ``slowly-varying'' may also be different.}
\edit{Figures \ref{fig:scan2d2}c,d show the influence modifying the modulation depth of the wakefield. Again, other parameters are as for Fig.~\ref{fig:scan2d}. In Fig.~\ref{fig:scan2d2}c, a 15\% modulation depth is used, corresponding to a peak accelerating field of 390~MV/m, near the saturation limit predicted in Ref.~\onlinecite{pwfa-pukhov-selfmodulationphase} for a self-modulating proton beam. As expected, the peak energy of electrons is larger than those obtained in Fig.~\ref{fig:scan2d} due to the larger accelerating fields. Although the trapping potential is larger, the transverse momentum of the electrons in the wake is also correspondingly increased, and so little difference is observed in the structure of the injection channels.}
\edit{The modulation depth is further increased to 40\% in Fig~\ref{fig:scan2d2}d. This is the maximum modulation depth observed in Ref.~\onlinecite{pwfa-lotov-twodimensionalequidistant}, and corresponds to a maximum accelerating field of $\sim1$~GV/m. It is important to note, however, that in that work, the wake quickly decays after reaching a maximum, while here we assume it is constant. Again, the larger accelerating field gives rise to an increased peak energy for the accelerated electrons. For this larger modulation depth, the wake is weakly nonlinear, resulting in a wakefield with anharmonic structure. The positively charged troughs of the wakefield become larger and shallower than the negatively charged peaks, which leads to a significantly larger trapping area than observed in Figs.~\ref{fig:scan2d} and \ref{fig:scan2d2}c. However, despite these differences, the injection channels outside the main trapping region remain.}
\begin{figure}
\includegraphics{length.eps}
\caption{Evolution of the energy against injection position over propagation distance. 5~MeV electrons are injected parallel to the driver into an 8.5\% modulated wake. The energies are shown (from lowest to highest) after 25, 50, 75 and 100~cm. The energies after 25~cm shown here equivalent to those in Fig.~\ref{fig:traj}a.}
\label{fig:scan1d_distance}
\end{figure}
For longer propagation distances, additional channels form closer to the centre of the wake, as shown in Fig.~\ref{fig:scan1d_distance}. This is because electrons closer to the wake centre experience a stronger accelerating field, and so take longer to dephase. As particles ejected at later times will have executed more oscillations in the trapping potential, the injection channels again become more tightly packed. Over time, the rate at which particles are ejected from the wake decreases, but the effect will occur as long as some particles continue to dephase.
\edit{In this work, we have considered the case of a wakefield with a constant phase velocity.} As the process relies on electrons dephasing, and ultimately being ejected from the wake, it is not itself useful as an acceleration technique. However, the result is still relevant to experiments in which external injection is used. Finding the optimal injection position in such schemes will require careful tuning of the angle and offset of the witness bunch relative to the driver. The presence of numerous ``false peaks'' in the parameter space topology makes this process more difficult than if there were only a single peak. Knowledge of this structure is therefore vital to facilitate such tuning.
These quasi-stable channels also allow particles to be accelerated to relatively high energies without becoming fully trapped in the wake. These electrons will be ejected from the wake throughout the acceleration process, and so precautions must be taken to avoid degradation of any focussing magnets.
\edit{The result may also prove significant in regimes where the phase velocity of the wake is not fixed. A decrease in the wake velocity could arise, for example, due to erosion of the driver head or the use of a plasma density gradient. In this case, it may be possible for electrons which have dephased to re-enter the trapping region of the wake. Such an investigation is beyond the scope of this work, but merits future study.}
\section{Conclusion}\label{conclusion}
The influence of injection position on the energy gain in a wakefield accelerator was investigated through the use of test-particle simulations. The fields themselves were generated using the quasistatic version of the PIC code VLPL. We observe, for the first time, the presence of complex structure in the parameter space, with narrow filaments in the injection position, away from the wake axis, in which relatively high energy may be achieved.
It is shown that these filaments correspond to quasi-stable injection channels. Electrons that are not trapped in the wake dephase, falling back to the defocussing phase of the wakefield. For a narrow range of initial positions, they approach the defocussing potential with an angle such that they are not immediately ejected from the wake. This allows them to remain in the wake for significantly longer than their neighbours, and so gain more energy.
The result provides significant insight into the process of external injection, and is relevant for the planning and optimisation of wakefield acceleration experiments in which external injection is used, such as the forthcoming AWAKE project.
\section{Acknowledgements}
JPF and AP would like to acknowledge funding from DFG TR18, EUCard$^2$ EU FP7 and BMBF. MWT would like to thank G\"otz Lehmann and Matthias Dellweg for helpful discussions.
|
1,108,101,565,466 | arxiv |
\section{Detailed setup}
The simplified scheme of the experimental setup shown in Fig.~2 in the main text does not show the implementation of the variable splitting-ratio beam-splitters, phase shifters, and the interferometer stabilization system. In Fig.~\ref{fig:detailed_setup} the detailed scheme of our setup can be found. The variable splitting-ratio beam-splitter is built from a polarizing beam-splitter and two half-wave plates. After each beam-splitter both beams possess horizontal polarization.
To phase-stabilize the interferometer in the setup, a light beam emitted from an additional laser at the central wavelength of 710~nm is injected into the unused input port of the beam-splitter. The two created spatial modes of light are recombined at the dichroic mirror and the resulting interfering beam is monitored by a fast photodiode. The detection signal is processed by a PID controller and a feedback signal is fed to the piezo-actuator-driven mirror in order to compensate for phase fluctuations. For comparison, in Fig.~\ref{fig:phase_stability} the time dependence of the coincidence count rate with and without stabilization is presented. This active feedback loop stabilization system also works as a fine phase shifter, for details see section ``Phase adjustment.'' The coarse adjustment of the phase can be done by a trombone system built in one arm of the interferometer.
The OAM mode shifter is inserted into the pump beam instead of the down-converted beams, as suggested by the principle scheme in Fig.~1 in the main text, for the following reason. Requirements on the temperature of nonlinear crystals and on the frequencies of down-converted photons did not allow for perfect collinearity of the generated pairs. In order to operate optimally the mode shifter has to be precisely centered with respect to the beam it acts upon. The failure to satisfy this condition for both photons of the pair leads to generation of undesirable higher-order OAM terms and a spread of the resulting OAM spiral spectrum.
\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{phase_stability}
\caption{\label{fig:phase_stability}Comparison of coincidence-count signal fluctuations for the case when the interferometer in the setup is actively stabilized (green solid line) and when it is not (orange dashed line). Photon pairs coming from crystals A and B, pumped with a beam having zero quanta of OAM, were collected in time steps of one second. Gray shaded areas correspond to one standard deviation region of collected data when Poissonian counting statistics is assumed.}
\end{figure}
\begin{figure}
\includegraphics[scale=1]{visibility_plot_Hz_rad}
\caption{\label{fig:interference}Interference of two SPDC processes in crystals A and B in the fundamental OAM mode $\ket{0,0}$. Coincidence counts are collected while the relative phase $\varphi_1$ is changed. Errors are determined assuming Poissonian counting statistics. The dashed line represents the fit of the displayed data. The obtained visibility is $97.1 \pm 0.5 \%$.}
\end{figure}
\begin{figure*}[htbp]
\includegraphics[width=1\textwidth]{detailed_setup}
\caption{\label{fig:detailed_setup}Detailed setup. For a description of the basic setup see Fig.~2 in the main text. The two interferometers are phase-locked by active feedback systems. An additional laser diode with central wavelength 710~nm is used to provide a locking-system beam that is injected into unused ports of beam-splitters and leaves the setup through dichroic mirrors (DM). After filtering out the pump beam by an interference filter (IF), the interference fluctuations of the locking-laser beam are detected by a fast photodiode (FP). The obtained signal is processed by a PID controller and a feedback signal is sent to a piezo actuator attached to one of the mirrors in the interferometer. The relative phases of the down-converted beams (denoted by $\varphi_1$ and $\varphi_2$ in the main text) can be adjusted by trombone systems (TS) and by a proper setting of polarization of the corresponding locking-system beam. This is accomplished by a series of two quarter-wave plates (QWP) and one half-wave plate (HWP) as is explained in the text. Magnitudes of individual terms in the quantum state are controlled by setting the splitting ratio of the beam-splitters. A variable splitting-ratio beam-splitter is implemented by a polarizing beam-splitter (PBS) with two half-wave plates. All three crystals are 10 mm long ppKTP non-linear crystals. Det -- single photon detector, SPP -- spiral phase plate, BPF -- band-pass filter, SLM -- spatial light modulator.}
\end{figure*}
\begin{figure}[!t]
\includegraphics[width=0.49\textwidth]{3d_density_matrices_ReIm}
\caption{Three selected three-dimensionally entangled states. Real and imaginary parts of density matrices are shown. Green and orange bars represent positive and negative values of reconstructed density matrices, respectively. Gray bars represent the theoretical expectation. Fidelities of the measured states with their reference states are $87.0 \pm 0.5 \%$, $89.0 \pm 0.4 \%$ and $84.8 \pm 0.8 \%$, respectively. (a) State $\ket{\psi_1} = 1/\sqrt{3}(\ket{0,0}+\ket{2,2}+\ket{-2,-2})$. (b) State $\ket{\psi_4} = 1/\sqrt{3}(\ket{0,0}-\ket{2,2}-\ket{-2,-2})$. (c) State $\ket{\psi_5} = 1/\sqrt{22}(2 \ket{0,0} + 3 \ket{2,2} + 3 \ket{-2,-2})$.}
\label{fig:3d_density_matrices_ReIm}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=0.49\textwidth]{2d_density_matrices_ReIm}
\caption{Two selected two-dimensionally entangled states. Real and imaginary parts are shown. Green and orange bars represent positive and negative values of reconstructed density matrices, respectively. Gray bars represent the theoretical expectation. (a) State $\ket{\Phi^+} = 1/\sqrt{2}(\ket{0,0}+\ket{2,2})$ with fidelity $90.4 \pm 0.5 \%$. (b) State $\ket{\Phi^-} = 1/\sqrt{2}(\ket{0,0}-\ket{2,2})$ with fidelity $89.1 \pm 0.5 \%$.}
\label{fig:2d_density_matrices_reim}
\end{figure}
\begin{figure}
\includegraphics[scale=.5]{spiralPlot}
\caption{\label{fig:spiralspectrum}%
Normalized coincidences for different projective measurements of OAM in the computational basis. Photon pairs produced in crystal A, pumped with a beam having zero quanta of OAM, were detected and coincidence counts collected for different choices of projections. On the two SLMs, see the detection system in Fig.~\ref{fig:detailed_setup}, the wavefronts corresponding to OAM modes $-2$, $\ldots$, $2$ were projected. In the ideal case, only diagonal entries would be nonzero. The coincidence rate for $\ket{0,0}$ mode is more than twenty times higher than the next highest coincidence rate.
}
\end{figure}
\section{Coherence conditions}
The generation of quantum states via the concept of entanglement by path identity requires coherent and indistinguishable photon-creation processes. To verify a sufficient level of coherence in our setup, the spiral phase plate was removed from the setup in Fig.~\ref{fig:detailed_setup} and the interference between different SPDC processes in the zero OAM mode was measured. The quality of the coherence is quantified by the interferometric visibility $V = (\text{Max}(D) - \text{Min}(D))/(\text{Max}(D) + \text{Min}(D))$, where $D$ is the coincidence count rate. Results for crystals A and B are shown in Fig.~\ref{fig:interference}. The observed visibility exceeds 97 \% in this case and the two SPDC processes in crystals A and B thus exhibit a high degree of coherence. Analogous results were also obtained for crystals B and C.
In general, the following relation has to be satisfied in order to observe interference for collinear SPDC processes. Let $L_{\mathrm{coh}}$ be the coherence length of the pump laser, which is in our case greater than 2 cm. Moreover, let $L_{\mathrm{p, A}}$ and $L_{\mathrm{p, B}}$ be the distances traveled by the pump beam from the beam splitter to crystals A and B, respectively. The physical conditions for coherence of corresponding SPDC processes are then given by \cite{coherencecond1,coherencecond2}
\begin{equation}
\abs{L_{\mathrm{p, B}} - L_{\mathrm{p, A}} - L_{\mathrm{SPDC}}} \leq L_{\mathrm{coh}},
\end{equation}
where $L_{\mathrm{SPDC}}$ is the propagation distance of down-converted photons from crystal A to crystal B. In other words, the optical path length difference between the two arms of the interferometer must be within the coherence length of the pump laser.
\section{State tomography results}
In Fig.~\ref{fig:3d_density_matrices_ReIm} the real and imaginary parts of density matrices are shown that correspond to states presented in Fig.~3 in the main text. Analogously, in Fig.~\ref{fig:2d_density_matrices_reim} the real and imaginary parts of two-dimensional states presented in Tab.~I in the main text are shown.
\section{Spiral spectrum}
Typically, the state of photon pairs $\ket{\psi}$ produced in an SPDC process contains a non-negligible admixture of higher-order OAM terms
\begin{multline}
\ket{\psi} = \alpha_0 \ket{0,0} + \alpha_1 (\ket{1,-1} + \ket{-1,1}) + \\ \alpha_2 (\ket{2,-2} + \ket{-2,2}) + \ldots
\end{multline}
Magnitudes of these contributions in general decrease for increasing OAM order. The precise relationship between the OAM order $k$ and its complex amplitude $\alpha_k$ is governed by several tunable parameters \cite{miatto}. In order for our scheme, presented in the main text, to work properly, these parameters have to be chosen such that all higher-order OAM terms coming from the SPDC processes are significantly suppressed. As shown in Fig.~\ref{fig:spiralspectrum} for crystal A, we were able to suppress the probability $|\alpha_1|^2$ of detecting the photons in the first OAM order below five percent of the probability of detecting them in the zero mode $|\alpha_0|^2$. Similar results were obtained for crystals B and C as well. These data justify our assumption in the main text that SPDC process produces photons only in their zero OAM mode.
\section{Phase adjustment}
Relative phases in generated quantum states can be tuned precisely by a series of \mbox{quarter-,} half-, and quarter-wave plates, henceforth referred to as the QHQ scheme, that are inserted into the locking-laser beam as shown in Fig.~\ref{fig:detailed_setup}. The QHQ scheme manipulates the local phase between the horizontal and vertical polarization components of the locking-laser beam. At the polarizing beam-splitter the relative phase between polarizations translates into relative phase between the two modes of propagation of the locking-laser beam through the interferometer. After recombination of the two paths at the dichroic mirror the intensity of the interfering beam is measured by a photodiode, which feeds the measured signal to the PID controller. The controller interprets the intensity change as unwanted fluctuation and offsets the piezo actuator to compensate for it. This way the phase change is imprinted into the pump beam and therefore into the down-converted photons as well.
When quarter-wave plates in the QHQ scheme are rotated correctly, the middle half-wave plate alone can be turned to adjust conveniently the phase in generated quantum states. In what follows, the working principle of QHQ scheme is explained.
In Jones matrix formalism, a quarter-wave plate (Q) and a half-wave plate (H), rotated by angle $\alpha$ with respect to the vertical direction, are represented by
\begin{eqnarray}
Q(\alpha) & = & R(\alpha) \, \begin{pmatrix}
1 & 0 \\
0 & i
\end{pmatrix} \, R(-\alpha), \\
H(\alpha) & = & R(\alpha) \ \sigma_Z\ R(-\alpha)
\end{eqnarray}
respectively, where $R(\alpha)$ is a rotation matrix and $\sigma_Z$ is Pauli-Z matrix. Their forms read
\begin{equation}
R(\alpha) = \begin{pmatrix}
\cos(\alpha) & -\sin(\alpha) \\
\sin(\alpha) & \cos(\alpha)
\end{pmatrix}, \
\sigma_Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.
\end{equation}
We first explain the working principle of the QHHQ scheme, where two half-wave plates are used, and then show that this scheme is equivalent to the QHQ scheme. It can be shown that a single Q and a single H can be used to transform any elliptical polarization into a linear polarization. Such a linear polarization can then be easily rotated by a half-wave plate independently of the input polarization. Finally, a quarter-wave plate rotated by $\pi/4$ transforms a linearly polarized state with polarization angle $\varphi$ into an equally-weighted superposition of horizontal and vertical polarization components as
\begin{equation}
Q\left(\frac{\pi}{4}\right) \begin{pmatrix}
\cos(\varphi) \\ \sin(\varphi)
\end{pmatrix} = \frac{e^{i(\frac{\pi}{4}-\varphi)}}{\sqrt{2}} \begin{pmatrix}
1 \\ e^{i (2 \varphi - \frac{\pi}{2})}
\end{pmatrix}.
\label{eq:qwaveplate}
\end{equation}
The polarization angle $\varphi$ is therefore transformed into a relative phase. In total, $Q(\frac{\pi}{4}) H(\alpha) H(\beta) Q(\gamma)$ scheme allows one to obtain a beam with polarization of the form $H + e^{i \omega} V$, where $\beta$ and $\gamma$ depend on the input polarization as generated by the locking laser and relative phase $\omega$ depends effectively only on the rotation angle $\alpha$ of the half-wave plate.\\[1ex]
It is straightforward to prove two useful relations $H(\alpha) H(\beta) = H(\alpha - \beta) \sigma_Z$ and $\sigma_Z Q(-\gamma) = Q(\gamma) \sigma_Z$, so that
\begin{equation}
Q \left( \frac{\pi}{4} \right) H(\alpha) H(\beta) Q(\gamma) = Q \left(\frac{\pi}{4} \right) H(\alpha - \beta) Q(-\gamma) \sigma_Z.
\label{eq:qhq}
\end{equation}
The extra $\sigma_Z$ merely shifts the relative phase of the incoming beam by $\pi$, which is corrected for by the proper setting of $\beta$ and $\gamma$. We thus showed that $Q(\frac{\pi}{4}) H(\alpha - \beta) Q(-\gamma)$ scheme can be used to adjust the relative phase in the state of the locking-laser beam by turning the half-wave plate appropriately.
|
1,108,101,565,467 | arxiv | \section{Introduction}
Studying the statistical fluctuations of a system is a powerful
method to characterise the thermodynamic properties of a system.
As a matter of fact, the presence of a phase transition is signalled
by an enhancement of the fluctuations in the system.
The observables appropriate to extract this kind of information are called
susceptibilities. Properly, the susceptibility describes the response of
a system to an applied field. Given an operator we can usually determine
a corresponding susceptibility that is the second derivative of the free
energy density.
In this paper we are interested in studying the quark number susceptibility
(QNS) $\chi=\frac{\partial n_q}{\partial \mu}$: it is the response of $n_q$,
the quark number density, to an infinitesimal change in the quark chemical
potential $\mu$.
The QNS has been studied so far at finite temperature and zero chemical
potential, see Refs.~\cite{Gottlieb:1987ac, Allton:2005gk, Gavai:2008zr}.
This is largely because QNS is directly related to experimental
measurements of fluctuations observed in heavy ion
collisions~\cite{Asakawa:2000wh}.
At non-zero density and zero (or low) temperature regime, QNS is not commonly
studied; an application to QCD of rainbow approximation of the
Dyson-Schwinger approach to study the QNS can be found in
Ref.~\cite{He:2008zzb}. One reason for the lack of studies in this
regime is the well known sign problem: it is not possible, using
standard tools, to simulate lattice QCD at non-zero density.
In this paper we study the QNS at non-zero density and at low temperature
in the context of $SU(2)$ gauge theory, {\it i.e.} where simulations
are feasible.
\section{Two color QCD}
In two color QCD, {\it i.e.} $SU(2)$ gauge theory, quarks and antiquarks
live in equivalent representations of the group; the physical consequence is
that $q\bar{q}$ mesons and $qq$, $\bar{q}\bar{q}$ baryons are contained in the
same hadron multiplet. In the limit when $m_\pi \ll m_\rho$, {\it i.e.}
when the pion mass is very small compared with the first non-Goldstone hadron,
it is possible to study the system using the chiral perturbation theory ($\chi$PT)
limit~\cite{Kogut:2000ek}. The fundamental result is that only for
$\mu \geq \mu_0 \equiv \frac{1}{2} m_\pi$ the quark number density $n_q$ becomes
different from zero and at the same onset value $\mu_0$ also a condensate
$\langle qq \rangle \neq 0$ develops, signalling the spontaneous breakdown
of the global $U(1)$ baryon number symmetry: a superfluid phase appears.
In this phase there are tightly bound scalar diquarks but the hadrons are
weakly interacting between them:
therefore, just above the onset, the system is very dilute and is
described as a Bose Einstein Condensate (BEC).
Using $\chi$PT it is possible to determine the behaviour of different
observables; in particular, we can write down the prediction for $n_q$ and
its susceptibility $\chi$, at zero temperature $T=0$ and in the limit
$\mu \to \mu_0$ (right-hand limit):
\begin{equation}
n^{{\chi}PT}_q \approx 32 N_f F^2 \left( \mu - \mu_0 \right)
\ , \quad
\chi^{{\chi}PT} \approx 32 N_f F^2 -192 N_f \frac{F^2}{m_\pi} \left( \mu - \mu_0
\right)
\ ,
\label{nchinCPT}
\end{equation}
where $F$ is the pion decay constant and $N_f$ the number of fermions.
The diquark condensate, calculated in the same limits,
is given in terms of the chiral condensate at zero chemical potential:
\begin{equation}
\langle q q \rangle \approx
2 \langle \bar{q}q \rangle_0 \sqrt{\frac{\mu}{\mu_0} - 1} \ .
\label{diqCPT}
\end{equation}
There are reasons to think that above a certain
value of the chemical potential, $\mu \ge \mu_Q$, a degenerate system of
weakly interacting quarks is more stable~\cite{Hands:2006ve}.
For an ideal gas (Stefan Boltzmann (SB) limit) of massless quarks and gluons,
at $T=0$, we have:
\begin{equation}
n^{SB}_q=\frac{N_f N_c}{3 \pi^2} \mu^3 \ ,
\quad
\chi^{SB}=\frac{N_f N_c}{\pi^2} \mu^2 \ ,
\label{nchinSB}
\end{equation}
where $N_c$ is the number of colours.
In this case the superfluidity is explained by a BCS condensation of Cooper
pairs within a layer of thickness $\Delta$ around the Fermi surface; the
diquark condensate is therefore given by:
\begin{equation}
\langle qq \rangle \propto \mu^2 \Delta \ .
\label{diqSB}
\end{equation}
Comparing Eq.s~(\ref{nchinCPT}),~(\ref{diqCPT}) with
Eq.s~(\ref{nchinSB}),~(\ref{diqSB}), we see clearly that the two phases
are characterised by two quite different behaviours.
We consider also the order parameter related to the confinement
property of the theory: the Polyakov loop $L$; as discussed in
Ref.~\cite{Hands:2010gd} the theory becomes deconfined only after
$\mu \ge \mu_D$. Surprisingly, there is a regime, $\mu_Q < \mu < \mu_D$,
where the theory is confined, {\it i.e.} $\langle L \rangle = 0$,
but the other observables seem to suggest non interacting fermions:
a confined BCS phase. This phase could be the so called quarkyonic phase
introduced in Ref.~\cite{McLerran:2007qj}; arguments against the extension
of this idea to the case of $N_c=2$ can be found in Ref.\cite{Lottini:2011zp}.
\section{Calculation of the observables}
The fermion action with $N_f=2$ and with a diquark source term $J$,
necessary to study the diquark condensate, is
given by:
\begin{equation}
S_f=\bar{\psi}_1 M(\mu) \psi_1+\bar{\psi}_2 M(\mu) \psi_2
-J \bar{\psi}_1 (C \gamma_5) \tau_2 \bar{\psi}_2^{tr}
+J \psi_2^{tr} (C \gamma_5) \tau_2 \psi_1 \ ,
\end{equation}
where $M(\mu)$ is the standard Wilson fermion matrix at non-zero
chemical potential and $C$ is the charge conjugation operator.
If we introduce the change of variables $\bar{\phi}= -\psi_2^{tr} C \tau_2 $,
$\phi= C^{-1} \tau_2 \bar{\psi}_2^{tr}$, $\psi=\psi_1$, $\bar\psi=\bar\psi_1$,
it is possible to rewrite the action as:
\begin{equation}
S_f= ( \bar\psi \bar\phi )
\left(
\begin{array}{cc}
M(\mu) & J \gamma_5 \\
-J \gamma_5 & M(-\mu)
\end{array}
\right)
\left(
\begin{array}{c}
\psi \\
\phi
\end{array}
\right)
\equiv \bar\Psi \mathcal{M} \Psi \ .
\end{equation}
It is worth mentioning that
$
\det{(\mathcal{M}^\dagger\mathcal{M})}=\left[ \det{(M^\dagger M+ J^2)}
\right]^2
$,
therefore we can take the square root analytically, {\it i.e.} there is no
square root problem. The partition function becomes:
\begin{equation}
Z= \int dU d\bar{\Psi} d\Psi e^{-S_g-\bar\Psi \mathcal{M} \Psi} \ .
\label{pfnew}
\end{equation}
It is now easy to write down an expression for $n_q$ using
the matrix $\mathcal{M}$:
\begin{equation}
n_q= \frac{T}{V_s} \frac{\partial \ln{Z}}{\partial \mu}
= \frac{T}{V_s} \sum_{\alpha,\beta}
\langle - \bar\Psi_\alpha \left( \frac{\partial \mathcal{M}}
{\partial \mu} \right)_{\alpha,\beta} \Psi_\beta \rangle
=
\frac{T}{V_s} \langle \mbox{Tr} \left\{
\mathcal{M}^{-1} \frac{\partial \mathcal{M}} {\partial \mu} \right\}
\rangle \ .
\end{equation}
Moreover, from the definition of QNS we have:
\begin{equation}
\chi=\frac{\partial n_q}{\partial \mu}
=\frac{T}{V_s} \left\{ - \langle \left[ -\bar\Psi \frac{\partial
\mathcal{M}}{\partial \mu}
\Psi \right] \rangle ^2
+
\langle \left[ -\bar\Psi \frac{\partial \mathcal{M}}{\partial \mu}
\Psi \right]^2 \rangle
+
\langle \left[ -\bar\Psi \frac{\partial^2 \mathcal{M}}{\partial \mu^2}
\Psi \right] \rangle \right\} \label{chi3terms} \ .
\end{equation}
From this equation we can identify four different terms:
\begin{eqnarray}
T1&=& - \langle \left[ -\bar\Psi \frac{\partial \mathcal{M}}{\partial \mu}
\Psi \right] \rangle ^2
= - \langle \mbox{Tr} \left[ \mathcal{M}^{-1} \frac{\partial \mathcal{M}}
{\partial \mu} \right] \rangle ^2 \label{T1} \\
T2&=& + \langle \left[ -\bar\Psi \frac{\partial \mathcal{M}}{\partial \mu} \Psi
\right]^2 \rangle_{disc}=\langle \mbox{Tr} \left[ \mathcal{M}^{-1}
\frac{\partial \mathcal{M}}{\partial \mu}\right] \cdot \mbox{Tr}
\left[ \mathcal{M}^{-1} \frac{\partial \mathcal{M}}{\partial \mu}\right]
\rangle \label{T2} \\
C1&=& + \langle \left[ -\bar\Psi \frac{\partial \mathcal{M}}{\partial \mu}
\Psi \right]^2 \rangle_{conn}=- \langle \mbox{Tr} \left[ \mathcal{M}^{-1}
\frac{\partial \mathcal{M}}{\partial \mu} \mathcal{M}^{-1} \frac{\partial
\mathcal{M}}{\partial \mu}\right] \rangle \label{C1} \\
T3&=& + \langle \left[ -\bar\Psi \frac{\partial^2 \mathcal{M}}{\partial \mu^2}
\Psi \right] \rangle= \langle \mbox{Tr} \left[\mathcal{M}^{-1} \frac{\partial^2
\mathcal{M}}{\partial \mu^2} \right] \rangle \label{T3} \ .
\end{eqnarray}
We see that from the second term of Eq.~(\ref{chi3terms}) we get two terms,
namely $T2$ and $C1$, because there are two ways to contract the spinors.
The calculation of the traces is done by unbiased estimators, introducing
$N_\eta$ complex noise vectors $\eta$ with the properties:
$\langle \eta_x \rangle =0$ and $\langle \eta_x \eta_y \rangle = \delta_{x y}$.
For example, the determination of the following trace, used for $T1$ and $T2$,
is based on the relation:
\begin{equation}
\mbox{Tr} \left[ \mathcal{M}^{-1} \frac{\partial \mathcal{M}}{\partial \mu}
\right] =
\frac{1}{N_\eta}\sum
\eta^*_{x \alpha i}\left( \frac{\partial \mathcal{M}}{\partial \mu}
\right)_{x \alpha i ; y \beta j} \mathcal{M}^{-1}_{y \beta j ; z \gamma k}
\eta_{z \gamma k} \ .
\label{trmmdm}
\end{equation}
Because for $T2$ we need two independent source vectors we refer to this term
as disconnected term; the other three terms need only one source vector and
we call them connected terms.
\section{Some numerical issues}
The source vector used to determine the traces can characterised by
different noise distributions.
The standard method is based on the introduction of gaussian complex noise
vector but it is possible also to use a $Z_2$ complex noise vector
(see Ref.~\cite{Dong:1993pk} where the potential advantage are discussed).
In the $Z_2$ case, the complex noise vectors $\eta$
takes one of the four values $\left\{ \pm 1, \pm i \right\}$, chosen
independently with equal probability.
In Table~\ref{tablenoise} we present an example where we used
only three noise vectors, with the following parameters:
$\beta=1.70$, $\kappa=0.178$, $j=0.04$, $\mu=0.25$, $8^3 \times 16$.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\phantom{.} & T1 & T2 & T3 & C1 & $\chi$ \\
\hline
Gaussian & 2.25(8.30)E-06 & 5.00(5.74)E-05 & 0.4010(27) & -0.3681(107) & 3.1(1.0) \\
\hline
$Z_2$ & 6.3(10.0)E-06 & 7.4(49.9)E-06 & 0.3994(32) & -0.3655(66) & 3.17(52) \\
\hline
\end{tabular}
\end{center}
\caption{Gaussian vs $Z_2$ noise vectors.}
\label{tablenoise}
\end{table}
From this simple analysis, we can see that there is no any particular
advantage in using a $Z_2$ noise, so in the following we continue to use only
gaussian noise distributions.
We also tried to see what happen when we increase the number of noise vectors;
in Table~\ref{tablenumbersources} we present an example with the
same parameters used above.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
noise vectors & T1 & T2 & T3 & C1 & $\chi$ \\
\hline
3 & 2.25(8.30)E-06 & 5.00(5.74)E-05 & 0.4010(27) & -0.3681(107) & 3.1(1.0) \\
\hline
300 & 1.50(17)E-05 & 1.53(12)E-05 & 0.4013(14) & -0.3612(20) & 4.04(11)E-02 \\
\hline
\end{tabular}
\end{center}
\caption{The effect of different number of noise vectors.}
\label{tablenumbersources}
\end{table}
From this result it is evident how increasing the number of noise vectors
has a strong effect mainly on observables with a small value; the effect
is very limited if the observable is clearly different from zero.
\begin{figure}[ht]
\center
\vspace{5mm}
\includegraphics[width=7cm]{plots/plot-terms.eps}
\hspace{5mm}
\includegraphics[width=7.2cm]{plots/plot-t1vst2.eps}
\caption{(Left) The four terms of the QNS are plotted against the
chemical potential. (Right) The ratio between T1 and T2 is plotted against
the chemical potential.}
\label{plot1}
\end{figure}
In Fig.~\ref{plot1} (Left) we plot the four terms (note the sign of $T1$)
vs the chemical potential. The connected contribution $C1$ gives
clearly an important contribution either at low and high values of the
chemical potential; note the changing of sign around $a\mu \approx 0.66$.
The terms $T1$ and $T2$ are equal in magnitude but with opposite sign,
therefore their contribution cancels everywhere, Fig.~\ref{plot1} (Right);
in other words the variance of the quantity in
Eq.~(\ref{trmmdm}) is equal to zero. We see very strong fluctuations,
apparently not compatible with one, for $\mu < \mu_0$, then smoothed
fluctuations comparable with one until $\mu_D$ and after this a stable
ratio equal to one.
\section{Numerical results}
In this contribution we are going to show some results obtained for $\beta=1.9$
and $\kappa=0.168$. From previous works, see Ref.~\cite{Hands:2010gd} and
the references therein, we know that for these parameters $m_\pi=0.68(1)$,
therefore $a\mu_0 \approx 0.34$; moreover, there are signals of a BEC phase
for $a\mu \lesssim 0.45$.
We obtained interesting results comparing the QNS with the other
observables; here we want to stress an interesting
effect we have observed studying the system at different temperatures.
Note that the results we are presenting should eventually be
extrapolated to $j=0.00$.
In Fig.~\ref{plot2} (Left) we plot the ratio $\chi/\mu^2$, for three different
temperatures, versus the chemical potential. For an ideal gas of
quarks this ratio would be a constant, see
Eq.~(\ref{nchinSB}), and we see that
a plateau is actually present for $a\mu \lesssim 0.55$; after this value
we can see a sharp increase of the QNS.
Moreover, it is evident from this plot that the QNS is quite independent from
the temperature, {\it i.e.} we do not see any drastic deviations in the
behaviour of the three curves increasing $\mu$.
This is in contrast with the Polyakov loop behaviour that shows deconfinement
for three different values of the chemical potential, correspondingly at
the three different temperatures, Fig.~\ref{plot2} (Right).
\begin{figure}[ht]
\center
\vspace{5mm}
\includegraphics[width=7.2cm]{plots/plot-suscett_vsmusq.eps}
\hspace{5mm}
\includegraphics[width=7.2cm]{plots/plot-polyak2.eps}
\caption{(Left) Ratio $\chi/\mu^2$ versus $\mu$.
The horizontal dotted line marks the SB value $4/\pi^2$ and the vertical
dashed one marks the position of $\mu_0$. (Right) Polyakov
loop versus $\mu$.}
\label{plot2}
\end{figure}
It is instructive to compare our lattice numerical results
with the equations corresponding to Eq.~(\ref{nchinSB}) but taking in account
the finite volume and the lattice discretisation.
In Ref.~\cite{Hands:2006ve}, see Eq.~(26), the expression for
the quark number density $n_q^{SBL}$ for free Wilson fermions
on the lattice is presented. It is then easy to obtain $\chi^{SBL}$.
In Fig.~\ref{plot3} we plot the ratio between the measured QNS and $\chi^{SBL}$
for two values of the fermion mass: in the determination of $\chi^{SBL}$
we have to fix a value for the mass of the free fermion; unfortunately we do
not know this value, therefore we consider massless fermions (note that this
is the same limit used in Eq.~(\ref{nchinSB})) and a value of the order
of $m_\pi/2$.
In this case we observe a different behaviour for $a\mu \lesssim 0.45$,
reminiscent of the BEC phase, followed by a flat region compatible with
ratio one,
{\it i.e.} the system is behaving as free fermions, and then again we see
an increase for higher values of $\mu$.
These plots again confirm the above scenario: we do not see any abrupt
change for QNS, for any of the values of $\mu$, where instead the Polyakov
loop becomes different from zero.
Note that the QNS is often taken as an alternative
signal for deconfinement in lattice studies of the thermal QCD transition
{\it i.e.} there is a strong connection between $\chi$ and Polyakov
Loop, see Refs.~\cite{Aoki:2006br, Bazavov:2009zn, Hands:2010vw}.
Our results shows that this relation is not replicated at low temperature
and high density.
\begin{figure}[ht]
\center
\vspace{5mm}
\includegraphics[width=7.2cm]{plots/plot-suscett_2.eps}
\hspace{5mm}
\includegraphics[width=7.2cm]{plots/plot-suscett_3.eps}
\caption{The ratio between the measured QNS and the ideal value
for \emph{lattice} free fermions for two values of the fermion mass:
$m=0.00$ and $m=0.34$.
The vertical dashed lines mark the position of $\mu_0$.}
\label{plot3}
\end{figure}
\section{Conclusion}
In this work we have studied the quark number susceptibility
at non-zero density and low temperature in the case of two color QCD with
two flavours. We have studied the contribution of the different terms,
connected and disconnected, which contribute to it and finally we have
shown its behavior at finite temperature. Surprisingly, we have seen that
in this case there is no relation between the quark number susceptibility
and the Polyakov loop, {\it i.e.} in this context it cannot be used as
a signal for deconfinement as it is usually done at finite temperature and
zero quark number density.
This observation could suggest that the deconfinement transition
is not characterised by a liberation of additional degrees of freedom;
if this phenomenon is exclusive to two color QCD or related
to the non small ratio $m_\pi/m_\rho = 0.80(1)$ or, somehow, connected
to the presence of a quarkyonic phase, is under study.
Clearly, there is much to learn about deconfinement from studying
a new physical environment.
|
1,108,101,565,468 | arxiv | \section{Introduction}
Let $G$ be a connected reductive group over $\mathbb{Q}$ and $\mathbb{A}$ the ring of adeles of $\mathbb{Q}$.
An equidistribution theorem for a family of automorphic representations of $G(\mathbb{A})$ is one of
recent topics in number theory and automorphic representations.
After Sauvageot's important results \cite{Sau},
Shin \cite{Shin} proved so called a limit multiplicity formula which shows that
the limit of an automorphic counting measure is the Plancherel measure. It implies the
equidistribution of Hecke eigenvalues or Satake parameters at a fixed prime in a family of cohomological automorphic forms on
$G(\mathbb{A})$.
A quantitative version of Shin's result is given by
Shin and Templier \cite{ST}. A different approach is discussed in \cite{FLM} for $G={\mathop{\mathrm{GL}}}_n$ or ${\mathop{\mathrm{SL}}}_n$
treating more general automorphic forms which are not necessarily cohomological.
Note that in the works of Shin and Shin-Templier, one needs to consider all cuspidal representations in the $L$-packets. Shin suggested in \cite[the second paragraph in p. 88]{Shin} that one can isolate just holomorphic discrete series at infinity. In \cite{KWY,KWY1}, we carried out his suggestion and established equidistribution theorems for holomorphic Siegel cusp forms of degree 2. We should also mention Dalal's work \cite{Dalal} (cf. Remark \ref{STD}). See also some related works \cite{KL,KSTs}.
In this paper we generalize several equidistribution theorems to holomorphic Siegel cusp forms of general degree. A main tool is Arthur's invariant trace formula as used in the previous
work but we need a more careful analysis in the computation of unipotent contributions.
Let us prepare some notations to explain our results.
Let $G={\mathop{\mathrm{Sp}}}(2n)$ be the symplectic group of rank $n$ defined over $\mathbb{Q}$.
For an $n$-tuple of integers ${\underline{k}}=(k_1,\ldots,k_n)$ with $k_1\ge \cdots \ge k_n>n+1$, let
$D^{{\rm hol}}_{\underline{l}}=\sigma_{\underline{k}}$ be the holomorphic discrete series representation of $G(\mathbb{R})$ with the Harish-Chandra parameter $\underline{l}=(k_1-1,\ldots,k_n-n)$ or
the Blattner parameter ${\underline{k}}$.
Let $\mathbb{A}$ (resp. $\mathbb{A}_f$) be the ring of (resp. finite) adeles of $\mathbb{Q}$, and $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$.
For $S_1$ a finite set of rational primes, let $S=\{\infty\}\cup S_1$, and $\mathbb{Q}_{S_1}=\prod_{p\in S_1}\mathbb{Q}_p$, and $\mathbb{A}^{S}$
be the ring of adeles outside $S$
and $\widehat{\mathbb{Z}}^{S}=\prod_{p\not\in S_1}\mathbb{Z}_p$.
We denote by $\widehat{G(\mathbb{Q}_{S_1})}$ the unitary dual of $G(\mathbb{Q}_{S_1})=\prod_{p\in S_1}G(\mathbb{Q}_p)$ equipped with the Fell topology.
Fix a Haar measure $\mu^{S}$ on $G(\mathbb{A}^{S})$ so that
$\mu^{S}(G(\widehat{\mathbb{Z}}^{S}))=1$, and let $U$ be a compact open subgroup of $G(\Bbb A^{S})$.
Consider the
algebraic representation
$\xi=\xi_{\underline{k}}$ of the highest weight ${\underline{k}}$ so that it is isomorphic to
the minimal $K$-type of $D^{\rm hol}_{\underline{l}}$. We choose a test function $f_{S}=f_\xi h_U\in C^\infty_c(G(\mathbb{R}))\otimes C^\infty_c(G(\mathbb{A}^{S}))$ such that $f_\xi$ is a pseudo-coefficient of $D^{\rm hol}_{\underline{l}}$ normalized as ${\rm tr}(\pi_\infty(f_\xi))=1$ and $h_U$ is the characteristic function of $U$.
Then we define a measure on $\widehat{G(\mathbb{Q}_{S_1})}$ by
\begin{equation}\label{mu}
\widehat{\mu}_{U,S_1,\xi,D^{\rm hol}_{\underline{l}}}:
=\frac{1}{{\rm vol}(G(\mathbb{Q}){\backslash} G(\mathbb{A}))\cdot {\rm dim}\, \xi} \sum_{\pi^0_{S_1}\in \widehat{G(\mathbb{Q}_{S_1})}}
\mu^S(U) \, m_{\rm cusp}(\pi^0_{S_1}; U,\xi,D^{\rm hol}_{\underline{l}}) \, \delta_{\pi^0_{S_1}},
\end{equation}
where $\delta_{\pi^0_{S_1}}$ is the Dirac delta measure supported at $\pi^0_{S_1}$ with respect to the Plancherel measure
$\widehat{\mu}^{{\rm pl}}_{S_1}$ on $\widehat{G(\mathbb{Q}_{S_1})}$
and for a given unitary representation $\pi^0_{S_1}$ of $G(\mathbb{Q}_{S_1})$,
\begin{equation}\label{mult}
m_{\rm cusp}(\pi^0_{S_1}; U,\xi,D^{\rm hol}_{\underline{l}})=
\sum_{\pi\in \Pi(G(\mathbb{A}))^0\atop \pi_{S_1}\simeq \pi^0_{S_1},\,\pi_\infty\simeq D^{\rm hol}_{\underline{l}}}m_{\rm cusp}(\pi) \, {\rm tr}(\pi^S(h_U)),
\end{equation}
where $\Pi(G(\mathbb{A}))^0$ stands for the isomorphism classes of all irreducible unitary cuspidal representations of
$G(\mathbb{A})$, and $\pi^S=\otimes_{p\notin S} \pi_p$.
To state the equidistribution theorem, we need to introduce the Hecke algebra $C^\infty_c(G(\mathbb{Q}_{S_1}))$ which is dense
under the map $h\mapsto \widehat{h}$, where $\widehat{h}(\pi_{S_1})=
{\rm tr}(\pi_{S_1}(h))$ is in
$\mathcal{F}(\widehat{G(\mathbb{Q}_{S_1})})$ consisting of suitable $\widehat{\mu}^{{\rm pl}}_{S_1}$-measurable functions
on $\widehat{G(\mathbb{Q}_{S_1})}$. (See \cite[Section 2.3]{Shin} for that space.)
Let $N$ be a positive integer. Put $S_N=\{p\ {\rm prime} :\ p|N\}$. We assume that $S_1\cap S_N=\emptyset$.
For each positive integer $N$, we denote by $K(N)$
the principal congruence subgroup of $G(\widehat{\mathbb{Z}})\cap G(\mathbb{A}^{S})$ of level $N$. For each rational prime $p$, let us consider
the unramified Hecke algebra $\mathcal{H}^{{\rm ur}}(G(\mathbb{Q}_p))\subset C^\infty_c(\mathbb{Q}_p)$ and
for each $\kappa >0$, $\mathcal{H}^{{\rm ur}}(G(\mathbb{Q}_p))^\kappa_{\le 1}$, the linear subspace of
$\mathcal{H}^{{\rm ur}}(G(\mathbb{Q}_p))$ consisting of all Hecke elements whose heights are less than $\kappa$
and complex values have absolute values less than 1. (See (\ref{height}).)
Our first main result is
\begin{thm}\label{main1} Fix ${\underline{k}}=(k_1,\ldots,k_n)$ satisfying $k_1\ge \cdots \ge k_n>n+1$. Fix a positive integer $\kappa$.
Then there exist constants $a,b$, and $c_0\neq 0$ depending only on $G$ such that
for each $h_1\in \otimes_{p\in S_1} \mathcal{H}^{{\rm ur}}(G(\mathbb{Q}_p))^\kappa_{\le 1}$,
$$\widehat{\mu}_{K(N),S_1,\xi,D^{\rm hol}_{{\underline{l}}}}(\widehat{h_1})=\widehat{\mu}^{{\rm pl}}_{S_1}\left(\widehat{h_1}\right)+
O\left(\left(\prod_{p\in S_1}p\right)^{a\kappa+b} N^{-n}\right),
$$
if $N\ge c_0 \prod_{p\in S_1}p^{2n\kappa}$ and $N\to\infty.$
\end{thm}
Let us apply this theorem to the vertical Sato-Tate theorem
and higher level density theorem for standard $L$-functions of holomorphic Siegel cusp forms.
Let $S_{\underline{k}}({\Gamma}(N))$ be the space of holomorphic Siegel
cusp forms of weight ${\underline{k}}$ with respect to $\Gamma(N)$ (see the next section for a precise definition), and let
$HE_{\underline{k}}(N)$ be a basis consisting of all Hecke eigenforms outside $N$. We can identify $HE_{\underline{k}}(N)$ with a basis of $K(N)$-fixed vectors in the set of cuspidal representations of ${\mathop{\mathrm{Sp}}}(2n,\Bbb A)$ whose infinity component is (isomorphic to) $D^{\rm hol}_{{\underline{l}}}$. (See Section 2 for the detail.)
Put $d_{\underline{k}}(N)=|HE_{\underline{k}}(N)|$. Then we have \cite{Wakatsuki}, for some constant $C_{\underline{k}}>0$,
\begin{equation}\label{dimension}
d_{{\underline{k}}}(N)=C_{{\underline{k}}} C_N N^{2n^2+n}+O_{{\underline{k}}}(N^{2n^2}),
\end{equation}
where $\displaystyle C_N=\prod_{p|N} \prod_{i=1}^n (1-p^{-2i})$. Note that $\displaystyle\prod_{i=1}^n \zeta(2i)^{-1}< C_N< 1$.
For each $F\in HE_{\underline{k}}(N)$, we denote by
$\pi_F=\otimes'_p\pi_{F,p}$ the corresponding automorphic cuspidal representation of $G(\mathbb{A})$.
Henceforth we assume that
\begin{equation}\label{suff-reg}
k_1>\cdots>k_n>n+1.
\end{equation}
Then the Ramanujan conjecture is true, namely, $\pi_{F,p}$ is tempered for any $p$
(see Theorem \ref{Ram}).
Unfortunately, this assumption forces us to exclude the scalar-valued Siegel cusp forms.
Let $\widehat{G(\mathbb{Q}_p)}^{{\rm ur,temp}}$ be the subspace of $\widehat{G(\mathbb{Q}_p)}$ consisting of
all unramified tempered classes. We denote by $(\theta_1(\pi_{F,p}),\ldots,\theta_n(\pi_{F,p}))$ the element of $\Omega$ corresponding to $\pi_{F,p}$
under the isomorphism
$\widehat{G(\mathbb{Q}_p)}^{{\rm ur,temp}}\simeq [0,\pi]^n/\mathfrak S_n=:\Omega$. Let $\mu_p$ be the measure on $\Omega$ defined in Section \ref{vertical}.
\begin{thm}\label{Sato-Tate-tm} Assume (\ref{suff-reg}). Fix a prime $p$.
Then the set $\left\{(\theta_1(\pi_{F,p}),\ldots,\theta_n(\pi_{F,p}))\in \Omega\ |\ F\in HE_{\underline{k}}(N) \right\}$ is
$\mu_p$-equidistributed in $\Omega$, namely, for each continuous function $f$ on $\Omega$,
$$\lim_{N\to \infty \atop (p,N)=1}\frac{1}{d_{\underline{k}}(N)}\sum_{F\in HE_{\underline{k}}(N)}f(\theta_1(\pi_{F,p}),\ldots,\theta_n(\pi_{F,p}))=
\int_{\Omega}f(\theta_1,\ldots,\theta_n)\mu_p.
$$
\end{thm}
By using Arthur's endoscopic classification, we have a more finer version of the above theorem.
Under the assumption (\ref{suff-reg}), the global $A$-parameter describing $\pi_F,\ F\in HE_{\underline{k}}(N)$,
is always semi-simple. (See Definition \ref{cap-endo}.) Let $HE_{\underline{k}}(N)^{{g}}$ be the subset of $HE_{\underline{k}}(N)$ consisting of $F$ such that the global $A$-packet
containing $\pi_F$ is associated to a simple global $A$-parameter. They are Siegel cusp forms which do not come from smaller
groups by Langlands functoriality in the Arthur's classification. In this paper, we call them genuine forms. Let $HE_{\underline{k}}(N)^{{ng}}$ be the subset of $HE_{\underline{k}}(N)$ consisting of $F$ such that the global $A$-packet
containing $\pi_F$ is associated to a non-simple global $A$-parameter, i.e., they are Siegel cusp forms which come from smaller
groups by Langlands functoriality in Arthur's classification. We call them non-genuine forms. We show that non-genuine forms are negligible. For this, we need some further assumptions on the level $N$.
\begin{thm}\label{finer-ver} Assume (\ref{suff-reg}). We also assume:
\begin{enumerate}
\item $N$ is an odd prime or
\item $N$ is odd and all prime divisors $p_1,\ldots,p_r\ (r\ge 2)$ of $N$ are
congruent to 1 modulo 4 such that
$\Big(\displaystyle\frac{p_i}{p_j}\Big)=1$ for $i\ne j$,
where $\Big(\displaystyle\frac{\ast}{\ast}\Big)$ denotes the Legendre symbol.
\end{enumerate}
Then
\begin{enumerate}
\item $\left|HE_{\underline{k}}(N)^{{g}}\right|=C_{{\underline{k}}}C_N N^{2n^2+n}+O_{n,{\underline{k}},\epsilon}\left(N^{2n^2+n-1+\epsilon}\right)$;
\item $\left|HE_{\underline{k}}(N)^{{ng}}\right|=O_{n,{\underline{k}},\epsilon}\left(N^{2n^2+n-1+\epsilon}\right)$ for any $\epsilon>0$;
\item for a fixed prime $p$, the set $\left\{(\theta_1(\pi_{F,p}),\ldots,\theta_n(\pi_{F,p}))\in \Omega\ |\ F\in HE_{\underline{k}}(N)^{{g}} \right\}$ is
$\mu_p$-equidistributed in $\Omega$.
\end{enumerate}
\end{thm}
The above assumptions on the level $N$ are necessary in order to estimate non-genuine
forms related to non-split but quasi split orthogonal groups in the Arthur's classification by using the transfer theorems for some Hecke elements in
the quadratic base change in the ramified case \cite{Yamauchi}. (See Proposition \ref{fixedV} for the detail.)
Next, we discuss $\ell$-level density ($\ell$ positive integer) for standard L-functions in the level aspect.
Let us denote by $ \Pi({\rm GL}_{n}(\mathbb{A}))^0$ the set of all isomorphism classes of
irreducible unitary cuspidal representations of ${\rm GL}_{n}(\mathbb{A})$.
Keep the assumption on ${\underline{k}}$ as in (\ref{suff-reg}) and the above assumption on the level $N$. Then $F$ can be described by a global $A$-parameter $\boxplus_{i=1}^r\pi_i$
with $\pi_i\in \Pi({\rm GL}_{m_i}(\mathbb{A}))^0$ and $\displaystyle\sum_{i=1}^rm_i=2n+1$.
Then we may define the standard $L$-function of $F\in HE_{\underline{k}}(N)$ by
$$L(s,\pi_F,{\rm St}):=\prod_{i=1}^r L(s,\pi_i)$$
which coincides with the classical definition in terms of Satake parameters of $F$ outside $N$.
Then we show unconditionally that the $\ell$-level density of the standard $L$-functions of the family
$HE_{\underline{k}}(N)$ has the symmetry type $Sp$ in the level aspect. (See Section \ref{r-level} for the precise statement. Shin and Templier \cite{ST} showed it under several hypotheses for a family which includes non-holomorphic forms.)
As a corollary, we obtain a result on the order of vanishing of $L(s,\pi_F,{\rm St})$ at $s=\frac 12$, the center of symmetry of the $L$-function by using the method of
\cite{ILS} for holomorphic cusp forms on ${\mathop{\mathrm{GL}}}_2(\mathbb{A})$ (see also \cite{Brumer} for another formulation related to Birch-Swinnerton-Dyer conjecture): Let $r_F$ be the order of vanishing of $L(s,\pi_F,{\rm St})$ at $s=\frac 12$.
Then we show that under GRH (generalized Riemann hypothesis), $\displaystyle\sum_{F\in HE_{\underline{k}}(N)} r_F\leq C d_{\underline{k}}(N)$ for some constant $C>0$. This would be the first result of this kind in Siegel modular forms.
We can also show a similar result for the degree 4 spinor $L$-functions of ${\mathop{\mathrm{GSp}}}(4)$.
Let us explain our strategy in comparison with the previous works.
Let $f$ be a Hecke element in Theorem \ref{main1}. A starting main equality is
$$I_{{\rm spec}}(f)=I(f)=I_{{\rm geom}}(f),
$$
where LHS (resp. RHS) is the spectral (resp. the geometric) side of the
Arthur's invariant trace $I(f)$.
Under the assumption on $k_n>n+1$, the spectral side becomes simple by
the results of \cite{Arthur} and \cite{Hiraga}, and it is directly
related to $S_{\underline{k}}(\Gamma(N))$ because of the choice of a pseudo-coefficient of $D^{{\rm hol}}_{\underline{l}}$.
Now the geometric side is given by:
\begin{equation}\label{geom}
I_{{\rm geom}}(f)=\sum_{M\in{\mathcal{L}}}(-1)^{\dim(A_M/A_G)}\frac{|W_0^M|}{|W_0^G|}
\sum_{\gamma\in (M(\mathbb{Q}))_{M,\tilde{S}}}a^M(\tilde{S},\gamma)\, I_M^G(\gamma,f_\xi)\, J_M^M(\gamma,h_P),
\end{equation}
($\tilde{S}=\{\infty\}\sqcup S_N \sqcup S_1$), where
$(M(\mathbb{Q}))_{M,\tilde{S}}$ denotes the set of $(M,\tilde{S})$-equivalence classes in $M(\mathbb{Q})$ (cf. \cite[p.113]{Arthur2})
which turns out to be a finite set; for each $M$ in a finite set ${\mathcal{L}}$ we choose a parabolic subgroup $P$ such that $M$ is a Levi subgroup of $P$. (See loc.cit. for details.)
Roughly speaking,
\begin{itemize}
\item the factor $a^M(\tilde{S},\gamma)$ is called a global coefficient and it is almost the volume of the centralizer of $\gamma$ in $M$ if $\gamma$ is semisimple. The general properties are unknown;
\item the factor $I_M^G(\gamma,f_\xi)$ is called an invariant weighted orbital integral, and as the notation shows, it strongly depends on the weight ${\underline{k}}$ of $\xi=\xi_{\underline{k}}$. Therefore, it is negligible when we consider the level aspect;
\item the factor $J_M^M(\gamma,h_P)$ is an orbital integral of $\gamma$ for $h={\rm vol}(K(N))^{-1}h_1h_{K(N)}$.
\end{itemize}
According to the types of conjugacy classes and $M$, the geometric side is divided into the following terms:
$$I_{{\rm geom}}(f)=I_1(f)+I_2(f)+I_3(f)+I_4(f)
$$
where
\begin{itemize}
\item $I_1(f)$: $M=G$ and $\gamma=1$;
\item $I_2(f)$: $M\neq G$ and $\gamma=1$;
\item $I_3(f)$: $\gamma$ is unipotent except for the case $\gamma\neq 1$;
\item $I_4(f)$: the other contributions.
\end{itemize}
The first term $I_1(f)$ is $f(1)$ up to constant factors and the Plancherel formula
$\widehat{\mu}^{{\rm pl}}_{S_1}(\widehat{f})=f(1)$ yields the first term of the equality in Theorem \ref{main1}.
The condition $N\ge c_0 \prod_{p\in S_1}p^{2n\kappa}$ in Theorem \ref{main1} implies that the non-unipotent contribution $I_4(f)$ vanishes by \cite[Lemma 8.4]{ST}.
Therefore, everything is reduced to studying the unipotent contributions $I_2(f)$ and $I_3(f)$.
An explicit bound for $I_2(f)$ was given by \cite[Proof of Theorem 9.16]{ST}.
However, as for $I_3(f)$, since the number of $(M,\tilde{S})$-equivalence classes in the geometric unipotent conjugacy class of each $\gamma$ is increasing when $N$ goes to be infinity, it is difficult to estimate $I_2(f)$ directly.
In the case of ${\mathop{\mathrm{GSp}}}(4)$, we computed unipotent contributions by using case by case analysis in \cite{KWY}.
Here we give a new uniform way to estimate all the unipotent contributions.
It is given by a sum of special values of zeta integrals with real characters for spaces of symmetric matrices (see Lemma \ref{lem:210513l2} and Theorem \ref{thm:tr}).
This formula is a generalization of the dimension formula (cf. \cite{Shintani, Wakatsuki}) to the trace formula of Hecke operators.
By using their explicit formulas \cite{Saito2} and analyzing Shintani double zeta functions \cite{KTW}, we express the geometric side as a finite sum of products of local integrals and special values of the Hecke $L$ functions with real characters, and then obtain the estimates of the geometric side (see Theorem \ref{thm:asy}).
This paper is organized as follows. In Section 2, we set up some notations.
In Section 3, we give key results (Theorem \ref{thm:tr}) and Theorem \ref{thm:asy})
in estimating trace formulas of Hecke elements. In Section 4, we study Siegel modular forms in terms of Arthur's classification and show that non-genuine forms are negligible. In Section 5, we give a notion of newforms which is necessary to estimate conductors.
Section 6 through 10 are devoted to proving the main theorems. Finally, in Appendix, we give
an explicit computation of the convolution product of some Hecke elements which is needed in
the computation of $\ell$-level density of standard $L$-functions.
\medskip
\textbf{Acknowledgments.} We would like to thank M. Miyauchi, M. Oi, S. Sugiyama, and M. Tsuzuki for helpful discussions. We thank KIAS in Seoul and RIMS in Kyoto for their incredible hospitality during this research.
\medskip
\section{Preliminaries}\label{sec:pre}
A split symplectic group $G={\mathop{\mathrm{Sp}}}(2n)$ over the rational number field $\mathbb{Q}$ is defined by
\[
G={\mathop{\mathrm{Sp}}}(2n)=\left\{g\in{\mathop{\mathrm{GL}}}_{2n} \Bigg|\ g\begin{pmatrix} O_n&I_n \\ -I_n & O_n \end{pmatrix} {}^t\!g=\begin{pmatrix} O_n&I_n \\ -I_n & O_n \end{pmatrix} \right\}.
\]
The compact subgroup
\[
K_\infty=\left\{ \begin{pmatrix}A&-B \\ B & A \end{pmatrix}\in G(\mathbb{R}) \right\}
\]
of $G(\Bbb R)$ is isomorphic to the unitary group ${\mathop{\mathrm{U}}}(n)$ via the mapping $ \begin{pmatrix}A&-B \\ B & A \end{pmatrix} \mapsto
A + i B$, where $i=\sqrt{-1}$. For each rational prime $p$,
we also set $K_p=G(\mathbb{Z}_p)$ and put $K=\prod_{p\le \infty} K_p$. The compact groups $K_v$ and $K$ are maximal in $G(\mathbb{Q}_v)$ and $G(\mathbb{A})$, resp.
Holomorphic discrete series of $G(\mathbb{R})$ are parameterized by $n$-tuples $\underline{k}=(k_1,\dots,k_n)\in \mathbb{Z}^n$ such that $k_1\geq\cdots\geq k_n>n$, which is called the Blattner parameter.
We write $\sigma_{\underline{k}}$ for the holomorphic discrete series corresponding to the Blattner parameter ${\underline{k}}=(k_1,\ldots,k_n)$. We also write $D^{{\rm hol}}_{\underline{l}}$ for one corresponding to the Harish-Chandra parameter ${\underline{l}}=(k_1-1,k_2-2,\ldots,k_n-n)$ so that $D^{{\rm hol}}_{\underline{l}}=\sigma_{{\underline{k}}}$.
Let ${\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p))$ denote the unramified Hecke algebra over $G(\mathbb{Q}_p)$, that is,
\[
{\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p))=\{\varphi\in C_c^\infty(G(\mathbb{Q}_p)) \mid \varphi(k_1xk_2)=\varphi(x) \quad (\forall k_1,k_2\in K_p , \;\; \forall x\in G(\mathbb{Q}_p)) \}.
\]
Let $T$ denote the maximal split $\mathbb{Q}$-torus of $G$ consisting of diagonal matrices.
We denote by $X_*(T)$ the group of cocharacters on $T$ over $\mathbb{Q}$.
An element $e_j$ in $X_*(T)$ is defined by
\begin{equation}\label{ejx}
e_j(x)={\mathop{\mathrm{diag}}}(\overbrace{1,\dots,1}^{j-1},x,\overbrace{1\dots,1}^{n-j+1},\overbrace{1,\dots,1}^{j-1},x^{-1},\overbrace{1,\dots,1}^{n-j+1})\in T \quad (x\in \mathbb{G}_m).
\end{equation}
Then, one has $X_*(T)=\langle e_1,\dots,e_n \rangle$.
By the Cartan decomposition, any function in ${\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p))$ is expressed by a linear combination of characteristic functions of double cosets $K_p\lambda(p)K_p$ $(\lambda\in X_*(T))$.
A height function $\|\; \|$ on $X_*(T)$ is defined by
\[
\left\|\prod_{j=1}^n e_j^{m_j}\right\|=\max\{ |m_j| \mid 1\leq j\leq n\} , \qquad (m_j\in\mathbb{Z}).
\]
For each $\kappa\in\mathbb{N}$, we set
\begin{equation}\label{height}
{\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p))^\kappa =\Big\{ \varphi\in {\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p)) \mid \mathrm{Supp}(\varphi) \subset \bigcup_{\mu\in X_*(T), \; \|\mu\|\leq \kappa } K_p\mu(p)K_p \Big\}.
\end{equation}
Choose a natural number $N$.
We set
\[
K_p(N)=\{x\in K_p \mid x\equiv I_{2n} \mod N\}, \quad K(N)= \prod_{p<\infty} K_p(N) .
\]
One gets a congruence subgroup $\Gamma(N)=G(\mathbb{Q})\cap G(\mathbb{R})K(N)$.
Let $\mathfrak{H}_n:=\{Z\in M_n(\mathbb{C}) \mid Z={}^t\!Z, \; \mathrm{Im}(Z)>0 \}$.
We write $S_{\underline{k}}(\Gamma(N))$ for the space of Siegel cusp forms of weight ${\underline{k}}$ for $\Gamma(N)$, i.e., $S_{\underline{k}}(\Gamma(N))$ consists of $V_{\underline{k}}$-valued smooth functions $F$ on $G(\mathbb{A})$ satisfying the following conditions:
\[
\begin{array}{ll}
\text{(i)}& F( \gamma gk_\infty k_f)=\rho_{\underline{k}}(k_\infty)^{-1}F(g), \; g\in G(\mathbb{A}) , \; \gamma\in G(\mathbb{Q}), \; k_\infty \in K_\infty, \; k_f\in K(N) \\
\text{(ii)}& \text{$ \rho_{\underline{k}}(g_\infty,iI_n) F|_{G(\mathbb{R})}(g_\infty)$ is holomorphic for $g_\infty\cdot iI_n \in \mathfrak{H}_n$,} \\
\text{(iii)}& \max_{g\in G(\mathbb{A})}\left|F(g)\right|\ll 1,
\end{array}
\]
where $\rho_{\underline{k}}$ denotes the finite dimensional irreducible polynomial representation of ${\mathop{\mathrm{U}}}(n)$ corresponding to ${\underline{k}}$ together with the representation space $V_{\underline{k}}$, and we set $\rho_{\underline{k}}(g,iI_n)=\rho_{\underline{k}}(iC+D)$ for
$g=\left(\begin{smallmatrix}A&B \\ C&D\end{smallmatrix}\right)\in G(\mathbb{R})$.
Let $\underline{m}=(m_1,...,m_n)$, $m_1| m_2|\cdots |m_n$, and $D_{\underline{m}}={\mathop{\mathrm{diag}}}(m_1,...,m_n)$.
Let $T(D_{\underline{m}})$ be the Hecke operator defined by the double coset $\Gamma(N)\begin{pmatrix} D_{\underline{m}}&0\\0&D_{\underline{m}}^{-1}\end{pmatrix}\Gamma(N)$.
In particular, for each prime $p$, let
$D_{p,\underline{a}}={\mathop{\mathrm{diag}}}(p^{a_1},...,p^{a_n})$, with $\underline{a}=(a_1,...,a_n), 0\leq a_1\leq\cdots\leq a_n$.
Let $F$ be a Hecke eigenform in $S_{{\underline{k}}}(\Gamma(N))$ with respect to the Hecke operator $T(D_{p,\underline{a}})$
for all $p\nmid N$. (See \cite[Section 2.2]{KWY} for Hecke eigenforms in the case of
$n=2$. One can generalize the contents there to $n\geq 3$.)
Then $F$ gives rise to an adelic automorphic form $\phi_F$ on ${\mathop{\mathrm{Sp}}}(2n,\Bbb Q)\backslash{\mathop{\mathrm{Sp}}}(2n,\Bbb A)$, and $\phi_F$ gives rise to a cuspidal representation $\pi_F$ which is a direct sum $\pi_F=\pi_1\oplus\cdots\oplus \pi_r$, where $\pi_i$'s are irreducible cuspidal representations of ${\mathop{\mathrm{Sp}}}(2n)$. Since $F$ is an eigenform, $\pi_i$'s are all near-equivalent to each other.
Since we do not have the strong multiplicity one theorem for ${\mathop{\mathrm{Sp}}}(2n)$, we cannot conclude that $\pi_F$ is irreducible. However, the strong multiplicity one theorem for ${\mathop{\mathrm{GL}}}_n$ implies that there exists a global $A$-parameter $\psi\in\Psi(G)$ such that $\pi_i\in\Pi_\psi$ for all $i$ \cite[p. 3088]{Schm}. (See Section \ref{Arthur} for the definition of the global $A$-packet.)
On the other hand, given a cuspidal representation $\pi$ of ${\mathop{\mathrm{Sp}}}(2n)$ with a $K(N)$-fixed vector and whose infinity component is holomorphic discrete series of lowest weight ${\underline{k}}$, there exists a holomorphic Siegel cusp form $F$ of weight ${\underline{k}}$ with respect to $\Gamma(N)$ such that $\pi_F=\pi$. (See \cite[p. 2409]{Schm1} for $n=2$. One can generalize the contents there to $n\geq 3$.)
So we define $HE_{\underline{k}}(N)$ to be a basis of $K(N)$-fixed vectors in the set of cuspidal representations of ${\mathop{\mathrm{Sp}}}(2n,\Bbb A)$ whose infinity component is holomorphic discrete series of lowest weight ${\underline{k}}$, and identify it with a basis consisting of all
Hecke eigenforms outside $N$. In particular, each $F\in HE_{\underline{k}}(N)$ gives rise to an irreducible cuspidal representation $\pi_F$ of ${\mathop{\mathrm{Sp}}}(2n)$.
Let $\mathcal F_{\underline{k}}(N)$ be the set of all isomorphism classes of cuspidal representations of ${\mathop{\mathrm{Sp}}}(2n)$ such that $\pi^{K(N)}\ne 0$ and $\pi_\infty\simeq \sigma_{\underline{k}}$.
Consider the map $\Lambda: HE_{\underline{k}}(N)\longrightarrow \mathcal F_{\underline{k}}(N)$, given by $F\longmapsto \pi_F$. It is clearly surjective.
For each $\pi=\otimes_v\pi_v\in \mathcal F_{\underline{k}}(N)$, set $\pi_f=\otimes_{p<\infty} \pi_p$. Then we get $|\Lambda^{-1}(\pi)|=\dim \pi_f^{K(N)}$, where $\pi_f^{K(N)}=\{ \phi\in \pi_f \mid \pi_f(k)\phi=\phi \,$ for all $k\in K(N)\}$.
\section{Asymptotics of Hecke eigenvalues}
For each function $h\in C_c^\infty(K(N)\backslash G(\mathbb{A}_f)/K(N))$, an adelic Hecke operator $T_h$ on $S_{\underline{k}}(\Gamma(N))$ is defined by
\[
(T_h F)(g)=\int_{G(\mathbb{A}_f)} f(g x) h(x) \, {\mathrm{d}} x, \qquad F\in S_{\underline{k}}(\Gamma(N)).
\]
See \cite[p.15--16]{KWY} for the relationship between the classical Hecke operators and adelic Hecke operators for $n=2$. One can generalize the contents there to $n\geq 3$ easily.
Let $f_{\underline{k}}$ denote the pseudo-coefficient of $\sigma_{\underline{k}}$ (cf. \cite{CD}).
\begin{lem}\label{lem:20210602sp}
Suppose $k_n>n+1$ and $h\in C_c^\infty(K(N)\backslash G(\mathbb{A}_f)/K(N))$.
The spectral side $I_\mathrm{spec}(f_{\underline{k}} h)$ of the invariant trace formula is given by
\[
I_\mathrm{spec}(f_{\underline{k}} h)=\sum_{\pi=\sigma_{\underline{k}}\otimes\pi_f,\; \mathrm{auto. \, rep. \, of} \, G(\mathbb{A})}m_\pi\, {\mathop{\mathrm{Tr}}}(\pi_f(h))= {\mathop{\mathrm{Tr}}}\left( T_h|_{S_{\underline{k}}(\Gamma(N))}\right),
\]
where $m_\pi$ means the multiplicity of $\pi$ in the discrete spectrum of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$.
\end{lem}
\begin{proof}
The second equality follows from \cite{Wallach}.
One can prove the first equality by using the arguments in \cite{Arthur} and the main result in \cite{Hiraga}.
\end{proof}
We choose two natural numbers $N_1$ and $N$, which are mutually coprime.
Suppose that $N_1$ is square free.
Set $S_1=\{ p\,:\, p|N_1\}$.
We write $h_N$ for the characteristic function of $\prod_{p\notin S_1\sqcup\{\infty\}} K_p(N)$.
For each automorphic representation $\pi=\pi_\infty\otimes\otimes_p' \pi_p$, we set $\pi_{S_1}=\otimes_{p\in S_1} \pi_p$.
\begin{lem}\label{lem:210513l1}
Take a test function $h$ on $G(\mathbb{A}_f)$ as
\begin{equation}\label{eq:testft}
h={\mathop{\mathrm{vol}}}(K(N))^{-1}\times h_1 \otimes h_N, \quad h_1\in C_c^\infty(G(\mathbb{Q}_{S_1})).
\end{equation}
Then
\[
I_\mathrm{spec}(f_{\underline{k}} h)=\sum_{\pi=\sigma_{\underline{k}}\otimes\pi_f,\; \mathrm{auto. \, rep. \, of} \, G(\mathbb{A})}m_\pi\, \dim\pi_f^{K(N)} \, {\mathop{\mathrm{Tr}}}(\pi_{S_1}(h_1))= {\mathop{\mathrm{Tr}}}\left( T_h|_{S_{\underline{k}}(\Gamma(N))}\right).
\]
\end{lem}
\begin{proof}
This lemma immediately follows from Lemma \ref{lem:20210602sp}.
\end{proof}
Let $V_r$ denote the vector space of symmetric matrices of degree $r$, and define a rational representation $\rho$ of the group ${\mathop{\mathrm{GL}}}_1\times {\mathop{\mathrm{GL}}}_r$ on $V_r$ by $x\cdot \rho(a,m)=a{}^t\!mxm$ $(x\in V_r$, $(a,m)\in {\mathop{\mathrm{GL}}}_1\times {\mathop{\mathrm{GL}}}_r)$.
The kernel of $\rho$ is given by $\mathrm{Ker}\rho=\{ (a^{-2} , aI_r) \mid a \in {\mathop{\mathrm{GL}}}_1 \}$, and we set $H_r=\mathrm{Ker}\rho\backslash ({\mathop{\mathrm{GL}}}_1 \times {\mathop{\mathrm{GL}}}_r)$.
Then, the pair $(H_r,V_r)$ is a prehomogeneous vector space over $\mathbb{Q}$.
For $1\leq r\leq n$ and $f\in C_c^\infty(G(\mathbb{A}))$ $\left(\text{resp.}\, f\in C_c^\infty(G(\mathbb{A}_f))\right)$, we define a function $\Phi_{f,r}\in C_c^\infty(V_r(\mathbb{A}))$ $\left(\text{resp.}\, \Phi_{f,r}\in C_c^\infty(V_r(\mathbb{A}_f))\right)$ as
\[
\Phi_{f,r}(x)=\int_K f(k^{-1}\begin{pmatrix} I_n & * \\ O_n & I_n \end{pmatrix}k) \, {\mathrm{d}} k \quad \left(\text{resp.} \;\; \int_{K_f} \right),\quad *=\begin{pmatrix} x&0 \\ 0 &0 \end{pmatrix}\in V_n.
\]
Let ${\tilde{f}}_{\underline{k}}$ denote the spherical trace function of $\sigma_{\underline{k}}$ with respect to $\rho_{\underline{k}}$ on $G(\mathbb{R})$ (cf. \cite[\S 5.3]{Wakatsuki}).
Notice that ${\tilde{f}}_{\underline{k}}$ is a matrix coefficient of $\sigma_{\underline{k}}$, and so it is not compactly supported.
Take a test function $h\in C_c^\infty(G(\mathbb{A}_f))$ and set ${\tilde{f}}={\tilde{f}}_{\underline{k}} h$.
Let $\chi$ be a real character on $\mathbb{R}_{>0}\mathbb{Q}^\times\backslash \mathbb{A}^\times$.
Define a zeta integral $Z_r(\Phi_{{\tilde{f}},r},s,\chi)$ by
\[
Z_r(\Phi_{{\tilde{f}},r},s,\chi)=\int_{H_r(\mathbb{Q})\backslash H_r(\mathbb{A})}|a^r\det(m)^2|^{s} \chi(a) \, \sum_{x\in V_r^0(\mathbb{Q})} \Phi_{{\tilde{f}},r}(x\cdot g) \, {\mathrm{d}} g \quad (g=\mathrm{Ker}\rho \,(a,m)),
\]
where $V_r^0=\{x\in V_r\mid \det(x)\neq 0\}$ and ${\mathrm{d}} g$ is a Haar measure on $H_r(\mathbb{A})$.
The zeta integral $Z_r(\Phi_{{\tilde{f}},r},s,\chi)$ is absolutely convergent for the range
\begin{equation}\label{eq:range0605}
k_n>2n, \quad \mathrm{Re}(s)>\frac{r-1}{2}, \quad \begin{cases} {\mathop{\mathrm{Re}}}(s)<\frac{k_n}{2} & \text{if $r=2$}, \\ \mathrm{Re}(s)<k_n-\frac{r-1}{2} & \text{otherwise}, \end{cases}
\end{equation}
see \cite[Proposition 5.15]{Wakatsuki}, and $Z(\Phi_{{\tilde{f}},r},s,\chi)$ is meromorphically continued to the whole $s$-plane (cf. \cite{Shintani,Wakatsuki,Yukie}).
The following lemma associates $Z(\Phi_{{\tilde{f}},r},s,\chi)$ with the unipotent contribution $I_\mathrm{unip}(f)=I_1(f)+I_2(f)+I_3(f)$ of the invariant trace formula.
\begin{lem}\label{lem:210513l2}
Let $S_0$ be a finite set of finite places of $\mathbb{Q}$.
Take a test function $h_{S_0}\in C_c^\infty (G(\mathbb{Q}_{S_0}))$, and let $h^{S_0}$ denote the characteristic function of $\prod_{p\notin S_0\sqcup\{\infty\}}K_p$.
Define a test function ${\tilde{f}}$ as ${\tilde{f}}={\tilde{f}}_{\underline{k}} h_{S_0}h^{S_0}$.
If $k_n$ is sufficiently large, then we have
\[
I_\mathrm{unip}(f_{\underline{k}} h_{S_0}h^{S_0})={\mathop{\mathrm{vol}}}_G\, h_{S_0}(1)\, d_{\underline{k}} + \frac{1}{2}\sum_{r=1}^n \sum_{\chi\in \mathscr{X}(S_0)} Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},\chi),
\]
where ${\mathop{\mathrm{vol}}}_G={\mathop{\mathrm{vol}}}(G(F)\backslash G(\mathbb{A}))$, $d_{\underline{k}}$ denotes the formal degree of $\sigma_{\underline{k}}$, and $\mathscr{X}(S_0)$ denotes the set consisting of real characters $\chi=\otimes_v \chi_v$ on $\mathbb{R}_{>0}\mathbb{Q}^\times\backslash \mathbb{A}^\times$ such that $\chi_v$ is unramified for any $v\notin S_0\sqcup\{\infty\}$.
Note that the point $s=n-\frac{r-1}{2}$ $(1\leq r\leq n)$ is contained in the range \eqref{eq:range0605}, and we have $Z_r(\Phi_{{\tilde{f}},r},s,\chi)\equiv 0$ for any real character $\chi\notin \mathscr{X}(S_0)$.
\end{lem}
\begin{proof}
To study $I_\mathrm{unip}(f_{\underline{k}} h_{S_0}h^{S_0})$, we need an additional zeta integral $\tilde{Z}_r(\Phi_{{\tilde{f}},r},s)$ defined by
\[
\tilde{Z}_r(\Phi_{{\tilde{f}},r},s)=\int_{{\mathop{\mathrm{GL}}}_r(\mathbb{Q})\backslash {\mathop{\mathrm{GL}}}_r(\mathbb{A})}|\det(m)|^{2s} \sum_{x\in V_r^0(\mathbb{Q})} \Phi_{{\tilde{f}},r}( {}^t\!m x m) \, {\mathrm{d}} m.
\]
The zeta integral $\tilde{Z}_r(\Phi_{{\tilde{f}},r},s)$ is absolutely convergent for the range \eqref{eq:range0605}, and $\tilde{Z}(\Phi_{{\tilde{f}},r},s)$ is meromorphically continued to the whole $s$-plane, see \cite{Shintani,Wakatsuki,Yukie}.
Applying \cite[Proposition 3.8, Proposition 3.11, Lemmas 5.10 and 5.16]{Wakatsuki} to $I_\mathrm{unip}(f)$, we obtain
\begin{equation}\label{eq:1030}
I_\mathrm{unip}(f_{\underline{k}} h_{S_0}h^{S_0})={\mathop{\mathrm{vol}}}_G\, h_{S_0}(1)\, d_{\underline{k}} + \sum_{r=1}^n \tilde{Z}_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2})
\end{equation}
for sufficiently large $k_n$.
Notice that $f_{\underline{k}}$ is changed to ${\tilde{f}}_{\underline{k}}$ in the right-hand side of \eqref{eq:1030}, and this change is essentially required for the proof of \eqref{eq:1030}.
By the same argument as in \cite[(4.9)]{HW}, we have
\[
\tilde{Z}_r(\Phi_{{\tilde{f}},r},s)=\frac{1}{2}\sum_\chi Z_r(\Phi_{{\tilde{f}},r},s,\chi)
\]
where $\chi$ runs over all real characters on $\mathbb{R}_{>0}\mathbb{Q}^\times\backslash\mathbb{A}^\times$.
If $\chi_p$ is ramified $(\chi=\otimes_v\chi_v)$ and
\begin{equation}\label{eq:210513b}
\Phi_{{\tilde{f}},r}(a_p x)=\Phi_{{\tilde{f}},r}(x) \quad (\forall a_p\in\mathbb{Z}_p^\times),
\end{equation}
holds, then one can get $Z_r(\Phi_{{\tilde{f}},r},s,\chi)\equiv 0$.
Hence, we have $Z_r(\Phi_{{\tilde{f}},r},s,\chi)\equiv 0$ for any $\chi\notin \mathscr{X}(S_0)$.
\end{proof}
\begin{remark}
The rational representation $\rho$ of $H_r$ on $V_r$ is faithful, but the representation $x\mapsto {}^t\!m x m$ of ${\mathop{\mathrm{GL}}}_r$ on $V_r$ is not.
Hence, $Z_r(\Phi_{{\tilde{f}},r},s,\chi)$ is suitable for Saito's explicit formula \cite{Saito2}, which we use in the proof of Theorem \ref{thm:asy}, but $\tilde{Z}_r(\Phi_{{\tilde{f}},r},s)$ is not.
This fact is also important for the study of global coefficients in the geometric side (cf. \cite{HW}).
\end{remark}
Let $\psi$ be a non-trivial additive character on $\mathbb{Q}\backslash\mathbb{A}$, and a bilinear form $\langle \; , \; \rangle$ on $V_r(\mathbb{A})$ is defined by $\langle x , y \rangle:={\mathop{\mathrm{Tr}}}(x y)$.
Let ${\mathrm{d}} x$ denote the self-dual measure on $V_r(\mathbb{A})$ for $\psi(\langle \; , \; \rangle)$.
Then, a Fourier transform of $\Phi\in C^\infty(V_r(\mathbb{A}))$ is defined by
\[
\widehat{\Phi}(y)=\int_{V_r(\mathbb{A})} \, \Phi(x)\, \psi(\langle x , y \rangle)\, {\mathrm{d}} x \qquad (y\in V_r(\mathbb{A})).
\]
For each $\Phi_0\in C^\infty_0(V_r(\mathbb{A}_f))$, we define its Fourier transform $\widehat{\Phi_0}$ in the same manner.
The zeta function $Z_r(\Phi_{{\tilde{f}},r},s,{\mathbbm{1}})$ satisfies the functional equation \cite{Shintani,Yukie}
\begin{equation}\label{eq:functional}
Z_r(\Phi_{{\tilde{f}},r},s,{\mathbbm{1}})=Z_r(\widehat{\Phi_{{\tilde{f}},r}},\tfrac{r+1}{2}-s,{\mathbbm{1}}),
\end{equation}
where ${\mathbbm{1}}$ denotes the trivial representation on $\mathbb{R}_{>0}\mathbb{Q}^\times\backslash\mathbb{A}^\times$.
Take a test function $\Phi_0\in C^\infty_0(V_r(\mathbb{A}_f))$ such that $\Phi_0({}^t\!k xk)=\Phi_0(x)$ holds for any $k\in \prod_{p<\infty}H_r(\mathbb{Z}_p)$ and $x\in V_r(\mathbb{A}_f)$, where $H_r(\mathbb{Z}_p)$ is identified with the projection of ${\mathop{\mathrm{GL}}}_1(\mathbb{Z}_p)\times {\mathop{\mathrm{GL}}}_r(\mathbb{Z}_p)$ into $H_r(\mathbb{A}_f)$.
We write $L_0$ for the subset of $V_r(\mathbb{Q})$ which consists of the positive definite symmetric matrices contained in the support of $\Phi_0$.
It follows from the condition of $\Phi_0$ that $L_0$ is invariant for $\Gamma=H_r(\mathbb{Z})$.
Put $\zeta_r(\Phi_0,s)=1$ for $r=0$.
For $r>0$, define a Shintani zeta function $\zeta_r(\Phi_0,s)$ as
\[
\zeta_r(\Phi_0,s)=\sum_{x\in L_0/\Gamma} \frac{\Phi_0(x)}{\#(\Gamma_x)\, \det(x)^s} .
\]
where $\Gamma_x=\{\gamma\in \Gamma\mid x\cdot \gamma=x\}$.
The zeta function $\zeta_r(\Phi_0,s)$ absolutely converges for ${\mathop{\mathrm{Re}}}(s)>\frac{r+1}{2}$, and is meromorphically continued to the whole $s$-plane. Furthermore, $\zeta_r(\Phi_0,s)$ is holomorphic except for possible simple poles at $s=1,\frac{3}{2},\dots \frac{r+1}{2}$.
\begin{lem}\label{lem:functzeta}
Let $1\leq r\leq n$, $k_n>2n$, $h\in C_c^\infty(G(\mathbb{A}_f))$ and take a test function ${\tilde{f}}$ as ${\tilde{f}}={\tilde{f}}_{\underline{k}} h$.
Then, there exists a rational function $\mathbf{C}_{n,r}(x_1,\dots,x_n)$ over $\mathbb{R}$ such that
\[
Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},{\mathbbm{1}})= \mathbf{C}_{n,r}({\underline{k}}) \times \zeta_r(\widehat{\Phi_{h,r}},r-n).
\]
Note that $\zeta_r(\widehat{\Phi_{h,r}},s)$ is holomorphic in $\{s\in\mathbb{C} \mid {\mathop{\mathrm{Re}}}(s)\leq 0\}$, and $\mathbf{C}_{n,r}(x_1,\dots,x_n)$ is explicitly expressed by the Gamma function and the partitions, see \cite[(5.17) and Lemma 5.16]{Wakatsuki}.
\end{lem}
\begin{proof}
This can be proved by the functional equation \eqref{eq:functional} and the same argument as in \cite[Proof of Lemma 5.16]{Wakatsuki}.
\end{proof}
We will use this lemma for the regularization of the range of ${\underline{k}}$.
The zeta integral $Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},{\mathbbm{1}})$ was defined only for $k_n>2n$, but the right-hand side of the equality in Lemma \ref{lem:functzeta} is available for any ${\underline{k}}$.
In addition, this lemma is necessary to estimate the growth of $I_\mathrm{unip}(f)$ with respect to $S=S_1\sqcup\{\infty\}$.
We later define a Dirichlet series $D_{m,u_S}^S(s)$, which appears in the explicit formula of $Z_r(\Phi,s,{\mathbbm{1}})$ when $r$ is even.
For the case that $r$ is even and $3<r<n$, it seems difficult to estimate the growth of its contribution to $Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},{\mathbbm{1}})$, but we can avoid such the difficulty by this lemma, since the special value $\zeta_r(\widehat{\Phi_{h,r}},r-n)$ excludes the part of $D_{m,u_S}^S(s)$.
\begin{thm}\label{thm:tr}
Suppose $k_n>n+1$. Let $h_1\in {\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_{S_1}))^\kappa=\otimes_{p\in S_1}{\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_p))^\kappa$ and let $h$ be a test function on $G(\mathbb{A}_f)$ given as \eqref{eq:testft}.
Then there exists a positive constant $c_0$ such that, if $N\geq c_0 N_1^{2n\kappa}$,
\begin{align}\label{eq:zeta}
{\mathop{\mathrm{Tr}}}\left( T_h|_{S_{\underline{k}}(\Gamma(N))}\right)=&\sum_{\pi=\sigma_{\underline{k}}\otimes\pi_f,\; \mathrm{auto. \, rep. \, of} \, G(\mathbb{A})}m_\pi\, \dim\pi_f^{K(N)} \, {\mathop{\mathrm{Tr}}}(\pi_{S_1}(h_1)) \\ \nonumber
=&{\mathop{\mathrm{vol}}}_G\, {\mathop{\mathrm{vol}}}(K(N))^{-1} \, h_1(1)\, d_{\underline{k}} + \frac{1}{2}\sum_{r=1}^n \mathbf{C}_{n,r}({\underline{k}}) \, \zeta_r(\widehat{\Phi_{h,r}},r-n).
\end{align}
\end{thm}
\begin{proof}
Let $f=f_{\underline{k}} h$ and ${\tilde{f}}={\tilde{f}}_{\underline{k}} h$.
By Lemma \ref{lem:210513l1}, it is sufficient to prove that the geometric side $I_\mathrm{geom}(f)$ equals the right-hand side of \eqref{eq:zeta}.
If one uses the results in \cite{Arthur} and applies \cite[Lemma 8.4]{ST} by putting $\Xi:G\subset {\mathop{\mathrm{GL}}}_m$, $m=2n$, $B_\Xi=1$, $c_\Xi=c_0$ in their notations, then one gets $I_\mathrm{geom}(f)=I_\mathrm{unip}(f)$.
Hence, by Lemma \ref{lem:210513l2} and putting $h_{S_0}h^{S_0}=h$, we have
\begin{equation}\label{eq:210513a}
{\mathop{\mathrm{Tr}}}\left( T_h|_{S_{\underline{k}}(\Gamma(N))}\right)={\mathop{\mathrm{vol}}}_G\, {\mathop{\mathrm{vol}}}(K(N))^{-1} \,h_1(1)\, d_{\underline{k}} + \frac{1}{2}\sum_{r=1}^n \sum_{\chi\in \mathscr{X}(S_0)} Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},\chi)
\end{equation}
for sufficiently large $k_n$.
Let $\mathscr{M}(a):={\mathop{\mathrm{diag}}}(\overbrace{1,\dots,1}^n,\overbrace{a,\dots,a}^n)$ for $a\in\mathbb{A}^\times$.
For any $a_p\in\mathbb{Z}_p^\times$, $b_p\in\mathbb{Q}_p^\times$, $\mu\in X_*(T)$, we have
\[
\mathscr{M}(a_p)^{-1}K_p(N)\mathscr{M}(a_p)=K_p(N), \quad \mathscr{M}(a_p)^{-1}\mu(b_p)\mathscr{M}(a_p)=\mu(b_p).
\]
Hence, \eqref{eq:210513b} holds for any $p<\infty$, and so $Z_r(\Phi_{{\tilde{f}},r},n-\tfrac{r-1}{2},\chi)$ vanishes for any $\chi\neq {\mathbbm{1}}$.
Therefore, by Lemma \ref{lem:functzeta} we obtain the assertion \eqref{eq:zeta} for sufficiently large $k_n$.
By the same argument as in \cite[Proof of Theorem 5.17]{Wakatsuki}, we can prove that this equality \eqref{eq:zeta} holds in the range $k_n>n+1$, because the both sides of \eqref{eq:zeta} are rational functions of ${\underline{k}}$ in that range, see Lemma \ref{lem:functzeta} and \cite[Proposition 5.3]{Wakatsuki}.
Thus, the proof is completed.
\end{proof}
Let $S$ denote a finite subset of places of $\mathbb{Q}$, and suppose $\infty\in S$.
For each character $\chi=\otimes_v \chi_v$ on $\mathbb{Q}^\times\mathbb{R}_{>0}\backslash \mathbb{A}^\times$, we set
\[
L^S(s,\chi)=\prod_{p\notin S}L_p(s,\chi_p) , \quad L(s,\chi)=\prod_{p<\infty}L_p(s,\chi_p),
\]
\[
\zeta^S(s)=L^S(s,{\mathbbm{1}})=\prod_{p\not\in S}(1-p^{-s})^{-1} \quad \text{and} \quad
\zeta(s)=L(s,{\mathbbm{1}}),
\]
where $L_p(s,\chi_p)=(1-\chi_p(p)p^{-s})^{-1}$ if $\chi_p$ is unramified, and $L_p(s,\chi_p)=1$ if $\chi_p$ is ramified.
\begin{lem}\label{lem:zeta}
Let $s\in\mathbb{R}$. For $s>1$, one has
\[
\zeta^S(s) \leq \zeta(s) \quad \text{and} \quad (\zeta^{S})'(s)\ll \frac{2s \zeta(s)}{s-1},
\]
where $(\zeta^{S})'(s)=\frac{{\mathrm{d}}}{{\mathrm{d}} s}\zeta^S(s)$.
For $s\leq -1$, one has
\[
|\zeta^S(s)| \leq N_1^{-s} |\zeta(s)|.
\]
\end{lem}
\begin{proof}
First of all, $(1-p^{-s})^{-1}\geq 1$ for $p\in S$. Hence $\zeta^S(s)\leq\zeta(s)$.
Let $\log \zeta^S(s) = \sum_{p\not\in S} \log(1-p^{-s})^{-1}$. Then
\[
\frac{(\zeta^{S})'(s)}{\zeta^S(s)}=\sum_{p\not\in S}\frac{-p^{-s}\log p}{1-p^{-s}}.
\]
If $s>1$, $1-p^{-s}\geq \frac{1}{2}$. Hence
\[
\left| \frac{(\zeta^{S})'(s)}{\zeta^S(s)} \right| \leq 2\sum_{p\not\in S} p^{-s}\log p \leq 2\sum_p p^{-s}\log p.
\]
By partial summation, $\sum_p p^{-s}\log p\leq \displaystyle\int_1^\infty (\sum_{p\leq x}\log p) s x^{-s-1} {\mathrm{d}} x\leq \displaystyle\int_1^\infty s x^{-s} {\mathrm{d}} x=\frac{s}{s-1}$.
Here we use the prime number theorem: $\sum_{p\leq x} \log p \sim x$.
Therefore, $(\zeta^{S})'(s)\ll \dfrac{2s \zeta(s)}{s-1}$.
\end{proof}
Set $\mathfrak{D}=\{ d(\mathbb{Q}^\times)^2 \mid d\in \mathbb{Q}^\times\}$.
For each $d\in \mathfrak{D}$, we denote by $\chi_d=\prod_v \chi_{d,v}$ the quadratic character on $\mathbb{Q}^\times\mathbb{R}_{>0}\backslash \mathbb{A}^\times$ corresponding to the quadratic field $\mathbb{Q}(\sqrt{d})$ via the class field theory.
If $d=1$, then $\chi_d$ means the trivial character ${\mathbbm{1}}$.
For each positive even integer $m$, we set
\[
\varphi_{d,m}^S(s)=\zeta^S(2s-m+1)\zeta^S(2s) \frac{L^S(m/2,\chi_d)}{L^S(2s-m/2+1,\chi_d)} N(\mathfrak{f}_d^S)^{(m-1)/2-s}
\]
where $\mathfrak{f}_d^S$ denotes the conductor of $\chi_d^S=\prod_{p\not\in S}\chi_{d,p}$.
For each $u_S\in \mathbb{Q}_S=\prod_{v\in S}\mathbb{Q}_v$, one sets
\[
\mathfrak{D}(u_S)=\left\{ d(\mathbb{Q}^\times)^2 \mid d\in \mathbb{Q}^\times ,\,\, d\in u_S(\mathbb{Q}_S^\times)^2 \right\}.
\]
We need the Dirichlet series
\[
D_{m,u_S}^S(s)=\sum_{d(\mathbb{Q}^\times)^2 \in \mathfrak{D}(u_S) }\varphi_{d,m}^S(s).
\]
The following proposition is a generalization of \cite[Proposition 3.6]{IS2}.
\begin{prop}\label{prop:1}
Let $m\geq 2$ be an even integer.
Suppose $(-1)^{m/2}u_\infty>0$ for $u_S=(u_v)_{v\in S}$ (namely the term of $d (\mathbb{Q}^\times)^2= (\mathbb{Q}^\times)^2$ does not appear in $D_{m,u_S}^S(s)$ if $(-1)^{m/2}=-1$).
The Dirichlet series $D_{m,u_S}^S(s)$ is meromorphically continued to $\mathbb{C}$, and is holomorphic at any $s\in \mathbb{Z}_{\leq 0}$.
\end{prop}
\begin{proof}
See \cite[Corollary 4.23]{KTW} for the case $m>3$.
For $m=2$, this statement can be proved by using \cite{HW,Yukie2}.
\end{proof}
\begin{thm}\label{thm:asy}
Fix a parameter ${\underline{k}}$ such that $k_n>n+1$.
Let $h_1\in {\mathcal H}^{\mathrm{ur}}(G(\mathbb{Q}_{S_1}))^\kappa$ and let $h\in C_c^\infty (G(\mathbb{A}_f))$ be a test function on $G(\mathbb{A}_f)$ given as \eqref{eq:testft}.
Suppose $\sup_{x\in G(\mathbb{Q}_{S_1})}|h_1(x)|\leq 1$.
Then, there exist positive constants $a$, $b$, and $c_0$ such that, if $N\geq c_0 N_1^{2n\kappa}$,
\begin{align*}
{\mathop{\mathrm{Tr}}}\left( T_h|_{S_{\underline{k}}(\Gamma(N))}\right)=&\sum_{\pi=\sigma_{\underline{k}}\otimes\pi_f,\; \mathrm{auto. \, rep. \, of} \, G(\mathbb{A})}m_\pi\, \dim\pi_f^{K(N)} \, {\mathop{\mathrm{Tr}}}(\pi_{S_1}(h_1)) \\
=&{\mathop{\mathrm{vol}}}_G\, {\mathop{\mathrm{vol}}}(K(N))^{-1}\, h_1(1)\, d_{{\underline{k}}} +{\mathop{\mathrm{vol}}}(K(N))^{-1}\, O(N_1^{a\kappa+b}N^{-n}).
\end{align*}
Here the constants $a$ and $b$ do not depend on $\kappa$, $N_1$ and $N$.
See Lemma \ref{lem:210513l2} for ${\mathop{\mathrm{vol}}}_G$ and $d_{\underline{k}}$.
\end{thm}
\begin{proof}
Set
\[
I({\tilde{f}},r)={\mathop{\mathrm{vol}}}(K(N)) \times \zeta_r(\widehat{\Phi_{h,r}},r-n) \qquad (1 \leq r \leq n).
\]
By Theorem \ref{thm:tr}, it is sufficient to prove $I({\tilde{f}},r)=O(N_1^{a\kappa+b}N^{-n})$.
Let $R$ be a finite set of places of $\mathbb{Q}$.
Take a Haar measure ${\mathrm{d}} x_\infty$ on $V_r(\mathbb{R})$, and for each prime $p$ we write ${\mathrm{d}} x_p$ for the Haar measure on $V_r(\mathbb{Q}_p)$ normalized by $\int_{V_r(\mathbb{Z}_p)}{\mathrm{d}} x_p=1$.
For a test function $\Phi_R\in C_c^\infty(V_r(\mathbb{Q}_R))$ and an $H_r(\mathbb{Q}_R)$-orbit $\mathscr{O}_R\in V_r^0(\mathbb{Q}_{R})/H_r(\mathbb{Q}_{R})$, we set
\[
Z_{r,R}(\Phi_R,s,\mathscr{O}_R)=c_R \int_{\mathscr{O}_R} \Phi_R(x)\, |\det(x)|_{R}^{s-\frac{r+1}{2}} \, {\mathrm{d}} x,
\]
where $c_R=\prod_{p\in R,\, p<\infty}(1-p^{-1})^{-1}$, $|\;|_R=\prod_{v\in R}|\;|_v$, and ${\mathrm{d}} x=\prod_{v\in R}{\mathrm{d}} x_v$.
It is known that $Z_{r,R}(\Phi_R,s,\mathscr{O}_R)$ absolutely converges for ${\mathop{\mathrm{Re}}}(s)\geq \frac{r+1}{2}$, and is meromorphically continued to the whole $s$-plane.
Suppose that $R$ does not contain $\infty$, that is, $R$ consists of primes.
Write $\eta_p(x)$ for the Clifford invariant of $x\in V_r^0(\mathbb{Q}_p)$, cf. \cite[Definition 2.1]{Ikeda}, and set $\eta_R((x_p)_{p\in R})=\prod_{p\in R}\eta_p(x_p)$.
For $\chi={\mathbbm{1}}_R$ (trivial) or $\eta_R$, we put $(\Phi_R\chi)(x)=\Phi_R(x)\, \chi(x)$.
It follows from the local functional equation \cite[Theorems 2.1 and 2.2]{Ikeda} over $\mathbb{Q}_p$ $(R=\{p\})$ that $Z_{r,p}(\Phi_p\chi,s,\mathscr{O}_p)$ is holomorphic in the range ${\mathop{\mathrm{Re}}}(s)<0$, and $Z_{r,p}(\Phi_p\chi,s,\mathscr{O}_p)$ possibly has a simple pole at $s=0$.
Hence, for any $R$, $Z_{r,R}(\Phi_R\chi,s,\mathscr{O}_R)$ does not have any pole in the area ${\mathop{\mathrm{Re}}}(s)<0$, but it may have a pole at $s=0$.
Let $\widehat{\Phi_{R}}$ denote the Fourier transform of $\Phi_{R}\in C_c^\infty(V_r(\mathbb{Q}_{R}))$ over $\mathbb{Q}_{R}$ for $\prod_{v\in R}\psi_v(\langle \; , \; \rangle)$, where $\psi_v=\psi|_{\mathbb{Q}_v}$.
Define $\Phi_{h_1,r}(x)= h_1(\begin{pmatrix}I_n & * \\ O_n & I_n \end{pmatrix})\in C_c^\infty(V_r(\mathbb{Q}_{S_1}))$, $*=\begin{pmatrix}x&0 \\ 0 &0 \end{pmatrix}\in V_n$.
Note that this definition is compatible with $\Phi_{{\tilde{f}},r}$ since $h_1$ is spherical for $\prod_{p\in S_1}K_p$.
Set
\[
\mathscr{Z}_r(S_1,h_1)= \sum_{\mathscr{O}_{S_1}\in V_r^0(\mathbb{Q}_{S_1})/H_r(\mathbb{Q}_{S_1})} \Big| Z_{r,S_1}(\widehat{\Phi_{h_1,r}}\chi_r,r-n,\mathscr{O}_{S_1}) \Big| ,
\]
where
\[
\chi_r=\begin{cases} {\mathbbm{1}}_{S_1} & \text{if ($r$ is odd and $r<n$) or $r=2<n$}, \\ \eta_{S_1} & \text{if $r$ is even and $2<r<n$}, \end{cases}
\]
and
\[
\mathscr{Z}_n(S_1,h_1)= \sum_{\mathscr{O}_{S_1}\in V_r^0(\mathbb{Q}_{S_1})/H_r(\mathbb{Q}_{S_1})}\Big| Z_{n,S_1}(\Phi_{h_1,n},\tfrac{n+1}{2},\mathscr{O}_{S_1}) \Big| \quad \text{if $r=n$.}
\]
It follows from Saito's formula \cite[Theorem 2.1 and \S 3]{Saito2} that the zeta function $\zeta_r(\widehat{\Phi_{h,r}},s)$ is expressed by a (finite or infinite) sum of Euler products of $Z_{r,p}(\Phi_p\chi_p,s,\mathscr{O}_p)$ $(\chi_p={\mathbbm{1}}_p$, $\eta_p)$ or its finite sums, and he explicitly calculated the local zeta function $Z_{r,p}(\Phi_p \chi_p,s,\mathscr{O}_p)$ in \cite[\S 2]{Saito} if $\Phi_p$ is a characteristic function of $V(\mathbb{Z}_p)$.
We shall prove $I({\tilde{f}},r)=O(N_1^{a\kappa+b}N^{-n})$ by using his results.
\vspace{2mm}
\noindent
(Case I) $r$ is odd and $r< n$.
In the following, we set $S=S_1\sqcup\{\infty\}$.
By Saito's formula we have
\begin{multline*}
I({\tilde{f}},r)=\text{(constant)}\times N^{\frac{r(r-1)}{2} - r n} \times \sum_{\mathscr{O}_{S_1}\in V_r^0(\mathbb{Q}_{S_1})/H_r(\mathbb{Q}_{S_1})} Z_{r,S_1}(\widehat{\Phi_{h_1,r}},r-n,\mathscr{O}_{S_1}) \\
\times \zeta^S(\tfrac{r+1}{2}-n)\times \prod_{l=2}^n\zeta^S(l)^{-1} \times \prod_{u=1}^{[r/2]}\zeta^S(2u) \zeta^S(2r-2n-2u+1).
\end{multline*}
Therefore, one has
\[
|I({\tilde{f}},r)| \ll N^{\frac{r(r-1)}{2} - r n}\times N_1^{2n^3} \times \mathscr{Z}_r(S_1,h_1)
\]
by using Lemma \ref{lem:zeta}.
\vspace{2mm}
\noindent
(Case II) $r$ is even and $3<r< n$.
By Saito's formula, Proposition \ref{prop:1}, and Lemma \ref{lem:zeta}, one can prove that $|I({\tilde{f}},r)|$ is bounded by
\begin{multline*}
N^{\frac{r(r-1)}{2} - r n} \times \mathscr{Z}_r(S_1,h_1) \times \Big| \zeta^S(\tfrac{r}{2})\times \prod_{l=2}^n\zeta^S(l)^{-1} \times \prod_{u=1}^{r/2-1}\zeta^S(2u) \times \prod_{u=1}^{r/2} \zeta^S(2r-2n-2u+1) \Big| \\
\ll N^{\frac{r(r-1)}{2} - r n}\times N_1^{2n^3} \times \mathscr{Z}_r(S_1,h_1)
\end{multline*}
up to constant.
Note that Proposition \ref{prop:1} was used for this estimate, since it is necessary to prove the vanishing of the term including $D^S_{r,u_{S}}(s)$ in the explicit formula \cite[Theorem 3.3]{Saito2}.
\vspace{2mm}
\noindent
(Case III) $r=n$. In this case, we should use a method different from (Case I) and (Case II) since $Z_{r,S_1}(\widehat{\Phi_{h_1,r}}\chi,s,\mathscr{O}_{S_1})$ may have a simple pole at $s=r-n=0$.
Take an $n$-tuple $\underline{l}=(l_1,\dots,l_n)$ $(l_1\geq \cdots\geq l_n>2n)$, and let $n(x)=\begin{pmatrix}I_n& x \\ O_n & I_n \end{pmatrix}\in G$ where $x\in V_n$.
Recall that ${\tilde{f}}_{\underline{l}}$ satisfies the following two properties:
\begin{itemize}
\item[(i)] ${\tilde{f}}_{\underline{l}}(k^{-1}gk)={\tilde{f}}_{\underline{l}}(g)$ $(\forall k\in K_\infty$, $g\in G(\mathbb{R}))$, see \cite[\S 5.3]{Wakatsuki}.
\item[(ii)] $\int_\mathbb{R} {\tilde{f}}_{\underline{l}}(g_1^{-1} n_1(t) g_2)\, {\mathrm{d}} t=0$ $(\forall g_1$, $g_2\in G(\mathbb{R}))$ where $n_1(t)=n((b_{ij})_{1\leq i,j\leq n})$, $b_{11}=t$ and $b_{ij}=0$ $(\forall (i,j)\neq(1,1))$, see \cite[Lemma 5.9]{Wakatsuki}.
\end{itemize}
By the property (i) we can define $\Phi_{{\tilde{f}}_{\underline{l}},n}(x)={\tilde{f}}_{\underline{l}}(n(x))$ $(x\in V_n(\mathbb{R}))$.
\begin{lem}\label{lem:20210602}
For each orbit $\mathscr{O}_\infty\in V_n^0(\mathbb{R}) H_n(\mathbb{R})$, we have $Z_{n,\infty}(\Phi_{{\tilde{f}}_{\underline{l}},n},\frac{n+1}{2},\mathscr{O}_{\infty})=0$.
\end{lem}
\begin{proof}
Let $\mathscr{O}_\infty \neq I_n\cdot H_n(\mathbb{R})$, and take a representative element $A$ of $\mathscr{O}_\infty$ as $A=\begin{pmatrix} 0& 0 & 1 \\ 0& \mathscr{A} &0 \\ 1&0&0 \end{pmatrix}$, $\mathscr{A}\in V_{n-2}^0(\mathbb{R})$.
The orbit $\mathscr{O}_\infty$ is decomposed into $A\cdot {\mathop{\mathrm{GL}}}_n(\mathbb{R}) \sqcup (-A)\cdot {\mathop{\mathrm{GL}}}_n(\mathbb{R})$.
The centralizer $H_{n(A)}$ of $n(A)$ in $H_n(\mathbb{R})$ is given by $H_{n(A)}=\{ m(h)n(y) \mid h\in {\mathop{\mathrm{O}}}_A(n) $, $y\in V_n(\mathbb{R}) \}$, where $m(h)=\begin{pmatrix} {}^t h^{-1} & O_n \\ O_n & h \end{pmatrix}$ and ${\mathop{\mathrm{O}}}_A(n)=\{h\in {\mathop{\mathrm{GL}}}_n \mid {}^t h A h=A\}$.
Hence, by the property (ii), we have
\begin{multline*}
Z_{n,\infty}(\Phi_{{\tilde{f}}_{\underline{l}},n},\tfrac{n+1}{2},\mathscr{O}_{\infty})=\sum_{A'=\pm A} \int_{{\mathop{\mathrm{O}}}_{A'}(n)\backslash {\mathop{\mathrm{GL}}}_n(\mathbb{R})} {\tilde{f}}_{\underline{l}}( m(h)^{-1} n(A') m(h) ) \, |\det(h)|^{n+1} \, d h \\
= \sum_{A'=\pm A} \int_{\mathscr{N}{\mathop{\mathrm{O}}}_{A'}(n)\backslash {\mathop{\mathrm{GL}}}_n(\mathbb{R})} \int_\mathbb{R} {\tilde{f}}_{\underline{l}}( m(h)^{-1} n(A') n_1(2t) m(h) ) \, |\det(h)|^{n+1}\, {\mathrm{d}} t \, d h=0,
\end{multline*}
where $\mathscr{N}=\{ (b_{ij}) \mid b_{jj}=1$ $(1\leq j\leq n)$, $b_{n1}\in \mathbb{R}$, and $b_{ij}=0$ otherwise$\}$.
In the case $s=\frac{n+1}{2}$, we note that $|\det(x)|$ vanishes in the integral of $Z_{n,\infty}(\Phi_{{\tilde{f}}_{\underline{l}},n},\tfrac{n+1}{2},\mathscr{O}_{\infty})$.
Hence, it follows from the property (ii) that
\[
\sum_{\mathscr{O}_\infty\in V_n^0(\mathbb{R})/H_n(\mathbb{R})} Z_{n,\infty}(\Phi_{{\tilde{f}}_{\underline{l}},n},\tfrac{n+1}{2},\mathscr{O}_\infty)= \int_{V_n(\mathbb{R})}\Phi_{{\tilde{f}}_{\underline{l}},n}(x) \, {\mathrm{d}} x=0,
\]
and so we also find $Z_{n,\infty}(\Phi_{{\tilde{f}}_{\underline{l}},n},\frac{n+1}{2}, I_n\cdot H_n(\mathbb{R}))=0$.
\end{proof}
By Lemmas \ref{lem:functzeta} and \ref{lem:20210602}, the residue formula \cite[Ch. 4]{Yukie} of $Z_n(\Phi,s,{\mathbbm{1}})$ and the same argument as in \cite[Proof of Theorem 4.22]{HW} we obtain
\begin{multline*}
\zeta_r(\widehat{\Phi_{h,r}},0)=\mathbf{C}_{n,n}(\underline{l})^{-1}\, Z_n(\Phi_{{\tilde{f}}_{\underline{l}}h,r} ,\tfrac{n+1}{2},{\mathbbm{1}})= \mathbf{C}_{n,n}(\underline{l})^{-1}\, {\mathop{\mathrm{vol}}}(H_n(\mathbb{Q})\backslash H_n(\mathbb{A})^1) \\
\times \int_{V(\mathbb{R})}\Phi_{{\tilde{f}}_{\underline{l}},n}(x_\infty) \, \log|\det(x_\infty)|_\infty \, {\mathrm{d}} x_\infty \, \int_{V(\mathbb{Q}_{S_1})} \Phi_{h_1,n}(x_{S_1}) \, {\mathrm{d}} x_{S_1} \, N^{-\frac{n(n+1)}{2}},
\end{multline*}
where $H_n(\mathbb{A})=\{(a,m)\in H_n(\mathbb{A})\mid |a^n\det(m)^2|=1\}$.
From this we have $|I({\tilde{f}},r)| \ll N^{-\frac{n(n+1)}{2}} \times \mathscr{Z}_r(S_1,h_1)$.
\vspace{2mm}
\noindent
(Case IV) $r=2<n$.
By Saito's formula \cite[Theorem 4.15]{HW} we have
\[
|I({\tilde{f}},r)| \ll N^{1 - 2 n} \times \mathscr{Z}_2(S_1,h_1) \times \Big| \zeta^S(2)^{-1} \zeta^S(3-2n)\Big| \times \max_{u_S\in \mathbb{Q}_S^\times/(\mathbb{Q}_S^\times)^2,\; u_\infty<0} |D_{2,u_S}^S(2-n)| .
\]
Hence, it is enough to give an upper bound of $|D_{2,u_S}^S(2-n)|$ for $u_\infty<0$.
Choose a representative element $u_S=(u_v)_{v\in S}$ satisfying $u_p\in \mathbb{Z}_p$ $(p\in S_1)$.
Take a test function $\Phi=\otimes_v \Phi_v$ such that the support of $\Phi_\infty$ is contained in $\{x\in V_2^0(\mathbb{R})\mid \det(x)>0\}$ and $\Phi_p$ is the characteristic function of ${\mathop{\mathrm{diag}}}(1,-u_p)+p^2 V_2(\mathbb{Z}_p)$ (resp. $V_2(\mathbb{Z}_p)$) for each $p\in S_1$ (resp. $p\not\in S$).
Let
\[
\Psi(y,yu)=\int_{K_2} \widehat\Phi({}^t\!k \begin{pmatrix} 0&y \\ y&yu\end{pmatrix}k)\, {\mathrm{d}} k ,\qquad K_2={\mathop{\mathrm{O}}}(2,\mathbb{R})\times \prod_p {\mathop{\mathrm{GL}}}_2(\mathbb{Z}_p),
\]
and we set
\[
T(\Phi,s)=\frac{{\mathrm{d}}}{{\mathrm{d}} s_1}T(\Phi,s,s_1)\Big|_{s_1=0}, \quad T(\Phi,s,s_1)=\int_{\mathbb{A}^\times}\int_\mathbb{A} |y^2|^s \|(1,u)\|^{s_1}\Psi(y,yu) \, {\mathrm{d}} u {\mathrm{d}}^\times y .
\]
By \cite[Lemma 1]{Shintani}, one obtains $Z_{2,\infty}(\widehat{\Phi_\infty},n-\tfrac{1}{2},\mathscr{O}_\infty)= 0$ for any orbit $\mathscr{O}_\infty$ in $V_2^0(\mathbb{R})$.
Therefore, from the functional equation \cite[Corollary (4.3)]{Yukie2} one deduces
\[
|N_1^{-6} D_{2,u_S}^S(2-n)| \ll |Z_{2,S}(\Phi_S,2-n,\mathscr{O}_S) \, D_{2,u_S}^S(2-n)| = \Big| 2^{-1}T(\widehat\Phi,n-\tfrac{1}{2}) \Big|.
\]
By \cite[Proposition (2.12) (2)]{Yukie2}, one gets
\[
|T(\widehat{\Phi},n-\tfrac{1}{2})|\ll N_1^{4n-2}\times \left\{ \zeta^S(2n-2) + | (\zeta^{S})'(2n-2) |+ \Big| \frac{(\zeta^{S})'(2n-1) \zeta^S(2n-2) }{\zeta^S(2n-1)} \Big| \right\} ,
\]
where $(\zeta^{S})'(s)=\frac{{\mathrm{d}}}{{\mathrm{d}} s}\zeta^S(s)$, because $\mathrm{Supp}(\widehat\Phi_p)\subset p^{-2}V(\mathbb{Z}_p)$ for any $p\in S_1$.
Therefore, one gets $|D_{2,u_S}^S(2-n)|\ll N_1^{4n+4}$ by Lemma \ref{lem:zeta}.
\vspace{2mm}
The final task is to prove $\mathscr{Z}_r(S_1,h_1) \ll N_1^{a\kappa+b}$ for some $a$ and $b$.
Using the local functional equations in \cite[Theorem 2.1]{Ikeda} (see also \cite{Sweet}), one gets
\[
\mathscr{Z}_r(S_1,h_1) \ll N_1^{c} \times \sum_{\mathscr{O}_{S_1}\in V_r^0(\mathbb{Q}_{S_1})/H_r(\mathbb{Q}_{S_1})}Z_{r,S_1}(|\Phi_{h_1,r}|,n-\tfrac{r-1}{2},\mathscr{O}_{S_1})
\]
for some $c\in \mathbb{N}$.
It is easy to prove
\[
\sum_{\mathscr{O}_{S_1}\in V_r^0(\mathbb{Q}_{S_1})/H_r(\mathbb{Q}_{S_1})}Z_{r,S_1}(|\Phi_{h_1,r}|,n-\tfrac{r-1}{2},\mathscr{O}_{S_1})\ll N_1^{a'\kappa+b'}
\]
for some $a',b'\in\mathbb{N}$ by using \cite[Lemma 2.1.1]{Assem1} and the results in \cite[Section 2]{Assem} (see also \cite{Saito}).
Thus, we obtain $I({\tilde{f}},r)=O(N_1^{a\kappa+b}N^{-n})$.
\end{proof}
\begin{remark}\label{STD}
We give some remarks on Shin-Templier's work \cite{ST} and Dalal's work \cite{Dalal}.
In the setting of \cite{ST}, they considered ``all" cohomological representations
as a family which exhausts an L-packet at infinity since they
chose the Euler-Poincar\'e pseudo-coefficient at the infinite place.
Then there is no contribution from non-trivial unipotent conjugacy classes.
Therefore, our work is different from Shin-Templier's work.
Shin suggested to consider a family of automorphic representations whose infinite type is any fixed discrete series representation. Dalal \cite{Dalal} carried it out in the weight aspect by using the stable trace formula.
The stabilization allows us to remove the contribution $I_3(f)$ (see the introduction), but instead of $I_3(f)$, the contributions of endoscopic groups have to enter.
Dalal obtained a good bound for them by using the concept of hyperendoscopy introduced by Ferrari \cite{Ferrari}.
In studying the level aspect, it seems difficult to get a sufficient bound for the growth of concerning hyperendoscopic groups, but our careful analysis (cf. the proof of Theorem \ref{thm:asy}) shows that estimating unipotent contributions is simpler than using stable trace formula.
In fact, some of zeta integrals $Z_r(\Phi_{{\tilde{f}},r},s,\chi)$ probably correspond to the contributions of endoscopic groups of ${\mathop{\mathrm{Sp}}}(2n)$, and so we still need similar computations even if we use the stable trace formula.
\end{remark}
\section{Arthur classification of Siegel modular forms}\label{Arthur}
In this section we study Siegel modular forms in terms of Arthur's classification
\cite{Ar-book} (see Section 1.4 and 1.5 of loc.cit.).
Recall $G={\mathop{\mathrm{Sp}}}(2n)/\mathbb{Q}$. We call a Siegel cusp form which comes from smaller
groups by Langlands functoriality ``a non-genuine form."
In this section, we estimate the dimension of non-genuine forms and
show that they are negligible.
Let $F\in HE_{\underline{k}}(N)$ (see Section \ref{sec:pre}), and $\pi=\pi_F$ be
the corresponding automorphic representation of $G(\mathbb{A})$. According to Arthur's classification, $\pi$ can be described by using the global
$A$-packets.
Let us recall some notations. A (discrete) global $A$-parameter is a symbol
$$\psi=\pi_1[d_1]\boxplus \cdots\boxplus \pi_r[d_r]
$$
satisfying the following conditions:
\begin{enumerate}
\item for each $i$ $(1\le i \le r)$, $\pi_i$ is an irreducible unitary cuspidal self-dual automorphic representation of ${\rm GL}_{m_i}(\mathbb{A})$.
In particular, the central character $\omega_i$ of $\pi_i$ is trivial or quadratic;
\item for each $i$, $d_i\in \mathbb{Z}_{>0}$ and $\displaystyle\sum_{i=1}^r m_id_i=2n+1$;
\item if $d_i$ is odd, $\pi_i$ is orthogonal, i.e., $L(s,\pi_i,{\rm Sym}^2)$ has a pole at $s=1$;
\item if $d_i$ is even, $\pi_i$ is symplectic, i.e., $L(s,\pi_i,\wedge^2)$ has a pole at $s=1$;
\item $\omega^{d_1}_1\cdots \omega^{d_r}_r={\mathbbm{1}}$;
\item if $i\neq j$, $\pi_i\simeq \pi_j$, then $d_i\neq d_j$.
\end{enumerate}
We say that two global $A$-parameters $\boxplus_{i=1}^r\pi_i[d_i]$ and $\boxplus_{i=1}^{r'}\pi'_i[d'_i]$ are
equivalent if $r=r'$ and there exists $\sigma\in \mathfrak S_r$ such that $d'_i=d_{\sigma(i)}$ and
$\pi'_i=\pi_{\sigma(i)}$.
Let $\Psi(G)$ be the set of equivalent classes of global $A$-parameters.
For each $\psi \in \Psi(G)$, one can associate a set $\Pi_\psi$ of equivalent classes of simple admissible $G(\mathbb{A}_f)\times (\frak g,K_\infty)$-modules (see \cite{Ar-book}).
The set $\Pi_\psi$ is called a global $A$-packet for $\psi$.
\begin{Def}\label{cap-endo}Let $\psi=\displaystyle\boxplus_{i=1}^r\pi_i[d_i]$ be a global $A$-parameter.
\begin{itemize}
\item $\psi$ is said to be semi-simple if $d_1=\cdots=d_r=1$; otherwise, $\psi$ is said to be non-semi-simple;
\item $\psi$ is said to be simple if $r=1$ and $d_1=1$.
\end{itemize}
\end{Def}
By \cite[Theorem 1.5.2]{Ar-book} (though our formulation is slightly different from the
original one), we have a following decomposition
\begin{equation}\label{AD}
L^2_{{\rm disc}}(G(\mathbb{Q}){\backslash} G(\mathbb{A}))\simeq \bigoplus_{\psi\in \Psi(G)}\bigoplus_{\pi\in \Pi_\psi}m_{\pi,\psi}\pi
\end{equation}
where $m_{\pi,\psi}\in \{0,1\}$ (cf. see \cite[Theorem 2.2]{Atobe} for $m_{\pi,\psi}$).
We have an immediate consequence of (\ref{AD}):
\begin{prop}\label{first-est} Let $1_{K(N)}$ be the characteristic function of $K(N)\subset G(\mathbb{A}_f)$. Then
$$S_{{\underline{k}}}(\Gamma(N))=
\bigoplus_{\psi\in \Psi(G)}\bigoplus_{\pi\in \Pi_\psi\atop \pi_\infty\simeq \sigma_{\underline{k}}}m_{\pi,\psi}\pi_f^{K(N)}
$$ and
\begin{equation}\label{characteristic}
|HE_{{\underline{k}}}(N)|={\rm vol}(K(N))^{-1}\sum_{\psi\in \Psi(G)}
\sum_{\pi\in \Pi_\psi \atop \pi_\infty\simeq \sigma_{\underline{k}}}m_{\pi,\psi}{\rm tr}(\pi_f(1_{K(N)})).
\end{equation}
\end{prop}
\begin{thm}\label{Ram} Assume (\ref{suff-reg}).
For a global $A$-parameter $\psi=\displaystyle\boxplus_{i=1}^r\pi_i[d_i]$, suppose that there exists $\pi\in \Pi_\psi$ with $\pi_\infty\simeq \sigma_{\underline{k}}$. Then $\psi$ is semi-simple, i.e., $d_i=1$ for all $i$, and each $\pi_i$ is regular algebraic and satisfies the Ramanujan conjecture, i.e., $\pi_{i,p}$
is tempered for any $p$.
\end{thm}
\begin{proof} By the proof of \cite[Corollary 8.5.4]{CL}, we see that $d_1=\cdots=d_r=1$. Hence $\psi$ is semi-simple.
Further, by comparing infinitesimal characters $c(\pi_\infty),\ c(\psi_\infty)$ of $\pi_\infty,\ \psi_\infty$
respectively, we see that each $\pi_i$ is regular algebraic by \cite[Corollary 6.3.6, p.164 and Proposition 8.2.10, p.196]{CL}.
It follows from \cite{Ca1} and \cite{Ca2} that $\pi_{i,p}$ is tempered for any $p$.
\end{proof}
Therefore, for each finite prime $p$,
the local Langlands parameter at $p$ of $\pi$ is described as one of
the isobaric sum $\boxplus_{i=1}^r\pi_{i,p}$ which is an admissible representation of ${\rm GL}_{2n+1}(\Bbb Q_p)$.
\begin{Def}
We denote by $HE_{{\underline{k}}}(N)^{ng}$ the subset of $HE_{{\underline{k}}}(N)$ consisting of all
forms which belong to
$$\bigoplus_{\psi\in \Psi(G)\atop \psi\text{:non-simple}}
\bigoplus_{\pi\in \Pi_\psi \atop \pi_\infty\simeq \sigma_{\underline{k}}}m_{\pi,\psi}\pi_f^{K(N)},
$$
under the isomorphism $(\ref{AD})$. A form in this space is called a non-genuine form.
Similarly, we denote by $HE_{{\underline{k}}}(N)^{g}$ the subset of $HE_{{\underline{k}}}(N)$ consisting of all forms which belong to
$$\bigoplus_{\psi\in \Psi(G)\atop \psi\text{:simple}}
\bigoplus_{\pi\in \Pi_\psi \atop \pi_\infty\simeq \sigma_{\underline{k}}}m_{\pi,\psi}\pi_f^{K(N)},
$$
under the isomorphism $(\ref{AD})$. A form in this space is called a genuine form.
\end{Def}
\begin{Def}\label{gln} Denote by $\Pi({\mathop{\mathrm{GL}}}_n(\Bbb R))^c$ the isomorphism classes of all irreducible cohomological admissible $(\mathfrak{gl}_n,O(n))$-modules.
For $\tau_\infty\in \Pi({\mathop{\mathrm{GL}}}_m(\Bbb R))^c$ and a quasi-character
$\chi:\mathbb{Q}^\times{\backslash}\mathbb{A}^\times{\longrightarrow} \mathbb{C}^\times$,
we define
$$L^{{\rm cusp},{\rm ort}}({\rm GL}_n(\mathbb{Q}){\backslash} {\rm GL}_n(\mathbb{A}),\tau_\infty,\chi):=
\bigoplus_{\pi:\text{orthogonal}\atop \pi_\infty\simeq \tau_\infty, \omega_\pi=\chi}m(\pi)\pi,\
m(\pi)\in\{0,1\}$$
and $$L^{{\rm cusp},{\rm ort}}(K^{{\rm GL}_n}(N),\tau_\infty,\chi):=
\bigoplus_{\pi:\text{orthogonal}\atop \pi_\infty\simeq \tau_\infty, \omega_\pi=\chi}
m(\pi)\pi^{K^{{\rm GL}_n}(N)}$$
where the direct sums are taken over the isomorphism classes of all orthogonal cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A})$
and $\omega_\pi$ stands for the central character of $\pi$. Here
$K^{{\rm GL}_{n}}(N)$ is the principal congruence subgroup of ${\rm GL}_{n}(\widehat{\mathbb{Z}})$ of level $N$.
Put
$$l^{{\rm cusp},{\rm ort}}(n,N,\tau_\infty,\chi):={\rm dim}_\mathbb{C}(L^{{\rm cusp},{\rm ort}}(K^{{\rm GL}_n}(N),\tau_\infty,\chi))
$$
for simplicity. Clearly, $l^{{\rm cusp},{\rm ort}}(1,N,\tau_\infty,\chi)=
|\widehat{\mathbb{Z}}^\times/(1+N\widehat{\mathbb{Z}})^\times|=\varphi(N)$ where $\varphi$ stands for
Euler's totient function.
\end{Def}
Let $P(2n+1)$ be the set of all partitions of $2n+1$ and
$P_{\underline{m}}$ be the standard parabolic subgroup of ${\rm GL}_{2n+1}$
associated to a partition $2n+1=m_1+\cdots+m_r$, and $\underline{m}=(m_1,...,m_r)$.
In order to apply the formula (\ref{characteristic}), it is necessary to study the transfer of Hecke elements in
the local Langlands correspondence established by \cite[Theorem 1.5.1]{Ar-book}.
We regard $G={\rm Sp}(2n)$ as a twisted elliptic endoscopic subgroup of ${\rm GL}_{2n+1}$ (cf. \cite{GV} or \cite{Oi}).
\begin{prop}\label{transfer} Let $N$ be an odd positive integer.
Put $S_N:=\{p\, \text{prime}\ :\ p\mid N\}$.
For the pair $({\rm GL}_{2n+1},G)$, the characteristic function of
${\rm vol}(K(N))^{-1}1_{K(N)}$ as an element of
$C^\infty_c(G(\mathbb{Q}_{S_N}))$ is transferred to
$${\rm vol}(K^{{\rm GL}_{2n+1}}(N))^{-1}1_{K^{{\rm GL}_{2n+1}}(N)}$$
as an element of $C^\infty_c({\rm GL}_{2n+1}(\mathbb{Q}_{S_N}))$.
\end{prop}
\begin{proof}It follows from \cite[Lemma 8.2.1-(i)]{GV}.
\end{proof}
Applying Proposition \ref{transfer}, we have the following.
\begin{prop}\label{second-est} Assume (\ref{suff-reg}) and $N$ is odd.
Then
$|HE_{{\underline{k}}}(N)^{ng}|$ is bounded by
$$
\frac{A_n(N)}{\varphi(N)}\sum_{\underline{m}=(m_1,\ldots,m_r)\in P(2n+1)}
\sum_{c(\boxplus_{i=1}^r \tau_i)=c(\sigma_{\underline{k}})\atop \tau_i\in\Pi({\rm GL}_{m_i}(\mathbb{R}))^c}
\sum_{\chi_i:\mathbb{Q}^\times{\backslash}\mathbb{A}^\times {\longrightarrow}\mathbb{C}^\times\atop \chi^2_i=1,\ c(\chi)|N}d_{P_{\underline{m}}}(N)\prod_{i=1}^r
l^{{\rm cusp},{\rm ort}}(m_i,N,\tau_{i},\chi_i),
$$
where for the second sum, it is indexed by all $r$-tuples $(\tau_1,\ldots,\tau_r)$ such that
$\tau_i\in\Pi({\rm GL}_{m_i}(\mathbb{R}))^c$ and
$c(\boxplus_{i=1}^r \tau_i)=c(\sigma_{{\underline{k}}})$ (the equality of the infinitesimal characters). Further $c(\chi)$ stands for the conductor of $\chi$ and
$\varphi(N)=|(\mathbb{Z}/N\mathbb{Z})^\times|$.
Here
\begin{enumerate}
\item $A_n(N):= 2^{(2n+1)\omega(N)}$ where $\omega(N):=|\{p\ prime\ :\ p|N\}|$;
\item $d_{P_{\underline{m}}}(N)=|P_{\underline{m}}(\mathbb{Z}/N\mathbb{Z}){\backslash} {\rm GL}_{2n+1}(\mathbb{Z}/N\mathbb{Z})|={\rm vol}({K^{{\rm GL}_{2n+1}}(N)})^{-1}/|P_{\underline{m}}(\mathbb{Z}/N\mathbb{Z})|$.
\end{enumerate}
\end{prop}
\begin{proof} Let $\pi=\pi_\infty\otimes\otimes'_p \pi_p$ be an element of $\Pi_\psi$ for
$\psi=\boxplus_{i=1}^r\pi_i$.
Let $\Pi_p$ be the local Langlands correspondence of $\pi_p$ to ${\rm GL}_{2n+1}(\mathbb{Q}_p)$
established by \cite[Theorem 1.5.1]{Ar-book} and $\mathcal{L}(\Pi_p):L_{\mathbb{Q}_p}{\longrightarrow} {\rm GL}_{2n+1}(\mathbb{C})$ be the local $L$-parameter of $\Pi_p$, where $L_{\mathbb{Q}_p}=W_{\mathbb{Q}_p}$ for each
$p<\infty$ and $L_{\mathbb{R}}=W_{\mathbb{R}}\times {\rm SL}_2(\mathbb{C})$.
Since the localization $\psi_p$ of the global $A$-parameter $\psi$ at $p$ is
tempered by Theorem \ref{Ram}, we see that $\mathcal{L}(\Pi_p)$ is equivalent to $\psi_p$.
Since $\mathcal{L}(\Pi_p)$ is independent of $\pi\in \Pi_\psi$ and multiplicity one for ${\rm GL}_{2n+1}(\mathbb{A})$ holds,
the isobaric sum $\psi=\boxplus_{i=1}^r \pi_i$ as an automorphic representation of ${\rm GL}_{2n+1}(\mathbb{A})$ gives rise to a unique
global L-parameter on $\Pi_\psi$. On the other hand, it follows from \cite[Theorem 1.5.1]{Ar-book} that
$|\Pi_{\psi_p}|\le 2^{2n+1}$ for the local A-packet $\Pi_{\psi_p}$ at $p$ if $p|N$, and $\Pi_{\psi_p}$ is a singleton if $p\nmid N$.
It yields that $|\Pi_\psi|\le 2^{(2n+1)\omega(N)}$.
Since the local Langlands correspondence $\pi_p\mapsto \Pi_p$ satisfies the character relation
by \cite[Theorem 1.5.1]{Ar-book}, it follows from Proposition \ref{transfer}
that for each $\pi\in \Pi_\psi$,
\begin{eqnarray*}
&&{\rm dim}(\pi^{K(N)}_f)={\rm vol}(K(N))^{-1} {\rm tr}(\pi(1_{K(N)}))=
{\rm vol}(K^{{\rm GL}_{2n+1}}(N))^{-1}{\rm tr}((\boxplus_{i=1}^r\pi_i)(1_{K^{{\rm GL}_{2n+1}}(N)}))\\
&&\phantom{xxxxxxxxx} ={\rm dim}((\boxplus_{i=1}^r\pi_i)_f^{{K^{{\rm GL}_{2n+1}}(N)}}),
\end{eqnarray*}
where we denote by $\pi_f=\otimes_{p<\infty}' \pi_p$ the finite part of the cuspidal representation $\pi$. Plugging this into Proposition \ref{first-est}, we have
\begin{eqnarray}
|HE_{{\underline{k}}}(N)^{ng}|&=&{\rm vol}(K(N))^{-1}
\sum_{\psi=\boxplus_{i=1}^r\pi_i\in \Psi(G),\ r\ge 2\atop c(\psi_\infty)=c(\sigma_{\underline{k}})} \sum_{\pi\in \Pi_\psi}m_{\pi,\psi}
{\rm tr}(\pi_f(1_{K(N)})) \nonumber \\
&\le &\frac{A_n(N)}{\varphi(N)}
\sum_{\psi=\boxplus_{i=1}^r\pi_i\in \Psi(G),\ r\ge 2\atop c(\psi_\infty)=c(\sigma_{\underline{k}})}
{\rm dim}((\boxplus_{i=1}^r\pi_i)_f^{{K^{{\rm GL}_{2n+1}}(N)}}). \nonumber
\end{eqnarray}
where
$\displaystyle\frac{1}{\varphi(N)}$ is inserted because of the condition on the central characters in global $A$-parameters. Here
$r\ge 2$ is essential to gain the factor $\displaystyle\frac{1}{\varphi(N)}$
(see Remark \ref{characters-exp}).
Next we describe ${\rm dim}((\boxplus_{i=1}^r\pi_i)_f^{{K^{{\rm GL}_{2n+1}(N)}}})$ in terms of
the data $(m_i,N,\tau_i,\chi_i)$ with $1\le i\le r$.
Since
$$P_{\underline{m}}(\mathbb{A}_f){\backslash} {\rm GL}_{2n+1}(\mathbb{A}_f)/K(N)\simeq
P_{\underline{m}}(\widehat{\mathbb{Z}}){\backslash} {\rm GL}_{2n+1}(\widehat{\mathbb{Z}})/K(N) \simeq P_{\underline{m}}(\mathbb{Z}/N\mathbb{Z}){\backslash} {\rm GL}_{2n+1}(\mathbb{Z}/N\mathbb{Z})
$$
and a complete system of the representatives can be taken
from elements in ${\rm GL}_{2n+1}(\widehat{\mathbb{Z}})$ and therefore they normalize $K(N)$.
Then a standard method for fixed vectors of
an induced representation shows that
$${\rm dim}((\boxplus_{i=1}^r\pi_i)_f^{{K^{{\rm GL}_{2n+1}}(N)}})
=d_{P_{\underline{m}}}(N) \prod_{i=1}^r
{\rm dim}(\pi^{{K^{{\rm GL}_{m_i}(N)}}}_{i,f}),
$$
Here if $\chi_i$ is the central character of $\pi_i$ and $\pi_{i,\infty}\simeq \tau_i$, then ${\rm dim}(\pi^{{K^{{\rm GL}_{m_i}(N)}}}_{i,f})=l^{{\rm cusp},{\rm ort}}(m_i,N,\tau_{i},\chi_i)$. Notice that the conductor of $\chi_i$ is a divisor of $N$.
Summing up, ${\rm dim}(S_{{\underline{k}}}({\Gamma}(N))^{ng})$ is bounded by
$$\frac{A_n(N)}{\varphi(N)}\sum_{\underline{m}=(m_1,\ldots,m_r)\in P(2n+1)\atop r\ge 2}
\sum_{c(\boxplus_{i=1}^r \tau_i)=c(\sigma_{\underline{k}})\atop \tau_i\in \Pi({\rm GL}_{m_i}(\mathbb{R}))^c}
\sum_{\chi_i:\mathbb{Q}^\times{\backslash}\mathbb{A}^\times {\longrightarrow}\mathbb{C}^\times\atop \chi^2_i=1,\ c(\chi)|N}
d_{P_{\underline{m}}}(N)\prod_{i=1}^r
l^{{\rm cusp},{\rm ort}}(m_i,N,\tau_{i},\chi_i).
$$
\end{proof}
\begin{remark}\label{characters-exp}Let $r\ge 2$. The group homomorphism $((\mathbb{Z}/N\mathbb{Z})^\times)^r{\longrightarrow} (\mathbb{Z}/N\mathbb{Z})^\times,\
(x_1,\ldots,x_r)\mapsto x_1\cdots x_r$ is obviously surjective and it yields
$$|\{(\chi_1,\ldots,\chi_r)\in \widehat{(\mathbb{Z}/N\mathbb{Z})^\times}^r\ |\ \chi_1\cdots\chi_r=1\}|
=\frac{|\widehat{(\mathbb{Z}/N\mathbb{Z})^\times}^r|}{\varphi(N)}.$$
This trivial equality explains the appearance of the factor $\displaystyle\frac{1}{\varphi(N)}$ in
Proposition \ref{second-est}.
\end{remark}
Next we study $l^{{\rm cusp},{\rm ort}}(n,N,\tau,\chi)$ for $\tau\in \Pi({\rm GL}_n(\mathbb{R}))^c$
and for $n\ge 2$. Now if $\pi$ is a cuspidal representation of ${\mathop{\mathrm{GL}}}_{2m+1}$ which is orthogonal, i.e., $L(s,\pi, {\rm Sym}^2)$ has a pole at $s=1$, then $\pi$ comes from a cuspidal representation $\tau$ on ${\mathop{\mathrm{Sp}}}(2m)$. In this case, the central character $\omega_{\pi}$ of $\pi$ is trivial.
If $\pi$ is a cuspidal representation of ${\mathop{\mathrm{GL}}}_{2m}$ which is orthogonal, i.e., $L(s,\pi, {\rm Sym}^2)$ has a pole at $s=1$, then
$\omega_\pi^2=1$; If $\omega_\pi=1$, $\pi$ comes from a cuspidal representation $\tau$ on the split orthogonal group ${\mathop{\mathrm{SO}}}(m,m)$;
If $\omega_\pi\ne 1$, then $\pi$ comes from a cuspidal representation $\tau$ on the quasi-split orthogonal group ${\mathop{\mathrm{SO}}}(m+1,m-1)$.
First we consider the case when $\chi$ is trivial in estimating
$l^{{\rm cusp},{\rm ort}}(2n+\delta, N,\tau,\chi)$ where $\delta=0$ or 1.
For a positive integer $n$, let $H=\begin{cases} {\mathop{\mathrm{SO}}}(n,n), &\text{if $G'={\mathop{\mathrm{GL}}}_{2n}$}\\
{\mathop{\mathrm{Sp}}}(2n), &\text{if $G'={\mathop{\mathrm{GL}}}_{2n+1}$}\end{cases}.$
We regard $H$ as a twisted elliptic endoscopic subgroup $G'$.
\begin{prop}\label{transfer-ort} Let $N$ be an odd
positive integer.
For the pair $(G',H)$, the characteristic function of
${\rm vol}(K^H(N))^{-1}1_{K^H(N)}$ as an element of
$C^\infty_c(H(\mathbb{Q}_{S_N}))$ is transferred to
$${\rm vol}(K^{G'}(N))^{-1} 1_{K^{G'}(N)}$$
as an element of $C^\infty_c(G'(\mathbb{Q}_{S_N}))$.
\end{prop}
\begin{proof}It follows from \cite[Lemma 8.2.1-(i)]{GV}.
\end{proof}
Each cuspidal representation $\pi$ of $G'(\mathbb{A})$ contributing $l^{{\rm cusp},{\rm ort}}(N,\tau,\chi)$ can be
regarded as a simple $A$-parameter and also
as a cuspidal representation it
strongly descend to a generic cuspidal representation $\Pi_\pi$ of $H(\mathbb{A})$ of which the $L$-parameter
$\mathcal{L}(\Pi_{\tau})$ at infinity of
$\Pi_{\pi}$ is same as one of $\pi_\infty$.
In this setting, by \cite[Proposition 8.3.2-(b), p.483]{Ar-book}, the problem is reduced to estimate
$$L^{{\rm cusp,gen}}(H,N,\mathcal{L}(\Pi_\tau),{\mathbbm{1}})
:=\bigoplus_{\pi\subset L^{{\rm cusp,generic},{\rm ort}}(H(\mathbb{Q}){\backslash} H(\mathbb{A}),\mathcal{L}(\Pi_{\tau}),{\mathbbm{1}})}
m(\pi)\pi^{K^{H}(N)},\, m(\pi)\in\{0,1,2\}
$$
where $\pi$ runs over all irreducible unitary, cohomological orthogonal cuspidal automorphic representations of $H(\mathbb{A})$ whose
$L$-parameter at infinity is isomorphic to $\mathcal{L}(\Pi_{\tau})$ with the
central character $\chi={\mathbbm{1}}$.
\begin{prop}\label{trivial-case} Keep the notations as above. Then
\begin{itemize}
\item $l^{{\rm cusp},{\rm ort}}(2n+\delta,N,\tau,\chi)\le C_n(N){\rm dim}(L^{{\rm cusp,gen}}(H,N,\mathcal{L}(\Pi_\tau),{\mathbbm{1}}))$, where
$C_n(N):=2^{(2n+\delta)\omega(N)}$
and $\delta=\begin{cases} 0, &\text{if $G'={\mathop{\mathrm{GL}}}_{2n}$}\\1, &\text{if $G'={\mathop{\mathrm{GL}}}_{2n+1}$}\end{cases}$.
\item ${\rm dim}(L^{{\rm cusp,gen}}(H,N,\mathcal{L}(\Pi_\tau),{\mathbbm{1}})) \ll c\cdot {\rm vol}(K^H(N))^{-1}\sim c N^{{\rm dim}(H)}$ for some $c>0$, when the infinitesimal character of $\mathcal{L}(\Pi_\tau)$ is fixed and $N\to\infty$.
\end{itemize}
\end{prop}
\begin{proof} The first claim follows from \cite[Proposition 8.3.2-(b), p.483]{Ar-book} with
a completely similar argument of Proposition \ref{second-est}.
The second claim follows from \cite{Savin}.
\end{proof}
Next we consider when $\chi$ is a quadratic character.
In this case, a cuspidal representation $\pi$
contributing to $L^{{\rm cusp},{\rm ort}}(K^{{\rm GL}_n}(N),\tau_\infty,\chi)$ comes from a
cuspidal representation of the quasi-split orthogonal group ${\mathop{\mathrm{SO}}}(m+1,m-1)$ defined over the quadratic extension associated to $\chi$.
However any transfer theorem for Hecke elements in $({\mathop{\mathrm{GL}}}_{2m},{\mathop{\mathrm{SO}}}(m+1,m-1))$ remains open.
To get around this situation, we make use of the transfer theorems for some Hecke elements in
the quadratic base change due to \cite{Yamauchi}. For this, we need the following assumptions on the level $N$:
\begin{enumerate}
\item $N$ is an odd prime or
\item $N$ is odd and all prime divisors $p_1,\ldots,p_r\ (r\ge 2)$ of $N$ are
congruent to 1 modulo 4 and
$\Big(\displaystyle\frac{p_i}{p_j}\Big)=1$ for $i\ne j$,
where $\Big(\displaystyle\frac{\ast}{\ast}\Big)$ denotes the Legendre symbol.
\end{enumerate}
These conditions are needed in order that for any quadratic extension $M/\mathbb{Q}$ with conductor
dividing $N$, there exists an integral ideal $\frak{N}$ of $M$ such that $\frak{N} \frak{N}^\theta
=(N)$ where $\theta$ is the generator of ${\rm Gal}(M/\mathbb{Q})$.
\begin{prop}\label{fixedV}
Keep the assumptions on $N$ as above.
Then
$$l^{{\rm cusp},{\rm ort}}(2m,N,\tau,\chi)\le 2^{2m\cdot\omega(N)}
{\rm vol}(K^{H}(N))^{-1},
$$
where $H={\rm SO}(m,m)$.
\end{prop}
\begin{proof}
Let $M/\mathbb{Q}$ be the quadratic extension associated to $\chi$ and $\mathcal{O}_M$
the ring of integers of $M$. Let $\theta$ be the generator of ${\rm Gal}(M/\mathbb{Q})$.
Let $K^{{\rm GL}_{2m}}_M(\frak{N})$ be the principal congruence subgroup of ${\mathop{\mathrm{GL}}}_{2m}(\widehat{\mathbb{Z}}\otimes_\mathbb{Z}
\mathcal{O}_M)$ of the level $\frak{N}$.
Clearly, $(K^{{\rm GL}_{2m}}_M(\frak{N}))^\theta=K^{{\rm GL}_{2m}}(N)$.
Applying \cite[Theorem 1.6]{Yamauchi}, we have
for a cuspidal representation $\pi$ of ${\mathop{\mathrm{GL}}}_{2m}(\mathbb{A})$ and its base change $\Pi:=
{\rm BC}_{M/\mathbb{Q}}(\pi)$
to ${\mathop{\mathrm{GL}}}_{2m}(\mathbb{A}_M)$,
$${\rm vol}(K^{{\rm GL}_{2m}}(N))^{-1}{\rm tr}(\pi(1_{K^{{\rm GL}_{2m}}(N)}))=
{\rm vol}(K^{{\rm GL}_{2m}}_M(\frak{N}))^{-1}
{\rm tr}(\Pi(1_{K^{{\rm GL}_{2m}}_M(\frak{N})})).
$$
Recall that our $\pi$ contributing
$L^{{\rm cusp},{\rm ort}}(2m,N,\tau,\chi)$ is orthogonal, namely,
$L(s,\pi, {\mathop{\mathrm{Sym}}}^2)$ has a pole at $s=1$.
Note that
$L(s,\Pi, {\mathop{\mathrm{Sym}}}^2)=L(s,\pi,{\mathop{\mathrm{Sym}}}^2)L(s,\pi, {\mathop{\mathrm{Sym}}}^2\otimes\chi)$.
Now $L(s,\pi \times (\pi\otimes\chi))=L(s,\pi,\wedge^2\otimes\chi)
L(s,\pi,{\mathop{\mathrm{Sym}}}^2\otimes\chi)$.
Suppose $\Pi$ is cuspidal. Then $\pi\not\simeq \pi\otimes\chi$.
So the left hand side has no zero at $s=1$,
and $L(s,\pi,{\mathop{\mathrm{Sym}}}^2\otimes\chi)$ has no zero at $s=1$.
Therefore, $L(s,\Pi,{\mathop{\mathrm{Sym}}}^2)$ has a pole at $s=1$.
If $\Pi$ is non-cuspidal, then by \cite{AC}, there exists a cuspidal representation $\tau$ of ${\mathop{\mathrm{GL}}}_m(\mathbb{A}_M)$ such that
$$\Pi=\tau\boxplus \tau^\theta.
$$
In such a case, if $m=2$, then $\pi={\rm AI}_M^\Bbb Q \tau$ for some cuspidal
representation $\tau$ of ${\mathop{\mathrm{GL}}}_2(\mathbb{A}_M)$; an automorphic induction from ${\mathop{\mathrm{GL}}}_2(\mathbb{A}_M)$ to ${\mathop{\mathrm{GL}}}_4(\mathbb{A}_\mathbb{Q})$. Since $\pi$ is cuspidal and orthogonal, $\tau$ has to be dihedral.
Such a $\pi$ is counted in \cite[Section 2.6]{KWY1} and it amounts to
$O(N^{\frac{11}{2}+\varepsilon})$ for any $\varepsilon>0$. This will be negligible because ${\rm vol}(K^H(N))\sim c N^{m(2m-1)}=c N^6$ for some constant $c>0$.
Assume $m\ge 3$.
It is easy to see that the dimension of
$\displaystyle\bigoplus_{\Pi:\text{non-cuspidal}}\Pi^{K^{{\rm GL}_{2m}}_M(\frak{N})}_f$
is bounded by
$$O(N^{m^2-1+\frac{m(m+1)}{2}})=O(N^{\frac{3}{2}m^2+\frac{m}{2}-1})
$$
where '$-1$' of $m^2-1$ in the exponent of LHS in the above equation is inserted because of the fixed central character.
Since ${\rm dim}\, {\mathop{\mathrm{SO}}}(m,m)=m(2m-1)$ and $m\ge 3$,
$\Pi^{K^{{\rm GL}_{2m}}_M(\frak{N})}_f$ for which $\Pi$ is non-cuspidal is negligible in the following
estimation.
Further, $\Pi$ is orthogonal with trivial central character. [The central character of $\Pi$ is $\chi\circ N_{M/\Bbb Q}=1$.]
Therefore, we can bound $l^{{\rm cusp},{\rm ort}}(2m,N,\tau,\chi)$ by
$$l^{{\rm cusp},{\rm ort}}(2m,\frak{N},{\rm BC}_{M_\infty/\mathbb{R}}(\tau),1),
$$
which is
similarly defined for cuspidal representations of ${\mathop{\mathrm{GL}}}_{2m}(\mathbb{A}_M)$.
Applying the argument of the proof of Proposition \ref{trivial-case} to
$({\mathop{\mathrm{GL}}}_{2m}/M,{\rm SO}(m,m)/M)$,
the quantity $l^{{\rm cusp},{\rm ort}}(2m,N,\tau,\chi)$ is bounded by
$2^{2m \omega(\frak{N})}{\rm vol}(K^{H_M}(\frak{N}))^{-1}$ where
$H_M:={\rm SO}(m,m)/M$ and $\omega(\frak{N})$ denotes the number of
prime ideals dividing $\frak{N}$.
The claim follows from $\mathcal{O}_M/\frak{N}\simeq \mathbb{Z}/N\mathbb{Z}$ since
${\rm vol}(K^{H_M}(\frak{N}))={\rm vol}(K^{H}(N))$ and clearly $\omega(\frak{N})=
\omega(N)$.
\end{proof}
Note that for any split reductive group $\mathcal{G}$ over $\mathbb{Q}$ and the principal congruence
subgroup $K^\mathcal{G}(N)$ of level $N$, we see easily that
${\rm vol}(K^\mathcal{G}(N))\sim c N^{{-\rm dim}\, \mathcal{G}}$ for some constant $c>0$ as $N\to\infty$. Furthermore $\omega(N)\ll \frac {\log N}{\log\log N}$. Hence
$2^{\omega(N)}\ll N^\epsilon$, and
$A_n(N)=O(N^\varepsilon)$ and $C_{m_i}(N)=O(N^\varepsilon)$ for each $1\le i\le r$.
\begin{thm}\label{non-simple} Assume (\ref{suff-reg}). Keep the assumptions on $N$ as in Proposition \ref{fixedV}.
Then $|HE_{{\underline{k}}}(N)^{ng}|=O_n(N^{2n^2+n-1+\varepsilon})$ for any $\varepsilon>0$. In particular,
$$\displaystyle\lim_{N\to \infty}\frac{|HE_{{\underline{k}}}(N)^{ng}|}{|HE_{{\underline{k}}}(N)|}=0.
$$
\end{thm}
\begin{proof} By Proposition \ref{second-est}, for each partition $\underline{m}=(m_1,\ldots,m_r)$ of $2n+1$,
we have only to estimate
$$\frac{A_n(N)}{\varphi(N)} d_{P_{\underline{m}}}(N)\prod_{i=1}^r
l^{{\rm cusp},{\rm ort}}(m_i,N,\tau_{i},\chi_i).
$$
By Proposition \ref{trivial-case} and Proposition \ref{fixedV},
$$l^{{\rm cusp},{\rm ort}}(m_i,N,\tau_{i},\chi_i)\ll N^{\frac{m_i(m_i-1)}{2}+\varepsilon},
$$
for any $\varepsilon>0$.
Further $d_{P_{\underline{m}}(N)}=O(N^{{\rm dim}\, P_{\underline{m}}\backslash {\mathop{\mathrm{GL}}}_{2n+1}})=O(N^{\sum_{1\le i<j\le r}m_im_j})$.
Note that $\varphi(N)^{-1}=O(N^{-1+\varepsilon})$ for any $\varepsilon>0$.
Since
$$\sum_{1\le i<j\le r}m_i m_j+\sum_{i=1}^r\frac{m_i(m_i-1)}{2}=
\frac{1}{2}\left(\sum_{1\le i,j\le r}m_i m_j \right)-\frac{1}{2}\sum_{i=1}^r m_i=\frac 12(2n+1)^2-\frac 12(2n+1)=2n^2+n,
$$
we have the first claim.
The second claim follows from the dimension formula (\ref{dimension}).
\end{proof}
\section{A notion of newforms in $S_{{\underline{k}}}(\Gamma(N))$}
In this section, we introduce a notion of a newform in $S_{\underline{k}}(\Gamma(N))$ with respect to
principal congruence subgroups.
Since any local newform theory for ${\mathop{\mathrm{Sp}}}(2n)$ is unavailable except for $n=1,2$, we
need a notion of newforms so that we can control a lower bound of analytic conductors for such newforms. This is needed in application to low lying zeros.
Recall the description $$S_{\underline{k}}(\Gamma(N))=\bigoplus_{\psi\in \Psi(G)}
\bigoplus_{\pi\in \Pi_\psi\atop \pi_\infty\simeq \sigma_{\underline{k}}}m_{\pi,\psi}\pi_f^{K(N)}$$
in terms of Arthur's classification.
\begin{Def}\label{newspace}
The new part of $S_{\underline{k}}(\Gamma(N))$ is defined by
$$S^{{\rm new}}_{\underline{k}}(\Gamma(N))=\bigoplus_{\psi\in \Psi(G)}
\bigoplus_{\pi=\pi_f\otimes \sigma_{\underline{k}}\in \Pi_\psi\atop
\pi^{K(N)}\neq 0 \text{ but } \pi^{K(d)}=0 \text{ for any }d\mid N,d\neq N}m_{\pi,\psi}\pi_f^{K(N)}.
$$
Let $HE^{{\rm new}}_{{\underline{k}}}(N)$ be a subset of $HE_{\underline{k}}(N)$ which is a basis of $S^{{\rm new}}_{\underline{k}}(\Gamma(N))$.
The orthogonal complement $S^{{\rm old}}_{\underline{k}}(\Gamma(N))$ of $S^{{\rm new}}_{\underline{k}}(K(N))$ in
$S_{\underline{k}}(K(N))$ with respect to Petersson inner product is said to be the old space.
\end{Def}
We denote by $T$ the maximal split torus in ${\mathop{\mathrm{Sp}}}(2n)$ and we write
$t={\mathop{\mathrm{diag}}}(t_1,\ldots,t_n,t^{-1}_1,\ldots,t^{-1}_n)\in T$.
\begin{lem}\label{linear-comb} Let $M$ be a divisor of $N$ with $M\neq N$ and $\pi$ be a cuspidal representation
appearing in $S_{\underline{k}}(\Gamma(N))$.
Let $F\in \pi^{K(N)}$ be a non-zero element and suppose that $\pi^{K(M)}\neq 0$.
Then there exists $F_1\in \pi^{K(M)}$ such that
$F$ is a linear combination of forms ${}^h\!F_1,\ h\in G(\mathbb{A}_f)$ for which $h$
satisfies the following: if we write
$h=k_1t k_2,\ k_1,k_2\in K=K(1),\ t={\mathop{\mathrm{diag}}}(t_1,\ldots,t_n,t^{-1}_1,\ldots,t^{-1}_n)\in T(\mathbb{A})$
(by Cartan decomposition), then $t_it_j\in \widehat{\mathbb{Z}}$ and it
divides $\frac{N}{M}$ for each $1\le i,j\le n$.
\end{lem}
\begin{proof}
Pick a non-zero element $F_1\in \pi^{K(M)}$. Since $F_1$ generates $\pi$, we can
write $F=\displaystyle\sum_{h\in I}c_h {}^h\!F_1,\ c_h\in \mathbb{C}$ for a finite subset $I\subset G(\mathbb{A}_f)$
(notice no contribution from $G(\mathbb{R})$ since $F$ is fixed to be holomorphic in the original coordinates).
By the strong approximation theorem for $G={\mathop{\mathrm{Sp}}}(2n)$, we write $h=\gamma\cdot k,\ \gamma\in G(\mathbb{Q}),\ k\in K(M)$.
Then ${}^h\!F_1(g)=F_1(gh)=F_1(g\gamma),\ g\in G(\mathbb{A})$ and
${}^h\!F_1$ is $\gamma^{-1}K(M)\gamma$-fixed since $F_1$ is $K(M)$-fixed.
By Cartan decomposition,
$\gamma=k_1tk_2$ for some $k_1,k_2\in K=K(1)$ and $t={\mathop{\mathrm{diag}}}(t_1,\ldots,t_n,t^{-1}_1,\ldots,t^{-1}_n)\in T(\mathbb{A})$.
Since $F$ is $K(N)$-fixed and $K(N)$ is normal in $K$, if ${}^h\!F_1$ for some $h\in I$
is fixed by a compact open subgroup $K'$ with $K\supset K'\supset K(N)$, then
by using trace operator $\displaystyle\frac{1}{[K':K(N)]}{\rm tr}_{K'/K(N)}$, we may assume that
each ${}^h\!F_1$ is $K(N)$-fixed.
If $t_it_j\in \widehat{\mathbb{Z}}$ and it divides $\displaystyle\frac{N}{M}$ for each $1\le i,j\le n$, then by direct computation $t^{-1}K(M)t\supset K(N)$
and it yields $\gamma^{-1}K(M)\gamma\supset K(N)$ since $k_1$ and
$k_2$ normalize the principal congruence subgroups. The converse is also true.
Hence we have the claim.
\end{proof}
\begin{lem}\label{dim-new}
For each $M|N$ with $M\neq N$, let $I(M,N)$ be the set of
all elements $k_1tk_2$ with $k_1,k_2\in K$ and
$t={\mathop{\mathrm{diag}}}(t_1,\ldots,t_n,t^{-1}_1,\ldots,t^{-1}_n)\in T(\mathbb{A})$ satisfying
$t_it_j\in \displaystyle\frac{N}{M}\widehat{\mathbb{Z}}$ for each $1\le i,j \le n$.
Put $V(M,N):=\langle {}^h\!F_1\ |\ h\in I(M,N), F_1\in S_{{\underline{k}}}({\Gamma}(M)) \rangle_\mathbb{C}$.
Then
$${\rm dim}\, V(M,N)\le\sigma_0(\tfrac{N}{M})^{n}\cdot
{\rm dim}\, S_{{\underline{k}}}({\Gamma}(M)),
$$
where $\sigma_0(N)$ is the number of divisors of $N$.
\end{lem}
\begin{proof}
Let $I_1(M,N)$ be the set of
all elements $t={\mathop{\mathrm{diag}}}(t_1,\ldots,t_n,t^{-1}_1,\ldots,t^{-1}_n)\in T(\mathbb{A})$ satisfying
$t_it_j\in \widehat{\mathbb{Z}}$ and it divides $\displaystyle\frac{N}{M}$ for each $1\le i,j \le n$.
Since $K/K(M)$ acts on $S_{{\underline{k}}}({\Gamma}(M))$, we can rewrite
$$V(M,N)=\langle \, {}^h\!F_1\ |\ h\in I_1(M,N), F_1\in S_{{\underline{k}}}({\Gamma}(M)) \, \rangle_\mathbb{C}.$$
The claim follows from this.
\end{proof}
\begin{thm}\label{newforms} Let $n\geq 2$. Then
$${\rm dim}\, S^{{\rm new}}_{\underline{k}}({\Gamma}(N))\geq \frac {2n^2-1-\prod_{i=1}^n \zeta(2i)}{2n^2-1}{\rm dim}\, S_{\underline{k}}({\Gamma}(N)).
$$
\end{thm}
Note that $\log\displaystyle\prod_{i=1}^\infty \zeta(2i)=\sum_{i=1}^\infty \log(1+(\zeta(2i)-1))<\sum_{i=1}^\infty (\zeta(2i)-1)=\sum_{k=2}^\infty \frac 1{k^2-1}=\frac 34$. Hence $\displaystyle\prod_{i=1}^n \zeta(2i)<\displaystyle\prod_{i=1}^\infty \zeta(2i)<2.118$ for all $n$.
\begin{proof}
By Lemma \ref{linear-comb},
$$S^{{\rm old}}_{\underline{k}}({\Gamma}(N))=\bigcup_{d|N,d\neq 1}V(\frac{N}{d},N),
$$
and by Lemma \ref{dim-new},
\begin{eqnarray*}
&& {\rm dim}\, S^{{\rm old}}_{\underline{k}}({\Gamma}(N))\le \sum_{d|N\atop d\neq 1}
\sigma_0(d)^n
\left( C_{\underline{k}} C_{\frac Nd} \Big(\frac{N}{d}\Big)^{2n^2+n}+O\Big(\frac Nd\Big)^{2n^2}\right) \\
&&\phantom{xxxxxxxxxxxx} \leq C_{\underline{k}} N^{2n^2+n}\sum_{d|N\atop d\neq 1}
\frac{\sigma_0(d)^n}{d^{2n^2+n}}+ O\Big(N^{2n^2}\sum_{d|N\atop d\neq 1} \frac {\sigma_0(d)^n}{d^{2n^2}}\Big).
\end{eqnarray*}
We have used the fact that $C_{\frac Nd}<1$.
Now $\sigma_0(d)\leq d$ for any $d\geq 1$, and for $r\geq 2$,
$$\sum_{d|N\atop d\neq 1} \frac 1{d^{r}}\leq \int_1^\infty t^{-r} \, dt=\frac 1{r-1}.
$$
Therefore, by using the fact that $C_N\geq \displaystyle\prod_{i=1}^n \zeta(2i)^{-1}$,
$$
{\rm dim}\, S^{{\rm old}}_{\underline{k}}({\Gamma}(N))\leq \frac {\prod_{i=1}^n \zeta(2i)}{2n^2-1}{\rm dim}\, S_{\underline{k}}({\Gamma}(N)).
$$
Hence our result follows.
\end{proof}
\section{Equidistribution theorem of Siegel cusp forms; proof of Theorem \ref{main1}}\label{ET}
We prove Theorem \ref{main1} for $S_1=\{p\}$. The general case is similar.
\begin{thm}\label{main} Let ${S_1}=\{p\}$ for $p\nmid N$, $\widehat{\mu}_{K(N),p,\xi_{{\underline{k}}},D_{\underline{l}}^{\rm hol}}=\widehat{\mu}_{K(N),\{ p \},\xi_{{\underline{k}}},D_{\underline{l}}^{\rm hol}}$ and $\widehat{\mu}^{{\rm pl}}_p=\widehat{\mu}^{{\rm pl}}_{\{p\}}$.
Fix ${\underline{k}}$ and $\xi=\xi_{\underline{k}}$ as in Section 1. Then for any $f\in H^{{\rm ur}}(G(\mathbb{Q}_p))^\kappa$,
$$\lim_{N\to \infty} \widehat{\mu}_{K(N),p,\xi_{{\underline{k}}},D_{\underline{l}}^{\rm hol}}(\widehat{f})=
\widehat{\mu}^{{\rm pl}}_p(\widehat{f}).
$$
Further, there exist positive constants $a$ and $b$ depending only on $G$ such that
if $N\gg p^{2n\kappa}$,
$$\widehat{\mu}_{K(N),p,\xi_{{\underline{k}}},D_{\underline{l}}^{\rm hol}}(\widehat{f})=\widehat{\mu}^{{\rm pl}}_p(\widehat{f})+
O(p^{a\kappa+b} N^{-n}).
$$
\end{thm}
\begin{proof}By definition, we see that
$$\widehat{\mu}_{K(N),p,\xi_{{\underline{k}}},D_{\underline{l}}^{\rm hol}}(\widehat{h})=\frac{{\rm Tr}(T_{h}|_{S_{{\underline{k}}}({\Gamma}(N))})}
{{\rm vol}(G(\mathbb{Q}){\backslash} G(\mathbb{A}))\cdot {\rm dim}\, \xi_{\underline{k}}}.
$$
Notice that ${\rm dim}\, \xi_{\underline{k}}=d_{{\underline{k}}}$ (under a suitable normalization of the measure).
Applying Theorem \ref{thm:asy} to $S_1=\{p\}$ we have the claim
with Plancherel formula of Harish-Chandra: $\widehat{\mu}^{{\rm pl}}_p(\widehat{f})=f(1)$.
\end{proof}
\section{Vertical Sato-Tate theorem for Siegel modular forms; Proofs of Theorem \ref{Sato-Tate-tm} and Theorem \ref{finer-ver}}\label{vertical}
Suppose that ${\underline{k}}=(k_1,\ldots,k_n)$ satisfies the condition (\ref{suff-reg}). Put $\mathbb{T}=\{z\in\mathbb{C}\ |\ |z|=1\}$.
For $F\in HE_{\underline{k}}(N)$, consider the cuspidal automorphic representation
$\pi=\pi_F=\pi_\infty\otimes \otimes_p' \pi_{F,p}$ of $G(\mathbb{A})$
associated to $F$. As discussed in the previous section, under the condition (\ref{suff-reg}),
the $A$-parameter $\psi$ whose $A$-packet contains $\pi$ is semi-simple and
$\pi_{F,p}$ is tempered for all $p$. Then if $p\nmid N$, $\pi_{F,p}$ is spherical, and
we can write $\pi_{F,p}$ as $\pi_{F,p}=Ind^{G(\mathbb{Q}_p)}_{B(\mathbb{Q}_p)}\chi_p$ where $B=TU$ is the upper Borel subgroup and
$\chi_p$ is a unitary character on $B(\mathbb{Q}_p)$. For each $1\le j\le n$, put $\alpha_{jp}(\chi_p):=\chi_p(e_j(p^{-1}))$
(see (\ref{ejx}) for $e_j(p^{-1})$) and by temperedness we may write
$\alpha_{jp}(\chi_p)=e^{\sqrt{-1}\theta_j},\ \theta_j\in [0,\pi]$. Let $\widehat{G}={\mathop{\mathrm{SO}}}(2n+1)(\mathbb{C})$ be the complex split orthogonal group over $\mathbb{C}$
associated to the anti-diagonal identity matrix.
Let $\mathcal{L}(\pi_p):W_{\mathbb{Q}_p}{\longrightarrow} {\mathop{\mathrm{SO}}}(2n+1)(\mathbb{C})$
be the local Langlands parameter given by
$$\mathcal{L}(\pi_p)({\rm Frob}_p)=(\alpha_{1p}(\chi_p),\ldots,\alpha_{np}(\chi_p),1,\alpha_{1p}(\chi_p)^{-1},
\ldots,\alpha_{np}(\chi_p)^{-1})$$
which is called to be $p$-Satake parameter.
Put $a^{(i)}(\chi_p)=a^{(i)}_{F,p}(\chi_p)=\frac 12(\alpha_{ip}(\chi_p)+\alpha_{ip}(\chi_p)^{-1})=\cos\theta_i$ for $1\le i\le n$.
Let $\widehat{G(\mathbb{Q}_p)}^{{\rm ur,temp}}$ be the isomorphism classes of unramified tempered representations of
$G(\mathbb{Q}_p)$.
By \cite[Lemma 3.2]{ST}, we have a topological isomorphism
$$\widehat{G(\mathbb{Q}_p)}^{{\rm ur,temp}}\stackrel{\sim}{{\longrightarrow}} [0,\pi]^n/\mathfrak S_n=:\Omega
$$
given by
$$\pi_p=Ind^{Sp_{2n}(\mathbb{Q}_p)}_{B(\mathbb{Q}_p)}\chi_p\mapsto
(\arg(a^{(1)}(\chi_p)),\ldots,\arg(a^{(n)}(\chi_p)))=:(\theta_1,\ldots,\theta_n).
$$
We denote by $(\theta_1(\pi_{F,p}),\ldots,\theta_n(\pi_{F,p}))\in\Omega$
the corresponding element to $\pi_{F,p}$ under the above isomorphism.
Let $\widehat{B}=\widehat{T}\widehat{U}$ be the upper Borel subgroup of $\widehat{G}={\mathop{\mathrm{SO}}}(2n+1)(\Bbb C)$.
Let $\Delta^+(\widehat{G})$ be the set of all positive roots in $X^\ast(\widehat{T})={\rm Hom}(\widehat{T},{\mathop{\mathrm{GL}}}_1)$
with respect to $\widehat{B}$.
We view $(\theta_1,\ldots,\theta_n)$ as
parameters of $\Omega$. Let $\mu^{{\rm pl,temp}}_p$ be the restriction of the Plancherel measure on
$\widehat{G(\mathbb{Q}_p)}$ to $\widehat{G(\mathbb{Q}_p)}^{{\rm ur,temp}}$ and by abusing the notation we denote by $\mu_p=\mu^{{\rm pl,temp}}_p$
its pushforward to $\Omega$.
Put
$$t:=(e^{\sqrt{-1}\theta_1},\ldots,e^{\sqrt{-1}\theta_n},1,e^{-\sqrt{-1}\theta_1},
\ldots,e^{-\sqrt{-1}\theta_n})
$$
for simplicity. By \cite[Proposition 3.3]{ST}, we have
$$\mu_p^{{\rm pl,temp}}(\theta_1,\ldots,\theta_n)=
W(\theta_1,\ldots,\theta_n)d\theta_1\cdots d\theta_n,$$
$$ W(\theta_1,\ldots,\theta_n)=\frac{\displaystyle\prod_{\alpha\in \Delta^+(\widehat{G})}|1-e^{\sqrt{-1}\alpha(t)}|^2}
{\displaystyle\prod_{\alpha\in \Delta^+(\widehat{G})}|1-p^{-1}e^{\sqrt{-1}\alpha(t)}|^2}=
\frac{\displaystyle\prod_{i=1}^n|1-e^{\sqrt{-1}\theta_i}|^2
\prod_{1\le i<j\le n\atop \varepsilon=\pm 1}|1-e^{\sqrt{-1}(\theta_i+\varepsilon \theta_j)}|^2}
{\displaystyle\prod_{i=1}^n|1-p^{-1}e^{\sqrt{-1}\theta_i}|^2
\prod_{1\le i<j\le n\atop \varepsilon=\pm 1}|1-p^{-1}e^{\sqrt{-1}(\theta_i+\varepsilon \theta_j)}|^2}.
$$
By setting $p\to\infty$, we recover the Sato-Tate measure
$$\mu^{{\rm ST}}_\infty=\lim_{p\to\infty}\mu^{{\rm pl,temp}}_p =\displaystyle\prod_{i=1}^n |1-e^{\sqrt{-1}\theta_i}|^2
\prod_{1\le i<j\le n\atop \varepsilon=\pm 1} |1-e^{\sqrt{-1}(\theta_i+\varepsilon \theta_j)}|^2.
$$
Then Theorem \ref{Sato-Tate-tm} and Theorem \ref{finer-ver} follow from Theorem \ref{main} and Theorem \ref{non-simple}.
\section{Standard $L$-functions of ${\mathop{\mathrm{Sp}}}(2n)$}\label{Standard-L}
Let ${\underline{k}}=(k_1,...,k_n)$ and $F\in HE_{\underline{k}}(N)$, and $\pi_F$ be a cuspidal representation of $G(\mathbb{A})$ associated to $F$.
Assume (\ref{suff-reg}) for ${\underline{k}}$.
By (\ref{AD}) and the observation there, the global $A$-packet $\Pi_\psi$ containing $\pi_F$ is associated to
a semi-simple global $A$ parameter $\psi=\boxplus_{i=1}^r\pi_i$
where $\pi_i$ is an irreducible cuspidal representation of ${\mathop{\mathrm{GL}}}_{m_i}(\mathbb{A})$.
Then the isobaric sum $\Pi:=\boxplus_{i=1}^r\pi_i$
is an automorphic representation of ${\mathop{\mathrm{GL}}}_{2n+1}(\mathbb{A})$.
Therefore we may define
$$L(s,\pi_F,{\rm St}):=L(s,\Pi)=\prod_{i=1}^rL(s,\pi_i).$$
Let $L_p(s,\pi_F,{\rm St}):=L(s,\Pi_p)=\prod_{i=1}^rL(s,\pi_{ip})$ be the local $p$-factor
of $L(s,\pi_F,{\rm St})$ for each rational prime $p$.
Let $\pi_F=\pi_\infty\otimes\otimes_p' \pi_p$.
For $p\nmid N$, $\pi_p$ is the spherical representation of $G(\mathbb{Q}_p)$ with the Satake parameter $\{\alpha_{1p},...,\alpha_{np},1,\alpha_{1p}^{-1},...,\alpha_{np}^{-1}\}$. Then
$$L_p(s,\pi_F,{\rm St})^{-1}=(1-p^{-s})\prod_{i=1}^n (1-\alpha_{ip}p^{-s})(1-\alpha_{ip}^{-1}p^{-s}).
$$
We define the conductor $q(F)$ of $F$ to be the product of the conductors $q(\pi_i)$ of $\pi_i$
($1\le i\le r$).
\begin{thm}
Let $F\in HE_{\underline{k}}(N)$. Then the standard L-function $L(s,\pi_F,{\rm St})$ has a meromorphic continuation to all of $\Bbb C$.
Let
$$\Lambda(s,\pi_F,{\rm St})=q(F)^\frac s2 \, L_\infty(s,\pi_F,{\rm St}) \, L(s,\pi_F,{\rm St}),
$$
where
$L_\infty(s,\pi_F,{\rm St})=\Gamma_{\Bbb R}(s+\epsilon) \Gamma_{\Bbb C}(s+k_1-1)\cdots \Gamma_{\Bbb C}(s+k_n-n)$,
$\epsilon=\begin{cases} 0, &\text{if $n$ is even}\\ 1, &\text{if $n$ is odd}\end{cases}$, and $\Gamma_\Bbb R(s)=\pi^{-\frac s2}\Gamma(\frac s2)$, $\Gamma_\Bbb C(s)=2(2\pi)^{-s}\Gamma(s)$.
Then
$$\Lambda(s,\pi_F,{\rm St})=\epsilon(F)\Lambda(1-s,\pi_F,{\rm St}),
$$
where $\epsilon(F)\in\{\pm 1\}$.
\end{thm}
\begin{proof} It follows from the functional equation of $L(s,\Pi)$ by noting that $\Pi$ is self-dual, and $L(s,\Pi_\infty)=L_\infty(s,\pi_F,{\rm St})$ is the local $L$-function attached to the holomorphic discrete series of the lowest weight ${\underline{k}}$ (cf. \cite{Kozima}).
\end{proof}
The epsilon factor $\epsilon(F)$ turns out to be always 1.
\begin{prop} Let $\pi_F$ be associated to a semi-simple $A$-parameter. Then
$\epsilon(F)=1$.
\end{prop}
\begin{proof} Recall the global $A$-parameter $\psi=\boxplus_{i=1}^r\pi_i$. Let $\omega_i$ be the central
character of $\pi_i$.
Since $\pi_i$ is orthogonal, its epsilon factor is $\omega_i(-1)$ by \cite[Theorem 1]{La}.
Hence $\epsilon(F)=\displaystyle\prod_{i=1}^r\omega_i(-1)=\Big(\prod_{i=1}^{r}\omega_i\Big)(-1)={\mathbbm{1}}(-1)=1$ by
the condition on the central character.
\end{proof}
\begin{thm}\label{stan-conductor}
For any $F\in HE_{\underline{k}}(N)$, the conductor $q(F)$ satisfies $q(F)\le N^{2n+1}$.
If $F\in HE^{{\rm new}}_{\underline{k}}(N)$, then $q(F)\ge \max\left\{N\displaystyle\prod_{p\mid N}p^{-(2n+1)},\ \prod_{p\mid N}p \right\}.$
So if $F\in HE^{{\rm new}}_{\underline{k}}(N)$, $q(F)\geq N^{\frac {1}{2n+2}}$.
\end{thm}
\begin{proof}
Let $\pi_F$ be associated to
a semi-simple global $A$ parameter $\psi=\boxplus_{i=1}^r\pi_i$
where $\pi_i$ is an irreducible cuspidal representation of ${\mathop{\mathrm{GL}}}_{m_i}(\mathbb{A})$, and let
$\Pi:=\boxplus_{i=1}^r\pi_i$.
Let $\Pi=\Pi_\infty\otimes\otimes_p' \Pi_p$. By Proposition \ref{transfer},
$\Pi$ has a non-zero fixed vector by $K^{GL_{2m+1}}(p^{e_p})$ where
$e_p={\rm ord}_p(N)$.
As in the proof of \cite[Lemma 8.1]{KWY},
it implies ${\rm depth}(\Pi_p)\leq e_p-1$. Hence $q(\Pi_p)\le p^{(2n+1)e_p}$.
Therefore, $q(F)\le N^{2n+1}$.
If $F\in HE^{{\rm new}}_{\underline{k}}(N)$,
it is not fixed by $K^{GL_{2m+1}}(p^{e_p-1})$. By \cite[Theorem 1.2]{MY},
we have $q(\Pi_p)\ge p^{m_i(e_p-1)}$ for some $i$.
In particular, $q(\Pi_p)\ge p^{e_p-1}$.
It is clear that $q(\Pi_p)\geq p$ if $p|N$. Hence
$$q(F)\ge
\max\left\{ N\cdot\prod_{p|N}p^{-(2n+1)},\, \prod_{p|N}p \right\}.
$$
Now, $q(F)^{2n+2}\geq q(F)\cdot q(F)^{2n+1}\geq N$. Hence our result follows.
\end{proof}
\begin{prop} \label{pole} Keep the assumptions on $N$ as in Proposition \ref{fixedV}.
Let $F\in HE_{\underline{k}}(N)$. Then $L(s,\pi_F,{\rm St})$ has a pole at $s=1$ if and only if $\pi_F$ is associated to
a semi-simple global $A$-parameter $\psi=1\boxplus \pi_1\boxplus\cdots \boxplus \pi_r$
where $\pi_i$ is an orthogonal irreducible cuspidal representation of ${\mathop{\mathrm{GL}}}_{m_i}(\mathbb{A})$, such that if $m_i=1$, $\pi_i$ is a non-trivial quadratic character.
Let $HE_{\underline{k}}(N)^0$ be the subset of $HE_{\underline{k}}(N)$ such that $L(s,\pi_F,{\rm St})$ has a pole at $s=1$. Then
$|HE_{\underline{k}}(N)^0|=O(N^{2n^2-n+\epsilon})$. So $\frac {|HE_{\underline{k}}(N)^0|}{|HE_{\underline{k}}(N)|}=O(N^{-2n+\epsilon})$.
\end{prop}
This proves \cite[Hypothesis 11.2]{ST} in our family.
\begin{proof} This follows from the proof of Theorem \ref{non-simple}, by noting that partitions $\underline{m}=(m_1,...,m_r)$ of $2n$ contribute to $HE_{\underline{k}}(N)^0$.
\end{proof}
In \cite{B1}, B\"ocherer gave the relationship between Hecke operators and $L$-functions for level one and scalar-valued Siegel modular forms and it is extended by Shimura \cite{Shimura} to more general setting.
Let $\underline{a}=(a_1,...,a_n)$, $0\leq a_1\leq \cdots\leq a_n$, and
$D_{p,\underline{a}}={\mathop{\mathrm{diag}}}(p^{a_1},...,p^{a_n})$.
Let $F$ be an eigenform in $HE_{{\underline{k}}}(N)$ with respect to the Hecke operator $\Gamma(N) \begin{pmatrix} D_{p,\underline{a}}&0\\0&D_{p,\underline{a}}^{-1}\end{pmatrix}\Gamma(N)$
for all $p\nmid N$,
and $\lambda(F,D_{p,\underline{a}})$ is the eigenvalue.
Then we have the following identity \cite[Theorem 2.9]{Shimura}:
\begin{equation}\label{Bo}
\sum_{\underline{a}} \lambda(F,D_{p,\underline{a}}) X^{\sum_{i=1}^n a_i}=\frac {(1-X)}{(1-p^n X)} \prod_{i=1}^n \frac {(1-p^{2i}X^2)}{(1-\alpha_{ip}p^n X)(1-\alpha_{ip}^{-1} p^n X)},
\end{equation}
where $\underline{a}=(a_1,...,a_n)$ runs over $0\leq a_1\leq \cdots\leq a_n$.
Let $\underline{m}=(m_1,...,m_n)$, $m_1|m_2|\cdots | m_n$, and $D_{\underline{m}}={\mathop{\mathrm{diag}}}(m_1,...,m_n)$, and let $\lambda(F,D_{\underline{m}})$ be the eigenvalue of the Hecke operator $T(D_{\underline{m}})$.
Let
$$L^N(s,F)=\sum_{\underline{m}, \, (m_n,N)=1} \lambda(F,D_{\underline{m}}) \det(D_{\underline{m}})^{-s}.
$$
Then
$$L^N(s,F)=\prod_{p\nmid N} L(s,F)_p,\quad L(s,F)_p=\sum_{\underline{a}} \lambda(F,D_{p,\underline{a}}) \det(D_{p,\underline{a}})^{-s}.
$$
It converges for ${\rm Re}(s)> 2n+\frac {k_1+\cdots+k_n}n+1$.
Hence we have
$$\zeta^N(s)\left[\prod_{i=1}^n \zeta^N(2s-2i)\right] L^N(s,F)=L^N(s-n,\pi_F, {\rm St}).
$$
The central value of $L^N(s,F)$ is at $s=n+\frac 12$, and $L^N(s,F)$ has a zero at $s=n+\frac 12$ since $L^N(s,\pi_F,{\rm St})$ is holomorphic at $s=\frac 12$. Theorem \ref{thm:asy} implies
\begin{thm}\label{lambda} For $\underline{m}=(m_1,...,m_n)$, $m_1|m_2|\cdots | m_n$
with $m_n>1$ and $(m_n,N)=1$,
$$\frac{1}{|HE_{{\underline{k}}}(N)|}\sum_{F\in HE_{{\underline{k}}}(N)} \lambda(F,D_{\underline{m}})=O(m_n^{a}N^{-n}),
$$
for some constant $a$.
\end{thm}
\begin{proof} Let $S_1$ be the set of all prime divisors of $m_n$.
Since $m_n>1$, $S_1$ is non-empty.
The main term of RHS in Theorem \ref{thm:asy} includes $h_1(1)$. Clearly, $h_1(1)=0$ because
the double coset defining the Hecke operator $h_1$ does not contain
any central elements.
Since the automorphic counting measure is supported on cuspidal representations,
Theorem \ref{thm:asy} implies the claim.
\end{proof}
Let $L^N(s,F)=\displaystyle\sum_{m=1\atop (m,N)=1}^\infty a_F(m)m^{-s}$, and $L(s,F)_p=\displaystyle\sum_{k=0}^\infty a_F(p^k) p^{-ks}$ for each prime $p\nmid N$. Here $a_F(p^k)=\sum_{\underline{a}} \lambda(F,D_{p,\underline{a}})$, where the sum is over all $\underline{a}=(a_1,...,a_n)$ such that $0\leq a_1\leq\cdots\leq a_n$, $a_1+\cdots+a_n=k$. Hence, for $k>0$, $p\nmid N$,
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} a_F(p^k)=O(p^{ka}N^{-n}).
$$
More generally,
\begin{cor}\label{a_F} For $m>1$, $(m,N)=1$,
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} a_F(m)=O(m^{a}N^{-n}).
$$
\end{cor}
\begin{proof} We have $a_F(m)=\sum_{\underline{m}} \lambda(F,D_{\underline{m}})$, where the sum is over all $\underline{m}=(m_1,...,m_n)$, $m_1|m_2|\cdots |m_n$, $m_1\cdots m_n=m$. Our assertion follows from Theorem \ref{lambda}.
\end{proof}
Let
$L^N(s,\pi_F,{\rm St})=\displaystyle\sum_{m=1\atop (m,N)=1}^\infty \mu_F(m)m^{-s}$.
Then from (\ref{Bo}), we have, for $p\nmid N$,
$$\mu_F(p)=(a_F(p)+1)p^{-n},\quad \mu_F(p^2)=1+p^{-2}+\cdots+p^{-2n}+(a_F(p^2)+a_F(p))p^{-2n}.
$$
More generally, for $p\nmid N$,
$$\mu_F(p^k)=\begin{cases} 1+p^{-2} h_k(p^{-2})+p^{-n}\displaystyle\sum_{i=1}^k h_{ik}(p^{-1})a_F(p^i), &\text{if $k$ is even}\\
p^{-n} h_k'(p^{-2})+p^{-n}\displaystyle\sum_{i=1}^k h_{ik}'(p^{-1}) a_F(p^i), &\text{if $k$ is odd},
\end{cases}
$$
where $h_k,h_k', h_{ik}, h_{ik}'\in\Bbb Z[x]$. Therefore, for $(m,N)=1$,
$$
\mu_F(m)=\prod_{p|m} (\delta_{p,m}+p^{-2}h_m^\delta(p^{-1}))+ \sum_{u|m\atop u>1} A_u a_F(u),
$$
where $A_u\in\Bbb Q$, $h_m^\delta\in\Bbb Z[x]$, and $\delta=\delta_{p,m}=\begin{cases} 1, &\text{if $v_p(m)$ is even}\\0, &\text{otherwise}\end{cases}$.
Therefore, by Corollary \ref{a_F}, we have
\begin{thm}\label{stan} Fix ${\underline{k}}=(k_1,...,k_n)$ and let $m=\displaystyle\prod_{p|m} p^{v_p(m)}$ which is
coprime to $N$. Then
$$\frac{1}{d_{{\underline{k}}}(N)}\sum_{F\in HE_{{\underline{k}}}(N)} \mu_F(m) = \prod_{p|m} \left(\delta_{p,m} + p^{-2}h_m^\delta(p^{-1})\right)+ O(N^{-n}m^c).
$$
\end{thm}
This proves \cite[Conjecture 6.1 in level aspect]{KWY1} for ${\mathop{\mathrm{Sp}}}(4)$ case.
\section{$\ell$-level density of standard $L$-functions}\label{r-level}
In this section, we assume (\ref{suff-reg}) and keep the assumptions on $N$ in Proposition \ref{fixedV}.
Then we show unconditionally that the $\ell$-level density ($\ell$ positive integer) of the standard $L$-functions of the family
$HE_{\underline{k}}(N)$ has the symmetry type $Sp$ in the level aspect. Shin and Templier \cite{ST} showed it under several hypotheses with a family which includes non-holomorphic forms.
Under assumption (\ref{suff-reg}), $F$ satisfies the Ramanujan conjecture, namely, $|\alpha_{ip}|=1$ for each $i$.
Let
$$-\frac {L'}L(s,\pi_F,{\rm St})=\sum_{m=1}^\infty \Lambda(m)b_F(m) m^{-s},
$$
where $b_F(p^m)=1+\alpha_{1p}^m+\cdots +\alpha_{np}^m+\alpha_{1p}^{-m}+\cdots+\alpha_{np}^{-m}$ when $\pi_p$ is spherical.
For $F\in HE_{{\underline{k}}}(N)$, let $\Pi$ be the Langlands transfer of $\pi_F$ to ${\rm GL}_{2n+1}$. If $F\in HE_{{\underline{k}}}(N)^g$, then $L(s,\Pi,\wedge^2)$ has no pole at $s=1$, and $L(s,\Pi,Sym^2)$ has a simple pole at $s=1$. Let $L(s,\Pi\times\Pi)=\sum \lambda_{\Pi\times\Pi}(n)n^{-s}, L(s,\Pi,\wedge^2)=\sum \lambda_{\wedge^2(\Pi)}(n)n^{-s}, L(s,\Pi,Sym^2)=\sum \lambda_{Sym^2(\Pi)}(n)n^{-s}$. Then $\mu_F(p^2)=\lambda_{Sym^2(\Pi)}(p)$, and
$\mu_F(p)^2=\lambda_{\Pi\times\Pi}(p)=\lambda_{\wedge^2(\Pi)}(p)+\lambda_{Sym^2(\Pi)}(p)$.
Note that $\mu_F(p)=b_F(p)$, and $b_F(p^2)=2\mu_F(p^2)-\mu_F(p)^2$.
Let $T(p,\underline{a})=\Gamma(N) \begin{pmatrix} D_{p,\underline{a}}&0\\0&D_{p,\underline{a}}^{-1}\end{pmatrix}\Gamma(N)$.
By Theorem \ref{main-appendix},
$T(p,(\overbrace{0,...,0}^{n-1},1))^2$ is a linear combination of
$$T(p,(\overbrace{0,...,0}^{n-1},2)),\
T(p,(\overbrace{0,...,0}^{n-2},1,1)),\ T(p,(\overbrace{0,...,0}^{n-1},1)),\
T(p,\overbrace{(0,...,0)}^{n})=\Gamma(N)I_{2n}\Gamma(N).
$$
Therefore,
by Theorem \ref{stan}, if $p\nmid N$,
$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} \mu_F(p)^2$ is of the form $1+p^{-1}g(p^{-1})+O(p^c N^{-n})$
for some polynomial $g\in\Bbb Z[x]$ and $c>0$.
Here the main term $1+p^{-1}g(p^{-1})$ comes from the coefficient of
$T(p,\overbrace{(0,...,0)}^{n})$ in the linear combination.
Hence we have
\begin{prop} \label{b_F} For some $a>0$, and $p\nmid N$,
\begin{eqnarray*}
&& \frac{1}{d_{{\underline{k}}}(N)}\sum_{F\in HE_{{\underline{k}}}(N)} b_F(p) = O(p^{-1})+O(p^a N^{-n}),\\
&&\frac{1}{d_{{\underline{k}}}(N)}\sum_{F\in HE_{{\underline{k}}}(N)} b_F(p^2) = 1+O(p^{-1})+O(p^a N^{-n}).
\end{eqnarray*}
\end{prop}
Denote the non-trivial zeros of $L(s,\pi_F,{\rm St})$ by $\sigma_F=\frac 12+\sqrt{-1} \gamma_F$, counting multiplicities.
Since we do not assume GRH, $\gamma_F$ can be a complex number.
Let $c(F)=q(F)(k_1\cdots k_n)^2$ be the analytic conductor, and
$\displaystyle \log c_{{\underline{k}},N}=\frac 1{d_{{\underline{k}}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} \log c(F)$.
From Theorems \ref{newforms} and \ref{stan-conductor}, we have
\begin{lem}\label{logN} $\log c_{k,N}\asymp\log N$, namely, $A_{\underline{k}}\log N\leq \log c_{k,N}\leq B_{\underline{k}}\log N$ for some constants $A_{\underline{k}}, B_{\underline{k}}>0$.
\end{lem}
This proves \cite[Hypothesis 11.4]{ST} in our family.
\begin{proof} By Theorem \ref{stan-conductor}, $q(F)\leq N^{2n+1}$. It gives rise to the upper bound.
If $F\in HE^{{\rm new}}_{\underline{k}}(N)$, $q(F)\geq N^{\frac {1}{2n+2}}$. By Theorem \ref{newforms}, $|HE^{{\rm new}}_{\underline{k}}(N)|\gg |HE_{\underline{k}}(N)|$.
Hence
$$\log c_{{\underline{k}},N}\geq \frac 1{d_{{\underline{k}}}(N)} \sum_{F\in HE^{\rm new}_{{\underline{k}}}(N)} \log c(F)\gg \log N.
$$
\end{proof}
Consider, for an even Paley-Wiener function $\phi$,
$$D(F,\phi)=\sum_{\gamma_F} \phi\left(\frac {\gamma_F}{2\pi}\log c_{{\underline{k}},N}\right).
$$
Then as in \cite[(9.1)]{KWY},
\begin{eqnarray*}
&& \frac 1{d_{{\underline{k}}}(N)} \sum_{F\in HE_{\underline{k}}(N)} D(F,\phi)=\widehat{\phi}(0)-\frac 12 \phi(0) \\
&& \phantom{xxxxxxxxxx} - \frac{2}{(\log c_{{\underline{k}},N}) d_{{\underline{k}}}(N)} \sum_{F\in HE_{\underline{k}}(N)} \sum_p \frac{b_F(p)\log p}{\sqrt{p}}\widehat{\phi}\left( \frac{\log p}{\log c_{{\underline{k}},N}}\right) \nonumber \\
&& \phantom{xxxxxxxxxx} -\frac{2}{(\log c_{{\underline{k}},N}) d_{{\underline{k}}}(N)} \sum_{F\in HE_{\underline{k}}(N)} \sum_p \frac{(b_F(p^2)-1)\log p}{p}\widehat{\phi}\left( \frac{2\log p}{\log c_{{\underline{k}},N}}\right) \\
&& \phantom{xxxxxxxxxx} +O\left(\frac {|HE_{\underline{k}}(N)^0|}{d_{{\underline{k}}}(N)}\right)+O\left( \frac{1}{\log c_{{\underline{k}},N}} \right),
\end{eqnarray*}
where $HE_{\underline{k}}(N)^0$ is in Proposition \ref{pole}. [In \cite[(9.4)]{KWY}, the term $O\left(\frac {|HE_{\underline{k}}(N)^0|}{d_{{\underline{k}}}(N)}\right)$ was omitted.]
By Proposition \ref{b_F}, we can show as in \cite{KWY} that for an even Paley-Wiener function $\phi$ such that the Fourier transform $\hat{\phi}$ of $\phi$ is supported in $(-\beta,\beta)$ ($\beta<1$ can be explicitly determined.),
\begin{equation}\label{one-level}
\frac 1{d_{{\underline{k}}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} D(F,\phi)=\widehat{\phi}(0)-\frac 12 \phi(0)
+O\left( \frac{1}{\log c_{{\underline{k}},N}} \right)
=\int_\Bbb R \phi(x)W(\text{Sp})(x)\, dx+O\left( \frac{\omega(N)}{\log N} \right),
\end{equation}
where $\omega(N)$ is the number of prime factors of $N$, and $W(\text{Sp})(x) = 1- \dfrac {\sin 2\pi x}{2\pi x}$. [When we exchange two sums, if $p\nmid N$, we use Proposition \ref{b_F}. If $p|N$, by the Ramanujan bound, $|b_F(p)|\leq n, |b_F(p^2)|\leq n$. Hence by the trivial bound, we would obtain $\sum_{p|N} \frac{b_F(p)\log p}{\sqrt{p}}\ll \omega(N)$ and $\sum_{p|N} \frac{b_F(p^2)\log p}p\ll \omega(N)$.]
For a general $\ell$, let
$$W(\text{Sp})(x) = \text{det}(K_{-1}(x_j,x_k))_{1\leq j\leq \ell\atop 1\leq k\leq \ell},
$$
where $K_{-1}(x,y)=\dfrac {\sin \pi(x-y)}{\pi(x-y)}- \dfrac {\sin \pi(x+y)}{\pi(x+y)}$.
Let $\phi(x_1,...,x_\ell)=\phi_1(x_1)\cdots \phi_\ell(x_\ell)$, where each $\phi_i$ is an even Paley-Wiener function and $\hat \phi(u_1,...,u_\ell)=\hat \phi_1(u_1)\cdots \hat\phi_\ell(u_\ell)$.
We assume that the Fourier transform $\hat{\phi_i}$ of $\phi_i$ is supported in $(-\beta,\beta)$ for $i=1,\dots,\ell$.
The $\ell$-level density function is
\begin{equation*}\label{n-level-st}
D^{(\ell)}(F, \phi)
={\sum}_{j_1,\cdots,j_\ell}^*\phi\left(\gamma_{j_1}\frac{\log c_{{\underline{k}},N}}{2 \pi},\gamma_{j_2}\frac{\log c_{{\underline{k}},N}}{2 \pi},\dots,\gamma_{j_\ell}\frac{\log c_{{\underline{k}},N}}{2 \pi}\right)
\end{equation*}
where $\sum_{j_1,...,j_\ell}^*$ is over $j_i=\pm 1,\pm 2,...$ with $j_{a}\ne \pm j_{b}$ for $a\ne b$.
Then as in \cite{KWY1}, using Theorem \ref{stan}, we can show
\begin{thm}\label{n-level-sp}
Let $\phi(x_1,...,x_\ell)=\phi_1(x_1)\cdots \phi_\ell(x_\ell)$, where each $\phi_i$ is an even Paley-Wiener function and $\hat \phi(u_1,...,u_\ell)=\hat \phi_1(u_1)\cdots \hat\phi_\ell(u_\ell)$. Assume the Fourier transform $\hat{\phi_i}$ of $\phi_i$ is supported in $(-\beta,\beta)$ for $i=1,\cdots,\ell$.
Then
$$
\frac 1{d_{{\underline{k}}}(N)} \sum_{F \in HE_{{\underline{k}}}(N)} D^{(\ell)}(F,\phi)=
\int_{\Bbb R^\ell} \phi(x)W({\rm Sp})(x)\, dx + O\left(\frac {\omega(N)}{\log N}\right).
$$
\end{thm}
\section{The order of vanishing of standard $L$-functions at $s=\frac 12$}\label{order}
In this section, we show that the average order of vanishing of standard $L$-functions at $s=\frac 12$ is bounded under GRH (cf. \cite{ILS, Brumer}). Under GRH of $L(s,\pi_F,{\rm St})$, its zeros are $\frac 12+\sqrt{-1}\gamma_F$ with $\gamma_F\in\Bbb R$.
\begin{thm} Assume the GRH. Let $r_F={\rm ord}_{s=\frac 12} L(s,\pi_F,{\rm St})$.
Then
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{{\underline{k}}}(N)} r_F\leq C,
$$
for some constant $C>0$.
\end{thm}
\begin{proof}
Choose $\displaystyle\phi(x)=\left(\frac {2\sin \frac {x \beta}2}x\right)^2$ for $x\in\Bbb R$, where
$\beta$ is from \eqref{one-level}. Then
$$\widehat\phi(x)=\begin{cases} \beta-|x|, &\text{if $|x|<\beta$}\\ 0, &\text{otherwise}\end{cases}.
$$
Since $\phi(x)\geq 0$ for $x\in\Bbb R$, from (\ref{one-level}),
we have
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{\underline{k}}(N)} r_F\phi(0)\leq \widehat\phi(0)-\frac 12\phi(0)+O\left(\frac 1{\log\log N}\right).
$$
Hence we have
$$
\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{\underline{k}}(N)} r_F\leq \frac 1{\beta}-\frac 12+O\left(\frac 1{\log\log N}\right).
$$
\end{proof}
We can show a similar result for the spinor $L$-function of ${\mathop{\mathrm{GSp}}}(4)$. Recall the following from \cite{KWY}:
\begin{prop} Assume $(N,11!)=1$.
\begin{enumerate}
\item (level aspect) Fix $k_1,k_2$.
Then for $\phi$ whose Fourier transform $\hat\phi$ has support in $(-u,u)$ for some $0<u<1$, as $N\to\infty$,
$$
\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{\underline{k}}(N)} D(\pi_F,\phi, {\rm Spin})=\hat\phi(0)+\frac 12 \phi(0)+O(\frac 1{\log\log N}).
$$
\item (weight aspect) Fix $N$. Then for $\phi$ whose Fourier transform $\hat\phi$ has support in $(-u,u)$ for some $0<u<1$, as $k_1+k_2\to\infty$,
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{\underline{k}}(N)} D(\pi_F,\phi, {\rm Spin})=\hat\phi(0)+\frac 12 \phi(0)+
O(\frac 1{\log ((k_1-k_2+2)k_1k_2)}).
$$
\end{enumerate}
\end{prop}
As in the above theorem, we have
\begin{thm} Assume the GRH, and let $r_F={\rm ord}_{s=\frac 12} L(s,\pi_F,{\rm Spin})$. Then
$$\frac 1{d_{\underline{k}}(N)} \sum_{F\in HE_{\underline{k}}(N)} r_F\leq C',
$$
for some constant $C'>0$.
\end{thm}
\section{Appendix}
In this appendix we compute the product $T(p,(\overbrace{0,...,0}^{n-1},1))^2$ in Section \ref{r-level}.
\begin{thm}\label{main-appendix}
\begin{eqnarray*}
&& T(p,(\overbrace{0,...,0}^{n-1},1))^2=T(p,(\overbrace{0,...,0}^{n-1},2))+(p+1) T(p,(\overbrace{0,...,0}^{n-2},1,1))+(p^n-1)T(p,(\overbrace{0,...,0}^{n-1},1)) \\
&&\phantom{xxxxxxxxxxxxx} +\left(p\sum_{i=0}^{2n-1}p^i\right) T(p,\overbrace{(0,...,0)}^{n}).
\end{eqnarray*}
\end{thm}
This agrees with \cite[(2.7), p.362]{KWY} when $n=2$. [Note that
the coefficient of $R_{p^2}$ there should be replaced with $p^4+p^3+p^2+p$.]
Since $p\nmid N$, we work on $K={\mathop{\mathrm{Sp}}}(2n,\mathbb{Z}_p)$ instead of ${\Gamma}(N)$. Put
$$T_{p,n-1}:=pT(p,,(0,\ldots,0,1))=K{\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,
\overbrace{p,\ldots,p}^{n-1})K\in {\mathop{\mathrm{GSp}}}(2n,\Bbb Q_p).
$$
It suffices to consider $T_{p,n-1}^2$.
Let us first compute the coset decomposition.
Put $\Lambda={\mathop{\mathrm{GL}}}_n(\mathbb{Z}_p)$ where the identity element is denoted by $1_n$.
For any ring $R$, let $S_n(R)$ be the set of
all symmetric matrices of size $n$ defined over $R$ and
$M_{m\times n}(R)$ be the set of matrices of size $m\times n$ defined over $R$.
Put $M_n(R)=M_{n\times n}(R)$ for simplicity. For each $D\in M_n(\mathbb{Z}_p)$ we define
$$B(D):=\{B\in M_n(\mathbb{Z}_p)\ |\ {}^tB D={}^tDB\}.$$
For each $B_1,B_2\in B(D)$, we write $B_1\sim B_2$ if there exists $M\in M_n(\mathbb{Z}_p)$
such that $B_1-B_2=MD$.
We denote by $B(D)/\sim$ the set of all equivalence classes of $B(D)$ by the relation $\sim$.
We regard $\mathbb{F}_p$ (resp. $\mathbb{Z}/p^2\mathbb{Z}$) as the subset $\{0,1,\ldots,p-1\}$
(resp. $\{0,1,\ldots,p^2-1\}$) of $\mathbb{Z}$.
Let $D_I$ be the set of the following matrices in $M_n(\mathbb{Z}_p)$:
$$D^I_{n-1}={\rm diag}(
\overbrace{p,\ldots,p}^{n-1},1),\ D^I_s=D^I_{s}(x):=
\left(\begin{array}{c|c|c}
p\cdot 1_s & & \\
\hline
& 1 & x \\
\hline
& &p\cdot 1_{n-1-s}
\end{array}
\right),\ 0\le s\le n-2,\ x\in M_{1\times(n-1-s)}(\mathbb{F}_p)
$$
where we fill out zeros in the blank blocks.
The cardinality of $D_I$ is $1+p+\cdots+p^{n-1}=\displaystyle\frac{p^n-1}{p-1}$ which is
equal to that of $\Lambda{\backslash} \Lambda d_{n-1}\Lambda$ where
$d_{n-1}={\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1})$.
Similarly, let $D_{II}$ be the set of the following matrices:
$$D^{II}_{n-1}={\rm diag}(p,
\overbrace{1,\ldots,1}^{n-1}),\ D^{II}_s=D^{II}_{s}(y):=
\left(\begin{array}{c|c|c}
1_s & y & \\
\hline
& p & \\
\hline
& &1_{n-1-s}
\end{array}
\right),\ 1\le s\le n-1,\ y\in M_{s\times 1}(\mathbb{F}_p)
$$
The cardinality of $D_{II}$ is $1+p+\cdots+p^{n-1}=\displaystyle\frac{p^n-1}{p-1}$ which is
equal to that of $\Lambda{\backslash} \Lambda d_{1}\Lambda$ where
$d_{1}={\mathop{\mathrm{diag}}}(\overbrace{1,\ldots,1}^{n-1},p)$.
Finally for each $M\in M_n(\mathbb{Z}_p)$ we denote by $r_p(M)$ the rank of $M$ mod $p\mathbb{Z}_p$.
\begin{lem}\label{DC}Assume $p$ is odd.
The right coset decomposition $T_{p,n-1}=\displaystyle\coprod_{\alpha\in J}K \alpha$
consists of the following elements:
\begin{enumerate}
\item (type I) $\alpha=\alpha_I(D,B)=
\begin{pmatrix}
p^2\cdot {}^tD^{-1} & B \\
0_n & D
\end{pmatrix}
$ where $D$ runs over the set $D_I$ and $B$ runs over complete representatives of
$B(D)/\sim$ such that $r_p(\alpha)=1$. Further, for each $D^I_s$, $B$ can be taken
over
\begin{itemize}
\item if $s\neq 0$, then $x\neq 0$ and $B=0$;
\item if $s=0$, then $x=0$ and $B=0$.
\end{itemize}
\item (type II)
$\alpha=\alpha_{II}(D,B)=
\begin{pmatrix}
p\cdot {}^tD^{-1} & B \\
0_n & pD
\end{pmatrix}
$
where $D$ runs over the set $D_{II}$ and $B$ runs over complete representatives of
$B(D)/\sim$ such that $r_p(\alpha)=1$. Further, for each $D^{II}_s$, $B$ can be taken
over
\begin{itemize}
\item if $s=0$, then $\begin{pmatrix}
B_{22} & B_{23}\\
p\cdot {}^t B_{23} & 0_{n-1}
\end{pmatrix}$ where $B_{22}$ runs over $\mathbb{Z}/p^2\mathbb{Z}$ and
$B_{23}$ runs over $M_{1\times (n-1)}(\mathbb{F}_p)$;
\item if $s\neq 0$, for $D^{II}_s(y),\ y\in M_{s\times 1}(\mathbb{F}_p)$,
$\left(\begin{array}{c|c|c}
0_s & p\cdot {}^tB_{21} & 0_{s\times (n-1-s)} \\
\hline
B_{21}& B_{22} & B_{23} \\
\hline
0_{(n-1-s)\times s} & p\cdot {}^t B_{23} &0_{n-1-s}
\end{array}
\right)$ where $B_{21},B_{22}$ and $B_{23}$ run over $M_{1\times s}(\mathbb{F}_p),\
\mathbb{Z}/p^2\mathbb{Z}$, and $M_{1\times t}(\mathbb{F}_p)$ respectively.
\end{itemize}
\item (type III) $\alpha=\alpha_{III}(B)=
\begin{pmatrix}
p1_n & B \\
0_n & p1_n
\end{pmatrix}
$ where $B$ runs over $S_n(\mathbb{F}_p)$ with $r_p(B)=1$.
The number of such $B$'s is $p^n-1$.
\end{enumerate}
\end{lem}
\begin{proof}
We just apply the formula \cite[(3.94), p. 98]{Andrianov}.
First we need to compute a complete system of representatives of
$\Lambda{\backslash} \Lambda t\Lambda \simeq (t^{-1}\Lambda t)\cap \Lambda{\backslash} \Lambda$
for each $t\in \{d_{n-1},d_1,p1_{n}\}$ where $d_{n-1}={\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1})$ and
$d_{1}={\mathop{\mathrm{diag}}}(\overbrace{1,\ldots,1}^{n-1},p)$. By direct computation, for $t=d_{n-1}$ (resp.
$t=d_1$), it is given by $D^I$
(resp. $D^{II}$). For $t=p\cdot 1_{n}$ it is obviously a singleton.
As for the computation of $B(D)/\sim$, we give details only for $D\in D^I$ and
the case of $D^{II}$ is similarly handled.
For each $D=D^I_s(x),\ 0\le s\le n-2 $, put
$A_s=\left(\begin{array}{c|c|c}
1_s & & \\
\hline
& 1 & -px \\
\hline
& &1_{n-1-s}
\end{array}
\right)$ so that $DA_s=\left(\begin{array}{c|c|c}
p\cdot 1_s & & \\
\hline
& 1 & \\
\hline
& &p\cdot 1_{n-1-s}
\end{array}
\right)$. Put $A_{n-1}=1_{2n}$ for $D=D^I_{n-1}$.
Then for each $D=D^I_s$ we have a bijection
$$B(D)/\sim\stackrel{\sim} {\longrightarrow} B(DA_s)/\sim,\quad B\mapsto BA_s.
$$
Therefore, we may compute $B(DA_s)/\sim$ and convert them by multiplying
$A^{-1}_s$ on the right.
We write $B\in B(DA_s)$ as a block matrix
$B=\left(\begin{array}{c|c|c}
\overbrace{B_{11}}^{s} & \overbrace{B_{12}}^{1} & \overbrace{B_{13}}^{n-1-s} \\
\hline
B_{21}& B_{22} & B_{23} \\
\hline
B_{31} & B_{32} &B_{33}
\end{array}
\right)$
with respect to the partition $s+1+(n-1-s)$ of $n$ where the column is also decomposed
as in the row. The relation yields
$$B=\left(\begin{array}{c|c|c}
B_{12 }&B_{12} & B_{13} \\
\hline
p\cdot{}^t B_{12}& B_{22} & p\cdot {}^t B_{32} \\
\hline
{}^t B_{13} & B_{32} &B_{33}
\end{array}
\right)$$
where $B_{11}\in S_{s}(\mathbb{Z}_p)$,\ $B_{22}\in \mathbb{Z}_p$, and $B_{33}\in S_{n-1-s}(\mathbb{Z}_p)$.
We write $X\in M_n(\mathbb{Z}_p)$ as
$\left(\begin{array}{c|c|c}
\overbrace{X_{11}}^{s} & \overbrace{X_{12}}^{1} & \overbrace{X_{13}}^{n-1-s} \\
\hline
X_{21}& X_{22} & X_{23} \\
\hline
X_{31} & X_{32} &X_{33}
\end{array}
\right)$
with respect to the partition $s+1+(n-1-s)$ of $n$ as we have done for $B$.
Then
$$XDA_s=
\left(\begin{array}{c|c|c}
pX_{11} &X_{12} & pX_{13} \\
\hline
pX_{21}& X_{22} & pX_{23} \\
\hline
pX_{31} & X_{32} &pX_{33}
\end{array}
\right)$$
Our matrix $B$ in $B(DA_s)/\sim$ is considered by taking modulo $XDA_s$
for any $X\in M_n(\mathbb{Z}_p)$. Hence $B$ can be, up to equivalence, of form
\begin{equation}\label{B(D)}B=\left(\begin{array}{c|c|c}
B_{11} &0_{s\times 1} & B_{13} \\
\hline
0_{1\times s}& 0 & 0_{1\times (n-1-s)} \\
\hline
{}^t B_{13} & 0_{(n-1-s)\times 1} & B_{33}
\end{array}
\right)
\end{equation}
where $B_{11},B_{33}$, and $B_{13}$ belong to $S_s(\mathbb{F}_p)$,
$S_{n-1-s}(\mathbb{F}_p)$, and $M_{s\times(n-1-s)}(\mathbb{F}_p)$ respectively.
Further, to multiply $A^{-1}_s$ on the right never change anything.
Therefore, (\ref{B(D)}) gives a complete system of representatives of $B(D)/\sim$ for
$D=D^I_s$. The condition $r_p(\alpha_I(D,B))=1$ and the modulo $K$ on the left yield the desired result.
For each $D\in D^{II}_s$, a similar computation shows
any element of $S(p\cdot D)/\sim$ is given by
$$\left(\begin{array}{c|c|c}
\overbrace{B_{11}}^{s} & \overbrace{p\cdot {}^t B_{21}}^{1} & \overbrace{B_{13}}^{n-1-s} \\
\hline
B_{21}& B_{22} & B_{23} \\
\hline
{}^t B_{13} & p\cdot {}^t B_{23} &B_{33}
\end{array}
\right)$$
modulo under the matrices of forms
$$\left(\begin{array}{c|c|c}
pX_{11} &p^2X_{12} & pX_{13} \\
\hline
pX_{21}& p^2X_{22} & pX_{23} \\
\hline
pX_{31} & p^2 X_{32} &pX_{33}
\end{array}
\right).$$
Therefore, $B_{11},B_{13},B_{21},B_{22},B_{23}$, and $B_{33}$ run over
$$M_{s}(\mathbb{F}_p),\ M_{s\times (n-1-s)(\mathbb{F}_p)},\ M_{1\times s(\mathbb{F}_p)},\ \mathbb{Z}/p^2\mathbb{Z},\
M_{1\times (n-1-s)(\mathbb{F}_p)},$$ and $M_{n-1-s}(\mathbb{F}_p)$ respectively.
The claim now follows from the rank condition $r_p(\alpha_{II}(D,B))=1$
and the modulo $K$ on the left again.
As for $D=p1_{n}$ in the case of type III, it is easy to see that
$S(D)/\sim$ is naturally identified with $S_n(\mathbb{F}_p)$.
Recall $p$ is an odd prime by assumption.
The number of matrices in $S_n(\mathbb{F}_p)$ of rank 1 is given in \cite[Theorem 2]{Mac}.
\end{proof}
Recall the right coset decomposition $T_{p,n-1}:=K{\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,
\overbrace{p,\ldots,p}^{n-1})K=\coprod_{\alpha\in J}K\alpha$.
For each $\alpha,\beta\in J$, we observe that any element of
$K\alpha \beta K$ is of mod $p$ rank at most two and has the similitude $p^4$.
Hence the double coset $K\alpha \beta K$ satisfies $K\alpha \beta K=
K\gamma K$, where $\gamma$ is one of the following 4 elements:
$$
\gamma_1:={\mathop{\mathrm{diag}}}(1,\overbrace{p^2,\ldots,p^2}^{n-1},p^4,
\overbrace{p^2,\ldots,p^2}^{n-1}),\ \gamma_2:={\mathop{\mathrm{diag}}}(p,p,\overbrace{p^2,\ldots,p^2}^{n-2},p^3,p^3,
\overbrace{p^2,\ldots,p^2}^{n-2}),$$
$$\gamma_3:={\mathop{\mathrm{diag}}}(p,\overbrace{p^2,\ldots,p^2}^{n-1},p^3,
\overbrace{p^2,\ldots,p^2}^{n-1}),\ \gamma_4:=p^2\cdot I_{2n}
$$
Here we use the Weyl elements in $K$ to renormalize the order of entries. Then
\begin{equation}\label{product}
T_{p,n-1}\cdot T_{p,n-1}=\sum_{i=1}^4 m(\gamma_i) K\gamma_i K
\end{equation}
where $m(\gamma_i)$ is defined by
\begin{equation}\label{m-gamma}
m(\gamma_i):=|\{(\alpha,\beta)\in J\times J\ |\ K\alpha\beta=K\gamma_i\}|
\end{equation}
for each $1\le i\le 4$ (cf. \cite[p.52]{Shimura-book}). Let us compute $m(\gamma_i)$ for each $\gamma_i$.
Let $J_I$ be the subset of $J$ consisting of the following elements
$$\alpha^s_I(x)=
\left(\begin{array}{ccc|ccc}
p\cdot 1_{s} & & &&& \\
& p^2 & & & & \\
& -p\cdot {}^t x & p\cdot 1_{n-1-s} &&& \\
\hline
&&& p\cdot 1_{s} & & \\
&&& & 1 & x\\
&&& & &p\cdot 1_{n-1-s}
\end{array}
\right),\ 0\le s\le n-2,\ x\in M_{1\times (n-1-s)}(\mathbb{F}_p)$$
and
$\alpha^{n-1}_I={\mathop{\mathrm{diag}}}(p^2,\overbrace{p,\ldots,p}^{n-1},1,\overbrace{p,\ldots,p}^{n-1})$.
Similarly, let $J_{II}$ be the subset of $J$ consisting of the following elements
$$\alpha^s_{II}(y,B_{21},B_{22},B_{33})=
\left(\begin{array}{ccc|ccc}
p\cdot 1_{s} & & & 0_s& p\cdot {}^t B_{21} & 0_{s\times (n-1-s)} \\
-{}^t y & 1 & & B_{21}& B_{22} &B_{23} \\
& & p\cdot 1_{n-1-s} & 0_{(n-1-s)\times s} &p\cdot {}^tB_{23} & 0_{n-1-s} \\
\hline
&&& p\cdot 1_{s} &py & \\
&&& & p^2 & \\
&&& & &p\cdot 1_{n-1-s}
\end{array}
\right)$$
where $1\le s\le n-1,\ y\in M_{s\times 1}(\mathbb{F}_p)$ and
$B_{21}, B_{23}$, and $B_{22}$ run over
$M_{1\times s}(\mathbb{F}_p),\ M_{1\times (n-1-s)}(\mathbb{F}_p)$, and $\mathbb{Z}/p^2\mathbb{Z}$
respectively.
In addition,
$$\alpha^0_{II}(C_{22},C_{23})=
\left(\begin{array}{cc|cc}
1 & & C_{22} & C_{23} \\
& p\cdot 1_{n-1} & p\cdot {}^tC_{23} & 0_{n-1} \\
\hline
& & p^2 & \\
& & & p\cdot 1_{n-1}
\end{array}
\right),\ C_{22}\in \mathbb{Z}/p^2\mathbb{Z},\ C_{23}\in M_{1\times (n-1)}(\mathbb{F}_p).$$
Finally, let $J_{III}$ be the subset of $J$ consisting of the following elements
$$\alpha_{III}(B)=\left(\begin{array}{c|c}
p\cdot 1_n & B \\
\hline
& p\cdot 1_n
\end{array}
\right),\ B\in S_n(\mathbb{F}_p)\ \text{with $r_p(B)=1$}.
$$
\begin{lem}\label{KalphaK}For the double coset $K\alpha K$ for $\alpha\in J$,
$$K\alpha K=K{\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,\overbrace{p,\ldots,p}^{n-1})K.
$$
and $${\rm vol}(K{\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,\overbrace{p,\ldots,p}^{n-1})K)
=p\sum_{i=0}^{2n-1}p^i$$
where the measure is normalized as ${\rm vol}(K)=1$.
\end{lem}
\begin{proof}
Except for the case of type III, it follows from elementary divisor theory.
For type III, it follows from \cite{Mac} that the action of ${\mathop{\mathrm{GL}}}_n(\mathbb{F}_p)$ on the set of
all matrices of rank 1 in $S_n(\mathbb{F}_p)$ given by $B\mapsto {}^tXBX,\ X\in {\mathop{\mathrm{GL}}}_n(\mathbb{F}_p)$ and
such a symmetric matrix $B$ has two orbits $O({\mathop{\mathrm{diag}}}(1,\overbrace{0,\ldots,0}^{n-1}))$ and
$O({\mathop{\mathrm{diag}}}(g,\overbrace{0,\ldots,0}^{n-1}))$ where $g$ is a generator of $\mathbb{F}^\times_p$.
The claim follow from this and elementary divisor theorem again.
For the latter claim, it is nothing but $|J|$ and we may compute the number of each type.
\end{proof}
\begin{remark}\label{WeylK}
Since $K={\mathop{\mathrm{Sp}}}_{2n}(\mathbb{Z}_p)$ contains Weyl elements,
\begin{eqnarray}
K{\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,\overbrace{p,\ldots,p}^{n-1})K
&=&K{\mathop{\mathrm{diag}}}(\overbrace{p,\ldots,p}^{i},1,\overbrace{p,\ldots,p}^{n-i-1},
\overbrace{p,\ldots,p}^{i},p^2,\overbrace{p,\ldots,p}^{n-i-1})K \nonumber \\
&=&K{\mathop{\mathrm{diag}}}(\overbrace{p,\ldots,p}^{i},p^2,\overbrace{p,\ldots,p}^{n-i-1},
\overbrace{p,\ldots,p}^{i},1,\overbrace{p,\ldots,p}^{n-i-1})K \nonumber
\end{eqnarray}
for $0\le i\le n-1$.
\end{remark}
Notice that $Kd_{n-1}(p)K=K(p^2\cdot d_{n-1}(p)^{-1})K$
where $d_{n-1}(p):={\mathop{\mathrm{diag}}}(1,\overbrace{p,\ldots,p}^{n-1},p^2,\overbrace{p,\ldots,p}^{n-1})$.
By definition and Lemma \ref{KalphaK} with Remark \ref{WeylK}, it is easy to see that
\begin{eqnarray}
m(\gamma_i)&=&|\{\beta\in J\ |\ \gamma_i\beta^{-1}\in
Kd_{n-1}(p)K\}| \nonumber \\
&=&|\{\beta\in J\ |\ \beta\cdot (p^2\cdot \gamma_i^{-1})\in
Kd_{n-1}(p)K\}| \nonumber \\
&=&|\{\beta\in J\ |\ \text{$\beta\cdot (p^2\cdot \gamma_i^{-1})$ is $p$-integral and } r_p(\beta\cdot (p^2\cdot \gamma_i^{-1}))=1\}| \nonumber
\end{eqnarray}
(see \cite[p.52]{Shimura-book} for the first equality).
We are now ready to compute the coefficients.
For $m(\gamma_1)$, we observe the $p$-integrality. We see that only $\alpha^0_{II}(C_{22},C_{23})$ with $C_{22}=0$ and
$C_{23}=0_{1\times(n-1)}$ can
contribute there. Hence $m(\gamma_1)=1$.
For $m(\gamma_2)$, we observe the $p$-integrality and
the rank condition. Then only $\alpha^0_{II}(0,0_{1\times (n-1)})$ and
$\alpha^1_{II}(y,0,0,0_{1\times (n-2)}),\ y\in \mathbb{F}_p$ can do there. Hence $m(\gamma_2)=1+p$.
For $m(\gamma_3)$, only $\alpha_{III}(B),\ B\in S_n(\mathbb{F}_p)$ with $r_p(B)=1$ contribute.
By Lemma \ref{DC}-(3), we have $m(\gamma_3)=p^n-1$.
Finally, we compute $m(\gamma_4)$.
Since $p^{-2}\gamma_4=I_4$, the condition is checked easily.
All members of $J=J_I\cup J_{II}\cup J_{III}$ can contribute there.
Therefore, we have only to count the number of each type. Hence, we have
$$m(\gamma_4)=
\overbrace{1+p+\cdots+p^{n-1}}^{{\rm type}\ I}+
\overbrace{p^{n+1}+p^{n+2}+\cdots+p^{2n}}^{{\rm type}\ II}+
\overbrace{p^n-1}^{{\rm type}\ III}
=p\sum_{i=0}^{2n-1}p^i$$
as desired. Note that $m(\gamma_4)$ is nothing but the volume of
$Kd_{n-1}(p)K$ (see Lemma \ref{KalphaK}).
Recalling $T_{p,n-1}:=pT(p,,(0,\ldots,0,1))$, we have
\begin{equation*}\label{final-form}
T(p,,(0,\ldots,0,1))^2=\sum_{i=1}^4 m(\gamma_i) K(p^{-2}\gamma_i)K.
\end{equation*}
Note that
$$K(p^{-2}\gamma_1)K=T(p,(\overbrace{0,...,0}^{n-1},2)),\quad
K(p^{-2}\gamma_2)K= T(p,(\overbrace{0,...,0}^{n-2},1,1)),
$$
$$
K(p^{-2}\gamma_3)K= T(p,(\overbrace{0,...,0}^{n-1},1)),\quad
K(p^{-2}\gamma_4)K= T(p,\overbrace{(0,...,0)}^{n})=K I_{2n}K.
$$
We can take $K$ back to ${\Gamma}(N)$ without changing anything since $p\nmid N$.
This proves Theorem \ref{main-appendix}.
\begin{remark} We would like to make corrections to \cite{KWY}.
\begin{enumerate}
\item In page 362, line 12-13, $T_{2,p}^2$ should be a linear combination of 4 double cosets $KMK$, where $M$ runs over
${\mathop{\mathrm{diag}}}(1,p^2,p^4,p^2), {\mathop{\mathrm{diag}}}(p,p,p^3,p^3), {\mathop{\mathrm{diag}}}(p,p^2,p^3,p^2), {\mathop{\mathrm{diag}}}(p^2,p^2,p^2,p^2)$.
\item In page 362, the coefficient of $R_{p^2}$ should be $p^4+p^3+p^2+p=p\displaystyle\sum_{i=0}^3p^i$
which is the volume of ${\mathop{\mathrm{Sp}}}(4,\mathbb{Z}_p){\mathop{\mathrm{diag}}}(1,p^2,p^4,p^2){\mathop{\mathrm{Sp}}}(4,\mathbb{Z}_p)$ explained in \cite[p.190, line -9 to -8]{RS}.
\item In page 403, Lemma 8.1, the inequality $q(F)\geq N$ is not valid. Similarly, in page 405, Lemma 8.3, the inequality $q(F)\geq N$ is not valid. We need to consider newforms as in Section 5 of this paper. Then for a newform, we obtain the inequality $q(F)\geq N^{\frac 16}$ and $\log c_{{\underline{k}},N}\asymp \log N$ is valid as in Lemma \ref{logN} of this paper.
\item In page 409, line 10, we need to add $-2(G(\frac 32)+G(-\frac 12))$, in order to account for the poles of $\Lambda(s,\pi_F,{\rm Spin})$, and the contour integral is over $Re(s)=2$. So we need to add $O\left(\frac {|HE_{\underline{k}}(N)^0|}{|HE_{\underline{k}}(N)|}\right)$ in (9.3).
However, only CAP forms give rise to a pole, and the number of CAP forms in $HE_{\underline{k}}(N)$ is $O(N^{8+\epsilon})$. So it is negligible.
In the case of standard $L$-functions, the non-CAP and non-genuine forms which give rise to poles are: $1\boxplus \pi$, where $\pi$ is an orthogonal cuspidal representation of $GL(4)$ with trivial central character, or $1\boxplus\pi_1\boxplus \pi_2$, where $\pi_i$'s are
dihedral cuspidal representations of $GL(2)$. In those cases, by Proposition \ref{trivial-case} and \cite[Theorem 2.9]{KWY1}, we can count such forms without extra conditions on $N$ in Proposition \ref{fixedV}. So our result is valid as it is written.
\end{enumerate}
\end{remark}
|
1,108,101,565,469 | arxiv | \section{Introduction}
The high-quality cosmological observational data (e.g. supernovae type
Ia, CMB, galaxy clustering, etc), accumulated during the last two decades,
have enabled cosmologists to gain substantial confidence that modern
cosmology is capable of quantitatively reproducing the details of
many observed cosmic phenomena, including the late time
accelerating stage of the Universe. Studies by many authors
have converged to a cosmic expansion history involving a spatially
flat geometry and a cosmic dark sector formed by cold dark matter and some
sort of dark energy, endowed with large negative pressure, in order to
explain the observed accelerating expansion of the Universe
\cite{Riess07,Spergel07,essence,Kowal08,komatsu08,Hic09,LJC09,BasPli10}.
In spite of that, the absence of a fundamental physical theory, regarding
the mechanism inducing the cosmic acceleration, have given rise to a
plethora of alternative cosmological scenarios.
Most are based either on the existence of new fields in nature (dark
energy) or in some modification of Einstein's general relativity,
with the present accelerating stage appearing as a sort of geometric effect.
The simplest dark energy candidate corresponds to a cosmological
constant, $\Lambda$ (see \cite{reviews} for reviews).
In the standard concordance cosmological ($\Lambda$CDM) model, the
overall cosmic fluid contains baryons, cold dark matter plus a
vacuum energy. This model fits accurately the current
observational data and it therefore provides an excellent scenario
to describe the observed universe. However, it is well known that
the concordance model suffers from, among others \cite{Peri08},
two fundamental problems:
{(i) {\it Fine tuning problem} -
the fact that the observed value of the vacuum energy density
($\rho_{\Lambda}=\Lambda c^{2}/8\pi G\simeq 10^{-47}\,GeV^4$) is
more than 120 orders of magnitude below the natural value
estimated using quantum field theory \cite{Weinberg89}.
(ii) {\it Coincidence problem} - the fact
that the matter and the vacuum energy densities are of the same
order just prior to the present epoch \cite{coincidence}.}
Such problems have inspired
many authors to propose alternative dark energy candidates such as
$\Lambda(t)$ cosmologies, quintessence, $k-$essence, vector
fields, phantom dark energy, tachyons, Chaplygin gas and the list
goes on (see \cite{Ratra88,Oze87,Lambdat,Bas09c,Wetterich:1994bg,
Caldwell98,Brax:1999gp,KAM,fein02,Caldwell,Bento03,chime04,Linder2004,
Brookfield:2005td,Grande06,Boehmer:2007qa} and references
therein). Naturally, in order to establish the evolution of the
dark energy equation of state (EoS), a realistic form of $H(a)$ is
required which should be constrained through a combination of
independent dark energy probes.
Nevertheless, there are other possibilities to explain the
present accelerating stage. For instance, one may consider
that the dynamical effects attributed to dark energy can be mimicked
by a nonstandard gravity theory. In other words, the present
accelerating stage of the universe can also be driven only by cold
dark matter under a modification of the nature of gravity.
Such a reduction of the so-called dark
sector is naturally obtained in the so-called $f(R)$ gravity
theories \cite{FR} (see, however, \cite{Nat10}).
On the other hand, general relativity predicts
that gravitational waves are non-dispersive and propagate with the
same vacuum light speed. These results lead to the common believe that
the graviton (the ``boson'' for general relativity),
must be a massless particle. However, massive gravitons are features
of some alternatives to general relativity as the one proposed by
Visser \cite{vis1998}. Such theories have motivated many experiments
and observations in order to detect a possible dispersive behavior
due to a non-zero graviton mass (see \cite {HL2010} and Refs. there in).
More recently, it was shown that the massive graviton approach proposed by Visser can be used
to build realistic cosmological models that can then be tested
against the available cosmological data \cite{Alves10}. One of the main advantages of such
massive graviton cosmology is the fact that it contains the same number of free
parameters as the concordance $\Lambda$CDM model, and, therefore, it does not
require the introduction of any extra fields in its dynamics. In this way, since the astronomical community is planning a
variety of large observational projects intended to test and constrain the standard $\Lambda$CDM
concordance model, as well as many of the proposed alternative models, it is timely and
important to identify and explore a variety of physical mechanisms (or substances) which
could also be responsible for the late-time acceleration of the Universe.
In what follows we focus our attention to a cosmological model
within Visser's massive graviton theory.
In particular we discuss how to differentiate the massive graviton
model from the concordance $\Lambda$CDM model. Initially, a joint
statistical analysis, involving the latest observational data
(SNIa, CMB shift parameter and BAO) is implemented. Secondly, we attempt to discriminate the MGCDM and
$\Lambda$CDM models by computing the halo mass function and the
corresponding redshift distribution of the cluster-size halos.
Finally, by using future X-ray and SZ surveys
we show that the evolution of the cluster abundances
is a potential discriminator between the MGCDM and $\Lambda$CDM
models. We would like to stress here that
the abundance of collapsed structures, as a
function of mass and redshift, is a key statistical test for studies
of the matter distribution in the universe, and, more importantly, it can
be accessed through observations \cite{Evra}. Indeed, the mass
function of galaxy clusters has been measured based on X-ray surveys
\cite{Borg01, Reip02, Vik09}, via weak and strong lensing studies
\cite{Bat98, Dahle06, Corl09}, using optical surveys, like the SDSS
\cite{Bah03, Wen10}, as well as, through Sunayev-Zeldovich (SZ)
effect \cite{Taub05}. In the last decade many authors have been
involved in this kind of studies and have found that the abundance
of the collapsed structures is affected by the presence of a dark energy component
\cite{Wein03,Liberato,manera,Abramo07,Fran08,Sch09,Mort09,Rap10,Pace10,Alam10,Khed10,BPL10,Lomb10}.
The paper is planned as follows.
{The basic elements of
Visser's theory are presented in section \ref{sec:two}, where we
also introduce the cosmological equations for a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry with massive gravitons.}
In section \ref{sec:three}, a joint statistical analysis based on SNe Ia, CMB and BAO
is used to constraint the massive graviton cosmological model
free parameter.
The linear growth factor of matter perturbations
is discussed in section \ref{sec:four}, while in \ref{sec:five}, we discuss
and compare the corresponding theoretical predictions regarding the evolution of
the cluster abundances. Finally, the main conclusions are
summarized in section \ref{sec:six}.
\section{\label{sec:two}Massive Gravitons Cold Dark Matter
(MGCDM) Cosmology: Basic Equations}
In this section we briefly present the main points of Visser's
massive gravity approach \cite{vis1998}.
The full action is given by (in what follows $\hbar = c=1$)
\begin{eqnarray}
\label{fullaction}
S=\int d^4x\left[\sqrt{-g}\frac{R(g)}{16\pi G}
+ {\cal{L}}_{mass_g}(g,g_0) +{\cal{L}}_{matter}(g)\right]
\end{eqnarray}
where besides the Einstein-Hilbert Lagrangian and the Lagrangian
of the matter fields, we have the bi-metric Lagrangian:
\begin{eqnarray}
{\cal{L}}_{mass}(g,g_0) = \frac{1}{2}{m_g}^2
\sqrt{-g_0}\bigg\{ ( g_0^{-1})^{\mu\nu}
( g-g_0)_{\mu\sigma}( g_0^{-1})^{\sigma\rho}
\\ \nonumber
\times ( g-g_0)_{\rho\nu}-\frac{1}{2}
\left[( g_0^{-1})^{\mu\nu}( g-g_0)_{\mu\nu}\right]^2\bigg\},
\end{eqnarray}
where $m_g$ is the graviton mass and
$(g_0)_{\mu\nu}$ is a general flat metric.
The field equations, which are obtained by variation of
(\ref{fullaction}), can be written as:
\begin{equation}\label{field-equations}
G^{\mu\nu} -\frac{1}{2}{m_g}^2 M^{\mu\nu} = -{8\pi G} T^{\mu\nu},
\end{equation}
where $G^{\mu\nu}$ is the Einstein tensor, $T^{\mu\nu}$ is the
energy-momentum tensor for perfect fluid, and the contribution of
the massive tensor to the field equations reads:
\begin{eqnarray}\label{massive tensor}
M^{\mu\nu} = (g_0^{-1})^{\mu\sigma}\bigg[ (g-g_0)_{\sigma\rho}
- \frac{1}{2}(g_0)_{\sigma\rho}(g_0^{-1})^{\alpha\beta}\\
\nonumber
\times(g-g_0)_{\alpha\beta} \bigg](g_0^{-1})^{\rho\nu} .
\end{eqnarray}
Note that if one takes the limit $m_g\rightarrow 0$ the standard
Einstein field equations are recovered.
Thus, from the construction of the Visser's theory, it can be classified as a
bimetric theory of gravitation. This kind of theory was first studied by N. Rosen \cite{Rosen1973}.
In the Rosen's concept the metric $g_{\mu\nu}$ describes the geometry of the spacetime in the same
way as in the context of the general relativity theory, and the second metric $(g_0)_{\mu\nu}$
(that Rosen denoted by $\gamma_{\mu\nu}$) refers to the flat spacetime and describes the inertial
forces. It is worth to mention that Rosen has shown that a bimetric theory satisfies the
covariance and the equivalence principles, a fact that was also pointed out by Visser (for more
discussion see \cite{Rham10}).
In this way, in order to follow the Rosen's approach we have constrained the background
metric to respect the Riemann-flat condition, that is, $R^\lambda_{\mu\nu\kappa}(g_0) \equiv 0$
in such a way that we have no ambiguity on the choice of $(g_0)_{\mu\nu}$, it will always be
chosen to be a flat metric, depending only on the particular coordinates we are dealing, of course.
Regarding the energy-momentum conservation we will follow the same
approach of Refs. \cite{Narlikar1984,deAraujo2007}.
Since the Einstein tensor satisfies the Bianchi identities
$\nabla_\nu G^{\mu\nu} = 0$, the energy conservation law is expressed as:
\begin{equation}\label{conservation}
\nabla_\nu T^{\mu\nu} = \frac{{m_g}^2}{16\pi G} \nabla_\nu M^{\mu\nu}.
\end{equation}
In the above framework, the global dynamics of a flat MGCDM cosmology
is driven by the following equations\footnote{In the present article we restrict our analysis
to the flat cosmologies in order to compare our results with those of the flat $\Lambda$CDM
model that is the most accepted cosmological model as shown, e.g., by the WMAP7 data \cite{komatsu08}.
A generalization of the model for a non spatially flat cosmology will appear in a forthcoming article.}:
\begin{equation}\label{eqfried1}
{8\pi G \rho} = 3\left( \frac{\dot{a}}{a}\right)^2 + \frac{3}{4}{m_g}^2(a^2 - 1),
\end{equation}
\begin{equation}\label{eqfried2}
{8\pi G p} = -2\frac{\ddot{a}}{a} - \left( \frac{\dot{a}}{a}\right)^2 - \frac{1}{4}{m_g}^2a^2(a^2-1),
\end{equation}
where $\rho$ is the energy density, $p$ is the
pressure and $a(t)$ is the scale factor.
From Eq. (\ref{conservation}) we get the evolution equation for
the energy density, namely:
\begin{equation}
\dot{\rho} + 3 H \left[ (\rho + p) + \frac{{m_g}^2}{32\pi G} (a^4 - 6a^2 + 3) \right] = 0,
\end{equation}
where $H = \dot{a}/a$. By integrating the above
equation for a matter dominated universe ($p= 0$)
one obtains:
\begin{equation}\label{rho-m-new}
\rho(a) = \frac{\rho_0}{a^3}- \frac{3{m_g}^2}{32\pi G}
\left( \frac{a^4}{7} - \frac{6a^2}{5} + 1 \right) \;,
\end{equation}
where $\rho_0$ is the present value of the energy density.
As expected, in the limiting case $m_g \rightarrow 0$ all
the standard FLRW expressions are recovered.
Now, inserting (\ref{rho-m-new}) in the modified Friedmann
equation (\ref{eqfried1}) we obtain the normalized Hubble parameter:
\begin{equation}\label{parHubMass}
E^{2}(a)=\frac{H^{2}(a)}{H_{0}^{2}}= \Omega_{m}a^{-3}+\delta H^{2},
\end{equation}
with
\begin{equation}\label{parHubMass1}
\delta H^{2}= \frac{1}{2}\Omega_{g} \left( 7a^2 -5a^4 \right),
\end{equation}
where $H_0$ is the Hubble constant, $\Omega_m$ is the matter
density parameter (for baryons and dark matter
$\Omega_i=\rho_{i0}/\rho_{c0}$, where $\rho_{c0}=3{H_0}^{2}/8\pi G$
is the critical density parameter), and $\Omega_{g}=\frac{1}{70}
(\frac{m_g}{H_0})^{2}$ is
the present contribution of the massive gravitons.
It should be stressed that the last term of the above
normalized Hubble function (\ref{parHubMass}) encodes
the correction to the standard FLRW expression.
In general, using the FLRW equations, one can express
the effective dark energy EoS parameter in terms of the
normalized Hubble parameter \cite{Saini00}
\begin{equation}
\label{eos22}
w_{\rm DE}(a)=\frac{-1-\frac{2}{3}a\frac{{d\rm ln}E}{da}}
{1-\Omega_{m}a^{-3}E^{-2}(a)}.
\end{equation}
After some simple algebra, it is also readily seen that
the effective (``geometrical'' in our case) dark energy EoS parameter is
given by (see \cite{Linjen03, Linder2004}):
\begin{equation}
\label{eos223}
w_{\rm DE}(a)=-1-\frac{1}{3}\;\frac{d{\rm ln}\delta
H^{2}}{d{\rm ln}a}.
\end{equation}
In our case, inserting Eq. (\ref{parHubMass1}) into Eq.
(\ref{eos223}) it is straightforward to obtain a simple analytical
expression for the geometrical dark energy EoS parameter:
\begin{equation}\label{eos221}
w_{\rm DE}(a) = - 1 - \frac{2}{3} \left(\frac{7 - 10a^2}{7 - 5a^2}\right).
\end{equation}
It thus follows that in the cosmological context, the modified
gravity theory as proposed by Visser can be
treated as an additional effective fluid with EoS parameter defined by (\ref{eos221}). Note also that the current Hubble function has
only two free parameters ($H_0$ and $\Omega_{m}$),
exactly the same number of free parameters as the conventional flat $\Lambda$CDM
model. Naturally, the value of $H_0$ is not predicted by any of the
models and it is set to its observational value of $H_0=70.4$ km
s$^{-1}$ Mpc$^{-1}$ \cite{komatsu08,freedman}.
\section{\label{sec:three}Likelihood Analysis}
Let us now discuss the statistical treatment of the observational data used
to constrain the MGCDM model presented in the previous section.
To begin with, we consider the {\em Constitution} supernovae Ia set of Hicken
et al. \cite{Hic09}, but in order to avoid possible problems
related to the local bulk flow, we use a subset of this sample
containing 366 SNe Ia all with redshifts $z>0.02$. The likelihood
estimator is determined by a $\chi^{2}_{\rm SNIa}$ statistics:
\begin{equation}
\label{chi22} \chi^{2}_{\rm SNIa}(\Omega_{m})=\sum_{i=1}^{366} \left[
\frac{ {\cal \mu}^{\rm th} (a_{i},{\Omega_{m}})-{\cal \mu}^{\rm
obs}(a_{i}) } {\sigma_{i}} \right]^{2},
\end{equation}
where $a_{i}=(1+z_{i})^{-1}$ is the scale factor of the Universe at
the observed redshift $z_{i}$, ${\cal \mu}$ is the distance modulus
${\cal \mu}=m-M=5{\rm log}d_{L}+25$ and $d_{L}$ is the
luminosity distance\footnote{Since
only the relative distances of the SNIa are accurate and not their
absolute local calibration, we always marginalize with respect to the
internally derived Hubble constant.}, $ d_{L}(a,\Omega_{m})=c{a}^{-1} \int_{a}^{1}
\frac{{\rm d}y}{y^{2}H(y)}$.
Now, from the likelihood analysis we find that
$\Omega_{m}=0.266\pm 0.016$ with
$\chi_{tot}^{2}(\Omega_{m})/dof\simeq 446.5/365$.
In addition to the SNe Ia
data, we also consider the BAO scale produced in the last
scattering surface by the competition between the pressure of the
coupled baryon-photon fluid and gravity. The resulting acoustic
waves leave (in the course of the evolution) an overdensity
signature at certain length scales of the matter distribution.
Evidence of this excess was recently found in the clustering
properties of SDSS galaxies (see \cite{Eis05,Perc10,Kazin10})
and it provides a suitable ``standard ruler'' for constraining dark energy
models. In this work we use
the measurement derived by Eisenstein et al. \cite{Eis05}. In particular,
we utilize the following estimator $A(\Omega_{m})=
\frac{\sqrt{\Omega_{m}}}{[z^{2}_{s}E(a_{s})]^{1/3}}
\left[\int_{a_{s}}^{1} \frac{da}{a^{2}E(a)} \right]^{2/3}$, measured
from the SDSS data to be $A=0.469\pm 0.017$, where $z_{s}=0.35$ [or
$a_{s}=(1+z_{s})^{-1}\simeq 0.75$]. Therefore, the corresponding
$\chi^{2}_{\rm BAO}$ function can be written as:
\begin{equation}
\chi^{2}_{\rm BAO}(\Omega_{m})=\frac{[A(\Omega_{m})-0.469]^{2}}{0.017^{2}}\;.
\end{equation}
The likelihood function peaks at $\Omega_{m}=0.306^{+0.026}_{-0.025}$.
Finally, a very interesting geometrical probe of
dark energy is provided by the angular scale of the sound horizon at
the last scattering surface. It is encoded in the location
of the first peak of the angular (CMB) power spectrum
\cite{Bond:1997wr,Nesseris:2006er}, and may be defined by the
quantity ${\cal R}=\sqrt{\Omega_{m}}\int_{a_{ls}}^1 \frac{da}{a^2 E(a)}$.
The shift parameter measured from the WMAP 7-years data
\cite{komatsu08} is ${\cal R}=1.726\pm 0.019$ at $z_{ls}=1091.36$ [or
$a_{ls}=(1+z_{ls})^{-1}\simeq 9.154\times 10^{-4}$]. In this case,
the $\chi^{2}_{\rm cmb}$ function reads
\begin{equation}
\chi^{2}_{\rm cmb}(\Omega_{m})=\frac{[{\cal R}(\Omega_{m})-1.726]^{2}}{0.018^{2}}.
\end{equation}
It should be stressed that for CMB shift parameter, the
contribution of the radiative component, ($\Omega_{R} a^{-4}$,
where $\Omega_{R}\simeq 4.174\times 10^{-5}h^{-2}$) needs also to
be considered \cite{komatsu08}. Note also that the measured CMB
shift parameter is somewhat model dependent but such
details of the models were not included in our analysis. For
example, such is the case when massive neutrinos are included.
The robustness of the shift parameter
has been tested and discussed in \cite{Elgaroy07}.
In this case the best fit value is: $\Omega_{m}=0.263 \pm 0.03$.
The derived $\Omega_m$ values from each individual probe appear to be
quite different, although within their mutual 2$\sigma$ uncertainty
range. Therefore, in order to put tighter constraints on the corresponding
parameter space of any cosmological model, the above probes are combined
through a joint likelihood analysis\footnote{Likelihoods are
normalized to their maximum values. In the present analysis we
always report $1\sigma$ uncertainties on the fitted parameters. Note
also that the total number of data points used here is
$N_{tot}=368$, while the associated degrees of freedom are: {\em
dof}$= 367$. Note that we sample $\Omega_{m} \in [0.1,1]$ in steps of
0.001.}, given by the product of the individual likelihoods
according to: ${\cal L}_{tot}(\Omega_{m})= {\cal L}_{\rm SNIa}\times
{\cal L}_{\rm BAO} \times {\cal L}_{\rm cmb}$, which translates in the
joint $\chi^2$ function in an addition:
$\chi^{2}_{tot}(\Omega_{m})=\chi^{2}_{\rm SNIa}+\chi^{2}_{\rm
BAO}+\chi^{2}_{\rm cmb}$.
Now, by applying our joint statistical procedure for both cosmologies,
we obtain the following best fit parameters:
\begin{itemize}
\item MGCDM model: $\Omega_{m}=0.276\pm 0.012$
with $\chi_{tot}^{2}(\Omega_{m})/dof \simeq
448.5/367$. Such results should be compared to those found
by Alves et al. \cite{Alves10}, namely: $\Omega_{m}=0.273\pm 0.015$
with a $\chi_{tot}^{2}(\Omega_{m})/dof \simeq
565.06/558$. This difference must be probably attributed
to the use of the {\em Union2} supernovae sample \cite{Union2} by the
latter authors.
\item $\Lambda$CDM model: $\Omega_{m}=0.280\pm 0.010$ with
$\chi_{tot}^{2}(\Omega_{m})/dof\simeq 439.5/367$,
which is in good agreement with recent studies
\cite{Riess07,Spergel07,essence,Kowal08,komatsu08,Hic09,LJC09,BasPli10}.
\end{itemize}
\begin{figure}[ht]
\mbox{\epsfxsize=8.5cm \epsffile{FIG1.ps}}
\caption{Normalized Hubble parameter as a function of redshift.
The solid line is the prediction of the MGCDM model.
For comparison, the dashed line corresponds to
the traditional $\Lambda$CDM model.}
\end{figure}
\begin{figure}[ht]
\mbox{\epsfxsize=8.5cm \epsffile{FIG2.ps}}
\caption{Expansion history. In the upper panel
we display the evolution of the dark energy effective
EoS parameter. In the lower panel we compare the
deceleration parameters of the MGCDM (solid line)
and the concordance $\Lambda$CDM (dashed line) models.
In the insert we show the relative deviation
$\Delta(q-q_{\Lambda})$ of the two deceleration parameters.}
\end{figure}
It should be mentioned here that using the BAO results
of Percival et al. \cite{Perc10}, does not change the previously
presented constraints.
\section{\label{sec:four}MGCDM versus $\Lambda$CDM cosmology}
\subsection{The cosmic expansion history}
In Figure 1 we plot the normalized MGCDM Hubble function (solid line)
as a function of redshift, which appears quite different both in amplitude
and shape with respect to the corresponding $\Lambda$CDM model expectations
(dashed-line).
In figure 2 (upper panel), we present the evolution of the MGCDM effective dark energy EoS
parameter.
One can divide the evolution of the cosmic expansion history in different
phases on the basis of the varying behavior of the
MGCDM and $\Lambda$CDM models. We will investigate such variations
in terms of the deceleration parameter, $q(a)=-(1+d{\rm ln}H/d{\rm
ln}a)$, which is plotted in the lower panel of figure 2. In the inset plot
we display the relative deviation of the deceleration parameter,
$\Delta(q-q_{\Lambda})$, between the two cosmological models.
We can divide the cosmic expansion history in the following phases:
\begin{itemize}
\item at early enough times $a\mincir 0.1$ the deceleration
parameters of both models are positive with $q\simeq
q_{\Lambda}$, which means that the two cosmological models
provide a similar expansion rate of the universe.
Note that by taking the limit $\displaystyle
\lim_{a \to 0} w_{DE}(a) = -5/3$ for the MGCDM model, while
we always have $w_{DE} = -1$ for the $\Lambda$CDM model;
\item for $0.1\le a\le 0.44$ the deceleration parameters are
both positive with $q>q_{\Lambda}$, which means that the
cosmic expansion in the MGCDM model is more rapidly
``decelerating'' than in the $\Lambda$CDM case;
\item between $0.44<a<0.52$ the deceleration parameters remain
positive but $q<q_{\Lambda}$;
\item for $0.52\le a\le 0.57$ the traditional $\Lambda$ model
remains in the decelerated regime ($q_{\Lambda}>0$) but
the MGCDM is starting to accelerate ($q<0$);
\item for $0.57<a\le 0.94$ the deceleration parameters are
both negative and since $q<q_{\Lambda}$, the
MGCDM model provides a stronger acceleration than in the
$\Lambda$CDM model (the opposite situation holds at
$0.85\le a\le 0.94$).
\end{itemize}
Interestingly, prior to the present epoch ($a>0.94$) the the
deceleration parameter of the MGCDM model becomes positive
and when $a = 1$ we have $w_{DE} = 0$, ie., the universe
becomes again matter dominated, implying that the late time
acceleration of the universe was a transient phase which has
already finished.
From the inset panel of figure 2 it becomes clear that the MGCDM
model reaches a maximum deviation from the $\Lambda$CDM cosmology
prior to $a \simeq 0.75$ and again at $a \simeq 1$. Finally, the
deceleration parameters at the present time are $q_{0}\simeq 0.50$
and $q_{0\Lambda}\simeq -0.58$. If we go further to the
future we find from Eq. (\ref{eos221}) that the state parameter as
well as the deceleration parameter diverges for $a = \sqrt{7/5}$.
This value sets the turning point after which the universe begins to
contract in the MGCDM model (for more details see
\cite{Alves2009}.)
\subsection{The growth factor and the rate of clustering}
It is well known that for small scales (smaller than the horizon)
the dark energy component (or ''geometrical'' dark energy)
is expected to be smooth and thus it is
fair to consider perturbations only on the matter component of the
cosmic fluid \cite{Dave02}. This assumption leads to the usual
equation for matter perturbations
\begin{equation}
\label{fluc01}
\ddot{\delta}_m+2H\dot{\delta}_m-4\pi G_{\rm eff} \rho_{m}\delta_m=0,
\end{equation}
where the effect of ''geometrical'' dark energy
is introduced via the expression of $G_{\rm eff}=G_{\rm eff}(t)$ [see \cite{Lue04},\cite{Tsu08}].
In the context of general relativity $G_{\rm eff}$ coincides with
the Newton's gravitational constant.
Now, for any type of dark energy
an efficient parametrization of the matter perturbations ($\delta_{m} \propto D$)
is based on the growth rate $f(a)\equiv d{\rm ln}D/d{\rm ln}a$
\cite{Peeb93}, which has the following functional form:
\be
\label{fzz221}
f(a)=\frac{d{\rm ln}D}{d{\rm ln}a}=\Omega^{\gamma}_{m}(a) \;\;,
\ee
where $D(a)$ is the linear growth factor,
$\Omega_{m}(a)=\Omega_{m}a^{-3}/E^{2}(a)$ and $\gamma$ is the so
called growth index (see Refs. \cite{Linder2004,Linjen03,Wang98,Lue04,Linder2007}).
Since the growth factor of a pure matter universe (Einstein de-Sitter)
has the form $D_{\rm EdS}=a$, one has to normalize the different
cosmological models such that $D\simeq a$ at
large redshifts due to the dominance of the non-relativistic matter component.
Using the latter condition we can easily integrate Eq. (\ref{fzz221})
to derive the growth factor \cite{Linder2004}
\be
\label{Dz221}
D(a)=a {\rm e}^{\int_{0}^{a} (dx/x) [\Omega_{m}^{\gamma}(x)-1]} \;.
\ee
In the present case we are working with a modification of
Einstein's gravity instead of an extra fluid, in such a way the
usual Poisson equation for the gravitational potential is modified
due the presence of the mass term. In the simple case of the
non-relativistic limit we have a Yukawa-like potential which
accomplishes corrections to the Newtonian potential to scales of
the order of the Compton wavelength of the graviton, $\lambda =
m^{-1}_{g}$. Using this kind of potential, the classic limit for the
graviton mass obtained from solar system dynamics observations is
$m_g < 7.68 \times 10^{-55}$g \cite{Talmadge1988}, but one of the
most stringent constraints is obtained by requiring the derived
dynamical properties of a galactic disk to be consistent with
observations \cite{deAraujo2007} thereby yielding $m_g < 10^{-59}$g. Now, by
considering the best fit value obtained here for $\Omega_g$ we
have $m_g \sim 10^{-65}$g, which is nearly 6 orders of magnitude
below to the previous bound.
This value gives a Compton wavelength of the order of the horizon. The
Compton wavelength can be seen as the physical length of graviton's perturbations which
implies that these perturbations play some role only close to the Hubble radius
and thus they will be negligible at sub-horizon scales. In other words,
Eqs. (\ref{fluc01}), (\ref{Dz221}) are both valid also in the MGCDM model.
\begin{table}[ht]
\caption[]{Data of the growth rate of clustering \cite{Ness08}. The
correspondence of the columns is as follows: redshift, observed
growth rate and references.} \tabcolsep 4.5pt
\begin{tabular}{ccc} \hline \hline
z& $f_{obs}$ & Refs. \\ \hline
0.15 & $0.51\pm 0.11$& \cite{Verde02,Hawk03}\\
0.35 & $0.70\pm 0.18$& \cite{Teg06} \\
0.55 & $0.75\pm 0.18$& \cite{Ross07}\\
1.40 & $0.90\pm 0.24$& \cite{daAng08}\\
3.00 & $1.46\pm 0.29$& \cite{McDon05}\\
\end{tabular}
\end{table}
Clearly in order to quantify the evolution of the growth
factor we need to know the growth index.
Since for the current graviton model there is yet no theoretically
predicted value of growth index, we attempt to provide a relevant value
by performing a standard $\chi^{2}$
minimization procedure (described previously) between the
observationally measured
growth rate (based on the 2dF and SDSS galaxy catalogs;
see Table I; \cite{Ness08})
and that expected in the MGCDM cosmological model, according to:
\be
\chi^{2}(\gamma)=\sum_{i=1}^{5} \left[ \frac{f_{obs}(z_{i})-
f_{\rm model}(z_{i},\gamma)}
{\sigma_{i}}\right]^{2} \;\;,
\ee
where $\sigma_{i}$ is the observed growth rate uncertainty.
Note that for comparison we perform the same analysis also
for the $\Lambda$CDM model.
\begin{figure}[ht]
\mbox{\epsfxsize=8.5cm \epsffile{FIG3.ps}} \caption{
{\it Upper Panel:}
Comparison of the observed (solid circles\,\cite{Ness08},
(see Table I) and theoretical evolution of the growth
rate of clustering $f(z)$.
The lines correspond to the MGCDM (solid curve) and
the $\Lambda$CDM (dashed curve) models.
{\it Bottom Panel:}
The evolution of the growth factor, with that
corresponding to
the MGCDM model ($\gamma=0.56$) showing a $\sim 1-4\%$
difference with respect to that of the $\Lambda$CDM model ($\gamma_{\Lambda}=0.62$), especially at
large redshifts ($z\ge 1$).
Errorbars are plotted only for the MGCDM model in order to avoid
confusion.}
\end{figure}
In Figure 3 (upper panel), we present
the measured $f_{obs}(z)$ (filled symbols)
with the estimated growth rate
function, $f(z)=\Omega^{\gamma}_{m}(z)$, for
the two considered cosmological models.
Notice, that for the MGCDM cosmological
model (solid line) we use $\Omega_{m}=0.276$
and for the $\Lambda$ case (dashed line) $\Omega_{m}=0.280$, which are
the values provided by our likelihood analysis of section 3.
In the inset panel of figure 3 we plot the variation of
$\Delta \chi^{2}=\chi^{2}(\gamma)-\chi^{2}_{\rm min}(\gamma)$
around the best $\gamma$ fit value. For the MGCDM model we find
$\gamma=0.56^{+0.15}_{-0.14}$ ($\chi^{2}/dof\simeq 0.69$),
while for the $\Lambda$CDM model we obtain
$\gamma_{\Lambda}=0.62^{+0.18}_{-0.15}$ ($\chi^{2}/dof\simeq 0.75$),
which is somewhat greater, but within $1\sigma$, of
the theoretically predicted value of $\gamma_{\Lambda}\simeq 6/11$.
Such a discrepancy between the theoretical $\Lambda$CDM and observationally fitted
value of $\gamma$ has also been found
by other authors. For example, Di Porto \& Amendola
\cite{Port08} obtained $\gamma=0.60^{+0.40}_{-0.30}$, while
Nesseris \& Perivolaropoulos \cite{Ness08},
based on mass fluctuations inferred from independent observations
at different redshifts, found $\gamma=0.67^{+0.20}_{-0.17}$.
If such a systematic difference between the measured and the theoretical
$\gamma$ $\Lambda$CDM values is due to observational uncertainties or
the method used to estimate the observed $\gamma$, then one may expect
a similar systematic difference to affect the measured $\gamma$ value
for the MGCDM model, pointing to a probably more realistic value for
this model of
$\gamma\simeq 0.49$. Since however this value is within the $1\sigma$
observational uncertainty, we will consider the originally fitted
MGCDM $\gamma$ value as the nominal one.
Using the above best fit $\gamma$ values we present, in
the lower panel of figure 3, the growth factor evolution derived
by integrating Eq. (\ref{Dz221}) for the two cosmological models
(MGCDM-solid and $\Lambda$CDM-dashed). The error bars correspond to
the 1$\sigma$ uncertainty of the fitted $\gamma$ values.
Note that the growth factors are normalized to unity
at the present time. The difference between the fitted growth factors
lies, at redshifts $z\ge 1$, in the interval $\sim 1-4\%$, while
when using the theoretically predicted $\Lambda$CDM
value of $\gamma_{\Lambda}\simeq 6/11$ the difference is less than $~1.5\%$.
For a consistent treatment of the two models and for the corresponding
comparison of their respective mass functions and halo redshift distributions
we will use, throughout the rest of the paper, the
observationally derived $\gamma$ values, ie., $\gamma_{\Lambda}\simeq
0.62$ and $\gamma_{\rm MGCDM}\simeq 0.56$.
\section{\label{sec:five}Compare the cluster Halo abundances}
It is important to define observational criteria that will enable us
to distinguish between the MGCDM model and the concordance $\Lambda$CDM
cosmology. An obvious choice, that has been extensively used, is to
compare the theoretically predicted cluster-size halo
redshift distributions and to use observational cluster data to
distinguish the models. Recently, the halo abundances predicted by a
large variety of DE models have been compared with those corresponding
to the $\Lambda$CDM model \cite{Bas09c, BPL10}. As a result, such analyses
suggest that many DE models explored in this study
(including some of modified gravity)
are clearly distinguishable from the $\Lambda$CDM cosmology.
We use the Press and Schecther \cite{press} (hereafter PSc) formalism,
based on random Gaussian fields,
which determines the fraction of matter that has formed bounded
structures as a function of redshift. Mathematical details of our
treatment can be found in \cite{BPL10}; here we only present the basic
ideas.
The number density of halos, $n(M,z)$,
with masses within the range $(M, M+\delta M)$ are given by:
\begin{equation}\label{MF}
n(M,z) dM = \frac{\bar{\rho}}{M} \frac{d{\rm \ln}\sigma^{-1}}{dM} f_{\rm
PSc}(\sigma) dM \;,
\end{equation}
where $f_{\rm PSc}(\sigma)=\sqrt{2/\pi} (\delta_c/\sigma)
\exp(-\delta_c^2/2\sigma^2)$, $\delta_{c}$ is the
linearly extrapolated density
threshold above which structures collapse \cite{eke}, while
$\sigma^2(M,z)$ is the mass variance of
the smoothed linear density field, extrapolated to redshift $z$ at which
the halos are identified. It depends on the power-spectrum of
density perturbations in Fourier space, $P(k)$, for which we use here the
CDM form according to \cite{Bard86}, and the values of the
baryon density parameter, the spectral slope and Hubble constant
according to the recent WMAP7 results \cite{komatsu08}.
Although the Press-Schecther formalism
was shown to provide a good first approximation to the
halo mass function provided by numerical simulations, it was later found
to over-predict/under-predict the number of low/high mass halos at the
present epoch \cite{Jenk01,LM07}.
More recently, a large number of works have provided better fitting
functions of $f(\sigma)$, some of them based on a phenomenological
approach. In the present
treatment, we adopt the one proposed by Reed et al. \cite{Reed}.
We remind the reader that it is traditional to parametrize the mass
variance in terms
of $\sigma_8$, the rms mass fluctuations on scales of $8 \;
h^{-1}$ Mpc at redshift $z=0$.
In order to compare the mass function predictions of the
different cosmological models, it is imperative to use
for each model the corresponding value of $\delta_c$ and $\sigma_8$.
It is well known that for the usual $\Lambda$ cosmology
$\delta_{c} \simeq 1.675$,
while Weinberg \& Kamionkowski \cite{Wein03} provide an accurate fitting
formula to estimate $\delta_{c}$ for any DE model
with a constant equation of state parameter.
Since for the current graviton cosmological vacuum model
the effective dark energy EoS parameter at the present
time is $w\simeq 0$ which implies that the Hubble parameter
is matter dominated it is fair to use the Einstein de-Sitter
value $\delta_{c} \simeq 1.685$ \cite{Wein03}.
Now, in order to estimate the correct model $\sigma_8$ power spectrum normalization,
we use the formulation developed in \cite{BPL10} which scales the
observationally determined $\sigma_{8, \Lambda}$ value to that of any
cosmological model. The corresponding MGCDM value is
$\sigma_{\rm 8, MGCDM}=0.828$ and it is based on $\sigma_{8, \Lambda}=0.804$ (as
indicated also in Table 1), derived from
an average of a variety of recent measurements (see also the corresponding discussion in
\cite{BPL10}) which are based on the WMAP7 results \cite{komatsu08}, on
a recent cluster
abundances analysis \cite{Rozo09}, on weak-lensing results \cite{Fu08}
and on peculiar velocities based analyses \cite{Wat09}.
\begin{table*}[h]
\tabcolsep 10pt
\vspace {0.2cm}
\begin{tabular}{|lcc|cc|ccc|} \hline \hline
Model & $\sigma_{8}$ & $\gamma$ &\multicolumn{2}{|c}{($\delta{\cal N}/{\cal N}_{\Lambda})_{\rm eROSITA}$} &
\multicolumn{3}{|c|}{$(\delta {\cal N}/{\cal N}_{\Lambda})_{\rm SPT}$} \\
& & & $z<0.3$& $0.6\le z <0.9$ & $z<0.3$ & $0.6\le z <0.9$ & $1.3\le z <2$ \\ \hline
$\Lambda$CDM & 0.804 & 0.62 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
MGCDM & 0.831 & 0.56 & -0.09 & 0.15$\pm 0.01$ & -0.09 & 0.11 & -0.03\\
MGCDM & 0.875 & 0.42 & 0.00 & 0.25$\pm 0.01$ & 0.00 & 0.18 & -0.01\\
MGCDM & 0.789 & 0.71 & -0.19 & 0.06$\pm 0.01$ & -0.19 & 0.05 & -0.05\\ \hline
\end{tabular}
\caption[]{Numerical results. The $1^{st}$ column indicates the
cosmological model. The $2^{rd}$ and $3^{rd}$ columns lists the
corresponding $\sigma_{8}$ and $\gamma$ values, respectively.
The remaining columns present the fractional relative difference of
the abundance of halos between the
MGCDM and the $\Lambda$CDM cosmology for two future
cluster surveys discussed in the text. The lower two rows show results
corresponding to the upper and lower $1\sigma$ range of the
observational $\gamma$ value uncertainty.
Errorbars are 2$\sigma$ Poisson
uncertainties and are shown only if they are larger than $10^{-2}$).}
\end{table*}
Given the halo mass function from Eq.(\ref{MF}) we can now derive an observable
quantity which is the redshift distribution of clusters, ${\cal
N}(z)$, within some determined mass range, say $M_1\le
M/h^{-1}M_{\odot}\le M_2=10^{16}$. This can be estimated by integrating, in mass,
the expected differential halo mass function, $n(M,z)$, according
to:
\be {\cal
N}(z)=\frac{dV}{dz}\;\int_{M_{1}}^{M_{2}} n(M,z)dM, \ee
where $dV/dz$ is the comoving volume element.
In order to derive observationally relevant
cluster redshift distributions and therefore test the possibility
of discriminating between the MGCDM and the $\Lambda$CDM
cosmological models, we will use the expectations of two realistic
future cluster surveys:
\noindent (a) the {\tt eROSITA} satellite X-ray survey, with a flux
limit of: $f_{\rm lim}=3.3\times 10^{-14}$ ergs s$^{-1}$ cm$^{-2}$,
at the energy band 0.5-5 keV and covering $\sim 20000$ deg$^{2}$ of
the sky,
\noindent (b) the South Pole Telescope SZ survey, with a limiting
flux density at $\nu_0=150$ GHz of $f_{\nu_0, {\rm lim}}=5$ mJy and
a sky coverage of $\sim 4000$ deg$^{2}$.
\begin{figure}[ht]
\mbox{\epsfxsize=8.8cm \epsffile{FIG4.ps}} \caption{The expected
cluster redshift distribution of the MGCDM (solid curve) and
$\Lambda$CDM (dashed curve) models
for the case of two future cluster
surveys (upper panels), and the corresponding fractional difference
with respect to the reference $\Lambda$CDM model (lower
panels). Errorbars are 2$\sigma$ Poisson uncertainties, while the
dashed lines in the lower panel bracket the range due to the uncertainty of
the observationally fitted value of $\gamma$.}
\end{figure}
To realize the predictions of the first survey we use the relation
between halo mass and bolometric X-ray luminosity, as a function of
redshift, provided in \cite{Fedeli}, ie:
\be\label{bolom}
L(M,z)=3.087 \times 10^{44} \left[\frac{M E(z)}{10^{15} h^{-1}
M_{\odot}} \right]^{1.554} h^{-2} \; {\rm erg s^{-1}} \;.
\ee
The limiting halo mass that can be observed at redshift $z$ is
then found by inserting in the above equation the limiting
luminosity, given by: $L=4 \pi d_L^2 f_{\rm lim}${\em c}$_b$, with
$d_L$ the luminosity distance corresponding to the redshift $z$ and
{\em c}$_b$ the band correction, necessary to convert the bolometric
luminosity of Eq.(\ref{bolom}) to the 0.5-5 keV band of {\tt
eROSITA}. We estimate this correction by assuming a Raymond-Smith
(1977) plasma model with a metallicity of 0.4$Z_{\odot}$, a typical
cluster temperature of $\sim 4$ keV and a Galactic absorption column
density of $n_{H}=10^{21}$ cm$^{-2}$.
The predictions of the second survey can be realized using again the
relation between limiting flux and halo mass from \cite{Fedeli}:
\be\label{sz} f_{\nu_0, {\rm lim}}= \frac{2.592 \times 10^{8} {\rm
mJy}}{d_{A}^{2}(z)} \left(\frac{M}{10^{15} h^{-1}M_{\odot}}\right)^{1.876}
E^{2/3}(z) \; \ee where $d_A(z) \equiv d_L/(1+z)^2$ is the angular
diameter distance out to redshift $z$.
In figure 4 (upper panels) we present the expected redshift
distributions above a limiting halo mass, which is $M_1 \equiv M_{\rm
limit}=\max[10^{14} h^{-1}M_{\odot}, M_f]$, with $M_f$ corresponding to
the mass related to the flux-limit at the different redshifts,
estimated by solving Eq.(\ref{bolom}) and Eq.(\ref{sz}) for $M$. In
the lower panels we present the fractional difference between the
MGCDM and $\Lambda$CDM.
The error-bars shown correspond to 2$\sigma$ Poisson
uncertainties, which however do not include cosmic variance
and possible observational systematic uncertainties, that would
further increase the relevant variance. A further source of
uncertainty that should be taken into account is related to the
uncertainty of the observationally derived value of $\gamma$ (see
section 4). The dashed lines in the lower panels of Fig.3 bracket
the corresponding number count relative model differences due to the
$1\sigma$ uncertainty in the value of $\gamma$, with the lower curve
corresponding to $(\gamma, \sigma_8)=(0.71, 0.787)$ and the upper to
$(\gamma, \sigma_8)=(0.42, 0.876)$.
The results (see also Table II) indicate that significant model differences should be
expected to be measured up to $z\mincir 1$ for the
case of the {\tt eROSITA} X-ray survey, and to much higher redshifts
for the case of the South Pole Telescope SZ survey. What is particularly
interesting is the differential difference between the $\Lambda$CDM
and MGCDM models, which is
negative locally ($z\mincir 0.3$), positive at intermediate redshifts
($0.4\mincir z \mincir 1$) and negative again for $z\magcir 1$.
This appears to be a unique signature of the MGCDM model, which
differentiates it from the behaviour of a large class of DE models (see
\cite{BPL10}) and makes it relatively easier to be distinguished.
In Table II, one may see a more compact presentation
of our results including the
relative fractional difference between the MGCDM model and the
$\Lambda$CDM model, in characteristic redshift bins and for both
future surveys.
\section{\label{sec:six}Conclusions}
In this work, the large and small scale dynamical properties of a flat
FLRW cold dark matter cosmology, endowed with massive gravitons
(MGCDM), were discussed from an analytical and a numerical viewpoints.
We find that the MGCDM can accommodate a ``dynamic phase transition"
from an early decelerating phase (driven only by cold dark matter) to a
late time accelerating expansion and a subsequent recent
re-deceleration phase.
Interestingly, the Hubble function of the MGCDM model contains
only two free parameters, namely $H_0$ and $\Omega_{m}$, which is the same
number of free parameters as the $\Lambda$CDM model.
Performing, a joint likelihood analysis using
the current observational data (SNIa, CMB shift parameter and BAOs),
we have provided
tight constraints on the main cosmological parameter
of the MGCDM model, i.e., $\Omega_{m}=0.276\pm 0.012$.
We then compared the MGCDM scenario with the conventional flat
$\Lambda$ cosmology regarding the rate of clustering as well as the
predicted halo redshift distribution.
The main conclusions of such a comparison are:
\begin{itemize}
\item At large redshifts the amplitude of the linear perturbation
growth factor of the MGCDM model is slightly different to the
$\Lambda$ solution (at a $1-4\%$ level), while the observationally determined
growth index of clustering ($\gamma\simeq 0.56$)
is smaller than the corresponding fit for
the $\Lambda$ model ($\gamma_{\Lambda}\simeq 0.62$), although within
their respective $1\sigma$ uncertainties.
\item
The shape and amplitude for the
redshift distribution of cluster-size halos predicted by the MGCDM model is quite
different from the one of a flat $\Lambda$CDM cosmology. Such a difference depends on redshift and has a characteristic
signature that can discriminate the current graviton model from
other contender DE models in the future cluster surveys.
\end{itemize}
\vspace {0.4cm}
\acknowledgments
MP acknowledges funding by
Mexican CONACyT grant 2005-49878, and JASL is partially supported by
CNPq and FAPESP under grants 304792/2003-9 and 04/13668-0,
respectively.
|
1,108,101,565,470 | arxiv | \section{Introduction}
\baselineskip=12pt
Back in 1962 Schwinger considered the possibility that a vector
gauge field can
imply a nonzero mass gauge particle. He showed this is indeed
the case for
an exactly solvable model,
namely two-dimensional Quantum Electrodynamics (QED$_2$) with
a massless fermion field
\cite{Schwinger}.
This simple, although non-trivial, field theory has become since
an arena to probe
different aspects
of quantum field theory \cite{general,ADAMS}. For instance,
the axial
anomaly, the charge screening and the bosonization presented by the
model,
together with alternative methods of solution,
have been studied by different authors
\cite{Manton,HetHo,Shifman,Link,IsoMurayama,HallinLiljenberg}. Of
particular
interest are
the results from
QED$_2$ that might shed light on the non-perturbative
features not only of QED$_4$ but also of QCD$_4$ and
quantum gravity in its gauge-theory-like formulation
\cite{G-P,RovelliSmolin,Gambini}.
Being QED$_4$ the gauge theory {\em par excellence} it was
considered worth adopting
loop variables techniques in its description
\cite{GambiniTrias,RovelliSmolin}
because
in this way gauge invariance could be explicitly implemented.
Lattice QED$_4$ has been successfully developed along these lines
and computational calculations have been improved
with respect to simulations \cite{FortGambini}.
Also, a four-dimensional (continuum) gravitational analogue
constructed out of gravity plus fermions
has been studied in
this framework \cite{HugoRovelli}. In every case the presence of
fermions is
automatically accounted for by including open curves, besides loops, to
parametrize operators and state vectors. The fermions necessarily
stand at the
end points of the open curves.
Moreover, the dynamics gets
geometrically coded
into the breaking, rejoining and rerouting of such open curves and
loops through
their
intersection
points. This is deeply significant for non-perturbative quantum gravity
where the role of diffeomorphism invariance is central and such a
description
naturally fits in \cite{HugoRovelli}. For the QED$_4$ case
this geometrical picture of dynamics led to useful criteria to
approximate the
strong coupling regime \cite{FortGambini}.
Wilson loops are computed in terms of the holonomy elements associated
to the parallel transport along closed curves. In the Maxwell case the
holonomies are elements of the U(1) group. They are not only invariant
under small gauge transformations generated by the Gauss law constraint,
but also under large gauge transformations. That is the reason why they
naturally describe compact electrodynamics, which is also characterized by the
property that the range of $A_1(x)$, in the Weyl gauge for the two-dimensional case, is the circle
instead of the real line as it is in the standard noncompact case. This fact is well known and
has far reaching consequences in higher dimensions. Polyakov \cite{Po}
has argued that it is necessary to decide, based on physical
grounds, what
version of QED is realized in nature. In particular, the fact that
in the
non abelian case the non compact version cannot be formulated on a
discrete lattice leads him to consider that, if QED arises as a
subgroup of
some nonabelian gauge theory, we are necessarily dealing with the
compact
version. Earlier discussions of compact QED in the lattice can be found in Refs. \cite{COMPACT}, for
example. The Schwinger model has been recently studied in the hamiltonian
\cite{FA} and lagrangean\cite{F} lattice loop representation. In these
papers, it is shown that the chiral symmetry is broken and the
$\theta$-dependence of the vacuum is not present.
In what concerns the case of higher dimensions, it has been shown that
monopoles arising in the compact abelian sector of 2+1 QCD play a
fundamental role in the confinement process. In Polyakov's analysis
loops are crucial to understand this process. In particular, he has
recently shown that the loop world sheets acquires string like degrees
of freedom due to the presence of a diluted gas of monopoles
\cite{Poly96}.
In this work, loop variables are
introduced for the compact Schwinger's model along the
lines of \cite{FortGambini,HugoRovelli}.
Starting with the
canonical analysis of QED$_2$, which yields the Gauss law first-class
constraint, loop variables are defined such that they have zero Poisson
brackets with it. They form a closed algebra and turn out to be enough to
describe the dynamics.
The Hamiltonian is reexpressed as a limit
of some of these loop variables when the curves shrink down to a point.
The quantum theory is defined such that the Poisson
algebra becomes a commutator algebra and the loop representation is built
by choosing one of these loop operators to create a state with an extra
loop (open curve) out of an arbitrary state and then using the operator
algebra.
Thus, it is possible to work entirely in the loop representation.
In order
to recover the standard (local) physical information, like the energy
spectrum for example, one has
to take the corresponding limit of the loops (open curves)
shrinking to a point.
Before determining the properties of the energy spectrum of the
full theory,
we found it convenient to study the semiclassical situation where
there is
a quantum fermion field
interacting with a classical electromagnetic field. Some technicalities
become more transparent if use is made of a bi-local Fourier transform
of the operators parametrized by open curves. Details are given in the
appendix.
This procedure yields a built-in {\em gauge-invariant} point-split
regularization for the fermion vacuum energy. In this semiclassical
context,
the vacuum to vacuum expectation value for the divergence of the
axial current
produces the known value for the axial anomaly in a straightforward
manner. Also, the vacuum expectation value of the so called
anomalous commutators are directly derived from the loop variables
algebra.
Remarkably
enough, the corresponding Poisson brackets algebra already contains the
relevant information. Using the separability of the full Schroedinger
equation for the system,
the zero mode sector of the spectrum is considered
next. The non-equally spaced results for the zero mode energy found in this case seem to
indicate that,
in the general compact case, the spectrum does not correspond to a
free massive boson. By considering the limit in which
the length of the $S^1$ spatial slice goes to zero one recovers the
typical harmonic oscillator spectrum
for the zero mass mode together with its free bosonic behavior, which is characteristic of the
non-compact Schwinger Model.
As we have previously emphasized, loop variables naturally describe
a compact
version of the electromagnetic interaction.
In order to recover the noncompact theory it is necessary to introduce
additional
angular variables, which are conjugated to the integer numbers
characterizing
the large
gauge transformations. A detailed discussion of this important
issue can be
found in Ref. \cite{GAMB5}.
The organization of the paper is as follows.
In section II we translate the local classical dynamics of QED$_2$ into
loop variables.
The corresponding algebra is displayed there. The quantum counterpart is
then exhibited in section III, where state functionals are loop/curve
parametrized; hence defining the loop representation. In section IV, the
analysis is carried out taking the fermion field as a quantum entity
evolving in
the external electromagnetic field. Hereby the fermion vacuum is
found. In section V, the
well known chiral anomaly coming from the non-conservation of the axial
current is computed. Section VI contains the calculation of the anomalous
commutators.
The zero mode sector of the theory is finally analyzed in section
VII based on the external electromagnetic field approach of the
previous section.
Finally, section VIII contains some general remarks on the
loop approach for QED$_2$
together with possible future developments. The appendix presents
the explicit relation between loop operators and their useful
bi-local Fourier transforms, making more transparent the analysis here
presented.
\section{Classical framework}
Our starting point is the {\em real} Lagrangian density
\begin{equation}
{\cal L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}
+\frac{\hbar}{2}\bar{\psi} \gamma^{\mu} \left(i\partial_{\mu}
- e A_{\mu}\right)\psi
-\frac{\hbar}{2}
\left[\left(i\partial_{\mu} + e
A_{\mu}\right)\bar{\psi}\right]
\gamma^{\mu} \psi
\label{LAG}
\end{equation}
where $F_{\mu\nu}=\partial_{\mu}A_{\nu} -\partial_{\nu}A_{\mu}$
and $\bar{\psi}=\psi^\dagger \gamma^0$ is a Grassmann valued
fermionic field.
Since space here is $S^1$, we will require periodic(antiperiodic)
boundary conditions for the fields
\begin{equation}
A_{\mu}(x+L) = A_{\mu}(x), \quad
\psi(x+L) = -\psi (x),
\label{BC}
\end{equation}
where $L=2\pi r$ is the length of the circle.
The gamma matrices are:
$\gamma^0=\sigma_1,\; \gamma^1=-i\sigma_2,\; \gamma^5=\gamma^0\gamma^1=
\sigma_3$, where $\sigma_i$ are the standard Pauli matrices.
We use the signature $(+,-),\;\; i.e.\;\; \eta_{00}=-\eta_{11}=1$.
The Lagrangian density (\ref{LAG}) is invariant under the
following gauge
transformations
\begin{equation} \psi \rightarrow e^{ie \alpha(x,t)} \psi,
\quad A_\mu \rightarrow A_\mu - \partial_\mu \alpha (x,t). \label{GGT}
\end{equation}
There are two families of gauge transformations: (i) those continuously
connected to the identity,
called small gauge transformations, characterized by the function
$\alpha = b(t) e^{i 2 \pi n x / L} $ which is periodic in $x$ and
preserves
the boundary conditions
(\ref{BC}). The second family corresponds to the so called large gauge
transformations, which is determined by the
non-periodic functions $\alpha={2 \pi
n \over e L}x , \ n=\pm 1, \pm 2, \dots $.
The boundary conditions (\ref{BC}) are also preserved in this case.
After the standard canonical analysis the Hamiltonian density
becomes
\begin{equation}
{\cal H} = \frac{1}{2} E^2
- \frac{i\hbar}{2} \psi^\dagger\sigma_3
\left(\partial_1 + i e A\right)\psi
+ \frac{i\hbar}{2} \left[\left(\partial_1-i e A\right)\psi^*\right]
\sigma_3 \psi
- A_0\,{\cal G}.
\label{HAM}
\end{equation}
Here $E=F_{01}$, $A=A_1$ and
\begin{equation}\label{GL}
{\cal G}=\partial_1 E - e\hbar \psi^*\psi
\end{equation}
is the Gauss law constraint.
The boundary term associated with the integration of $\partial_x(EA_0)$
yields no contribution
to the Hamiltonian because of the boundary conditions on the
vector potential $A_{\mu}$.
In what follows $\psi = (\psi_1,\psi_2)^\top$,
$\top$ denoting transposition. The charge density is
given by $\rho(x)= e (\psi_1^*\psi_1 + \psi_2^*\psi_2$).
We are working in units such that $c=1$, $\hbar \neq 1$ and we take
mass $[g]$ and length $[cm]$ as the basic ones. In this way the
corresponding
dimensions are: $ \hbar = [g \ cm],\ E=[\sqrt{g\over cm}],\
A=[ \sqrt{g \ cm}], \
eA=[{1\over cm}], \ \psi_1^* \psi_1=[{1\over cm}],
\ \psi_2^* \psi_2=[{1\over cm}]$ and $ \hbar e^2 L^2 $ is a
dimensionless
quantity. Both the combinations ${\hbar \over L }$ and $ e
\hbar^{3\over2}$
have the dimensions of mass.
The resulting Poisson brackets algebra at equal times is
\begin{eqnarray}
\left\{ A(x), E(y) \right\} &=& \delta(x,y) \nonumber \\
\left\{ \psi_{\alpha}(x), \psi^*_{\beta}(y) \right\}
&=& -\frac{i}{\hbar} \delta_{\alpha\beta}
\delta(x,y)\;\;\;\;\;\alpha=1,2\,.
\label{PB}
\end{eqnarray}
\subsection{Loop variables}
In order to take into account the Gauss law {\it ab initio},
one can adopt the following
gauge invariant non local variables
\cite{FortGambini}:
\begin{eqnarray}
T^0(\gamma) &=& \exp\{ie\oint_{\gamma}dx\,A(x)\}, \label{T0} \\
\Pi_0(\eta_x{}^y) &=& \psi_1^*(x) U(\eta_x{}^y) \psi_2(y), \label{PI)} \\
\Pi_1(\eta_x{}^y) &=& \psi_1^*(x) U(\eta_x{}^y) \psi_1(y),
\;\label{PI1} \\
\Pi_2(\eta_x{}^y) &=& \psi_2^*(x) U(\eta_x{}^y) \psi_2(y), \label{PI2} \\
\Pi_3(\eta_x{}^y) &=& \psi_2^*(x) U(\eta_x{}^y) \psi_1(y), \label{PI3}
\end{eqnarray}
with $U(\eta_x{}^y)=\exp\{ie\int_{\eta_x{}^y}dz\,A(z)\}$
and, of course, $E(x)$ which is gauge invariant by construction.
The open paths ${\eta}_x^y$ are always arcs of circumference starting
at the point $x$ and ending at the point $y$.
No independent loop variable is obtained by considering
a gauge invariant non local variable containing $E(x)$
as an insertion.
It is important to recall, once again, that in this
approach all the information about the theory is encoded in terms of
loop variables, which are gauge invariant under small and large gauge
transformations. That means, in particular, that the electromagnetic
information is encoded in the elements of the U(1) group $U(\eta_x^y)$
and consequently the loop representation naturally describes compact
electrodynamics. In other words, since the basic electromagneric variable
is ${\rm exp} \ ie \int_0^L dx A(x) $, it is enough to restrict $ \int_0^L dx A(x)$ to the interval
$[0, \frac{2 \pi}{e}]$.
The induced non zero Poisson brackets
among the loop variables are
\begin{eqnarray}
\left\{ T^0(\gamma), E(x) \right\} &=&
ie \oint_{\gamma}dz\,\delta(x,z)\, T^0(\gamma), \label{eq:te}\\
\left\{ \Pi_i(\eta_x{}^y), E(z) \right\} &=&
ie \int_{\eta_x{}^y} du\, \delta(z,u)\,
\Pi_i(\eta_x{}^y)\,,\;\;\;i=0,1,2,3, \\
\left\{ \Pi_0(\alpha_x{}^y), \Pi_1(\eta_u{}^v) \right\} &=&
{ i\over \hbar}
\delta(x,v)\, \Pi_0 \left( (\eta\circ\alpha)_u{}^y \right), \\
\left\{ \Pi_0(\alpha_x{}^y), \Pi_2(\eta_u{}^v) \right\} &=&
-{i\over \hbar}
\delta(u,y)\, \Pi_0 \left( (\alpha\circ\eta)_x{}^v \right), \\
\left\{ \Pi_0(\alpha_x{}^y), \Pi_3(\eta_u{}^v) \right\} &=&
{i\over \hbar} \delta(x,v)\, \Pi_2 \left(
(\eta\circ\alpha)_u{}^y \right),
\nonumber \\
&& -{i\over \hbar}
\delta(y,u)\, \Pi_1 \left((\alpha\circ\eta)_x{}^v \right), \\
\left\{ \Pi_i(\alpha_x{}^y), \Pi_i(\eta_u{}^v) \right\} &=&
- {i\over \hbar}
\delta(y,u)\, \Pi_i \left( (\alpha\circ\eta)_x{}^v \right) +
\nonumber\\
&&{i\over \hbar}
\delta(x,v)\, \Pi_i \left((\eta\circ\alpha)_u{}^y \right),
i=1,2, \\
\left\{ \Pi_1(\alpha_x{}^y), \Pi_3(\eta_u{}^v) \right\} &=&
{i\over \hbar} \delta(x,v)\, \Pi_3 \left(
(\eta\circ\alpha)_u{}^y \right),
\\
\left\{ \Pi_2(\alpha_x{}^y), \Pi_3(\eta_u{}^v) \right\} &=&
- {i\over \hbar}
\delta(y,u)\, \Pi_3 \left( (\alpha\circ\eta)_x{}^v \right).
\label{eq:p2p3}
\end{eqnarray}
Since it will be crucial later on, we stress here that the line integral
$\int_{\eta_x{}^y} du\, \delta(u,z) $ is more conveniently expressed
as \cite{HallinLiljenberg}
\begin{equation}
\theta(x,y;z)\equiv \int_{\eta_x{}^y} du\, \delta(u,z)
= \frac{1}{L} \left( y-x + \sum_{k\ne 0} \frac{1}{ik}
\left( e^{ik(y-z)} - e^{ik(x-z)} \right)
\right)\, ,
\label{THETA}
\end{equation}
where $k=\frac{2\pi n}{L} , \ n=\pm 1 , \pm 2 , \dots$.
We observe that
$\lim_{x\rightarrow y} \theta(x,y;z) = 0$. We can verify also that
\begin{equation}
\frac{\partial\theta(x,y;z)}{\partial z} =
\delta(x,z) - \delta(y,z) \;\; .
\end{equation}
The representation of the delta function is given by
\begin{equation}
\delta(x,y)={1\over L} \sum_{n=-\infty}^{+ \infty}
e^{{2 \pi i n \over L}(x-y)}
\label{DELTA}
\end{equation}
and the following properties can be directly obtained
\begin{equation}
\delta(x,y)=\delta(y,x), \quad \delta(x,0)=\delta(x,L).
\label{PROPDEL}
\end{equation}
To proceed further we rewrite ${\cal H, G}, \rho$ in terms
of loop variables
\begin{eqnarray}
{\cal H}(x) &=& \frac{1}{2} E^2(x)
+ \frac{i\hbar}{2} \lim_{y\rightarrow x}
(\partial_x - \partial_y)(\Pi_2 (\alpha_y{}^x) - \Pi_1 (\alpha_y{}^x))
\label{eq:dham} \\
{\cal G}(x) &=& \partial_x E(x) - e \hbar \lim_{y\rightarrow x}
\left( \Pi_1 (\alpha_y{}^x) + \Pi_2 (\alpha_y{}^x) \right)
\label{eq:gauss} \\
\rho(x) &=& \lim_{y\rightarrow x}\,e\,
\left( \Pi_1 (\alpha_y{}^x) + \Pi_2 (\alpha_y{}^x) \right)
\label{eq:dnum}
\end{eqnarray}
Although the quantities ${\cal H}, \rho$ are gauge invariant by
construction
in terms of the local fields,
one can directly verify that their Poisson brackets with the
Gauss constraint are identically zero using the above algebra, i.e.
Eqs. (\ref{eq:te}-\ref{eq:p2p3}).
Because it is illuminating we next show that the charge
$Q=\int_0^L dw \rho(w)$ is a constant of motion
\begin{eqnarray}
\left\{ Q, H\right\} &=& e \int_0^L dw \int_0^L dy
\lim_{\stackrel{\scriptstyle z\rightarrow w}
{x \rightarrow y}}
\left[ E(y) ie \theta(z,w;y) ( \Pi_1(\eta_z{}^w)
+\Pi_2(\eta_z{}^w) ) \right. \nonumber\\
& & + i\partial_y
( -i \delta(w,x)\, \Pi_2((\eta\circ\alpha)_z{}^y)
+i \delta(z,y)\, \Pi_2((\alpha\circ\eta)_x{}^w) ) \nonumber \\
& & \left. - i\partial_y
( -i \delta(w,x)\, \Pi_1((\eta\circ\alpha)_z{}^y)
+i \delta(z,y)\, \Pi_1((\alpha\circ\eta)_x{}^w) )\right] =0
\,.
\label{eq:nh}
\end{eqnarray}
The Hamiltonian is $H =\int_0^L dx \ {\cal H}$. Note that the
first line of
(\ref{eq:nh}) is zero because of
$\lim_{z\rightarrow w} \theta(z,w;y)=0$, provided
$\lim_{z\rightarrow w} \Pi_+(z,w)$ is finite.
That the second line is also zero can be readily seen as follows:
after taking the limits $x\rightarrow y\,,\,z\rightarrow w$
in the term $\partial_y\delta(z,y)\, \Pi_2((\alpha\circ\eta)_x{}^w)$ and
integrating by parts
the first term
$\delta(w,y)\, \partial_y\Pi_2(\alpha_w{}^y)$ gets cancelled.
The remaining boundary term is zero after performing the second integral.
The third line is zero for analogous reasons.
\section{Quantum framework}
Our general procedure of quantization will be to promote the
observables $A,B$ to operators $\hat A,\hat B$ and
let their Poisson brackets $\left\{A,B\right\}=C$
go over (anti)commutators
$[\hat{A},\hat{B}]_{\pm} = i\hbar
\hat C $ according to the standard prescription.
\subsection{Local field representation}
We define this representation by choosing the configuration
variables as \
$\hat{A}(x), \ \hat{\psi}_1^*(x), \\ \hat{\psi}_2(x)$
acting multiplicatively on a wave function
$\Psi(A(x), \psi_1^*(x), \psi_2(x))$.
The remaining variables will be represented by:
\begin{equation}
\hat{E}(x)=-i\hbar \frac{\delta}{\delta A(x)},\;\;\;
\hat{\psi}_1= \frac{\delta}{\delta \psi_1^*(x)},\;\;\;
\hat{\psi}_2^*= \frac{\delta}{\delta \psi_2(x)}\;.
\end{equation}
In this representation, the state of nothing $|0 \rangle$ which
does not contain any field excitation,
is given by a constant functional.
\subsection{Loop representation}
This representation will be constructed as a quantum realization of
the loop variables Poisson algebra given by
Eqs.(\ref{eq:te}-\ref{eq:p2p3}).
To this end, it is convenient to first characterize the state of nothing
$|0 \rangle$, which is not to be confused with the physical vacuum of
the theory. The former is characterized by $\langle A, \psi^*_1,
\psi_2|0
\rangle= {\rm const.}$ in the previous representation. This leads
us to define the state of nothing as
\begin{equation}
\hat{E}(x)|0\rangle=0,
\quad \hat{\Pi}_i(x,y)|0\rangle=0, \quad i=1,2,3.
\end{equation}
Note that these conditions enforce the gauge
invariance of this state. The above definitions imply that, in the
connection representation, the functional derivative operator must always
go to the right. In this way, we choose the following
representation for the
gauge invariant operators $ {\hat \Pi}_i(\eta_x{}^y)$
\begin{eqnarray}
{\hat \Pi}_0(\eta_x{}^y) &=&\psi_1^*(x) U(\eta_x{}^y)
\psi_2(y), \\
{\hat \Pi}_1(\eta_x{}^y) &=& \psi_1^*(x) U(\eta_x{}^y)
{\delta\over \delta \psi_1^*(y)}, \\
{\hat \Pi}_2(\eta_x{}^y) &=&-\psi_2(y) U(\eta_x{}^y){\delta \over
\delta \psi_2(x)}, \\
{\hat \Pi}_3(\eta_x{}^y) &=&{\delta \over \delta \psi_2(x)}
U(\eta_x{}^y)
{\delta\over \delta \psi_1^*(y)},
\end{eqnarray}
where the minus sign in ${\hat \Pi}_2$ has been introduced to recover the
classical
limit of Eq.(\ref{PI2}). Note that
${\hat \Pi}_3^{\dagger}(x,y) = {\hat \Pi}_0(y,x)$ in the standard scalar
product of Grassmann variables..
The resulting non-zero commutators
are
\begin{eqnarray}
{[{{\hat \Pi}_{0}} (u,v),{{\hat \Pi}_{1}} (x,y)]} &=&
- \delta (u,y) {{\hat \Pi}_{0}} (x,v),
\label{eq:p0p1}\\
{[{{\hat \Pi}_{0}} (u,v),{{\hat \Pi}_{2}} (x,y)]} &=&
\delta (x,v){{\hat \Pi}_{0}} (u,y),
\label{eq:p0p2}\\
{[{{\hat \Pi}_{0}} (u,v),{{\hat \Pi}_{3}} (x,y)]} &=&
\delta (x,v){\hat \Pi}_{1} (u,y)
- \delta (u,y) {\hat \Pi}_{2} (x,v)\nonumber \\
& & -\delta(u,y)\delta(x,v),
\label{eq:p0p3}\\
{[{\hat \Pi}_{i} (u,v),{\hat \Pi}_{i} (x,y)]} &=&
\delta (x,v) {\hat \Pi}_{i} (u,y) -
\delta (u,y){\hat \Pi}_{i} (x,v), \quad i=1,2
\label{eq:ppmppm}\\
{[{\hat \Pi}_+ (u,v),{\hat \Pi}_+ (x,y)]} &=&
\delta (x,v) {\hat \Pi}_+ (u,y) - \delta (u,y) {\hat \Pi}_+ (x,v),
\label{eq:p+p+}\\
{[{\hat \Pi}_- (u,v),{\hat \Pi}_- (x,y)]} &=&
\delta (x,v) {\hat \Pi}_+ (u,y) - \delta (u,y) {\hat \Pi}_+ (x,v),
\label{eq:p-p-}\\
{[{\hat \Pi}_+ (u,v),{\hat \Pi}_- (x,y)]} &=&
\delta (x,v) {\hat \Pi}_- (u,y) - \delta (u,y) {\hat \Pi}_- (x,v),
\label{eq:p+p-}\\
{[{\hat \Pi}_3(u,v),{\hat \Pi}_1(x,y)]} &=&
\delta(x,v){\hat \Pi}_3(u,y),
\label{eq:p3p+}\\
{[{\hat \Pi}_3(u,v),{\hat \Pi}_2(x,y)]} &=&
- \delta(u,y) {\hat \Pi}_3(x,v),
\label{eq:p3p-}\\
{[{\hat \Pi}_i(u,v),{\hat E}(x)]} &=&
- e \hbar \theta(u,v ; x) {\hat \Pi}_i(u,v), \ \ i =0,1,2,3.
\label{eq:pie}
\end{eqnarray}
where the c-number contribution in Eq.(\ref{eq:p0p3}) arises from the
ordering of the operators. In the above equations we have
introduced the notation
\begin{equation}
{\hat \Pi}_{\pm}(x,y):={\hat \Pi}_1(x,y) \pm
{\hat \Pi}_2(x,y)
\label{PIMM}
\end{equation}
and from now on we denote
${\hat \Pi}_i(\eta_x^y)$ simply by ${\hat \Pi}_i(x,y) $.
An heuristic application of the loop transform
shows that the operators ${\hat T}^0(\gamma)$
together with ${\hat \Pi}_0(\eta_x{}^y)$ acting
on the state of nothing $|0>$, create states with closed and open curves
respectively \cite{GambiniTrias,HugoRovelli}. This
can be formally stated as
\begin{eqnarray}
{\hat T}^0(\gamma_1)\dots{\hat T}^0(\gamma_q) |0\rangle&=&
|\gamma_1,\dots, \gamma_q \rangle, \label{eq:t0t0} \\
{\hat \Pi}_0(x_1,y_1)\dots{\hat \Pi}_0(x_m, y_m)
|0\rangle&=&|x_1, y_1,\dots, x_m, y_m \rangle.
\label{PARES}
\end{eqnarray}
Hence, the commutators of the operators with $\hat \Pi_0$
will provide the action of such operators upon generic states.
For simplicity we will consider
states like $|\gamma\rangle $ and
$|\dots, x, y, \dots \rangle$ separately.
The basic idea
is to apply the corresponding commutator to the state of nothing.
Suppose we
want to calculate $\hat{E}(x)|\gamma \rangle$. To this end let us
consider
\begin{eqnarray}
[\hat{E}(x), \hat{T}^0(\gamma)]|0\rangle
&=& e\hbar \oint_{\gamma}du\,\delta(x,u)
\hat{T}^0(\gamma)|0\rangle, \nonumber \\
\hat{E}(x)|\gamma\rangle &=& e\hbar \oint_{\gamma}du\delta(x,u)\,
|\gamma\rangle=e\hbar n|\gamma\rangle,
\end{eqnarray}
where $n$ is the winding number of the closed curve $\gamma$.
In a completely analogous way we obtain
\begin{equation}
\hat{E}(z)|x_1, y_1,\dots,x_m, y_m\rangle
= e\hbar \left( \sum_{k=1}^m \theta(x_k,y_k;z)\right)
|x_1, y_1,\dots,x_m, y_m\rangle.
\end{equation}
Let us observe that in spite of Eq.(\ref{eq:t0t0}), which would demand
$q$ labels $n_q$, we need only one label counting the total
winding number $n=n_1+\dots+n_q$. This is because
$\hat E(x) | n_1, n_2\rangle= e\hbar (n_1+n_2)| n_1, n_2\rangle
\approx \hat E(x) | n_1 + n_2\rangle$, which allow us to identify the
states
$| n_1, n_2\rangle $ and $| n_1 + n_2\rangle $ up to a phase. Thus,
we identify
$|\gamma_1,\dots, \gamma_q \rangle =| n \rangle$.
The remaining operators $\hat{\Pi}_i, \ i=1,2,3$, leave invariant
the state associated to the closed curve $\gamma$, while some examples of
their
action upon states defined by open curves are
\begin{eqnarray}
\hat{\Pi}_1(w, u)| x , y\rangle &=&
\delta(x,u) | w, y \rangle, \nonumber \\
\hat{\Pi}_2(w, u)|x, y\rangle &=&
- \delta(y,w) | x, u \rangle, \nonumber \\
{\hat \Pi}_3(u,v)|x,y\rangle &=& \delta(x,v) \delta(u,y) |0\rangle \,,
\nonumber \\
\hat{\Pi}_3( u, v)| x, y, w, z\rangle &=&
- \delta(x,v)\delta(z,u)
| w, y \rangle - \delta(y,u) \delta(w,v)
| x, z \rangle\nonumber\\
&& + \delta(u,y)\delta(x,v)| w, z \rangle +\delta(u,z)\delta(v,w)
| x, y \rangle
\end{eqnarray}
In this way, we consider our Hilbert space to be spanned by the set
of all
vectors
\begin{equation}\label{BDEF}
|n; x_1y_1, \dots, x_a , y_a \rangle \equiv |n\rangle \otimes
|x_1y_1, \dots, x_a , y_a \rangle, \ -\infty< n< +\infty, \
a=0,1,2 \dots,
\infty.
\end{equation}
These vectors satisfy the following orthogonality and closure
properties
\begin{eqnarray}\label{ORTH}
\langle n; x_1, y_1, \dots,x_a, y_a && |m ; u_1v_1, \dots, u_b ,
v_b \rangle =
\delta_{m,n} \delta_{a,b} \nonumber \\
&& \sum_{q_i,p_j} \epsilon_{q_1q_2 \dots q_a}\epsilon_{p_1p_2 \dots p_a}
\delta( x_{1} - u_{q_1}) \delta( y_{1} - v_{p_1})
\dots
\delta( x_{a} - u_{q_a}) \delta( y_{a} - v_{p_a}),
\end{eqnarray}
\begin{eqnarray}\label{CLOS}
\sum_{n=-\infty}^{+\infty}\ \sum_{a=0}^{+\infty} \int dx_1dy_1
\dots dx_ady_a \
{1\over {a !}^2}
| n; x_1, y_1, \dots, x_a, y_a \rangle \langle n; x_1, y_1,
\dots,x_a, y_a |
= 1,
\end{eqnarray}
which are a direct consequence of the basic algebra
(\ref{eq:p0p1})-(\ref{eq:p3p-}).
They also satisfy the following exchange properties
\begin{eqnarray}
|n;\dots,x_i,y_i,\dots,x_j,y_j, \dots x_ay_a\rangle &=&
|n;\dots,x_j,y_j,\dots,x_i,y_i, \dots, x_a,y_a\rangle,\label{IJIJ} \\
|n;\dots,x_i,y_i,\dots,x_j,y_j, \dots x_ay_a\rangle&=&
-|n;\dots,x_j,y_i,\dots,x_i,y_j, \dots x_ay_a\rangle,\label{XIJ} \\
|n;\dots,x_i,y_i,\dots,x_j,y_j, \dots x_ay_a\rangle &=&
-|n;\dots,x_i,y_j,\dots,x_j,y_i, \dots x_ay_a\rangle.\label{YIJ}
\end{eqnarray}
Equation (\ref{IJIJ}) is a direct consequence of the definition in Eq.
(\ref{PARES}) and the fact that
$[\hat{\Pi}_0(x_i,y_i),\hat{\Pi}_0(x_j,y_j)]=0$ , while Eqs.
(\ref{XIJ}) and (\ref{YIJ}) arise from the anticommuting property of the
fermion operators together with the abelian composition rule
\begin{equation}\label{PRODU}
U(u,v)U(x,y)=U(u,y)U(x,v).
\end{equation}
In the loop representation, a general wave
function $|\Psi\rangle$ can be written as
\begin{equation}
|\Psi\rangle= \sum_{a=0}^{\infty} \sum_{n=-\infty}^{+ \infty}\int
dx_1dy_1
\dots dx_ady_a {1\over a !} \Psi^a(n; x_1, y_1, \dots, x_a, y_a )
|n; x_1y_1,
\dots, x_a , y_a \rangle, \label{eq:wf}
\end{equation}
where
\begin{equation}
\langle n; x_1, y_1, \dots,x_a, y_a| \Psi \rangle = \Psi^a(n; x_1, y_1,
\dots, x_a, y_a )
\label{COMP}
\end{equation}
are the corresponding components. They inherit the exchange properties
(\ref{IJIJ}), (\ref{XIJ}) and (\ref{YIJ}) of the basis vectors and also
satisfy the following boundary conditions
\begin{eqnarray}
\Psi^a(n;\dots,x_k,y_k+mL,\dots) &=&
e^{im\pi} \Psi^a (n+m;\dots,x_k,y_k,\dots),\label{eq:yp} \\
\Psi^a(n;\dots,x_k+sL,y_k,\dots) &=&
e^{-is\pi} \Psi^a (n-s;\dots,x_k,y_k,\dots),\label{eq:xp}
\end{eqnarray}
which arise from the property
$\Pi_0(x,y+mL) = e^{im\pi} \psi_1^*(x) U(x,y)T^0 (m)
\psi_2(y)$, in virtue of the boundary conditions
(\ref{BC}). As a consequence of the symmetry properties
(\ref{eq:yp}) and (\ref{eq:xp}) we will assume from here on that
$0 \leq x_k, y_k < L$.
The scalar product is given by
\begin{eqnarray}\label{PRODI}
\langle \Phi | \Psi \rangle = \sum_{a=0}^{\infty}
\sum_{n=-\infty}^{+ \infty}
\sum_{i_a}\int dx_1dy_1 \dots dx_ady_a &&(\Phi^a(n; x_1, y_1,
\dots, x_a, y_a
))^*\nonumber \\
&&\times \Psi^a(n; x_1, y_{i_1}, \dots, x_a, y_{i_a} ),
\end{eqnarray}
where the functions $ \Phi^a(n; x_1, y_1, \dots, x_a, y_a )$ are the
components of the wave function $ | \Phi\rangle$.
In order to find the associated Schroedinger equation we need to
compute
the action of the operators appearing in the Hamiltonian density
(\ref{eq:dham}) upon the basis vectors. We
obtain
\begin{eqnarray}
\hat{E}^2(x) |n;x_1,y_1,\dots,x_a,y_a \rangle &=&
e^2 {\hbar^2} \left[ n+\sum_{k=1}^a \theta(x_k,y_k,x)\right]^2
|n;x_1,y_1,\dots,x_a,y_a\rangle, \label{eq:e2} \\
\hat{\Pi}_1(x,y) |n;x_1,y_1,\dots,x_a,y_a\rangle &=&
\hbar\sum_{k=1}^a \delta(x_k,y)
|n;\dots,x_{k-1},y_{k-1},x,y_k,x_{k+1},y_{k+1},\dots,
x_a,y_a\rangle, \nonumber\\
& & \label{eq:pi1xy} \\
\hat{\Pi}_2(x,y) |n;x_1,y_1,\dots,x_a,y_a\rangle &=&
-\hbar\sum_{k=1}^a \delta(y_k,x)
|n;\dots,x_{k-1},y_{k-1},x_k,y,x_{k+1},y_{k+1},\dots, x_a,
y_a\rangle. \nonumber\\
& & \label{eq:pi2xy}
\end{eqnarray}
Using the above actions, we have explicitly verified that the basis
vectors
(\ref{BDEF}) are
annihilated by the Gauss law constraint (\ref{eq:gauss}).
The Hamiltonian $\hat{H}$ is block-diagonal in the subspace of
fixed number
of pairs and fixed ${n}$. Thus we look for solutions of the Schroedinger
equation, ${\hat H} |\Psi\rangle = E |\Psi\rangle$, which are of the
form
\begin{equation}\label{WFSS}
|\Psi\rangle_{a,n}= \int dx_1dy_1 \dots dx_ady_a \Psi^a(n; x_1,
y_1, \dots,
x_a, y_a ) |n; x_1y_1, \dots, x_a , y_a \rangle,
\end{equation}
The action upon the components of the wave function is
\begin{eqnarray}
\langle n;x_1,y_1, &&\dots,x_a,y_a|{\hat H}|\Psi\rangle_{a,n} =
{ e^2 {\hbar^2} \over 2}
\left( \int_0^{L}dx\left[ n+\sum_{k=1}^a
\theta(x_k,y_k,x)\right]^2 \right) \Psi^a
(n;x_1,y_1,\dots,x_a,y_a)\nonumber
\\
& & + i \frac{\hbar}{2} \int_0^{L} dx
\sum_{k=1}^a \partial_x\delta(x_k,x)
\Psi^a(n;\dots,x_{k-1},y_{k-1},x,y_k,x_{k+1},y_{k+1},\dots, x_a,
y_a)\nonumber
\\
& & + i \frac{\hbar}{2}\int_0^{L} dx \sum_{k=1}^a \delta(y_k,x)
\partial_x
\Psi^a(n;\dots,x_{k-1},y_{k-1},x_k,x,x_{k+1},y_{k+1},\dots, x_a,
y_a )
\nonumber \\
& & -i \frac{\hbar}{2} \int_0^{L} dx
\sum_{k=1}^a \partial_x\delta(y_k,x)
\Psi^a(n;\dots,x_{k-1},y_{k-1},x_k,x,x_{k+1},y_{k+1},\dots, x_a,
y_a)\nonumber \\
& & -i \frac{\hbar}{2}\int_0^{L} dx \sum_{k=1}^a \delta(x_k,x)
\partial_x
\Psi^a(n;\dots,x_{k-1},y_{k-1},x,y_k,x_{k+1},y_{k+1},\dots, x_a,
y_a).
\nonumber\\
& & \label{eq:ham1}
\end{eqnarray}
Integrating by parts the term containing the derivative of the
delta function in (\ref{eq:ham1})
we obtain the final result
\begin{eqnarray}
\langle n;x_1,y_1, \dots,x_a,y_a|{\hat H}|\Psi\rangle_{a,n} &=&
{ e^2 {\hbar^2} \over 2}
\left( \int_0^{L}dx\left[ n+\sum_{k=1}^a
\theta(x_k,y_k,x)\right]^2 \right)
\Psi^a(n;x_1,y_1,\dots,x_a,y_a)\nonumber \\
& & -i\hbar \sum_{k=1}^a \left(\frac{\partial}{\partial x_k}
- \frac{\partial}{\partial y_k} \right)
\Psi^a(n;x_1,y_1,\dots,x_k,y_k,\dots, x_a,y_a).\nonumber \\
\label{eq:ham2}
\end{eqnarray}
\section {External field analysis}
As a first step in the quantization of the full system we consider the
quantization of the fermionic
fields in a background
electromagnetic field. According to Ref. \cite{IsoMurayama}, the
fermionic
field operators are given by
\begin{equation}
\psi_1(x,t)=\sum_{n} a_n \phi_n(x)e^{-{i\over \hbar}\epsilon_n t}, \quad
\psi_2(x,t)=\sum_{n} b_n^{\dagger} \phi_n(x)
e^{{i\over \hbar}\epsilon_n t},\label{psi12}
\end{equation}
where we have slightly changed the notation in the second equation
(\ref{psi12}).
The operators $a_n, b_n$ are standard fermionic annihilation operators
satisfying the non-zero anticommutators: \ $\{a_n, a_m^\dagger\}=
\delta_{mn}=\{b_m, b_n^\dagger\}$.
The basic wave functions $\phi_n$ , together with the
eigenvalues of the energy are given by
\begin{equation}
\phi_n(x)={1\over\sqrt L}e^{{i\over \hbar}\epsilon_nx -ie\int_0^x
A(z)dz},\quad
{1\over \hbar}\epsilon_n=
{2\pi\over L}\left(n+{1\over2} +{eL\over 2\pi}c\right)\equiv {2\pi
n\over L} + \theta,
\label{VFP}
\end{equation}
where
\begin{equation}
c={1\over L}\oint A(z)dz, \quad \theta={\pi\over L}+ ec .
\label{cteta}
\end{equation}
The energy eigenvalues are invariant under small gauge transformations
in such a way that the corresponding eigenfunctions transform covariantly
according to (\ref{GGT}). The same structure is kept in the case
of large gauge transformations. To see this, it is enough to
recall that
in the compact U(1) case under consideration
$c$ lies in the interval $[ 0, {2 \pi \over e L}]$. This means that
$c$ is
invariant under large gauge transformations.
In the scalar product
\begin{equation}
(\phi, \xi)=\int_0^L dz \ \left[\phi(z)\right]^*\xi(z),
\label{PE}
\end{equation}
the functions defined in (\ref{VFP}) are
orthonormal, i.e. \ $(\phi_m, \phi_n)=\delta_{mn}$.
It is convenient to introduce a bi-local Fourier transform of the loop
space operators $\hat\Pi(x,y)$ in
the following way
\begin{equation}
\Pi^{mn}=\int {dx dy\over L} e^{{i\over \hbar}\epsilon_m x}
e^{-{i\over \hbar}\epsilon_n y}\hat\Pi(x,y),
\label{FT}
\end{equation}
together with its inverse
\begin{equation}
\hat\Pi(x,y)={1\over L}\sum_{m,n} e^{-{i\over \hbar}\epsilon_m x}
e^{{i\over \hbar}\epsilon_n y}\Pi^{mn}.
\label{IFT}
\end{equation}
Also, we define the Fourier transform of the electric field
operator $E_b, \
$
as
\begin{equation}\label{EMODES}
E(x) = \sum_{b=-\infty }^{\infty} E_b \ { e}^{ -{2\pi i b \over L} x}, \quad E^\dagger_b
=E_{-b}, \quad b=0,\pm1, \pm 2, \dots
\end{equation}
which leads to the following inverse transformations
\begin{equation}\label{INVEMOD}
E_b= {1\over L} \int_0^L
dx E(x) \ { e}^{ {2\pi i b \over L} x} .
\end{equation}
The commutator algebra (3.7)-(3.13) can be directly rewritten in
terms of the
Fourier-transformed
operators $\Pi^{mn}$ and $E_b$. The result
is
\begin{eqnarray}
\left[ \Pi_0^{mn} , \ \Pi_1^{kl} \ \right]&=& - \delta^{ml} \Pi_0^{kn},
\label{eq:crf1}\\
\left[ \Pi_0^{mn} , \ \Pi_2^{kl} \ \right]&=& \ \ \delta^{kn}
\Pi_0^{ml}
, \label{eq:crf2}\\
\left[ \Pi_i^{mn} , \ \Pi_i^{kl} \ \right]&=& \ \delta^{kn}
\Pi_i^{ml} -
\delta^{ml} \Pi_i^{kn} , \quad i=1,2, \label{eq:crf3}\\
\left[ \Pi_+^{mn} , \ \Pi_-^{kl} \ \right]&=& \ \delta^{kn}
\Pi_-^{ml} -
\delta^{ml} \Pi_-^{kn} , \label{eq:crf4}\\
\left[ \Pi_3^{mn} , \ \Pi_1^{kl} \ \right]&=& \ \ \delta^{kn}
\Pi_3^{ml}, \label{eq:crf5}\\
\left[ \Pi_3^{mn} , \ \Pi_2^{kl} \ \right]&=& - \delta^{ml} \Pi_3^{kn},
\label{eq:crf6}\\
\left[ \Pi_0^{mn} , \ \Pi_3^{kl} \ \right]&=& \ \delta^{kn}
\Pi_1^{ml} -
\delta^{ml} \Pi_2^{kn}
-\delta^{kn} \delta^{ml}\,, \label{eq:crf7}
\end{eqnarray}
where $\Pi_3^{mn\,\dagger}=\Pi_0^{nm}$.
The commutators involving the electric modes are
\begin{eqnarray}\label{CEM}
\left[ E_a, \ E_b \right] &=&0, \quad \left[ c, \ E_b \right] =
{i\hbar\over L} \delta_{b0},
\nonumber \\
\left[ T^0(n), \ E_b \right] &=&- {en\hbar} \ T^0(n)
\delta_{b0}, \quad
\left[ \Pi_i^{kl}, \ E_0 \right] =0, \quad i= 0, 1,2,3, \nonumber \\
\left[ \Pi_i^{kl}, \ E_b \right]&=& {ie\hbar\over 2\pi b }
\left( \Pi_i^{k \ l-b} - \Pi_i^{k+ b \ l} \right),
b\neq 0.
\end{eqnarray}
\subsection{Vacuum state in a background electromagnetic field}
The Hamiltonian density that describes the external field
approximation is
given by the
the second term in the RHS of (\ref{eq:dham}). When written in the
momentum
space, the corresponding Hamiltonian is
\begin{equation}\label{mopmham}
H_{D}= \sum_m \epsilon_m \Pi_-^{mm}.
\end{equation}
The calculation of the commutators
\begin{equation}
[ \Pi_0^{kl}, \ H_{D}]=-(\epsilon_k + \epsilon_l) \ \Pi_0^{kl}, \ \
[ \Pi_3^{kl}, \ H_{D}]=+(\epsilon_k + \epsilon_l) \ \Pi_3^{kl},
\end{equation}
implies that $ \Pi_0^{kl}$ is an energy raising operator, i.e.
$\Pi_0^{kl} \ |
E\rangle \sim | E +
\epsilon_k + \epsilon_l \rangle$, for any Hamiltonian eigenstate $ | E
\rangle$. For analogous reasons, $ \Pi_3^{kl}$ is the
corresponding lowering
operator. The charge operator $Q$ commutes with $ \Pi_0^{kl}$,
showing that
$ \Pi_0^{kl}$ creates a zero charge pair of
particles having the corresponding energies $\epsilon_k$ and
$\epsilon_l$.
Thus we can identify
each superindex of $ \Pi_0^{kl}$ with a definite and opposite
charge label.
In this way we have that $ \Pi_0^{kl} \Pi_0^{kr} | E\rangle \sim | E +
2\epsilon_k + \epsilon_l + \epsilon_r \rangle $.
Since in one spatial dimension the momentum is proportional to the
energy, we
conclude that the state $| E + 2\epsilon_k + \epsilon_l +
\epsilon_r \rangle$
contains two fermions having the same quantum numbers and
therefore must be
zero according to the Pauli principle. Since the eigenstates of the
Hamiltonian
provide a basis for the Hilbert space, we must have the operator
identity
\begin{equation}\label{PAPRI}
\Pi_0^{kl} \Pi_0^{kr} =0
\end{equation}
and analogously, when the repeated indices are those in the right.
The vacuum state $|0\rangle_D$ corresponds to a filled Dirac sea
with zero
charge
which can be defined as
\begin{equation}
| 0 \rangle_D=\prod_{k=-\infty}^{N-1} \Pi_0^{kk}|0\rangle,\quad
{}_D\langle 0|= \langle 0|\prod_{k=-\infty}^{N-1}\Pi_3^{kk},
\label{VAC}
\end{equation}
This means that all energy levels below $\epsilon_N$ are completely
filled. Provided that $-(N+{1\over 2}) \leq {eL\over 2\pi}c \leq
-(N-{1\over
2})$,
we have that $ \epsilon_{N}\geq 0 $ and the above
construction includes an infinite set of negative-energy states
$\epsilon_{N-1}\leq 0 $.
{}From now on we will use the convention that all indices ranging from
$-\infty$ to $N-1$ will
be denoted by capital letters from the beginning of the alphabet
$(A, B, C,
\dots)$, while those
going from $N$ to $+\infty$ will be denoted by lower case greek
letters from
the beginning of the
alphabet $( \alpha, \beta, \gamma, \dots)$.
The commutation relations (\ref{eq:crf1}), (\ref{eq:crf2})
imply that
\begin{eqnarray}
\Pi_1^{kk} \left(\Pi_0^{mm}\right)|0\rangle &=&
\delta^{mk} \left(\Pi_0^{mm}\right)|0\rangle \\
\Pi_2^{kk} \left(\Pi_0^{mm}\right)|0\rangle &=&
-\delta^{mk} \left(\Pi_0^{mm}\right)|0\rangle,
\end{eqnarray}
which allow us to prove that the vacuum satisfies
\begin{equation}\label{VAC1}
\Pi_1^{AA}| 0 \rangle_D= |0 \rangle_D\,,\quad
\Pi_2^{AA}|0\rangle_D = -|0\rangle_D, \quad
\label{vac1}
\end{equation}
\begin{equation}\label{VAC2}
\Pi_1^{\alpha\alpha}|0\rangle_D=0= \Pi_2^{\alpha\alpha} |0\rangle_D.\,
\label{vac2}
\end{equation}
Next we discuss some additional properties of the vacuum. To begin
with let us
calculate
$\Pi_+^{AB}|0\rangle_D$.
We have already shown that
$\Pi_+^{AA}|0\rangle_D = 0\,, \forall A$.
Now, for the case $A\ne B$, we have
\begin{eqnarray}
\Pi_+^{AB}|0\rangle_D &=& \Pi_+^{AB} \prod_{C=-\infty}^{N-1}
\Pi_0^{CC}|0\rangle
= \left[\Pi_+^{AB}\,, \prod_{C=-\infty}^{N-1} \Pi_0^{CC} \right]
|0\rangle
\nonumber \\
&=& \sum_{D=-\infty}^{N-1} \prod_{s=-\infty}^{D-1} \Pi_0^{CC}
\left[\Pi_+^{AB}\,, \Pi_0^{DD}\right] \prod_{G=D+1}^{N-1}
\Pi_0^{GG}|0\rangle.
=0
\end{eqnarray}
The last equality holds by virtue of
the commutation relation
\begin{equation}
[ \Pi_+^{AB}, \Pi_0^{CD}]= \delta^{BC}\Pi_0^{AD}-\delta^{AD}\Pi_0^{BC},
\end{equation}
together with the fact that for $A\ne B$ one will always find
products of the type $\Pi_0^{AA}\Pi_0^{AB}$, $\Pi_0^{AA}\Pi_0^{BA}$,
which are identically zero, according to the property
(\ref{PAPRI}). Thus, we
have proved
that $\Pi_+^{AB}|0\rangle_D =0$. Analogously one can show that
$\Pi_+^{\alpha\beta}|0\rangle_D=0 $. Nevertheless, both states
$\Pi_+^{A\beta}|0\rangle_D $
together with $\Pi_+^{\alpha B}|0\rangle_D $ are different from
zero and are in
fact orthogonal
to $|0\rangle_D$. Thus, we have
\begin{equation}
{}_D\langle 0|\Pi_+(x,y)|0\rangle_D =0.\label{pi+vac}
\end{equation}
Next, we repeat the calculation for $\Pi_-(x,y)$. Again, we start
from the "momentum-space" formulation. The properties (\ref{VAC1}) ,
(\ref{VAC2})
imply
\begin{equation}
\Pi_-^{AA}| 0 \rangle_D= 2|0\rangle_D, \quad
\qquad
\Pi_-^{\alpha\alpha}|0\rangle_D=0,
\end{equation}
for the diagonal terms.When $ A\neq B$, following
analogous steps to the previous case, we find
\begin{equation}
\Pi_-^{AB}|0\rangle_D = \sum_{C=-\infty}^{N-1} \prod_{D=-\infty}^{C-1}
\Pi_0^{DD} \left[\Pi_-^{AB}\,, \Pi_0^{CC}\right] \prod_{G=C+1}^{N-1}
\Pi_0^{GG}|0\rangle.\label{pi-vacmom}
\end{equation}
The corresponding commutator here is
\begin{equation}
[ \Pi_-^{AB}, \Pi_0^{CD}]= \delta^{BC}\Pi_0^{AD}+\delta^{AD}\Pi_0^{BC}.
\end{equation}
Once more, we obtain $\Pi_-^{AB}|0\rangle_D=0, A \neq B $ because of the
presence of products of $\Pi_0 's$ having a repeated index. In
analogous way
one obtains the remaining actions leading to
\begin{equation}
\Pi_-^{AB}| 0 \rangle_D= 2 \delta^{AB}|0\rangle_D, \quad
\Pi_-^{\alpha\beta}| 0 \rangle_D=0, \quad \Pi_-^{A\beta}| 0
\rangle_D\neq 0,
\quad \Pi_-^{\alpha B}| 0 \rangle_D \neq 0.
\label{PI-VAC}
\end{equation}
Again, the non-zero vectors resulting from the last two actions in
the above
equation are orthogonal to the Dirac vacuum. In this way,
going back to the coordinate representation we obtain
\begin{equation}
{}_D\langle0|\Pi_-(x,y)| 0 \rangle_D=2 e^{-i\theta (x-y)}
{1 \over L} \sum_{A=-\infty}^{N-1} e^{-{2\pi iA\over
L}(x-y)}{}_D\langle 0
| 0 \rangle_D ={2\over L} e^{-i\theta (x-y)} F(x,y){}_D\langle 0| 0
\rangle_D.
\label{pi-vac}
\end{equation}
In the above equation we have introduced the
function $F$ defined as
\begin{equation}
F(x,y)=\sum_{A =-\infty}^{N-1} e^{-{2\pi iA\over L}(x-y)}=
{e^{-{2\pi i (N-1) \over L}(x-y)}
\over (1-e^{{ 2\pi i\over L}(x-y)})}, \label{FN}
\end{equation}
where the summation can be calculated because it is a geometric series.
\subsection { The vacuum energy}
Let us recall that the Hamiltonian is given by
\begin{equation}
H_D= {i\hbar \over 2}\int _0^Ldx \lim_{y\rightarrow x}
\left[ (\partial_y - \partial_x)\Pi_-(y,x) \right].
\label{HR1}
\end{equation}
According to the relations (\ref{PI-VAC}) and (\ref{FN}) the action of
$\Pi_-(y,x)$ on the Dirac vacuum can be written as
\begin{equation}
\Pi_-(x,y)| 0 \rangle_D={2\over L} e^{-i\theta (x-y)} F(x,y)| 0
\rangle_D+ | x,
y; -\rangle,
\end{equation}
where the state $ | x, y; -\rangle$ does not contributes in the limit of
Eq.(\ref{HR1}).
The corresponding vacuum energy in the external field is given by
\begin{equation}
E_D(\epsilon)={\langle 0|\hat H_D|0\rangle_D \over
\langle 0| 0\rangle_D}.
\end{equation}
The
function $E_D(\epsilon)$
will have an expansion in powers of $\epsilon$ of the form
$E_D(\epsilon)= {a\over \epsilon^2}+b\epsilon^0 + O(\epsilon)$. We
will take
$b$ as the regularized expression for the vacuum energy $E_D$.
The resulting term is
\begin{equation}
E_D(\epsilon) =2i\hbar \partial_{\epsilon}\left[
e^{-i\theta\epsilon}{e^{-{2 \pi i(N-1) \over L }\epsilon} \over
1- e^{{2 \pi i \over L }\epsilon}} \right].
\end{equation}
The finite part of the above equation, when $\epsilon \rightarrow 0$,
is
\begin{equation}\label{VACEN}
E_D=\hbar \left[ {2 \pi N^2 \over L }-{2 \pi N \over L }+ 2N \theta -
\theta + {\pi \over 3L } + {\theta^2 L \over 2 \pi}\right],
\end{equation}
which coincides with the result of Ref.\cite{IsoMurayama}. In the
sequel we
choose $N=0$, in such a way that $ -\pi \leq ecL
\leq +
\pi$, which reinforces the fact that $ ec$ is a compact degree of
freedom.
\section{ The Axial Anomaly}
The vector and axial currents are defined as:
\begin{eqnarray}
J_{V}{}^{\mu}(x)&:=& \lim_{y\rightarrow x}
e\overline{\psi}(y)\gamma^{\mu} U(y,x) \psi(x), \\
J_A{}^{\mu}(x)&:=& \lim_{y\rightarrow x}
e\overline{\psi}(y)\gamma^{\mu}\gamma_5 U(y,x) \psi(x).
\end{eqnarray}
Thus, in terms of loop variables, their components become
\begin{eqnarray}
J_V{}^0 &=& \lim_{y\rightarrow x} e \Pi_+(y,x)
\;,\;\;\;\;
J_V{}^1 = \lim_{y\rightarrow x} e \Pi_-(y,x),
\\
J_A{}^0 &=& \lim_{y\rightarrow x} e \Pi_-(y,x)
\;,\;\;\;\;
J_A{}^1 = \lim_{y\rightarrow x} e \Pi_+(y,x).
\end{eqnarray}
Next we calculate the relevant commutators in order to
determine $[\hat Q_{A,V},\hat H]$, where
\begin{equation}
\hat Q_V:=\int_0^L dx \lim_{y\rightarrow x}
e \hat\Pi_+(y,x)\;\;,\;\;\;\;\;\;
\hat Q_A:=\int_0^L dx \lim_{y\rightarrow x}
e \hat\Pi_-(y,x)\,
\end{equation}
and $\hat H$ is the full Hamiltonian.
To do so we start by looking at the following commutators
\begin{eqnarray}
[\hat\Pi_{\pm}(y,x), \hat H] =
-\frac{e\hbar}{2} \int_x^y dw (\hat
E(w)\hat\Pi_{\pm}(y,x)+\hat\Pi_{\pm}(y,x)
\hat E(w))
-i\hbar (\partial_x+\partial_y)\hat\Pi_{\mp}(y,x), \label{eq:cp1h}
\end{eqnarray}
In obtaining the above commutators, we have used the property
\begin{equation}
\delta(x,L)\hat\Pi_{\pm}(y,L) -\delta(x,0)\hat\Pi_{\pm}(y,0) =0=
\delta(y,L)\hat\Pi_{\pm}(L,x) -\delta(y,0)\hat\Pi_{\pm}(0,x),
\end{equation}
according to Eq. (\ref{DELTA}).
Let us consider now $\int_0^L dx\lim_{y\rightarrow x}
[\hat\Pi_1(y,x),\hat H]$.
One can readily see that the integral
\begin{equation}
\int_0^L dx
\lim_{y\rightarrow x}(\partial_x + \partial_y) \hat\Pi_{\pm}(y,x)
= \lim_{\epsilon\rightarrow 0} \int_0^L dx
\partial_x\hat\Pi_{\pm}(x+\epsilon,x)
= \hat\Pi_{\pm}(L,L) - \hat\Pi_{\pm}(0,0) =0\,.
\end{equation}
The only term left is
\begin{equation}
\int_0^Ldx \lim_{y\rightarrow x} [\hat\Pi_{\pm}(y,x),\hat H]
= -\frac{e\hbar}{2}\lim_{\epsilon\rightarrow 0}
\int_0^L dx \int_x^{x+\epsilon} dw
[\hat
E(w)\hat\Pi_{\pm}(x+\epsilon,x)+\hat\Pi_{\pm}(x+\epsilon,x)\hat
E(w)]\, .
\label{eq:lcp1h}
\end{equation}
Thus, the above limit would yield zero provided $\hat \Pi_{\pm}(x,x)$
is finite. Nevertheless we expect $\hat\Pi_{\pm}(x,x)$ to be divergent
so that the limit must be carefully calculated.
In order to obtain the divergence of the vector current we
start from
\begin{equation}
\partial_0 \hat J_V{}^0 = {i \over \hbar}[\hat H,\hat J_V{}^0] \,.
\end{equation}
The calculation of the commutator leads to
\begin{equation}
\partial_0 \hat J_V{}^0 =
\lim_{y\rightarrow x}{ie^2\over 2}\int_x^y dw
(\hat E(w)\hat\Pi_+(y,x)+\hat\Pi_+ (y,x)\hat E(w))-
\lim_{y\rightarrow x} (\partial_x+\partial_y)
e\hat\Pi_-(y,x).
\label{eq:d0j0}
\end{equation}
After taking the limit,
the second term in the r.h.s. in (\ref{eq:d0j0}) can be
identified as $-\partial_1 \hat J_V{}^1$. This can be shown by
going to the
momentum representation: on one hand we have that
\begin{equation}
\partial_1 \hat J_V{}^1=\partial_x \lim_{y\rightarrow x} e
\hat \Pi_-(y,x)={e\over \hbar L}\sum_{m,n}i(\epsilon_m-\epsilon_n)
e^{{i\over \hbar}(\epsilon_m-\epsilon_n)x}\Pi_-^{mn}. \label{d1j1}
\end{equation}
On the other hand, the limit appearing in Eq.(\ref{eq:d0j0})
is calculated as
\begin{equation}
\lim_{y\rightarrow x} (\partial_x+\partial_y)
e\hat\Pi_-(y,x)=\lim_{y\rightarrow x}{e\over \hbar L}\sum_{m,n}i
(\epsilon_m-\epsilon_n)
e^{{i\over \hbar}\epsilon_m x}e^{-{i\over \hbar} \epsilon_n y
}\Pi_-^{mn},
\end{equation}
which reduces exactly to Eq.(\ref{d1j1}) after taking the limit
$y\rightarrow x$.
Hence the final result in the calculation of Eq.(\ref{eq:d0j0}) is
\begin{equation}
\partial_{\mu}\hat J_V{}^{\mu}=
\lim_{y\rightarrow x}{ie^2\over 2}\int_x^y dw
(\hat E(w)\hat\Pi_+(y,x)+\hat\Pi_+(y,x)\hat E(w)) \;.
\label{vectcons}
\end{equation}
In analogous way we obtain
\begin{equation}
\partial_{\mu}\hat J_A{}^{\mu}=
\lim_{y\rightarrow x}{ie^2\over 2}\int_x^y dw
(\hat E(w)\hat\Pi_-(y,x)+\hat\Pi_-(y,x)\hat E(w)) \;.
\label{avectcons}
\end{equation}
Let us remark that Eqs. (\ref{vectcons}), (\ref{avectcons}) are full operator relations which
describe the exact non-perturbative behavior of the divergences of the corresponding currents.
In order to recover the standard form of the axial anomaly, we assume
that only the fermions are quantized, and calculate the vacuum
expectation
values of the above divergences of the currents regarding $E(x)$ as an
external field, together with the vacuum given by Eqs.(\ref{vac1},
\ref{vac2}). Using
Eq.(\ref{pi+vac}) and Eq.(\ref{pi-vac}) we obtain
\begin{equation}
{}_D\langle 0|\partial_{\mu}\hat J_V{}^{\mu} |0 \rangle_D =0,
\end{equation}
\begin{eqnarray}
\langle 0|\partial_{\mu}\hat J_A{}^{\mu} |0 \rangle_D &=&
\lim_{y\rightarrow x}ie^2
E(x)(y-x){}_D\langle 0|\Pi_-(y,x)|0 \rangle_D, \nonumber \\
{\langle 0|\partial_{\mu}\hat J_A{}^{\mu} |0 \rangle_D \over \langle 0|0
\rangle_D}&=&
{2\over L} ie^2 E(x) \lim_{y\rightarrow x}\left(
{(y-x) e^{{2\pi i \over L}(y-x)}
\over (1-e^{{2\pi i\over L}(y-x)})}
\right)=-{e^2 \over \pi} E(x).
\end{eqnarray}
\section{ Anomalous commutators}
It is a general property of quantum field theory that, if a current is
conserved, the equal time commutator of its spatial and temporal
components cannot vanish \cite{SCHCOM}. However, by a naive use of the
canonical
commutation relations one finds that this commutator is equal to
zero. To obtain the correct result it is necessary to introduce a
regularization. The non-locality of our formalism provides a natural
regularization and the algebra (\ref{eq:p0p1}
-\ref{eq:p3p-}) produces directly the anomalous commutators. For
example, we have
\begin{eqnarray}
\left[ J^0_V (u), J^1_V(x)\right] &=& \left[ \lim_{v\to u} e\hat
\Pi_+(v,u), \lim_{y\to x} e\hat\Pi_-(y,x) \right] \\
&=& e^2 \lim_{v\to u} \lim_{y\to x} \left( \delta(y-u) \hat \Pi_{-}(v,x)
- \delta(v-x) \hat \Pi_-(y,u) \right).
\end{eqnarray}
Taking the limits $v=u+\epsilon$ and $y=x+\epsilon$, with $\epsilon\to
0$, the vacuum expectation value of the commutator has the form
\begin{equation}
_D \langle 0| [ J_V^0 (u), J_V^1(x)] |0\rangle_D = {ie^2 \over \pi}
\delta^\prime (x-u).
\end{equation}
Also, from the relations (\ref {eq:pie}), it follows
that the
commutator of the currents and the
electric field is different of zero, which agree with the Gauss law
(\ref{eq:gauss}). In this case we have
\begin{equation}
\left[ J_V^0 (x), E(z) \right] = \left [ \lim_{y\to x} e \hat
\Pi_+(x,y) , \hat E (z) \right] = -e^2 \lim_{y\to x} \theta(x,y;z) \hat
\Pi_+(x,y),
\end{equation}
\begin{equation}
\left[ J_V^1 (x), E(z) \right] = \left [ \lim_{y\to x} e \hat
\Pi_-(x,y) , \hat E (z) \right] = -e^2 \lim_{y\to x} \theta(x,y;z) \hat
\Pi_-(x,y),
\end{equation}
{}From the above expressions we obtain the vacuum expectation value
of these commutators
\begin{equation}
_D \langle 0| [ J_V^0 (x), E(z) ]|0\rangle_D = 0,
\end{equation}
\begin{equation}
_D \langle 0| [ J_V^1 (x), E(z) ] |0\rangle_D = {ie^2\hbar \over \pi}
\delta (x-z).
\end{equation}
\section{ The zero mode}
The simplest state with energy above the Dirac sea is the one associated
to the only degree of freedom of the electromagnetic field
which cannot be gauged away: the zero mode. Such a state can be
neatly described in the present formalism as follows.
Recalling that $c=\frac{1}{L}\int_{0}^{L} dz A(z)$,
and $-\frac{\pi}{eL}\leq c\leq \frac{\pi}{eL}$ it is useful to
consider the transformation
\begin{equation}
|c,x_1,y_1,\dots,x_a,y_a\rangle =
\sum_n {\rm e}^{-iec\left[nL +\sum_k(y_k-x_k)\right]}
|n, x_1,y_1,\dots,x_a,y_a\rangle\, .
\end{equation}
Accordingly, the action of the relevant operators becomes
\begin{eqnarray}
T^0 [\gamma] |c,x_1,y_1,\dots,x_a,y_a\rangle &=&
{\rm e}^{ie{n_{\gamma}}cL} |c,x_1,y_1,\dots,x_a,y_a\rangle \\
\Pi_0 (x,y) |c,x_1,y_1,\dots,x_a,y_a\rangle &=&
{\rm e}^{iec(y-x)} |c, x, y, x_1,y_1,\dots,x_a,y_a\rangle \\
\Pi_1(x,y) |c,x_1,y_1,\dots,x_a,y_a\rangle &=&
{\rm e}^{iec(y-x)} \sum_i \delta (y-x_i)
|c,x_1,y_1,\dots,x,y_i,\dots,x_a,y_a\rangle \\
\Pi_2(x,y) |c,x_1,y_1,\dots,x_a,y_a\rangle &=&
- {\rm e}^{iec(y-x)} \sum_i \delta (x-y_i)
|c,x_1,y_1,\dots,x_i,y,\dots,x_a,y_a\rangle \\
E(z) |c,x_1,y_1,\dots,x_a,y_a\rangle &=&
\frac{i\hbar}{L} \frac{\partial{}}{\partial c}
|c,x_1,y_1,\dots,x_a,y_a\rangle \nonumber\\
&+& e\hbar \left[ \sum_k \theta(x_k,y_k,z)
- \sum_k \frac{(y_k-x_k)}{L}\right] |c,x_1,y_1,\dots,x_a,y_a\rangle
\label{EREP}
\end{eqnarray}
Translation of the boundary conditions in this representation yields
\begin{eqnarray}
|c,\dots,x_i,y_i+pL\dots\rangle
&=& {\rm e}^{ip\pi}\sum_n {\rm
e}^{iec\left[(n+p)L+\sum_k(y_k-x_k)\right]}
|n+p,\dots,x_k,y_k\dots\rangle \nonumber \\
&=& {\rm e}^{ip\pi}|c,\dots,x_i,y_i\dots\rangle.\,
\end{eqnarray}
The Hamiltonian can be rewritten now as
\begin{eqnarray}
&H& \Psi(c,x_1,y_1,\dots,x_a,y_a) = \nonumber \\
& & \frac{e^2\hbar^2}{2}
\left\{ \int_0^L dx
\left[ \frac{i}{eL} \frac{\partial~}{\partial c}
+ \sum_k \left(\theta(x_k,y_k;x)-\frac{(y_k-x_k)}{L}\right)
\right]^2
\right\} \Psi(c,x_1,y_1,\dots,x_a,y_a) \nonumber\\
&-& i\hbar \lim_{\epsilon\rightarrow 0}
\frac{\partial~}{\partial\epsilon}
\sum_i \left\{ {\rm e}^{iec\epsilon}
\left[\Psi(c,\dots,x_i,y_i-\epsilon,\dots)
+ \Psi(c,\dots,x_i+\epsilon,y_i,\dots)\right]
\right\}\,.
\label{zmodeH}
\end{eqnarray}
Now it is convenient to disentangle the term
\begin{equation}
X= \int_0^L dx
\left[ \frac{i}{eL} \frac{\partial~}{\partial c}
+ \sum_k \left(\theta(x_k,y_k;x)-\frac{(y_k-x_k)}{L}\right)
\right]^2 \equiv X_1 + X_2,
\label{SQR}
\end{equation}
in Eq. (\ref{zmodeH}), where we recall that
\begin{equation}
{\bar \theta}(x_k,y_k;x)\equiv \theta(x_k,y_k;x)- \frac{y_k-x_k}{L}
=\frac{1}{L} \sum_{p\ne 0} \frac{1}{ip}
\left( e^{ip y_k} - e^{ip x_k}
\right) e^{-ipx}\, ,
\end{equation}
according to Eq.(\ref{THETA}). Since the summation in the
above equation
is over $ p\neq 0$, the integration of the crossed term in the
square of Eq.
(\ref{SQR}) is zero. The integration of the first square term is
immediate,
leading to
\begin{equation}
X_1= -\frac{1}{e^2 L} \frac{\partial^2}{\partial c ^2}.
\end{equation}
The integration of the second square term reduces to
\begin{eqnarray}
X_2&=& \int_0^L dx \sum_{k,l} {\bar \theta}(x_k,y_k;x){\bar
\theta}(x_l,y_l;x)\nonumber \\
&= &L \sum_{k,l} \left( V(y_k-y_l) + V(x_k-x_l) - V(y_k-x_l) -
V(x_k-y_l)
\right),
\end{eqnarray}
where
\begin{equation}
V(z) = \sum_{n\neq 0} \frac{1}{ 4 \pi^2 n^2} e^{ \frac{2 \pi i n
}{L} z}.
\label{POTEN}
\end{equation}
The main conclusion of the above calculation is the fact the the
solutions of
the whole problem
satisfy the separability condition
\begin{equation}
\Psi(c, \dots x_{ I },y_{ I}, \dots) = \Phi(c) \
\Theta( \dots, x_{I},y_{I},\dots)\,.
\end{equation}
The external field analysis performed in Section IV corresponds to
neglecting the quadratic (electric field) term in (\ref{zmodeH})
with respect
to the fermionic contribution involving the $\epsilon\rightarrow
0$ limit.
The lowest energy state should be associated to the Dirac sea and the
corresponding
energy is $E(c)= -\frac{\hbar \pi}{6L} + \frac{\hbar L}{2\pi} e^2 c^2$,
$-\frac{\pi}{eL}\leq c\leq \frac{\pi}{eL}$.
The next step is to consider the zero-mode sector of the theory, which
consists in
taking
$E(z) \rightarrow E_0= {1\over L} \int dz E(z) $ in the Hamiltonian
(\ref{zmodeH}). It is important to remark that the separability of Eq.
(\ref{zmodeH}) in
the form
$\Psi(c,...x_I,y_I...)=\Phi(c)\Theta(...x_I,y_I...)$,
implies that, in the compact case, the zero mode
spectrum is a part of the exact spectrum of the full
Schwinger model,
in complete analogy with the noncompact case.
From the
general expression (\ref{EREP}), and recalling Eq. (\ref{THETA}) we
readily
verify that
\begin{equation}
E_0= {i\hbar\over L}{\partial \over \partial c},
\label{E0}
\end{equation}
in this representation. It is a general property that $E_0 |
0\rangle_D=0$,
which means that
$\Theta_D= \langle c,\dots,x_I,y_I \dots | 0\rangle_D$ is independent
of the
coordinate $c$.
Now, the contribution of the
zero mode to the exact solution of the problem, corresponds to the
choice
\begin{equation}
\Psi(c, \dots x_{ I },y_{ I}, \dots) = \Phi(c) \
\Theta_{D}( \dots, x_{I},y_{I},\dots)\,.
\end{equation}
Recalling that the fermionic part of the Hamiltonian acts like a
derivative,
the zero mode contribution $\Phi(c)$ fulfills the equation
\begin{equation}
\left[-\frac{\hbar^2}{2L} \frac{\partial^2~}{\partial c^2}
+ \frac{\hbar L}{2\pi} {e^2}{c^2} \right] \Phi(c) = E_1 \Phi(c) \,
\label{zeromode}
\end{equation}
where the constant $-\frac{\hbar\pi}{6L}$ coming from the Dirac sea energy
$E(c)$ was reabsorbed in $E_1$.
The inner product and the boundary conditions
for the zero mode contributions are:
\begin{eqnarray}
\langle\phi|\psi\rangle &=& \int_{-\frac{\pi}{eL}}^{\frac{\pi}{eL}} dc\
\phi^{\ast}(c)\psi(c), \nonumber \\
\phi(-\frac{\pi}{eL})&=&
\phi(\frac{\pi}{eL}), \quad
\phi'(-\frac{\pi}{eL})=
\phi'(\frac{\pi}{eL}), \quad
-\frac{\pi}{eL} \leq c \leq \frac{\pi}{eL},
\end{eqnarray}
in such a way that the electric field and the
Hamiltonian are hermitian. Here the prime means the derivative of the
corresponding
function with respect to the argument.
The solution to the eigenvalue equation
(\ref{zeromode}) can be expressed in terms of cylindrical parabolic
functions
\cite{Abramowitz} upon the change of variables
\begin{eqnarray}
&&x:=\sqrt{ \frac{2eL}{\sqrt{\pi\hbar}} } c, \quad
E_1=-\sqrt{\frac{\hbar^3}{\pi}}e a,
\nonumber \\
&&\psi(c)\rightarrow
y(x), \quad
\quad - x_M \leq x\leq x_M, \quad
x_M\equiv \sqrt{ \frac{2}{eL}\sqrt{\frac{\pi^3}{\hbar}} },
\label{CHANGVAR}
\end{eqnarray}
where $x$ and $a$ are dimensionless quantities. This yields the equation
\begin{equation}
y''-\left(\frac{1}{4} x^2 + a\right)y =0.
\label{ZMODEQ}
\end{equation}
The periodic boundary conditions on $y(x)$ are
\begin{equation}\label{PBCG}
y(x_M)=y(-x_M), \quad y'(x_M)=y'(-x_M).
\end{equation}
The general solution of Eq.(\ref{ZMODEQ}) is
\begin{equation}
y(x)=A \ {\rm e}^{-\frac{x^2}{4}}\,
{\rm M}\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2}, \frac{x^2}{2}\right)
+ B \ x \ {\rm e}^{-\frac{x^2}{4}}\,
{\rm M}\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},
\frac{x^2}{2}\right) \,,
\end{equation}
where ${\rm M}(A,B,z)$ is the confluent
hypergeometric function. It will be convenient to introduce the new
label
\begin{equation}\label{NEWVAR}
{\tilde l} := {2 \pi^{3/2}
\over {x_M}^2}= eL \hbar^{1/2}:= { l} \pi^{3/2}.
\end{equation}
Now we separately discuss the even and odd solutions:
(i) Even solutions: in this case the periodic boundary conditions
(\ref{PBCG})
are
automatic on $y$ and
imply
\begin{equation}
y'(x_M)=y'(-x_M)=0.
\end{equation}
This eigenvalue condition for Eq. (\ref{ZMODEQ})
will determine the
energy $E_1(a)$ as a function of $l$ . The above condition can
be written as
\begin{equation}
{\rm M}\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2},
\frac{{x_M}^2}{2}\right)=
(2a +1) {\rm M}\left(\frac{a}{2}+\frac{5}{4},\frac{3}{2},
\frac{{x_M}^2}{2}\right)
\end{equation}
and it defines the function $a = a (l)$. This function can only be
determined
numerically for arbitrary $l$ and it is shown in Fig. 1.
The novel properties are:
\begin{enumerate}
\item $\lim_{L\rightarrow 0} a(l)= a(0) = -\frac{1}{2} -2n, \quad
n=0,1,2,3\dots $ which
reproduces the even subset of the standard
U(1) non compact case (i.e. $-\infty\leq c\leq\infty)$. We will label the
even functions
$a(l)$ by the integer $2n$: $a_{2n}(l)$. In Fig. 1. we use the
notation $ En$,
i.e. $E0, E1, E2, \dots, $ for the corresponding solutions.
\item From the numerical calculation we find that $a_0$ is
monotonously
increasing
and also that $\lim_{L\rightarrow \infty} a_0(l)= a_0(\infty) =
0$. This
last property
is consistent with the fact that if $a$ remains finite when $l\rightarrow
\infty$, then $a=0$.
\item The behavior of $a_{2n}( l)$ for ${
l}\rightarrow0$ is
\begin{equation}
a_{2n}( l)=-{1\over 2} - 2n + { 2 {\rm e}^{ -{1\over
l}} \over
{ l}{}^{(2n + {1\over 2})} \ n! \ \Gamma(n+{1\over2})}.
\end{equation}
{}From the above equation we conclude that $a_{2n}'(
l)|_{ l =
0}=0$ and also that
$a_{2n}( l)$ is an increasing function near $ l =0$.
\item The behavior for negative $a_{2n}$, with large absolute value (
$|a_{2n}|>>1$), is given by
\begin{equation}
a_{2n}( l)=-{\pi^2\over 2} n^2 l, \quad n= 1, 2, 3 \dots,
\end{equation}
which can be readily observed in Fig.1.
\end{enumerate}
(ii) Odd solutions: in this case the periodic boundary conditions
(\ref{PBCG})
are
automatic on $y'$ and
imply
\begin{equation}
y(x_M)=y(-x_M)=0.
\end{equation}
This eigenvalue condition is
\begin{equation}
{\rm M}\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},
\frac{{x_M}^2}{2}\right)=0
\end{equation}
and it defines the odd sector of the function $a = a (l)$. Again, this
function can only be determined numerically for arbitrary $l$ and
it is shown
in Fig. 1. The properties are:
\begin{enumerate}
\item $\lim_{L\rightarrow 0} a(l)= a(0) = -\frac{1}{2} -(2n-1), \quad
n=1,2,3\dots $, which
reproduces the odd subset of the standard
U(1) non compact case (i.e. $-\infty\leq c\leq\infty)$. We will label the
functions
$a(l)$ by the integer $2n-1$: $a_{2n-1}(l)$. In Fig.1. we denote by
$On$, i.e.
$O1, O2, \dots, $
the corresponding solutions.
\item From the numerical calculation we find that $a_{2n-1}(l)$ are
monotonously decreasing functions.
\item The behavior of $a_{2n-1}( l)$ for $ l\rightarrow0$ is
\begin{equation}
a_{2n-1}( l)=-{1\over 2} - (2n-1) - { 2 {\rm e}^{ -{1\over
l}}
\over { l}{}^{(2n - {1\over 2})} \ (n-1)! \ \Gamma(n+{1\over2})}.
\end{equation}
{}From the above equation we conclude that $a_{2n-1}'(
l)|_{ l =
0}=0$ and also that
$a_{2n-1}( l)$ is a decreasing function near $ l =0$.
\item The behavior for negative $a_{2n-1}$ with large absolute value (
$|a_{2n-1}|>>8$) is given by
\begin{equation}
a_{2n-1}( l)=-{\pi^2\over 2} n^2 l, \quad n= 1, 2, 3 \dots,
\end{equation}
which again can be readily observed in the figure.
\end{enumerate}
Let us notice that the asymptotic behavior (large ${ l}$)
of $a_{2n}$
and $a_{2n-1}$
coincide for $n=1,2, \dots$
Finally, we notice that all the states in the compact case are
invariant under large gauge transformations. In particular, that is the
case for the vacuum, and consequently the $\theta$-dependence does not
appear.
\section{Concluding remarks}
In this paper we have analysed the two dimensional compact
electrodynamics using the loop representation.
The loop variables introduced here adapt naturally to the study of the compact version
of the Schwinger model in the continuum, which, up to our knowledge, has not been previously discussed in the literature. These variables provide gauge invariant and intrinsically regularized non-local fields to describe the model. In terms of them, the quantum commutator algebra is constructed in an unambiguous manner. In particular, the current-current and electric field-current commutators are directly recovered. The choice of our basic non-local variables is such that the algebra involved in the anomalous commutators is already realized in terms of the classical Poisson brackets . We also obtain full operator expressions for the divergences of the charged and axial currents, which still need to be analyzed in the Hilbert space of the problem. This work also reports the spectrum of the zero mode sector of the compact Schwinger model, which is an exact piece of the full spectrum, due to the separability of the Schroedinger
equation. The zero mode energy turns out to be completely different from the standard equally-spaced oscillator spectrum of the non-compact case.
Even though bosonization seems to be still present, the zero mode
spectrum suggests that the compact system will not behave as a free massive
scalar field.
The invariance of the
theory under large gauge transformations insures the uniqueness of the
vacuum. The axial anomaly is still present, however as it was noticed
by Jackiw
\cite{Ja}, the presence of an axial anomaly is not necessarily related
with a non trivial topological structure. Further studies are required
to determine that, either the chiral charge may be redefined in a gauge
invariant way, or the conservation of the axial current cannot be
restored in the compact case.
Our main obstacle to proceed with the solution of the full compact model is to obtain a complete set of solutions of the Schroedinger equation arising from the Hamiltonian (\ref{zmodeH}), which is of the many-body type, with a linear kinetic energy term and pairwise interactions given by the non-trivial potential (\ref{POTEN}).There remains also the task of giving a complete discussion
of the structure of the Hilbert space. These
problems are now under investigation.
Again, we emphasize that the choice between the compact or
noncompact version
of any
gauge theory is just a mater of convenience, which should be
ultimately
decided in terms
of the experimental consequences of each model. In this spirit, what is
really missing in the vast literature regarding the Schwinger
model is, up
to our knowledge, a study of its compact version using one's
favorite
method of solution. The main difference between the compact and the
noncompact
version is expected to appear
in the boundary conditions satisfied by the corresponding wave
functions, as
opposed to the form of the (functional) differential equations
involved, which
should be the same in both cases. This is in complete analogy with
the case of
a free particle in a line (noncompact case) compared with the free
one-dimensional rotator (compact case), both of which are governed by
the same
Schroedinger equation.
An alternative route in understanding the compact Schwinger model
is been
pursued in Ref.\cite{LMUV},
following the
method of Ref.\cite{IsoMurayama}.
Some preliminary results are the following. Since the Gauss law
still implies
that the wave function is independent of the excited modes of the vector
potential in the Weyl gauge, there are no boundary condition
modifications
related to these variables arising from the compactification. The wave
function depends only upon the zero mode of the vector potential
and the full
Hamiltonian is still separable into zero plus excited modes with
the same
Schroedinger equations
as in the noncompact case. The zero mode
wave function satisfies new boundary conditions, imposed by the
compactification of the zero mode vector potential, leading to the same
non-linearly spaced spectrum derived in the present
work. This piece of the spectrum cannot be interpreted as a collection of particles
with mass $
\frac{e}{\sqrt \pi}$ at zero momentum. The study of the complete spectrum and the full Hilbert space of the compact
Schwinger model, using this approach, is also in progress.\cite{LMUV}
\acknowledgments
Partial support from CONACyT grant 3141-PE to HAMT, LFU and JDV is
acknowledged.
HAMT would also like to thank
CLAF-M\'exico, the organizers
of ``Quantum Gravity in the Southern Cone"
and ICTP-Trieste for hospitality and financial support at
different stages of
the present work.
HAMT and LFU are also indebted to R. Gambini for the
hospitality extended
to them at the
Departamento de F\'{\i}sica
de la Universidad de la Rep\'ublica, Uruguay. LFU and JDV are
supported by
the
grant DGAPA-IN100694 and DGAPA-IN100397. We thank Rom\'an Linares for verifying
some of the
calculations. RG acknowledges useful conversations with H. Fort regarding
compact electrodynamics.
\section*{Appendix A}
Using the scalar product given in Eq.(\ref{PE}), we are able to
solve for the creation-annihilation operators in terms of the fields
$\psi_1,\psi_2$
\begin{equation}
a_m=e^{{i\over \hbar}\epsilon_m t}(\phi_m,\ \psi_1),\quad
a_m^\dagger=e^{-{i\over \hbar}\epsilon_m t}(\phi_m^*,\ \psi_1^*),
\end{equation}
\begin{equation}
b_m=e^{{i\over \hbar}\epsilon_m t}(\phi_m^*,\ \psi_2^*),\quad
b_m^\dagger=e^{-{i\over \hbar}\epsilon_m t}(\phi_m,\ \psi_2).
\end{equation}
In this way, all bilinear expressions containing creation-annihilation
operators can be written in terms of the gauge-invariant operators
$\Pi(x,y)$
defined at $t=0$.
A direct calculation shows that
\begin{eqnarray}
\Pi_0^{mn}&=& a_m^\dagger b_n^\dagger, \\
\Pi_1^{mn}&=& \,a_m^\dagger a_n, \\
\Pi_2^{mn}&=& -b_n^\dagger b_m, \\
\Pi_3^{mn}&=& b_m a_n\;.
\end{eqnarray}
The fermionic character of the operators involved imply the following
important symmetry properties of the above operators
\begin{equation}
\Pi_a^{mn}\Pi_a^{mk}=0=\Pi_a^{mn}\Pi_a^{km},\quad a=0,3,
\end{equation}
which is reflected through $\Pi_a(x,y)\Pi_a(x,z)=0=\Pi_a(x,y)\Pi_a(z,y)$
in coordinate space. Another important related symmetry of the external
field is $U(u,v)U(x,y)=U(u,y)U(x,v)$, which results because of the
abelian character of the $U(1)$ connection.
|
1,108,101,565,471 | arxiv |
\section{Introduction}
\label{SecIntro}
Regularization is an essential mechanism in Machine Learning that usually refers to the set of techniques that attempt to improve the estimates by biasing them away from their sample-based values towards values that are deemed to be more ``physically plausible''~\cite{Friedman1989}.
In practice, it is often used to avoid over-fitting, apply some prior knowledge about the problem at hand or induce some desirable properties over the resulting learning machine.
One of these properties is the so called sparsity, which can be roughly defined as expressing the learning machines using only a part of the training information. This has advantages in terms of the interpretability of the model and its manageability, and also preventing the over-fitting.
Two representatives of this type of models are the Support Vector Machines (SVM~\cite{Cortes1995}) and the Lasso model~\cite{Tibshirani1996}, based on inducing sparsity at two different levels.
On the one hand, the SVMs are sparse in their representation in terms of the training patterns, which means that the model is characterized only by a subsample of the original training dataset.
On the other hand, the Lasso models induce sparsity at the level of the features, in the sense that the model is defined only as a function of a subset of the inputs, hence performing an implicit feature selection.
Recently, Jaggi~\cite{Jaggi2014} showed an equivalence between the optimization problems corresponding to a classification $\lpg{2}$-SVM and a constrained regression Lasso.
As explored in this work, this connection can be useful to transfer ideas from one field to the other. In particular, and looking for sparser SVMs, in this paper the reweighted Lasso approach of Cand\`es \emph{et al.}~\cite{Candes2008} is taken as the basis to define first a weighted $\lpg{2}$-SVM, and then to propose a simple way of adjusting iteratively the weights that leads to a Modified Frank--Wolfe algorithm.
This adaptation of the weights does not add an additional cost to the algorithm. Moreover, as shown experimentally the proposed approach needs less iterations to converge than the standard Frank--Wolfe, and the resulting SVMs are sparser and much more robust with respect to changes in the regularization hyper-parameter, while retaining a comparable accuracy.
In summary, the contributions of this paper can be stated as follows:
\begin{enumerate}[(i)]
\item The definition of a new weighted SVM model, inspired by the weighted Lasso and the connection between Lasso and SVM. This definition can be further extended to a re-weighted SVM, based on an iterative scheme to define the weights.
\item The proposal of a modification of the Frank--Wolfe algorithm based on the re-weighting scheme to train the SVM. This algorithm results in a sparser SVM model, which coincides with the model obtained using a standard SVM training algorithm over only an automatically-selected subsample of the original training data.
\item The numerical comparison of the proposed model with the standard SVM over a number of different datasets. These experiments show that the proposed algorithm requires less iterations while providing a sparser model which is also more stable against modifications of the regularization parameter.
\end{enumerate}
The remaining of the paper is organized in the following way. \Ref{SecPre} summarizes some results regarding the connection of SVM with Lasso. The weighted and re-weighted SVM are introduced in \ref{SecWSVM}, whereas the proposed modified Frank--Wolfe algorithm is presented in \ref{SecMFW}. The performance of this algorithm is tested through some numerical experiments in \ref{SecExp}, and \ref{SecConc} ends the paper with some conclusions and pointers to further work.
\subsubsection*{Notation}
$N$ denotes the number of training patterns, and $D$ the number of dimensions. The data matrix is denoted by $\ensuremath{\mathbf{X}} = \prn{\mxp{1}, \mxp{2}, \dotsc, \mxp{N}}^\intercal \in \R^{\npat \times \ndim}$, where each row correspond to the transpose of a different pattern $\mxp{i} \in \R^\ndim$. The corresponding vector of targets is $\ensuremath{\mathbf{y}} \in \R^\npat$, where $\vyi{i} \in \set{-1, +1}$ denotes the label of the $i$-th pattern. The identity matrix of dimension $N$ is denoted by $\iden{N} \in \R^{\npat \times \npat}$.
\section{Preliminaries}
\label{SecPre}
This section covers some preliminary results concerning the Support Vector Machine (SVM) formulation, its connection with the Lasso model, and the re-weighted Lasso algorithm, which are included since they are the basis of the proposed algorithm.
\subsection{SVM Formulation}
The following $\lpg{2}$-SVM classification model (this model is described for example in~\cite{Keerthi2000}), crucial in~\cite{Jaggi2014}, will be used as the starting point of this work:
\begin{equation*}
\label[problem]{EqProbSVMP}
\minpc{\ensuremath{\mathbf{w}}, \rho, \xi}{\frac{1}{2} \nt{\ensuremath{\mathbf{w}}}^2 - \rho + \frac{C}{2} \sum_{i = 1}^N {\xi_i^2}}{\ensuremath{\mathbf{w}}^\intercal \mzi{i} \ge \rho - \xi_i} \eeq{,}
\end{equation*}
where $\mzi{i} = \vyi{i} \mxp{i}$.
Straightforwardly, the corresponding Lagrangian dual problem can be expressed as:
\begin{equation}
\label[problem]{EqProbSVMD}
\minpc{\ensuremath{\boldsymbol{\alpha}} \in \R^\npat}{\ensuremath{\boldsymbol{\alpha}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}}{0 \le \vai{i} \le 1 ,\sepcon \sum_{i=1}^{N} \vai{i} = 1} \eeq{,}
\end{equation}
where $\ensuremath{\hat{\mk}} = \ensuremath{\mathbf{Z}} \ensuremath{\mathbf{Z}}^\intercal + \frac{1}{C} \iden{N}$.
A non-linear SVM can be considered simply by substituting $\ensuremath{\mathbf{Z}} \ensuremath{\mathbf{Z}}^\intercal$ by the (labelled) kernel matrix $\ensuremath{\mathbf{K}} \circ \ensuremath{\mathbf{y}} \ensuremath{\mathbf{y}}^\intercal$ (where $\circ$ denotes the Hadamard or component-wise product).
It should be noticed that the feasible region of \ref{EqProbSVMD} is just the probability simplex, and the objective function is simply a quadratic term.
\subsection{Connection between Lasso and SVM}
\label{SecEquiv}
There exists an equivalence between the SVM dual \ref{EqProbSVMD} and the following problem, which corresponds to a constrained Lasso regression model:
\begin{equation}
\label[problem]{EqProbLasso}
\minpc{\ensuremath{\mathbf{w}} \in \R^\ndim}{\nt{\ensuremath{\mathbf{X}} \ensuremath{\mathbf{w}} - \ensuremath{\mathbf{y}}}^2}{\no{\ensuremath{\mathbf{w}}} \le 1} \eeq{,}
\end{equation}
where in this case the vector $\ensuremath{\mathbf{y}} \in \R^\npat$ does not need to be binary. In particular, a problem of the form of \ref{EqProbSVMD} can be rewritten in the form of \ref{EqProbLasso} and vice-versa~\cite{Jaggi2014}.
This relation is only at the level of the optimization problem, which means that an $\lpg{2}$-SVM model can be trained using the same approach as for training the Lasso model and the other way around (as done in~\cite{Alaiz2015}), but it cannot be extended to a prediction phase, since the Lasso model is solving a regression problem, whereas the SVM solves a classification one. Moreover, the number of dimensions and the number of patterns flip when transforming one problem into the other.
Nevertheless, and as illustrated in this paper, this connection can be valuable by itself to inspire new ideas.
\subsection{Re-Weighted Lasso}
The re-weighted Lasso (RW-Lasso{}) was proposed as an approach to approximate the \lpg{0}{} norm by using the \lpg{1}{} norm and a re-weighting of the coefficients~\cite{Candes2008}.
In particular, this approach was initially designed to approximate the problem
\begin{equation*}
\label[problem]{EqProbRWA}
\minpc{\ensuremath{\mathbf{w}} \in \R^\ndim}{\nz{\ensuremath{\mathbf{w}}}}{\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{X}} \ensuremath{\mathbf{w}}} \eeq{,}
\end{equation*}
by minimizing weighted problems of the form:
\begin{equation}
\label[problem]{EqProbRWB}
\minpc{\ensuremath{\mathbf{w}} \in \R^\ndim}{\sum_{i = 1}^{D} \vti{i} \abs{\vwi{i}}}{\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{X}} \ensuremath{\mathbf{w}}} \eeq{,}
\end{equation}
for certain weights $\vti{i} > 0$, $i = 1, \dotsc, D$. An iterative approach was proposed, where the previous coefficients are used to define the weights at the current iterate:
\begin{equation}
\label{EqRWLWeights}
\vtik{i}{k} = \frac{1}{\abs{\vwik{i}{k-1}} + \epsilon} \eeq{,}
\end{equation}
what results in the following problem at iteration $k$:
\begin{equation*}
\label[problem]{EqProbRWC}
\minpc{\vwk{k} \in \R^\ndim}{\sum_{i = 1}^{D} \frac{1}{\abs{\vwik{i}{k-1}} + \epsilon} \abs{\vwik{i}{k}}}{\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{X}} \vwk{k}} \eeq{.}
\end{equation*}
The idea is that if a coefficient is small, then it could correspond to zero in the ground-truth model, and hence it should be pushed to zero. On the other side, if the coefficient is large, it most likely will be different from zero in the ground-truth model, and hence its penalization should be decreased in order not to bias its value.
This approach is based on a constrained formulation that does not allow for training errors, since the resulting model will always satisfy $\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{X}} \ensuremath{\mathbf{w}}$. A possible implementation of the idea of \ref{EqProbRWB} without such a strong assumption is the following:
\begin{equation*}
\label[problem]{EqProbRWD}
\minpc{\ensuremath{\mathbf{w}} \in \R^\ndim}{\nt{\ensuremath{\mathbf{X}} \ensuremath{\mathbf{w}} - \ensuremath{\mathbf{y}}}^2}{\sum_{i = 1}^{D} \vti{i} \abs{\vwi{i}} \le 1} \eeq{,}
\end{equation*}
where the errors are minimized and the weighted $\lpg{1}$ regularizer is included as a constraint (equivalently, the regularizer could be also added to the objective function~\cite{Zou2006}). The iterative procedure to set the weights can still be the one explained above, where the weights at iteration $k$ are defined using \ref{EqRWLWeights}.
\subsection{Towards a Sparser SVM}
One important remark regarding the RW-Lasso{} is that the re-weighting scheme breaks the equivalence with the SVM explained in \ref{SecEquiv}, i.e., one cannot simply apply the RW-Lasso{} approach to solve the SVM problem in order to get more sparsity in the dual representation (i.e.\ fewer support vectors).
Instead, an analogous scheme will be directly included in the SVM formulation in the section below.
More specifically, and as shown in \ref{FigScheme}, the connection between Lasso and SVM suggests to apply a weighting scheme also for SVM. In order to set the weights, an iterative procedure (analogous to the RW-Lasso{}) seems to be the natural step, although this would require to solve a complete SVM problem at each iteration. Finally, an online procedure to determine the weights, that are adapted directly at the optimization algorithm, will lead to a modification of the Frank--Wolfe algorithm.
{
\renewcommand{}[1]{{\color{box}$\blacksquare$}}
\begin{mfigure}{\label{FigScheme} Scheme of the relation between the proposed methods and the inspiring Lasso variants.}
\legend{\showlegendcolour{box}~State-of-the-art Methods}{\showlegendcolour{box}~State-of-the-art Methods\qquad\showlegendcolour{boxo}~Proposed Methods}
\includetikzna{Scheme}
\end{mfigure}
}
It should be stated that a weighted SVM has been already proposed in~\cite{Lapin2014}, but that model differs from the approach described here. In particular, the weighing of \cite{Lapin2014} refers to the primal problem (through different regularization parameters associated to each pattern) whereas in this work the weighting refers directly to the dual problem. As explained in \ref{SSecWSVM}, both models are not equivalent.
\section{Weighted and Re-Weighted SVM}
\label{SecWSVM}
In this section the weighted SVM model is proposed. Furthermore, a re-weighting scheme to define iteratively the weights is sketched.
\subsection{Weighted SVM}
\label{SSecWSVM}
In order to transfer the weighting scheme of RW-Lasso{} to an SVM framework, the most natural idea is to directly change the constraint of \ref{EqProbSVMD} to introduce the scaling factors $\vti{i}$. This results in the following Weighted-SVM (W-SVM{}) dual optimization problem:
\begin{equation}
\label[problem]{EqProbWSVMD}
\minpc{\ensuremath{\boldsymbol{\alpha}} \in \R^\npat}{\ensuremath{\boldsymbol{\alpha}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}}{0 \le \vai{i} ,\sepcon \sum_{i=1}^{N} {\vti{i} \vai{i}} = 1} \eeq{,}
\end{equation}
for a fixed vector of weights $\ensuremath{\mathbf{t}}$. This modification relates with the primal problem as stated in the proposition below.
\begin{mprop}
The W-SVM{} primal problem corresponding to \ref{EqProbWSVMD} is:
\begin{equation}
\label[problem]{EqProbWSVMP}
\minpc{\ensuremath{\mathbf{w}}, \rho, \xi}{\frac{1}{2} \nt{\ensuremath{\mathbf{w}}}^2 - \rho + \frac{C}{2} \sum_{i = 1}^N {\xi_i^2}}{\ensuremath{\mathbf{w}}^\intercal \mzi{i} \ge \vti{i} \rho - \xi_i} \eeq{.}
\end{equation}
\end{mprop}
\begin{proof}
The Lagrangian of \ref{EqProbWSVMP} is:
\begin{equation*}
\mathcal{L}\prn{\ensuremath{\mathbf{w}}, \rho, \xi ; \ensuremath{\boldsymbol{\alpha}}} = \frac{1}{2} \nt{\ensuremath{\mathbf{w}}}^2 - \rho + \frac{C}{2} \sum_{i = 1}^N {\xi_i^2} + \sum_{i = 1}^N {\vai{i} \prn{- \ensuremath{\mathbf{w}}^\intercal \mzi{i} + \vti{i} \rho - \xi_i}} \eeq{,}
\end{equation*}
with derivatives with respect to the primal variables:
\begin{align*}
\deriv{\mathcal{L}}{\ensuremath{\mathbf{w}}} &= \ensuremath{\mathbf{w}} - \ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}} = 0 && \implies && \ensuremath{\mathbf{w}} = \ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}} \eeq{;} \\
\deriv{\mathcal{L}}{\rho} &= - 1 + \sum_{i = 1}^N {\vti{i} \vai{i}} = 0 && \implies && \sum_{i = 1}^N {\vti{i} \vai{i}} = 1 \eeq{;} \\
\deriv{\mathcal{L}}{\xi} &= C \xi - \ensuremath{\boldsymbol{\alpha}} = 0 && \implies && \xi = \frac{1}{C} \ensuremath{\boldsymbol{\alpha}} \eeq{.}
\end{align*}
Substituting into the Lagrangian, the following objective function for the dual problem arises:
\begin{equation*}
\frac{1}{2} \nt{\ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}}}^2 - \rho + \frac{C}{2 C^2} \nt{\ensuremath{\boldsymbol{\alpha}}}^2 - \nt{\ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}}}^2 + \rho \sum_{i = 1}^N {\vti{i} \vai{i}} - \frac{1}{C} \nt{\ensuremath{\boldsymbol{\alpha}}}^2 = - \frac{1}{2} \nt{\ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}}}^2 - \frac{1}{2 C} \nt{\ensuremath{\boldsymbol{\alpha}}}^2 \eeq{.}
\end{equation*}
Hence, the resulting dual problem is:
\begin{equation*}
\minpc{\ensuremath{\boldsymbol{\alpha}} \in \R^\npat}{\nt{\ensuremath{\mathbf{Z}} \ensuremath{\boldsymbol{\alpha}}}^2 + \frac{1}{C} \nt{\ensuremath{\boldsymbol{\alpha}}}^2}{0 \le \vai{i} ,\sepcon \sum_{i = 1}^N {\vti{i} \vai{i}} = 1} \eeq{,}
\end{equation*}
which coincides with \ref{EqProbWSVMD}.
\end{proof}
Therefore, the effect of increasing the scaling factor $\vti{i}$ in the W-SVM{} dual formulation is equivalent to increasing the margin required for the $i$-th pattern in the primal formulation. Thus, intuitively an increase of $\vti{i}$ should facilitate the $i$-th pattern to become a support vector.
This influence is numerically illustrated in \ref{FigWSVMEvo}, where the value of one weight $\vti{i}$ is varied to analyse its influence over the corresponding multiplier $\vai{i}$ in a binary classification problem with $N = 100$ and $D = 2$. The other weights are just fixed equal to one, but before solving the problem all the vector $\ensuremath{\mathbf{t}}$ is normalized so that its maximum is still equal to one in order to preserve the scale. This experiment is done for three different values of $C$ (\num{e-3}, \num{1e0} and \num{e3}) and for the weights corresponding to the maximum, minimum and an intermediate value of the multiplier of the standard (unweighted) SVM. Clearly $\vti{i}$ and $\vai{i}$ present a proportional relationship, so the larger $\vti{i}$ is, the larger the obtained multiplier $\vai{i}$ becomes (until some point of saturation), confirming the initial intuition.
\begin{mfigure}{\label{FigWSVMEvo} Evolution of the SVM coefficient $\vai{}$ with respect to the weight $\vti{}$, for $C$ equal to \num{e-3} (first row), \num{e0} (second row) and \num{e3} (third row), and for the patterns corresponding to the maximum (first column), an intermediate (second column) and the minimum (third column) initial value of $\vai{}$.}
\colorlet{mycolor1}{graphic1}%
\tikzwidth{0.3\textwidth}%
\begin{footnotesize}
\begin{tabular}{ccc}
\includetikz{ExampleEvolutionMax1e-03} &
\includetikz{ExampleEvolutionInt1e-03} &
\includetikz{ExampleEvolutionMin1e-03} \\
\includetikz{ExampleEvolutionMax1e+00} &
\includetikz{ExampleEvolutionInt1e+00} &
\includetikz{ExampleEvolutionMin1e+00}\\
\includetikz{ExampleEvolutionMax1e+03} &
\includetikz{ExampleEvolutionInt1e+03} &
\includetikz{ExampleEvolutionMin1e+03}
\end{tabular}
\end{footnotesize}
\end{mfigure}
As another illustration, \ref{FigWSVMFeasible} shows a small toy example of three patterns, which allows to represent the feasible set in two dimensions as the convex hull of the three vertices. The value of one weight $\vti{i}$ is changed in the set $\set{\num{e-2}, \num{e-1}, \num{e0}, \num{e1}, \num{e2}}$, whereas the other two weights are kept fixed to $1$. As before, increasing the weight pushes the solution towards the corresponding pattern.
Moreover, the last row in \ref{FigWSVMFeasible} shows the same example but with a three dimensional representation, so that it is more clear the effect of decreasing $\vti{1}$ in the feasible set, basically lengthening the triangle and increasing its angle with respect to the horizontal plane, until the point where the triangle becomes an unbounded rectangle ($\vti{1} = 0$) completely vertical. Taking into consideration that the solution of the unconstrained problem (for $C \neq \infty$) is the origin, decreasing $\vti{1}$ is moving away the first vertex from the unconstrained solution, thus making less likely to assign a non-zero coefficient to that point unless it really decreases the objective function.
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\begin{mfigurec}{\label{FigWSVMFeasible} Example of the feasible region and the solution for a problem with three patterns, for different values of the weighting vector $\ensuremath{\mathbf{t}}$. For each plot, the value of $\ensuremath{\mathbf{t}}$ is shown above in boldface. The three rows correspond to changes in $\vti{1}$, $\vti{2}$ and $\vti{3}$ respectively, and the weighted probability simplex is represented as the convex hull of the three vertices. The fourth row corresponds again to changes in $\vti{1}$ but with a $3$-dimensional representation keeping the same aspect ratio for all the axis, and also including the limit case $\vti{1} = 0$ where $\vai{1}$ is not upper bounded. The solution of the constrained optimization problem is shown with a red dot~\showlegend{0}.}
\tikzwidth{0.125\textwidth}%
\scriptsize
\ifusetikz
\begin{tabular}{*5{@{\hspace{10pt}}c@{\hspace{10pt}}}}
\else
\begin{tabular}{*5c}
\fi
\toprule
\tabformathrow{$\prns{1, 1, \num{e-2}}$} & \tabformathrow{$\prns{1, 1, \num{e-1}}$} & \tabformathrow{$\prns{1, 1, \num{e0}}$} & \tabformathrow{$\prns{1, 1, \num{e1}}$} & \tabformathrow{$\prns{1, 1, \num{e2}}$} \\\midrule
\includetikz{FeasibleSet-1-1} & \includetikz{FeasibleSet-2-1} & \includetikz{FeasibleSet-3-1} & \includetikz{FeasibleSet-4-1} & \includetikz{FeasibleSet-5-1} \\[5pt]\toprule
\tabformathrow{$\prns{1, \num{e-2}, 1}$} & \tabformathrow{$\prns{1, \num{e-1}, 1}$} & \tabformathrow{$\prns{1, \num{e0}, 1}$} & \tabformathrow{$\prns{1, \num{e1}, 1}$} & \tabformathrow{$\prns{1, \num{e2}, 1}$} \\\midrule
\includetikz{FeasibleSet-1-2} & \includetikz{FeasibleSet-2-2} & \includetikz{FeasibleSet-3-2} & \includetikz{FeasibleSet-4-2} & \includetikz{FeasibleSet-5-2} \\\toprule
\tabformathrow{$\prns{\num{e-2}, 1, 1}$} & \tabformathrow{$\prns{\num{e-1}, 1, 1}$} & \tabformathrow{$\prns{\num{e0}, 1, 1}$} & \tabformathrow{$\prns{\num{e1}, 1, 1}$} & \tabformathrow{$\prns{\num{e2}, 1, 1}$} \\\midrule
\includetikz{FeasibleSet-1-3} & \includetikz{FeasibleSet-2-3} & \includetikz{FeasibleSet-3-3} & \includetikz{FeasibleSet-4-3} & \includetikz{FeasibleSet-5-3} \\\toprule
\tabformathrow{$\prns{\num{4}, 1, 1}$} & \tabformathrow{$\prns{\num{2}, 1, 1}$} & \tabformathrow{$\prns{\num{1}, 1, 1}$} & \tabformathrow{$\prns{\num{0.5}, 1, 1}$} & \tabformathrow{$\prns{0, 1, 1}$} \\\midrule
\includetikz{FeasibleSetUnboundedEvo-1} & \includetikz{FeasibleSetUnboundedEvo-2} & \includetikz{FeasibleSetUnboundedEvo-3} & \includetikz{FeasibleSetUnboundedEvo-4} & \includetikz{FeasibleSetUnboundedEvo-5} \\\bottomrule
\end{tabular}
\end{mfigurec}
}
It is mandatory to state the differences between the W-SVM{} proposed here and the previous model proposed in~\cite{Lapin2014}. First of all, the formulations over which both approach are based are different.
But, even if the same starting SVM model were used, both weighting schemes are essentially different:
\begin{itemize}
\item Lapin \emph{et al.} propose a modification of the primal SVM formulation so that the cost associated is different for each pattern. This means that the loss associated to that pattern is multiplied by a constant.
\item The W-SVM{} proposed here introduces the weights directly into the dual problem, what results into a modification of the margin required for each pattern in the primal problem. Considering again the loss associated to each pattern, the ``insensitivity'' zone (the region of predictions that are associated to a zero loss) is widened or narrowed according to a constant.
\end{itemize}
Hence, both approaches are fundamentally different, and the effects that they produce are not equivalent.
\subsection{Re-Weighted SVM}
Once \ref{EqProbWSVMD} has been defined, and provided that the scaling factors seem to influence the sparsity of the solution (as illustrated in \ref{FigWSVMEvo,FigWSVMFeasible}), a procedure to set the weighting vector $\ensuremath{\mathbf{t}}$ is needed.
In parallelism with the original RW-Lasso{}, but considering that in this case the relation between the weight $\vti{i}$ and the corresponding optimal multiplier $\vai{i}$ is directly proportional, the following iterative approach, namely Re-Weighted SVM (RW-SVM{}), arises naturally:
\begin{enumerate}
\item At iteration $k$, the following W-SVM{} problem is solved:
\begin{equation}
\label[problem]{EqProbRWSVMD}
\vaok{k} = \argminpcd{\ensuremath{\boldsymbol{\alpha}} \in \R^\npat}{\ensuremath{\boldsymbol{\alpha}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}}{0 \le \vai{i} \le 1 ,\sepcon \sum_{i=1}^{N} {\vtik{i}{k} \vai{i}} = 1 }
\end{equation}
\item The weighting vector for the next iteration, $\vtk{k}$, is updated as:
\begin{equation*}
\vtik{i}{k + 1} = \ensuremath{f_\text{mon}}\prn{\vaiok{i}{k}} \eeq{,}
\end{equation*}
where $\ensuremath{f_\text{mon}}: \mathbb{R} \to \mathbb{R}$ is some monotone function.
\end{enumerate}
This approach has two main drawbacks.
The first one is how to select the function $\ensuremath{f_\text{mon}}$. This also implies selecting some minimum and maximum values to which the weights $\vtik{i}{k}$ should saturate, so it is not a trivial task, and it can greatly influence the behaviour of the model.
The second drawback is that this approach requires to solve \ref{EqProbRWSVMD} at each iteration, which means training completely a W-SVM{} model (with a complexity that should not differ from that of training a standard SVM) on each iteration, and hence the overall computational cost can be much larger. Although this is in fact an affordable drawback if the objective is solely to approach the \lpg{0}{} norm as it was the case in the original paper of RW-Lasso{}~\cite{Candes2008}, in the case of the SVM the aim is to get sparser models in order to reduce their complexity and to improve the performance specially in large datasets, and hence it does not make sense to need for this several iterations.
As a workaround, the next section proposes an online modification of the weights that leads to a simple modification of the training algorithm for SVMs.
\section{Modified Frank--Wolfe Algorithm}
\label{SecMFW}
This section proposes a training algorithm to get sparser SVMs, which is based on an online modification of the weighting vector $\ensuremath{\mathbf{t}}$ of a W-SVM{} model.
In particular, the basis of this proposal is the Frank--Wolfe optimization algorithm.
\subsection{Frank--Wolfe Algorithm}
The Frank--Wolfe algorhtm (FW{};~\cite{Frank1956}) is a first order optimization method for constrained convex optimization. There are several versions of this algorithm, in particular the basis of this work is the Pairwise Frank--Wolfe~\cite{Jaggi2013,Lacoste-Julien2015}.
Roughly speaking, it is based on using at each iteration a linear approximation of the objective function to select one of the vertices as the target towards which the current estimate of the solution will move (the \emph{forward} node), and another vertex as that from which the solution will move away (the \emph{away} node), and then updating the solution in the direction that goes from the \emph{away} node to the \emph{forward} one using the optimal step length. At the end, the linear approximation boils down to selecting the node corresponding to the smallest partial derivative as the \emph{forward} node, and that with the largest derivative as the \emph{away} node.
This general algorithm can be used in many different contexts, and in particular it has been succesfully applied to the training of SVMs~\cite{Gaertner2009,Ouyang2010,Frandi2013}. Specifically, for the case of \ref{EqProbSVMD}, the following definitions and results are employed.
Let $\ensuremath{f}$ denote the (scaled) objective function of \ref{EqProbSVMD} (or \ref{EqProbWSVMD}), with gradient and partial derivatives:
\begin{align}
\ensuremath{f}\prn{\ensuremath{\boldsymbol{\alpha}}} &= \frac{1}{2} \ensuremath{\boldsymbol{\alpha}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}} \eeq{;} \label{Eqfobj} \\
\nabla \ensuremath{f}\prn{\ensuremath{\boldsymbol{\alpha}}} &= \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}} \eeq{;} \nonumber \\
\frac{\partial \ensuremath{f}}{\partial \vai{i}}\prn{\ensuremath{\boldsymbol{\alpha}}} &= \mkhi{i}^\intercal \ensuremath{\boldsymbol{\alpha}} \eeq{,} \label{Eqgobj}
\end{align}
where $\mkhi{i}^\intercal$ is the $i$-th row of $\ensuremath{\hat{\mk}}$.
Let $\ensuremath{\mathbf{d}}$ denote the direction in which the current solution will be updated. The optimal step-size can be computed by solving the problem:
\begin{equation}
\label[problem]{EqProbStep}
\minp{\ensuremath{\gamma}}{\ensuremath{f}\prn{\ensuremath{\boldsymbol{\alpha}} + \ensuremath{\gamma} \ensuremath{\mathbf{d}}}} \eeq{,}
\end{equation}
and truncating the optimal step, if needed, in order to remain in the convex hull of the nodes, i.e., to satisfy the constraints of \ref{EqProbSVMD} (or, equivalently, of \ref{EqProbWSVMD}).
Straightforwardly, \ref{EqProbStep} can be solved simply taking the derivative with respect to $\ensuremath{\gamma}$ and making it equal to zero:
\begin{align*}
\frac{\partial \ensuremath{f}}{\partial \ensuremath{\gamma}}\prn{\ensuremath{\boldsymbol{\alpha}} + \ensuremath{\gamma} \ensuremath{\mathbf{d}}} &= \ensuremath{\mathbf{d}}^\intercal \ensuremath{\hat{\mk}} \prn{\ensuremath{\boldsymbol{\alpha}} + \ensuremath{\gamma} \ensuremath{\mathbf{d}}} = 0 \\
\implies \ensuremath{\gamma} &= - \frac{\ensuremath{\mathbf{d}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}}{\ensuremath{\mathbf{d}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\mathbf{d}}} \eeq{.}
\end{align*}
It should be noticed that $\ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}$ is the gradient of $\ensuremath{f}$ at the point $\ensuremath{\boldsymbol{\alpha}}$, and thus there is no need to compute it again (indeed, the gradient times the direction is minus the FW{} gap, that can be used as a convergence indicator). Moreover, if the direction $\ensuremath{\mathbf{d}} = \sum_{i \in \ensuremath{\mathcal{U}}}{\vdiri{i} \bas{i}}$ is sparse, then $\ensuremath{\hat{\mk}} \ensuremath{\mathbf{d}} = \sum_{i \in \ensuremath{\mathcal{U}}} \vdiri{i} \mkhi{i}$ only requires to compute the columns of the kernel matrix corresponding to the set of updated variables $\ensuremath{\mathcal{U}}$. In particular, in the Pairwise FW{} only the columns of the \emph{forward} and \emph{away} nodes are used to determine $\ensuremath{\gamma}$ and to keep the gradient updated.
The whole procedure for applying FW{} to the SVM training is summarized in \ref{AlgFW}.
\begin{malgorithm}{\label{AlgFW} Pairwise Frank--Wolfe algorithm for SVM.}
\begin{algorithmic}[1]
\Procedure{TrainSVM}{$\ensuremath{\hat{\mk}}, \epsilon$} \Comment{\textbullet~Kernel $\ensuremath{\hat{\mk}} \in \R^{\npat \times \npat}$. \\\textbullet~Precision $\epsilon \in \mathbb{R}$.}
\algorithmiccommentb{Initialization.}
\Take $i_0 \in \set{1, \dotsc, N}$ \Comment{Initial vertex.}
\State $\ensuremath{\boldsymbol{\alpha}} \gets \bas{i_0}$ \Comment{Initial point.}
\State $\ensuremath{\mathbf{g}} \gets \vkhi{i_0}$ \Comment{Initial gradient.}
\Repeat \Comment{Main Loop.}
\algorithmiccommentb{Update of Coefficients.}
\State $s \gets \argmin_{i}{\vgi{i}}$ \Comment{Select forward node.}
\State $v \gets \argmax_{i}{\vgi{i}}$ \Comment{Select away node.}
\State $\ensuremath{\mathbf{d}} \gets \bas{s} - \bas{v}$ \Comment{Build update direction.}
\State $\ensuremath{\delta} \gets - \ensuremath{\mathbf{g}} \cdot \ensuremath{\mathbf{d}}$ \Comment{FW{} gap.}
\State $\ensuremath{\gamma} \gets \minme{\maxme{\ensuremath{\delta}/\prns{\ensuremath{\mathbf{d}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\mathbf{d}}}, 0}, \vai{v}}$ \Comment{Compute step length.}
\State $\ensuremath{\boldsymbol{\alpha}} \gets \ensuremath{\boldsymbol{\alpha}} + \ensuremath{\gamma} \ensuremath{\mathbf{d}}$ \Comment{Point update.}
\State $\ensuremath{\mathbf{g}} \gets \ensuremath{\mathbf{g}} + \ensuremath{\gamma} \mkhi{s} - \ensuremath{\gamma} \mkhi{v}$ \Comment{Gradient update.}
\Until{$\ensuremath{\delta} \le \epsilon$} \Comment{Stopping criterion.}
\EndProcedure
\end{algorithmic}
\end{malgorithm}
\subsection{Modified Frank--Wolfe Algorithm}
\label{SWSVM}
The idea of the proposed Modified Frank--Wolfe (M-\fw{}) is to modify the weights $\vti{i}$, i.e., the margin required for each training pattern, directly on each inner iteration of the algorithm, hence with an overall cost similar to that of the original FW{}.
In particular, and since according to \ref{FigWSVMEvo,FigWSVMFeasible} the relation between each weight and the resulting coefficient seems to be directly proportional, an incremental procedure with binary weights is defined, leading to a new training algorithm for SVM. Specifically, the training vectors will be divided into two groups, the working vectors, with a weight $\vti{i} = 1$, and the idle vectors, with a weight $\vti{i} = 0$.
The proposed M-\fw{} will start with only one initial working vector, and at each iteration, the idle vector with the smaller negative gradient (if there is any) will be add to the working set. After that, the coefficients of the working vectors will be updated by using a standard FW{} pair-wise step.
The intuition behind this algorithm is the following.
The standard FW{} algorithm applied to the SVM training will activate (make non-zero) the coefficient of a certain vector if its partial derivative is better (smaller) than that of the already active coefficients, i.e., if that vector is ``less bad'' than the others.
On the other side, the M-\fw{} will only add a coefficient to the working set if its partial derivative is negative (hence, that coefficient would also be activated without the simplex constraint), i.e., the vector has to be somehow ``good'' by itself.
In what follows, the M-\fw{} algorithm is described in more detail.
\subsubsection{Preliminaries}
The set of working vectors is denoted by $\ensuremath{\mathcal{W}} = \setst{i}{1 \le i \le N, \vti{i} = 1}$, and that of idle vectors as $\ensuremath{\bar{\act}} = \setst{i}{1 \le i \le N, \vti{i} = 0}$.
The dual problem becomes:
\begin{equation*}
\label[problem]{EqProbWSVMB}
\minpc{\ensuremath{\boldsymbol{\alpha}} \in \R^\npat}{\ensuremath{\boldsymbol{\alpha}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\boldsymbol{\alpha}}}{0 \le \vai{i} ,\sepcon \sum_{i \in \ensuremath{\mathcal{W}}} {\vai{i}} = 1} \eeq{.}
\end{equation*}
Thus, the coefficients for the points in $\ensuremath{\mathcal{W}}$ have to belong to the probability simplex of dimension $\abs{\ensuremath{\mathcal{W}}}$, whereas the coefficients for $\ensuremath{\bar{\act}}$ only have a non-negative constraint.
\subsubsection{Algorithm}
The proposed M-\fw{} algorithm to train an SVM is summarized in \ref{AlgMFW}.
This algorithm is very similar to \ref{AlgFW}, except for the initialization and control of the working set in \ref{LNactini,LNactBeg,LNacta,LNactb,LNactc,LNactd,LNacte,LNactf,LNactEnd}, the search for the \emph{forward} and \emph{away} nodes of \ref{LNaway,LNforward} (which is done only over the working set) and the stopping criterion of \ref{LNstop} (which requires both that the dual gap is small enough and that no new vertices have been activated).
\begin{malgorithm}{\label{AlgMFW} Modified Frank--Wolfe algorithm for SVM.}
\begin{algorithmic}[1]
\Procedure{TrainSVM\textsuperscript{M-\fw{}}{}}{$\ensuremath{\hat{\mk}}, \epsilon$} \Comment{\textbullet~Kernel $\ensuremath{\hat{\mk}} \in \R^{\npat \times \npat}$. \\\textbullet~Precision $\epsilon \in \mathbb{R}$.}
\algorithmiccommentb{Initialization.}
\Take $i_0 \in \set{1, \dotsc, N}$ \Comment{Initial vertex.}
\State $\ensuremath{\mathcal{W}} \gets \set{i_0}$ \Comment{Initial working set.}\label{LNactini}
\State $\ensuremath{\boldsymbol{\alpha}} \gets \bas{i_0}$ \Comment{Initial point.}
\State $\ensuremath{\mathbf{g}} \gets \vkhi{i_0}$ \Comment{Initial gradient.}
\Repeat \Comment{Main Loop.}
\algorithmiccommentb{Activation of Coefficients.}
\If {$\abs{\ensuremath{\mathcal{W}}} < N$}\label{LNactBeg}
\State $b_\text{chng} \gets \text{false}$ \Comment{Flag for changes.}\label{LNacta}
\State $u \gets \argmin_{i \in \ensuremath{\bar{\act}}}{\vgi{i}}$ \Comment{Select node.}\label{LNactb}
\If {$\vgi{u} < 0$}\label{LNactc}
\State $\ensuremath{\mathcal{W}} \gets \ensuremath{\mathcal{W}} \cup \set{u}$ \Comment{Activate node.}\label{LNactd}
\State $b_\text{chng} \gets \text{true}$ \Comment{Mark change.}\label{LNacte}
\EndIf\label{LNactf}
\EndIf\label{LNactEnd}
\algorithmiccommentb{Update of Working Coefficients.}
\State $s \gets \argmin_{i \in \ensuremath{\mathcal{W}}}{\vgi{i}}$ \Comment{Select forward node.}\label{LNforward}
\State $v \gets \argmax_{i \in \ensuremath{\mathcal{W}}}{\vgi{i}}$ \Comment{Select away node.}\label{LNaway}
\State $\ensuremath{\mathbf{d}} \gets \bas{s} - \bas{v}$ \Comment{Build update direction.}
\State $\ensuremath{\delta} \gets - \ensuremath{\mathbf{g}} \cdot \ensuremath{\mathbf{d}}$ \Comment{FW{} gap.}
\State $\ensuremath{\gamma} \gets \minme{\maxme{\ensuremath{\delta}/\prns{\ensuremath{\mathbf{d}}^\intercal \ensuremath{\hat{\mk}} \ensuremath{\mathbf{d}}}, 0}, \vai{v}}$ \Comment{Compute step length.}
\State $\ensuremath{\boldsymbol{\alpha}} \gets \ensuremath{\boldsymbol{\alpha}} + \ensuremath{\gamma} \ensuremath{\mathbf{d}}$ \Comment{Point update.}
\State $\ensuremath{\mathbf{g}} \gets \ensuremath{\mathbf{g}} + \ensuremath{\gamma} \mkhi{s} - \ensuremath{\gamma} \mkhi{v}$ \Comment{Gradient update.}
\Until{$\ensuremath{\delta} \le \epsilon$ and not $b_\text{chng}$} \Comment{Stopping criterion.}\label{LNstop}
\EndProcedure
\end{algorithmic}
\end{malgorithm}
\subsubsection{Convergence}
Regarding the convergence of the M-\fw{} algorithm, the following theorem states that this algorithm will provide a model that is equivalent to a standard SVM model trained only over a subsample\footnote{Due to its sparse nature, an SVM is expressed only in terms of the support vectors. Nevertheless, the proposed M-\fw{} provides an SVM trained over a subsample of the training set, although not all the vectors of this subsample have to become support vectors.} of the training patterns.
\begin{mtheor}
\label{TheoConvergence}
\Ref{AlgMFW} converges to a certain vector $\opt{\ensuremath{\boldsymbol{\alpha}}} \in \R^\npat$. In particular:
\begin{enumerate}[(i)]
\item The working set converges to a set $\ensuremath{\opt{\act}}$.
\item The components of $\opt{\ensuremath{\boldsymbol{\alpha}}}$ corresponding to $\ensuremath{\opt{\act}}$ conform the solution of the standard SVM \ref{EqProbSVMD} posed over the subset $\ensuremath{\opt{\act}}$ of the set of training patterns. The remaining components $\opt{\vai{i}}$, for $i \notin \ensuremath{\opt{\act}}$, are equal to zero.
\end{enumerate}
\end{mtheor}
\begin{proof}
{\quad}
\begin{enumerate}[(i)]
\item Let $\actk{k}$ denote the working set at iteration $k$. At iteration $k + 1$, the set $\actk{k + 1}$ will be either equal to $\actk{k}$ or equal to $\actk{k} \cup \set{u}$ for some $u \notin \actk{k}$. Hence, $\actk{k} \subseteq \actk{k + 1}$ for all $k$. Moreover, $\actk{k}$ is always a subset of the whole set of training vectors $\ensuremath{\mathcal{T}} = \set{1, \cdots, N}$, i.e. $\actk{k} \subseteq \ensuremath{\mathcal{T}}$ for all $k$. Since $\set{\actk{k}}$ is a monotone nondecreasing sequence of subsets of a finite set $\ensuremath{\mathcal{T}}$, then $\actk{k} \uparrow \ensuremath{\opt{\act}} \subseteq \ensuremath{\mathcal{T}}$, as proved next. Let $\set{k_1, \dotsc, k_{\npat'}}$ be those iterations in which the working set grows, $\actk{k_i} \subset \actk{k_i + 1}$. Obviously, the number of such $\npat'$ iterations is finite with $\npat' \le N$ since no more than $N$ elements can be added to the working set. Therefore, $\forall k \ge k_{\npat'}$, $\actk{k} = \actk{k_\npat'} = \ensuremath{\opt{\act}} \subseteq \ensuremath{\mathcal{T}}$.
\item Provided that $\actk{k} \subseteq \ensuremath{\opt{\act}}$ for all $k$, then for $i \notin \ensuremath{\opt{\act}}$ the corresponding coefficients will never be updated (they cannot be selected in \ref{LNforward,LNaway} of \ref{AlgMFW}), so they would conserve they initial value, i.e. $\vaik{i}{k} = 0$ for all $i \notin \ensuremath{\opt{\act}}$ and for all $k$.
With respect to the convergence of $\vaik{i}{k}$ for $i \in \ensuremath{\opt{\act}}$, it suffices to consider the iterations after the convergence of the working set, $k \ge k_{\npat'}$. Let $\vak{k}_{\ensuremath{\opt{\act}}} \in \mathbb{R}^{\npat'}$ be the vector composed by the coefficients of the working patterns. Using \ref{Eqfobj} and since the coefficients of idle vectors are equal to zero (proved above):
\begin{equation*}
\ensuremath{f}\prn{\vak{k}} = \sum_{i = 1}^N \sum_{j = 1}^N \vkhi{ij} \vaik{i}{k} \vaik{j}{k} = \sum_{i \in \ensuremath{\opt{\act}}} \sum_{j \in \ensuremath{\opt{\act}}} \vkhi{ij} \vaik{i}{k} \vaik{j}{k} = \ensuremath{f}_{\ensuremath{\opt{\act}}}\prn{\vak{k}_{\ensuremath{\opt{\act}}}} \eeq{,}
\end{equation*}
where $\ensuremath{f}_{\ensuremath{\opt{\act}}}$ denotes the objective function of \ref{EqProbSVMD} posed only over the subset $\ensuremath{\opt{\act}}$ of the original training set. A similar result can be obtained for the components of the gradient using \ref{Eqgobj}:
\begin{equation*}
\frac{\partial \ensuremath{f}}{\partial \vai{i}}\prn{\vak{k}} = \sum_{j = 1}^N \vkhi{ij} \vaik{j}{k} = \sum_{j \in \ensuremath{\opt{\act}}}^N \vkhi{ij} \vaik{j}{k} = \frac{\partial \ensuremath{f}_{\ensuremath{\opt{\act}}}}{\partial \vai{i}}\prn{\vak{k}_{\ensuremath{\opt{\act}}}} \eeq{.}
\end{equation*}
Therefore, once the working set has converged both the objective function and the partial derivatives of the working set computed in \ref{AlgMFW} are equal to those computed in \ref{AlgFW} when this algorithm is applied only over the vectors of the working set. Hence, in the remaining iterations M-\fw{} reduces to the standard FW{} algorithm but considering only the vertices in $\ensuremath{\opt{\act}}$, which converges to the solution of \ref{EqProbSVMD} over the subset $\ensuremath{\opt{\act}}$~\cite{Lacoste-Julien2015}.
\end{enumerate}
\end{proof}
It is worth mentioning that, although the proposed M-\fw{} algorithm converges to an SVM model trained over a subsample $\ensuremath{\opt{\act}}$ of the training data, this subsample will (as shown in \ref{SecExp}) depend on the initial point of the algorithm.
\section{Experiments}
\label{SecExp}
In this section the proposed M-\fw{} algorithm will be compared with the standard FW{} algorithm over several classification tasks.
In particular, the binary datasets that will be used for the experiments are described in \ref{TabDatasets}, which includes the size of the training and test sets, the number of dimensions and the percentage of the majority class (as a baseline accuracy). All of them belong to the LibSVM repository~\cite{Chang2011} except for \formatd{mgamma}{} and \formatd{miniboone}{}, which belong to the UCI repository~\cite{Lichman2015}.
\begin{mtable}{\label{TabDatasets} Description of the datasets.}
\begin{tabular}{c
S[table-format=5]
S[table-format=5]
S[table-format=3]
S[table-format=2.1]
}
\toprule
\tabformathrow{Dataset} & \tabformathrow{Tr. Size} & \tabformathrow{Te. Size} & \tabformathrow{Dim.} & \tabformathrow{Maj. Class (\%)} \\
\midrule
\input{TableDatasets}
\bottomrule
\end{tabular}
\end{mtable}
\subsection{Preliminary Experiments}
The first experiments will be focused on the first two datasets of \ref{TabDatasets}, namely \formatd{ijcnn}{} and \formatd{mgamma}{}, which are the largest ones except for \formatd{miniboone}{}.
\subsubsection{Set-Up}
The standard SVM model trained using FW{} (\formatm{SVM}{}) and the model resulting from the proposed M-\fw{} algorithm (denoted by \formatm{\isvmname}{}, which as shown in \ref{TheoConvergence} is just an SVM trained over a subsample $\ensuremath{\opt{\act}}$ of the original training set) will be compared in terms of their accuracies, the number of support vectors and the number of iterations needed to achieve the convergence during the training algorithm. Two different kernels will be used, the linear and the RBF (or Gaussian) ones.
With respect to the hyper-parameters of the models, the value of both $C$ and the bandwidth $\sigma$ (in the case of the RBF kernel) will be obtained through \num{10}-fold Cross Validation (CV) for \formatd{mgamma}{}, whereas for the largest dataset \formatd{ijcnn}{} only $C$ will be tuned, and $\sigma$ will be fixed as $\sigma = 1$ in the RBF kernel (this value is similar to the one used for the winner of the IJCNN competition~\cite{Chang2001}).
Once the hyper-parameters are tuned, both models will be used to predict over the test sets.
The stopping criterion used is $\epsilon = \num{e-5}$.
\subsubsection{Results}
The test results are summarized in \ref{TabResultsIncLarge}.
Looking first at the accuracies, both models \formatm{SVM}{} and \formatm{\isvmname}{} are practically equivalent in three of the four experiments, where the differences are insignificant, whereas for \formatd{ijcnn}{} with the linear kernel the accuracy is higher in the case of \formatm{\isvmname}{}.
Regarding the number of support vectors, \formatm{\isvmname}{} gets sparser models for \formatd{ijcnn}{} with linear kernel and \formatd{mgamma}{} with RBF kernel, whereas for the other two experiments both models get a comparable sparsity.
Finally, and with respect to the convergence of the training algorithms, \formatm{\isvmname}{} shows an advantage when dealing with linear kernels, whereas for the RBF ones both approaches are practically equivalent.
\begin{mtable}{\label{TabResultsIncLarge} Test results for the larger datasets.}
\sisetup{output-exponent-marker=\textsc{e},exponent-product={},retain-explicit-plus}
\renewcommand{}{\widthof{$\num{9.99e+9}$}}
\begin{tabular}{l@{}c*6{c}}
\toprule
\multirow{2}{*}{\tabformathrow{Data}} & \multirow{2}{*}{\tabformathrow{K.}} & \multicolumn{2}{c}{\tabformathrow{Accuracy ($\%$)}} & \multicolumn{2}{c}{\tabformathrow{Number SVs}} & \multicolumn{2}{c}{\tabformathrow{Number Iters.}} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8}
& & \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} & \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} & \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} \\
\midrule
\input{TableIncSVM2}
\bottomrule
\end{tabular}
\end{mtable}
It should be noticed that, for these larger datasets, only one execution is done per dataset and kernel, and hence it is difficult to get solid conclusions.
Hence, it can be interesting to analyse the performance of the models during the CV phase, as done below.
\subsubsection{Robustness w.r.t. Hyper-Parameter \texorpdfstring{$C$}{C}}
The evolution with respect to the parameter $C$ of the accuracy, the number of support vectors and the number of training iterations is shown in \ref{FigResultsParIJCNN} for both \formatm{SVM}{} and the proposed \formatm{\isvmname}{}. For the RBF kernel, the curves correspond to the optimum value of $\sigma$ for \formatm{SVM}{}.
Observing the plots of the accuracy, \formatm{\isvmname}{} turns out to be much more stable than \formatm{SVM}{}, getting an accuracy almost optimal and larger than that of \formatm{SVM}{} in a wide range of values of $C$. Moreover, this accuracy is achieved with a smaller number of support vectors and with less training iterations. At some point, when the value of $C$ is large enough, both \formatm{SVM}{} and \formatm{\isvmname}{} perform the same since all the support vectors of \formatm{SVM}{} also become working vectors during the training of \formatm{\isvmname}{}, and both algorithms FW{} and M-\fw{} provide the same model.
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\begin{mfigurel}{\label{FigResultsParIJCNN} Evolution of the validation results for \formatd{ijcnn}{} and \formatd{mgamma}{}, using both the linear and the RBF kernel for the optimum $\sigma$ of \formatm{SVM}{}, both for the standard \formatm{SVM}{} and the proposed \formatm{\isvmname}{}. The striped regions represent the range between minimum and maximum for the \num{10} partitions, whereas the lines in the middle represent the average values.}
\legend{\showlegenddouble{1}{3}~\formatm{\isvmname}{}}{\showlegenddouble{0}{2}~\formatm{SVM}{}\qquad\showlegenddouble{1}{3}~\formatm{\isvmname}{}}
\tikzwidth{0.49\textwidth}%
\pgfplotsset{yticklabel pos=left}%
\subfloat[\label{FigResultsParIJCNNlin} Linear kernel for \formatd{ijcnn}{}.]{\shortstack[r]{%
\pgfplotsset{xticklabels={,,}, ylabel={Accuracy ($\%$)}, ymin=6.92e+01, ymax=9.89e+01}\includetikz{demoIncSVMDataset-ijcnn-lin-01A}\\
\pgfplotsset{xticklabels={,,}, ylabel={Number of SVs}, ymin=4.42e+03, ymax=4.50e+04}\includetikz{demoIncSVMDataset-ijcnn-lin-01N}\\
\pgfplotsset{xlabel=$C$, ylabel={Iterations}, ymin=9.42e+03, ymax=4.02e+05}\includetikz{demoIncSVMDataset-ijcnn-lin-01I}%
}}\ifusetikz\hfill\else\,\fi%
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\subfloat[\label{FigResultsParMGAMMAlin} Linear kernel for \formatd{mgamma}{}.]{\shortstack[r]{%
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}}\ifusetikz\hfill\else\,\fi%
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\end{mfigurel}
}
The stability of \formatm{\isvmname}{} concerning the value of the regularization parameter suggests to fix $C$ beforehand in order to get rid of a tuning parameter. This option will be explored in the next bunch of experiments.
\subsection{Exhaustive Experiments}
\label{SecExpEx}
In the following experiments, the smaller \num{9} datasets of the second block of \ref{TabDatasets} will be used to compare exhaustively three models: \formatm{SVM}{}, the proposed \formatm{\isvmname}{}, and an alternative \formatm{\isvmname}{} model with a fixed regularization parameter (denoted as \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}), in particular $C = 1$ (normalized).
\subsubsection{Set-Up}
As in the previous experiments, the hyper-parameters will be obtained using \num{10}-fold CV (except for \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}, where $C$ is fixed and only $\sigma$ will be tuned for the RBF kernel). The stopping criterion is again $\epsilon = \num{e-5}$. Once trained, the models will be compared over the test set.
Furthermore, in order to study the significance of the differences between the models, the whole procedure, including the CV and the test phase, will be repeated \num{10} times for different training/test partitions of the data (with a proportion \SI{90}{\percent}/\SI{10}{\percent}).
\subsubsection{Results}
The results are detailed in \ref{TabResultsInc}, which includes for each of the three models the mean and standard deviation of the accuracy, the number of support vectors and the number of training iterations over the \num{10} partitions. The colours represent the rank of the models for each dataset and kernel, where the same rank is used if there is no significant difference between the models\footnote{Using a Wilcoxon signed rank test for zero median, with a significance level of $5\%$.}.
\begin{mtablel}{\label{TabResultsInc} Test results for the exhaustive experiments (\num{10} repetitions). The colour indicates the rank (the darker, the better).}
\sisetup{output-exponent-marker=\textsc{e},exponent-product={},retain-explicit-plus}
\renewcommand{}{\widthof{$\num{9.99e+9}\pm\num{9.9e+9}$}}
\begin{tabular}{l@{}c@{\enspace}*3{c@{\,}}}
\toprule
\tabformathrow{Data} & \tabformathrow{K.} & \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} & \tabformathrow{\formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}} \\
\midrule & & \multicolumn{3}{c}{\tabformathrow{Accuracy ($\%$)}} \\ \cmidrule{3-5}
\input{TableIncSVMA}
\midrule & & \multicolumn{3}{c}{\tabformathrow{Number SVs}} \\ \cmidrule{3-5}
\input{TableIncSVMN}
\midrule & & \multicolumn{3}{c}{\tabformathrow{Number Iters.}} \\ \cmidrule{3-5}
\input{TableIncSVMI}
\bottomrule
\end{tabular}
\end{mtablel}
The results are averaged as a summary in \ref{TabResultsPercentage}, where they are included as a percentage with respect to the reference \formatm{SVM}{}.
This table shows that \formatm{\isvmname}{} allows to reduce the number of support vectors, and of training iterations, to a \SI{30.1}{\percent} and a \SI{26.5}{\percent}, whereas the accuracy only drops to a \SI{99.8}{\percent}.
Moreover, using the \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{} approach allows to avoid tuning $C$, while reducing the support vectors and iterations to a \SI{26.0}{\percent} and a \SI{8.0}{\percent}, with a drop of the accuracy to only the \SI{99.7}{\percent} of the \formatm{SVM}{} accuracy.
\begin{mtable}{\label{TabResultsPercentage} Geometric mean of the test results as a percentage with respect to \formatm{SVM}{} for the exhaustive experiments.}
\sisetup{detect-all}
\begin{tabular}{l *3{S[table-format=3.2]}}
\toprule
& \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} & \tabformathrow{\formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}} \\
\midrule
\input{PercentageIncSVM}
\bottomrule
\end{tabular}
\end{mtable}
\subsection{Evolution over a Large Dataset}
This section shows the evolution of the training algorithms over a larger dataset, namely the \formatd{miniboone}{} shown in \ref{TabDatasets}, for the three approaches \formatm{SVM}{}, \formatm{\isvmname}{}, and \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}.
\subsubsection{Set-Up}
In this experiment the only kernel used is the RBF one. In order to set the hyper-parameters $C$ and $\sigma$, \num{10}-fold CV is applied over a small subsample of \num{5000} patterns. Although this approach can seem quite simplistic, it provides good enough parameters for the convergence comparison that is the goal of this experiment. In the case of \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}, $C$ is fixed as $C = 1$, and the optimal $\sigma$ of \formatm{SVM}{} is directly used instead of tuning it, so that no validation is done for this model.
Once $C$ and $\sigma$ are selected, the models are trained over the whole training set during \num{40000} iterations. During this process, intermediate models are extracted every \num{5000} iterations, simulating different selections of the stopping criterion $\epsilon$. These intermediate models (trained using \num{5000}, \num{10000}, \num{15000}... iterations) are used to predict over the test set, and thus they allow to analyse the evolution of the test accuracy as a function of the number of training iterations.
\subsubsection{Results}
The results are shown in \ref{FigResultsLarge}, which includes the evolution of the number of support vectors and the test accuracy.
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\begin{mfigure}{\label{FigResultsLarge} Evolution of the training for \formatd{miniboone}{} with RBF kernel, for the standard \formatm{SVM}{}, the proposed \formatm{\isvmname}{} and the parameter free \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}. The accuracy corresponds to the test set.}
\legend{\showlegend{2}~\formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}}{\showlegend{0}~\formatm{SVM}{}\qquad\showlegend{1}~\formatm{\isvmname}{}\qquad\showlegend{2}~\formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}}
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\end{mfigure}
}
It can be observed that the standard \formatm{SVM}{} starts with the higher accuracy, but it is rapidly matched by \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{}, and later by \formatm{\isvmname}{}. Nevertheless, all of the models get finally a comparable and stable accuracy, and they reach it at approximately the same number of iterations (around \num{15000}).
The main difference can be seen in the evolution of the number of support vectors. In the first iterations, all the models introduce a new support vector at each iteration, but first \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{} and second \formatm{\isvmname}{} saturate this number presenting a final almost flat phase. On the contrary, although \formatm{SVM}{} reduces slightly the rate of growth of the number of support vectors, it continues adding more patterns to the solution during the whole training. This means that, if the stopping criterion is not carefully chosen for \formatm{SVM}{}, this model will use much more support vectors than needed, with the corresponding increase in its complexity. On the other side, \formatm{\isvmname}{} and \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{} (both models trained with M-\fw{}) limit successfully the number of support vectors, providing sparser models with the same accuracy as \formatm{SVM}{}.
As a remark, it should be noticed that for \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{} no validation phase was needed, since $C$ is fixed beforehand, and for $\sigma$ the optimal of \formatm{SVM}{} was used.
This suggests again that \formatm{SVM$^\text{M-\fw{}}_\text{FP}$}{} can be applied successfully with $C = 1$ and only tuning $\sigma$ if the RBF kernel is to be used.
\subsection{Dependence on the Initialization}
Another aspect of the proposed algorithm is its dependence on the initialization. Whereas the standard SVM is trained by solving a convex optimization problem with unique solution in the non-degenerate case, the proposed method summarized in \ref{AlgMFW} starts with an initial working vector that influences the resulting model, since it will determine the final subset of working vectors $\ensuremath{\opt{\act}}$.
\subsubsection{Set-Up}
A comparison of the models obtained using different initial working vectors will be done to study the variability due to the initialization.
In particular, for all \num{9} smaller datasets of \ref{TabDatasets} and in this case only for the linear kernel with the parameters obtained in \ref{SecExpEx} (no CV process is repeated), one model per possible initial point will be trained, so that at the end there will be as many models as training patterns for each partition.
\subsubsection{Results}
A first measure for the dependence on the initialization are the differences between the sets of support vectors of the models. \Ref{TabInitialOverlap} shows in the second column the average overlap between these sets of support vectors for every pair of models with different initializations, quantified as the percentage of support vectors that are shared on both models over the total number of support vectors\footnote{In particular, there are $\sfrac{N \prn{N - 1}}{2}$ measures per each one of the \num{10} repetitions, since there are $N$ different possible initializations (as many as training patterns).}. The two easiest datasets, \formatd{iris}{} and \formatd{mushrooms}{}, show the smallest overlaps (around \SI{30}{\percent}) and hence the highest dependence on the initialization. This is not surprising, since for example in the \formatd{iris}{} dataset there are many hyperplanes that separate both classes perfectly. The remaining datasets show an overlap above \SI{80}{\percent}, and there are \num{4} datasets above \SI{95}{\percent}. Therefore, the influence on the initialization will depend strongly on the particular dataset.
\renewcommand{\datasettitle}[1]{\truncate[]{25pt}{#1}}
\begin{mtable}{\label{TabInitialOverlap} Results for the initialization dependence, including the overlap of the different sets of support vectors for \formatm{\isvmname}{}, and the accuracies of \formatm{SVM}{}, \formatm{\isvmname}{} and \formatm{\isvmname}{} considering all possible initializations.}
\sisetup{detect-all}
\begin{tabular}{l*4{c}}
\toprule
\multirow{2}{*}{\tabformathrow{Data}} & \tabformathrow{SVs Overlap ($\%$)} & \multicolumn{3}{c}{\tabformathrow{Accuracy ($\%$)}} \\
\cmidrule(lr){2-2} \cmidrule(lr){3-5}
& \tabformathrow{\formatm{\isvmname}{} Ini.} & \tabformathrow{\formatm{SVM}{}} & \tabformathrow{\formatm{\isvmname}{}} & \tabformathrow{\formatm{\isvmname}{} Ini.} \\
\midrule
\input{TableInitialDependence}
\bottomrule
\end{tabular}
\end{mtable}
Nevertheless, looking at the accuracies included in \ref{TabInitialOverlap}, and specifically comparing the results of \formatm{\isvmname}{} when considering only one or all the possible initializations (columns \num{4} and \num{5}), it seems that there is no noticeable difference between them. In particular, and reducing the table to a single measure, the average error is \SI{86.05}{\percent} for \formatm{SVM}{}, \SI{85.99}{\percent} for \formatm{\isvmname}{} and \SI{85.91}{\percent} for \formatm{\isvmname}{} considering all the initializations.
Moreover, as an additional experiment \ref{FigResultsParInitialDependence} shows the results of an extra \num{10}-fold CV for the \formatd{heart}{} dataset with linear kernel, including the results of \formatm{\isvmname}{} with all the possible initializations. It can observed that \formatm{\isvmname}{} performs basically the same in average when changing the initial vector, in terms of all three the accuracy, the number of support vectors and the number of iterations, although obviously the distance between minimum and maximum value for each $C$ (striped region in the plots) increases since more experiments are included.
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\begin{mfigure}{\label{FigResultsParInitialDependence} Evolution of the validation results for \formatd{heart}{} with linear kernel, for the standard \formatm{SVM}{}, the proposed \formatm{\isvmname}{} and \formatm{\isvmname}{} considering all possible initializations. The striped regions represent the range between minimum and maximum for the \num{10} partitions (\num{10} times the number of training patterns when considering all the possible initializations), whereas the lines in the middle represent the average values.}
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Therefore, it can be concluded that, although the proposed method can depend strongly on the initialization for some datasets, it seems that the resulting models are comparable in terms of accuracy, number of support vectors and required training iterations.
On the other side, it should be noticed that trying to establish a methodology to initialize in a clever way the algorithm would probably need of a considerable overhead, since the computational advantage of Frank--Wolfe and related methods is that they compute the gradient incrementally because the changes only affect a few coordinates. A comparison between all the possible initial vertices, leaving aside heuristics, would require the use of the whole kernel matrix, what could be prohibitive for large datasets.
\section{Conclusions}
\label{SecConc}
The connection between Lasso and Support Vector Machines (SVMs) has been used to propose an algorithmic improvement in the Frank--Wolfe (FW{}) algorithm used to train the SVM. This modification is based on the re-weighted Lasso to enforce more sparsity, and computationally it just requires an additional conditional check at each iteration, so that the overall complexity of the algorithm remains the same.
The convergence analysis of this Modified Frank--Wolfe (M-\fw{}) algorithm shows that it provides exactly the same SVM model that one would obtain applying the original FW{} algorithm only over a subsample of the training set.
Several numerical experiments have shown that M-\fw{} leads to models comparable in terms of accuracy, but with a sparser dual representation, requiring less iterations to be trained, and much more robust with respect to the regularization parameter, up to the extent of allowing to fix this parameter beforehand, thus avoiding its validation.
Possible lines of extension of this work are to explore other SVM formulations, for example based on the \lpg{1}{} loss, which should allow for even more sparsity. The M-\fw{} algorithm could also be applied to the training of other machine learning algorithms such as non-negative Lasso, or even to general optimization problems that permit a certain relaxation of the original formulation.
\section*{Acknowledgments}
The authors would like to thank the following organizations.
\begin{itemize*}
\item EU: The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC AdG A-DATADRIVE-B (290923). This paper reflects only the authors' views, the Union is not liable for any use that may be made of the contained information.
\item Research Council KUL: GOA/10/09 MaNet, CoE PFV/10/002 (OPTEC), BIL12/11T; PhD/Postdoc grants.
\item Flemish Government:
\begin{itemize*}
\item FWO: G.0377.12 (Structured systems), G.088114N (Tensor based data similarity); PhD/Postdoc grants.
\item IWT: SBO POM (100031); PhD/Postdoc grants.
\end{itemize*}
\item iMinds Medical Information Technologies SBO 2014.
\item Belgian Federal Science Policy Office: IUAP P7/19 (DYSCO, Dynamical systems, control and optimization, 2012-2017).
\item Fundaci\'on BBVA: project FACIL--Ayudas Fundaci\'on BBVA a Equipos de Investigaci\'on Cient\'ifica 2016.
\item UAM--ADIC Chair for Data Science and Machine Learning.
\end{itemize*}
\section*{References}
\bibliographystyle{elsarticle-num}
|
1,108,101,565,472 | arxiv | \section{Introduction}\label{sec-into}
Cyclic codes are an important class of linear codes. Due to their desirable algebraic properties and efficient algorithms for encoding and decoding processes, cyclic codes have been widely used in many areas such as communication and data storage system. They can also be used to construct other interesting structures such as quantum codes \cite{qc}, frequency hopping sequences \cite{fhs} and so on.
Let $p$ be a prime number, $l \ge 1, q=p^l$ and ${\mathrm{GF}}(q)$ be the finite field of order $q$. A cyclic code $\mathcal{C}$ of length $n$ over ${\mathrm{GF}}(q)$ (assume $(n,q)=1$), by the one-to-one correspondence
$$\begin{array}{cccl}
\sigma:& \mathcal{C}&\rightarrow &R:={\mathrm{GF}}(q)[x]/(x^n-1)\\
&(c_0,c_1,\cdots ,c_{n-1})&\mapsto&c_0+c_1x+\cdots +c_{n-1}x^{n-1},
\end{array}$$
can be identified with an ideal of $R$. There exists a unique monic polynomial $g(x)$ with least degree such that $\sigma(\mathcal{C})=g(x)R$ and $g(x)\mid (x^n-1)$. The $g(x)$ is called the \textit{generator polynomial} and $h(x):=(x^n-1)/g(x)$ is called the \textit{parity-check polynomial} of $\mathcal{C}$. If $h(x)$ has $t$ irreducible factors over ${\mathrm{GF}}(q)$, we follow the literature and say that ``the dual of $\mathcal{C}$ has $t$ zeroes''. (Note that this is different from \cite{XLZD} in which we call ``$\mathcal{C}$ has $t$ zeroes'' instead.) $\mathcal{C}$ is called \emph{irreducible} if $t=1$ and \emph{reducible} if $t \ge 2$.
Denote by $A_i$ the number of codewords of $\mathcal{C}$ with Hamming weight $i$, where $0 \le i \le n$. The study of the weight distribution $(A_0,A_1,\cdots ,A_n)$ or equivalently the weight enumerator $1+A_1Y+A_2Y^2+\cdots+A_nY^n$ is important in both theory and application, because the weight distribution gives the minimum distance and thus the error correcting capability of the code, and the weight distribution allows the computation of the probability of error detection and correction with respect to some algorithms \cite{Klov}. Moreover, the weight distribution is related to interesting and challenging problems in number theory (\cite{cal,Schroof}).
In a recent paper \cite{gegeng2}, the authors constructed some classes of cyclic codes whose duals have two Niho type zeroes and obtained the weight distribution. These classes of cyclic codes are quite interesting because they contain some optimal cyclic codes (among linear codes) and in general have only three or four non-zero weights. This beautiful work was recently extended to more general classes of cyclic codes whose duals may have arbitrary number of Niho type zeroes (see \cite{XLZD}). The purpose of this paper is to extend these works in yet two other directions. It is interesting to note that this not only vastly generalizes the construction of \cite{gegeng2,XLZD}, but also yields many optimal or almost optimal cyclic codes with very few non-zero weights, none of which was present in \cite{gegeng2,XLZD} (See Examples \ref{1:ex1}--\ref{2:ex2}, Tables \ref{Table:ex1} and \ref{Table:ex2} in Section \ref{sec-2}). We study the weight distribution of these cyclic codes, by employing similar ideas from \cite{gegeng2,XLZD} and by carrying out some quite subtle analysis of certain exponential sums.
In recent years, for many families of cyclic codes the weight distribution problem has been solved. We only mention here that most of the results are for cyclic codes whose duals have no more than three zeroes (see for example \cite{AL06,BM72,BM73,fit,McE72,Rao10,schmidt,van,Vega1,Vega2,wol,D-Y12} and \cite{Ding2,FL08,Feng12,F-M12,holl,luo2,luo3,luo4,Ding1,M09,Mois09,
Vega12,Tang12,Xiong1,Xiong2,Xiong3,zeng}). There are only a few results for cyclic codes whose duals may have arbitrary number of zeroes (\cite{gegeng, Y-X-D12,Y-X-X14,XLZD}). The duals of the cyclic codes considered in this paper may also have arbitrary number of zeroes.
The paper is organized as follows. In Section \ref{sec-2}, for any prime $p$, we introduce the cyclic codes ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ and ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ and the main results (Theorems \ref{1:thm1} and \ref{1:thm2}). The special cases of 3-weight and 4-weight cyclic codes are presented in Corollaries \ref{cor:ca} and \ref{cor:cb}. Then we provide numerical examples of optimal or almost optimal cyclic codes over ${\mathrm{GF}}(4)$ and ${\mathrm{GF}}(8)$ and compute the weight distribution. We also compile a list of such codes over other finite fields. In Section \ref{sec-00} we prove a simple lemma which will be used later. For $p=2$, Statements (i) of Theorems \ref{1:thm1} and \ref{1:thm2} are proved in Section \ref{sec-3}, and Statements (ii) of Theorems \ref{1:thm1} and \ref{1:thm2} are proved in Section \ref{sec-4}. For $p \ge 3$, Statements (i) and (ii) of Theorems \ref{1:thm1},\ref{1:thm2} are proved in Sections \ref{sec-32} and \ref{sec-42} respectively. As was noted in \cite{gegeng2,XLZD}, we remark here that the proofs for $p=2$ and $p \ge 3$ are quite different. In Section \ref{sec-conclusion} we conclude the paper.
\section{Cyclic codes ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ and ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$} \label{sec-2}
From what follows, let $p$ be a prime number, and $l,m$ be positive integers. Let $q=p^l$ and $r=q^m$. Denote by ${\mathrm{GF}}(r^2)$ the finite field of order $r^2$. Let $\gamma$ be a primitive element of ${\mathrm{GF}}(r^2)$.
\subsection{Cyclic code ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$}
For any integers $h,f$, define
\begin{eqnarray} \label{2:e1} e=(h,r+1), \quad \delta=\left((r+1)f,(r-1)e\right), \quad n=(r^2-1)/\delta. \end{eqnarray}
Let $t$ be an integer and assume that
\begin{itemize}
\item[(a).] $1 \le t < \frac{r+1}{2e}$,
\item[(b).] $\left(f,\frac{r-1}{q-1}\right)=1$,
\item[(c).] if $p$ is odd, then $m$ is odd, or $m$ and $h$ are both even.
\end{itemize}
Let $d_0,d_1,\ldots,d_t$ be integers such that
\begin{eqnarray} \label{2:da1}
d_j \equiv (jh+f) (r-1)+2 f \pmod{r^2-1}, \quad 0 \le j \le t\, .
\end{eqnarray}
A positive integer $d$ is called a {\em Niho exponent} if $d \equiv q^j\pmod{r-1}$ for some $j$. The Niho exponents were originally introduced by Niho \cite{Niho-PhD} who investigated the cross-correlation between an $m$-sequence and its decimation. Since then, Niho exponents were further studied and had been used in other research topics. Here $d_j \equiv 2f \pmod{r-1}\, \forall j$ and $\left(f,\frac{r-1}{q-1}\right)=1$, the $d_j$'s are called the ``generalized Niho exponents'', and the ${\gamma}^{-d_j}$'s are called the ``generalized Niho type zeroes''.
It can be seen that $(d_0,d_1,\ldots,d_t,r^2-1)=\delta$. The $q$-ary cyclic code ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ of length $n$ consists of elements $c(\vec{a})$ given by
\begin{equation}\label{2:ca}
\begin{array}{l}
c(\vec{a})=\left({\mathrm{Tr}}_{r/q}\left(a_0 \gamma^{d_0 i}\right)+{\mathrm{Tr}}_{r^2/q}\left(\sum_{j=1}^t a_j \gamma^{d_j i} \right)\right)_{i=0}^{n-1},
\end{array}\end{equation}
where $\vec{a}=(a_0,a_1,\ldots,a_t)$ for any $a_1,\cdots,a_{t}\in {\mathrm{GF}}(r^2)$ and $a_0 \in {\mathrm{GF}}(r)$. Here ${\mathrm{Tr}}_{r/q}$ and ${\mathrm{Tr}}_{r^2/q}$ denote the trace map from ${\mathrm{GF}}(r)$ and ${\mathrm{GF}}(r^2)$ to ${\mathrm{GF}}(q)$ respectively. It will be seen that the dimension of ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ is always $(2t+1)m$ and the dual of ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ has the $(t+1)$ zeroes ${\gamma}^{-d_0},\ldots, {\gamma}^{-d_t}$.
We remark that a similar ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ over ${\mathrm{GF}}(p)$ was constructed in \cite{XLZD}, but it required $(2f,r-1)=1$, which is valid only when $p=2$. Here we consider ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ over ${\mathrm{GF}}(q)$ for any $p$ under more flexible conditions. Note that (b) reduces to $(2f,r-1)=1$ only if $q=p=2$. As it turns out, the weight distribution of the new ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ is very similar to that in \cite{XLZD}. However, the proofs are much more involved. For the sake of completeness, we describe the weight distribution as follows.
Let $N_0=1,N_1=0$ and define
\begin{eqnarray} \label{2:nr}
N_k:=k! e^k \sum_{\substack{\lambda_2,\lambda_3,\ldots,\\
\sum_{j \ge 2} j \lambda_j=k}} \binom{\frac{r+1}{e}}{\sum_{j } \lambda_j} \left(\sum_{j } \lambda_j\right)! \prod_{j } \frac{\left(B_j/j!\right)^{\lambda_j}}{(\lambda_j)!},\quad \forall \, k \ge 2. \end{eqnarray}
Here the summation is over all non-negative integers $\lambda_2,\lambda_3,\ldots$ such that $\sum_{j \ge 2}j \lambda_j=k$ and
\begin{eqnarray*}
B_j:=r^{-1}(r-1)^j+(-1)^j(1-r^{-1}),\end{eqnarray*}
and $\binom{u}{v}$ is the standard binomial coefficient ``$u$-choose-$v$''. It is easy to compute that $N_2=e(r^2-1),N_3=e^2(r-2)(r^2-1)$, $N_4=e^2(r^2-1)\left\{(e+3)r^2-6er+6e-3\right\}$, etc. We prove the following.
\begin{thm} \label{1:thm1}
(i). For $p=2$ or $p$ being an odd prime, under assumptions (\ref{2:e1})--(\ref{2:ca}) and (a)--(c), the code ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}} $ is a $q$-ary cyclic code of length $n=(r^2-1)/\delta$ and dimension $(2t+1)m$, with at most $(2t+1)$ non-zero weights, each of which is given by
\begin{eqnarray} \label{2:wj} w_j=\frac{q-1}{q \delta} \cdot \left(r^2-(je-1)r\right), \, 0 \le j \le 2t. \end{eqnarray}
(ii). Let $\mu_j$ be the frequency of the weight $w_j$ for each $j$. Define $\vec{\mu}=(\mu_0,\mu_1,\ldots,\mu_{2t})^T$, and $\vec{b}=(b_0,b_1,\ldots,b_{2t})^T$ where $b_i=r^{2t+1}N_i-\left(r^2-1\right)^i$. Then
\[\vec{\mu}=\left(M^{(1)}_t\right)^{-1}\vec{b}. \]
Here $M_t^{(1)}=[m_{ij}]_{0 \le i,j \le 2t}$ is an invertible Vandermonde matrix whose entry is given by $m_{ij}=\left(jer-r-1\right)^i$.
\end{thm}
By using computer algebra such as {\bf Mathematica}, Theorem \ref{1:thm1} can be used easily to compute the weight distribution of ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ explicitly for any $t$, though the results are quite complicated to be written down even for $t =2$. When $t=1$ which is the most interesting, it is a 3-weight cyclic code, whose weight distribution can be described as follows.
\begin{cor} \label{cor:ca} Under assumptions (\ref{2:e1})--(\ref{2:ca}) and (a)--(c) for $t=1$, the code ${\mathcal{C}_{(d_0,d_1)}^{(1)}}$ is a 3-weight cyclic code of length $n=(r^2-1)/\delta$ and dimension $3m$. The weight distribution is given by Table \ref{Table1}.
\end{cor}
\begin{table}[ht]
\caption{The weight distribution of ${\mathcal{C}_{(d_0,d_1)}^{(1)}}$}\label{Table1}
\begin{center}{
\begin{tabular}{|l|l|}
\hline
Weight & Frequency \\
\hline
\hline
$0$ & once \\
\hline
$\frac{q-1}{q \delta} \left(r^2+r\right)$ & $\frac{-1+3e - 2 e^2 - q + 2 e q + r^2 - 3 e r^2 + r^3 -
2 e r^3 + 2 e^2 r^3}{2e^2}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(e-1)r\right)$ & $\frac{1 - 2 e + r - e r - r^2 + 2 e r^2 - r^3 + e r^3}{e^2}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(2e-1)r\right)$ & $\frac{-1 + e - r + r^2 - e r^2 + r^3}{2e^2}$\\
\hline
\end{tabular}}
\end{center}
\end{table}
From what follows we present some interesting examples of cyclic codes from ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ over ${\mathrm{GF}}(4)$ and ${\mathrm{GF}}(8)$ respectively which we find optimal or almost optimal by checking ``Bounds on the minimum distance of linear codes'' provided by the website http://www.codetables.de/. Note that these cyclic codes have only a few none-zero weights. Here we omit optimal cyclic codes which could be obtained from \cite{gegeng2,XLZD} and hence only consider the case that $(f,r-1) >1$.
\begin{exam} \label{1:ex1} Let $q=4,m=2,r=16,h=1,f=3$. Then $e=1,(f,r-1)=3$.
\begin{itemize}
\item[(1).] $t=1$: $(d_0,d_1)=(51,66)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(51,66)}^{(1)}$ is a three-weight cyclic code with the weight enumerator
\[1+2040Y^{60}+255Y^{64}+1800Y^{68}.\]
This is a $[85,6,60]$ code over ${\mathrm{GF}}(4)$ which is optimal among linear codes.
\item[(2).] $t=2$: $(d_0,d_1,d_2)=(51,66,81)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(51,66,81)}^{(1)}$ is a five-weight cyclic code with the weight enumerator
\[1+35700Y^{52}+30600Y^{56}+250920Y^{60}+377655Y^{64}+353700Y^{68}.\]
This is a $[85,10,52]$ code over ${\mathrm{GF}}(4)$. It is known that for optimal linear codes of length 85 and dimension 10 over ${\mathrm{GF}}(4)$, the minimal distance satisfies $52 \le d \le 56$.
\item[(3).] $t=3$: $(d_0,d_1,d_2,d_3)=(51,66,81,96)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(51,66,81,96)}^{(1)}$ is a seven-weight cyclic code with the weight enumerator
$1+185640Y^{44}+464100Y^{48}+4641000Y^{52}+17646000Y^{56}+
54396600Y^{60}+101483115Y^{64}+89619000Y^{68}$. This is a $[85,14,44]$ code over ${\mathrm{GF}}(4)$. It is known that for optimal linear codes of length 85 and dimension 14 over ${\mathrm{GF}}(4)$, the minimal distance satisfies $48 \le d \le 53$.
\end{itemize}
\end{exam}
\begin{exam} \label{1:ex2} Let $q=8,m=1,r=8,h=1,f=7,r^2=64$. Then $e=1,(f,r-1)=7$.
\begin{itemize}
\item[(1).] $t=1$: $(d_0,d_1)=(63,70)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(63,70)}^{(1)}$ is a three-weight cyclic code with the weight enumerator
\[1+252Y^{7}+63Y^{8}+196Y^{9}.\]
This is a $[9,3,7]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\item[(2).] $t=2$: $(d_0,d_1,d_2)=(63,70,77)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(63,70,77)}^{(1)}$ is a five-weight cyclic code with the weight enumerator
\[1+882Y^{5}+1764Y^{6}+7812Y^{7}+12411Y^8+9898Y^9.\]
This is a $[9,5,5]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\item[(3).] $t=3$: $(d_0,d_1,d_2,d_3)=(63,70,77,84)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(63,70,77,84)}^{(1)}$ is a seven-weight cyclic code with the weight enumerator
\[1+588Y^{3}+4410Y^{4}+33516Y^{5}+154056Y^{6}+463428Y^7+810621Y^8+630532Y^9.\]
This is a $[9,7,3]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\end{itemize}
\end{exam}
Now we present a table of cyclic codes from ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ over ${\mathrm{GF}}(3),{\mathrm{GF}}(9),{\mathrm{GF}}(5)$ and ${\mathrm{GF}}(7)$ respectively which we find optimal or almost optimal by checking ``Bounds on the minimum distance of linear codes'' provided by the website http://www.codetables.de/. For simplicity, only the parameters of the codes are listed. None of these cyclic codes can be obtained from \cite{gegeng2,XLZD}.
\begin{table}[ht]
\caption{Optimal or almost optimal cyclic codes from ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$}\label{Table:ex1}
\begin{center}{
\begin{tabular}{|l|l|c|c|c|}
\hline
$q$ & $r$ & $(t=,h=,f=)$ & code parameters & Optimal (?)\\
\hline
\hline
$3$ & $27$ & $(1,2,1)$ & $[182,9,108]$ & $111 \le d \le 115$ is optimal \\
\hline
$9$ & $9$ & $(1,1,2)$ & $[20,3,16]$ & $16 \le d \le 17$ is optimal\\
\hline
$9$ & $9$ & $(1,1,4)$ & $[10,3,8]$ &Y\\
\hline
$9$ & $9$ & $(2,1,4)$ & $[10,5,6]$ &Y\\
\hline
$9$ & $9$ & $(3,1,4)$ & $[10,7,4]$ &Y\\
\hline
$9$ & $9$ & $(4,1,4)$ & $[10,9,2]$ &Y\\
\hline
$9$ & $9$ & $(1,2,8)$ & $[5,3,3]$ &Y\\
\hline
\hline
$5$ & $5$ & $(1,1,1)$ & $[12,3,8]$ &Y\\
\hline
$5$ & $5$ & $(1,1,2)$ & $[6,3,4]$ &Y\\
\hline
$5$ & $5$ & $(2,1,2)$ & $[6,5,2]$ &Y\\
\hline
\hline
$7$ & $7$ & $(1,1,2)$ & $[24,3,18]$ &$d=19$ is optimal\\
\hline
$7$ & $7$ & $(1,1,3)$ & $[8,3,6]$ &Y\\
\hline
$7$ & $7$ & $(2,1,3)$ & $[8,5,4]$ &Y\\
\hline
$7$ & $7$ & $(3,1,3)$ & $[8,7,2]$ &Y\\
\hline
$7$ & $7$ & $(1,2,3)$ & $[4,3,2]$ &Y\\
\hline
\end{tabular}}
\end{center}
\end{table}
\subsection{Cyclic code ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$}
For any integers $h,f$ and $t \ge 1$, define
\begin{eqnarray} \label{2:e2} e=(h,r+1), \quad \delta=\left\{\begin{array}{ll}
\left(\frac{h+f}{2}(r-1)+f,r^2-1\right) & \mbox{ if } t=1\\
\left(\frac{h+f}{2}(r-1)+f,(r-1)e\right) & \mbox{ if } t \ge 2 \end{array} \right., \quad n=(r^2-1)/\delta. \end{eqnarray}
Assume that
\begin{itemize}
\item[(a').] $1 \le t \le \frac{r+1}{2e}$,
\item[(b').] if $p=2$, then $\left(f,\frac{r-1}{q-1}\right)=1$,
\item[(c').] if $p \ge 3$, then
\begin{itemize}
\item[(c1').] $h \equiv f \pmod{2}$ and $\left(f,\frac{r-1}{q-1}\right)=1$, or
\item[(c2').] $h \equiv f \equiv 0 \pmod{2}$ and $\left(\frac{f}{2},\frac{r-1}{q-1}\right)=1$.
\end{itemize}
\end{itemize}
Let $\widetilde{d}_1,\ldots,\widetilde{d}_t$ be integers such that
\begin{eqnarray} \label{2:db}
\widetilde{d}_j \equiv \left(j \cdot h+\frac{f-h}{2}\right)(r-1)+f \pmod{r^2-1}, \quad 1 \le j \le t.
\end{eqnarray}
Here if $p=2$, the number $\frac{1}{2}$ shall be interpreted as an integer which is the multiplicative inverse of $2 \pmod{r-1}$. It can be seen that $(\widetilde{d}_1,\ldots,\widetilde{d}_t,r^2-1)=\delta$. The $q$-ary cyclic code ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ of length $n$ consists of elements $\widetilde{c}(\vec{a})$ given by
\begin{equation}\label{2:cb}
\begin{array}{l}
\widetilde{c}(\vec{a})=\left({\mathrm{Tr}}_{r^2/q}\left(\sum_{j=1}^t a_j \gamma^{\widetilde{d}_j i} \right)\right)_{i=0}^{n-1},
\end{array}\end{equation}
where $\vec{a}=(a_1,\ldots,a_t)$ for any $a_1,\cdots,a_{t}\in {\mathrm{GF}}(r^2)$. Note that the dual of ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ has the $t$ zeroes $\gamma^{-\widetilde{d}_1},\cdots,\gamma^{-\widetilde{d}_t}$. Here $\widetilde{d}_j \equiv f \pmod{r-1} \, \forall j$, the ${\gamma}^{-\widetilde{d}_j}$'s are call the ``generalized Niho type zeroes''.
We remark that a similar ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ over ${\mathrm{GF}}(p)$ was constructed in \cite{XLZD} under the condition $(f,r-1)=1$ (see \cite[Theorem 1]{XLZD}). Clearly the conditions (b')(c') are more general and provide more flexible parameters. The weight distribution of ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ can be described as follows.
\begin{thm} \label{1:thm2} (i). For $p=2$ or $p$ being an odd prime, under assumptions (\ref{2:e2})--(\ref{2:cb}) and (a')--(c'), the code ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}} $ is a $q$-ary cyclic code of length $n=(r^2-1)/\delta$ and dimension $2tm$, with at most $2t$ non-zero weights, each of which is given by
\[\widetilde{w}_j=\frac{q-1}{q\delta} \cdot \left(r^2-(je-1)r\right), \, 0 \le j \le 2t-1. \]
(ii). Let $\widetilde{\mu}_j$ be the frequency of the weight $\widetilde{w}_j$ for each $j$. Define $\vec{\widetilde{\mu}}=(\widetilde{\mu}_0,\widetilde{\mu}_1,\ldots,\widetilde{\mu}_{2t-1})^T$, and $\vec{\widetilde{b}}=(\widetilde{b}_0,\widetilde{b}_1,\ldots,\widetilde{b}_{2t-1})^T$ where $\widetilde{b}_i=r^{2t}N_i-\left(r^2-1\right)^i$. Then
\[\vec{\widetilde{\mu}}=\left(M^{(2)}_t\right)^{-1}\vec{\widetilde{b}}. \]
Here $M_t^{(2)}=[m_{ij}]_{0 \le i,j \le 2t-1}$ is a $2t \times 2t$ Vandermonde matrix whose entry is given by $m_{ij}:=\left(jer-r-1\right)^i$.
\end{thm}
\begin{cor} \label{cor:cb} (i). Under assumptions (\ref{2:e2})--(\ref{2:cb}) and (a')--(c') for $t=1$, the code ${\mathcal{C}_{(\widetilde{d}_1)}^{(2)}}$ is a $q$-ary cyclic code of length $n=(r^2-1)/\delta$ and dimension $2m$ with most two non-zero weights. The weight distribution is given by Table \ref{Table2}. Note that it is a 1-weight code if and only if $e=1$.
(ii). Under assumptions (\ref{2:e2})--(\ref{2:cb}) and (a')--(c') for $t=2$, the code ${\mathcal{C}_{(\widetilde{d}_1,\widetilde{d}_2)}^{(2)}}$ is a 4-weight cyclic code of length $n=(r^2-1)/\delta$ and dimension $4m$. The weight distribution is given by Table \ref{Table3}.
\end{cor}
Since ${\mathcal{C}_{(\widetilde{d}_1)}^{(2)}}$ is irreducible, (i) of Corollary \ref{cor:cb} should be known to researchers in the field. We collect the result here only for the sake of completeness. However, the dual of ${\mathcal{C}_{(\widetilde{d}_1,\widetilde{d}_2)}^{(2)}}$ has two zeroes, and (ii) of Corollary \ref{cor:cb} is new.
\begin{table}[ht]
\caption{The weight distribution of ${\mathcal{C}_{(\widetilde{d}_1)}^{(2)}}$} \label{Table2}
\begin{center}{
\begin{tabular}{|l|l|}
\hline
Weight & Frequency \\
\hline
\hline
$0$ & once \\
\hline
$\frac{q-1}{q \delta} \left(r^2+r\right)$ & $\frac{(e-1)(r^2-1)}{e}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(e-1)r\right)$ & $\frac{r^2-1}{e}$\\
\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}[ht]
\caption{The weight distribution of ${\mathcal{C}_{(\widetilde{d}_1,\widetilde{d}_2)}^{(2)}}$}\label{Table3}
\begin{center}{
\begin{tabular}{|l|l|}
\hline
Weight & Frequency \\
\hline
\hline
$0$ & once \\
\hline
$\frac{q-1}{q \delta} \left(r^2+r\right)$ & $\frac{1 - 6 e + 11 e^2 - 6 e^3 + 2 r - 9 e r + 9 e^2 r +
3 e r^2 - 5 e^2 r^2 - 2 r^3 + 9 e r^3 - 9 e^2 r^3 -
r^4 + 3 e r^4 - 6 e^2 r^4 + 6 e^3 r^4}{6e^3}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(e-1)r\right)$ & $\frac{-1 + 5 e - 6 e^2 - 2 r + 7 e r - 4 e^2 r - 3 e r^2 +
4 e^2 r^2 + 2 r^3 - 7 e r^3 + 4 e^2 r^3 + r^4 -
2 e r^4 + 2 e^2 r^4}{2e^3}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(2e-1)r\right)$ & $\frac{1 - 4 e + 3 e^2 + 2 r - 5 e r + e^2 r + 3 e r^2 -
3 e^2 r^2 - 2 r^3 + 5 e r^3 - e^2 r^3 - r^4 + e r^4}{2e^3}$\\
\hline
$\frac{q-1}{q \delta} \left(r^2-(3e-1)r\right)$ & $\frac{-1 + 3 e - 2 e^2 - 2 r + 3 e r - 3 e r^2 + 2 e^2 r^2 +
2 r^3 - 3 e r^3 + r^4}{6e^3}$\\
\hline
\end{tabular}}
\end{center}
\end{table}
From what follows we present some interesting examples of cyclic codes from ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ over ${\mathrm{GF}}(4)$ and ${\mathrm{GF}}(8)$ respectively which we find optimal or almost optimal by checking ``Bounds on the minimum distance of linear codes'' provided by the website http://www.codetables.de/. Note that these cyclic codes have only a few none-zero weights. Here we omit optimal cyclic codes which could be obtained from \cite{gegeng2,XLZD} and hence only consider the case that $(f,r-1) >1$.
\begin{exam} \label{2:ex1}Let $q=4,m=2,r=16,h=2,f=6$. Then $e=1,(f,r-1)=3$.
\begin{itemize}
\item[(1).] $t=1$: $(\widetilde{d}_1)=(66)$. Both Theorem \ref{1:thm2} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(66)}^{(1)}$ is a one-weight cyclic code with the weight enumerator
\[1+255Y^{64}.\]
This is a $[85,4,64]$ code over ${\mathrm{GF}}(4)$ which is optimal among linear codes.
\item[(2).] $t=2$: $(\widetilde{d}_1,\widetilde{d}_2)=(66,96)$. Both Theorem \ref{1:thm2} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(66,96)}^{(1)}$ is a four-weight cyclic code with the weight enumerator
\[1+10200Y^{56}+4080Y^{60}+30855Y^{64}+20400Y^{68}.\]
This is a $[85,8,56]$ code over ${\mathrm{GF}}(4)$. It is known that for optimal linear codes of length 85 and dimension 8 over ${\mathrm{GF}}(4)$, the minimal distance satisfies $56 \le d \le 59$.
\item[(3).] $t=3$: $(\widetilde{d}_1,\widetilde{d}_2,\widetilde{d}_3)=(66,96,126)$. Both Theorem \ref{1:thm2} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(66,96,126)}^{(1)}$ is a six-weight cyclic code with the weight enumerator
$1+92820Y^{48}+142800Y^{52}+1285200Y^{56}+3272160Y^{60}+
6390555Y^{64}+5593680Y^{68}$. This is a $[85,12,48]$ code over ${\mathrm{GF}}(4)$. It is known that for optimal linear codes of length 85 and dimension 12 over ${\mathrm{GF}}(4)$, the minimal distance satisfies $48 \le d \le 55$.
\end{itemize}
\end{exam}
\begin{exam} \label{2:ex2} Let $q=8,m=1,r=8,h=2,f=14,r^2=64$. Then $e=1,(f,r-1)=7$.
\begin{itemize}
\item[(1).] $t=1$: $(\widetilde{d}_1)=(70)$. Both Theorem \ref{1:thm1} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(70)}^{(2)}$ is a one-weight cyclic code with the weight enumerator
\[1+63Y^{8}.\]
This is a $[9,2,8]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\item[(2).] $t=2$: $(\widetilde{d}_1,\widetilde{d}_2)=(70,84)$. Both Theorem \ref{1:thm2} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(70,84)}^{(2)}$ is a four-weight cyclic code with the weight enumerator
\[1+588Y^{6}+504Y^{7}+1827Y^{8}+1176Y^9.\]
This is a $[9,4,6]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\item[(3).] $t=3$: $(\widetilde{d}_1,\widetilde{d}_2,\widetilde{d}_3)=(70,84,98)$. Both Theorem \ref{1:thm2} and numerical computation by {\bf Magma} show that $\mathcal{C}_{(70,84,98)}^{(2)}$ is a six-weight cyclic code with the weight enumerator
\[1+882Y^{4}+3528Y^{5}+19992Y^{6}+57456Y^{7}+101493Y^8+78792Y^9.\]
This is a $[9,6,4]$ code over ${\mathrm{GF}}(8)$ which is optimal among linear codes.
\end{itemize}
\end{exam}
Now we present a table of cyclic codes from ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ over ${\mathrm{GF}}(3),{\mathrm{GF}}(9),{\mathrm{GF}}(5)$ and ${\mathrm{GF}}(7)$ respectively which we find optimal or almost optimal by checking ``Bounds on the minimum distance of linear codes'' from the website http://www.codetables.de/. For simplicity, only the parameters of the code are listed. Note that none of these cyclic codes can be obtained from \cite{gegeng2,XLZD}.
\begin{table}[ht]
\caption{Optimal or almost optimal cyclic codes from ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$}\label{Table:ex2}
\begin{center}{
\begin{tabular}{|l|l|c|c|c|}
\hline
$q$ & $r$ & $(t=,h=,f=)$ & code parameters & Optimal (?)\\
\hline
\hline
$3$ & $27$ & $(1,4,2)$ & $[91,6,54]$ & $57 \le d \le 58$ is optimal \\
\hline
\hline
$9$ & $9$ & $(1,2,8)$ & $[5,2,4]$ & Y \\
\hline
$9$ & $9$ & $(2,2,8)$ & $[5,4,2]$ & Y \\
\hline
\hline
$7$ & $7$ & $(1,1,3)$ & $[16,2,14]$ & Y \\
\hline
$7$ & $7$ & $(2,1,3)$ & $[16,4,10]$ & $d=11$ is optimal \\
\hline
\end{tabular}}
\end{center}
\end{table}
\section{Preliminaries}\label{sec-00}
Following notation from Section \ref{sec-2}, let $p$ be a prime, $q=p^l, r=q^m$ and let $\gamma$ be a primitive element of ${\mathrm{GF}}(r^2)$. Define ${\mathrm{GF}}(r^2)^*:={\mathrm{GF}}(r^2)-\{0\}$. For any integer $d$, let $h_d(x) \in {\mathrm{GF}}(q)[x]$ be the minimal polynomial of $\gamma^{-d}$ over ${\mathrm{GF}}(q)$. We first prove the following lemma, the proof of which is similar to \cite[Lemma 5]{XLZD}.
\begin{lemma} \label{3:lem33} Suppose $\left(\triangle,\frac{r-1}{q-1}\right)=1$ or $\left(\frac{\triangle}{2},\frac{r-1}{q-1}\right)=1$ if $\triangle$ is even.
\begin{itemize}
\item[(i).] If $d=s(r-1)+\triangle$, then $\deg h_d(x)=\left\{\begin{array}{lll}
m &:& \mbox{ if } \triangle \equiv 2s \pmod{r+1},\\
2m &:& \mbox{ if } \triangle \not \equiv 2s \pmod{r+1}. \end{array}\right.$
\item[(ii).] If $d=s(r-1)+\triangle$ and $d'=s'(r-1)+\triangle$, then
$h_{d}(x) = h_{d'}(x)$ if and only if $s \equiv s' \pmod{r+1}$ or $s+s' \equiv \triangle \pmod{r+1}$.
\end{itemize}
\end{lemma}
\begin{proof}
(i). $\deg h_d(x)$ is the least positive integer $k$, $1 \le k \le 2m$ such that
$d q^k \equiv d \pmod{r^2-1}$. Since $d \equiv \triangle \pmod{r-1}$, we have $(q^m-1)|\triangle(q^k-1)$. Dividing $q-1$ on both sides, we find that $(q^m-1)|2(q^k-1)$. Let $\nu=(m,k)$ and $\lambda=\frac{q^m-1}{q^{\nu}-1}$. Then $\left(\lambda,\frac{q^k-1}{q^{\nu}-1}\right)=1$, and we have $\lambda|2$. If $\lambda=2$, then $m \ge 2$ and $\nu<m$, hence $\nu \le \frac{m}{2}$. We have $q^m-1=2(q^{\nu}-1) \le 2(q^{m/2}-1)$. This implies that $q \le q^{m/2}<2$, contradiction. So we must have $\lambda=\frac{q^m-1}{q^{\nu}-1}=1$, that is, $\nu=m$, hence $k=m$ or $2m$.
If $k=m$, this is equivalent to $d(r-1) \equiv 0 \pmod{r^2-1}$, that is $d \equiv 0 \pmod{r+1}$, and hence $(r+1)|(\triangle -2s)$. If $(r+1) \nmid (\triangle-2s)$, we must have $k=2m$.
(ii). $h_{d}(x) = h_{d'}(x)$ if and only if there exists an integer $k$, $1 \le k \le 2m$ such that $dq^k \equiv d' \pmod{r^2-1}$. Reducing the equation modulo $r-1$, by similar argument we find that $k=m$ or $2m$. If $k=2m$, then obviously $s \equiv s' \pmod{r+1}$. Otherwise $k=m$, we have $\left(s(r-1)+\triangle\right)r \equiv s'(r-1)+\triangle \pmod{r^2-1}$. This is equivalent to $\triangle \equiv s+s' \pmod{r+1}$ by simple computation. This completes the proof of Lemma \ref{3:lem33}.
\end{proof}
From Lemma \ref{3:lem33} we immediately obtain the following.
\begin{lemma} \label{3:lem2} (1). For ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$, let assumptions be as in Theorem \ref{1:thm1}. Then
\begin{itemize}
\item[(i).] $\deg h_{d_0}(x)=m$ and $\deg h_{d_i}(x)=2m, \forall \, 1 \le i \le t$.
\item[(ii).] $h_{d_i}(x) \ne h_{d_j}(x)$ for any $0 \le i \ne j \le t$.
\end{itemize}
\noindent (2). For ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$, let assumptions be as in Theorem \ref{1:thm2}. Then
\begin{itemize}
\item[(i).] $\deg h_{\widetilde{d}_i}(x)=2m, \forall \, 1 \le i \le t$.
\item[(ii).] $h_{\widetilde{d}_i}(x) \ne h_{\widetilde{d}_j}(x)$ for any $1 \le i \ne j \le t$.
\end{itemize}
\end{lemma}
\section{$p=2$: Proofs of (i) of Theorems \ref{1:thm1} and \ref{1:thm2}}\label{sec-3}
We first prove Statement (i) of Theorem \ref{1:thm1}. By Delsarte's Theorem \cite{dels} and Lemma \ref{3:lem2}, ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ is a cyclic code of length $n$ with parity-check polynomial given by $\prod_{i=0}^t h_{d_i}(x)$, which is of degree $(2t+1)m$, hence ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ has dimension $(2t+1)m$ over ${\mathrm{GF}}(q)$. Since
\[\delta=\left(d_0,\ldots,d_t,r^2-1\right)=\left((r+1)f,(r-1)e\right),\]
we see that the Hamming weight of a codeword $c(\vec{a})$ can be expressed as
\begin{eqnarray*} \delta \omega_H(c(\vec{a}))&=&r^2-\#\left\{x \in {\mathrm{GF}}(r^2): {\mathrm{Tr}}_{r/q}\left(a_0x^{d_0}\right)+{\mathrm{Tr}}_{r^2/q}\left(\sum_{j=1}^ta_jx^{d_j}\right)=0 \right\}\\
&=&r^2-\frac{1}{q}\sum_{x \in {\mathrm{GF}}(r^2)} \sum_{\lambda \in {\mathrm{GF}}(q)} \psi_q \left\{\lambda {\mathrm{Tr}}_{r/q} \left( a_0x^{d_0}\right)+\lambda {\mathrm{Tr}}_{r^2/q}\left( \sum_{j=1}^ta_jx^{d_j}\right)\right\} \\
&=&r^2\left(1-\frac{1}{q}\right)-\frac{S(\vec{a})}{q},
\end{eqnarray*}
where $\psi_q:{\mathrm{GF}}(q) \to \mathbb{C}^*$ is the standard additive character given by $\psi_q(x)=\zeta_p^{{\mathrm{Tr}}_{q/p} (x)}$ for any $x \in {\mathrm{GF}}(q)$, $\zeta_p=\exp\left(2 \pi \sqrt{-1}/p\right)$, and
\begin{eqnarray} \label{3:sa} S(\vec{a}):=(q-1)+\sum_{\lambda \in {\mathrm{GF}}(q)^*}\sum_{x \in {\mathrm{GF}}(r^2)^*} \psi_q \left\{\lambda{\mathrm{Tr}}_{r/q}\left(a_0x^{d_0}\right)+\lambda {\mathrm{Tr}}_{r^2/q}\left(\sum_{j=1}^ta_jx^{d_j}\right)\right\}.\end{eqnarray}
Since $(r-1,r+1)=1$, we can write each $x \in {\mathrm{GF}}(r^2)^*$ uniquely as $x=yz$ for $y \in {\mathrm{GF}}(r)^*$ and $z \in U:=\{\omega \in {\mathrm{GF}}(r^2): \bar{\omega} \omega=\omega^{r+1}=1\}$. Here we denote $\bar{x}:=x^r$. Note that $U$ is a cyclic subgroup of ${\mathrm{GF}}(r^2)^*$ generated by $\gamma^{r-1}$. Since $y^r=y$ for any $y \in {\mathrm{GF}}(r)$, from (\ref{2:da1}) we have
\[x^{d_j}=y^{d_j}z^{d_j}=y^{2f} z^{-2jh}, \forall j, \]
and
\[x^{rd_j}=y^{2rf}z^{-2rjh}=y^{2f} z^{2jh}, \forall j. \]
Hence
\[S(\vec{a})=(q-1)+\sum_{z \in U}\sum_{y \in {\mathrm{GF}}(r)^*} \sum_{\lambda \in {\mathrm{GF}}(q)^*} \psi_r\left\{\lambda y^{2f}\left(a_0 +\sum_{j=1}^t a_j z^{-2jh}+\bar{a}_jz^{2jh}\right)\right\}.\]
Here $\psi_r: {\mathrm{GF}}(r) \to \{\pm 1\}$ is the standard additive character. Since $\left(2f,\frac{r-1}{q-1}\right)=\left(f,\frac{r-1}{q-1}\right)=1$, we observe that as $\lambda$ runs over ${\mathrm{GF}}(q)^*$ and $y$ runs over ${\mathrm{GF}}(r)^*$ respectively, the value $\lambda y^{2f}$ will run over each element of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times. Hence we obtain
\begin{eqnarray*} \label{3:saq} S(\vec{a})=(q-1)+(q-1) \sum_{z \in U} \sum_{y \in {\mathrm{GF}}(r)^*} \psi_r \left\{ y \left( a_0 +\sum_{j=1}^t a_j z^{-2jh}+ \bar{a}_j
z^{2jh} \right) \right\}. \end{eqnarray*}
Clearly $S(\vec{a})=(q-1)r(N-1)$, where $N$ is the number of $z \in U$ such that
\[a_0+\sum_{j=1}^t a_j z^{-2jh}+\bar{a}_j z^{2jh}=0. \]
Letting $u=z^{-2h}$ and multiplying $u^{t}$ on both sides, we find
\[a_0u^t+\sum_{j=1}^ta_ju^{t+j}+\bar{a}_j u^{t-j}=0. \]
This is a polynomial of degree at most $2t$, it may have $0,1,\ldots$, or $2t$ solutions for $u$, and for each such $u \in U$, the number of $z \in U$ such that $z^{-2h}=u$ is always $e=(2h,r+1)=(h,r+1)$. Hence the possible values of $N$ are $je, \forall 0 \le j \le 2t$. This indicates that $S(\vec{a})$ and $\omega_H(c(\vec{a}))$ take at most $(2t+1)$ distinct values. This proves (i) of Theorem \ref{1:thm1}.
Statement (i) of Theorem \ref{1:thm2} can be proved similarly by using the above idea and by modifying the proof of \cite[Theorem 1]{XLZD} for ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ accordingly. We omit the details. \qquad $\square$
\section{$p=2$: Proofs of (ii) of Theorems \ref{1:thm1} and \ref{1:thm2}} \label{sec-4}
Since it is proved that there are only a few non-zero weights in ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ and ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$, a standard procedure to determine the weight distribution is to compute power moment identities.
We now prove Statement (ii) of Theorem \ref{1:thm1}. Let $\mu_j$ be the frequency of weight $w_j$ for each $j$. Obviously $S(\vec{a})=(q-1)r^2$ if and only if $\vec{a}=\vec{0}$. We have
\begin{eqnarray} \label{4:id0} r^{1+2t}=1+\sum_{j=0}^{2t} \mu_j, \end{eqnarray}
and for any positive integer $k$,
\begin{eqnarray} \label{4:idr} \sum_{\substack{a_0 \in {\mathrm{GF}}(r)\\
a_j \in {\mathrm{GF}}(r^2), 1 \le j \le t}} \left(S(\vec{a})-(q-1)\right)^k=(q-1)^k\left(r^{2}-1\right)^k+\sum_{j=0}^{2t} (q-1)^k\left(jer-r-1\right)^k \mu_j. \end{eqnarray}
On the other hand, by the orthogonal relation
\[\frac{1}{r^2} \sum_{x \in {\mathrm{GF}}(r^2)} \psi_q\left\{{\mathrm{Tr}}_{r^2/q}(xa)\right\}=\left\{\begin{array}{ccl}
0&:& \mbox{ if } a \in {\mathrm{GF}}(r^2)^*\\
1&:& \mbox{ if } a =0, \end{array}\right.\]
we find easily that
\begin{eqnarray} \label{4:idr2} \sum_{\substack{a_0 \in {\mathrm{GF}}(r)\\
a_j \in {\mathrm{GF}}(r^2), 1 \le j \le t}} \left(S(\vec{a})-1\right)^k=r^{1+2t}M_k, \end{eqnarray}
where $M_k$ denotes the number of solutions $(\lambda_1,\ldots,\lambda_k) \in \left({\mathrm{GF}}(q)^*\right)^k$ and $(x_1,\ldots,x_k) \in \left({\mathrm{GF}}(r^2)^*\right)^k$ that satisfy the equations
\begin{eqnarray} \label{4:nra}
\left\{\begin{array}{ccc}
\lambda_1 x_1^{d_0}+\lambda_2x_2^{d_0}+\cdots+\lambda_kx_k^{d_0} &=&0, \\
\lambda_1x_1^{d_1}+\lambda_2x_2^{d_1}+\cdots+\lambda_kx_k^{d_1} &=&0, \\
\cdots \cdots & & \\
\lambda_1x_1^{d_t}+\lambda_2x_2^{d_t}+\cdots+\lambda_kx_k^{d_t} &=&0.
\end{array}\right.
\end{eqnarray}
Lemma \ref{4:lem1} which we will prove below states that $M_k=(q-1)^kN_k$ for any $1 \le k \le 2t$, where $N_k$ is given by the formula (\ref{2:nr}). Combining this with identities (\ref{4:id0}), (\ref{4:idr}) and (\ref{4:idr2}) for $1 \le k \le 2t$, we obtain the matrix equation
\[M_t^{(1)} \cdot \vec{\mu}=\vec{b}, \]
where $M_t^{(1)}, \vec{\mu}$ and $\vec{b}$ are explicitly defined in Theorem \ref{1:thm1}. Since $M_t^{(1)}$ is invertible, we obtain $\vec{\mu}=\left(M_t^{(1)}\right)^{-1} \cdot \vec{b}$, as claimed by (ii) of Theorem \ref{1:thm1}. Now we prove the technical lemma.
\begin{lemma} \label{4:lem1} $M_k=(q-1)^kN_k$ for any $1 \le k \le 2t$, where $N_k$ is given by the formula (\ref{2:nr}).
\end{lemma}
\begin{proof}
Using the same notation as before, we may write each $x_i \in {\mathrm{GF}}(r^2)^*$ as
\begin{eqnarray} \label{6:xi} x_i=y_i z_i, \quad y_i \in {\mathrm{GF}}(r)^*, z_i \in U. \end{eqnarray}
Since
\[x_i^{d_j}=y_i^{2f} z_i^{-2jh}, \, \forall i,j, \]
The equations (\ref{4:nra}) can be written as
\begin{eqnarray} \label{4:nraa} \sum_{i=1}^k \lambda_i \cdot y_i^{2f} z_i^{-2jh}=0, \forall 0 \le j \le t. \end{eqnarray}
Since $\left(2f,\frac{r-1}{q-1}\right)=1$, $\lambda_i \cdot y_i^{2f}$ takes each value of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times as $\lambda_i$ and $y_i$ run over the sets ${\mathrm{GF}}(q)^*$ and ${\mathrm{GF}}(r)^*$ respectively. So $M_{k}=(q-1)^kM_{k,1}$ where $M_{k,1}$ counts the number of $y_i \in {\mathrm{GF}}(r)^*, z_i \in U \, \forall i$ such that
\begin{eqnarray} \label{4:nrb} \sum_{i=1}^k y_i z_i^{-2jh}=0, \forall 0 \le j \le t. \end{eqnarray}
In \cite{XLZD} we have used a combinatorial method to obtain the number of solutions to equations (\ref{4:nrb}). Roughly speaking, let $u_i=z_i^{-2h} \in U^e$ where $e=(2h,r+1)=(h,r+1)$. Using $y_i,u_i$'s, we can write (\ref{4:nrb}) as a matrix equation ${\bf A} \cdot \vec{y}=\vec{0}$ where
\begin{eqnarray*}
{\bf A}=\left[\begin{array}{rrrr}
1 & 1 & \cdots &1\\
u_1 & u_2 & \cdots &u_k \\
u_1^2 & u_2^2 & \cdots &u_k^2 \\
\vdots & \vdots & \ddots &\vdots \\
u_1^{t} &u_2^{t} & \cdots &u_k^{t}
\end{array} \right], \quad \vec{y}=
\left[\begin{array}{c}
y_1\\
y_2\\
\vdots\\
y_k\end{array}\right]. \end{eqnarray*}
We observe that ${\bf A}$ is a Vandermonde matrix. We may take the $r$-th power of each equation to obtain additional $(t-1)$ equations. It turns out that for any $1 \le k \le 2t$ there are only ``trivial'' solutions which can be counted exactly by using combinatorial argument. We conclude that $M_{k,1}=N_k$, which is given by the formula (\ref{2:nr}). Interested readers may review \cite{XLZD} for details. Therefore we obtain $M_k=(p-1)^kN_k$ as desired.
Statement (ii) of Theorem \ref{1:thm2} can be proved similarly, by using the above idea and by modifying the proof of \cite[Theorem 2]{XLZD} for ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ accordingly. We omit the details. \end{proof}
\section{$p \ge 3$: Proofs of (i) of Theorems \ref{1:thm1} and \ref{1:thm2}}\label{sec-32}
We first prove Statement (i) of Theorem \ref{1:thm1}. Similar to the case that $p=2$, ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ is a cyclic code of length $n$ with parity-check polynomial given by $\prod_{i=0}^t h_{d_i}(x)$, which is of degree $(2t+1)m$, hence ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}}$ has dimension $(2t+1)m$ over ${\mathrm{GF}}(q)$, and the Hamming weight of a codeword $c(\vec{a})$ can be expressed as
\begin{eqnarray*} \delta \omega_H(c(\vec{a}))&=&r^2\left(1-\frac{1}{q}\right)-\frac{S(\vec{a})}{q},
\end{eqnarray*}
where
\begin{eqnarray} \label{32:sa} S(\vec{a}):=(q-1)+\sum_{\lambda \in {\mathrm{GF}}(q)^*}\sum_{x \in {\mathrm{GF}}(r^2)^*} \psi_q \left\{\lambda{\mathrm{Tr}}_{r/q}\left(a_0x^{d_0}\right)+\lambda {\mathrm{Tr}}_{r^2/q}\left(\sum_{j=1}^ta_jx^{d_j}\right)\right\}.\end{eqnarray}
\subsection{Case 1: $m$ is odd.} We write each $x \in {\mathrm{GF}}(r^2)^*$ uniquely as $x=yw$ for $y \in {\mathrm{GF}}(r)^*$ and $w \in \Omega=\{ 1,\gamma,\gamma^2,\ldots,\gamma^r\}$ (see also \cite[Lemma 2]{gegeng2}). We may observe that $U:=\{\alpha^{r-1}: \alpha \in \Omega \}$ is a cyclic subgroup of ${\mathrm{GF}}(r^2)^*$ generated by $\gamma^{r-1}$. Since $y^r=y$ for any $y \in {\mathrm{GF}}(r)$, from (\ref{2:da1}) we have
\[x^{d_j}=y^{d_j}\omega^{d_j}=y^{2f} \omega^{d_j}, \forall j, \]
and
\[x^{rd_j}=y^{2fr}\omega^{rd_j}=y^{2f} \overline{\omega}^{d_j}, \forall j. \]
Here we denote $\bar{x}:=x^r$. Hence
\[S(\vec{a})=(q-1)+\sum_{\omega \in \Omega}\sum_{y \in {\mathrm{GF}}(r)^*} \sum_{\lambda \in {\mathrm{GF}}(q)^*} \psi_q \left\{{\mathrm{Tr}}_{r/q}\left(\lambda y^{2f}\left\{a_0\omega^{d_0} +\sum_{j=1}^ta_j\omega^{d_j}+\bar{a}_j\overline{\omega}^{d_j}\right\}\right)\right\}.\]
Since $\frac{r-1}{q-1} \equiv m \pmod{2}$ is odd, $\left(2f,\frac{r-1}{q-1}\right)=\left(f,\frac{r-1}{q-1}\right)=1$. We observe that as $\lambda$ runs over ${\mathrm{GF}}(q)^*$ and $y$ runs over ${\mathrm{GF}}(r)^*$ respectively, the value $\lambda y^{2f}$ will run over each element of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times. Hence we obtain
\begin{eqnarray*} \label{32:saq} S(\vec{a})=(q-1)+(q-1)\sum_{\omega \in \Omega}\sum_{y \in {\mathrm{GF}}(r)^*} \psi_q \left\{{\mathrm{Tr}}_{r/q}\left(y\left\{a_0\omega^{d_0} +\sum_{j=1}^ta_j\omega^{d_j}+\bar{a}_j
\overline{\omega}^{d_j}\right\}\right)\right\}. \end{eqnarray*}
Clearly $S(\vec{a})=(q-1)r(N-1)$, where $N$ is the number of $\omega \in \Omega$ such that
\[a_0\omega^{d_0} +\sum_{j=1}^ta_j\omega^{d_j}+\bar{a}_j\omega^{rd_j}=0. \]
Dividing $\omega^{d_0}$ on both sides and writing $z=\omega^{r-1} \in U$, the equation becomes
\[a_0+\sum_{j=1}^ta_j z^{jh}+\bar{a}_jz^{-jh}=0. \]
Letting $u=z^{h}$ and multiplying $u^{t}$ on both sides, we find
\[a_0u^t+\sum_{j=1}^ta_ju^{t+j}+\bar{a}_j u^{t-j}=0. \]
This is a polynomial of degree at most $2t$, so possibly it may have $0,1,\ldots$, or $2t$ solutions for $u$, and for each such $u \in U$, the number of $z \in U$ such that $z^{h}=u$ is always $e=(h,r+1)$. Hence the possible values of $N$ are $je, \forall 0 \le j \le 2t$. This indicates that $S(\vec{a})$ and $\omega_H(c(\vec{a}))$ take at most $(2t+1)$ distinct values. This proves (i) of Theorem \ref{1:thm1} when $m$ is odd.
\subsection{Case 2: $m$ and $h$ are both even.} For this case we use a different strategy. Since
\[{\mathrm{GF}}(r)^* \bigcap U=\{-1,1\}, \]
we may write each $x \in {\mathrm{GF}}(r^2)^*$ as
\begin{eqnarray} \label{32:xsplit} x=y z \epsilon, \quad x \in {\mathrm{GF}}(r)^*,\, z \in U,\, \epsilon \in \{\xi,1\}, \end{eqnarray}
where $\xi \in {\mathrm{GF}}(r^2)^*$ is a fixed non-square, $U$ is the cyclic subgroup of ${\mathrm{GF}}(r^2)^*$ generated by $\gamma^{r-1}$. We may choose $\xi=\gamma^{(r+1)/2}$ as $r =q^m \equiv 1 \pmod{4}$. It is clear that as $y,z,\epsilon$ run over the sets ${\mathrm{GF}}(r)^*, U$ and $\{\xi,1\}$ respectively, the value $x$ will run over each element of ${\mathrm{GF}}(r^2)^*$ exactly twice. Also observing
\[x^{d_j}=(yz\epsilon)^{d_j}= \left(y^2\epsilon^{r+1}\right)^f z^{-2jh}\epsilon^{(r-1)jh}, \forall j,\]
\[x^{rd_j}=(yz\epsilon)^{rd_j}= \left(y^2\epsilon^{r+1}\right)^f z^{2jh}\epsilon^{-(r-1)jh}, \forall j,\]
and
\[\xi^{(r-1)h}=\left(\gamma^{(r^2-1)}\right)^{h/2}=1, \]
we can rewrite (\ref{32:sa}) as
\[S(\vec{a})=(q-1)+\frac{1}{2}\sum_{\lambda \in {\mathrm{GF}}(q)^*}\sum_{\substack{y \in {\mathrm{GF}}(r)^*\\
z \in U\\
\epsilon \in \{1,\xi\}}} \psi_q \left\{{\mathrm{Tr}}_{r/q}\left(\lambda\left(y^2\epsilon^{r+1}\right)^f \left\{ a_0+
\sum_{j=1}^ta_jz^{-2jh}+\bar{a}_j z^{2jh}\right\} \right)\right\}.\]
Next we observe that $\xi^{r+1}=\left(\gamma^{r+1}\right)^{(r+1)/2}$ is a non-square in ${\mathrm{GF}}(r)^*$, so as $y$ runs over ${\mathrm{GF}}(r)^*$ and $\epsilon$ runs over $\{\xi,1\}$ respectively, the value $y^2 \epsilon^{r+1}$ will run over each element of ${\mathrm{GF}}(r)^*$ exactly twice. Therefore we obtain
\[S(\vec{a})=(q-1)+\sum_{z \in U} \sum_{\lambda \in {\mathrm{GF}}(q)^*} \sum_{y \in {\mathrm{GF}}(r)^*} \psi_q \left\{{\mathrm{Tr}}_{r/q}\left(\lambda y^f \left\{ a_0+
\sum_{j=1}^ta_jz^{-2jh}+\bar{a}_j z^{2jh}\right\} \right)\right\}.\]
Since $\left(f,\frac{r-1}{q-1}\right)=1$, $\lambda y^f$ will take each value of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times as $y,\lambda$ runs over ${\mathrm{GF}}(r)^*$ and ${\mathrm{GF}}(q)^*$ respectively. So
\begin{eqnarray*} S(\vec{a})&=&(q-1)+ (q-1) \sum_{z \in U} \sum_{y \in {\mathrm{GF}}(r)^*} \psi_q \left\{{\mathrm{Tr}}_{r/q}\left(y \left\{ a_0+
\sum_{j=1}^ta_jz^{-2jh}+\bar{a}_j z^{2jh}\right\} \right)\right\}\\
&=&(q-1)r(N-1),\end{eqnarray*}
where $N$ is the number of $z \in U$ such that
\[a_0+\sum_{j=1}^ta_j z^{-2jh}+\bar{a}_jz^{2jh}=0. \]
Letting $u=z^{-2h}$ and multiplying $u^{t}$ on both sides, we find
\[a_0u^t+\sum_{j=1}^ta_ju^{t+j}+\bar{a}_j u^{t-j}=0. \]
This yields at most $2t$ solutions for $u$, and for each such $u$, the number of $z \in U$ such that $z^{-2h}=u$ is exactly $(-2h,r+1)=(h,r+1)=e$ because $r \equiv 1 \pmod{4}$ and $2|h$. Hence $N \in \{je: 0 \le j \le 2t\}$. This indicates again that $S(\vec{a})$ and $\omega_H(c(\vec{a}))$ take at most $(2t+1)$ distinct values. This concludes the case for both $m$ and $h$ being even. Now (i) of Theorem \ref{1:thm1} is proved.
Statement (i) of Theorem \ref{1:thm2} can be proved similarly by using the above idea and by modifying the proof of \cite[Theorem 1]{XLZD} for ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ accordingly. We omit the details. \qquad $\square$
\section{$p \ge 3$: Proofs of (ii) of Theorems \ref{1:thm1} and \ref{1:thm2}} \label{sec-42}
We now prove Statement (ii) of Theorem \ref{1:thm1}. Let $\mu_j$ be the frequency of weight $w_j$ for each $j$. Similar to the case $p=2$, we have
\begin{eqnarray} \label{42:id0} r^{1+2t}=1+\sum_{j=0}^{2t} \mu_j, \end{eqnarray}
and for any positive integer $k$,
\begin{eqnarray} \label{42:idr} \sum_{\substack{a_0 \in {\mathrm{GF}}(r)\\
a_j \in {\mathrm{GF}}(r^2), 1 \le j \le t}} \left(S(\vec{a})-(q-1)\right)^k=(q-1)^k\left(r^{2}-1\right)^k+\sum_{j=0}^{2t} (q-1)^k \left(jer-r-1\right)^k \mu_j, \end{eqnarray}
and
\begin{eqnarray} \label{42:idr2} \sum_{\substack{a_0 \in {\mathrm{GF}}(r)\\
a_j \in {\mathrm{GF}}(r^2), 1 \le j \le t}} \left(S(\vec{a})-1\right)^k=q^{1+2t}M_k, \end{eqnarray}
where $M_k$ denotes the number of solutions $(\lambda_1,\ldots,\lambda_k) \in \left({\mathrm{GF}}(q)^*\right)^k$ and $(x_1,\ldots,x_k) \in \left({\mathrm{GF}}(r^2)^*\right)^k$ that satisfy the equations
\begin{eqnarray} \label{42:nra}
\left\{\begin{array}{ccc}
\lambda_1 x_1^{d_0}+\lambda_2x_2^{d_0}+\cdots+\lambda_kx_k^{d_0} &=&0, \\
\lambda_1x_1^{d_1}+\lambda_2x_2^{d_1}+\cdots+\lambda_kx_k^{d_1} &=&0, \\
\cdots \cdots & & \\
\lambda_1x_1^{d_t}+\lambda_2x_2^{d_t}+\cdots+\lambda_kx_k^{d_t} &=&0.
\end{array}\right.
\end{eqnarray}
Lemma \ref{43:lem1} which we will prove below states that that $M_k=(q-1)^kN_k$ for any $1 \le k \le 2t$, where $N_k$ is given by the formula (\ref{2:nr}). Combining this with identities (\ref{42:id0}), (\ref{42:idr}) and (\ref{42:idr2}) for $1 \le k \le 2t$, we obtain the matrix equation
\[M_t^{(1)} \cdot \vec{\mu}=\vec{b}, \]
where $M_t^{(1)}, \vec{\mu}$ and $\vec{b}$ are explicitly defined in Theorem \ref{1:thm1}. Since $M_t^{(1)}$ is invertible, we obtain $\vec{\mu}=\left(M_t^{(1)}\right)^{-1} \cdot \vec{b}$, as claimed by (ii) of Theorem \ref{1:thm1}. Now we prove the technical lemma.
\begin{lemma} \label{43:lem1} $M_k=(q-1)^kN_k$ for any $1 \le k \le 2t$, where $N_k$ is given by the formula (\ref{2:nr}).
\end{lemma}
\begin{proof}
Using the same notation as before, we may write each $x_i \in {\mathrm{GF}}(r^2)^*$ as
\begin{eqnarray} \label{6:xi} x_i=y_i z_i \epsilon_i, \quad y_i \in {\mathrm{GF}}(r)^*, z_i \in U, \epsilon_i \in \{\xi,1\}, \end{eqnarray}
where $\xi \in {\mathrm{GF}}(r^2)^*$ is a fixed non-square. As $y_i,z_i,\epsilon_i$ run over the sets ${\mathrm{GF}}(r)^*,U$ and $\{\xi,1\}$ respectively, $x_i=y_iz_i\epsilon_i$ will run over each element of ${\mathrm{GF}}(r^2)^*$ exactly twice. So $M_k=2^{-k}M_{k,1}$ where $M_{k,1}$ is the number of $\lambda_i \in {\mathrm{GF}}(q)^*, y_i \in {\mathrm{GF}}(r)^*,z_i \in U,\epsilon_i \in \{\xi,1\}, 1 \le i \le r$ such that $\lambda_i, x_i=y_iz_i\epsilon_i \, \forall i$ satisfy the equations (\ref{42:nra}) simultaneously. Since
\[x_i^{d_j}=\left(y_i^2\epsilon_i^{r+1}\right)^f \left(z_i^{-2}\epsilon_i^{r-1}\right)^{jh}, \, \forall j, \]
The equations (\ref{42:nra}) can be written as
\begin{eqnarray} \label{42:nraa} \sum_{i=1}^k \lambda_i \cdot \left(y_i^2\epsilon_i^{r+1}\right)^f \left(z_i^{-2}\epsilon_i^{r-1}\right)^{jh}=0, \forall 0 \le j \le t. \end{eqnarray}
\subsection{Case 1: $m$ is odd} Then $\frac{r-1}{q-1} \equiv m \equiv 1 \pmod{2}$, we may take $\xi=\gamma^{(r-1)/(q-1)}$. Then $\xi^{r+1} =\gamma^{(r^2-1)/(q-1)} \in {\mathrm{GF}}(q)^*$. For each fixed $\epsilon_i$, denoting $\lambda_i'=\lambda_i \epsilon_i^{(r+1)f} \in {\mathrm{GF}}(q)^*$, so to find $M_{k,1}$, it is equivalent to count the number of $\lambda_i' \in {\mathrm{GF}}(q)^*, y_i \in {\mathrm{GF}}(r)^*, z_i \in U, \epsilon_i \in \{\xi,1\} \, \forall i$ such that
\begin{eqnarray*} \sum_{i=1}^k \lambda_i' \cdot y_i^{2f} \left(z_i^{-2}\epsilon_i^{r-1}\right)^{jh}=0, \forall 0 \le j \le t. \end{eqnarray*}
Since $\left(2f,\frac{r-1}{q-1}\right)=1$, $\lambda_i' \cdot y_i^{2f}$ takes each value of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times as $\lambda_i'$ and $y_i$ run over the sets ${\mathrm{GF}}(q)^*$ and ${\mathrm{GF}}(r)^*$ respectively. Moreover, $\xi^{r-1}=\left(\gamma^{r-1}\right)^{(r-1)/(q-1)}$ is a non-square in $U$, hence $z_i^{-2} \epsilon_i^{r-1}$ takes each value of $U$ exactly twice as $z_i$ and $\epsilon_i$ run over the sets $U$ and $\{\xi,1\}$ respectively. So $M_{k,1}=2^k(q-1)^kM_{k,2}$ where $M_{k,2}$ counts the number of $y_i \in {\mathrm{GF}}(r)^*, z_i \in U \, \forall i$ such that
\begin{eqnarray*} \label{42:nrb} \sum_{i=1}^k y_i z_i^{jh}=0, \forall 0 \le j \le t. \end{eqnarray*}
Similar to (\ref{4:nrb}), the above equations can be solved completely by using a combinatorial method of \cite{XLZD}. We conclude that $M_{k,2}=N_k$, which is given by the formula (\ref{2:nr}). Interested readers may review \cite{XLZD} for details. Therefore we obtain $M_k=(q-1)^kN_k$ as desired.
\subsection{Case 2: $m$ and $h$ are both even}
In this case $r \equiv 1 \pmod{4}$, we may take $\xi=\gamma^{(r+1)/2}$. Hence $\xi^{(r-1)h}=1$ as $2|h$ and $\xi^{r+1}$ is a non-square in ${\mathrm{GF}}(r)^*$. So $y_i^2 \epsilon_i^{r-1}$ takes each value of ${\mathrm{GF}}(r)^*$ exactly twice as $y_i$ and $\epsilon_i$ run over ${\mathrm{GF}}(r)^*$ and $\{\xi,1\}$. Hence (\ref{42:nraa}) can be reduced to
\begin{eqnarray*} \sum_{i=1}^k \lambda_i \cdot y_i^f z_i^{-2jh}=0, \forall 0 \le j \le t. \end{eqnarray*}
Since $\left(f,\frac{r-1}{q-1}\right)=1$, $\lambda_i y_i^f$ will take each value of ${\mathrm{GF}}(r)^*$ exactly $(q-1)$ times as $\lambda_i$ and $y_i$ run over ${\mathrm{GF}}(q)^*$ and ${\mathrm{GF}}(r)^*$ respectively. We have $M_{k,1}=2^k(q-1)^kM_{k,2}$, where $M_{k,2}$ is the number of solutions $y_i \in {\mathrm{GF}}(r)^*, z_i \in U \, \forall i$ such that
\begin{eqnarray*} \sum_{i=1}^k y z_i^{-2jh}=0, \forall 0 \le j \le t. \end{eqnarray*}
Again by using combinatorial argument as in \cite{XLZD} we can obtain that $M_{k,2}=N_k$ for any $1 \le k \le 2t$ which is given by (\ref{2:nr}). Hence we conclude $M_k=(q-1)^kN_k$ as desired. This completes the proof of Lemma \ref{43:lem1}.
\end{proof}
Statement (ii) of Theorem \ref{1:thm2} can be proved similarly by using the above idea and by modifying the proof of \cite[Theorem 2]{XLZD} for ${\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ accordingly. We omit the details. \qquad $\square$
\section{Conclusions}\label{sec-conclusion}
In this paper we extended \cite{gegeng2,XLZD} further in two directions, that is, for any prime $p$, $q=p^l$ and $r=q^m$, we determined the weight distribution of the cyclic codes ${\mathcal{C}_{(d_0,d_1,\cdots,d_t)}^{(1)}},{\mathcal{C}_{(\widetilde{d}_1,\cdots,\widetilde{d}_t)}^{(2)}}$ over ${\mathrm{GF}}(q)$ whose duals have $t+1$ and $t$ generalized Niho type zeroes respectively for any $t$ (see Theorems \ref{1:thm1} and \ref{1:thm2}). Numerical examples show that the classes considered contain many optimal linear codes which were not presented in \cite{gegeng2,XLZD}.
\section*{Acknowledgement}
M. Xiong's research was supported by the Hong Kong Research Grants Council under Grant Nos. 609513 and 606211. N. Li's research was supported by the Norwegian Research Council.
|
1,108,101,565,473 | arxiv | \section{Introduction}
An {\em $r$-graph} is an $r$-uniform hypergraph, namely a set of
sets called {\em edges}, all of size $r$. For a hypergraph $H$ we
denote by $\nu(H)$ the largest size of a matching (set of disjoint
edges) in $H$, and by $\tau(H)$ the minimal size of a cover (a set
of vertices meeting all edges). Since the union of all edges in a
maximal matching is a cover, in an $r$-graph $\tau \le r\nu$. Ryser
conjectured that in $r$-partite hypregraphs $\tau \le
(r-1)\nu$. The conjecture appeared in a Ph.D thesis of his student, Henderson. At about the same time Lov\'asz \cite{lovasz} made the stronger conjecture
that in a non-empty $r$-partite hypergraph $H$ there exists a set $S$ of $r-1$ vertices such that $\nu(H-S) \le \nu(H)-1$.
The case $r=2$ of Ryser's conjecture is K\"onig's theorem \cite{konig}. The
conjecture was proved for $r=3$ in \cite{ryser3}. The Lov\'asz version is open even in this case. The fractional
version, $\tau^* \le (r-1)\nu$, is a corollary of a famous theorem
of F\"uredi \cite{furedi}.
Of particular interest is the case $\nu=1$, namely that of
intersecting hypergraphs. As defined in the abstract, given a vector
$\vec{a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{i \le
p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be {\em
$\vec{a}$-partitioned} if $V(H)=\bigcup_{i \le p}V_i$, where the
sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i
\le p$. We denote by $t(\vec{a})$ the maximum of $\tau(H)$ over all
$\vec{a}$-partitioned intersecting hypergraphs. (Using a common abbreviated notation, we shall write $t(a_1, \ldots ,a_m)$ for $t((a_1, \ldots ,a_m))$.) So, the $\nu=1$ case of Ryser's
conjecture is that $t(\vec{1})\le r-1$. This is known to be sharp
for $r$ for which there exists an $r$-uniform projective plane. In
\cite{abu, ronjanosian} the conjecture was shown to be sharp also for the first value
of $r$ for which an $r$-uniform projective plane does not exist, namely $r=7$. It
is plausible that the conjecture is sharp for all $r$.
A natural question is whether weaker conditions than $r$-partiteness
suffice to guarantee $\tau \le r-1$. In the fractional case, this is indeed true: by F\"uredi's
theorem $\tau^* \le (r-1)\nu$ for all
$\vec{a}$-partitioned hypergraphs, for every non-trivial partitioning $\vec{a}$ of $r$. But it is likely that
$t(\vec{a})=r-1$ is quite rare. In \cite{tuza} the following was
suggested:
\begin{conjecture} \label{tuza}
If $r>2$ then $t(\vec{a})=r$ for all
two-components vectors $\vec{a}$. \end{conjecture}
Our main result is:
\begin{theorem}\label{main}
If $a\neq b$ then $t(a,b)=a+b$.
\end{theorem}
\section{Proof of Theorem \ref{main}}
Let $m \ge 2$.
A family $(H_1, H_2, \ldots, H_m)$ of $a$-graphs is called {\em cross intersecting} if any two edges $e,f$
taken from distinct $H_i$s have nonempty intersection. If all $H_i$ are $a$-uniform, then this implies
\begin{enumerate}
\item
$\tau(H_i)\le a$ (any edge from $H_j,~j \neq i$ is a cover for $H_i$)
\item
$\tau\left(\bigcup_{i\leq m} H_i\right) \le 2a-1$ ( the union of two edges, taken from distinct $H_i$s,
is a cover for $\bigcup_{i \le m}H_i$.)
\end{enumerate}
If $\tau(H_i)=a$ for all $i \le m$ and $\tau\left(\bigcup_{i\leq m} H_i\right) = 2a-1$ then the family is called {\em evasive}.
\begin{observation}\label{largetau}
If $(H_1, H_2, \ldots, H_m)$ (where $m\ge 2$) is an evasive system of cross intersecting $a$-graphs then $\tau(H_i)=a$ for all $i \le m$.
\end{observation}
The reason - if there exists a cover $C$ of $H_i$ of size smaller than $a$, then for any edge $e \in H_i$ the set $C \cup e$ is a cover of size smaller than $2a-1$.
\begin{lemma}\label{evasive} \nopagebreak \hfill
\begin{enumerate}
\item For every $a$ there exists an evasive cross intersecting family of three $a$-graphs.
\item If $a=p^s$ for some prime $p$ and integer $s$ then there exists an evasive cross intersecting family of $a+1$ $a$-graphs.
\end{enumerate}
\end{lemma}
\begin{proof}
(1)~~ Consider the set of vertices
$V=\{v_{ij}: 1\leq i,j\leq a\}$. Let $H_1$ be the matching
consisting of the rows of this grid, namely the edges of $H_1$ are $R_i:=\{v_{ij} \mid j \le a\}$, and let $H_2$
be the matching consisting of the columns of the grid, namely its
edges are $C_j:=\{v_{ij} \mid i \le a\}$. Let $H_3$ be the set of all
edges of the form
$e_\sigma = \{v_{i,\sigma(i)}: 1\leq i \leq a\}$ where $\sigma$ is a permutation of
$[a]$.
Clearly, $(H_1,H_2,H_3)$ is cross-intersecting. To show evasiveness, consider a cover $C$ of $H_1\cup H_2\cup H_3$. Since $C$
covers $H_3$, by Hall's theorem there are sets $S,T\subseteq [a]$
such that $M := \{v_{s,t}: s\in S, t\in T\} \subseteq C$ and
$|S|+|T|>a$. Let $d=\min(|S|,|T|)$. Since $C$ covers $H_1\cup H_2$, we have $|C \setminus M| \ge a - d$.
thus $|C| \geq |S|\cdot |T| +d =a(a-d)+(a-d)=a(a-d+1)$, and since $a-d+1\ge 2$ we have $|C| \geq 2a-1$, as required. \\
(2)~~In this case there exists an $a+1$-uniform Desarguian projective plane (the projective plane ordinarily constructed, using vector spaces, is Desarguian). Take a line $v_1v_2\ldots v_{a+1}$ in this plane, and let $H_i ~(i \le a+1)$ be the sets of edges containing $v_i$, with $v_i$ deleted. The system of $H_i$s thus obtained is called an {\em affine plane}, in this case - a Desarguian affine plane. Clearly, the system $(H_1, \ldots, H_{a+1})$ is cross intersecting. A well known result of Jamison, Brower and Schrijver \cite{brouwerschrijver,jamison} states that it is also evasive.
\end{proof}
\begin{lemma}\label{mainlemma}\nopagebreak \hfill
Let $F$ be any intersecting $b$-graph with
$\tau(F)=b$.
If there
exists an evasive cross intersecting family $(H_1,H_2, \ldots ,H_m)$
of $a$-graphs, and $\frac{a}{m-1}\leq b < a$, then there exists an $(a,b)$-partitioned $a+b$-graph $G$ in which
the $b$-side consists of $m$ disjoint copies of $F$, and $\tau(G)=a+b$.
\end{lemma}
\begin{proof}
Let $F_1,\dots, F_m$ be $m$ disjoint copies of $F$.
Define $G = \{h\cup f: h\in H_i, f\in F_i, 1\leq i\leq m\}$. Clearly, $G$ is intersecting and is $(a,b)$-partitioned.
We shall show that $\tau(G)=a+b$. Let $C$ be a cover of $G$.
If $C$ covers all $H_i$ then by the evasiveness property $|C| \ge 2a-1 \ge a+b$.
For every $i \le m$, if $C$ does not cover $H_i$ then it covers $F_i$. Hence, if $C$ does not cover any $H_i$ then $|C| \ge mb \ge a+b$.
There remains the case that $C$ covers some $H_i$ but does not cover some $H_j$. In this
case by Observation \ref{largetau} $|C \cap H_i| \ge a$. Since also $|C \cap F_j| \ge b$,
we have $|C| \ge a+b$.
\end{proof}
Note that $F$ as in the lemma exists - for example the collection of all $b$-subsets of $[2b-1]$.
Hence each part of the next lemma follows by combining Lemma \ref{mainlemma} with the corresponding part of Lemma \ref{evasive}:
\begin{lemma}\label{club}
Let $b<a$ be integers. If either \begin{enumerate} \item $\frac{a}{2}\leq b$, or:
\item $a=p^s$ for some prime $p$ and integer $s$
\end{enumerate}
then $t(a,b)=a+b$.
\end{lemma}
\begin{lemma}\label{bminusk}
If $k<b$ and both $(a,b)$ and $(b-k,k)$ satisfy either $(1)$ or $(2)$ then $t(a,b-k,k)=a+b$.
\end{lemma}
\begin{proof}
The assumption that $(b-k,k)$ satisfies either $(1)$ or $(2)$ implies by
Lemma \ref{club} that there exists an intersecting
$(b-k,k)$-partitioned $b$-graph $G$, with $\tau(G)=b$. Putting $F=G$
in Lemma \ref{mainlemma} yields the present lemma.
\end{proof}
We now combine the results of Lemmas \ref{club} and \ref{bminusk} to prove Theorem \ref{main}.
\begin{proof}
As before, we shall assume that $b<a$. The only pair $(a,b)$ with $b<a\leq 8$ that does not satisfy either condition in
Lemma \ref{club} is $(a,b)=(6,2)$.
Since $(5,3)$ and $(2,1)$ both satisfy $(1)$ of Lemma \ref{club}, by Lemma \ref{bminusk}, $t(6,2)=t(5,2,1)=8$.
Thus, we may assume that $a>8$ and $2b<a$. We proceed by induction on $a+b$.
Then there exists a pair of
numbers
$(u,v)$ satisfying \\ (i)~$\frac{u}{2}\leq v < u <a$~~\\ (ii)~ $u+v=a+b$ and\\
~~(iii)~$v\neq 2b$ (we shall use $v-b \neq b$).\\
Indeed, at least one of the pairs $(u,v)=\left( \left\lceil \frac{a+b+3}{2}
\right\rceil,\left\lfloor \frac{a+b-3}{2} \right\rfloor \right)$ and
~~ $~~ (u,v)=\left( \left\lceil \frac{a+b+1}{2} \right\rceil, \left\lfloor \frac{a+b-1}{2} \right\rfloor
\right)$ satisfies all three conditions:
the assumption that $a+b>9$ implies (i) for both pairs, and (iii) is satisfied by at least one of these pairs.
By the induction hypothesis $t(v-b,b)=v$, which by Lemma
\ref{bminusk} implies that $t(u,v-b,b)=a+b$, which in turn implies
that
$t(u+v-b,b)=a+b$, namely $t(a,b)=a+b$.
\end{proof}
\section{Other values of $\vec{a}$}
\subsection{An example showing
$t(2,2)=4$.}
\begin{example}
Let $U=\{u_i: i\in \mathbb{Z}_5\}$, $W=\{w_i: i\in \mathbb{Z}_5\}$, and arrange each of $U$ and $W$ in a pentagon.
We construct a hypergraph $H$ with $10$ edges, $\nu=1$ and $\tau=4$, as follows.
Five edges, $e_1, \ldots ,e_5$, consist each of an edge in the $U$-pentagon and a diagonal parallel to it in the $W$-pentagon
(so, say, $e_i=\{u_i, u_{i+1}, w_{i-1},w_{i+2}\}$).
The other five edges, $f_1, \ldots, f_5$, consist each of an edge of the $W$ pentagon, and the parallel diagonal of the $U$-pentagon, shifted by $1$ (so, say,
$f_i=\{u_{i-1}, u_{i+1}, w_{i}, w_{i+1}\}$). Every two $e_i$s meet, because a diagonal in $W$ is disjoint only from the diagonals shifted $+1$ and $-1$ with respect to it, and then the corresponding parallel edges in $W$ meet. The $f_i$s meet for a similar reason. Each $e_i$ meets each $f_j$ because of the shifting. Thus $\nu(H)=1$.
\end{example}
\begin{assertion} $\tau(H) = 4$.
\end{assertion}
We have to show that any $S\subset V$ of size $3$ is not a cover.
If $S \subset U$ then it misses an edge of the complete graph on $U$, and hence by itself it is not a cover for $H$. Similarly, if $S \subset W$ then it is not a cover. Thus we may assume that (say)
$|S\cap U|=2$ and $|S\cap W| = 1$.
Let $E'$ be the set of edges that do not intersect $S\cap U$.
There are distinct $i,j$ such that $\{e_i, e_j\}\subset E'$ or $\{f_i,f_j\}\subset E'$.
For any $i,j$, $|e_i\cap e_j|=1$.
By the choice of the edges $e_i,e_j$ this intersection point is in $U\backslash S$.
Thus, $e_i\cap W$ and $e_j\cap W$ are disjoint, so $S$ cannot cover both $e_i$ and $e_j$.
The argument is similar if $\{f_i,f_j\}\subset E'$.
Thus $S$ is not a cover, which completes the proof that $\tau(H)=4$.
\subsection{An example showing $t(2,2,2,2)=8$}
Let $T$ be the truncated $4$-uniform projective plane (namely, the projective plane of order $3$, with a vertex deleted). Then $T$ is a $3$-regular $4$-partite hypergraph, with $9$ edges and $12$ vertices partitioned into parts of size $3$.
Denote its sides by $V_1,V_2,V_3,V_4$.
For each $v\in V(T)$, let $G_v$ be a copy of $K_4$.
For each $f=(f_1,f_2,f_3,f_4) \in E(T)$ and each $v=f_i \in F$
choose arbitrarily a perfect matching $m(f,v)=\{m(f,v,0),m(f,v,1)\} $ in $G_v$. This can be done in such a way that for every $v \in V(T)$, denoting the three edges of $T$ going through $v$ by $f,g,k$,
the three matchings $m(f,v), m(g,v), m(k,v)$ are distinct.
Let $B$ be the set of all binary strings $\vec{\beta}=(\beta_1, \beta_2, \beta_3, \beta_4)$ such that $\beta_1=0$ if $\beta_3=\beta_4$ and $\beta_1=1$ if $\beta_3\neq \beta_4$. Clearly, $|B|=8$.
For every pair $(\beta, f) \in B \times E(T)$ let $h=h(\beta,f)$ be the set $\bigcup_{i \le 4} \bigcup m(f,f_i,\beta_i)$. So, $\vec{\beta}$ decides for each $i$ which of the two edges in the matching $m(f,f_i)$ is chosen by $h$.
Let $H$ be $\{h(\beta,f) \mid \beta \in B,~f \in E(T)\}$. Clearly, $H$ is $(2,2,2,2)$-partitioned, with sides
$S_i = \bigcup_{v\in V_i} V(G_v)~~(i=1,2,3,4)$. It has $72$ edges.
\begin{assertion} $\nu(H) = 1$.
\end{assertion}
\begin{proof}
For any $\vec{\beta},~\vec{\gamma} \in B$, if $\beta_3\neq \gamma_3$ and $\beta_4\neq \gamma_4$, then $\beta_1=\gamma_1$.
Thus for any $f\in E(T)$ it is true that $h(f,\beta) \cap h(f,\gamma)\neq \emptyset$.
Consider next two edges $h(\beta,f)$ and $h(\gamma,g)$ in $H$ for $f \neq g$. By the intersection property of $P$, there exists a vertex $v \in f \cap g$, say $v=f_i=g_i$. By the construction $m(f,v)\neq m(g,v)$. Hence the two edges in $S_i$ representing $f$ and $g$ in $h(\beta,f)$ and $h(\gamma,g)$ intersect.
\end{proof}
\begin{assertion} $\tau(H) = 8$.
\end{assertion}
\begin{proof}
Assume for contradiction that there exists a cover $C$ of size $7$.
For each edge $e$, fix a vertex $x_e\in C$ contained in $e$. For each $e$, any pair $(y,e)$ where $y\in (C \cap e)\setminus \{x_e\}$
is said to constitute a {\em waste}. Since
$\sum_{v \in C}deg_H(v) = 7 \times 12 =84$ and $|E(H)|=72$, we are allowed at most $84-72=12$
wastes.
Note that the common degree of two vertices belonging to different sides $S_i$ is $2$,
while the common degree of two vertices belonging to the same $G_v$ is $4$.
Suppose first that
there exist two vertices in $C$ that meet a side in distinct
$G_v$s. Then all other vertices, apart from possibly
one that may belong to the
third copy of $G_v$ in the same side, contributes at
least
$4$ wastes, and thus the total number of wastes is more than $12$.
Thus we may assume that each side of $H$ contains at
most one $G_v$ containing points from $C$. Fixing a
vertex $x$, every other vertex contributes with $x$
at least $2$
wastes, while two vertices belonging to the
same $G_v$ (which must exist) contribute $4$
wastes, so again there are more than $12$ wastes.
\end{proof}
\begin{corollary} $t(4,4)=8$. \end{corollary}
The first value of the function $t$ that is not settled is $t(3,3)$.
{\bf Acknowledgement} We are grateful to Aart Blokhuis for useful information.
|
1,108,101,565,474 | arxiv | \section*{Introduction}
Working symbolically with matrices requires \emph{non-commuting}
variables and thus non-com\-mu\-tative (nc) rational expressions.
Although the (algebraic) construction of \emph{free fields},
that is, universal fields of fractions of \emph{free associative algebras},
is available due to Paul M.~Cohn since 1970
\cite[Chapter~7]{Cohn2006a
, its practical application in terms of \emph{free fractions}
\cite{Schrempf2018c2
\ ---building directly on Cohn and Reutenauer's \emph{linear representations}
\cite{Cohn1999a
--- in computer algebra systems is only at the very beginning.
The main difficulty for arithmetic
---or rather \emph{lexetic} from the non-existing Greek word
\selectlanguage{greek}%
lexhtikos \selectlanguage{english}%
(from \selectlanguage{greek}%
lexis \selectlanguage{english}%
for \emph{word}) as analogon to \selectlanguage{greek}%
arijmhtikos \selectlanguage{english}%
(from \selectlanguage{greek}%
arijmos \selectlanguage{english}%
for \emph{number})---
was the construction of \emph{minimal} linear representations
\cite{Schrempf2018a9
, that is, the \emph{normal form} of Cohn and Reutenauer
\cite{Cohn1994a
.
Here we will show that free fractions also provide a
framework for ``free'' derivation, in particular of nc polynomials.
The construction we provide generalizes the univariate (commutative)
case we are so much used to, for example
\begin{displaymath}
f = f(x) = x^3 + 4x^2+3x+5\quad\text{with}\quad
f' = \tsfrac{\dd}{\dd x} f(x) = 3x^2 + 8x+3.
\end{displaymath}
The coefficients are from a \emph{commutative} field
$\field{K}$ (for example the rational $\numQ$, the real $\numR$, or
the complex number field $\numC$),
the (non-commuting) variables from a (finite)
alphabet $\alphabet{X}$, for example $\alphabet{X} = \{ x, y, z \}$.
\medskip
For simplicity we focus here (in this motivation)
on the \emph{free associative algebra}
$R := \freeALG{\field{K}}{\alphabet{X}}$,
aka ``algebra of nc polynomials'', and recall the
properties of a (partial) \emph{derivation} $\ncpd_x : R \to R$
(for a fixed $x \in \alphabet{X}$), namely
\begin{itemize}
\item $\ncpd_x (\alpha) = 0$ for $\alpha \in \field{K}$, and more general,
$\ncpd_x(g) =0$ for $g \in \freeALG{\field{K}}{\alphabet{X}\!\setminus\! \{x\}}$,
\item $\ncpd_x (x) = 1$, and
\item $\ncpd_x (fg) = \ncpd_x(f)\,g + f\, \ncpd_x(g)$.
\end{itemize}
In \cite{Rota1980a
\ this is called \emph{Hausdorff derivative}.
For a more general (module theoretic) context we refer to
\cite[Section~2.7]{Cohn2003b
\ or \cite{Bergman1975b
.
\medskip
Before we continue we should clarify the wording: To avoid
confusion we refer to the \emph{linear operator} $\ncpd_x$ (for
some $x \in \alphabet{X}$) as \emph{free partial derivation} (or just
\emph{derivation}) and call $\ncpd_x f \in R$ the (free partial)
\emph{derivative} of $f \in R$,
sometimes denoted also as $f_x$ or $f'$ depending on the context.
In the commutative, we usually do not distinguish too much between
algebraic and analytic concepts. But in the (free) non-commutative
setting, analysis is quite subtle \cite{Kaliuzhnyi2014a
. There are even concepts like ``matrix convexity''
and symbolic procedures to determine (nc) convexity
\cite{Camino2003a
. However, a systematic treatment of the underlying algebraic
tools was not available so far. We are going to close this gap in the following.
\medskip
After a brief description of the setup
(in particular that of \emph{linear representations} of elements
in free fields) in Section~\ref{sec:fd.intro},
we develop the formalism for \emph{free derivations}
in Section~\ref{sec:fd.der} with the main
result, Theorem~\ref{thr:fd.ncpd}.
To be able to state a (partial) ``free'' chain rule
(in Proposition~\ref{pro:fd.chainrule})
we derive a language for the
\emph{free composition} (and illustrate
how to ``reverse'' it) in Section~\ref{sec:fd.comp}.
And finally, in Section~\ref{sec:fd.newton},
we show how to develop a meta algorithm
``nc Newton'' to find matrix-valued roots
of a non-commutative rational equation.
\medskip
\begin{notation}
The set of the natural numbers is denoted by $\numN = \{ 1,2,\ldots \}$,
that including zero by $\numN_0$.
Zero entries in matrices are usually replaced by (lower) dots
to emphasize the structure of the non-zero entries
unless they result from transformations where there
were possibly non-zero entries before.
We denote by $I_n$ the identity matrix (of size $n$)
respectively $I$ if the size is clear from the context.
By $v^{\!\top}$ we denote the transpose of a vector $v$.
\end{notation}
\section{Getting Started}\label{sec:fd.intro}
We represent elements (in free fields) by
\emph{admissible linear systems} (Definition~\ref{def:fd.als}),
which are just a special form of \emph{linear representations}
(Definition~\ref{def:fd.rep}) and ``general'' \emph{admissible systems}
\cite[Section~7.1]{Cohn2006a
. Rational operations
(scalar multiplication, addition, multiplication, inverse) can be
easily formulated in terms of linear representations
\cite[Section~1]{Cohn1999a
. For the formulation on the level of admissible linear systems
and the ``minimal'' inverse we refer to
\cite[Proposition~1.13]{Schrempf2017a9
\ resp.~\cite[Theorem~4.13]{Schrempf2017a9
.
Let $\field{K}$ be a \emph{commutative} field,
$\aclo{\field{K}}$ its algebraic closure and
$\alphabet{X} = \{ x_1, x_2, \ldots, x_d\}$ be a \emph{finite} (non-empty) alphabet.
$\freeALG{\field{K}}{\alphabet{X}}$ denotes the \emph{free associative
algebra} (or \emph{free $\field{K}$-algebra})
and $\field{F} = \freeFLD{\field{K}}{\alphabet{\alphabet{X}}}$ its \emph{universal field of
fractions} (or ``free field'')
\cite{Cohn1995a
,
\cite{Cohn1999a
. An element in $\freeALG{\field{K}}{\alphabet{X}}$ is called (non-commutative or nc)
\emph{polynomial}.
In our examples the alphabet is usually $\alphabet{X}=\{x,y,z\}$.
Including the algebra of \emph{nc rational series}
\cite{Berstel2011a
\ we have the following chain of inclusions:
\begin{displaymath}
\field{K}\subsetneq \freeALG{\field{K}}{\alphabet{X}}
\subsetneq \field{K}^{\text{rat}}\langle\!\langle \alphabet{X}\rangle\!\rangle
\subsetneq \freeFLD{\field{K}}{\alphabet{X}} =: \field{F}.
\end{displaymath}
\begin{definition}[Inner Rank, Full Matrix \protect{%
\cite[Section~0.1]{Cohn2006a
}, \cite{Cohn1999a
]\label{def:fd.full}
Given a matrix $A \in \freeALG{\field{K}}{\alphabet{X}}^{n \times n}$,
the \emph{inner rank}
of $A$ is the smallest number $k\in \numN$
such that there exists a factorization $A = C D$ with
$C \in \freeALG{\field{K}}{\alphabet{X}}^{n \times k}$ and
$D \in \freeALG{\field{K}}{\alphabet{X}}^{k \times n}$.
The matrix $A$ is called \emph{full} if $k = n$,
\emph{non-full} otherwise.
\end{definition}
\begin{theorem}[\protect{%
\cite[Special case of Corollary~7.5.14]{Cohn2006a
}]\label{thr:fd.uff}
Let $\alphabet{X}$ be an alphabet and $\field{K}$ a commutative field.
The free associative algebra $R = \freeALG{\field{K}}{\alphabet{X}}$
has a universal field of fractions $\field{F}=\freeFLD{\field{K}}{\alphabet{X}}$
such that every full matrix over $R$ can be inverted over $\field{F}$.
\end{theorem}
\begin{remark}
Non-full matrices become singular under a homomorphism into some
field \cite[Chapter~7]{Cohn2006a
.
In general (rings), neither do full matrices need to be invertible,
nor do invertible matrices need to be full. An example for the former
is the matrix
\begin{displaymath}
B =
\begin{bmatrix}
. & z & -y \\
-z & . & x \\
y & -x & .
\end{bmatrix}
\end{displaymath}
over the \emph{commutative} polynomial ring $\field{K}[x,y,z]$
which is \emph{not} a Sylvester domain
\cite[Section~4]{Cohn1989b
. An example for the latter are rings
without \emph{unbounded generating number} (UGN)
\cite[Section~7.3]{Cohn2006a
.
\end{remark}
\begin{definition}[Linear Representations, Dimension, Rank
\cite{Cohn1994a,Cohn1999a
]\label{def:fd.rep}
Let $f \in \field{F}$.
A \emph{linear representation} of $f$ is a triple $\pi_f = (u,A,v)$ with
$u^{\!\top}, v \in \field{K}^{n \times 1}$, full
$A = A_0 \otimes 1 + A_1 \otimes x_1 + \ldots + A_d \otimes x_d$
with $A_\ell \in \field{K}^{n\times n}$ for all $\ell \in \{ 0, 1, \ldots, d \}$
and $f = u A^{-1} v$.
The \emph{dimension} of $\pi_f$ is $\dim \, (u,A,v) = n$.
It is called \emph{minimal} if $A$ has the smallest possible dimension
among all linear representations of $f$.
The ``empty'' representation $\pi = (,,)$ is
the minimal one of $0 \in \field{F}$ with $\dim \pi = 0$.
Let $f \in \field{F}$ and $\pi$ be a \emph{minimal}
linear representation of $f$.
Then the \emph{rank} of $f$ is
defined as $\rank f = \dim \pi$.
\end{definition}
\begin{definition}[Left and Right Families
\cite{Cohn1994a
]\label{def:fd.family}
Let $\pi=(u,A,v)$ be a linear representation of $f \in \field{F}$
of dimension $n$.
The families $( s_1, s_2, \ldots, s_n )\subseteq \field{F}$
with $s_i = (A^{-1} v)_i$
and $( t_1, t_2, \ldots, t_n )\subseteq \field{F}$
with $t_j = (u A^{-1})_j$
are called \emph{left family} and \emph{right family} respectively.
$L(\pi) = \linsp \{ s_1, s_2, \ldots, s_n \}$ and
$R(\pi) = \linsp \{ t_1, t_2, \ldots, t_n \}$
denote their linear spans (over $\field{K}$).
\end{definition}
\begin{proposition}[\protect{%
\cite[Proposition~4.7]{Cohn1994a
}]\label{pro:fd.cohn94.47}
A representation $\pi=(u,A,v)$ of an element $f \in \field{F}$
is minimal if and only if both, the left family
and the right family are $\field{K}$-linearly independent.
In this case, $L(\pi)$ and $R(\pi)$ depend only on $f$.
\end{proposition}
\begin{remark}
The left family $(A^{-1} v)_i$ (respectively the right family $(u A^{-1})_j$)
and the solution vector $s$ of $As = v$ (respectively $t$ of $u = tA$)
are used synonymously.
\end{remark}
\begin{definition}[Admissible Linear Systems, Admissible Transformations
\cite{Schrempf2017a9
]\label{def:fd.als}
A linear representation $\als{A} = (u,A,v)$ of $f \in \field{F}$
is called \emph{admissible linear system} (ALS) for $f$,
written also as $A s = v$,
if $u=e_1=[1,0,\ldots,0]$.
The element $f$ is then the first component
of the (unique) solution vector $s$.
Given a linear representation $\als{A} = (u,A,v)$
of dimension $n$ of $f \in \field{F}$
and invertible matrices $P,Q \in \field{K}^{n\times n}$,
the transformed $P\als{A}Q = (uQ, PAQ, Pv)$ is
again a linear representation (of $f$).
If $\als{A}$ is an ALS,
the transformation $(P,Q)$ is called
\emph{admissible} if the first row of $Q$ is $e_1 = [1,0,\ldots,0]$.
\end{definition}
\section{Free Derivation}\label{sec:fd.der}
Before we define the concrete (partial) \emph{derivation}
and (partial) \emph{directional derivation}, we start with
a (partial) \emph{formal derivation}
on the level of admissible linear systems
and show the basic properties with respect to the
represented elements, in particular that
the (formal) derivation does \emph{not} depend on the ALS
in Corollary~\ref{cor:fd.forder}.
In other words: Given some letter $x \in \alphabet{X}$
and an \emph{admissible linear system} $\als{A}=(u,A,v)$,
there is an \emph{algorithmic} point of view
in which the (free) derivation $\ncpd_x$ defines the
ALS $\als{A}' = \ncpd_x \als{A}$. (Alternatively one
can identify $x$ by its index $\ell \in \{ 1, 2, \ldots, d \}$
and write $\als{A}' = \ncpd_{\ell} \als{A}$.)
Written in a sloppy way, we show that
$\ncpd_x(\als{A} + \als{B}) = \ncpd_x \als{A} + \ncpd_x \als{B}$ and
$\ncpd_x(\als{A} \cdot \als{B})
= \ncpd_x \als{A} \cdot \als{B} + \als{A} \cdot \ncpd_x \als{B}$,
yielding immediately the \emph{algebraic} point of view
(summarized in Theorem~\ref{thr:fd.ncpd})
by taking the respective first component of the
\emph{unique} solution vectors $f = (s_{\als{A}})_1$
and $g = (s_{\als{B}})_1$.
\begin{remark}
The following definition is much more general then usually needed.
One gets the ``classical'' (partial) derivation with respect to
some letter $x = x_{\ell} \in \alphabet{X}$
(with $\ell \in \{ 1, 2, \ldots, d \}$) for $k=0$
resp.~(the empty word) $a = 1 \in \alphabet{X}^*$.
\end{remark}
\begin{definition}[Formal Derivative]
\label{def:fd.alsfor}
Let $\als{A}=(u,A,v)$ be an \emph{admissible linear system}
of dimension $n \ge 1$ for some element in the free field
$\field{F} = \freeFLD{\field{K}}{\alphabet{X}}$
and $\ell \neq k \in \{ 0, 1, \ldots, d \}$.
The ALS
\begin{displaymath}
\ncpd_{\ell|k} \als{A} = \ncpd_{\ell|k} (u,A,v) =
\left(
\begin{bmatrix}
u & .
\end{bmatrix},
\begin{bmatrix}
A & A_{\ell}\otimes x_k \\
. & A
\end{bmatrix},
\begin{bmatrix}
. \\ v
\end{bmatrix}
\right)
\end{displaymath}
(of dimension $2n$)
is called (partial) \emph{formal derivative} of $\als{A}$,
(with respect to $x_{\ell},x_k \in \{ 1 \} \cup \alphabet{X}$).
For $x, a \in \{ 1 \} \cup \alphabet{X}= \{ 1, x_1, x_2, \ldots, x_d \}$
with $x \neq a$ we write also
$\ncpd_{x|a} \als{A}$, having the indices $\ell \neq k \in \{ 0, 1, \ldots, d \}$
of $x$ resp.~$a$ in mind.
\end{definition}
\begin{lemma}\label{lem:fd.alstoele}
Let $\als{A}_f = (u_f,A_f,v_f)$ and $\als{A}_g = (u_g,A_g,v_g)$
be admissible linear systems of dimension $\dim \als{A}_f \ge 1$
resp.~$\dim \als{A}_g \ge 1$.
Fix $x, a \in \{ 1 \} \cup \alphabet{X}$
such that $x \neq a$. Then
$\ncpd_{x|a}(\als{A}_f + \als{A}_g) = \ncpd_{x|a}\als{A}_f + \ncpd_{x|a}\als{A}_g$.
\end{lemma}
\begin{proof}
Let $\ell \neq k \in \{ 0, 1, \ldots, d \}$ be the indices of
$x$ resp.~$a$. We write $A_f^{\ell}$ for the coefficient matrix
$A_{\ell}$ of $A_f$ resp.~$A_g^{\ell}$ for $A_{\ell}$ of $A_g$.
Taking the sum from
\cite[Proposition~1.13]{Schrempf2017a9
\ we have
\begin{align*}
\ncpd_{x|a} & (\als{A}_f + \als{A}_g) = \left(
\begin{bmatrix}
u_f & .
\end{bmatrix},
\begin{bmatrix}
A_f & -A_f u_f^{\!\top} u_g \\
. & A_g
\end{bmatrix},
\begin{bmatrix}
v_f \\ v_g
\end{bmatrix}
\right) \\
&= \left(
\begin{bmatrix}
u_f & . & . & .
\end{bmatrix},
\begin{bmatrix}
A_f & -A_f u_f^{\!\top} u_g & A_f^{\ell}\otimes a & -A_f^{\ell} u_f^{\!\top} u_g \otimes a \\
. & A_g & . & A_g^{\ell}\otimes a \\
. & . & A_f & -A_f u_f^{\!\top} u_g \\
. & . & . & A_g
\end{bmatrix},
\begin{bmatrix}
. \\ . \\ v_f \\ v_g
\end{bmatrix}
\right) \\
&= \left(
\begin{bmatrix}
u_f & . & . & .
\end{bmatrix},
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a & -A_f u_f^{\!\top} u_g & -A_f^{\ell} u_f^{\!\top} u_g \otimes a \\
. & A_f & . & -A_f u_f^{\!\top} u_g \\
. & . & A_g & A_g^{\ell}\otimes a \\
. & . & . & A_g
\end{bmatrix},
\begin{bmatrix}
. \\ v_f \\ . \\ v_g
\end{bmatrix}
\right) \\
&= \left(
\begin{bmatrix}
u_f & . & . & .
\end{bmatrix},
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a & -A_f u_f^{\!\top} u_g & 0\\
. & A_f & . & 0 \\
. & . & A_g & A_g^{\ell}\otimes a \\
. & . & . & A_g
\end{bmatrix},
\begin{bmatrix}
. \\ v_f \\ . \\ v_g
\end{bmatrix}
\right) \\
&= \left(
\begin{bmatrix}
u_f & .
\end{bmatrix},
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a \\
. & A_f
\end{bmatrix},
\begin{bmatrix}
. \\ v_f
\end{bmatrix}
\right)
+ \left(
\begin{bmatrix}
u_g & .
\end{bmatrix},
\begin{bmatrix}
A_g & A_g^{\ell}\otimes a \\
. & A_g
\end{bmatrix},
\begin{bmatrix}
. \\ v_g
\end{bmatrix}
\right) \\
&= \ncpd_{x|a}\als{A}_f + \ncpd_{x|a}\als{A}_g.
\end{align*}
The two main steps are swapping block rows~2 and~3 and
block columns~2 and~3,
and eliminating the \emph{single} non-zero (first) column in
$-A_f^{\ell} u_f^{\!\top} u_g \otimes a$ and $-A_f u_f^{\!\top} u_g$
(in block column~4)
using the first column in block column~2.
\end{proof}
\begin{corollary}\label{cor:fd.forder}
Let $f,g \in \field{F}$ be given by the admissible linear systems
$\als{A}_f = (u_f,A_f,v_f)$ and $\als{A}_g = (u_g,A_g,v_g)$ of
dimensions $n_f,n_g \ge 1$ respectively.
Fix $x, a \in \{ 1 \} \cup \alphabet{X}$ such that $x\neq a$.
Then $f=g$ implies that $\ncpd_{x|a}\als{A}_f - \ncpd_{x|a}\als{A}_g$
is an ALS for $0 \in \field{F}$.
\end{corollary}
\begin{definition}[Formal Derivative]
\label{def:fd.forder}
Let $f \in \field{F}$ be given by the ALS $\als{A}=(u,A,v)$
and fix $x,a \in \{ 1 \} \cup \alphabet{X} = \{ 1, x_1, x_2, \ldots, x_d \}$
such that $x \neq a$.
Denote by $\ncpd_{x|a} f$ the element defined by the ALS $\ncpd_{x|a} \als{A}$.
The map $\ncpd_{x|a} : \field{F} \to \field{F}$,
$f \mapsto \ncpd_{x|a} f$ is called
(partial) \emph{formal derivation},
the element $\ncpd_{x|a} f$ (partial) \emph{formal derivative} of $f$.
\end{definition}
\begin{corollary}\label{cor:fd.linear}
For each $x,a \in \{ 1 \} \cup \alphabet{X}$
with $x \neq a$,
the formal derivation $\ncpd_{x|a} : \field{F} \to \field{F}$
is a \emph{linear} map.
\end{corollary}
Now we are almost done.
Before we show the \emph{product rule}
in the following Lemma~\ref{lem:fd.product},
we have a look into the left family of the ALS of the
(formal) derivative of a polynomial. Let $p = x^3 \in \field{F}$
(and $a=1$).
A (minimal) ALS for $\ncpd_{x|1} p$ is given by
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & . & 0 & -1 & . & . \\
. & 1 & -x & . & . & 0 & -1 & . \\
. & . & 1 & -x & . & . & 0 & -1 \\
. & . & . & 1 & . & . & . & 0 \\
. & . & . & . & 1 & -x & . & . \\
. & . & . & . & . & 1 & -x & . \\
. & . & . & . & . & . & 1 & -x \\
. & . & . & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
0 \\ 0 \\ 0 \\ 0 \\ . \\ . \\ . \\ 1
\end{bmatrix},
\quad
s =
\begin{bmatrix}
3 x^2 \\ 2x \\ 1 \\ 0 \\ x^3 \\ x^2 \\ x \\ 1
\end{bmatrix}.
\end{displaymath}
Notice that the first four entries in the left family of $\ncpd_{x|1} \als{A}$
are $s_i = \ncpd_{x|1} s_{i+4}$.
\begin{lemma}[Product Rule]\label{lem:fd.product}
Let $f,g \in \field{F}$ be given by the
admissible linear systems $\als{A}_f = (u_f, A_f, v_f)$
and $\als{A}_g = (u_g, A_g, v_g)$ of dimension $n_f,n_g \ge 1$
respectively.
Fix $x \in \alphabet{X}$ and $a \in \{ 1 \} \cup \alphabet{X} \setminus \{ x \}$.
Then $\ncpd_{x|a} (fg) = \ncpd_{x|a} f \,g + f \, \ncpd_{x|a} g$.
\end{lemma}
\begin{proof}
Let $\ell \neq k \in \{ 0, 1, \ldots, d \}$ be the indices of
$x$ resp.~$a$. We write $A_f^{\ell}$ for the coefficient matrix
$A_{\ell}$ of $A_f$ resp.~\raisebox{0pt}[0pt][0pt]{$A_g^{\ell}$}
for $A_{\ell}$ of $A_g$.
We take the sum and the product from
\cite[Proposition~1.13]{Schrempf2017a9
\ and start with the ALS from the right hand side,
\begin{displaymath}
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a & 0 & -A_f u_f ^{\!\top} u_g & . & . \\
. & A_f & -v_f u_g & . & . & . \\
. & . & A_g & . & . & . \\
. & . & . & A_f & -v_f u_g & . \\
. & . & . & . & A_g & A_g^{\ell}\otimes a \\
. & . & . & . & . & A_g
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ v_g \\ . \\ . \\ v_g
\end{bmatrix},
\end{displaymath}
subtract block row~6 from block row~3, add block column~3 to block
column~6 and remove block row/column~3 to get the ALS
\begin{displaymath}
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a & -A_f u_f ^{\!\top} u_g & . & . \\
. & A_f & . & . & -v_f u_g \\
. & . & A_f & -v_f u_g & . \\
. & . & . & A_g & A_g^{\ell}\otimes a \\
. & . & . & . & A_g
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ v_g
\end{bmatrix}.
\end{displaymath}
Now we can add block row~3 to block row~1 and
eliminate the remaining columns in block $(1,3)$ by
the columns $\{ 2, 3, \ldots, n_f \}$ from block $(1,1)$, remove
block row/column~3 to get the ALS
\begin{displaymath}
\begin{bmatrix}
A_f & A_f^{\ell}\otimes a & -v_f u_g & . \\
. & A_f & . & -v_f u_g \\
. & . & A_g & A_g^{\ell}\otimes a \\
. & . & . & A_g
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ v_g
\end{bmatrix}.
\end{displaymath}
Swapping block rows~2 and~3 and block columns~2 and~3 yields the ALS
\begin{displaymath}
\begin{bmatrix}
A_f & -v_f u_g & A_f^{\ell}\otimes a & 0 \\
. & A_g & . & A_g^{\ell}\otimes a \\
. & . & A_f & -v_f u_g \\
. & . & . & A_g
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ v_g
\end{bmatrix}
\end{displaymath}
of the left hand side $\ncpd_{x|0} (fg)$.
Notice the upper right zero in the system matrix
which is because of $x \neq 1$.
\end{proof}
\begin{definition}[Partial Derivative]\label{def:fd.ncpd}
Let $f \in \field{F}$, $x \in \alphabet{X}$ and $a \in \alphabet{X} \setminus \{ x\}$.
The element $\ncpd_{x} f := \ncpd_{x|1} f$ is called
\emph{partial derivative} of $f$.
The element $\ncpd_{x|a} f$ is called
(partial) \emph{directional derivative} of $f$ (with respect to $a$).
\end{definition}
\begin{theorem}[Free Derivation]\label{thr:fd.ncpd}
Let $x \in \alphabet{X}$. Then the (partial) \emph{free derivation}
$\ncpd_x: \field{F} \to \field{F} = \freeFLD{\field{K}}{\alphabet{X}}$
is the \emph{unique} map with the properties
\begin{itemize}
\item $\ncpd_x h = 0$ for all
$h \in \freeFLD{\field{K}}{\alphabet{X}\!\setminus\!\{ x \}}$,
\item $\ncpd_x x = 1$, and
\item $\ncpd_x (fg) = \ncpd_x f \, g + f\, \ncpd_x g$ for all
$f,g \in \field{F} = \freeFLD{\field{K}}{\alphabet{X}}$.
\end{itemize}
\end{theorem}
\begin{proof}
Let $h$ be given by the ALS $\als{A} = (u,A,v)$ and let
$\ell \in \{ 1, 2, \ldots, d \}$ such that $x = x_{\ell}$.
We just need to recall
the ALS for $\ncpd_x h$,
\begin{displaymath}
\begin{bmatrix}
A & A_{\ell}\otimes 1 \\
. & A
\end{bmatrix}
\begin{bmatrix}
s' \\ s''
\end{bmatrix}
=
\begin{bmatrix}
. \\ v
\end{bmatrix}
\end{displaymath}
and observe that $A_{\ell} = 0$ and thus $As'=0$,
in particular the first component of $s'$.
Therefore $\ncpd_x h = 0$.
For $\ncpd_x x = 1$ we need to minimize
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & -1 \\
. & 1 & . & . \\
. & . & 1 & -x \\
. & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
And the product rule
$\ncpd_x (fg) = \ncpd_x f \, g + f\, \ncpd_x g$
is due to Lemma~\ref{lem:fd.product}.
For the uniqueness we assume that there exists another
$\ncpd'_x : \field{F} \to \field{F}$ with the same properties.
From the product rule we obtain
$\ncpd_x(x f) = f + x\, \ncpd_x f = f + x\, \ncpd'_x f = \ncpd'_x(x f)$,
that is, $x(\ncpd_x f - \ncpd'_x f) = 0$ for \emph{all} $f \in \field{F}$,
thus $\ncpd_x = \ncpd'_x$.
\end{proof}
\begin{corollary}[Hausdorff Derivation
\cite{Rota1980a
]\label{cor:fd.hausdorff}
Let $x \in \alphabet{X}$. Then
$\ncpd_x \kappa =0$ for all $\kappa \in \field{K}$,
$\ncpd_x y=0$ for all $y \in \alphabet{X} \setminus \{ x \}$,
$\ncpd_x x = 1$, and
$\ncpd_x(fg) = \ncpd_x f\, g + f\, \ncpd_x g$
for all $f,g \in \freeALG{\field{K}}{\alphabet{X}}$.
\end{corollary}
\begin{remark}
More general
\cite[Theorem~7.5.17]{Cohn2006a
: ``Any derivation of a Sylvester domain extends to a derivation
of its universal field of fractions.''
Recall however that the \emph{cyclic derivative}
is \emph{not} (from) a derivation
\cite[Section~1]{Rota1980a
. For a discussion of cyclic derivatives of nc algebraic power series
we refer to \cite{Reutenauer1983a
.
\end{remark}
\begin{proposition}
Let $f \in \field{F}$, $x,y \in \alphabet{X}$
and $a,b \in \alphabet{X} \setminus \{ x, y \}$
with $a=b$ if and only if $x=y$.
Then $\ncpd_{x|a} (\ncpd_{y|b} f) = \ncpd_{y|b} (\ncpd_{x|a} f)$,
that is, the (partial) derivations $\ncpd_{x|a}$ and $\ncpd_{y|b}$
\emph{commute}.
\end{proposition}
\begin{proof}
Let $l,k \in \{ 1, 2, \ldots, d \}$ the indices of $x$ resp.~$y$.
There is nothing to show for the trivial case $x=y$,
thus we can assume $l \neq k$.
Let $f$ be given by the admissible linear system $\als{A} = (u,A,v)$.
Then the ``left'' ALS $\ncpd_{x|a} (\ncpd_{y|b} \als{A})$ is
\begin{displaymath}
\begin{bmatrix}
A & A_k \otimes b & A_l \otimes a & 0 \\
. & A & 0 & A_l \otimes a \\
. & . & A & A_k \otimes b \\
. & . & . & A
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ v
\end{bmatrix}.
\end{displaymath}
Swapping block rows/columns~2 and~3 yields
the ``right'' ALS $\ncpd_{y|b}(\ncpd_{x|a} \als{A})$:
\begin{displaymath}
\begin{bmatrix}
A & A_l \otimes a & A_k \otimes b & 0 \\
. & A & 0 & A_k \otimes b \\
. & . & A & A_l \otimes a \\
. & . & . & A
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ v
\end{bmatrix}.
\end{displaymath}
\end{proof}
Since there is no danger of ambiguity, we can
define the (free) ``higher'' derivative of $f \in \field{F}$
as $\ncpd_w f$ for each word $w$ in the \emph{free monoid}
$\alphabet{X}^*$ with the ``trivial'' derivative
$\ncpd_{(1)} f = f$.
Let $w \in \alphabet{X}^*$ and $\sigma(w)$ denote
any permutation of the letters of $w$.
Then $\ncpd_w f = \ncpd_{\sigma(w)} f$.
The proof for nc formal power series in
\cite[Proposition~1.8]{Popescu2006a
\ is based on words (monomials), that is,
$\ncpd_x (\ncpd_y w ) = \ncpd_y (\ncpd_x w) \in \freeALG{\field{K}}{\alphabet{X}}$.
Recall that one gets the (nc) \emph{rational} series by
intersecting the (nc) series and the \emph{free field}
\cite[Section~9]{Reutenauer2008a
:
\begin{displaymath}
\field{K}\langle\!\langle \alphabet{X}\rangle\!\rangle
\cap \freeFLD{\field{K}}{\alphabet{X}}
= \field{K}^{\text{rat}}\langle\!\langle \alphabet{X}\rangle\!\rangle.
\end{displaymath}
Overall, (free) nc derivation does not appear that often
in the literature. And when there is some discussion
it is (almost) always connected with ``not simple''
\cite[Section~1]{Rota1980a
, ``complicated''
\cite[Section~14.3]{Hackbusch2009a
, etc. This is however \emph{not} due to the Hausdorff derivation
but to the use of (finite) formal series as representation
(for nc polynomials). Using \emph{linear representations}
in the sense of Cohn and Reutenauer
\cite{Cohn1994a
\ for elements in the free field $\field{F}$ can even
reveal additional structure, as indicated in Example~\ref{ex:fd.matfact}
(below).
For the somewhat more ``complicated'' example
\begin{align*}
\tilde{p}
&= 3 c y x b + 3 x b y x b + 2 c y x a x + c y b x b - c y a x b
- 2 x b y x a x + 4 x b y b x b\\
&\quad - 3 x b y a x b + 3 x a x y x b - 3 b x b y x b + 6 a x b y x b
+ 2 x a x y x a x + x a x y b x b\\
&\quad - x a x y a x b - 2 b x b y x a x - b x b y b x b + b x b y a x b
+ 5 a x b y b x b - 4 a x b y a x b
\end{align*}
from \cite[Section~8.2]{Camino2006a
\ we refer to \cite[Example~3.7]{Schrempf2019a
.
\begin{Example}\label{ex:fd.matfact}
Let $p = xyzx$. A (minimal) polynomial ALS for $p$ is
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & . & . \\
. & 1 & -y & . & . \\
. & . & 1 & -z & . \\
. & . & . & 1 & -x \\
. & . & . & . & 1
\end{bmatrix}
s=
\begin{bmatrix}
. \\ . \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
Then $\ncpd_x p = xyz + yzx$ admits a factorization into matrices
\cite[Section~3]{Schrempf2019a
:
\begin{displaymath}
\ncpd_x p =
\begin{bmatrix}
x & y
\end{bmatrix}
\begin{bmatrix}
y & . \\
. & z
\end{bmatrix}
\begin{bmatrix}
z \\ x
\end{bmatrix}.
\end{displaymath}
A \emph{minimal} ALS for $\ncpd_x p$ is
\begin{displaymath}
\begin{bmatrix}
1 & -x & -y & . & . & . \\
. & 1 & . & -y & 0 & . \\
. & . & 1 & 0 & -z & . \\
. & . & . & 1 & . & -z \\
. & . & . & . & 1 & -x \\
. & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
\end{Example}
\medskip
\begin{example}
We will use the \emph{directional derivative}
later in Section~\ref{sec:fd.newton} for the (nc)
Newton iteration. For $q = x^2$ we get
the ``Sylvester equation'' $\ncpd_{x|a} q = xa + ax$
which is \emph{linear} in $a$.
\end{example}
\medskip
In (the next) Section~\ref{sec:fd.comp}
we have a look on the
``free'' chain rule which will turn out to be
very elegant.
We avoid the term ``function'' here since one needs
to be careful with respect to evaluation
(domain of definition), e.g.~$f=(xy-yx)^{-1}$ is
not defined for diagonal matrices.
For the efficient evaluation of polynomials (by matrices)
one can use \emph{Horner Systems}
\cite{Schrempf2019a
. A generalization to elements in free fields
is considered in future work.
If there is a ``compositional structure'' available
(in admissible linear systems) it could be used to
further optimize evaluation.
\begin{remark}
For details on minimization (of linear representations)
we refer to \cite{Schrempf2018a9
. Notice in particular that the construction
(of the formal derivative) in Definition~\ref{def:fd.alsfor}
\emph{preserves} refined pivot blocks.
Therefore, if $\als{A}$ is refined,
\emph{linear} (algebraic) techniques
suffice for minimization of $\ncpd_x \als{A}$.
\end{remark}
\medskip
Last but not least, given the alphabet
$\alphabet{X} = \{ x_1, x_2, \ldots, x_d \}$,
we can define the ``free'' (canonical) \emph{gradient}
$\nabla f = [\ncpd_1 f, \ncpd_2 f, \ldots, \ncpd_d f]^{\!\top}
= [\ncpd_{x_1} f, \ncpd_{x_2} f, \ldots, \ncpd_{x_d} f]^{\!\top} \in \field{F}^d$
for some $f \in \freeFLD{\field{K}}{\alphabet{X}}$.
\emph{Cyclic} gradients are discussed in
\cite{Voiculescu2000b
.
And for a ``vector valued'' element $\mathbf{f} = (f_1,f_2,\ldots,f_d)$
we can define the \emph{Jacobian matrix}
$J(\mathbf{f}) = (\ncpd_j f_i)_{i,j=1}^d = (\ncpd_{x_j} f_i)_{i,j=1}^d$.
For a discussion of non-commutative Jacobian matrices
on the level of nc formal power series
we refer to \cite{Reutenauer1992a
.
\newpage
\section{Free Composition}\label{sec:fd.comp}
\begin{figure}
\begin{center}
$\xymatrix@C=0em{ &
{\als{A}_g \sim g \in \field{F}_{\alphabet{Y}}}
\ar[dl]_{\circ \mathbf{f}}
\ar[dr]^{\ncpd_{\mathbf{y}|\mathbf{y}'}} \\
{\tilde{\als{A}}_h \sim h \in \field{F}_{\alphabet{X}}}
\ar[d]_{\text{lin.}} & &
{\als{A}'_g \sim g' \in \field{F}_{\alphabet{Y} \cup \alphabet{Y}'}}
\ar[d]^{\circ (\mathbf{f},\mathbf{f}')} \\
{\als{A}_h \sim h \in \field{F}_{\alphabet{X}}}
\ar[dr]_{\ncpd_x } & &
{\tilde{\als{A}}'_h \sim h' \in \field{F}_{\alphabet{X}}}
\ar[dl]^{\text{lin.}} \\
& {\ncpd_x \als{A}_h \sim \ncpd_x h = h' \sim \als{A}_h'}
}$
\end{center}
\caption{Let $\mathbf{f}=(f_1,\ldots,f_d)$ with
$f_i \in \field{F}_{\alphabet{X}}=\freeFLD{\field{K}}{\alphabet{X}}$
given by the $d$-tuple of admissible linear systems
$\als{A}_{\mathbf{f}} = (\als{A}_1, \ldots, \als{A}_d)$
and $g \in \field{F}_{\alphabet{Y}}$ given by $\als{A}_g=(u_g,A_g,v_g)$
such that the system matrix $A_g$ remains \emph{full} when we replace
each letter $y_i \in \alphabet{Y}$ by the respective element $f_i$,
written as $A_g \circ \mathbf{f}$.
Then $h = g \circ \mathbf{f} \in \field{F}_{\alphabet{X}}$ is defined by the
\emph{admissible system} $\tilde{\als{A}}_h = (u_g, A_g \circ \mathbf{f}, v_g)$
which we \emph{linearize} to obtain an ALS $\als{A}_h$ for $h$
before we apply the (partial) derivation $\ncpd_x$ (left path).
On the other hand, we can apply the (directional) derivation
$\ncpd_{\mathbf{y}|\mathbf{y}'}$ by going over to the free field
$\field{F}_{\alphabet{Y} \cup \alphabet{Y}'}
= \freeFLD{\field{K}}{\alphabet{Y} \cup \alphabet{Y}'}$
with an extended alphabet (with ``placeholders'' $y_i'$), yielding
$g' = \ncpd_{y_1|y_1'} g + \ldots + \ncpd_{y_d|y_d'} g$
given by some ALS $\als{A}'_g=(u'_g,A'_g,v'_g)$.
Then $h' = g' \circ (\mathbf{f},\mathbf{f}') \in \field{F}_{\alphabet{X}}$
is defined by the \emph{admissible system}
$\tilde{\als{A}}_h' = \bigl(u'_g,A'_g \circ (\mathbf{f},\mathbf{f}'),v'_g\bigr)$,
where also each letter $y_i'\in \alphabet{Y}'$ is replaced by the respective
element $f_i' = \ncpd_x f_i \in \field{F}_{\alphabet{X}}$.
After linearization we get an ALS $\als{A}'_h$ such that
$\ncpd_x \als{A}_h - \als{A}'_h = 0$,
that is, $\ncpd_x h = h'$ (right path).
}
\label{fig:fd.chain}
\end{figure}
To be able to formulate a (partial) ``free'' \emph{chain rule} in an elegant way,
we need a suitable notation. It will turn out that Cohn's
\emph{admissible systems} \cite[Section~7.1]{Cohn2006a
\ provide the perfect framework for the ``expansion'' of
letters by elements from another free field.
First we recall the ``classical'' (analytical) chain rule:
Let $X$, $Y$ and $Z$ be (open) sets, $f=f(x)$ differentiable on $X$,
$g=g(y)$ differentiable on $Y$ and $h=h(x)=g\bigl(f(x)\bigr)$.
Then $\frac{\dd}{\dd x} h = \frac{\dd}{\dd f}g\,\frac{\dd}{\dd x} f$
resp.~$h'(x) = g'\bigl(f(x)\bigr) f'(x)$.
\begin{displaymath}
\xymatrix{
{X} \ar[r]^{f} \ar@/^-3ex/[rr]_{h}
& {Y} \ar[r]^{g}
& {Z}
}
\end{displaymath}
\begin{notation}
For a fixed $d \in \numN$ let
$\alphabet{X} = \{ x_1, x_2, \ldots, x_d \}$,
$\alphabet{Y} = \{ y_1, y_2, \ldots, y_d \}$ and
$\alphabet{Y}' = \{ y_1', y_2', \ldots, y_d' \}$
be \emph{pairwise disjoint} alphabets, that is,
$\alphabet{X} \cap \alphabet{Y} =
\alphabet{X} \cap \alphabet{Y}' =
\alphabet{Y} \cap \alphabet{Y}' = \emptyset$.
By $\field{F}_{\alphabet{Z}}$ we denote the free field
$\freeFLD{\field{K}}{\alphabet{Z}}$.
Let $A \in \freeALG{\field{K}}{\alphabet{Y}}^{n \times n}$ be a \emph{linear full}
matrix and $\mathbf{f}=(f_1,f_2,\ldots,f_d)$ a $d$-tuple of elements
$f_i \in \field{F}_{\alphabet{X}}$. By $A \circ \mathbf{f}$
we denote the (not necessarily full) $n \times n$ matrix over
$\field{F}_{\alphabet{X}}$ where each letter $y_i \in \alphabet{Y}$
is replaced by the corresponding $f_i \in \field{F}_{\alphabet{X}}$,
that is,
\begin{displaymath}
A \circ \mathbf{f}
= A_0 \otimes 1 + A_1 \otimes f_1 + \ldots + A_d \otimes f_d
\quad\in \field{F}_{\alphabet{X}}^{n \times n}.
\end{displaymath}
We write $A \circ \als{A}_{\mathbf{f}}
= A \circ (\als{A}_{f_1}, \als{A}_{f_2}, \ldots, \als{A}_{f_d})$
for a \emph{linearized} version induced by the
$d$-tuple of admissible linear systems $\als{A}_{f_i} = (u_{f_i},A_{f_i},v_{f_i})$.
\end{notation}
\medskip
Now let $g \in \field{F}_{\alphabet{Y}}$ be given by the ALS
$\als{A}_g = (u_g,A_g,v_g)$ and
$\mathbf{f}=(f_1,f_2,\ldots,f_d)\in \field{F}_{\alphabet{X}}^d$
such that $A_g \circ \mathbf{f}$ is \emph{full}.
Then (the \emph{unique} element)
$h = g \circ \mathbf{f} \in \field{F}_{\alphabet{X}}$ is
defined by the \emph{admissible system}
$\tilde{\als{A}}_h = (u_g, A_g\circ \mathbf{f}, v_g)$
and we write
\begin{displaymath}
\als{A}_h = (u_h,A_h,v_h) =
\bigl(u_g \circ \als{A}_{\mathbf{f}},
A_g \circ \als{A}_{\mathbf{f}},
v_g \circ \als{A}_{\mathbf{f}}\bigr)
=: \als{A}_g \circ \als{A}_{\mathbf{f}}
\end{displaymath}
for a \emph{linearized} version using
``linearization by enlargement''
\cite[Section~5.8]{Cohn2006a
.
(For details we refer to the proof of Proposition~\ref{pro:fd.chainrule} below.)
Fixing some $x \in \alphabet{X}$,
we get the (partial) derivative $\ncpd_x h$ of $h$
via the derivative $\ncpd_x \als{A}_h$
\begin{displaymath}
\begin{bmatrix}
A_h & A_h^x \otimes 1 \\
. & A_h
\end{bmatrix}
s =
\begin{bmatrix}
. \\ v_h
\end{bmatrix}.
\end{displaymath}
To give a meaning to the right hand side of
$\ncpd_x h = \ncpd_x (g \circ \mathbf{f})$,
we introduce the ``total'' (directional) derivative
$g' := \ncpd_{\mathbf{y}|\mathbf{y}'} g
= \ncpd_{y_1|y_1'} g + \ncpd_{y_2|y_2'} g + \ldots + \ncpd_{y_d|y_d'} g
\in\field{F}_{\alphabet{Y} \cup \alphabet{Y}'}$
given by the ALS
\begin{displaymath}
\als{A}_g' :=
\ncpd_{\mathbf{y}|\mathbf{y}'} \als{A}_g = \left(
\begin{bmatrix}
u_g & .
\end{bmatrix},
\begin{bmatrix}
A_g & \sum_{i=1}^d A_g^{(i)}\otimes y_i' \\
. & A_g
\end{bmatrix},
\begin{bmatrix}
. \\ v_g
\end{bmatrix}
\right)
\end{displaymath}
with letters $y_i' \in \alphabet{Y}'$.
Using a similar notation for the derivatives
\begin{displaymath}
\mathbf{f}' = (f_1',f_2',\ldots,f_d')
:= (\ncpd_x f_1, \ncpd_x f_2, \ldots, \ncpd_x f_d) = \ncpd_x \mathbf{f},
\end{displaymath}
we can write
\begin{equation}\label{eqn:fd.chain}
h' := \ncpd_x(g \circ \mathbf{f})
= g' \circ (\mathbf{f},\mathbf{f}')
= \ncpd_{\mathbf{y}|\mathbf{y}'} g \circ (\mathbf{f},\mathbf{f}')
\end{equation}
given by the \emph{admissible system}
$\als{A}_g' \circ (\mathbf{f},\mathbf{f}')
= \ncpd_{\mathbf{y}|\mathbf{y}'} \als{A}_g \circ (\mathbf{f},\mathbf{f}')$.
After an illustration in the following example,
we show in Proposition~\ref{pro:fd.chainrule} that indeed
\begin{displaymath}
\ncpd_x h = h' = \ncpd_x(g \circ \mathbf{f})
= \ncpd_{\mathbf{y}|\ncpd_x} g \circ \mathbf{f}
\end{displaymath}
using an abbreviation for the right hand side of \eqref{eqn:fd.chain}.
For an overview see Figure~\ref{fig:fd.chain}.
\medskip
\begin{remark}
Cohn writes an \emph{admissible system} $\als{A} = (u,A,v)$
as ``block'' $[A,v]$ with not necessarily scalar column~$v$
\cite[Section~7.1]{Cohn2006a
. A ring homomorphism which preserves fullness of matrices is
called \emph{honest} \cite[Section~5.4]{Cohn2006a
.
\end{remark}
\begin{remark}
It is \emph{crucial} that $A_g \circ \mathbf{f}$ is \emph{full}
to be able to define the composition.
On a purely algebraic level this is sufficient to
define the (partial) chain rule.
For a $d$-tuple $\mathbf{g}=(g_1,g_2,\ldots,g_d)$ given by
admissible linear systems
$\als{A}_{\mathbf{g}}=(\als{A}_{g_1},\als{A}_{g_2},\ldots,\als{A}_{g_d})$
with $\als{A}_{g_i} =(u_{g_i},A_{g_i},v_{g_i})$ we
need $A_{g_i} \circ \mathbf{f}$ \emph{full}
for \emph{all} $i \in \{ 1, 2, \ldots, d \}$.
\end{remark}
\begin{Example}
Here we take $\alphabet{X} = \{ x,y \}$,
$\alphabet{Y} = \{ f, p \}$ and abuse notation.
Let $f = (x^{-1} +y)^{-1} \in \field{F}_{\alphabet{X}}$,
$p = xy \in \field{F}_{\alphabet{X}}$ and
$g = pfp \in \field{F}_{\alphabet{Y}}$.
Then $h = g \circ (f,p) \in \field{F}_{\alphabet{X}}$
is given by the (minimal) ALS
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & . & . & . \\
. & 1 & -y & . & . & . \\
. & . & 1 & -x & . & . \\
. & . & y & 1 & -x & . \\
. & . & . & . & 1 & -y \\
. & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix},
\end{displaymath}
and $\ncpd_x h = \ncpd_x (pfp) = \ncpd_x p \, fp + p\,\ncpd_x f\, p + pf\,\ncpd_x p$ by
\begin{equation}\label{eqn:fd.chain.1}
\left[\!\!
\begin{array}{cccccc|cccccc}
\rule[-0.5ex]{0pt}{2.8ex}
1 & -x & . & . & . & . & . & -1 & . & . & . & . \\
. & 1 & -y & . & . & . & . & . & . & . & . & . \\
. & . & 1 & -x & . & . & . & . & . & -1 & . & . \\
. & . & y & 1 & -x & . & . & . & . & . & -1 & . \\
. & . & . & . & 1 & -y & . & . & . & . & . & . \\
. & . & . & . & . & 1 & . & . & . & . & . & . \\\hline\rule[-0.5ex]{0pt}{2.8ex}
& & & & & & 1 & -x & . & . & . & . \\
& & & & & & . & 1 & -y & . & . & . \\
& & & & & & . & . & 1 & -x & . & . \\
& & & & & & . & . & y & 1 & -x & . \\
& & & & & & . & . & . & . & 1 & -y \\
& & & & & & . & . & . & . & . & 1
\end{array}
\!\!\right]
s =
\left[\!\!
\begin{array}{c}
. \\ . \\ . \\ . \\ . \\ . \\\hline\rule[-0.5ex]{0pt}{2.8ex}
. \\ . \\ . \\ . \\ . \\ 1
\end{array}
\!\!\right].
\end{equation}
On the other hand,
$\ncpd_x (\ncpd_f g + \ncpd_p g) = \ncpd_x (\ncpd_f g) + \ncpd_x (\ncpd_p g)$
is given by (the admissible system)
\begin{displaymath}
\begin{bmatrix}
1 & -p & . & . & . & -\ncpd_x p & . & . \\
. & 1 & -f & . & . & . & -\ncpd_x f & . \\
. & . & 1 & -p & . & . & . & -\ncpd_x p \\
. & . & . & 1 & . & . & . & . \\
. & . & . & . & 1 & -p & . & . \\
. & . & . & . & . & 1 & -f & . \\
. & . & . & . & . & . & 1 & -p \\
. & . & . & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\. \\ . \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
The summands $\ncpd_x p \, fp$ and $pf\,\ncpd_x p$ are easy
to read off in the ALS~\eqref{eqn:fd.chain.1}.
(Alternatively one could add row~8 to row~1 resp.~row~11 to row~4.)
To read off $p\,\ncpd_x f \, p = - p \, (x^{-1} + y)^{-2} \,p$,
we just need to recall the (minimal) ALS
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & -1 \\
y & 1 & . & . \\
& & 1 & -x \\
& & y & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
In other words and with $g \in \field{F}_{\alphabet{Y}}$
given by the ALS $\als{A}_g = (u_g,A_g,v_g)$ of dimension~$n$: $\ncpd_x h$
is given by the \emph{admissible system}
$\ncpd_{\mathbf{y}|\ncpd_x} \als{A}_g$,
\begin{displaymath}
\begin{bmatrix}
A_g & \ncpd_x A_g \\
. & A_g
\end{bmatrix}
s =
\begin{bmatrix}
. \\ v
\end{bmatrix}
\end{displaymath}
of dimension~$2n$. Notice that $\ncpd_x A_g$ is understood here
in a purely symbolic way, that is, over $\field{F}_{\alphabet{Y} \cup \alphabet{Y}'}$
with additional letters ``$\ncpd_x y$'' in $\alphabet{Y}'$ for each $y\in \alphabet{Y}$.
In this sense $\ncpd_{\mathbf{y}|\ncpd_x} \als{A}_g$ is actually \emph{linear}.
\end{Example}
\begin{proposition}[Free Chain Rule]\label{pro:fd.chainrule}
Let $\alphabet{X} = \{ x_1, x_2, \ldots, x_d \}$,
$\alphabet{Y} = \{ y_1, y_2, \ldots, y_d \}$ and
$\alphabet{Y}' = \{ y'_1, y'_2, \ldots, y'_d \}$
be \emph{pairwise disjoint} alphabets
and fix $x \in \alphabet{X}$.
For $g \in \freeFLD{\field{K}}{\alphabet{Y}}$
given by the ALS $\als{A}_g = (u_g,A_g,v_g)$
and the $d$-tuple
$\mathbf{f}=(f_1,f_2,\ldots,f_d)\in \freeFLD{\field{K}}{\alphabet{X}}^d$
such that $A_g \circ \mathbf{f}$ is \emph{full}.
Denote by $\mathbf{f}'$ the $d$-duple
$(\ncpd_x f_1, \ncpd_x f_2, \ldots, \ncpd_x f_d)$.
Then
\begin{displaymath}
\ncpd_x (g \circ \mathbf{f}) =
\ncpd_{\mathbf{y}|\mathbf{y}'} g \circ (\mathbf{f},\mathbf{f}')
=: \ncpd_{\mathbf{y}|\ncpd_x} g \circ \mathbf{f}.
\end{displaymath}
\end{proposition}
\begin{proof}
For $i \in \{ 1, 2, \ldots, d \}$ let
$f_i \in \field{F}_{\alphabet{X}}$ be given by
the admissible linear systems $\als{A}_{f_i} = (u_{f_i},A_{f_i},v_{f_i})$
respectively. In the following we assume ---without loss of generality---
$d=3$, and decompose $A_g \circ \mathbf{f}$ into
$\bigl[\begin{smallmatrix} A_{11} & A_{12} \\ A_{12} & a \end{smallmatrix}\bigr]$
of size~$n$
with $a = \alpha_0 + \alpha_1 f_1 + \alpha_2 f_2 + \alpha_3 f_3$.
A generalization of the ``linearization by enlargement''
\cite[Section~5.8]{Cohn2006a
\ is then to start with the \emph{full} matrix
$A_g \circ \mathbf{f} \oplus A_{f_1} \oplus \ldots \oplus A_{f_d}$,
add \raisebox{0pt}[0pt][0pt]{$u_{f_i} A_{f_i}^{-1}$}
from the corresponding block row to row~$n$, and
$-\alpha_i A_{f_i}^{-1} v_{f_i}$ from the corresponding
block column to column~$n$ in the upper left block
(these transformations preserve fullness):
\begin{displaymath}
\begin{bmatrix}
A_{11} & A_{12} & . & . & . \\
A_{21} & \alpha_0 & u_{f_1} & u_{f_2} & u_{f_3} \\
. & -\alpha_1 v_{f_1} & A_{f_1} & . & . \\
. & -\alpha_2 v_{f_2} & . & A_{f_2} & . \\
. & -\alpha_3 v_{f_3} & . & . & A_{f_3}
\end{bmatrix}.
\end{displaymath}
A \emph{partially} linearized system matrix for
$\ncpd_x (\als{A}_g \circ \mathbf{f})$ is
\begin{equation}\label{eqn:fd.comp.1}
\left[\!\!
\begin{array}{ccccc|ccccc}
\rule[-0.5ex]{0pt}{2.8ex}
A_{11} & A_{12} & . & . & . & * & * & . & . & . \\
A_{21} & \alpha_0 & u_{f_1} & u_{f_2} & u_{f_3} & * & 0 & 0 & 0 & 0 \\
. & \makebox[2.2em]{$-\alpha_1 v_{f_1}$} & A_{f_1} & . & .
& . & 0 & \makebox[2.9em]{$A_{f_1}^x\otimes 1$} & . & . \\
. & \makebox[2.2em]{$-\alpha_2 v_{f_2}$} & . & A_{f_2} & .
& . & 0 & . & \makebox[2.9em]{$A_{f_2}^x\otimes 1$} & . \\
. & \makebox[2.2em]{$-\alpha_3 v_{f_3}$} & . & . & A_{f_3}
& . & 0 & . & . & \makebox[2.9em]{$A_{f_3}^x\otimes 1$} \\\hline\rule[-0.5ex]{0pt}{2.8ex}
& & & & & A_{11} & A_{12} & . & . & . \\
& & & & & A_{21} & \alpha_0 & u_{f_1} & u_{f_2} & u_{f_3} \\
& & & & & . & \makebox[2.1em]{$-\alpha_1 v_{f_1}$} & A_{f_1} & . & . \\
& & & & & . & \makebox[2.1em]{$-\alpha_2 v_{f_2}$} & . & A_{f_2} & . \\
& & & & & . & \makebox[2.1em]{$-\alpha_3 v_{f_3}$} & . & . & A_{f_3}
\end{array}
\!\!\right].
\end{equation}
On the other hand, the system matrix of $\ncpd_{y_i|y_i'} \als{A}_g$ is
\begin{displaymath}
\begin{bmatrix}
A_{11} & A_{12} & A_{11}^{y_i}\otimes y_i' & A_{12}^{y_i}\otimes y_i' \\
A_{21} & a & A_{21}^{y_i}\otimes y_i' & \alpha_i y_i' \\
& & A_{11} & A_{12} \\
& & A_{21} & a
\end{bmatrix},
\end{displaymath}
the corresponding \emph{partially} linearized system matrix
of $(\ncpd_{y_i|y_i'} \als{A}_g)\circ (\mathbf{f},f_i')$ is
\begin{displaymath}
\left[\!\!
\begin{array}{ccccc|ccccc}
\rule[-0.5ex]{0pt}{2.8ex}
A_{11} & A_{12} & . & . & . & A_{11}^{y_i}\otimes f_i' & A_{12}^{y_i}\otimes f_i' & . & . & . \\
A_{21} & \alpha_0 & u_{f_1} & u_{f_2} & u_{f_3} & A_{21}^{y_i}\otimes f_i' & \fbox{$\alpha_i f_i'$} & . & . & . \\
. & -\alpha_1 v_{f_1} & A_{f_1} & . & . & . & . & . & . & . \\
. & -\alpha_2 v_{f_2} & . & A_{f_2} & . & . & . & . & . & . \\
. & -\alpha_3 v_{f_3} & . & . & A_{f_3} & . & . & . & . & . \\\hline\rule[-0.5ex]{0pt}{2.8ex}
& & & & & A_{11} & A_{12} & . & . & . \\
& & & & & A_{21} & \alpha_0 & u_{f_1} & u_{f_2} & u_{f_3} \\
& & & & & . & -\alpha_1 v_{f_1} & A_{f_1} & . & . \\
& & & & & . & -\alpha_2 v_{f_2} & . & A_{f_2} & . \\
& & & & & . & -\alpha_3 v_{f_3} & . & . & A_{f_3}
\end{array}
\!\! \right].
\end{displaymath}
To eliminate the boxed entry $\alpha_i f_i'$ we just need to recall
the derivative $\ncpd_x \als{A}_{f_i}$ of $\als{A}_{f_i}=(u_{f_i},A_{f_i},v_{f_i})$,
the invertible matrix $Q$ swaps block columns~1 and~2:
\begin{displaymath}
\left[\!\!
\begin{array}{c|cc}
0 & u_{f_i} & . \\\hline\rule[-0.5ex]{0pt}{2.8ex}
. & A_{f_i} & A_{f_i}^x \otimes 1 \\
-\alpha_i v_{f_i} & . & A_{f_i}
\end{array}
\!\!\right]
Q =
\begin{bmatrix}
u_{f_i} & \fbox{$0$} & . \\
A_{f_i} & . & A_{f_i}^x \otimes 1 \\
. & -\alpha_i v_{f_i} & A_{f_i}
\end{bmatrix}.
\end{displaymath}
Thus, after summing up (over $i \in \{ 1, 2, \ldots, d \}$),
we get the system matrix \eqref{eqn:fd.comp.1}
and hence
\begin{displaymath}
\ncpd_x \als{A}_h = \ncpd_x (\als{A}_g \circ \als{A}_f) =
\ncpd_{\mathbf{y}|\mathbf{y}'} A_{g} \circ \als{A}_{\mathbf{f},\mathbf{f}'},
\end{displaymath}
that is, $\ncpd_x h = \ncpd_x (g \circ \mathbf{f})
= \ncpd_{\mathbf{y}|\ncpd_x \mathbf{f}} g \circ \mathbf{f}$.
\end{proof}
\newpage
Free (non-commutative) \emph{de}composition is important in
\emph{control theory}
\cite[Section~6.2.2]{deOliveira2006a
:
``The authors do not know how to fully implement the decompose operation.
Finding decompositions by hand can be facilitated with the use of certain
type of collect command.''
Let $g = f a b f + c f + d e$ and $f = x y + z$,
that is, $f$ solves a Riccati equation.
Given $h = g \circ f = (xy+z)ab(xy+z) + c(xy+z) +de$ by
the (minimal) admissible \emph{linear} system
$\als{A}_h=(u_h,A_h,v_h)$,
\begin{displaymath}
\begin{bmatrix}
1 & -x & -z & . & -d & -c & . & . \\
. & 1 & -y & . & . & . & . & . \\
. & . & 1 & -a & . & . & . & . \\
. & . & . & 1 & . & -b & . & . \\
. & . & . & . & 1 & . & . & -e \\
. & . & . & . & . & 1 & -x & -z \\
. & . & . & . & . & . & 1 & -y \\
. & . & . & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix},
\end{displaymath}
one can read off $f = xy+z$ \emph{directly} since
$A_h$ has the form
\begin{displaymath}
\begin{bmatrix}
1 & \fbox{$-f$} & . & -d & -c & . \\
. & 1 & -a & . & . & . \\
. & . & 1 & . & -b & . \\
. & . & . & 1 & . & -e \\
. & . & . & . & 1 & \fbox{$-f$} \\
. & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
So, starting with a \emph{minimal} ALS $\tilde{\als{A}}_h$ for $h$
one ``just'' needs to find appropriate (invertible)
transformation matrices $P,Q$ such that $\als{A}_h = P \tilde{\als{A}}_h Q$
using (commutative) Gröbner bases techniques
similar to \cite[Theorem~4.1]{Cohn1999a
\ or the refinement of pivot blocks
\cite[Section~3]{Schrempf2018a9
. Although this is quite challenging using brute force
methods, in practical examples it is rather easy using
\emph{free fractions}, that is, \emph{minimal} and \emph{refined}
admissible linear systems, as a ``work bench'' where one
can perform the necessary row and column operations manually.
``The challenge to computer algebra is to start with an expanded version of
$h = g \circ f$, which is a mess that you would likely see in a computer
algebra session, and to automatically motivate the selection of $f$.''
\cite[Section~6.2.1]{deOliveira2006a
Working with \emph{free fractions} is simple, in particular
with nc polynomials. The main difficulty in working with (finite)
formal power series is that the number of words can grow exponentially
with respect to the \emph{rank}, the minimal dimension of a linear
representation \cite[Table~1]{Schrempf2019a
. In this case, for
\begin{displaymath}
h = g \circ f = x y a b x y + x y a b z + z a b x y + z a b z + c x y + c z + d e,
\end{displaymath}
the main ``structure'' becomes (almost) visible already after
\emph{minimization} of the corresponding ``polynomial''
admissible linear system (in upper triangular form with ones in the diagonal).
\newpage
\section{Application: Newton Iteration}\label{sec:fd.newton}
To compute the third root $\sqrt[3]{z}$ of a (positive) real number $z$,
say $z=2$,
we just have to find the roots of the polynomial $p = x^3 - z$,
for example by using the Newton iteration
$x_{k+1} := x_k - p(x_k)/p'(x_k)$, that
---given a ``good'' starting value $x_0$--- yields
a (quadratically) convergent sequence $x_0, x_1, x_2, \ldots$
such that $\lim_{k \to \infty} x_k^3 = z$.
Using \textsc{FriCAS} \cite{FRICAS2019
, the first 6~iterations for $x_0=1$ are
\begin{center}
\begin{tabular}{rd{8}d{16}}
$k$ & \multicolumn{1}{c}{$\lvert x_k - x_{k-1}\rvert$}
& \multicolumn{1}{c}{$x_k$} \\\hline\rule[-0.5ex]{0pt}{3.2ex}
1 & 3.333\cdot 10^{-1} & 1.3333333333333333 \\
2 & 6.944\cdot 10^{-2} & 1.2638888888888888 \\
3 & 3.955\cdot 10^{-3} & 1.259933493449977 \\
4 & 1.244\cdot 10^{-5} & 1.2599210500177698 \\
5 & 1.229\cdot 10^{-10} & 1.2599210498948732 \\
6 & 0 & 1.2599210498948732
\end{tabular}
\end{center}
Detailed discussions are available in every book on
numerical analysis, for example
\cite{Henrici1964a
. As a starting point towards current research,
one could take \cite{Schleicher2017a
.
\medskip
Now we would like to compute the third root $\sqrt[3]{Z}$
of a real (square) matrix $Z$ (with positive eigenvalues).
We (still) can use
$X_{k+1} := \tsfrac{2}{3} X_k + \tsfrac{1}{3} X_k^{-2} Z$
if we choose a starting matrix $X_0$
such that $X_0 Z = Z X_0$
because then all $X_k$ \emph{commute} with $Z$
\cite[Section~7.3]{Higham2008a
.
For $X_0=I$ and
\begin{displaymath}
Z =
\begin{bmatrix}
47 & 84 & 54 \\
42 & 116 & 99 \\
9 & 33 & 32
\end{bmatrix}
\quad
\text{respectively}
\quad
\sqrt[3]{Z} =
\begin{bmatrix}
3 & 2 & 0 \\
1 & 4 & 3 \\
0 & 1 & 2
\end{bmatrix},
\end{displaymath}
some iterations are (with $\fronorm{.}$ denoting the Frobenius norm)
\begin{center}
\begin{tabular}{rd{8}d{8}}
$k$ & \multicolumn{1}{c}{$\lVert X_k - X_{k-1}\rVert_{\text{F}}$}
& \multicolumn{1}{c}{$\lVert X_k - \sqrt[3]{Z} \rVert_{\text{F}}$} \\\hline\rule[-0.5ex]{0pt}{3.2ex}
4 & 9.874 & 23.679 \\
5 & 6.511 & 13.818 \\
6 & 4.109 & 7.309 \\
7 & 2.254 & 3.200 \\
8 & 8.295\cdot 10^{-1} & 9.455\cdot 10^{-1} \\
9 & 1.140\cdot 10^{-1} & 1.160\cdot 10^{-1} \\
10 & 2.044\cdot 10^{-3} & 2.044\cdot 10^{-3} \\
11 & 1.115\cdot 10^{-6} & 6.489\cdot 10^{-7}
\end{tabular}
\end{center}
Does this sequence of matrices $X_0,X_1,\ldots$ converge?
Some more iterations reveal immediately that something goes wrong,
visible in Table~\ref{tab:fd.newton}.
The problem is that due to rounding errors (in finite precision
arithmetics) commutativity of $X_k$ (with $Z$) is lost,
that is, $\fronorm{X_k Z - Z X_k}$ \emph{increases} with every
iteration. This happens even for a starting value close
to the solution $X_0 = \sqrt[3]{Z} + \varepsilon I$.
For a detailled discussion of matrix roots
and how to overcome such problems
we refer to \cite[Section~7]{Higham2008a
, for further information about numerics
to \cite{Demmel1997a
. A classical introduction to matrix functions
is \cite[Chapter~V]{Gantmacher1965a
. For the (matrix) square root and discussions about
the stability (of Newton's method) we recommend
\cite{Higham1986b
.
\begin{table}[ht]
\begin{center}
\begin{tabular}{rd{8}d{8}d{8}}
$k$ & \multicolumn{1}{c}{$\fronorm{X_k - X_{k-1}}$}
& \multicolumn{1}{c}{$\fronorm{X_k - \sqrt[3]{Z}}$}
& \multicolumn{1}{c}{$\fronorm{X_k Z - Z X_k}$} \\\hline\rule[-0.5ex]{0pt}{3.2ex}
1 & 65.836 & 5.385 & 2.274\cdot 10^{-13}\\
2 & 22.279 & 60.756 & 7.541\cdot 10^{-13}\\
3 & 14.845 & 38.500 & 1.110\cdot 10^{-12}\\
4 & 9.874 & 23.679 & 2.031\cdot 10^{-12}\\
5 & 6.511 & 13.818 & 7.725\cdot 10^{-12}\\
6 & 4.109 & 7.309 & 7.010\cdot 10^{-11}\\
7 & 2.254 & 3.200 & 9.451\cdot 10^{-10}\\
8 & 8.295\cdot 10^{-1} & 9.455\cdot 10^{-1} & 1.716\cdot 10^{-8}\\
9 & 1.140\cdot 10^{-1} & 1.160\cdot 10^{-1} & 3.548\cdot 10^{-7}\\
10 & 2.044\cdot 10^{-3} & 2.044\cdot 10^{-3} & 7.473\cdot 10^{-6}\\
11 & 1.115\cdot 10^{-6} & 6.489\cdot 10^{-7} & 1.574\cdot 10^{-4}\\
12 & 1.979\cdot 10^{-5} & 8.971\cdot 10^{-7} & 3.314\cdot 10^{-3}\\
13 & 4.168\cdot 10^{-4} & 1.890\cdot 10^{-5} & 6.978\cdot 10^{-2}\\
14 & 8.777\cdot 10^{-3} & 3.979\cdot 10^{-4} & 1.469 \\
15 & 1.848\cdot 10^{-1} & 8.379\cdot 10^{-3} & 3.094\cdot 10^{+1}\\
16 & 3.892 & 1.764\cdot 10^{-1} & 6.515\cdot 10^{+2}\\
17 & 8.195\cdot 10^{+1} & 3.715 & 1.372\cdot 10^{+4}\\
18 & 1.726\cdot 10^{+3} & 7.823\cdot 10^{+1} & 2.889\cdot 10^{+5}
\end{tabular}
\end{center}
\vspace{-2ex}
\caption{The first 18 Newton iterations to find $\sqrt[3]{Z}$
for $X_0=I$. The values in column~2 (and~3) for the first 11 Newton
steps seem to indicate convergence. However, the (Frobenius) norm
of the \emph{commutator} $X_k Z - Z X_k$ (column~4) increases steadily and
that causes eventually a \emph{diverging} sequence $(X_k)$.
}
\label{tab:fd.newton}
\end{table}
\begin{remark}
To ensure commutation of the iterates $X_k$ with $Z$
one could use an additional correction step solving
the Sylvester equation $Z \Delta X_k - \Delta X_k Z= X_k Z - Z X_k$
for an update $\Delta X_k$.
Since this is expensive the benefit of using
the ``commutative'' Newton iteration would be lost.
For fast solutions of Sylvester equations we
refer to \cite{Kirrinnis2001a
.
\end{remark}
\medskip
Although what we are going to introduce now
as ``\emph{non-commutative} (nc) \emph{Newton iteration}'' is even
more expensive, it can be implemented as \emph{black box} algorithm,
that is, \emph{without} manual computation of the non-commutative
derivative and any individual implementation/programming.
Furthermore, there is absolutely no restriction for the
initial iterate.
\medskip
To get the nc Newton method for solving $f(x)=0$ we just need to
``truncate'' the Hausdorff \emph{polarization operator}
\cite[Section~4]{Rota1980a
\begin{displaymath}
f(x+b) = f(x) + \ncpd_{x|b} f(x)
+ \tsfrac{1}{2}\ncpd_{x|b}\bigl(\ncpd_{x|b} f(x) \bigr) + \ldots
\end{displaymath}
as analogon to Taylor's formula. Thus, for $f(x) = x^3 - z$ we
have to solve
\begin{displaymath}
0 = \underbrace{x^3 - z}_{f} + \underbrace{bx^2 + xbx + x^2b}_{\ncpd_{x|b} f}
\quad \in \freeFLD{\numR}{\alphabet{X}} =: \field{F}
\end{displaymath}
with respect to $b$. In terms of (square) matrices $X,Z,B$ this
is just the generalized Sylvester equation
$B X^2 + X B X + X^2 B = Z - X^3$
which is \emph{linear} in $B$
\cite[Section~7.2]{Higham2008a
. Taking a (not necessarily with $Z$ commuting) starting matrix $X_0$,
we can compute $B_0$ and get $X_1 := X_0 + B_0$ and iteratively
$X_{k+1} := X_k + B_k$.
Minimal admissible linear systems for $f=x^3-z$ and
$\ncpd_{x|b} f=bx^2 + xbx + x^2b$ are given by
\vspace{-3ex}
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & z \\
. & 1 & -x & . \\
. & . & 1 & -x \\
. & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ 1
\end{bmatrix}
\quad\text{and}\quad
\begin{bmatrix}
1 & -x & . & -b & . & . \\
. & 1 & -x & . & -b & . \\
. & . & 1 & . & . & -b \\
. & . & . & 1 & -x & . \\
. & . & . & . & 1 & -x \\
. & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix}
\end{displaymath}
respectively. A \emph{minimal} ALS
$\als{A}=(u,A,v)$ of dimension $n=6$
for $g = f+ \ncpd_{x|b}f$ is immediate:
\begin{displaymath}
\begin{bmatrix}
1 & -x & . & -b & . & z \\
. & 1 & -x & . & -b & . \\
. & . & 1 & . & . & -b-x \\
. & . & . & 1 & -x & . \\
. & . & . & . & 1 & -x \\
. & . & . & . & . & 1
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ . \\ . \\ . \\ 1
\end{bmatrix}.
\end{displaymath}
Since the system matrix is just a linear matrix pencil
$A = A_1 \otimes 1 + A_b \otimes b + A_x \otimes x + A_z \otimes z$
we can plug in $m \times m$ matrices using the Kronecker product
\begin{displaymath}
A(B,X,Z)
= A_1 \otimes I + A_b \otimes B + A_x \otimes X + A_z \otimes Z.
\end{displaymath}
In this case, for $v = e_n = [0, \ldots, 0, 1]^{\!\top}$,
the evaluation of $g: M_m(\numR)^3 \to M_m(\numR)$
is just the upper right $m \times m$ block of $A(B,X,Z)^{-1}$.
For how to efficiently evaluate (nc) polynomials by matrices
using \emph{Horner systems} we refer to
\cite{Schrempf2019a
.
Here, $A(B,X,Z)$ is invertible for arbitrary $B$, $X$ and $Z$.
Given $Z$ and $X$,
to find a $B$ such that $g(B,X,Z)=0$
we have to look for transformation
matrices $P=P(T_{ij})$ and $Q=Q(U_{ij})$ of the form
\begin{displaymath}
P =
\begin{bmatrix}
I & . & . & T_{1,1} & T_{1,2} & 0 \\
. & I & . & T_{2,1} & T_{2,2} & 0 \\
. & . & I & T_{3,1} & T_{3,2} & 0 \\
. & . & . & I & . & . \\
. & . & . & . & I & . \\
. & . & . & . & . & I
\end{bmatrix},
\quad
Q =
\begin{bmatrix}
I & . & . & 0 & 0 & 0 \\
. & I & . & U_{2,1} & U_{2,2} & U_{2,3} \\
. & . & I & U_{3,1} & U_{3,2} & U_{3,3} \\
. & . & . & I & . & . \\
. & . & . & . & I & . \\
. & . & . & . & . & I
\end{bmatrix}
\end{displaymath}
such that the upper right block of size $3m \times 3m$ of
$P A(B,X,Z) Q$ is zero, that is,
we need to solve a \emph{linear} system of equations
(with $6m^2+6m^2+m^2 = 13m^2$ unknowns)
similar to the word problem
\cite[Theorem~2.4]{Schrempf2017a9
. Table~\ref{tab:fd.ncnewton} shows the nc
Newton iterations for the starting matrix
\begin{equation}\label{eqn:fd.ncnewton.x0}
X_0 =
\begin{bmatrix}
1 & 0 & 2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{equation}
using \textsc{FriCAS} \cite{FRICAS2019
\ and the least squares solver DGELS from
\cite{LAPACK2018
.
\begin{table}
\begin{center}
\begin{tabular}{rd{8}d{8}d{8}}
$k$ & \multicolumn{1}{c}{$\fronorm{B_k}$}
& \multicolumn{1}{c}{$\fronorm{X_k - \sqrt[3]{Z}}$}
& \multicolumn{1}{c}{$\fronorm{X_k Z - Z X_k}$} \\\hline\rule[-0.5ex]{0pt}{3.2ex}
0 & 46.877 & 5.745 & 113.842 \\
1 & 16.081 & 42.298 & 5552.242 \\
2 & 10.768 & 26.374 & 3659.788 \\
3 & 7.320 & 15.912 & 2395.971 \\
4 & 4.971 & 9.358 & 1534.892 \\
5 & 2.934 & 6.122 & 912.700 \\
6 & 2.651 & 4.389 & 414.201 \\
7 & 1.380 & 1.846 & 86.771 \\
8 & 3.878\cdot 10^{-1} & 4.875\cdot 10^{-1} & 5.638 \\
9 & 9.378\cdot 10^{-2} & 1.023\cdot 10^{-1} & 2.652\cdot 10^{-1} \\
10 & 8.432\cdot 10^{-3} & 8.506\cdot 10^{-3} & 6.737\cdot 10^{-3} \\
11 & 7.407\cdot 10^{-5} & 7.408\cdot 10^{-5} & 1.951\cdot 10^{-5} \\
12 & 5.895\cdot 10^{-9} & 5.895\cdot 10^{-9} & 5.106\cdot 10^{-10} \\
13 & 1.825\cdot 10^{-14} & 1.521\cdot 10^{-14} & 1.491\cdot 10^{-12}
\end{tabular}
\end{center}
\vspace{-2ex}
\caption{The first 13 (nc) Newton iterations to find $\sqrt[3]{Z}$
for $X_0$ from \eqref{eqn:fd.ncnewton.x0}.
In the beginning, the iterates $X_k$ do not commute with $Z$
(column~4).
}
\label{tab:fd.ncnewton}
\end{table}
\medskip
In general, depending on the initial iterate $X_0$,
there is \emph{no} guarantee that one ends up in some
prescribed solution $X$ since there can be several
(matrix) roots.
For the \emph{principal} $p$-th root (and further discussion)
we refer to \cite[Theorem~7.2]{Higham2008a
.
For a discussion of Taylor's theorem (for matrix functions)
one should have a look in
\cite{Deadman2016a
.
\medskip
The goal of this section was mainly for illustration
(of the use of the free derivative).
How could we attack analysis of convergence (for special classes
of rational functions) in this context?
``Classical'' interval Newton is discussed in
\cite[Chapter~6.1]{Rump2010a
. What's about ``nc interval Newton''?
The next natural step would be \emph{multi-dimensional} nc Newton,
say to find a root $G(X,Y,Z)=0$ for
\begin{displaymath}
G =
\begin{bmatrix}
g_1 \\ g_2 \\ g_3
\end{bmatrix}
=
\begin{bmatrix}
x^3 + z -d \\
\bigl(x(1-yx)\bigr)^{-1} -e \\
xzx - yz + z^2 - f
\end{bmatrix}
\end{displaymath}
with (matrix-valued) parameters $d,e,f$.
Although very technical, one can set up a ``joint'' linear
system of equations to solve
\begin{displaymath}
\begin{pmatrix}
g_1 + \ncpd_{x\mid a} g_1 & g_1 + \ncpd_{y\mid b} g_1 & g_1 + \ncpd_{z\mid c} g_1 \\
g_2 + \ncpd_{x\mid a} g_2 & g_2 + \ncpd_{y\mid b} g_2 & g_2 + \ncpd_{z\mid c} g_2 \\
g_3 + \ncpd_{x\mid a} g_3 & g_3 + \ncpd_{y\mid b} g_3 & g_3 + \ncpd_{z\mid c} g_3
\end{pmatrix} = 0
\end{displaymath}
for matrices $A$, $B$ and $C$
using the previous approach to find tuples of ``individual''
transformation matrices $P_{ij}$ and $Q_{ij}$ to create
respective upper right blocks of zeros.
The problem however is, that this system is \emph{overdetermined}
in general, and starting arbitrary close to a root
leads to a residual that causes divergence.
How can one overcome that?
\section*{Acknowledgement}
I am very grateful to Tobias Mai for making me aware of cyclic derivatives
respectively the work of Rota, Sagan and Stein
\cite{Rota1980a
\ in May 2017 in Graz and for a
discussion in October 2018 in Saarbrücken.
I also use this opportunity to thank Soumyashant Nayak
for continuous ``non-commutative'' discussions,
in particular about free associative algebras
and (inner) derivations.
And I thank the referees for the feedback and a hint on the
literature.
\bibliographystyle{alpha}
|
1,108,101,565,475 | arxiv | \section{I. Introduction}
Although supersymmetry has so far eluded discovery at the LHC \cite{ATLAS20,CMS20}, it remains an important component for building models
beyond the Standard Model, particularly when going to grand unified scales and beyond. It is well known that supersymmetry can provide for the naturalness of the weak scale \cite{Maiani:1979cx}, the unification of gauge couplings \cite{Ellis:1990zq}, and a cold dark matter candidate \cite{gold,ehnos}.
As supergravity~\cite{cremmer,susy} is the extension of supersymmetry which includes gravity, it is the
natural framework for any cosmological scenario involving supersymmetry.
Thus, the gravitino also becomes a component in post-inflationary cosmology.
The gravitino has long since been a cosmological headache \cite{prob,sw,eln,ego}.
It has the potential to overclose the universe if it is stable
(and hence a dark matter candidate), or lead to an overabundance of the lightest supersymmetric particle if it decays. While inflation may dilute any primordial gravitinos \cite{eln}, they are reproduced during reheating after inflation \cite{nos,ehnos,kl,ekn,oss,Juszkiewicz:gg,mmy,Kawasaki:1994af,Moroi:1995fs,enor,Giudice:1999am,bbb,cefo,kmy,stef,Pradler:2006qh,ps2,rs,kkmy,egnop,Garcia:2017tuj,Eberl:2020fml}. Decaying gravitinos may also upset the concordance of big bang nucleosynthesis \cite{foyy} and these constraints result in limits on the gravitino abundance which ultimately translate into limits on the reheat temperature \cite{bbn,cefo,ceflos175,Kawasaki:1994af,stef}.
In addition, there is also a non-thermal contribution to the gravitino abundance stemming from the non-perturbative decay of the inflaton when supersymmetry is broken during inflation \cite{Kallosh:1999jj,Giudice:1999yt,Giudice:1999am,Kallosh:2000ve,Nilles:2001ry,Nilles:2001fg,Ema:2016oxl}. However, it was shown that what is produced in this process is the goldstino, that is, the combination of fermions contributing to supersymmetry breaking at the time of their production.
At the inflationary scale, the goldstino is typically dominated by the inflatino, the fermionic partner of the inflaton. The longitudinal component of the gravitino at low energy is associated with the true goldstino which may be unrelated to the inflatino produced in reheating \cite{Nilles:2001ry,Nilles:2001fg}.
The production of inflatinos may be problematic, particularly if the mass of the inflatino is less than the mass of the inflaton, but may be suppressed and the final gravitino abundance is model dependent \cite{Nilles:2001my}.
Recently, it has been claimed that if the sound speed, $c_s$, for gravitinos
were to vanish, there would be a catastrophic non-thermal production of slow gravitinos \cite{Kolb:2021xfn,Kolb:2021nob}.
This is due to the lack of suppression in the production of high momentum modes in the helicity 1/2 spectrum.
While $c_s$ may indeed vanish in certain models of inflation derived from supergravity, as in the case of the non-thermal production of the longitudinal mode, mixing with other fermionic fields whose scalar partners have non-vanishing time derivatives precludes the runaway production of gravitinos.
Nevertheless, in some realizations of non-linear supergravity with constrained fields, the inflatino may not be present in the spectrum, and indeed catastrophic production may occur.
In this paper, we consider the conditions which lead to $c_s = 0$. We look at several
models of inflation in supergravity theories
and distinguish those for which $c_s = 0$ at some point in the post-inflationary oscillation period of the inflaton. We stress that $c_s = 0$ is not a generic feature of supergravity inflation models. Further we argue that even if $c_s = 0$, in the absence of constrained fields, catastrophic gravitino production does {\em not} occur. Indeed even in constrained models with a nilpotent field (eliminating a scalar from the spectrum) \cite{Rocek:1978nb,Komargodski:2009rz,Antoniadis:2014oya,Ferrara:2014kva,Kallosh:2014via,DallAgata:2014qsj,Dudas:2015eha,Ferrara:2015gta,Ferrara:2015cwa}, there is no catastrophic production. Only when a second, orthogonal constraint \cite{Ferrara:2015tyn,Carrasco:2015iij,DallAgata:2015zxp,Kallosh:2016hcm,DallAgata:2016syy} which eliminates, the pseudoscalar, fermionic and auxiliary partners of the inflaton, can catastrophic production occur when $c_s = 0$. In \cite{hase}, only models with both constraints were considered leading to a truncated equation of motion for the longitudinal component of the gravitino \cite{Ferrara:2015tyn} and explosive particle production. While the nilpotent constraint can easily be viewed as the infrared limit of a heavy sgoldstino (the scalar field associated with supersymmetry breaking), the orthogonal constraint appears to require higher derivative operators in the action \cite{DallAgata:2016syy}. Indeed we would argue that the constraints from gravitino production signal a further problem for the orthogonal constraint rather than imposing conditions on general models of inflation in supergravity.
In what follows, we first consider in Section \ref{sec:theta} the calculation of the sound speed in supergravity theories. It is easy to see that in models with a single chiral field, $c_s = 1$. We next consider in Section \ref{sec:slow} several models of inflation with supersymmetry breaking showing that $c_s$ is quite model dependent.
In some cases $c_s = 0$ occurs at some specific points during the oscillation of the inflaton, while in other often studied models, $c_s \simeq 1$ always. In Section \ref{sec:theta-Y}, we reconsider the equations of motion for the longitudinal component of the gravitino. We show clearly that even when $c_s = 0$, there is no enhanced production of gravitinos. The nilpotency conditions are discussed in Section \ref{sec:nil}. With a single nilpotent field, our conclusions are left unchanged. Only when a second orthogonal condition is imposed, and the inflatino is eliminated from the spectrum, is there the potential for explosive gravitino production. A discussion and our conclusions are given in Section \ref{sec:conclusions}.
\section{Gravitino sound speed in supergravity}
\label{sec:theta}
It is argued in \cite{Kolb:2021xfn,Kolb:2021nob} that the vanishing of the gravitino sound speed is accompanied by a catastrophic gravitino overproduction during the first set of inflaton oscillations. The mathematical reason for this statement can be understood from the linearized theory of an uncoupled longitudinal gravitino. The gravitino has a dispersion relation of the type $\omega_k^2=c_s^2 k^2 + a^2 m^2(t)$, $a$ being the scale factor and $k$ the comoving momentum. As is well known from studies of preheating \cite{preheating,stb, kt,Greene:1997fu}, a non-adiabatic change of the frequency, $\omega_k' / \omega_k^2 \gg 1$ (where prime denotes derivative with respect to conformal time), results in significant particle production. Normally, the $k^2$ term ensures that the variation is adiabatic in the deep UV, so that modes of high momenta remain in their vacuum state. This is not the case if $c_s^2 =0$. In fact, the production obtained in \cite{hase,Kolb:2021xfn,Kolb:2021nob} is formally divergent in the UV, and it is presumably regulated by higher-order operators. This would in any case result in the production of an unacceptably large gravitino abundance.
As we shall see in Section \ref{sec:theta-Y}, this problem is typically not present in multi-field models, since in this case the longitudinal gravitino is coupled already at the linearized level to a linear combination $\Upsilon$ of the fermions in the theory \cite{Kallosh:2000ve,Nilles:2001fg}. The gravitino momentum term is then replaced by a matrix in field space, that is non-singular when $c_s^2 = 0$. Therefore, the physical eigenvalues of the system do not vanish, and the catastrophic production in the UV is avoided.
We begin by first computing $c_s$ in supergravity.
The equations for the transverse and longitudinal gravitino component presented in this work follow the formalism of Ref.~\cite{Kallosh:2000ve}. In that work, the FLRW line element is chosen according to $ds^2 = a^2 \left( \tau \right) \left[ - d \tau^2 + d \vec{x}^2 \right]$, where $\tau$ is the conformal time. We denote by $\gamma^\mu$ the gamma matrices in flat spacetime. We work in units of $M_p =1$, where $M_p$ is the reduced Planck mass.
The equation of motion for the transverse gravitino component is
\begin{eqnarray}
\left( \gamma^0 \partial_0 + i \gamma^i k_i + a m_{3/2} \right) \psi^T = 0
\label{eq-psi32}
\end{eqnarray}
where $\vec{k}$ is the comoving momentum, and $\partial_0$ is the derivative with respect to the conformal time. The propagation of the
transverse mode of the gravitino
is not directly affected by the supersymmetry breaking mechanism, and obeys a classical Dirac equation with a speed of sound $c_s=1$. The gravitino mass is given by
\begin{equation}
m_{3/2} = {\rm e}^{K/2} \, \left\vert W \right\vert \;,
\end{equation}
where $K$ and $W$ denote the K\"ahler potential and the superpotential, respectively.
The equation for the longitudinal gravitino will be presented and discussed in Section \ref{sec:theta-Y}. Here we discuss one important difference with Eq.~(\ref{eq-psi32}) for the longitudinal component. Namely, the momentum-dependent term in the equation for the longitudinal component is multiplied by a function of the background scalar fields, that, once squared, is identified as the square of the longitudinal gravitino sound speed $c_s^2$,
\begin{equation}
c_s^2 = \frac{\left(p-3 m_{3/2}^{2}\right)^{2}}{\left(\rho+3 m_{3/2}^{2} \right)^{2}}+\frac{4 \dot{m}_{3/2}^2}{\left(\rho+3 m_{3/2}^{2} \right)^{2}} \, ,
\label{Eq:generic32}
\end{equation}
where $p$ is the pressure, $\rho$ is the energy density, and dot denotes the derivative with respect to cosmological time.
Ref.~\cite{Kolb:2021xfn} provides a rather compact expression for this quantity in a supergravity model, namely
\begin{equation}
c_s^2 = 1 - \frac{4}{
\left(\vert\dot{\varphi}\vert^2 + \vert F \vert^2 \right)^2} \,
\left\{
|\dot{\varphi}|^2 |F|^2 -
\left\vert \dot{\varphi} \cdot F^* \right\vert^2 \right\} \;,
\label{vs2}
\end{equation}
where $\varphi$ is the multiplet of scalar fields in the model, and the $F$-term is
\begin{equation}
F^i \equiv {\rm e}^{K/2} K^{ij^*} \, D_{j^*} W^* \;,
\label{DWF}
\end{equation}
where, using standard supergravity notation, $K^{ij^*}$ is the inverse of the K\"ahler metric
\begin{equation}
K_{ij^*} \equiv \frac{\partial^2 K}{\partial \varphi^i \, \partial \varphi^{j*}} \;,
\label{Kij}
\end{equation}
while
\begin{equation}
D_i W \equiv \frac{\partial W}{\partial \varphi^i} + \frac{\partial K}{\partial \varphi^i} \, W \,.
\end{equation}
The dot operator in Eq.~(\ref{vs2}) denotes a scalar product with the K\"ahler metric (\ref{Kij}), namely $\vert \dot{\varphi} \vert^2 = \dot{\varphi}^i \, K_{ij^*} \, \dot{\varphi}^{j*}$, and analogously for the other terms.
Notice that due to the Cauchy-Schwarz type inequality
$|\dot{\varphi}|^2 |F|^2 \geq
\left\vert \dot{\varphi} \cdot F^* \right\vert^2$, causality $c_s \leq 1$ is always guaranteed to hold.
In the case of a single chiral superfield, the two terms in the curly bracket in Eq. (\ref{vs2}) are equal to each other, and therefore $c_s=1$.
Thus, $c_s^2 \simeq 1$ is expected whenever a single field dominates the kinetic energy and supersymmetry breaking in the model.
Ref. \cite{Kolb:2021xfn} considered the case in which multiple fields are relevant, and they conspire to give a vanishing or nearly vanishing $c_s^2$.
This can be achieved if
\begin{equation}
\dot{\varphi} \cdot F^* = 0 \;\;{\rm and}\;\;
\dot{\varphi} \cdot \dot{\varphi}^* =
F \cdot F^* \;\; \Rightarrow \;\;
c_s^2 = 0 \;.
\label{vs0}
\end{equation}
Note that the first of these conditions, $\dot{\varphi} \cdot F^* = 0$, is realized whenever the gravitino mass is constant.
These conditions can be satisfied during inflaton oscillations after inflation. Typically, the inflaton dominates the kinetic energy, so the condition $\dot{\varphi} \cdot F^* \simeq 0 $ generically requires that the $F$-term associated with the inflaton is small. Barring cancellations, this would typically require that both $W$ and $\frac{\partial W}{\partial \phi}$, where $\phi$ denotes the inflaton field, are small. We note that the potential energy is given by
\begin{equation}
V = F \cdot F^* - 3 \, {\rm e}^K \, \left\vert W \right\vert^2 \;,
\end{equation}
which we can approximate by $V \simeq F \cdot F^*$ if $W$ is small. Then the second condition in (\ref{vs0}) simply demands that the kinetic and the potential energy are equal to each other, which happens twice per period of the inflaton oscillations. In Section \ref{sec:slow}, we discuss several supergravity models of inflation where these conditions are and are not achieved.
\section{Gravitino sound speed in specific models}
\label{sec:slow}
In this section we consider several specific supergravity models where $c_s^2$ is very small, or is of order one, to emphasize what aspects of the model can lead to a slow gravitino.
We start our discussion from a model constructed in \cite{Kolb:2021xfn}, where $\Phi$ is the inflaton superfield while $S$ is a superfield responsible for supersymmetry breaking. Ref. \cite{Kolb:2021xfn} imposes that this field is nilpotent, $S^2=0$. To study the relevance of this assumption we instead use a strong stabilization mechanism for $S$ \cite{ekn3,strongpol,dlmmo,ADinf,ego,eno7,egno4,Ellis:2020xmk} (see also Section \ref{sec:nil} below):
\begin{eqnarray}
K &=& - \frac{\left( \Phi - {\bar \Phi} \right)^2}{2} + \frac{S \, {\bar S}}{ 1 + \frac{m^2}{M^2} \left\vert \Phi \right\vert^2}
- \frac{\left( S \, {\bar S} \right)^2}{\Lambda^2} \;, \nonumber\\
W &=& M \, S + W_0 \;.
\label{kolb-Lambda}
\end{eqnarray}
The resulting potential is extremized along the real directions $\Phi = {\bar \Phi} = \frac{\phi}{\sqrt{2}} \;,\; S = {\bar S} = \frac{s}{\sqrt{2}}$. The minimum of the potential with respect to $s$ is $\phi-$dependent and given by:
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{\Lambda^2}{\sqrt{6} \left( \frac{m^2 \phi^2}{2 M^2}+1\right)^2} + \mathcal{O} \left( \Lambda^4 \right) \;.
\label{smin-Kolb}
\end{equation}
We see that, as is typical for strong stabilization, $\langle s \rangle_\phi$ vanishes in the limit $\Lambda \rightarrow 0$. Inserting this into the potential, leads to %
\begin{equation}
V = \frac{m^2 \phi^2}{2} + \frac{M^2}{3} \left( 1 - \frac{1}{\left(\frac{m^2 \phi^2}{2 M^2}+1\right)^2} \right) \Lambda^2 + \mathcal{O} \left( \Lambda^4 \right) \;.
\label{Vmin-Kolb}
\end{equation}
In both Eqs.~(\ref{smin-Kolb}) and (\ref{Vmin-Kolb}), the parameter $W_0$ has been set to $W_0 = \frac{M}{\sqrt{3}} \left( 1 - \frac{\Lambda^2}{6} + \mathcal{O} \left( \Lambda^4 \right) \right)$, so to have a vanishing potential in the minimum at $\phi =0$. In the minimum, the gravitino mass is given by
\begin{equation}
m_{3/2} \Big\vert_{\phi = 0} = \frac{M}{\sqrt{3}} + \mathcal{O} \left( \Lambda^2 \right) \;.
\label{m32-min-Kolb}
\end{equation}
From Eqs.~(\ref{Vmin-Kolb}) and (\ref{m32-min-Kolb}) we see that $m$ corresponds to the inflaton mass, while (assuming gravity mediation), $M$ to the supersymmetry breaking scale in the model. Therefore, the ratio $(m/M)^2 \sim (m/m_{3/2})^2$
appearing in the K\"ahler potential is typically much greater than 1 in models with weak-scale supersymmetry breaking.
Evaluating the quantities entering in Eq.~(\ref{vs2}) leads to
\begin{eqnarray}
\dot{\varphi}^i &\simeq& \frac{\dot{\phi}}{\sqrt{2}} \left\{ 1 ,\,
- \frac{8 \sqrt{2} \Lambda^2 M^4}{\sqrt{3} m^4 \phi^5} \right\} \;,
\nonumber\\
F^i &\simeq& \left\{ \sqrt{\frac{2}{3}} \, \frac{2 M^5 \Lambda^2}{m^4 \phi^5} ,\, \frac{m^2 \phi^2}{2 M} \right\} \;,
\label{dF-Kolb}
\end{eqnarray}
where $i=1,2$ corresponds to the field $\Phi,\, S$, respectively. These expressions hold for $\Lambda \ll 1$ and $M^2 \ll m^2 \phi^2$, and only the dominant term has been retained in each entry.
From Eq.~(\ref{dF-Kolb}) we see that the model approximately meets the conditions (\ref{vs0}) for a vanishing sound speed. The scale of the superpotential, $M$ is much smaller than the inflationary scale, $m$, and $dW/d\phi = 0$. In particular, the inflaton provides a negligible contribution to supersymmetry breaking.
Applying Eq.~(\ref{vs2}), we find
\begin{equation}
c_s^2 = \left( \frac{\frac{\dot{\phi}^2}{2} -\frac{m^2 \phi^2}{2} - M^2}{\frac{\dot{\phi}^2}{2} +\frac{m^2 \phi^2}{2} + M^2} \right)^2 + \mathcal{O} \left( \Lambda^2 \right) \;,
\end{equation}
which indeed vanishes to $\mathcal{O} \left( \Lambda^0 \right)$ when the inflaton kinetic and potential energy are equal to each other (disregarding the subdominant $M^2$ correction).
Expanding the various terms to higher order in $\Lambda$ we find that, when the $\mathcal{O} \left( \Lambda^0 \right)$ vanishes, the minimum sound speed is
\begin{equation}
c_{s,{\rm min}}^2 = \frac{256 \, \Lambda^4}{3 \phi^2} \, \left( \frac{M}{m \phi} \right)^{10} \left[ 1 + \mathcal{O} \left( \frac{M^2}{m^2 \phi^2} \right) \right] + \mathcal{O} \left( \Lambda^6 \right) \;.
\label{vsmin-Kolb}
\end{equation}
The strong stabilization condition $\Lambda \ll 1$ provided us with an expansion parameter to organize the study of the model and present a simple analytical result. However, we see that it (or the nilpotency condition) is not origin for the near vanishing of $c_s$, which is instead mostly due to the absence of $\phi$ from the superpotential and from the smallness of the mass scale $M$.
To emphasize this, let us consider a different model with strong stabilization, which does not lead to a small gravitino sound speed. The model \cite{Dudas:2016eej} is also characterized characterized by the two superfields $\Phi$ and $S$, and by
\begin{eqnarray}
K &=& - \frac{\left( \Phi - {\bar \Phi} \right)^2}{2} + \left\vert S \right\vert^2
- \frac{\left\vert S \right\vert^4}{\Lambda^2} \;, \nonumber\\
W &=& f \left( \Phi \right) \left( 1 + \delta \, S \right)
\;\;,\;\; f \left( \Phi \right) = f_0 + \frac{m}{2} \, \Phi^2 \;.
\label{mod-Emilian}
\end{eqnarray}
The K\"ahler potential in this model is relatively simple, and the origin of the inflationary potential resides in the superpotential.
Supersymmetry breaking in the vacuum is due to the constant $f_0$, though supersymmetry breaking gets a contribution from the inflaton during inflation and in its subsequent oscillations.
We can anticipate that the condition (\ref{vs0}), and in particular,
$\dot{\varphi}\cdot F^* = 0$ will not be satisfied.
As in the previous example considered, this model has real solutions. The field $s$ is stabilized to a $\phi-$ dependent $\mathcal{O} \left( \Lambda^2 \right)$ value, while the parameter $\delta$ can be set to $\delta = \sqrt{3} + \frac{\Lambda^2}{2 \sqrt{3}} + \mathcal{O} \left( \Lambda^4 \right)$ to ensure that the vacuum energy vanishes in the minimum at $\phi =0$. We then find that the potential and the gravitino mass are given by
\begin{eqnarray}
V \left( \phi \right) &=& V^{(0)} \left( \phi \right) + \mathcal{O} \left( \Lambda^2 \right) \;, \nonumber \\
m_{3/2} \left( \phi \right) & = & m_{3/2}^{(0)} \left( \phi \right) + \mathcal{O} \left( \Lambda^2 \right) \;,
\end{eqnarray}
with
\begin{equation}
V^{(0)} \left( \phi \right) =
\left\vert f' \left( \phi \right) \right\vert^2 = \frac{m^2 \phi^2}{2} \;,
\end{equation}
and
\begin{equation}
m_{3/2}^{(0)} \left( \phi \right) = \left\vert f \left( \phi \right) \right\vert =
f_0 + \frac{m \, \phi^2}{4} \;,
\end{equation}
and we see that $m$ is the inflaton mass, while $f_0$ is the gravitino mass in the vacuum. In terms of the zeroth order potential and gravitino mass, the $\phi-$dependent minimum value of $s$ is given by the compact expression
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{2 m_{3/2}^{(0)2} - V^{(0)}}{m_{3/2}^{(0)2}} \, \frac{\Lambda^2}{2 \, \sqrt{6}} + \mathcal{O} \left( \Lambda^4 \right) \,.
\end{equation}
For this model, we then have
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \frac{\dot{\phi}}{\sqrt{2}} ,\, 0 \right\} + \mathcal{O} \left( \Lambda^2 \right) \;, \nonumber\\
F^i &=& \left\{ \frac{m \phi}{\sqrt{2}} ,\,
\sqrt{3} \left( f_0 + \frac{m \, \phi^2}{4} \right) \right\} + \mathcal{O} \left( \Lambda^2 \right) \;.
\label{dF-Emilian}
\end{eqnarray}
We see that $F^\phi$ is not suppressed, and hence we do not expect a small sound speed. Indeed, we obtain
\begin{eqnarray}
c_s^2 &=& 1 - \frac{6 \, \dot{\phi}^2 \, m_{3/2}^{(0)2}}{\left[
\frac{1}{2} \dot{\phi}^2 + V^{(0)} + 3 \, m_{3/2}^{(0)2} \right]^2} + \mathcal{O} \left( \Lambda^2 \right) \nonumber\\
&=&
\frac{\left[\phi^2 \left( 1 + \frac{3}{8} \, \phi^2 \right) + \frac{\dot{\phi}^2}{m^2} \right]^2 - \frac{3}{2} \frac{\dot{\phi}^2}{m^2} \phi^4}{\left[ \phi^2 \left( 1 + \frac{3}{8} \, \phi^2 \right) + \frac{\dot{\phi}^2}{m^2} \right]^2} \nonumber\\
& + & \mathcal{O} \left( \frac{f_0}{m} ,\, \Lambda^2 \right) \;,
\end{eqnarray}
which is always of $\mathcal{O} \left( 1 \right)$.
The comparison of the two models (\ref{kolb-Lambda}) and (\ref{mod-Emilian}) shows that, as already remarked, the main cause for the suppressed sound speed is the smallness of the inflaton $F-$term. In the model (\ref{kolb-Lambda}) this happens because of the rather peculiar fact that $\Phi$ is absent from the superpotential, particularly when combined with the lack of mobility of $S$ ($S=0$ in the nilpotent limit $\Lambda \rightarrow 0$). That is, $\dot{\varphi}$ and $F^*$ are nearly orthogonal ($\dot \varphi^i\sim\{\dot \phi,0 \}$, $F^i \sim \{0,F_S \}$). In the model of (\ref{mod-Emilian}), that is clearly not the case, as there is a large contribution to supersymmetry breaking from $\Phi$ as long
as $\Phi$ is displaced from its minimum at $\Phi = 0$.
Thus, with this latter simple example, we see that
$c_s \rightarrow 0$ is not a general consequence for supergravity models of inflation, nor for the restricted set of examples with a nilpotent field.
The above two examples, lead to chaotic inflation models of the type $V = m^2 \phi^2$ \cite{Linde:1983gd}. As this model is disfavored due to the upper limit on the tensor-to-scalar ratio (the model predicts $r = 0.13$, whereas the experimental upper limit is $r < 0.06$ \cite{planck18,rlimit}), we next consider an alternative set of models leading to Starobinsky-like inflation \cite{Staro}. These models are based on no-scale supergravity \cite{no-scale,LN} which can be derived from string models as their effective low-energy theories \cite{Witten}.
Further, there is strong relation between no-scale supergravity and higher derivative theories of gravity \cite{Cecotti,DLT,eno9}.
Starobinsky inflation models in no-scale supergravity generally require two chiral superfields, one of which is associated with the volume modulus of string theory. Depending on a choice of basis, this or the second field may serve as the inflaton \cite{eno7,enov1,building}. In the former, it is necessary to include a third chiral multiplet for supersymmetry breaking. In the remainder of this section, we consider an example of each type.
In the first Starobinsky-like model we consider, the volume modulus serves as the inflaton \cite{FeKR,eno7}. As noted above, to include supersymmetry breaking, we need to add a third chiral superfield
\cite{egno4,Dudas:2017kfz,Ellis:2020xmk} which we denote as $Z$ and plays the role of a Polonyi field \cite{pol}.\footnote{The model described here is the untwisted Polonyi field model of ref. \cite{egno4,Dudas:2017kfz,Ellis:2020xmk}. Analogous result are obtained for the twisted Polonyi field model of that work. To keep a notation as close as possible to that of the models (\ref{kolb-Lambda}) and (\ref{mod-Emilian}) discussed above, we have relabeled as $\Phi$ and as $S$ the inflaton and the stabilizer field that were denoted, respectively, as $T$ and $\phi$ in those papers.} The K\"ahler potential and superpotential can be written as
\begin{eqnarray}
K &=& - 3 \, \ln \left( \Phi + {\bar \Phi} - \frac{1}{3} \left\vert S \right\vert^2 + \frac{\left\vert S \right\vert^4}{\Lambda_S^2} - \frac{1}{3} \left\vert Z \right\vert^2 + \frac{\left\vert Z \right\vert^4}{\Lambda^2} \right) \;, \nonumber\\
W &=& m \left[ \sqrt{3} \, S \left( \Phi - \frac{1}{2} \right) + \delta \left( Z + b \right) \right] \; . \label{staro1}
\end{eqnarray}
As we will see, Starobinsky inflation is obtained for $S$ and $Z$ near 0.
As in the previous cases, the model admits real solutions, and we parametrize as $\Phi = {\bar \Phi} = \phi ,\, S = {\bar S} = s ,\, Z = {\bar Z} = z $. The potential is minimized by $\left\langle \phi \right\rangle = \frac{1}{2} + \frac{\delta^2}{3} + \mathcal{O} \left( \delta^4 ,\, \Lambda^2 \right)$, by $\left\langle s \right\rangle = \delta + \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right)$, and by $\left\langle z \right\rangle = \frac{\Lambda^2}{6 \sqrt{3}} + \mathcal{O} \left( \delta^2 \Lambda^2 ,\, \Lambda^4 \right)$, where the coefficient $b$ has been set to $b = \frac{1}{\sqrt{3}} \left( 1 - \frac{\delta^2}{6} \right) + \mathcal{O} \left( \delta^4 ,\, \Lambda^2 \right)$ so to have a vanishing potential in the minimum. In the minimum the gravitino mass is
\begin{equation}
m_{3/2} = \frac{m \, \delta}{\sqrt{3}} + m \times \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right) \;.
\end{equation}
The extremization of the potential with respect to $s$ and $z$ when $\phi$ is away from the minimum gives instead
\begin{eqnarray}
\left\langle s \right\rangle_\phi &=& \frac{4 \Lambda_S^2 \phi}{36 \left( 1 - 2 \phi \right)^2 \phi + \Lambda_S^2 \left( 1 + 4 \phi - 4 \phi^2 \right)} \, \delta + \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right) \;, \nonumber\\
\left\langle z \right\rangle_\phi &=& \frac{2 \Lambda^2 \Lambda_S^2 \phi}{3 \sqrt{3} \left[ 36 \left( 1 - 2 \phi \right)^2 \phi + \Lambda_S^2 \left( 1 + 4 \phi - 4 \phi^2 \right) \right]} \nonumber\\
&&\quad\quad + \mathcal{O} \left( \delta^2 \Lambda^2 ,\, \Lambda^4 \right) \;.
\end{eqnarray}
The potential along this direction is
\begin{equation}
V = \frac{3m^2}{16} \, \frac{\left( 1 - 2 \phi \right)^2}{\phi^2}
+ m^2 \times \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) \;.
\end{equation}
Note that unlike the previous examples discussed, $\phi$
does not have a properly normalized kinetic term.
For small $s$ and $z$, the canonical field is given by
\begin{equation}
\label{canrho}
\rho = \sqrt{\frac32} \ln \left( 2 \phi \right) \, ,
\end{equation}
so that the potential becomes
\begin{equation}
V = \frac{3 m^2}{4} \left(1-e^{-\sqrt{2/3}\rho} \right)^2 \, .
\label{staropot}
\end{equation}
The inflaton mass is $m$ and hence we see that $\delta \ll 1$ is the ratio of the gravitino mass to the inflaton mass and justifies the limit that we have adopted in our analysis.
Along this direction, labeling as $1,2,3$ the fields $\Phi ,\, S ,\, Z$, respectively, we find
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \dot{\phi} ,\, m \times \mathcal{O} \left( \delta ,\, \Lambda^2 \right) ,\, m \times \mathcal{O} \left( \Lambda^2 \right) \right\} \;, \nonumber\\
F^i &=& m \Bigg\{ \mathcal{O} \left( \delta ,\, \Lambda^2 \right) ,\,- \frac{\sqrt{3} \left( 1 - 2 \phi \right)}{\sqrt{2 \phi}} + \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) ,\, \nonumber\\
&& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mathcal{O} \left( \delta ,\, \Lambda^2 \right) \Bigg\} \;.
\end{eqnarray}
We see that $\dot{\varphi} \cdot F$ is suppressed, leading to the possibility of a small gravitino sound speed. We indeed find
\begin{equation}
c_s^2 = \left( \frac{4 \dot{\phi}^2-m^2\left(1-2 \, \phi \right)^2}{4 \dot{\phi}^2+m^2\left(1-2 \, \phi \right)^2} \right)^2 + \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) \;,
\end{equation}
and the leading term can vanish during the oscillations of the inflaton.
We conclude this section with a second no-scale supergravity model, based on a simple Wess-Zumino form for the inflaton superpotential \cite{eno6}. In this case, the volume modulus is strongly stabilized \cite{eno7}. This simple model further emphasizes that the suppression of the sound speed is by no means a generic feature of supergravity models. The model does not require an additional field for supersymmetry breaking and can be written as \cite{egno4,Ellis:2020xmk}
\begin{eqnarray}
K \!\!&=&\!\! - 3 \, \ln \left( S + {\bar S} + \frac{\left( S + {\bar S} - 1 \right)^4}{\Lambda^2} + \frac{d \left( S - {\bar S} \right)^4}{\Lambda^2} - \frac{\left\vert \Phi \right\vert^2}{3} \right) \;, \nonumber\\
W \!\!&=&\!\! m \, \left(\frac{\Phi^2}{2} - \frac{\Phi^3}{3 \sqrt{3}}\right) + \lambda \, m^3 \;,
\end{eqnarray}
where $d$ is a constant of $\mathcal{O}(1)$, and $\lambda$ sets the scale of supersymmetry breaking.
The model admits real solutions that we parametrize as in the previous models, $\Phi = {\bar \Phi} = \phi ,\, S = {\bar S} = s$. In the minimum, $\left\langle \phi \right\rangle = 0$ and $\left\langle s \right\rangle = \frac{1}{2}$. For these values the potential vanishes, while the gravitino mass is $m_{3/2} = \lambda \, m^3$.
Extremizing the potential with respect to $s$, with the inflaton away from the minimum, we find
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{1}{2} +
\frac{\phi^2}{2 \left( 6 \lambda m^2 + \phi^2 \right)^2} \, \frac{\sqrt{3}-\phi}{\sqrt{3}+\phi} \Lambda^2 + \mathcal{O} \left( \Lambda^4 \right) \;,
\end{equation}
leading to the potential
\begin{equation}
V = \frac{3 m^2 \phi^2 }{\left( \sqrt{3}+\phi\right)^2} + \mathcal{O} \left( \Lambda^2 \right) \;.
\end{equation}
As in the previous no-scale example,
$\phi$ is not canonical and writing
\begin{equation}
\phi = \sqrt{3} \tanh \left( \frac{\rho}{\sqrt{6}} \right)
\end{equation}
leads to the same potential given in Eq.~(\ref{staropot}).
We then find, using the same notation as for the previous models,
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \dot{\phi} ,\, m \times \mathcal{O} \left( \Lambda^2 \right) \right\} \;, \nonumber\\
F^i &=& \left\{
m \phi \, \sqrt{\frac{\sqrt{3}-\phi}{\sqrt{3}+\phi}} ,\,
- \frac{m \left( 3 \lambda m^2 + \phi^2 \right)}{2 \sqrt{3} \sqrt{3-\phi^2}} \right\} \nonumber \\
&& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad + m \times \mathcal{O} \left( \Lambda^2 \right) \; .
\end{eqnarray}
We see that $\dot{\varphi} \cdot F$ is not suppressed, and we do not expect a suppression in the gravitino sound speed. A direct inspection of the $\mathcal{O} \left( \Lambda^0 \right)$ term indicates that this is indeed the case, though the full expression is not particularly illuminating. In the $\sqrt{\lambda} m \ll \phi ,\, \frac{\dot{\phi}}{m} \ll 1$ regime we obtain
\begin{equation}
c_s^2 \simeq 1 - \frac{m^2 \phi^4 \dot{\phi}^2}{6 \left( m^2 \phi^2 + \dot{\phi}^2 \right)^2} \;,\;\; \Lambda \ll 1 \;,\; \sqrt{\lambda} m \ll \phi ,\, \frac{\dot{\phi}}{m} \ll 1 \;.
\end{equation}
\section{Complete linearized equation for the longitudinal gravitino}
\label{sec:theta-Y}
So far, we have seen that while it is possible to find models with $c_s \simeq 0$, it is by no means a generic feature of supergravity inflation models. Next, we point out that even
if $c_s = 0$, one can not conclude that there is any enhanced production of gravitinos.
We show that this is the case by considering the equation of motion for the longitudinal gravitino, as obtained in \cite{Kallosh:2000ve} and then studied in \cite{Nilles:2001fg}.
In the unitary gauge, the dynamical variable encoding the longitudinal gravitino is the combination $\theta = \gamma^i \psi_i$, where $\psi$ is the gravitino field. In a nontrivial background, the longitudinal gravitino is coupled to another fermionic combination $\Upsilon$ at the linearized level. In a cosmological background, where the scalar fields depend on time,
\begin{equation}
\Upsilon = K_{ij^*} \left( \chi^i \partial_0 \varphi^{j^*} + \chi^{j^*} \partial_0 \varphi^i \right) \;,
\end{equation}
where the indices run over the number of chiral superfields in the model, $\left( \varphi^i ,\, \chi^i \right)$ are the scalar and fermion components of a chiral complex multiplet (where
$\chi^i$ is a left-handed fermion), while $\left( \varphi^{i^*} ,\, \chi^{i^*} \right)$ are their conjugates.
For simplicity, we consider the case of two chiral superfields, although this can be immediately generalized to an arbitrary number. In this case, the coupled equations for $\theta$ and $\Upsilon$ form a closed system. To write these equations, we introduce the following quantities (that we express through the scalar product defined above)
\begin{eqnarray}
\alpha &\equiv& \rho + 3 m_{3/2}^2 = \dot{\varphi} \cdot \dot{\varphi}^* + F \cdot F^* \;, \nonumber\\
\alpha_1 &\equiv& p - 3 m_{3/2}^2 = \dot{\varphi} \cdot \dot{\varphi}^* - F \cdot F^* \;, \nonumber\\
\alpha_2 &\equiv& 2 \frac{\partial}{\partial t}
\left[ {\rm e}^{K/2} \, W \, P_L + {\rm e}^{K/2} \, W^* \, P_R \right] \nonumber\\
&=& 2 \left( \dot{\varphi} \cdot F^* \, P_L +
\dot{\varphi}^* \cdot F \, P_R \right) \nonumber\\
&& \quad\quad + {\rm e}^{K/2} \left( \dot{\varphi}^i \frac{\partial K}{\partial \varphi^i} - \dot{\varphi}^{*i} \frac{\partial K}{\partial \varphi^{*i}} \right) \left( W^* \, P_R - W \, P_L \right) \;. \nonumber\\
\end{eqnarray}
In these expressions $\rho$ and $p$ denote, respectively, the energy density and pressure of the background scalars, while $P_{L/R}$ are the left- and right-handed chiral fermion projection operators. These expressions considerably simplify along real solutions for the scalar fields
\begin{eqnarray}
{\rm real \; scalars}: \;&& \alpha =
\dot{\varphi} \cdot \dot{\varphi} + F \cdot F \;, \nonumber\\
&& \alpha_1 = \dot{\varphi} \cdot \dot{\varphi} - F \cdot F \;, \nonumber\\
&& \alpha_2 = 2 \, \dot{\varphi} \cdot F \;.
\label{alphas-real}
\end{eqnarray}
These expressions apply to the models considered in the previous sections, that have real background solutions. In the following we employ the simplified expressions (\ref{alphas-real}).
The closed system of equations for $\theta$ and $\Upsilon$ is
\begin{equation}
\left( \gamma^0 \, \partial_0 + i \gamma^i \, k_i \, N + M \right) X = 0 \;\;\;\;\;,\;\;\;\;\; X = \left( \begin{array}{c}
{\tilde \theta} \\ {\tilde \Upsilon}
\end{array} \right) \;,
\label{eqs-theta-Y}
\end{equation}
where ${\tilde \theta}$ and ${\tilde \Upsilon}$, are canonically-normalized fields, related to the original fields by
\begin{equation}
\theta \equiv \frac{2 i \gamma^i \, k_i}{\left( \alpha \, a^3 \right)^{1/2}} \, {\tilde \theta} \;\;\;,\;\;\;
\Upsilon \equiv \frac{\Delta}{2} \left( \frac{\alpha}{a} \right)^{1/2} \, {\tilde \Upsilon } \;,
\end{equation}
(with $\Delta$ to be defined shortly), and where
\begin{equation}
N = \left( \begin{array}{cc}
- \frac{\alpha_1}{\alpha} - \gamma^0 \, \frac{\alpha_2}{\alpha} & - \gamma^0 \, \Delta \\
- \gamma^0 \, \Delta & - \frac{\alpha_1}{\alpha} + \gamma^0 \, \frac{\alpha_2}{\alpha}
\end{array} \right) \;,
\label{N}
\end{equation}
while the expression for $M$ is not important for the present discussion, and can be found in \cite{Nilles:2001fg}.
Disregarding the presence of the field $\Upsilon$ amounts to the system studied in
\cite{Kolb:2021xfn,Kolb:2021nob}. The square of the gravitino sound speed would then be given by the square of the $N_{11}$ element (we note that, due to the signature we have chosen, $\gamma^0$ is anti-hermitian),
\begin{equation}
N_{11} \, N_{11}^\dagger = \frac{\left\vert \alpha_1 \right\vert^2 + \left\vert \alpha_2 \right\vert^2}{\alpha^2} = c_s^2 \;.
\end{equation}
Namely, using the expressions (\ref{alphas-real}) leads precisely to the sound speed (\ref{vs2}). The complete system however has also off-diagonal elements, with %
\begin{equation}
\Delta = \sqrt{1-c_s^2} \;.
\end{equation}
Therefore, when the coefficient $c_s^2$ vanishes,
\begin{equation}
c_s^2 = 0 \;\; \rightarrow \;\; \Delta = 1 \;\;,\;\; N = \left( \begin{array}{cc}
0 & - \gamma^0 \\
- \gamma^0 & 0
\end{array} \right) \;,
\end{equation}
leading to a non-singular matrix $N$, and therefore to a nonvanishing sound speed for the physical eigenstates of the system.
Consequently {\em none} of the models discussed in Section 3 have catastrophic production of gravitinos. Problems can only arise if $\Upsilon$ can be ignored as is the case when a second, orthogonal nilpotency condition, is applied as discussed in the next section. Then, the only problematic models would be those defined in Eqs.~(\ref{kolb-Lambda}) and (\ref{staro1}), in the case where the inflaton multiplet $\Phi$ is subject to the additional orthogonal constraint $S ({\Phi} - \overline{\Phi}) = 0$. Indeed, this additional constraint removes the inflatino from the spectrum (hence $\Upsilon = 0$) and the speed of sound in these models hits zero at some point during the inflationary evolution. As we will see, such models seem suspicious from the viewpoint of a fundamental theory of gravity.
\section{Models with Orthogonal Nilpotent Superfields}
\label{sec:nil}
In the cases considered in this paper, the goldstino $G$ belongs to a chiral multiplet and has a scalar superpartner (the sgoldstino), which, once supersymmetry is broken, acquires a non-supersymmetric mass.
Decoupling the sgoldstino by giving it an {\it infinite} mass leads to a non-linear realization of supersymmetry.
A particularly simple way non-linear realization can be obtained is by imposing the nilpotent constraint
\cite{Rocek:1978nb,Komargodski:2009rz,Antoniadis:2014oya,Ferrara:2014kva,Kallosh:2014via,DallAgata:2014qsj,Dudas:2015eha,Ferrara:2015gta,Ferrara:2015cwa}
\begin{equation}
\label{X2}
S^2 = 0 \, .
\end{equation}
When supersymmetry is broken by means of a non-trivial $F^S \neq 0$, the constraint is solved by
\begin{eqnarray}
S = \frac{G^2}{2F^S} + \sqrt 2 \theta G + \theta^2 F^S \, .
\label{nil1}
\end{eqnarray}
Here and in what follows, we discuss the constraints at the level of global supersymmetry for simplicity. The generalization to supergravity can be found in the literature \cite{Antoniadis:2014oya}-\cite{DallAgata:2016syy}.
The constraint (\ref{X2}) can be interpreted as the infrared limit of a very heavy sgoldstino. This can be obtained starting from a microscopic Lagrangian of the type \cite{Komargodski:2009rz}:
\begin{equation}
K = |S|^2 - \frac{1}{\Lambda^2} |S|^4 \, , \qquad \ W = W_0 + W_1 S \, , \label{nil2}
\end{equation}
in the limit $\Lambda \rightarrow 0$. Indeed, the sgoldstino mass
$m_S^2 = 4 \frac{{F^S}^2}{\Lambda^2}$ is sent to infinity in this limit, leading to a nonlinear realization of supersymmetry in the IR. The limit $\Lambda \rightarrow 0$ has its limitations
\cite{Dudas:2016eej}, since it implies field-theory dynamics in some heavy sector, which after decoupling, leaves behind the ``strong stabilization" term $\frac{1}{\Lambda^2} |S|^4$. Modulo these subtleties, the UV Lagrangian (\ref{nil2}) contains only two derivatives and is pretty standard.
The situation is different for the orthogonal constraint on the chiral superfield ${\Phi}$ that removes the imaginary part of the scalar, the fermion, and the auxiliary field \cite{Komargodski:2009rz,Ferrara:2015tyn,Carrasco:2015iij,DallAgata:2015zxp,Kallosh:2016hcm,DallAgata:2016syy}:
\begin{equation}
S ({\Phi} - \overline{\Phi}) = 0 \ . \label{nil3}
\end{equation}
It was shown in \cite{DallAgata:2016syy} that (\ref{nil3}) is equivalent to the following set of constraints
\begin{eqnarray}\label{nil4}
|S|^2 ({\Phi} - \overline{\Phi} )&=& 0 \, , \\[2mm]
\label{nil5}
|S|^2 \overline D_{\dot \alpha} \overline{\Phi} &=& 0\, , \\[2mm]
\label{nil6}
|S|^2 \overline D^2 \overline{\Phi} &=& 0 \, .
\end{eqnarray}
Each constraint above eliminates one component field:
Eq.~(\ref{nil4}) eliminates the scalar, Eq.~(\ref{nil5}) eliminates the fermion, whereas Eq.~(\ref{nil6}) eliminates the auxiliary field in the $\Phi$ multiplet.
The constraint \eqref{nil3} can be obtained starting from a microscopic Lagrangian containing three additional terms \cite{DallAgata:2016syy}, which generate non-supersymmetric masses for the component fields that we remove:
\begin{eqnarray}
&& \int \! d^4 \theta \Big{[} \frac{m^2_b}{2f^2} |S|^2 ({\Phi} - \overline{\Phi} )^2
- \frac{g_{F^{\Phi}}}{f^2} |S|^2 D^2 {\Phi} \overline D^2 \overline {\Phi } \Big{]} \nonumber \\
&& - \frac{m_\zeta}{2 f^2} \int \! d^4 \theta \Big{[} |S|^2 D^{\alpha} {\Phi} D_\alpha {\Phi} + c.c. \Big{]} \ . \label{nil7}
\end{eqnarray}
In (\ref{nil7}), $f$ can be taken to be (by convention) the supersymmetry breaking scale in the vacuum $f= \langle F^S \rangle$, $m_b$ and $m_{\zeta}$ are mass parameters for the decoupling scalar and fermion, respectively, $g_{F^{\Phi}}$
is a dimensionless coupling, $D_\alpha$ denotes a covariant derivative in superspace and $D^2 =D^\alpha D_\alpha$, see e.g. \cite{Wess:1992cp}.
Notice that only the first term in (\ref{nil7}) is a standard correction to the K\"ahler potential, whereas the second and the third terms contain higher-derivatives.
The limit $m_b \rightarrow \infty$, $g_{F^{\Phi}} \rightarrow \infty$ and $m_\zeta \rightarrow \infty$ generate infinitely large masses to the scalar and fermion and an infinite coefficient to the auxiliary field $F^{\Phi}$. In this limit, the superspace equations of motion for the chiral superfield ${\Phi}$ are dominated by
\begin{eqnarray}
&& \overline D^2 \left\{ \frac{m^2_b}{f^2} |S|^2 ({\Phi} - \overline{\Phi} )
+\frac{m_\zeta}{f^2} D^\alpha ( |S|^2 D_\alpha {\Phi} ) \right.
\nonumber \\
&& \quad\quad \left. - \frac{g_{F^{\Phi}}}{f^2} D^2 (|S|^2 \overline D^2 \overline {\Phi}) \right\} = 0 \ .\label{nil8}
\end{eqnarray}
This indeed reproduces the constraints (\ref{nil4})-(\ref{nil6}) and therefore (\ref{nil3}): multiplication with $S \overline S$ leads to (\ref{nil6}); the multiplication with $D_\beta S \overline S$ gives (\ref{nil5}); and finally, using (\ref{nil6}) and (\ref{nil5}) in (\ref{nil8}) then multiplying with $\overline S$ leads to (\ref{nil4}).
As emphasized already, in contrast to the nilpotent UV Lagrangian
(\ref{nil2}), the action (\ref{nil7}) contains higher derivatives. As such, it is qualitatively different and the orthogonal constraint cannot therefore be obtained, to our knowledge, starting from a standard two-derivative action in the UV. In particular, if the chiral superfield $S$ was not nilpotent, the second term in Eq. (\ref{nil7}) proportional to $g_{F^\Phi}$ could introduce ghosts into the theory. This is a special property of the decoupling procedure, and could signal that such constraints could (but not necessarily) come from a sick UV theory.
This could explain the problems with the slow gravitinos in some inflationary models using the orthogonal constraint.
We conclude this section with some general properties and implications of supersymmetric models with a second degree nilpotency condition, $S^2 = 0$, and the orthogonality constraint, $S(\Phi - \bar{\Phi}) = 0$. It follows from these constraints that if we choose to work in the unitary gauge the goldstino and inflatino fields are absent in these models. If we consider a supergravity model with a K\"ahler potential and superpotential of the form
\begin{equation}
K = -\frac{\left(\Phi - \bar{\Phi} \right)^2}{2} + S \bar{S} \, ,
\end{equation}
and
\begin{equation}
\label{sup1}
W (\Phi, S) \; = \; f(\Phi) S + g(\Phi) \, ,
\end{equation}
so that the K\"ahler potential possesses a shift symmetry, and vanishes when the field constraints are imposed, $K(S, \bar{S}; \Phi, \bar{\Phi})|_{S = \bar{S} = \Phi - \bar{\Phi} = 0} = 0$. The orthogonality constraint ensures that the covariant derivative $D_{\Phi}W$ is absent in the effective scalar potential, which is then given by \cite{Ferrara:2015tyn,Carrasco:2015iij}
\begin{equation}
\label{pot1}
V \; = \; e^{K} \left( K^{S S^{*}} |D_S W|^2 - 3|W|^2 \right) = f^2 - 3 m_{3/2}^2 \, ,
\end{equation}
where the gravitino mass is
\begin{equation}
\label{gravmass}
m_{3/2} \; = \; |g(\Phi)| \, .
\end{equation}
Note that the sound speed given in Eq.~(\ref{vs2}) should not be used in this case, since $F^\phi = 0$ for the bosonic part, while $D_\phi W \ne 0$. Rather, one can use
Eq.~(\ref{Eq:generic32}) and the potential in Eq.~(\ref{pot1}) with Eq.~(\ref{gravmass}) to obtain
\begin{eqnarray}
c_s^2 & = & \frac{\left(f(\Phi)^2 -\dot{\Phi}^2 \right)^2}{\left(f(\Phi)^2 + \dot{\Phi}^2 \right)^2} + \frac{(2 g'(\Phi) \dot{\Phi})^2}{\left(f(\Phi)^2 +\dot{\Phi}^2 \right)^2} \nonumber \\
& = & 1- 4 \frac{\dot{\Phi}^2}{(f^2 + \dot{\Phi}^2)^2} \left( f^2 - g'^2 \right)
\, ,
\label{soundspeed}
\end{eqnarray}
where $g'(\Phi) = \partial g/ \partial \Phi$. If we want to ensure that the sound speed $c_s > 0$ does not vanish, we must impose the constraint
\begin{equation}
\label{conspos}
\left( f(\Phi)^2 - \Dot{\Phi}^2 \right)^2 + \left(2 g'(\Phi) \dot{\Phi} \right)^2 > 0 \, .
\end{equation}
The above quantity is non-negative at all times in moduli space and this condition is only violated when $f(\Phi) = \pm \dot{\Phi}$ and $g'(\Phi) = 0$. If the gravitino mass is constant at all times, this implies that $g'(\Phi) = 0$ and the constraint~(\ref{conspos}) will be violated when the first term vanishes.
While it is straightforward to construct models for which the gravitino mass is not constant,
additional problems quickly ensue. Consider, for example, a superpotential defined by
\begin{equation}
\label{funcs1}
f = f_0 + \sqrt{3}g, \qquad g = \frac{1}{2\sqrt{3}} \left(a f_0 + \tilde{m} \right) \, ,
\end{equation}
where $\tilde{m}$ is associated with the weak scale, and the parameter $a$ is an arbitrary positive number. At the minimum of the potential, the gravitino mass is of order ${\tilde m}$. Any variation in a quantity this small will still lead to a sound speed which is also very small. For $a \ne 0$, the variation of the gravitino mass may be comparable to the potential. The effective scalar potential~(\ref{pot1}) becomes
\begin{equation}
\label{pot2}
V = (a + 1) f_0^2 + \tilde{m} f_0 \, .
\end{equation}
If we choose
\begin{equation}
f_0 = \frac{\sqrt{3}}{2 \sqrt{a + 1}} m \left(1 - e^{-\frac{2}{\sqrt{3}}\Phi} \right) \, ,
\end{equation}
and use the canonically-normalized field $\Phi = \phi/\sqrt{2}$, we recover approximately, for $\tilde m \ll m$, the Starobinsky inflationary potential~(\ref{staropot}).
However, during each oscillation, when $\Phi = 0$,
$f^2 - g'^2 = -a^2/12(1+a)$.
As a result the sound speed exceeds 1, signalling a sickness in the theory.
The Cauchy-Schwarz inequality noted above is no longer
guaranteed to hold as the sound speed is now
\begin{equation}
c_s^2 = 1 - \frac{4}{
\left(\vert\dot{\Phi}\vert^2 + \vert F^S \vert^2 \right)^2} \,
\left\{
|\dot{\Phi}|^2 |F^S|^2 -
\left\vert \dot{\Phi} e^{K/2} D_{\Phi} W \right\vert^2 \right\} \;,
\label{vs3}
\end{equation}
and there is no such inequality between $|\dot{\varphi}|^2 |F|^2 = |\dot{\Phi}|^2 |F^S|^2$ and $\left\vert \dot{\varphi} \cdot F^* \right\vert^2 = |e^{K/2} \dot{\Phi} D_\Phi W|^2 $,
since we can no longer use Eq.~(\ref{DWF}) to relate $F^\Phi$ and $D_\Phi W$.
Thus pathologies might arise whenever $|e^{K/2} D_\Phi W| > |F^S|$.
Such behavior was also seen in \cite{Kolb:2021xfn}. We are not sure if this behavior is a remnant of the lack of two derivative UV origin stemming from the orthogonal constraint.
\section{Conclusions}
\label{sec:conclusions}
If supersymmetry is realized below the Planck scale, it is natural to construct models of inflation in the context of supersymmetry/supergravity. Particle production and reheating after the period of exponential expansion is an essential component of any model of inflation, and in the context of
supersymmetry, the production of gravitinos must be considered.
If the gravitino is not the lightest supersymmetric particle, it is expected to be unstable. Because of its gravitational coupling, its lifetime can be quite long (for weak scale gravitino masses), causing a host of potential problems \cite{sw}.
In general, the production of gravitinos following inflation is complicated by the possibility that the field content of the longitudinal mode comprised of the goldstino may be different during inflation from that in the vacuum. Indeed,
supersymmetry might be broken during inflation,
and the longitudinal component of the gravitino may be identified with the inflatino. Thus, models
with substantial non-perturbative particle production \cite{Kallosh:1999jj,Giudice:1999yt,Giudice:1999am,Kallosh:2000ve}, may be producing an inflatino rather than a gravitino \cite{Nilles:2001ry,Nilles:2001fg}.
The inflatino problem is generally model dependent
and is non-existent in certain constructions \cite{Nilles:2001my}.
An additional problem associated with gravitinos has been discussed recently \cite{hase,Kolb:2021xfn,Kolb:2021nob} in connection with a vanishing sound speed for gravitinos.
It was argued that when $c_s \rightarrow 0$, the non-perturbative production
of high-momentum modes in the helicity 1/2 mode is unsuppressed. Indeed, there are a variety of well-studied models where the gravitino sound speed does vanish. We have discussed several examples for which $c_s \rightarrow 0$ and several for which it does not. That is, while a vanishing sound speed may occur in supergravity models of inflation, it is by no means a necessary consequence. Furthermore, we have shown that even in the models where $c_s \rightarrow 0$, runaway gravitino production is blocked by mixing with the inflatino in a time-dependent background. This is the case in models with and without a nilpotent field associated with supersymmetry breaking.
However, the problem of catastrophic gravitino production may occur in models with a second orthogonal constraint for which the inflatino is removed from the particle spectrum. It is not clear how these models can be extracted from a UV theory involving no more than two derivatives. This argument along with the problem of a vanishing sound speed (or superluminal sound speed when $c_s \ne 0$) may be a constraint on models with constrained fields rather than supergravity models of inflation in general.
\vskip.1in
{\bf Acknowledgments:}
\noindent
The authors want to thank T. Gherghetta, H.P. Nilles, and L. Sorbo for helpful discussions. This project has received support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk$\lslash$odowska-Curie grant agreement No 860881-HIDDeN, by the ANR grant Black-dS-String ANR-16-CE31-0004-01, the CNRS PICS MicroDark, and the IN2P3 Master Project UCMN.
The work of K.A.O.~was supported in part by DOE grant DE-SC0011842 at the University of
Minnesota. The work of MG was supported by the Spanish Agencia Estatal de Investigaci\'on through Grants No.~FPA2015-65929-P (MINECO/FEDER, UE) and No.~PGC2018095161-B-I00, IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and Red Consolider MultiDark FPA2017-90566-REDC.
|
1,108,101,565,476 | arxiv | \section{Introduction}\label{sec:introduction}}
\IEEEPARstart{S}{tate} machine replication~\cite{DBLP:journals/csur/Schneider90}
is a common technique for
implementing distributed, fault-tolerant services. Commonly,
replicated state machine (RSM) implementations are centred
around the use of a consensus protocol, as replicas
must sequentially apply the same commands in the same order to prevent divergence.
Existing consensus protocols such as Paxos~\cite{lamport2001paxos,DBLP:journals/tocs/Lamport98},
Raft~\cite{DBLP:conf/usenix/OngaroO14}, or variations
thereof~\cite{DBLP:conf/osdi/MaoJM08,DBLP:conf/dsn/JunqueiraRS11,DBLP:conf/sosp/MoraruAK13} that can
be used to build an RSM are based on the idea of a command log.
Once a replica learns one or multiple commands by consensus, it appends them
to its persistent local command log. Several practical systems~\cite{DBLP:conf/osdi/CorbettDEFFFGGHHHKKLLMMNQRRSSTWW12,
DBLP:conf/osdi/Burrows06,DBLP:conf/cidr/BakerBCFKLLLLY11} follow this general approach.
However, the implementation of such a command log incurs additional challenges
such as log truncation, snapshotting, and log recovery. In case of Paxos, these
problems have to be addressed separately on top of the consensus algorithm. This
is a challenging and error-prone task, as noted by Chandra et al.~\cite{DBLP:conf/podc/ChandraGR07}. Other
consensus solutions, e.g. Raft, consider some of these issues as part of the
core protocol while sacrificing the ability to make consensus decisions without an
elected leader. In either case, implementing consensus sequences requires extensive state
management.
A command log is worth its overhead when the commands are small
compared to the managed state. However, aggregating largely independent data
into a bigger managed state, such as multiple key-value pairs in a key-value
store, to compensate for the log overhead is counterproductive because the log
would then unnecessarily order commands targeting different keys.
Managing each key-value pair separately would be ideal, but this is
unpractical when using a log due to the implied overhead and challenges.
In this paper, we present a novel approach called \emph{Read-Modify-Write
Paxos} (/) where the state
of an RSM is managed `in-place'. Instead of replicating a command log as an
intermediate step, / replicates the latest state directly. A new command is processed
by applying it to the current state and proposing the result as the next value
in a sequence of consensus decisions. Thereby, it is possible to
use a fixed set of state variables for all decisions, which avoids the
state management issues. At the same time, distributed consensus
can be used on a finer granularity than before and it becomes trivial to use an arbitrary number
of parallel consensus instances. This allows the fault-tolerant implementation of
ubiquitous primitives like counter, locks, or sets. In addition to existing
use cases like key-value stores, we believe that such fault-tolerant, fine-granular
RSM usage might become more and more relevant with the rise of byte-addressable
non-volatile memory and RDMA-capable low latency interconnects.
Before presenting /, we introduce the notion of a \emph{consensus sequence register}, an
obstruction-free multi-writer, multi-reader register that performs any submitted write operation at-least-once.
Writes are expressed in the
form of update commands applied on an opaque object. Instead of explicitly agreeing
on a sequence of commands, such register agrees on the sequence of object states that
result from the submitted update commands. By adhering to the safety properties
of consensus, reads are guaranteed to observe the latest consistent state.
Strengthening the register to apply writes exactly-once results in /---a fault-tolerant general atomic read-modify-write (RMW) register.
The main contributions of this paper are:
\begin{itemize}[topsep=0pt]
\item We introduce the abstractions of a \emph{consensus sequence
register} and strengthen it to provide an \emph{atomic RMW register}
(\secref{sec:problem_statement}). These abstractions can be used to
implement RSMs. If updates are idempotent the \emph{consensus
sequence register} suffices to build an RSM.
Otherwise, the atomic RMW register is required (\secref{sec:RSM}).
\item We provide a new implementation of a fault-tolerant atomic
\emph{write-once} register by modifying the Paxos algorithm. In particular,
we enhance Paxos by using the concept of \emph{consistent quorums}---a set of replies containing identical answers---to reduce contention
in read-heavy workloads. Once a consistent quorum is detected, the
consensus decision is known. This allows learning the register's value
in two message delays and prevents concurrent reads from blocking each other (\secref{sec:cmd_sequence}).
\item By further exploiting consistent quorums, we extend the atomic
write-once register to a multi-write register that is a \emph{consensus sequence register}. Here, a
consistent quorum indicates the most recent consensus decision. This
makes it possible to propose a follow-up value in-place, i.e. without
a command log or multiple independent consensus instances (\secref{sec:cmd_sequence}).
If there is
only a single writer, follow-up decisions can be made in two
message delays (\secref{sec:optiseqwrite}) without electing a leader.
\item The \emph{consensus sequence register} applies submitted updates
from multiple writers \emph{at-least} once, which is sufficient when
updates manipulate the opaque object (or parts of it) in an
idempotent way (like adding a member to a set). We show that by
using ordered links, \emph{exactly-once} semantics can be achieved to
build an \emph{atomic RMW register}, called /
(\secref{sec:atomic_cmd_sequence}).
\end{itemize}
\section{System Model} \label{sec:system_model}
We consider an asynchronous distributed system with processes
that communicate by message passing. Processes work at arbitrary speed,
may crash, omit messages and may recover with their internal state intact (a recovering process
is indistinguishable from one experiencing omission failures). We do
not consider Byzantine failures. A process is \emph{correct} if
it does not crash or recovers from crashes in finite time with its
(possibly outdated) state intact. We assume that every process can be identified
by its process ID (PID).
In the first part of this paper, processes send messages to each other via direct unreliable
communication links. Links may lose or delay messages indefinitely or
deliver them out-of-order. While a fair-loss property~\cite{DBLP:books/daglib/0025983}
is desirable to support progress, it is not formally necessary. In \secref{sec:atomic_cmd_sequence}, we strengthen this and require reliable
in-order message delivery. Such reliable links can easily be constructed on
top of unreliable fair-loss links~\cite{DBLP:books/daglib/0025983}. In practice,
TCP is often used as reliable communication protocol~\cite{DBLP:conf/dsn/JunqueiraRS11}.
\section{The Consensus Problem} \label{sec:paxos_background}
The consensus problem describes the agreement of several processes
on a common value in a distributed system. We differentiate between
\newterm{proposer} processes that propose values and \newterm{learner}
processes that must agree on a single proposed value. In practice, a process can also
implement both roles. A correct solution to the consensus
problem must satisfy the following \newterm{safety}
properties~\cite{lamport2005generalized}:
\begin{description}
\item[C-Nontriviality.] Any learned value must have been proposed.
\item[C-Stability.] A learner can learn at most one value.
\item[C-Consistency.] Two different learners cannot learn different values.
\end{description}
In addition to safety, the \newterm{liveness} property requires that some value is eventually
learned if a sufficient number of processes are correct. However, guaranteeing
liveness while satisfying the safety properties of consensus is impossible in
an asynchronous system with one faulty process~\cite{DBLP:journals/jacm/FischerLP85}.
\section{Problem Statement} \label{sec:problem_statement}
We define a fault-tolerant register that is replicated on $N$
processes and tolerates the crashes of a minority of replicas. The
register holds a value $v$. Its initial value is $v=\bot$.
Any number of clients can read or modify $v$ by submitting
\emph{commands} to any replica. The primary motivation of our work is to provide
a register abstraction that allows the implementation of a replicated
state machine. For that, we start with a simpler abstraction, which we then extend.
\begin{description}[listparindent=\parindent, itemindent=\parindent, leftmargin=0cm]
\item[Write-Once Atomic Register.]
Commands submitted to the register either write a value or read its
current value. Read commands return either $\bot$ or a value $v_w$ that
has been submitted by some write.
The register is linearisable~\cite{DBLP:journals/toplas/HerlihyW90}, i.e.
all commands appear to take effect instantaneously at some time between
their submission and the corresponding completion response from the register. Thus,
once a read returns $v_w$, then all subsequent reads must return $v_w$ as well. However,
an arbitrary number of reads is allowed to return $\bot$ beforehand if no value was written yet. This is achieved by satisfying the
safety properties stated in \secref{sec:paxos_background}.
\item[Consensus Sequence Register.]
We extend the write-once atomic register by allowing multiple clients to submit
\emph{update} commands that change the register's value.
We say that a value $v$ is the result of
\emph{update sequence} $s(v) = u_1, \ldots, u_n$, iff $v$ equals $u_n \circ \dots \circ u_1$
applied on $\bot$ ($\circ$~being function composition).
The register ensures that reads return values with growing update sequences.
For that, we extend the safety properties of consensus for
consensus sequences.
\begin{description}[leftmargin=\leftmargini]
\item[CS-Nontriviality.] Any read value is the result of applying a sequence of
submitted updates.
\item[CS-Stability.] For any two subsequent reads returning values
$v_1$ and $v_2$, $s(v_1)$ is a prefix of $s(v_2)$.
\item[CS-Consistency.] For any two reads
(including concurrent ones) returning values $v_1$ and $v_2$,
$s(v_1)$ is a prefix of $s(v_2)$ or vice versa.
\end{description}
The prefix relation on update sequences is reflexive. Every
update sequence is its own prefix. Update sequences are merely
a tool to argue about the register's properties. The actual register implementation
does not explicitly store them. It simply keeps the value resulting
from the latest update.
For updates, we also require the following properties:
\begin{description}[leftmargin=\leftmargini]
\item[CS-Update-Visibility.] Any completed update is
included at least once in the update sequence of all
values returned by subsequent reads.
\item[CS-Update-Stability.] For any two subsequent updates
$u_1$ and $u_2$, $u_1$ appears before $u_2$ in the update sequence of
any returned value that includes both $u_1$ and $u_2$.
\end{description}
\item[Atomic Read-Modify-Write Register.]
To satisfy linearisability, we strengthen CS-Update-Visibility by requiring
that every completed update is included \emph{exactly-once} in the update sequence
of all values returned by subsequent reads. This results in a general atomic
read-modify-write (RMW) register~\cite{DBLP:books/mk/Lynch96}.
Unlike specialised RMW registers that
can perform a single type of RMW operation like test-and-set or fetch-and-add,
this register can atomically execute arbitrary computations on its previous value.
\end{description}
As liveness is impossible in our system model, wait-freedom~\cite{DBLP:journals/toplas/Herlihy91}
cannot be provided. However, we require obstruction-freedomness~\cite{DBLP:conf/icdcs/HerlihyLM03}
for a valid implementation of the registers. If wait-freedom is still required,
an obstruction-free implementation can be extended by a leader oracle
assuming a $\Diamond\mathcal{W}$ failure detector~\cite{DBLP:journals/jacm/ChandraHT96}.
\section{In-Place Consensus Sequence} \label{sec:implementation}
In this section, we present our protocols that
satisfy the properties of the register abstractions introduced
in \secref{sec:problem_statement}. The write-once atomic register makes use of the
principles of Paxos consensus~\cite{DBLP:journals/tocs/Lamport98,lamport2001paxos}
and adopts the concept of \emph{consistent
quorums}~\cite{consistent-quorum}.
These concepts are then extended for the more powerful abstractions
to allow a sequence of multiple consensus decisions `in-place', i.e.
on the same set of state variables by overwriting the previous consensus.
A more detailed, albeit more informal description of a previous version
is given by Skrzypczak~\cite{Skrzypczak2017}. We discuss how to build an
RSM with our register in \secref{sec:RSM}.
\subsection{Paxos Overview}
Our approach is derived from the Paxos
protocol. In addition to proposers and
learners, Paxos introduces the role of \newterm{acceptor} processes that coordinate
concurrent proposals by \newterm{voting} on them. If a sufficient number of
acceptors have voted for the same proposal, the proposal's value can be learned
by a learner. Such a set of acceptors is called a \newterm{quorum}. A
proposal is \newterm{chosen} if it has acquired a quorum of votes. The value of
a chosen proposal is a chosen value. The
size of quorums depends on the application and Paxos variant in
use~\cite{DBLP:journals/dc/Lamport06,DBLP:conf/opodis/HowardMS16,DBLP:conf/sosp/MoraruAK13}.
However, it is generally required that any two quorums have a
non-empty intersection to prevent two disjoint quorums that voted for different
values (as this would allow two learners to learn different values).
For Paxos to learn a value, a quorum of acceptors, a learner, and
the proposer that has proposed the value, must be correct during the execution
of the protocol. For simplicity, we consider any majority of acceptors to be
a quorum. Thus, a system with $2F + 1$ acceptors can tolerate at
most $F$ acceptor failures.
If enough processes are correct, then Paxos is obstruction-free~\cite{DBLP:conf/icdcs/HerlihyLM03},
i.e. an isolated proposer without concurrent access succeeds in a finite number
of steps. However, concurrent proposals can invalidate each other repeatedly,
thereby preventing learners from learning any value. This scenario is known
as \newterm{duelling proposers}.
\subsection{Consistent Quorums}\label{sec:consistent-quorum}
Similar to Paxos, our approach structures the communication between
proposers and acceptors into phases. In each phase, a proposer sends a
message to all acceptors and waits for a minimal quorum of replies.
The seen quorum is
\newterm{consistent} if the indicated state by the acceptors in the
quorum is identical, otherwise, it is \newterm{inconsistent}
(see \figref{fig:cons_quorum}). Not waiting for more replies than
necessary ensures tolerating a minority of failed acceptors without
delaying progress.
\begin{figure}[!t]
\centering
\scalebox{0.9}{\includegraphics{fig_cons_quorum}}
\vspace*{2mm}
\caption{Consistent/inconsistent quorum with 7 acceptors. A quorum
view $Q$ for a system using $n$ acceptors consists of
$|Q| = \lfloor\frac{n}{2}\rfloor+1$ elements (here 4).}
\label{fig:cons_quorum}
\end{figure}
If a proposer $p$ observes an inconsistent quorum, it cannot infer
which of the seen values is or will be chosen and learned. For example, if $p$
receives the quorum depicted in the right part of \figref{fig:cons_quorum}, it cannot
decide if \oldstate\ or \newstate\ exists in a majority since it has no
information about the state of acceptors $1$--$3$.
In contrast, it is trivial for $p$ to deduce the chosen
value with a consistent quorum (\figref{fig:cons_quorum} left).
Existing Paxos variants do not distinguish consistent or inconsistent
quorums. As we will see, detecting a consistent quorum allows the
proposer to terminate the protocol early in the single-decree case.
Furthermore, the consistent state can be used as the basis for follow-up proposals
if multiple consensus decisions are needed in sequence.
\subsection{Paxos Write-Once Atomic Register} \label{sec:write-once}
In the following, we present our modifications to the original
single-decree Paxos protocol for implementing a write-once atomic
register. Its pseudocode is depicted in Algorithm~1.
We note that no separate learner role exists, as each proposer also
implements the functionality of a learner in our implementation.
To make the algorithm easier to understand,
we provide an execution example in Figure~\ref{fig:flow}.
We discuss differences to Paxos in \secref{sec:comparison-paxos}. Before
proceeding to the algorithm description, we first cover some general concepts and conventions.
\begin{description}[listparindent=\parindent, itemindent=\parindent, leftmargin=0cm]
\item[Rounds.]
Concurrent proposals are ordered by so-called \newterm{rounds}
(analogue to `proposals numbered $n$' in~\cite{lamport2001paxos} and
`ballot numbers' in~\cite{DBLP:journals/tocs/Lamport98}). A round is a tuple $(n, id)$,
where $n$ is a non-negative integer and $id$ some globally unique identifier.
Rounds are partially ordered. $r_1 < r_2$ iff $r_1.n < r_2.n$.
Furthermore, $r_1 = r_2$ iff $r_1.n = r_2.n \land r_1.id =
r_2.id$. Newer proposals are indicated by higher rounds. Rounds with
the same $n$ but different $id$ cannot be ordered.
\item[Acceptor State.]
Acceptors act as the distributed, fault-tolerant storage. Each acceptor
manages three values (cf. Algorithm~1, line 24): (1) the
highest round \ensuremath{r_\mathit{ack}}\ it has acknowledged,
(2) the last value \ensuremath{\mathit{val}}\ it has voted for, and (3) round \ensuremath{r_\mathit{voted}}\
in which the proposal including the value was proposed in. By acknowledging a round,
acceptors promise not to vote for lower-numbered proposals in the future.
\begin{figure*}[t]
\includegraphics[width=\textwidth]{code_once}
\vspace{-1.3em}
\end{figure*}
\item[Pseudocode Conventions.]
For brevity's sake, we use the following conventions when handling
sets of reply messages: Let a process receive a set of reply messages
$S$. Each message in $S$ is an $n$-tuple denoted as \msg{t, e_1,\dots,e_{n-1}}.
We make use of pattern matching techniques commonly found in functional programming. The
type $t$ of the message is matched to ensure it has the correct format. Its
payload is stored in tuple elements $e_1$ to $e_{n-1}$. Since messages in $S$ may hold different
values in the same tuple element, we define the following functions:
$\ensuremath{\mathit{cons_S}}(e_i)$ returns the value of $e_i$ if it is equal for all messages in $S$,
or \ensuremath{\mathit{false}}\ otherwise;
$\ensuremath{\mathit{max_S}}(e_i)$ returns the largest value of $e_i$;
$\ensuremath{\mathit{max_S}}(e_i, e_j)$ returns the value of $e_j$ from the message with the largest value of $e_i$.
We furthermore assume that processes can keep track of multiple concurrent requests and
know to which outstanding request a received reply belongs.
\end{description}
\subsubsection{Protocol Description}
The protocol has two phases. In the
first phase, a proposer checks for concurrently proposed values and prepares
acceptors to deny outdated proposals. In the second phase, a proposer
proposes either its own or a value seen in the first phase. To eventually
learn a value, both phases must be passed without interruption by other proposers.
The protocol begins with proposer $p$ receiving a request from a client (line~1).
The request is either a \ensuremath{\mathit{write}}\ that tries to set the register to a value \ensuremath{\mathit{val}},
or a \ensuremath{\mathit{read}}\ that returns the register's current value (here, $\ensuremath{\mathit{val}} = \bot$).
The request of the client is handled asynchronously. The client will be notified by
a \ensuremath{\mathit{DONE}}\ message once the request has been processed.
Proposer $p$ starts the first phase by choosing a round ID and
sending it in a \ensuremath{\mathit{PREPARE}}\ message along with the request type to all acceptors (line~2--4).
Any acceptor $\mathcal{A}$ that receives a \ensuremath{\mathit{write}}\ request from $p$ acknowledges this
by incrementing \ensuremath{r_\mathit{ack}}\ and updating its ID. Thereby, $\mathcal{A}$
promises $p$ to not vote for any lower-numbered proposals in the future (line~28--29).
If $\mathcal{A}$ received a \ensuremath{\mathit{read}}\ request, then it does not increment \ensuremath{r_\mathit{ack}}\
as $p$ does not intend to modify the register's value by submitting a proposal.
Letting the state untouched when processing reads reduces their interference with
other ongoing requests and is not part of canonical Paxos.
After processing the request, $\mathcal{A}$ replies with its current state and
indicates if its \ensuremath{r_\mathit{ack}}\ round was incremented (lines~29,~32).
The second phase begins as soon as $p$ has received replies from a quorum $Q$ of
acceptors. Depending on the replies, $p$ proceeds in one of the following ways:
(1) If all acceptors in $Q$ have voted for the same proposal (same \ensuremath{r_\mathit{voted}}), then $p$
knows that consensus was already reached and that the proposal's value is chosen.
Thereby, $p$ has learned the register's value and returns it to the client.
Similarly, $p$ can be certain that consensus was not reached if no acceptor in
$Q$ has voted for any proposal yet. Thus, it can return an empty value
if it is processing a \ensuremath{\mathit{read}}\ (line~6--8).
(2) If all acceptors in $Q$ incremented their rounds and responded with a consistent
\ensuremath{r_\mathit{ack}}\ round, then $p$ can propose a value. If at least one of the acceptors has voted
for a past proposal, $p$ receives an inconsistent quorum as shown in \figref{fig:cons_quorum}.
It cannot decide if the proposal's value is already established or not. In order
to not violate safety, $p$ must propose the value seen in the
highest round. If no acceptor has voted for any proposal yet, $p$ can choose its
own value. The proposal is sent in a \ensuremath{\mathit{VOTE}}\ message to all acceptors using the
acknowledged round \ensuremath{r_\mathit{ack}}\ (line~9--16).
(3) In all other cases, $p$ has to retry the first phase. This happens if
acceptors are currently in an inconsistent state, e.g. because of an ongoing
proposal, lost messages, or a crashed proposer. As $p$ has already knowledge
about the current state of the acceptors, it can choose an explicit round
number that is higher than all rounds observed so far, which is then included
in \ensuremath{\mathit{PAXOS\_PREP}}\ messages (line~17--20).
An example of this is depicted in \figref{fig:flow}.
Each acceptor that has received a proposal by $p$ (case~(2)), votes
for the proposal if they have not given a promise for a higher round during
a (concurrent) phase 1 and notifies $p$ of its vote (line~40--43).
Otherwise, the acceptor ignores the proposal or may optionally
notify $p$ that its proposal is outdated (not shown).
Once $p$ has received a quorum of positive replies, it knows that
its proposed value is chosen and notifies the client on the established consensus
(line~22). This concludes the protocol.
\begin{figure*}[t]
\centering
\resizebox{.88\textwidth}{!}{%
\includegraphics{fig_example_msg_flow}
}
\captionsetup{justification=centering,margin=1cm}
\caption{Example message exchange of a write, starting with inconsistent acceptor states. Time moves from left to right.
Acceptor states are shown as ($r_{ack}$, $\textit{val}$, $r_{voted}$). Round IDs are omitted for simplicity.}
\label{fig:flow}
\vspace{-1em}
\end{figure*}
\subsubsection{Comparison to Paxos}\label{sec:comparison-paxos}
Our write-once atomic register is based on the same mechanism for
safety as Paxos, but differs from the canonical single-decree
Paxos~\cite{lamport2001paxos} in several aspects:
\begin{description}[listparindent=\parindent, itemindent=\parindent, leftmargin=0cm]
\item[Consistent Quorums.] In canonical Paxos, all proposals must
complete both phases of the algorithm even if a value was already
chosen. This effectively serialises concurrent reads and causes
unnecessary state changes in acceptors (their round numbers). Our
protocol, instead, terminates early and returns the result after the
first phase, when a proposer observes a consistent quorum. This
prevents (1) state modifications by reads, (2) allows termination in two
message delays and (3) prevents live-locks caused by duelling proposers
once all correct acceptors have agreed on a proposal. This is
possible because once a proposal with value $v$ is chosen, any
proposal made in a higher round will contain $v$ (see
\secref{sec:proof}). As the value of the register cannot
change any more, it is needless to execute the second phase.
\item[Distinguishing between reads and writes.] In canonical Paxos,
to read the state of a consensus it is necessary to propose a
value for consensus when no proposal was seen yet, i.e. actually
performing a write, which is unintended. For a read, a client can either
(1) initiate the protocol as a proposer and---in accordance to the
protocol---has to propose a (dummy) value itself when no value was
chosen yet or (2) it can ask a learner. However, a learner that has
not learned a value also has to propose a (dummy) value to
ensure its answer is up-to-date. As this dummy value might be
written, the read semantic is violated.
Drawing from the concept of consistent quorums, we support reads
without the risk to change the register's value and are also able to
reliably recognise an empty register. A read acts like a write
only when an ongoing, partially accepted proposal is seen that may need help
to fully establish.
However, no value will be proposed that was not already proposed by a write.
\item[Incremental round number negotiation.]
Proposers have to choose a high enough round number for their
proposal to succeed. Canonically, a proposer chooses the round number itself. If
it is too low, the proposer's attempt fails and it has to try
again with a higher round. This works
well when a leader makes the proposals, as it knows the
previous used round number. Without a leader, however, the first guess of
a proposer is likely to fail, costing a round trip even without concurrent access.
Instead, we let the acceptors increment their round on an initial
round-less attempt and retrieve the `assigned' round from the
replies when they form a consistent quorum. Otherwise, we calculate a
higher round number from the replies and retry like in Paxos.
Using incremental rounds is optional. If a proposer can determine a
round number that likely succeeds, it can also start with that
without violating the protocol's safety.
\item[Single learner per request.]
In canonical Paxos, acceptors send their \ensuremath{\mathit{VOTED}}\ messages to a set of learner
processes, which learn the value once they have received a quorum of votes for a
proposal. Therefore, the number of messages sent is the product of the number of
acceptors and the number of learners. In our approach, the proposer that has received
a request acts as its sole learner. Thus, every acceptor sends only a
single \ensuremath{\mathit{VOTED}}\
message.
\end{description}
\subsubsection{Sketched Proof of Safety} \label{sec:proof}
In this section, we provide a proof sketch for our Paxos atomic write-once register.
We show that the safety requirements of \secref{sec:paxos_background}, as well as
linearisability are satisfied.
Since our protocol has a close resemblance to canonical Paxos, we can use analogue
arguments and invariants as described by Lamport~\cite{lamport2001paxos} to prove safety.
\begin{prop} \label{prop:1}
If a proposal $p$ was learned in round $r$, then there exists a
quorum of acceptors $Q$ such that any acceptor in $Q$ has given a
vote for $p$ (i.e. the proposal must have been chosen).
\end{prop}
\begin{proofsketch}
For any two acceptors $a_1$, $a_2$, which have voted for proposal $p_1$ and $p_2$
respectively in the same round $r$, it holds that $p_1=p_2$ because rounds are
uniquely identified by their ID. To learn a value, a proposer must either (a)
receive a consistent quorum of \ensuremath{r_\mathit{voted}}\ rounds from acceptors at the beginning
of phase 2, or (b) receive a quorum of \ensuremath{\mathit{VOTED}}\ messages. For (a) to be possible, a
quorum with $r=\ensuremath{r_\mathit{voted}}$ must exist. For (b), a quorum of acceptors
must have voted for the proposer's proposal. Since a proposal is issued for
a specific round, all replying acceptors have voted for a proposal in the
same round.
\end{proofsketch}
C-Nontriviality is trivial to proof using proposition~\ref{prop:1}
since acceptors can only vote for any value
that was previously proposed by a proposer. C-Stability and C-Consistency hold by
satisfying the following invariant:
\begin{prop} \label{prop:2}
If a proposal with value $v_c$ and round $r_c$ is chosen, then every proposal
issued with round $r > r_c$ by any proposer has also value $v_c$.
\end{prop}
\begin{proofsketch}
By proposition~\ref{prop:1}, there is a quorum $Q$ that has voted for $v_c$ in $r_c$.
Since any two quorums have a non-empty intersection, any proposer $p$ will receive
at least one \ensuremath{\mathit{ACK}}\ reply of an acceptor included in $Q$. Furthermore, no acceptor has
voted for a proposal valued $v'$ with $v' \neq v$ in round $r'$ with $r' > r_c$.
This would imply the existence of a quorum $Q'$ for which every acceptor has
acknowledged round $r'$ before voting for the proposal in round $r_c$. This contradicts
the existence of $Q$ since acceptors cannot vote for a lower round than they have
previously acknowledged. Therefore, the proposal with the highest round that $p$
receives has value $v_c$. Thus, $p$ issues a proposal with $v_c$.
\end{proofsketch}
Proposition~\ref{prop:2} assumes that rounds can be totally ordered.
However, they are only partially ordered due to our modified negotiation mechanism.
Thus, we must show:
\begin{prop}
For any round number $n$, at most one proposal is issued.
\end{prop}
\begin{proofsketch}
A proposer can only issue a proposal in a round with round number $n$ once it has
received an acknowledgement from a quorum of acceptors with consistent and increased
\ensuremath{r_\mathit{ack}}\ with round number $n$. Any acceptor can send at
most one \ensuremath{\mathit{ACK}}\ message in which it has also increased its \ensuremath{r_\mathit{ack}}\ to
have round number $n$. Thus, at most one proposer can receive such a quorum to make
a proposal. If incremental rounds are not used, proposers have to choose their own
unique round numbers (cf. canonical Paxos).
\end{proofsketch}
\begin{prop} \label{prop:write-once-linearizable}
The Paxos-based write-once atomic register is linearisable.
\end{prop}
\begin{proofsketch}
Proposition~\ref{prop:1} and~\ref{prop:2} show that all writes return value $v_c$
of the first chosen proposal as their result. Reads differ from writes in
that they can return the initial value $\bot$, but only if no value is chosen since a consistent quorum
is required. Since a proposer must have learned a value before any write (or read
not returning $\bot$) can complete, any subsequent read will return $v_c$.
\end{proofsketch}
\subsection{Consensus Sequence Register} \label{sec:cmd_sequence}
The typical approach to learn a sequence of consensus values is to
chain multiple consensus instances on separate resources~\cite{lamport2001paxos,DBLP:conf/podc/ChandraGR07}.
In contrast, we aim to operate on the same set of resources. For that, we extend
our fault-tolerant write-once atomic register to support a sequence of updates.
The interface of our extended register changes slightly. Instead of including a
specific value \ensuremath{\mathit{val}}\ (see Algorithm~1, line~1)
in a write request, clients include an update command \ensuremath{\mathit{cmd}}, which transforms
the current value of the register to the next value. The required changes of
the proposer's second phase are highlighted in Algorithm~2.
The behaviour of the acceptors remains unchanged.
We introduced the concept of consistent quorums in our write-once register
to detect if the current value is chosen or not (see \secref{sec:write-once}).
We can use this information to handle a sequence of updates:
A proposer is allowed to propose a new value if the current value is
chosen. Otherwise, it must complete the unfinished consensus by proposing an
existing value. We refer to the former as a
\newterm{successor proposal} and to the latter as a \newterm{write-through proposal}.
Line~5--13 shows how a proposer submits
a successor proposal. It first applies the update command \ensuremath{\mathit{cmd}}\ it has received from the
client on the current established value. If the update is a valid operation,
the proposer can send the result to all acceptors. Sometimes, the update reduces
to a no-op as it cannot be applied to the current value, for instance, if
it includes compare-and-swap semantics or requires a write lock that is missing.
The proposer does not have to complete the second phase as the update has no effect
and can therefore immediately return to the client.
The submission of a write-through proposal (line~14--23)
is equivalent to proposing a value using our write-once register. The proposer
proposes the value seen in the highest round. Afterwards, it must re-execute
the protocol to process the received write request as a successor proposal.
\begin{description}[listparindent=\parindent, topsep=0pt, leftmargin=0cm]
\item[Safety.]
Intuitively, the register behaves as if executing multiple single-decree Paxos
instances in sequence, with each instance using the previously chosen
proposal and its round as initial state. Updates are applied on top of
a chosen value, which is ensured by observing a consistent quorum. Thus, for
any two values $v_1$ and $v_2$ that are chosen in this order, $s(v_1)$ is the
prefix of $s(v_2)$. By an argument analogous to
proposition~\ref{prop:write-once-linearizable}, reads always return the
latest chosen value. Thus, CS-Stability and CS-Consistency are satisfied.
An update $u$ can only complete if a value that includes $u$ in its update sequence
is chosen, as a quorum of \ensuremath{\mathit{VOTED}}\ messages is required. As only chosen values
are returned, CS-Update-Visibility is guaranteed.
No proposer applies $u$ on any chosen value after $u$ is
completed. Thus, every subsequent update appears after the last occurrence of $u$
in $s(v)$ of a subsequently chosen value $v$ (CS-Update-Stability).
\end{description}
\begin{figure}[t]
\vspace{0.36em}
\hspace{-0.15cm}
\includegraphics[width=\linewidth+0.2cm]{code_multiple}
\end{figure}
\subsection{/: Atomic Read-Modify-Write Register} \label{sec:atomic_cmd_sequence}
\begin{figure*}[t]
\includegraphics[width=\textwidth]{code_atomic}
\end{figure*}
The consensus sequence register presented in the previous section is not atomic,
as it is possible that an update command submitted by a client is proposed and
applied multiple times by the same proposer. For example, consider the following scenario:
Proposer $p_1$ completes phase 1 and submits a successor proposal. However, it only gets a
minority of acceptor votes, as some concurrent proposer $p_2$ already increased the
\ensuremath{r_\mathit{ack}}\ rounds of a quorum of acceptors. In this case, $p_2$ may observe an
inconsistent quorum and therefore executes a write-through of $p_1$'s proposal.
If it succeeds, then $p_1$'s proposal was effectively accepted because the value
proposed by $p_1$ is chosen. However, $p_1$ does not know this and retries, potentially
executing the command twice.
For atomicity, we must ensure that a proposer does not re-submit a successor proposal
once the proposed value of a previous attempt is chosen. For that, we assume
reliable in-order message delivery (see \secref{sec:system_model}).
This can be provided by reliable communication protocols such as TCP.
Note, that messages can be lost if a TCP connection fails
and is later re-established during the processing of a request.
To solve this issue, processes can be treated as crashed until the request
is completed. Now, the protocol can be modified as follows (cf. Algorithm~3):
For every write request that proposer $p$ receives, it generates a request ID (ReqID) consisting
of its PID and some locally unique value (line~2). Every acceptor holds the ReqID of the last
proposal it voted for and includes it in all phase~1 \ensuremath{\mathit{ACK}}\ messages it sends.
If a proposer submits a successor proposal, it includes its own ReqID as \ensuremath{\mathit{req}}\ensuremath{_\mathit{cur}}\ and the ReqID
received in phase 1 as \ensuremath{\mathit{req}}\ensuremath{_\mathit{prev}}\ in its \ensuremath{\mathit{VOTE}}\ messages
(line~14). Here, \ensuremath{\mathit{req}}\ensuremath{_\mathit{prev}}\ indicates the last successor proposal that was
chosen by the register. If the proposer submits a write-through, it includes
the ReqID received in phase~1 as \ensuremath{\mathit{req}}\ensuremath{_\mathit{cur}}. Since the last chosen proposal is now known,
\ensuremath{\mathit{req}}\ensuremath{_\mathit{prev}}\ remains empty (line~20).
Each time an acceptor votes for a new proposal, it updates $\ensuremath{\mathit{req}}$ to $\ensuremath{\mathit{req}}\ensuremath{_\mathit{cur}}$ (line~53).
If $\ensuremath{\mathit{req}}\ensuremath{_\mathit{prev}}$ is non-empty, it sends a \ensuremath{\mathit{LEARNED}}\ message to the respective proposer
(line~52). Receiving a \ensuremath{\mathit{LEARNED}}\ message guarantees that the corresponding proposal
was chosen. A proposer that retries a request with some ReqID can stop the protocol if (1) it observes
a consistent quorum with this ReqID (line~8), or (2) it receives a \ensuremath{\mathit{LEARNED}}\ message with it
(line~30). In both cases, it notifies the client
that its write request succeeded.
We note that it is easy to avoid sending values in \ensuremath{\mathit{LEARNED}}\ and \ensuremath{\mathit{VOTED}}\ messages back
to the proposer if the proposer keeps track of its proposed values locally. By extension, it is not
necessary to include \ensuremath{\mathit{val}}\ensuremath{_\mathit{prev}}\ in \ensuremath{\mathit{VOTE}}\ messages.
For simplicity, this is not shown in Algorithm~3.
\begin{description}[listparindent=\parindent, topsep=0pt, leftmargin=0cm]
\item[Safety.]
Assume a write request with ReqID $r$ and update command $u$ is processed by proposer $p$.
Assume that $p$'s attempt failed, but its proposed value is chosen (e.g. due to a write-through).
Proposer $p$ does not propose $u$ as the direct successor of its own proposed value because
it would observe a consistent quorum with ReqID $r$ beforehand. Thus, assume
that some successor value proposed by a different proposer is chosen.
This means that \ensuremath{\mathit{LEARNED}}\ messages with ReqID $r$ are sent to $p$ by some quorum $Q$.
Let $p$ retry its request. In order to apply $u$ and
propose a new value, $p$ must observe a consistent quorum $Q'$.
As $Q \cap Q' \neq \emptyset$ and reliable ordered links are used, $p$ receives
a \ensuremath{\mathit{LEARNED}}\ message before receiving a consistent quorum.
Thus, $p$ does not apply $u$ on a value whose update sequence already includes $u$.
\end{description}
\subsection{State Machine Replication}\label{sec:RSM}
By using /, we can build a fault-tolerant
replicated state machine using a fixed set of storage resources.
The state is stored in the register and state changes are done by the
corresponding update commands. If updates are idempotent, the
consensus sequence register suffices. One way to achieve this is by
using transactional semantics such as compare-and-swap.
In log-based approaches like Multi-Paxos~\cite[Sect.~3]{lamport2001paxos},
acceptors accept commands, i.e. state transitions of the state
machine. In our approach, in contrast, the acceptors accept the complete state. This has several implications. First, a dedicated set of learner processes is no longer required.
Any process that wishes to learn the current state of the RSM can do so by
executing a read. This process then acts as the sole learner in the context of
this command. In contrast, Multi-Paxos requires multiple learners in order to
have access to the state in a fault-tolerant manner. Since every learner must also
learn every command to make progress, $n*m$ \ensuremath{\mathit{VOTED}}\ messages are required
in a setup with $n$ acceptors and $m$ learners.
Our approach requires only $n$ \ensuremath{\mathit{VOTED}}\ messages.
Second, by keeping the full state in acceptors, a sequence of commands can now be applied
to the RSM in-place using the same set of acceptors. Thus, it is not necessary to
allocate and free storage resources. This simplifies the
protocol's complexity and its implementation. Due to the absence of any state management
overhead, it is trivial to use arbitrary many / instances in parallel, allowing
a more fine-granular use of the RSM paradigm. This is especially useful if the
state can be split into many independent partitions, as
it is often the case in key-value structured data.
\subsection{Liveness}
Reads and writes are obstruction-free~\cite{DBLP:conf/icdcs/HerlihyLM03} as long
as a quorum of acceptors and the proposer receiving the requests are correct.
Wait- or lock-freedom~\cite{DBLP:journals/toplas/Herlihy91} cannot be guaranteed without
further assumptions, as postulated by the FLP result~\cite{DBLP:journals/jacm/FischerLP85}.
A common assumption is the existence of a stable leader to which all
requests are forwarded. The leader then acts as the sole proposer of the system.
To handle leader failures, a $\Diamond\mathcal{W}$ failure detector~\cite{DBLP:journals/jacm/ChandraHT96}
is necessary.
\subsection{Optimisations} \label{sec:optiseqwrite}
There are several ways to optimise the basic protocol.
\begin{description}[listparindent=\parindent, topsep=0pt, leftmargin=0cm]
\item[Fast Writes.] Handling writes requires a proposer to complete both phases
of the protocol. That means that at least four message delays are needed. By
using a mechanism similar to Multi-Paxos~\cite{lamport2001paxos}, the first phase
can be skipped by a proposer that processes multiple writes uninterrupted
by other proposers. We refer to such writes as \newterm{fast writes}.
The modification is simple. Whenever an acceptor votes for a proposal made in
round $r$, it sets \ensuremath{r_\mathit{ack}}\ to $(r.n + 1, r.\mathit{id})$
(cf. Algorithm~1 line~40). By doing
so, it effectively behaves as if receiving a \ensuremath{\mathit{PREPARE}}\ message from
the same proposer immediately after voting. Therefore, this proposer can skip
the first phase when making its next proposal.
This optimisation is useful for single-writer settings or scenarios in
which a proposer must execute multiple writes within a short period.
As no locking or lease mechanism is used, an ongoing fast write sequence can
be interrupted at any time by other proposers. Thus, we avoid the costs
and unavailability associated with a leader and its (re-)election.
\item[Fewer concurrency conflicts caused by reads.]
If a read observes a consistent quorum after the first phase, it returns
a result without interfering with any concurrent request because acceptors
do not modify their rounds. If a read observes an inconsistent quorum, a write-through
is triggered, which can cause interference. Write-throughs cannot be prevented
completely, as a crashing proposer can cause a proposal to be only partially established.
Therefore, we adopt the idea of contention management~\cite{DBLP:conf/icdcs/HerlihyLM03,DBLP:conf/podc/SchererS05}
to unreliably detect a crashed writer:
When a reading proposer observes an inconsistent quorum, it stores the highest
round it has received. Then, it retries phase 1 without an explicit round. If the quorum is
again inconsistent, it checks whether progress was made by comparing the received
rounds with the rounds from the previous iteration. If they remain unchanged,
then it is possible that the write crashed and a write-through must be triggered.
Otherwise, the reader can try again. The proposer can keep collecting replies
from its previous attempts as it is possible to reach a consistent quorum with delayed
replies.
To prevent a read from starving due to a continuous stream of writes, we define
an upper limit on the number of retry attempts. Its effects are
evaluated in \secref{sec:practical-evaluation}.
\item[Batching.] Batching is a commonly used engineering technique to reduce bandwidth
and contention by bundling multiple commands in a single request at the cost
of higher response latency. Every proposer manages separate batches for read and update commands.
A batch is processed at regular intervals by starting the protocol. For write
batches, all update commands of the batch are applied in-order on the old value before
proposing the resulting new value. When processing a read batch, the read value is simply
returned to all clients. The size of all messages remains constant, independent
of the number of batched commands. This shifts the performance bottleneck from
internal communication to the processing speed of the respective proposers.
\end{description}
\begin{figure*}[t!]
\centering
\captionsetup{justification=centering}
\begin{subfigure}[t]{\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_throughput_compare_load}
\caption{Throughput comparison with an increasing number of clients}
\label{fig:tp_compare_load}
\end{subfigure}%
\\ \vspace{1em}
\begin{subfigure}[t]{\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_throughput_compare_size}
\caption{Throughput comparison with increasing value sizes (using 512 clients)}
\label{fig:tp_compare_size}
\end{subfigure}%
\vspace{2mm}
\caption{Comparing the throughput of / with Raft and Multi-Paxos
using three replicas.}
\end{figure*}
\section{Analysis}\label{sec:evaluation}
In this section, we focus on additional aspects that might be beneficial for practical
deployments. An experimental evaluation can be found in \secref{sec:practical-evaluation}.
Compared to canonical Paxos and Multi-Paxos our registers require a
similar number of 2--4 message delays per consensus in the conflict-free case.
Two additional message delays are needed by canonical Paxos when a
valid round number is not known yet and by our registers when a read
using incremental rounds has to help to establish a
consensus. Reading a stable, established consensus with our approach
only needs 2 message delays, no concurrency control and does not cause
acceptor state changes, which is costly if their state must be persisted.
Furthermore, our approach
works on a fixed set of resources which makes dynamic resource
allocation, pruning, and deallocation needless. This makes our
register applicable on a more fine-granular level than other
consensus-based approaches that rely on a command log.
Relying on consistent quorums does not harm robustness nor
performance. Like in canonical Paxos, a single replica with the
highest round seen in an inconsistent quorum will suffice to propose
its value. But on a \emph{consistent quorum}, we can (a) terminate a
read operation early by not needing to write and re-learn the
consensus and (b) can base the next consensus in our consensus
sequence on that.
Not requiring an explicit leader provides more continuous availability. In
our approach, any proposer can issue requests to the register at any
time. When a proposer fails, other proposers can immediately proceed
and do not need to wait for or elect a new leader. Still, a proposer
submitting many requests in sequence without any interference of other
proposers can perform each write to the register in just two message
delays (no batching), like a leader.
\section{Experimental Evaluation}
\label{sec:practical-evaluation}
We implemented / in Scalaris~\cite{DBLP:conf/erlang/SchuttSR08}, a distributed
key-value store written in Erlang. The correctness of our implementation was tested using a protocol
scheduler~\cite{scalaris_proto_sched}, which forces random interleavings of incoming messages.
We detected no safety violations using this approach.
The primary focus of the evaluation is to show the scalability of
our approach under different workloads, as absolute performance is highly
dependent on the available hardware environment and engineering efforts that are
independent of the actual approach. Our register aims to be a general
primitive. Thus, we consider use-case dependent techniques
that optimise network traffic and concurrent access, e.g. request batching,
being out-of-scope of this paper.
All benchmarks were performed on a cluster with two Intel Xeon E5-2670
v3, 2.40\,GHz per node. All nodes are fully-connected with 10\,Gbit/s
links. Each cluster node hosts a single replica,
which is a Scalaris node that encapsulates one proposer and one acceptor process. Load generation
was performed on up to two separate cluster nodes using the benchmarking tool Basho
Bench~\cite{github_basho_bench}, which was modified to enable workloads with heterogeneous
client processes. In all experiments, Basho Bench clients were distributed evenly
across the load generating nodes. All clients submit their requests sequentially,
i.e. each client waits for a response before issuing the next request.
All shown measurements ran for 10 minutes with request data aggregation
in one-second intervals. We show the mean with 99\,\% confidence intervals (CI)
and 99th percentile latencies. In almost all cases, the CI lies within two percent of the reported median.
\begin{figure*}[t!]
\centering
\captionsetup{justification=centering}
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_single_writer_xread_compare}
\caption{Read throughput}
\label{fig:single_tp_read}
\end{subfigure}%
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_single_writer_xwrite_compare}
\caption{Write throughput}
\label{fig:single_tp_write}
\end{subfigure}
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_hanging_writer_lat_64}
\caption{99th pctl. latency (64 readers, \textsf{X}=10)}
\label{fig:single_lat_hanging}
\end{subfigure}
\vspace{1mm}
\caption{Single-writer performance of / with five replicas.}
\label{fig:single_writer_multiple_reader}
\end{figure*}
\begin{figure*}[t!]
\vspace{-3mm}
\centering
\captionsetup{justification=centering}
\begin{subfigure}[t]{0.319\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_multi_writer_mixed}
\caption{Single register throughput}
\label{fig:multi_single_reg}
\end{subfigure}%
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_multi_reg_read_heavy}
\caption{Multi-register read-heavy throughput}
\label{fig:multi_reg_read}
\end{subfigure}
~
\begin{subfigure}[t]{0.322\textwidth}
\centering
\includegraphics[draft=false,width=\textwidth]{bench_multi_reg_write_heavy}
\caption{Multi-register write-heavy throughput}
\label{fig:multi_reg_write}
\end{subfigure}
\vspace{1mm}
\caption{Leaderless multi-writer performance of / with five replicas.}
\label{fig:multiple_writer_multiple_reader}
\vspace{-1mm}
\end{figure*}
\subsection{Comparison with Raft and Multi-Paxos}
First, we compare the performance of / with open-source implementations of
Multi-Paxos~\cite{github_riak_ensemble, lamport2001paxos} and
Raft~\cite{github_raft, DBLP:conf/usenix/OngaroO14}, two commonly used state-of-the-art protocols.
To minimise the performance impact of the IO subsystem, we configured both approaches
to write their data to RAM disk. In /, data is stored by using
Erlang's build-in term storage~\cite{erlang_ets}. All approaches use three replicas.
As both Multi-Paxos and Raft make use of a leader, we simulate a leader by randomly selecting
one node to which all requests are forwarded to in the case of /. As any leader election
protocol can be implemented on top of /, we consider leader election to be
out-of-scope.
We measured the throughput of all approaches in scenarios: First, a counter
that is accessed by an increasing number of clients (\figref{fig:tp_compare_load}).
Second, a binary value of increasing size accessed by a fixed number of clients (\figref{fig:tp_compare_size}).
All three approaches handle requests in a single round-trip between leader and
a quorum of following nodes. Thus, the observed differences can largely be attributed
to their different strategies in handling the data locally. Due to the absence of
any state management, / consistently outperforms both the Raft and Multi-Paxos implementation
for small state sizes. For the latter two, overhead caused by reading/writing data to the local file system
increases request latency, which in turn negatively affects throughput. In addition, the Multi-Paxos and
Raft implementations use mechanisms such as checksum validation to protect against disk corruption.
We note that Multi-Paxos has a higher throughput than / in read-heavy
workloads with few clients. We attribute this to our method of load generation. As clients submit requests
sequentially, both approaches do not reach full capacity. Here, we observe a slightly
lower mean read latency for Multi-Paxos (0.6ms vs 0.8ms), which is likely caused by
implementation-specific overhead.
For values smaller or equals to 4kB, all approaches exhibit nearly constant read performance. However,
the throughput of / decreases for larger values. This is because the full value is always transferred
from a quorum of nodes to the proposer when executing a read. This causes high communication costs
in settings where individual objects have moderate or large size. However, analysis of existing large-scale key-value
stores have shown a heavy skew towards small values of less than a kilobyte~\cite{DBLP:conf/sigmetrics/AtikogluXFJP12, DBLP:conf/nsdi/NishtalaFGKLLMPPSSTV13}.
In contrast to /, the Raft and Multi-Paxos implementations include optimisations to keep data
transfer costs between nodes constant when executing a read if the leader is stable.
In Raft's case, an empty heartbeat log entry must be appended to the command logs to
ensure that the data of the leader is up-to-date.
This introduces a slight overhead when reading entries.
\subsection{Leaderless Performance} \label{sec:leaderless}
/ is derived from Paxos. Thus, it does not depend on the existence of a leader
to satisfy the safety properties of consensus, in contrast to protocols like Raft, which
do not work without a single leader. However, a leader
is beneficial for progress because it prevents the duelling proposer problem.
For /, we can alleviate the need for a leader as it is trivial
to deploy an arbitrary number of concurrent / instances. This way,
load on a single instance can be greatly reduced, depending on the workload.
We examined both single-writer (\figref{fig:single_writer_multiple_reader})
and multi-writer (\figref{fig:multiple_writer_multiple_reader}) workloads, as
previous work in the design of data structures has shown that supporting
concurrent modifications often inhibits their performance~\cite{DBLP:conf/podc/SchererS05}.
To better illustrate the effects of concurrent requests, we increased the system
size to five replicas (acceptors).
\begin{description}[topsep=0pt, leftmargin=0cm, listparindent=\parindent]
\item[Single-Writer.] To evaluate single-writer performance, we used
one writing client and up to 1024 concurrent readers
with a different number of read retries (parameter \textsf{X}). The results are depicted
in \figref{fig:single_writer_multiple_reader}.
We observed that even a single retry (\textsf{X}=1) improves both read and write
throughput greatly compared to disabling this optimization (\textsf{X}=0). In the latter
case, the register was overloaded due to concurrent write-through attempts by the readers
if more than 64 readers where used, dropping throughput to 0 at some times.
As these results are not stable, they are not shown in \figref{fig:single_tp_read}. Choosing
a value for \textsf{X} larger than 2 has only a minor impact on the read throughput.
As acceptors must handle more messages with an increasing number of clients, their response
latency increases. This leads to a consistent decline of the write throughput, as shown
in \figref{fig:single_tp_write}.
Since the load is distributed more evenly across all replicas, the maximum observed
throughput increased by roughly $70\,\%$ compared to our leader-based experiments (cf.
\figref{fig:tp_compare_load}), even though the system size increased from 3 to 5 replicas.
\figref{fig:single_lat_hanging} shows the latency impact of using
read retries. Read latency only increases by approx. 0.5\,ms in the presence of a concurrent
writer. This may contradict the expectation that some reads require multiple
round trips as they can observe an inconsistent quorum initially. However, proposers
can continue collecting replies from the initial attempt and return a result once they
observe a consistent quorum. As there is only a single writer, such a quorum always exists,
at the latest after receiving a reply from every acceptor. This also
means that reads trigger no write-throughs. Thus, both reads and writes succeed
after a single round trip in a single writer setup as long as no acceptor fails. Note that
writes exhibit a slightly lower latency as they always succeed with a quorum of replies,
wheres reads must potentially wait for all replies in some cases.
\item[Multi-Writer, Single-Register.]
All clients sent a uniform mix of read and write requests for
the evaluation of multiple writers. \figref{fig:multi_single_reg} compares
the throughput of a read-heavy workload (5\,\% writes) with a write-heavy
workload (50\,\% writes)~\cite{DBLP:conf/cloud/CooperSTRS10}. Performance degradation caused by duelling proposers
can be observed for both workloads. The throughput of the read-heavy workload scales
up until four concurrent clients. Afterwards, clients begin to invalidate each
other's proposals repeatedly. In write-heavy workloads,
even two concurrent clients are enough to have a negative impact on the system's
performance. As shown in the previous experiments, a leader at the application
level helps to handle write concurrency effectively.
\item[Multi-Writer, Multi-Register.]
All previous measurements focused on a single register. As highlighted in \secref{sec:RSM},
the absence of state management overhead easily allows for arbitrary many registers to be
used. We benchmarked configurations using up to $10^6$ register instances and 512 concurrent
clients. The registers were accessed according to a Pareto distribution~\cite{Newman_2005} with
$\alpha\approx1.16$ ($80\,\%$ of requests targeted $20\,\%$ of registers).
Figures \ref{fig:multi_reg_read} and \ref{fig:multi_reg_write} show the results
for read-heavy (5\,\% writes) and write-heavy (50\,\% writes), respectively. The results
are as expected. More concurrent clients can be handled without performance degradation
due to duelling proposers by increasing the number of parallel registers. The load is evenly
distributed across all replicas, as no leader is used. In addition, contention is low
in settings with a large number of parallel registers. This results in a higher achievable
throughput than it is possible with the use of a leader (cf. \figref{fig:tp_compare_load}).
/ performs consistently better under read-heavy workloads, which coincides
with the results from the single-register evaluation. We used
the read-write ratios of YCSB~\cite{DBLP:conf/cloud/CooperSTRS10}, a benchmarking framework
that aims to simulate real-world use-cases. Studies of large-scale distributed systems
have shown an even higher skew towards reads, reporting read-write ratios of
up to 450:1~\cite{DBLP:conf/sigmetrics/AtikogluXFJP12,DBLP:conf/osdi/CorbettDEFFFGGHHHKKLLMMNQRRSSTWW12, wikimedia_stats}.
\end{description}
\subsection{Impact of Replication Degree and Failures}
We investigated the impact of the number of replicas on the response latency
of /, as well as its ability to tolerate replica failures.
For that, we used different deployment strategies from our previous experiments:
(1) A single register accessed by a leader, (2) a single register accessed by
a single writer and multiple readers, and (3) a 10.000 register setup accessed
by multiple writers and readers with the Pareto distribution used in Section 7.2.
We will refer to them as the leader, single-writer, and multi-register strategy,
respectively. All measurements were executed using 64 clients. Clients used a
read-heavy workload (5\% writes, 95\% reads) in the leader and multi-register deployment.
The results are shown in \figref{latcompare}.
\begin{figure}[t]
\begin{subfigure}[t]{\columnwidth}
\centering
\includegraphics[draft=false,width=\columnwidth]{bench_size_lat}
\caption{Latency with a growing number of replicas.}
\label{latsize}
\end{subfigure}%
\\
\begin{subfigure}[t]{\columnwidth}
\centering
\includegraphics[draft=false,width=\columnwidth]{bench_crash_lat}
\caption{Latency with five replicas and process failures.}
\label{latcrash}
\end{subfigure}%
\caption{99th percentile latency comparison using 64 clients.}
\label{latcompare}
\end{figure}
When using a leader, the number of messages the leader must process increases
with a growing number of replicas. This results in an increasing response latency
as shown in \figref{latsize}. In contrast, the load is distributed evenly among all
node in both the single-writer and multi-register setup. Assuming a constant
throughput, the number of messages each proposer is sending is independent of the
system size. As only replies from a quorum of replicas is needed, fewer messages
must be received in total by each proposer to answer all requests. This results
in a slightly lower response latency of these strategies with growing system sizes.
To measure the impact of failures, we let one replica crash after every three minutes.
Overall, all latencies with the exception of the read latency of the single-writer
strategy remained fairly consistent as long as a sufficient number of replicas is
available. We observed only a slight increase
for each new failure, as proposers must potentially wait for the replies of slower
acceptor processes. However, to ensure that reads can be processed in the
single-writer setup, answers from all replicas are necessary
(see single-writer evaluation in \secref{sec:leaderless}). If a replica fails,
proposers do not always observe a consistent quorum after all remaining acceptors
reply. They must therefore retry their request. This is more likely to happen as
more replicas fail.
\subsection{Leader Load and Applicability to NVM}
Our results show potential for future improvements. First, we aim to improve
the issue of high write contention on a single register while alleviating the
bottleneck caused by a leader. As a single register is able to handle high read
concurrency (see \secref{sec:leaderless}), only writes have to be forwarded to the leader.
This can be coupled with a dynamic leader allocation for only highly contentious registers,
which further reduces the load placed on the leader.
Second, we believe that the fine-granular nature of our approach is a promising fit
for the use in combination with byte-addressable, non-volatile main memory.
With the recent availability of NVRAM, along with the current work
in NVMe over Fabrics~\cite{minturn2015under}, we believe that our approach can
leverage these technologies in the future.
\section{Related Work}
Starting with Lamport's work on the discovery of the Paxos
algorithm~\cite{lamport2001paxos,DBLP:journals/tocs/Lamport98},
numerous Paxos extensions~\cite{DBLP:conf/sosp/MoraruAK13,DBLP:conf/cloud/WangJCYC17,DBLP:journals/dc/GafniL03,DBLP:conf/dsn/MarandiPSP10}
have been proposed---most of them following the design of using multiple Paxos instances
to learn a sequence of commands. As a notable exception, Generalized Paxos~\cite{lamport2005generalized}
and its derivatives~\cite{DBLP:conf/srds/SutraS11}, only use a
single Paxos instance but require keeping track of an ever-growing set of commands
in its messages. In all cases, pruning in some form must be implemented to prevent
unbounded memory consumption, which introduces a considerable amount of complexity to
the system. This is identified by Chandra et al.~\cite{DBLP:conf/podc/ChandraGR07} as one of the main challenges
for using Paxos-based designs in practical systems. Despite numerous efforts of making
Paxos more approachable~\cite{kirsch2008paxos,DBLP:journals/sigact/BoichatDFG03,DBLP:journals/csur/RenesseA15},
reliable state management with Paxos is seldom discussed in detail.
Only a few practical Paxos-based systems exist to this date such as
Chubby~\cite{DBLP:conf/osdi/Burrows06}, Spanner~\cite{DBLP:conf/osdi/CorbettDEFFFGGHHHKKLLMMNQRRSSTWW12},
Megastore~\cite{DBLP:conf/cidr/BakerBCFKLLLLY11}, and Scalaris~\cite{DBLP:conf/erlang/SchuttSR08}.
In recent years, various proposals were made to alleviate the dependence on a single leader.
Mencius~\cite{DBLP:conf/osdi/MaoJM08} evenly shares the leader's responsibilities by
assigning individual consensus instances
to single replicas. In Egalitarian Paxos~\cite{DBLP:conf/sosp/MoraruAK13}, the replica receiving
a command is regarded as its command leader. Each replica can act as a
leader simultaneously for a subset of
commands. This is achieved by decoupling command commit and application from
each other and making use of the dependency constraints of each
command. In contrast, we do not need an explicit leader depending on workload
and load-distribution.
As of today, few consensus protocols, which are not Paxos-based, exist. Most prominently
Raft~\cite{DBLP:conf/usenix/OngaroO14} and the closely related Zab protocol~\cite{DBLP:conf/dsn/JunqueiraRS11}.
Both are based on the idea of a central command log. Furthermore, they
require a \emph{strong} leader, meaning that at most a single leader is allowed to
exist at any given time. In contrast, we perform updates on a distributed state in-place and
do not need a strong leader.
To the best of our knowledge, we present the first Paxos-based approach that does not rely on
additional state management without requiring a leader to satisfy the safety properties
of consensus by implementing an atomic RMW register.
The register by Li et al.~\cite{DBLP:conf/srds/LiCAA07} only recasts the original Paxos
without modification and provides a regular
write-once register. The round-based register proposed by Boichat et al.~\cite{DBLP:journals/sigact/BoichatDFG03} is
not atomic and only write-once. It is similar to the approach of Li et al. and modular
to build several, known Paxos variants such as Multi-Paxos or Fast Paxos~\cite{DBLP:journals/dc/Lamport06}.
CASPaxos~\cite{DBLP:journals/corr/abs-1802-07000} provides a Paxos-based linearisable
multi-reader multi-writer register by letting clients submit a user-defined function
instead of a value. However, when handling concurrent writes it is not guaranteed that
all (or any) writes are processed by the register due to duelling proposers, which makes
it unsuitable to implement basic primitives like counters. The key-value consensus
algorithm Bizur~\cite{DBLP:journals/corr/HochBLV17} is based on a set
of single-writer multi-reader registers and therefore
relies on electing a strong leader.
The use of consistent quorums in conjunction with Paxos is
first introduced by Arad et al.~\cite{consistent-quorum} in the context
of group membership reconfigurations. In this context, a consistent quorum expresses
a consistent \emph{view} of the system in terms of group memberships. Skrzypczak et al.~\cite{DBLP:conf/podc/SkrzypczakSS19}
use consistent quorums to provide linearisable access to CRDTs. While this approach
is similar to the protocol presented here, it heavily relies on the mathematical properties
of CRDTs and can therefore not be used for general state machine replication.
Shared register abstractions were first formalized by
Lamport~\cite{DBLP:journals/dc/Lamport86a}. Among them, the atomic register
provides the strongest guarantees by being linearisable. Numerous implementations
exist today. In particular, the multi-writer generalisation~\cite[p.~25ff.]{DBLP:series/synthesis/2012Vukolic}
of ABD~\cite{DBLP:journals/jacm/AttiyaBD95} has the greatest resemblance to our approach.
However, the properties of atomic registers alone do not suffice to solve consensus,
as not every completed write is necessarily applied to the register when being confronted
with concurrent access. Moreover, only fixed values can be written.
Our register abstractions provide arbitrary value transformations based
on the register's previous value and ensure that completed writes are applied
at-least-once (consensus sequence register) or exactly-once (/).
\section{Conclusion}
In this paper, we introduced register abstractions that satisfy the safety
properties of consensus and allow consensus sequences.
We provided implementations extending
the principles of Paxos consensus, to allow a sequence of consensuses `in-place'
using a single set of storage resources, instead of a separate instance
for every consensus decision.
Additionally, read operations in /
do not interfere with each other (are not serialised with each other) and do not modify any state in the
acceptors when the register is stable, i.e. no write operation is induced. This improves the parallel read throughput and saves
unnecessary, potentially costly state changes of persistent storage
for reads. When reads detect ongoing writes, they can either hope the
writer will finish soon and mitigate the chance of duelling proposers
by just retrying the read, or can start to support the writing
themselves as the writer might have crashed. As we show in our
evaluation (\secref{sec:practical-evaluation}), the trade-off between
both strategies and how often one should retry the read before helping the writer depends on the system
deployment, the number of expected concurrent readers and writers, etc.
Avoiding the need for costly state management and complex protocols
for state pruning, providing fast writing in two message delays and
supporting concurrent readers without serialisation opens a
wide new spectrum of use-cases for Paxos based fault-tolerance. The protocols
we provide are beneficial and applicable on a more fine-grained level than
Multi-Paxos or similar approaches, as they have low system overhead and
provide good scalability.
\medskip
\begin{description}[listparindent=\parindent, topsep=0pt, leftmargin=0cm]
\item[Code Availability.]
The source code for our / implementation~\cite{scalaris_approach_impl}
and the protocol scheduler~\cite{scalaris_proto_sched} can be found on GitHub
under the Apache License 2.0.
\end{description}
\section*{Acknowledgements}
We thank Alexander Reinefeld and anonymous reviewers for their dedicated comments and valuable
discussions that helped to improve this manuscript. This work
received funding from the German Research Foundation (DFG) under grant
RE~1389 as part of the DFG priority program SPP~2037. We thank ZIB's
core facilities unit for providing us the machines and infrastructure
for the evaluation.
\bibliographystyle{IEEEtran}
|
1,108,101,565,477 | arxiv | \section{Introduction}
The present paper is comprised of a geometric model-theorist's attempts to do noncommutative algebraic geometry. The latter endeavour takes as its starting point the possibility of extending the anti-equivalence of categories $\mathsf{CRing}$ and $\mathsf{AffSch}$ (which denote the categories of commutative unital rings and affine schemes respectively) to arbitrary rings and putative `noncommutative schemes'; namely establishing that the following diagram commutes:
\[ \xymatrix{ \mathsf{CRing}^{op} \ar@<2pt>[r] \ar@{^{(}->}[d] & \mathsf{AffSch} \ar@<2pt>[l] \ar@{^{(}->}[d] \\ \mathsf{Ring}^{op} \ar@<2pt>[r] & \mathsf{NSch} \ar@<2pt>[l] } \] where $\mathsf{NSch}$ is a candidate for the category of noncommutative schemes. The author has adopted the viewpoint that geometric model theory can only interact with noncommutative algebraic geometry via the theory of Zariski structures; and indeed does so most naturally. This is a viewpoint he now wishes to justify.
\\
\\
Firstly, an investigation of important applications of geometric model theory to questions of \textit{commutative} algebraic geometry (specifically diophantine geometry) indicates that a difference of approach is currently needed if our endeavours are to succeed. The methodology of such applications can be summarized as follows. One selects appropriate structures (e.g. algebraically closed fields, differentially closed fields, separably closed fields), establishes what `stability class' the structures belong to and deduces results about definable sets by applying the relevant abstract model-theoretic tools associated with this stability class \footnote{The articles \cite{Mac03} and \cite{Hru98} contain an introduction to methods of geometric model theory; \cite{B99} discusses the specific application of these methods to Mordell-Lang.}. Crucially, the language and techniques of geometric model theory, with its emphasis on stability and appropriate generalizations of this notion, independence and ranks, working in a universal domain etc, is closer in spirit and language to Weil's foundations than scheme theory. One certainly does not work at the level of generality of an arbitrary commutative ring. For any hope of applying model theory to noncommutative algebraic geometry, it seems that one should remain (at the very least) in a suitably geometric setting and look for geometric counterparts to suitable classes of noncommutative $k$-algebras, where $k$ is an algebraically closed field. But at the same time, there seems to be no reason for suspecting that there is a nice structure whose definable subsets can be regarded as coordinate rings of a sufficiently interesting and large class of noncommutative $k$-algebras. It is not possible to do any `naive' noncommutative algebraic geometry in the manner that one can work with varieties as subsets of affine or projective space. The language of schemes and category-theoretic generalizations of it are indispensable for most of the popular existing approaches to noncommutative algebraic geometry (\cite{Mah06}, \cite{Ros95}).
\\
\\
Rather, we are forced to
\begin{itemize}
\item find a systematic means of associating a structure to a given noncommutative $k$-algebra, suitably axiomatized in an appropriate language.
\item ask whether these structures share any common geometry.
\end{itemize} An association of structures to algebras should be functorial if it is to be systematic; thus we must work with a geometric category of structures not necessarily all defined in the same language. If we are aiming for an extension of commutative algebraic geometry then a basic intersection theory resembling the commutative case should exist, i.e. some well-behaved notion of dimension should exist for a large class of subsets of each noncommutative structure. It transpires that these rather basic requirements lead us to stipulate that the associated structures are Zariski structures. We work with the definition of \cite{Zil10}:
\begin{definition}
\label{definition - Zariski structure - Noetherian}
Let $\mathbf{X}$ be a set. A \textbf{Zariski structure} \footnote{Technically, according to the terminology of \cite{Zil10} we shall be defining Noetherian Zariski structures as opposed to analytic Zariski structures. Because we do not deal with the latter, for the purposes of this thesis the adjective `Noetherian' can be dropped.} on $\mathbf{X}$ consists of a Noetherian topology on $\mathbf{X}^{n}$ for every $n > 0$ and an $\mathbb{N}$-valued dimension function $\dim$ on non-empty projective subsets (finite unions of projections of closed subsets) satisfying the following properties:
\begin{enumerate}
\item The dimension of a point is $0$.
\item $\dim(\mathbf{P}_{1} \cup \mathbf{P}_{2}) = \max\{\dim \mathbf{P}_{1}, \dim \mathbf{P}_{2}\}$ for all projective subsets $\mathbf{P}_{1}, \mathbf{P}_{2}$.
\item For $\mathbf{C}$ closed and irreducible in $\mathbf{X}^{n}$ and $\mathbf{C}_{1}$ a closed subset of $\mathbf{C}$, if $\mathbf{C}_{1} \neq \mathbf{C}$ then $\dim \mathbf{C}_{1} < \dim \mathbf{C}$.
\item For $\mathbf{C}$ irreducible and closed in $\mathbf{X}^{n}$, if $\pi: \mathbf{X}^{n} \rightarrow \mathbf{X}^{m}$ is a projection then
\[ \dim \mathbf{C} = \dim \pi(\mathbf{C}) + \min_{a \in \pi(\mathbf{C})} \dim (\pi^{-1}(a) \cap \mathbf{C}) \]
\item For any irreducible closed $\mathbf{C}$ in $\mathbf{X}^{n}$ and projection map $\pi: \mathbf{X}^{n} \rightarrow \mathbf{X}^{m}$, there is a subset $\mathbf{V}$ relatively open in $\pi(\mathbf{C})$ such that
\[ \min_{a \in \pi(\mathbf{C})} \dim(\pi^{-1}(a) \cap \mathbf{C}) = \dim(\pi^{-1}(v) \cap \mathbf{C}) \] for every $v \in \mathbf{V} \cap \pi(\mathbf{C})$.
\end{enumerate} Moreover, projections must be semi-proper, i.e. for any closed irreducible subset $\mathbf{C}$ of $\mathbf{X}^{n}$ and projection map $\pi: \mathbf{X}^{n} \rightarrow \mathbf{X}^{m}$, there is a proper closed subset $\mathbf{D}$ of $\overline{\pi \mathbf{C}}$ such that $\overline{\pi \mathbf{C}} \setminus \mathbf{D} \subseteq \pi \mathbf{C}$. A Zariski structure is said to be \textbf{presmooth} if for any closed irreducible subsets $\mathbf{C}_{1}, \mathbf{C}_{2}$ of $\mathbf{X}^{n}$ the dimension of any irreducible component of $\mathbf{C}_{1} \cap \mathbf{C}_{2}$ is greater than or equal to
\[ \dim \mathbf{C}_{1} + \dim \mathbf{C}_{2} - \dim \mathbf{X}^{n} \]
\end{definition} A natural candidate for a morphism $f: \mathbf{X} \rightarrow \mathbf{Y}$ of Zariski structures is a function inducing a continuous map on $\mathbf{X}^{n}$ for every $n$. Thus we have a category of Zariski structures with these morphisms, which we denote by $\mathsf{Zar}$. Some familiarity with algebraic geometry (in particular results on the dimensions of fibers, \cite{Har77}, II, Exercise 3.22) will allow one to conclude that varieties are Zariski structures, and are presmooth if the varieties are smooth. Hence the category of algebraic varieties is a subcategory of $\mathsf{Zar}$. Moreover, like schemes, Zariski structures have the advantage of being abstractly given and not as sitting in some ambient structure.
\\
\\
However, Zariski structures were not introduced to fulfill the purpose of being a model-theorists' answer to algebraic manifolds. Rather, they first appeared in \cite{HZ96} as a response to the failure of Zilber's trichotomy conjecture. Roughly speaking, the trichotomy conjecture proposed that the geometry of certain subsets of models (the so-called strongly minimal sets) fell into three mutually exclusive classes; such geometries were either trivial, linear, or that of an algebraically closed field. After the ingenious refutation of this conjecture by Hrushovski, it was natural to ask whether there was a natural class of structures for which the trichotomy conjecture did hold. One-dimensional Zariski structures \footnote{The definition of Zariski structures appearing in \cite{HZ96} is less general than Definition \ref{definition - Zariski structure - Noetherian} because it stipulates that the underlying set $\mathbf{X}$ is one-dimensional in a suitable model-theoretic sense. When $\mathbf{X}$ is one-dimensional, both definitions coincide. We shall be dealing with Zariski structures where $\mathbf{X}$ has dimension $> 1$. Such Zariski structures will be referred to as higher-dimensional.} turned out to be such a class. For our purposes, two aspects of the work in \cite{HZ96} are particularly important. Firstly, as already mentioned, projective algebraic curves are Zariski structures. Secondly, there are one-dimensional Zariski structures which are demonstrably not projective curves but are certain finite covers of them. These structures, rather than turning out to be mathematical pathologies, can be taken to be geometric objects corresponding to certain noncommutative algebras. In this regard, we mention the paper \cite{ZS09} as providing an example of such a one-dimensional Zariski structure corresponding to a physically important algebra, namely the Heisenberg algebra. In short, Zariski structures corresponding to noncommutative algebras do exist that can be distinguished from projective curves by their geometry not being reducible to them.
\\
\\
Given that there are one-dimensional \textbf{non-classical} Zariski structures (those not arising from algebraic curves) and that these correspond to certain noncommutative algebras, it is natural to expect that there are higher-dimensional Zariski structures corresponding to other noncommutative algebras. The paper \cite{Zil06} establishes exactly this: that non-classical Zariski structures can be associated to a class of noncommutative algebras, described in the paper as `quantum algebras at roots of unity'. The definition of such algebras can be simplified with some knowledge of ring theory and the results of \cite{Zil06} shall be discussed in due course. The results of \cite{ZS09} and \cite{Zil06} provide sufficient evidence to propose the following conjecture.
\begin{conjecture}
\label{conjecture - Zilber program}
Let $k$ be an algebraically closed field. Then there is a commutative diagram of functors
\[ \xymatrix{ (\mathsf{CAlg}(k)_{fg, int})^{op} \ar[r] \ar@{^{(}->}[d] & \mathsf{Zar}^{c} \ar@{^{(}->}[d] \\ \mathsf{Alg}(k)^{op} \ar[r] & \mathsf{Zar} } \]
\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c|c|}
$\mathsf{CAlg}(k)_{fg, int}$ & Finitely generated, commutative $k$-algebras that are domains \\
$\mathsf{Alg}(k)$ & $k$-algebras \\
$\mathsf{Zar}^{c}$ & Classical Zariski structures \\
\end{tabular}
\end{center}
\end{table} where the functor $\mathsf{Alg}(k)^{op} \rightarrow \mathsf{Zar}$ is an equivalence of categories.
\end{conjecture} The conjectural functor is, of course, $\mathsf{Alg}(k)^{op} \rightarrow \mathsf{Zar}$ and the work in this paper has the construction of this functor as a focal point. To date, a general means of constructing a suitable such functor has not been found. As far as the author's work is concerned, the most fruitful modus operandi (both conceptually and pragmatically) has been the following:
\begin{enumerate}
\item Rather than attempting to construct a general functor $\mathsf{Alg}(k)^{op} \rightarrow \mathsf{Zar}$, isolate an interesting subcategory of $k$-algebras $\mathsf{A}$ that contains a suitably large subcategory $\mathsf{B}$ of the category of affine commutative $k$-algebras that are domains.
\item Constrain the algebraic characterisation of $\mathsf{A}$ by those additional assumptions necessary to associate an $\mathcal{L}_{A}$-structure $\nSpec A$ to every object $A$ of $\mathsf{A}$ (where the language $\mathcal{L}_{A}$ depends on the object $A$). The structure $\nSpec A$ should be a moduli space for certain representations of $A$, preferably those $A$-modules that `generate' an interesting subcategory of the category of all left $A$-modules, ${}_{A} \mathsf{Mod}$.
\item Carry out an analysis of the definable subsets of $\nSpec A$ and conclude that $\nSpec A$ is a Zariski structure.
\item Extend the correspondence $A \mapsto \nSpec A$ to a functor $\nSpec: \mathsf{A}^{op} \rightarrow \mathsf{Zar}$ and verify that the following diagram commutes:
\[ \xymatrix{ \mathsf{B}^{op} \ar[r] \ar@{^{(}->}[d] & \mathsf{Zar}^{c} \ar@{^{(}->}[d] \\ \mathsf{A}^{op} \ar[r]_{\nSpec} & \mathsf{Zar} } \]
\item Finally analyze the relationship between $\nSpec A$ and ${}_{A} \mathsf{Mod}$ for every object $A$ of $\mathsf{A}$.
\end{enumerate} It is appropriate to be a little bit more specific about syntax and related issues at this juncture. Let $\mathsf{A}'$ be our category of $k$-algebras obtained after appropriate constraints are introduced in 2. Then to each algebra $A$ in $\mathsf{A}'$ we associate an $\mathcal{L}_{A}$-theory $T_{A}$ that is first-order axiomatizable. The structure $\nSpec A$ is then taken to be a large saturated model (universal domain) of $T_{A}$. In much the same way that the language of rings naturally axiomatizes the theory of algebraically closed fields, the language $\mathcal{L}_{A}$ is chosen to be natural for $T_{A}$. Moreover, the Zariski structure obtained on $\nSpec A$ should respect the theory $T_{A}$, in the sense that it arises from a suitable quantifier-elimination result. This particular methodology is uniquely model-theoretic and results in a topology that is rather descriptive. Crucial to this is the insistence in 2 on the structure $\nSpec A$ being a moduli space for a class of $A$-modules. Thus $\nSpec A$ incorporates the internal structure of the modules explicitly into the geometry.
\\
\\
We now summarize the contents of this paper. We deal with a class of algebras which are described as \textbf{equivariant}. The choice of terminology here is motivated by important structures appearing in geometric representation theory; namely those line bundles $L$ over a variety $V$ endowed with an action of an algebraic group $G$, such that
\[ \mbox{for all $g \in G$, $g(L_{x}) = L_{gx}$ and $g: L_{x} \rightarrow L_{gx}$ is a linear isomorphism}\] where $L_{x}$ denotes the fiber of $L$ at $x \in V$. Such line bundles are said to be \textbf{$G$-equivariant} (see \cite{RTT07}). The structure corresponding to the Heisenberg algebra introduced in \cite{ZS09} looked, at least superficially, to be an equivariant line bundle. However, further examination revealed some crucial differences. Firstly, there was no claim on local triviality. Secondly, whereas certain operators ($\mathbf{a}$ and $\mathbf{a}^{\dagger}$ for those familiar with the paper) did move between fibers in a manner that introduced an action of a group on the base, these two operators themselves didn't generate a group because they were not mutually inverse. It is the author's contention (and no doubt that of B. Zilber also) that such phenomena are characteristic of `quantum' objects. Additional examples worked out in a similar vein (the quantum $2$-torus by Zilber, $U_{q}(\mathfrak{sl}_{2}(k))$ for generic $q$ by the author) suggested that an appropriate formalism could be found that treated all of these examples (and more) collectively. We discuss these examples in Section 2 and the category of equivariant algebras (denoted $\mathsf{Equiv}(k)$) is defined in Section 3. It is not a full subcategory of $\mathsf{Alg}(k)$ and an appropriate notion of a morphism in this category is given. We also show that given an object $A$ of $\mathsf{Equiv}(k)$, we can associate a first-order $\mathcal{L}_{A}$-theory $T_{A}$ to $A$.
\\
\\
Sections 4 and 5 are devoted to the model theory of $T_{A}$ under an additional technical assumption on $T_{A}$ ($\Gamma$-rigidity), which the key examples mentioned in Section 1 are shown to satisfy. Uncountable categoricity and quantifier elimination results are established thus leading to the expected consequences for the category of definable subsets; namely that every definable subset is constructible for an appropriate topology on models. With this topology, an appropriate dimension theory turns each model into a Zariski structure. The method of technical analysis is that of \cite{Zil06}. It is worth remarking that the condition of $\Gamma$-rigidity encapsulates precisely what is required for $T_{A}$ to possess a rich structure theory, i.e. it is only for $\Gamma$-rigid $T_{A}$ that models are Zariski structures.
\\
\\
The final section concludes our excursion into equivariant algebras and their associated Zariski structures with the expected construction of a functor $\nSpec$. Appendices are provided summarizing relevant background material for Lie algebras and Hopf algebras. To the author's knowledge, the structures $\nSpec A$ for general equivariant $A$ have no precedent. They are also unusual in being able to assign to certain noncommutative algebras parametrized at a generic parameter a bone fide topological space, in contrast to the approaches to noncommutative algebraic geometry surveyed.
\section{Some examples}
Three examples of noncommutative algebras are discussed, occupying a central place in physics, the theory of quantum groups and noncommutative geometry respectively.
\subsection{Weyl Algebra}
Recall that for a commutative ring $R$, the \textbf{$n$-th Weyl algebra} $A_{n}(R)$ (for $n > 0$) is defined to be
\[ R\langle x_{1}, \dots x_{n}, \partial_{1}, \dots, \partial_{n} \rangle/I \] where $I$ is the ideal generated by
\[ \partial_{i} x_{j} - x_{j} \partial_{i} - \delta_{ij} \qquad x_{i}x_{j} - x_{j}x_{i} \qquad \partial_{i}\partial_{j} - \partial_{j} \partial_{i} \qquad \mbox{ for } 1 \leq i,j \leq n \] We shall concentrate on the first Weyl algebra $A_{1}(k)$ for $k$ an algebraically closed field of characteristic $0$. Firstly, we note that $A_{1}(k)$ can be redefined in terms of three operators $H, \mathbf{a}, \mathbf{a}^{\dagger}$:
\[ \mathbf{H} = \frac{1}{2}(x^{2} - \partial_{x}^{2}) \qquad \mathbf{a} = \frac{1}{\sqrt{2}}(x + \partial_{x}) \qquad \mathbf{a}^{\dagger} = \frac{1}{\sqrt{2}}(x - \partial_{x}) \] Because we are working in an arbitrary algebraically closed field, $\sqrt{2}$ represents an element that squares to $2$. The operator $\mathbf{a}^{\dagger}$ is a formal adjoint to $\mathbf{a}$, as an element of the differential ring $A_{1}(k)$.
\begin{proposition}
\label{proposition - first Weyl algebra - Hamiltonian form}
The following relations hold between $H, \mathbf{a}, \mathbf{a}^{\dagger}$:
\begin{enumerate}
\item $\mathbf{a}^{\dagger}\mathbf{a} = \mathbf{H} - 1/2$, $\mathbf{a}\mathbf{a}^{\dagger} = \mathbf{H} + 1/2$.
\item $[\mathbf{a}, \mathbf{a}^{\dagger}] = \mathbf{a}\mathbf{a}^{\dagger} - \mathbf{a}^{\dagger}\mathbf{a} = 1$.
\item Putting $\mathbf{N} = \mathbf{H} - 1/2$, we have $[\mathbf{N}, \mathbf{a}^{\dagger}] = \mathbf{a}^{\dagger}$ and $[\mathbf{N}, \mathbf{a}] = - \mathbf{a}$. Thus we also have
\[ [\mathbf{H}, \mathbf{a}^{\dagger}] = \mathbf{a}^{\dagger} \qquad [\mathbf{H}, \mathbf{a}] = -\mathbf{a} \]
\end{enumerate}
\end{proposition}
\begin{proof} $1$ and $2$ are easy verification. For $3$, use the fact that for any three operators $A,B,C$, we have the relation $[A,BC] = [A,B]C+ B[A,C]$.
\end{proof} In \cite{ZS09}, a Zariski structure was associated to the Heisenberg algebra $k\langle \mathbf{P}, \mathbf{Q}\rangle/I$ where $I$ is generated by $[\mathbf{P},\mathbf{Q}] + i$. Similarly this algebra was also re-expressed in terms of operators $\mathbf{H}, \mathbf{a}, \mathbf{a}^{\dagger}$ defined slightly differently to the above, namely by
\[ \mathbf{H} = \frac{1}{2}(\mathbf{P}^{2} + \mathbf{Q}^{2}) \qquad \mathbf{a} = \frac{1}{\sqrt{2}}(\mathbf{P} - i\mathbf{Q}) \qquad \mathbf{a}^{\dagger} = \frac{1}{\sqrt{2}}(\mathbf{P} + i \mathbf{Q}) \] The relations satisfied by these are, however, the same as in Proposition \ref{proposition - first Weyl algebra - Hamiltonian form}. The structure we define below is different to that of \cite{ZS09}; indeed the following structure originally appeared as a quotient of an initial (also Zariski) structure in that paper. The latter was important insofar as it provided another example of a one-dimensional Zariski geometry (a finite cover of the projective line) not definable in an algebraically closed field. For our purposes, we can start directly with the quotient.
\begin{definition}
\label{definition - first Weyl algebra - associated theory}
We consider a two-sorted language $\mathcal{L}_{A_{1}} = (k, L, \pi, \mathbf{E}, \mathbf{H}, \mathbf{a}, \mathbf{a}^{\dagger})$ where
\begin{enumerate}
\item $\pi: L \rightarrow k$ and $\mathbf{H}, \mathbf{a}, \mathbf{a}^{\dagger}: L \rightarrow L$ are maps.
\item $\mathbf{E} \subseteq L \times k$ is a relation.
\item The sort $k$ has the language of rings. The sort $L$ comes equipped with
\begin{itemize}
\item a map $+: L \times L \rightarrow L$ which is interpreted as addition of elements in each $\pi^{-1}(x)$ for $x \in k$
\item a map $\cdot: k \times L \rightarrow L$ which is interpreted as scalar multiplication in each fiber $\pi^{-1}(x)$.
\end{itemize}
\end{enumerate}The $\mathcal{L}_{A_{1}}$-theory $T_{A_{1}}$ says the following:
\begin{enumerate}
\item $k$ is an algebraically closed field of characteristic $0$.
\item $\pi: L \rightarrow k$ is a surjective map; each fiber $\pi^{-1}(x)$ for $x \in k$ is a one-dimensional $k$-vector space.
\item For each $x \in k$, the subset $\mathbf{E}(L,x)$ is non-empty and $\mathbf{E}(L,x) \subseteq \pi^{-1}(x)$.
\item We fix an $l \in \mathbb{Z}$, $l > 0$, $l$ even. If $\Gamma[l]$ denotes the group of $l$-th roots of unity of $k$, then there is a free and transitive action of $\Gamma[l]$ on $\mathbf{E}(L,x)$ induced by the vector space action on the fiber $\pi^{-1}(x)$.
\item The map $\mathbf{H}$ is linear on each fiber and satisfies the following axiom
\[ (\forall v \in \pi^{-1}(x))(\mathbf{H}v = xv) \]
\item The maps $\mathbf{a}$ and $\mathbf{a}^{\dagger}$ are linear and move between fibers according to the following axiom:
\[ (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow (\exists v' \in \pi^{-1}(x+1))(\exists y \in k)(y^{2} = x \wedge \mathbf{a}^{\dagger}v = yv' \wedge \mathbf{a}v' = yv))\]
\end{enumerate}
\end{definition} Let $(L,k) \models T_{A_{1}}$. Then $L$ is a `line bundle' over the base $k$, though we do not claim local triviality. Each fiber $\pi^{-1}(x)$ is an $x$-eigenspace for $\mathbf{H}$. The elements $\mathbf{E}(L,x) \subseteq \pi^{-1}(x)$ are to be regarded as normal basis elements of the fiber $\pi^{-1}(x)$ which can be permuted by the group of $l$-th roots of unity $\Gamma[l]$. This setup serves as a discrete (and algebraic) model for a well-known phenomenon encountered when dealing with normed vector spaces. If $V$ is a normed vector space over $\mathbb{C}$ and $v \in V$ is an element of norm $1$, then so is $\alpha v$ for any $\alpha \in \mathbb{C}$ such that $|\alpha| = 1$. Of course, in this case there is an infinite group $S^{1}$ acting on the elements in $V$ of norm $1$. It will be seen, in the next chapter, that being able to permute the normal basis elements by a finite group is also essential for a decent structure theory for $T_{A_{1}}$.
\begin{proposition}
\label{proposition - first Weyl algebra - models are representations}
Let $(L,k) \models T_{A_{1}}$. Then $(L,k)$ is a representation of $A_{1}(k)$.
\end{proposition}
\begin{proof} Let $e \in \pi^{-1}(x)$ such that $\mathbf{E}(e,x)$ holds. Then there is $e' \in \pi^{-1}(x+1)$ such that
\[ \mathbf{H}\mathbf{a}^{\dagger}e = y\mathbf{H}e' = (x+1)ye' \qquad y^{2} = x \] But
\[ (x+1)ye' = (x+1)\mathbf{a}^{\dagger}e = \mathbf{a}^{\dagger}\mathbf{H}e + \mathbf{a}^{\dagger}e \] Thus $[\mathbf{H}, \mathbf{a}^{\dagger}]e = \mathbf{a}^{\dagger}e$. Similarly, we obtain that $[\mathbf{H}, \mathbf{a}]e = -\mathbf{a}e$. Now
\[ \mathbf{a}\mathbf{a}^{\dagger}e = y\mathbf{a}e' = xe \] whereas
\[ \mathbf{a}^{\dagger}\mathbf{a}e = z\mathbf{a}^{\dagger}e'' = (x-1)e \] where $z^{2} = x-1$ and $e'' \in \pi^{-1}(x-1)$. Thus $[\mathbf{a},\mathbf{a}^{\dagger}]e = e$ as required.
\end{proof}
\subsection{$U_{q}(\mathfrak{sl}_{2}(k))$ for generic $q$}
Let $k$ be an algebraically closed field of characteristic $0$, $q \in k$ with $q \neq 0, \pm 1$. The \textbf{quantized enveloping algebra} of $sl_{2}(k)$, denoted $U_{q}(sl_{2}(k))$, is defined to be the $k$-algebra with generators $E,F, K^{\pm 1}$ subject to the following relations
\[ KEK^{-1} = q^{2}E \qquad KFK^{-1} = q^{-2}F \qquad EF - FE = \frac{K - K^{-1}}{q - q^{-1}} \] along with $KK^{-1} = K^{-1}K = 1$. We associate a structure to this algebra when $q$ is a generic parameter; namely when $q$ is not a root of unity.
\begin{definition}
\label{definition - quantum sl2 generic - associated theory}
Consider the two-sorted language $\mathcal{L}_{q} = (k, L, \pi, \mathbf{E}, E, F, K^{\pm 1},q)$ where
\begin{enumerate}
\item $\pi: L \rightarrow k$ and $E,F,K^{\pm 1}: L \rightarrow L$ are maps.
\item $\mathbf{E} \subseteq L \times k^{*}$ is a relation.
\item $q$ is a constant from the sort $k$.
\end{enumerate} The sorts $k,L$ are equipped with the same language as in condition 3 of Definition \ref{definition - first Weyl algebra - associated theory}. The first-order $\mathcal{L}_{q}$-theory $T_{q}$ states the following:
\begin{enumerate}
\item $k$ is an algebraically closed field of characteristic $0$.
\item $q^{n} \neq 1$ for every $n \in \mathbb{N}$.
\item The map $\pi: L \rightarrow k^{*}$ is surjective and each fiber $\pi^{-1}(x)$ for $x \in k^{*}$ is a one-dimensional $k$-vector space.
\item For each $x \in k^{*}$, the set $\mathbf{E}(L,x)$ is non-empty and $\mathbf{E}(L,x) \subseteq \pi^{-1}(x)$.
\item Fix an $l \in \mathbb{Z}$, $l > 0$, $l$ even. If $\Gamma[l]$ denotes the group of $l$-th roots of unity of $k$, then there is a free and transitive actions of $\Gamma[l]$ on $\mathbf{E}(L,x)$ induced by the vector space action on the fiber $\pi^{-1}(x)$.
\item The $K^{\pm 1}$ act on each fiber according to the following axiom:
\[ (\forall v \in \pi^{-1})(\overline{x}) (Kv = xv \wedge K^{-1}v = x^{-1}v ) \] The maps $K^{\pm 1}$ are linear.
\item The linear maps $E$ and $F$ move between the fibers according to the following axiom:
\[ \begin{array}{ll} (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow & (\exists v' \in \pi^{-1}(q^{2}x)(\exists y \in k) \\ & (y^{2} = x \wedge Ev = \lambda(y)v' \wedge Fv' = -\lambda(qx)v)) \end{array}\] where $\lambda: L \rightarrow k$ is defined by
\[ \lambda(x) = \frac{y^{-1} + y}{q - q^{-1}} \]
\end{enumerate}
\end{definition} If $(L,k) \models T_{q}$ then each fiber $\pi^{-1}(x)$ is an eigenspace for $K$ with eigenvalue $x$. Each eigenspace contains a finite set of normal basis elements selected by $\mathbf{E}$, and permuted by $\Gamma[l]$.
\begin{proposition}
\label{proposition - quantum sl2 generic - models are representations}
Let $(L,k) \models T_{q}$. Then $(L,k)$ is a representation of $U_{q}(\mathfrak{sl}_{2}(k))$.
\end{proposition}
\begin{proof} Consider $e \in \pi^{-1}(x)$ such that $\mathbf{E}(e,x)$ holds. Then there are $y$ such that $y^{2} = x$ and $e' \in \pi^{-1}(q^{2}x)$ such that
\[ KEe = \lambda(y) Ke' = \lambda(y)q^{2}xe' \] But
\[ EKe = xEe = x\lambda(y)e' \] Thus $KEe = q^{2}EKe$. Similarly $KFe = q^{-2}FKe$. We shall now adopt the more intuitive notation $x^{1/2}$ for the element $y$ such that $Ee = \lambda(y)e'$. Thus
\[ Ee = \frac{x^{-1/2} + x^{1/2}}{q - q^{-1}}e' \] and by the linearity of $F$,
\[ FEe = - \frac{(x^{-1/2} + x^{1/2})(q^{-1}x^{-1/2} + qx^{1/2})}{(q - q^{-1})^{2}} e \] Whereas applying $F$ first,
\[ Fe = - \frac{x^{-1/2} + x^{1/2}}{q - q^{-1}}e'' \qquad e'' \in \pi^{-1}(q^{-2}x) \] hence
\[ EFe = - \frac{(x^{-1/2} + x^{1/2})(qx^{-1/2} + q^{-1}x^{1/2})}{(q - q^{-1})^{2}}e \] After some expansion and rearrangement,
\[ (EF - FE)e = \frac{x - x^{-1}}{q - q^{-1}}e \] as required.
\end{proof}
\subsection{Quantum torus}
Our final example will be a certain multi-parameter quantum torus $\mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$ where the parameters $\mathbf{q}$ will depend on some generic $q$. Recall that this is the $k$-algebra with generators $\{\mathbf{U}_{i}^{\pm 1}: 1 \leq i \leq n\}$ subject to the relations
\[ \mathbf{U}_{i}\mathbf{U}_{i}^{-1} = \mathbf{U}_{i}^{-1}\mathbf{U}_{i} = 1 \qquad \mathbf{U}_{i} \mathbf{U}_{j} = q_{ij} \mathbf{U}_{j}\mathbf{U}_{i} \qquad \mbox{ for } i < j \] We shall consider the following specific parameters
\[ q_{ij} = q^{j-i} \] which are best visualized as being upper triangular elements of a multiplicatively anti-symmetric $n \times n$ matrix:
\[ \left( \begin{array}{lllll} * & q & q^{2} & \dots & q^{n-1}
\\ & * & q& \dots & q^{n-2}
\\ \vdots & & \ddots & & q
\\ & & & & * \end{array} \right) \] The base of our line bundle will parametrize eigenvalues of $\mathbf{U}_{1}$ and the remaining operators will move between fibers. We eliminate some linguistic preliminaries from the following definition, which should now be clear by referring to Definitions \ref{definition - first Weyl algebra - associated theory} and \ref{definition - quantum sl2 generic - associated theory}.
\begin{definition}
\label{definition - quantum torus generic - associated theory}
We work with a two-sorted language $\mathcal{L}_{q} = (k,L,\pi, q, \mathbf{E}, \mathbf{U}_{i}^{\pm 1}: 1 \leq i \leq n)$. The $\mathcal{L}_{q}$-theory $T_{q}$ says the following:
\begin{enumerate}
\item $k$ is an algebraically closed field of characteristic $0$.
\item $q^{n} \neq 1$ for every $n \in \mathbb{N}$.
\item $\pi: L \rightarrow k^{*}$ is a surjective map and each fiber $\pi^{-1}(x)$ is a one-dimensional vector space for $x \in k$.
\item For each $x \in k$, $\mathbf{E}(L,x)$ is non-empty and $\mathbf{E}(L,x) \subseteq \pi^{-1}(x)$.
\item Let $l$ be any positive integer (not necessarily odd). Then there is a free and transitive action of $\Gamma[l]$ on each $\mathbf{E}(L,x)$ induced by the vector space structure on $\pi^{-1}(x)$.
\item $\mathbf{U}_{1}^{\pm 1}$ are linear and we have
\[ (\forall v \in \pi^{-1}(x))(\mathbf{U}_{1}v = xv \wedge \mathbf{U}^{-1}_{1}v = x^{-1}v ) \]
\item The linear maps $\mathbf{U}_{2}^{\pm 1}, \dots \mathbf{U}_{n}^{\pm 1}$ move between fibers according to
\[ (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow \bigwedge_{i = 2}^{n} (\exists v_{i} \in \pi^{-1}(q^{i-1}x))(\mathbf{E}(v, q^{i-1}x) \wedge \mathbf{U}_{i}v = xv_{i} \wedge \mathbf{U}_{i}^{-1}v_{i} = x^{-1}v)) \]
\item For each $i < j$ with $i \neq 1$ we have the following axiom:
\[ (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow \mathbf{U}_{i}\mathbf{U}_{j} v = q^{j-i}\mathbf{U}_{j} \mathbf{U}_{i} v) \]
\end{enumerate}
\end{definition} There are some points of difference with the previous examples worth noting. The first is that we have allowed our group $\Gamma[l]$ to be finite and cyclic of any order. The reasons for this are again model-theoretic. The second point is axiom $8$ stipulating explicitly some good behaviour of basis elements with respect to the relations satisfied. This good behaviour was actually coded into the definitions of how the operators act between fibers in the previous two examples.
\begin{proposition}
\label{proposition - quantum torus generic - models are representations}
Let $(L,k) \models T_{q}$. Then $(L,k)$ is a representation of $\mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$.
\end{proposition}
\begin{proof} Let $e \in \pi^{-1}(x)$ for some $x \in k^{*}$. By the axioms there is a $e_{i} \in \pi^{-1}(q^{i-1}x)$ such that $\mathbf{U}_{i}e = xe_{i}$. Thus
\[ \mathbf{U}_{1} \mathbf{U}_{i}e = x\mathbf{U}_{1}e_{i} = x^{2}q^{i-1}e_{i} \] whereas
\[ \mathbf{U}_{i}\mathbf{U}_{1}e = x\mathbf{U}_{i}e = x^{2}e_{i} \] But $q^{i-1} = q_{1i}$, thus $\mathbf{U}_{1}\mathbf{U}_{i}e = q_{1i}\mathbf{U}_{i}\mathbf{U}_{1}e$. For $i < j$ and $i \neq 1$, we have $\mathbf{U}_{i}\mathbf{U}_{j} e = q^{j-i}\mathbf{U}_{j}\mathbf{U}_{i}e$ by definition.
\end{proof}
\begin{remark} Note that for $i < j$ and $i \neq 1$, if $\mathbf{E}(e,x)$ holds then
\[ \mathbf{U}_{i}\mathbf{U}_{j}e = x\mathbf{U}_{i}e_{j} = q^{j - 1}x^{2}e_{ji} \] for some $e_{j} \in \pi^{-1}(q^{j-1}x)$ and $e_{ji} \in \pi^{-1}(q^{i+j - 2}x)$. On the other hand
\[ \mathbf{U}_{j}\mathbf{U}_{i}e = x\mathbf{U}_{j}e_{i} = q^{i-1}x^{2}e_{ij} \] where $e_{i} \in \pi^{-1}(q^{i-1}x)$ and $e_{ij} \in \pi^{-1}(q^{i+j - 2}x)$. Thus the stipulation that $\mathbf{U}_{i}\mathbf{U}_{j} e = q^{j-i}\mathbf{U}_{j}\mathbf{U}_{i}e$ implies that $e_{ij} = e_{ji}$.
\end{remark}
\section{Equivariant Algebras}
We now define the class of equivariant algebras over an algebraically closed field $k$ of characteristic $0$. As indicated in the introduction, there are two parts to this definition. Firstly, we have to isolate a suitable class of algebras with an algebraic characterization, a class which we call `semi-equivariant'. An equivariant algebra is then defined to be a semi-equivariant algebra satisfying some additional (though rather cumbersome) assumptions. The reader is referred to Appendix \ref{appendix - Hopf algebras} for basic definitions and notations concerning Hopf algebras.
\begin{definition}
\label{definition - semi-equivariant algebras}
A prime $k$-algebra $A$ is said to be \textbf{semi-equivariant} if
\begin{enumerate}
\item There is a maximal commutative affine $k$-subalgebra $H$ of $A$ that is a Hopf algebra.
\item $A$ is generated as a $k$-algebra by the generators of $H$ and finitely many eigenvectors $\mathbf{U}_{1}, \dots \mathbf{U}_{n}$ of the left adjoint action of $H$ on $A$; namely the action defined by
\[ h \cdot a = \sum_{(h)} h'aS(h'') \qquad \mbox{ for all $a \in A$ and $h \in H$} \]
\item There are generators $h_{1}, \dots h_{m}$ of $H$ such that $A$ has a presentation in terms of the $h_{i}$ and $\mathbf{U}_{j}$ and finitely many relations between them. All relations not expressing the adjoint action of $h_{i}$ on $\mathbf{U}_{j}$ have the form
\[ c \prod_{k=0}^{p -1} \mathbf{U}_{i_{p -k}} - d \prod_{k=0}^{q-1} \mathbf{U}_{i_{q -k}} = f(h_{1}, \dots, h_{m}) \] where $c,d \in k$ and $f$ is a polynomial over $k$.
\end{enumerate}
\end{definition} If $H$ and $\mathbf{U}_{i}$ exist as above, then $A$ is said to be semi-equivariant with respect to $H$ and the elements $\mathbf{U}_{1}, \dots \mathbf{U}_{n}$. It should be noted that although one typically defines the adjoint action of a Hopf algebra on itself, the definition makes sense in the current setting, thus turning $A$ into a $H$-module.
\\
\\
Definition \ref{definition - semi-equivariant algebras} takes its inspiration from the basic result that a finite dimensional complex semisimple Lie algebra $\mathfrak{g}$ possesses a Cartan decomposition (see Appendix \ref{appendix - Lie theory}). Recall that if $\mathfrak{g}$ is such a Lie algebra then there is a direct sum decomposition
\[ \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_{\alpha} \] where $\mathfrak{h}$ is an abelian Lie subalgebra (the Cartan subalgebra) and the $\mathfrak{g}_{\alpha}$ are eigenspaces of the adjoint action of $\mathfrak{g}$ on itself. Regarding $U(\mathfrak{g})$ (the universal enveloping algebra of $\mathfrak{g}$) as a Hopf algebra, the Lie algebra and Hopf algebra adjoint actions agree. Thus, heuristically, Definition \ref{definition - semi-equivariant algebras} says that a semi-equivariant algebra $A$ has a `generalized Cartan decomposition' where the eigenvectors of the adjoint action satisfy some manageable relations amongst themselves.
\subsection{Towards equivariant algebras}
Let $A$ be a semi-equivariant $k$-algebra. Because $A$ is prime, $H$ is a domain and by the assumption that $H$ is an affine $k$-algebra, it is therefore the coordinate ring of an affine variety $V$. Suppose that $V \subseteq k^{m}$. Clearly there is a bijective correspondence between points of $V$ and characters ($k$-algebra homomorphisms) on $H$, and we denote the character corresponding to $x \in V$ by $\chi_{x}$.
Let $L_{x}$ be a one-dimensional $k$-vector space endowed with the structure of a $H$-module by the character $\chi_{x}$, i.e.
\[ hv = \chi_{x}(h)v \qquad \mbox{ for all $h \in H$ and $v \in L_{x}$} \] The $H$-modules $L_{x}$ will form the fibers of a line bundle over $V$; thus we form the disjoint union
\[ L = \coprod_{x \in V} L_{x} \] and define the surjective map $\pi: L \rightarrow V$ by $\pi(v) = x$ if $v \in L_{x}$.
\begin{lemma} $V$ is a group.
\end{lemma}
\begin{proof} This is a consequence of $H$ being a Hopf algebra. First note that if $\chi_{x}, \chi_{y}: H \rightarrow k$ are the characters corresponding to the points $x,y \in V$ then $\chi_{x} \otimes \chi_{y}$ is a character on $H \otimes H$ by
\[ (\chi_{x} \otimes \chi_{y})(h_{1} \otimes h_{2}) = \chi_{x}(h_{1})\chi_{y}(h_{2}) \qquad h_{1}, h_{2} \in H\] extended to an algebra homomorphism. Now $\chi = (\chi_{x} \otimes \chi_{y}) \circ \Delta$ is a character on $H$ and the kernel of $\chi$ is a maximal ideal of $H$, thus corresponding to a point $z \in V$. This allows us to define a map $\cdot: (y,z) \mapsto z$. The coassociativity of $\Delta$ easily implies that $\cdot$ is associative. Similarly, we define $\chi_{x^{-1}} = \chi_{x} \circ S$ and put $x^{-1}$ as the point in $V$ corresponding to the kernel of $\chi_{x^{-1}}$.
\end{proof} Because $A$ is semi-equivariant, we have
\[ h \cdot \mathbf{U}_{i} = \chi_{i}(h) \mathbf{U}_{i} \qquad \mbox{ for all } h \in H \] for some characters $\chi_{i}: H \rightarrow k$. In particular, we obtain constants $\alpha_{ji} = \chi_{i}(h_{j})$ such that $h_{j} \cdot \mathbf{U}_{i} = \alpha_{ji} \mathbf{U}_{i}$ for all $1 \leq j \leq m$.
\begin{lemma}
\label{lemma - semi-equivariant algebras - characters}
Let $v \in L_{x}$. Then
\[ \sum_{(h)} \chi_{x^{-1}}(h''_{j})h'_{j} \mathbf{U}_{i} v = \alpha_{ji} \mathbf{U}_{i} v \]
\end{lemma}
\begin{proof} By $h_{j} \cdot \mathbf{U}_{i} = \alpha_{ji} \mathbf{U}_{i}$ we obtain that
\[ \sum_{(h)} h'_{j}\mathbf{U}_{i}S(h''_{j})v = \alpha_{ji}\mathbf{U}_{i}v \] But $S(h''_{j})$ acts on $v$ by the scalar $\chi_{x^{-1}}(h''_{j})$ and the result follows.
\end{proof}
\begin{corollary}
\label{corollary - semi-equivariant algebras - characters}
Suppose that for each $1 \leq j \leq m$, $\Delta(h_{j}) = \sum_{(h)} h'_{j} \otimes h''_{j}$ is such that
\begin{enumerate}
\item $h'_{j} = h_{j}$ or $1$ for every element $h'_{j}$ in the sum.
\item There is at least one $h'_{j}$ with $h_{j}' = h_{j}$.
\end{enumerate} Then there are regular maps $r_{j},s_{j}: V \rightarrow k$ such that for each $v \in L_{x}$,
\[ h_{j}\mathbf{U}_{i}v = \frac{\alpha_{ji} - r_{j}(x)}{s_{j}(x)} \mathbf{U}_{i} v \]
\end{corollary}
\begin{proof} Immediate from Lemma \ref{lemma - semi-equivariant algebras - characters}.
\end{proof} Assuming that the coproduct satisfies the assumptions of Corollary \ref{corollary - semi-equivariant algebras - characters}, we have functions $\eta_{ji}(x) = (\alpha_{ji} - r_{j}(x))/s_{j}(x)$. Given $x \in V$, put $\eta_{i}(x) = (\eta_{1i}(x), \dots, \eta_{mi}(x))$ for each $1 \leq i \leq n$. Then for each $i$, $\eta_{i}: V \rightarrow V$ because $\eta_{i}(x)$ defines a character on $H$ for each $x \in V$. We obtain that $\mathbf{U}_{i}: L_{x} \rightarrow L_{\eta_{i}(x)}$ for every $x \in V$ and $1 \leq i \leq n$. Let $\Pi$ be the semigroup generated by the $\eta_{i}$ (under composition). Thus $\Pi$ acts on $V$ in an obvious way. We shall, henceforth, treat the elements of $\Pi$ as functions on $V$ and adopt the usual convention for composition of functions; namely for $\eta_{1}, \eta_{2} \in \Pi$, $\eta_{1}\eta_{2} = \eta_{1} \circ \eta_{2}$ (apply $\eta_{2}$ first, then $\eta_{1}$).
\\
\\
We can now define the class of equivariant algebras. As stated previously, the definition has the sole purpose of narrowing down the class of semi-equivariant algebras to those which have a suitably geometric first-order definable space which is a representation of $A$.
\begin{definition}
\label{definition - equivariant algebras}
We define a semi-representable algebra $A$ to be \textbf{equivariant} if
\begin{enumerate}
\item For each $1 \leq j \leq m$, $\Delta(h_{j}) = \sum_{(h)} h'_{j} \otimes h'_{j}$ is such that
\begin{enumerate}
\item $h'_{j} = h_{j}$ or $1$ for each element in the sum.
\item $h'_{j} = h_{j}$ for at least one $h'_{j}$.
\end{enumerate}
\item $\Pi$ is a group such that for each $1 \leq i \leq n$, $\eta_{i}^{-1} = \eta_{j}$ for some $j \leq n$.
\item $V$ is defined over $\mathbb{Q}$.
\item There exist regular functions $\lambda_{i}: V \rightarrow k$ and polynomials
\[ P_{i}(x,y):= y^{n_{i}} - \mu_{i}(x) \qquad n_{i} \in \mathbb{N}, n_{i} > 0 \qquad \mu_{i} \in \Pi \] for each $1 \leq i \leq n$ such that for each relation of the form
\begin{equation} c\prod_{k = 0}^{p-1} \mathbf{U}_{i_{p-k}} - d\prod_{k=0}^{q-1} \mathbf{U}_{j_{q-k}} = f(h_{1}, \dots, h_{m}) \end{equation} we have
\begin{enumerate}
\item $\eta_{i_{p}}\dots \eta_{i_{1}} = \eta_{j_{q}} \dots \eta_{j_{1}}$ and $\eta_{j_{q}} \dots \eta_{j_{1}}= 1$ if $f \neq 0$.
\item
\begin{equation} c \prod_{k=0}^{p-1}\lambda_{i_{p-k}}(y_{i_{p-k}}) - d \prod_{k=0}^{q-1}\lambda_{j_{q-k}}(y_{j_{q-k}}) = f(x) \end{equation} holding for every $x \in V$, where
\begin{enumerate}
\item $y_{i_{1}}$ (respectively $y_{j_{1}}$) is a solution to the polynomial equation $P_{i_{1}}(x,y_{i_{1}}) = 0$ (respectively $P_{j_{1}}(x,y_{j_{1}}) = 0$).
\item $y_{i_{p-k}}$ (respectively $y_{j_{q-k}}$) for $k < p-1$ is a solution to the polynomial equation
\[ P_{i_{p-k}}((\prod_{r > k} \eta_{i_{p-r}})(x), y_{i_{p-k}}) = 0 \qquad \left( P_{j_{q-k}}((\prod_{r>k}\eta_{j_{q-r}})(x), y_{j_{q-k}}) = 0 \right) \]\end{enumerate} \end{enumerate} Moreover, the roots $y_{i_{k}}$ can be chosen compatibly for all conditions of the form $(4.2)$ and for all $x \in V$.
\item The parameters appearing in all $\lambda_{i}, \eta_{i}$ and $f$, along with the constants $c,d$, are solutions to types over $\mathbb{Q}$.
\item The maps $\lambda_{i}$ are $\Gamma$-linear, where $\Gamma$ is the group of roots of unity of order $l$ for some $l > 0$ such that $n_{i} | l$ for all $1 \leq i \leq n$.
\end{enumerate}
\end{definition} The significance of many of these conditions will only become apparent when we commence doing some model theory, although $4$ can be presently motivated. If an algebra $A$ is semi-equivariant then the relations of the form $(4.1)$ may not have much to do with the adjoint action of $H$ on $A$. Condition $4$ effectively remedies this. We already know that $\mathbf{U}_{i}: L_{x} \rightarrow L_{\eta_{i}(x)}$ for every $x \in V$, $1 \leq i \leq n$. So $4$ (combined with $5$) states that we can define the action of the $\mathbf{U}_{i}$, respecting the way they move between fibers, in such a way that all of these relations are satisfied regardless of what fiber $L_{x}$ we start at. The use of the polynomials $P_{i}$ and the functions $\lambda_{i}$ is to allow a certain amount of definitional flexibility. For two of the preliminary examples considered (the Weyl algebra and $U_{q}(\mathfrak{sl}_{2}(k))$, this extra flexibility is necessary. We now define the category $\mathsf{Equiv}(k)$.
\begin{definition}
\label{definition - morphisms of equivariant algebras}
Let $A$ and $B$ be two equivariant $k$-algebras. A $k$-algebra homomorphism $\varphi: A \rightarrow B$ is defined to be \textbf{equivariant} if for any Hopf algebra $H$ such that $A$ is equivariant with respect to $H$ and elements $\mathbf{U}_{11}, \dots \mathbf{U}_{1n_{1}}$ of $A$, there is a Hopf subalgebra $H'$ of $B$ such that
\begin{enumerate}
\item $B$ is equivariant with respect to $H'$ and $\mathbf{U}_{21}, \dots, \mathbf{U}_{2n_{2}} \in B$.
\item $\varphi(\mathbf{U}_{1i})$ is a monomial in $\mathbf{U}_{21}, \dots \mathbf{U}_{2n_{2}}$ for each $i$.
\item $\varphi|_{H}: H \rightarrow H'$.
\end{enumerate}
\end{definition} Thus the category $\mathsf{Equiv}(k)$ is defined to consist of equivariant $k$-algebras and equivariant morphisms. We conclude this subsection with the following observation.
\begin{proposition}
\label{proposition - semi-equivariant algebras - epimorphic images}
Let $A$ be semi-equivariant. Then there is an equivariant $A'$ such that $A$ is an epimorphic image of $A'$.
\end{proposition}
\begin{proof} Say $A$ is equivariant with respect to $H$ (generated by $h_{1}, \dots, h_{m}$) and $\mathbf{U}_{1}, \dots \mathbf{U}_{n}$. Define $A'$ to have the same generators, to satisfy the relations of $A$ expressing the adjoint action of $H$ on the $\mathbf{U}_{i}$, and only those additional relations that satisfy equivariance. The result is now immediate.
\end{proof}
\subsection{Application to initial examples}
\label{subsection - initial examples are equivariant}
We now show that our initial examples are equivariant algebras.
\subsubsection{First Weyl algebra}
We consider $A = A_{1}(k)$. Put $H = k[\mathbf{H}]$ and endow $H$ with the Hopf algebra structure associated to universal enveloping algebras, namely
\[ \Delta(\mathbf{H}) = 1 \otimes \mathbf{H} + \mathbf{H} \otimes 1 \qquad \epsilon(\mathbf{H}) = 0 \qquad S(\mathbf{H}) = -\mathbf{H} \] The variety corresponding to $H$ is the affine line $k$. For each $x \in k$, we have the one-dimensional $H$-module $L_{x}$ given by the character $\chi_{x}: H \rightarrow k$ with $\chi_{x}(\mathbf{H}) = x$. Of course, the module $L_{x}$ is merely an $x$-eigenspace for $\mathbf{H}$. It is easy to see that $V$ is the group $(k,+)$. Indeed
\[ \chi_{xy}(\mathbf{H}) = [(\chi_{x} \otimes \chi_{y}) \circ \Delta](\mathbf{H}) = \chi_{x}(1)\chi_{y}(\mathbf{H}) + \chi_{x}(\mathbf{H})\chi_{y}(1) = x + y \] and
\[ \chi_{x^{-1}}(\mathbf{H}) = -\chi_{x}(\mathbf{H}) = -x \] For any $a \in A$,
\[ \mathbf{H} \cdot a = \mathbf{H}a - a\mathbf{H} = [\mathbf{H}, a] \] But $[\mathbf{H}, \mathbf{a}^{\dagger}] = \mathbf{a}^{\dagger}$ and $[\mathbf{H}, \mathbf{a}] = -\mathbf{a}$, thus $\mathbf{a}^{\dagger}$ and $\mathbf{a}$ are eigenvectors for the adjoint action of $H$ on $A$. Moreover, we have the additional relation
\[ \mathbf{a}\mathbf{a}^{\dagger} - \mathbf{a}^{\dagger} \mathbf{a} = 1 \] which is of the required form. Hence $A$ is semi-equivariant. By inspection, the coproduct has the required form. Now we determine $\Pi$. Suppose that $v \in L_{x}$. By Corollary \ref{corollary - semi-equivariant algebras - characters} we obtain
\[ \mathbf{H}\mathbf{a}^{\dagger}v = (x+1)\mathbf{a}^{\dagger}v \qquad \mathbf{H}\mathbf{a}v = (x-1)\mathbf{a}v \] Thus the semigroup $\Pi$ is generated by two functions:
\[ \eta_{\dagger}(x) = x+1 \qquad \eta(x) = x-1 \] Hence $\Pi = \mathbb{Z}$ is a group such that the inverse of $\eta_{\dagger}$ is $\eta$ (and vice versa). Furthermore, the parameters appearing $\eta_{\dagger}$ and $\eta$ are integral. For the relation $[\mathbf{a}, \mathbf{a}^{\dagger}] = 1$, we note that for every $x \in k$,
\[ (x^{1/2})^{2} - ((x-1)^{1/2})^{2} = 1 \] where $x^{1/2}$ denotes some $y$ such that $y^{2} = x$. So we put $\lambda(y) = \lambda_{\dagger}(y) = y$ and
\[ P(x,y) = y^{2} - \eta(x) \qquad P_{\dagger}(x,y) = y^{2} - x = P(\eta(x),y) \] It is then clear that all of the roots $y$ of $P, P_{\dagger}$ can be chosen compatibly for all $x \in k$ (for any $x \in k$ we just pick, once and for all, any $y$ such that $P(x,y) = 0$ and everything works). We can take $\Gamma$ to be the group of roots of unity of order $l$ for any even $l$. Trivially, $\lambda_{1}$ and $\lambda_{2}$ are $\Gamma$-linear.
\subsubsection{$U_{q}(\mathfrak{sl}_{2}(k))$}
Put $A = U_{q}(\mathfrak{sl}_{2}(k))$ and consider $H = k[K,K^{-1}]$. Then $V = k^{*}$ (which is definable over $\mathbb{Q}$) and we endow $H$ with the group Hopf algebra structure, namely
\[ \Delta(K) = K \otimes K \qquad \epsilon(K) = 1 \qquad S(K) = K^{-1} \] with analogous relations for $K^{-1}$. For $x \in k^{*}$, $L_{x}$ is therefore an $x$-eigenspace for $K$. Now $V$ is the group $(k^{*}, \cdot)$. To see this, note that
\[ \chi_{xy} = [(\chi_{x} \otimes \chi_{y}) \circ \Delta](K) = \chi_{x}(K)\chi_{y}(K) = xy \] and
\[ \chi_{x^{-1}} = \chi_{x}(K^{-1}) = x^{-1} \] Now for any $a \in A$,
\[ K \cdot a = KaK^{-1} \] But $KEK^{-1} = q^{2}E$ and $KFK^{-1} = q^{-2}F$, thus $E$ and $F$ are eigenvectors of the adjoint action of $H$ on $A$. We have the additional relation
\[ EF - FE = \frac{K - K^{-1}}{q - q^{-1}} \] which of the required form, thus giving that $A$ is semi-equivariant. Let $v \in L_{x}$. Then by Corollary \ref{corollary - semi-equivariant algebras - characters},
\[ KEv = q^{2}xEv \qquad KFv = q^{-2}xFv \] Thus $\Pi$ is generated by the functions
\[ \eta_{E}(x) = q^{2}x \qquad \eta_{F}(x) = q^{-2}x \] hence $\Pi = q^{2 \mathbb{Z}} = \{q^{l}: l \in 2\mathbb{Z}\}$ (also a group) and $\eta_{E}$, $\eta_{F}$ are mutually inverse. The parameter $q$ appearing in the definition of $\eta_{E}$ and $\eta_{F}$ satisfies the type
\[ \{x^{n} \neq 1: n \in \mathbb{N}, n > 0\} \] By reference to Proposition \ref{proposition - quantum sl2 generic - models are representations}, we take
\[ \lambda_{E}(y) = -\lambda_{F}(y) = \frac{y^{-1} + y}{q - q^{-1}} \] and
\[ P_{E}(x,y) = P_{F}(x,y) = y^{2} - x \] By the calculation performed in Proposition \ref{proposition - quantum sl2 generic - models are representations}, we obtain
\[ \lambda_{E}(y_{2})\lambda_{F}(y_{1}) - \lambda_{F}(z_{2})\lambda_{E}(z_{1}) = \frac{x - x^{-1}}{q- q^{-1}} \] for appropriate $y_{i}, z_{i}$. By contrast with the previous example, not any $y_{i}, z_{i}$ will do and we have to be careful about picking them compatibly. For this purpose, we partition $k^{*}$ into cosets of $q^{2 \mathbb{Z}}$:
\[ k^{*} = \bigcup_{x \in \Lambda} q^{2 \mathbb{Z}}x \] where $\Lambda$ is a set of representatives. Given $x \in \Lambda$, choose any square root $y$ of $x$. For any other $z \in q^{2\mathbb{Z}}x$, there is $l \in \mathbb{Z}$ such that $z = q^{2l} x$ and we choose the square root $q^{l}y$ of $z$. Now repeat this for every coset representative in $\Lambda$. The result is a compatible set of square roots. Clearly our polynomial $P_{E}$ satisfies the required conditions involving parameters. Consider $\Gamma = \{\pm 1\}$. Then $\lambda_{E}$ is $\Gamma$-linear, for
\[ \lambda_{E}(-y) = \frac{-y^{-1} - y}{q - q^{-1}} = -\lambda_{E}(y) \] hence so is $\lambda_{F}$.
\subsubsection{The multiparameter quantum torus}
Put $A = \mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$. We take the same Hopf algebra $H$ as for $U_{q}(\mathfrak{sl}_{2}(k))$, hence we obtain a line bundle $L$ over the base $(k^{*}, \cdot)$. Again, for any $a \in A$ we have
\[ \mathbf{U}_{1} \cdot a = \mathbf{U}_{1}a\mathbf{U}_{1}^{-1} \] But $\mathbf{U}_{1}\mathbf{U}_{i}\mathbf{U}_{1}^{-1} = q^{i-1}\mathbf{U}_{i}$ for $i > 1$. Moreover, the remaining relations are
\[ \mathbf{U}_{i} \mathbf{U}_{j} - q^{j-i}\mathbf{U}_{j} \mathbf{U}_{i} = 0 \qquad i < j \] giving that $A$ is semi-equivariant. By Corollary \ref{corollary - semi-equivariant algebras - characters}, for $v \in L_{x}$,
\[ \mathbf{U}_{1}\mathbf{U}_{i}v = xq^{i-1}\mathbf{U}_{i}v \] Thus we have functions
\[ \eta_{i}(x) = q^{i-1}x \qquad i > 1 \] and $\Pi$ is generated by these $\eta_{i}$ and their inverses (hence $\Pi$ is a group). Again, the single parameter $q$ appearing in the definition of the $\eta_{i}$ is generic, hence satisfies $\{x^{n} \neq 1: n \in \mathbb{N}, n > 0\}$. We take $\lambda_{i}(y) = y$ and $P_{i}(x,y) = y - x$ for all $i > 1$ and obtain what is required. On this occasion, the $P_{i}$ are linear and we do not have to worry about roots. Because the $\lambda_{i}$ are also linear, they are automatically $\Gamma$-linear for any group of roots of unity $\Gamma$.
\subsection{Associating a theory to an equivariant algebra}
Let $A$ be equivariant. First, we establish some notation. Let $\mathbf{i} \in n^{< \omega}$ be a finite sequence of elements from $\{1, \dots, n\}$. Say $\mathbf{i} = (i_{1}, \dots, i_{p})$. Put
\[ \eta_{\mathbf{i}} = \eta_{i_{p}} \dots \eta_{i_{1}} \qquad \mathbf{U}_{\mathbf{i}} = \mathbf{U}_{i_{p}} \dots \mathbf{U}_{i_{1}} \]
\begin{definition}
\label{definition - theory of an equivariant algebra}
We consider the three-sorted language
\[ \mathcal{L}_{A} = \{L,V, \Gamma, k, \pi, \mathbf{E}, \mathbf{U}_{i}, h_{j}, C: 1 \leq j \leq m, 1 \leq i \leq n) \] where
\begin{enumerate}
\item $C$ is a finite set of constants.
\item $\pi: L \rightarrow V$ and $\mathbf{U}_{i}, h_{j}: L \rightarrow L$ are functions.
\item $\mathbf{E} \subseteq L \times V$ is a relation.
\item $L,V, k$ are sorts, $k$ has the language of rings and $L$ comes equipped with
\begin{itemize}
\item a map $+: L \times L \rightarrow k$ which is interpreted as addition of elements in each $\pi^{-1}(x)$ for $x \in V$.
\item a map $\cdot: k \times L \rightarrow L$ which is interpreted scalar multiplication in each fiber $\pi^{-1}(x)$.
\end{itemize}
\end{enumerate} The $\mathcal{L}_{A}$-theory $T_{A}$ says the following:
\begin{enumerate}
\item $k$ is an algebraically closed field of characteristic $0$.
\item $\Sigma_{c}(c)$ holds for each $c \in C$, where $\Sigma_{c}$ is the type that $c$ satisfies.
\item $V = \phi(k)$ where $\phi$ is the formula over $\mathbb{Q}$ defining $V$.
\item $\pi: L \rightarrow V$ is a surjective map and each fiber $\pi^{-1}(x)$ is a one-dimensional $k$-vector space for each $x \in V$.
\item $\mathbf{E}(L,x)$ is non-empty for each $x \in V$ and $\mathbf{E}(L,x) \subseteq \pi^{-1}(x)$.
\item Let $\Gamma$ be the group of $l$-th roots of unity for some $l$ satisfying condition $4$ of the definition for representable algebras. Then $\Gamma$ acts faithfully and transitively on $\mathbf{E}(L,x)$.
\item The operators $h_{j}$ are linear and act on each fiber according to the following axiom:
\[ (\forall v \in \pi^{-1}(x))\left(\bigwedge_{j = 1}^{m} h_{j}v = x_{j}v \right) \]
\item The linear operators $\mathbf{U}_{i}$ act according to
\[ (\forall v \in \pi^{-1}(x))\exists v_{i}(\exists y_{i} \in k)(\mathbf{E}(v,x) \rightarrow \left( \bigwedge_{i=1}^{n} \mathbf{E}(v_{i}, \eta_{i}x) \wedge \mathbf{U}_{i}v = \lambda_{i}(y_{i})v_{i} \wedge P_{i}(y_{i},x) = 0 \right) \] We denote the conjunct appearing in the big parentheses by $\varphi(v,v_{i},y_{i})$.
\item Enumerate the relations not expressing the adjoint action of $H$ on the $\mathbf{U}_{i}$ as
\[ c_{i} \mathbf{U}_{\mathbf{j}_{i}} - d_{i} \mathbf{U}_{\mathbf{k}_{i}} = f_{i}(h_{1}, \dots, h_{m}) \qquad 1 \leq i \leq r\] where $\mathbf{j}_{i}, \mathbf{k}_{i} \in n^{< \omega}$ for each $i$. For any $\mathbf{i} = (i_{1}, \dots, i_{p}) \in n^{< \omega}$ we define the formula $\phi_{\mathbf{i}}(e_{0}, e,y)$ to be
\[ \left( \bigwedge_{k=1}^{p} \mathbf{E}(e^{k}, \eta_{i_{k}} \pi(e^{k-1})) \wedge \mathbf{U}_{i_{k}} e^{k-1} = \lambda_{i_{k}}(y_{k})e^{k} \wedge P_{i_{k}}(y_{k}, \pi(e^{k-1})) = 0 \right) \] where $e^{0} = e_{0}$. Then the following axiom holds:
\[ \begin{array}{ll} (\forall v \in \pi^{-1}(x))(\forall_{l=1}^{r} v^{(l)} \in \pi^{-1}(\eta_{l}x))( & \\ \quad \mathbf{E}(v,x) \wedge \exists a^{(l)}\varphi(v,v^{(l)},a^{(l)}) \rightarrow & \forall_{i=1}^{r} v_{i} \forall_{i=1}^{r} w_{i} \forall_{i=1}^{r} y_{i} z_{i}(\bigwedge_{i=1}^{r} \phi_{\mathbf{j}_{i}}(v,v_{i}, y_{i}) \\ & \wedge \phi_{\mathbf{k}_{i}}(v,w_{i}, z_{i}) \wedge \theta_{1} \wedge \psi) \wedge \\ & \forall_{s} v^{(l)}_{s} \forall_{s} w^{(l)}_{s} \forall y^{(l)}_{s} z^{(l)}_{s}(\bigwedge_{l=1}^{r} \bigwedge_{s=1}^{r} \phi_{\mathbf{j}_{s}}(v^{(l)},v^{(l)}_{s}, y^{(l)}_{s}) \\ & \wedge \phi_{\mathbf{k}_{s}}(v^{(l)}, w^{(l)}_{s}, z^{(l)}_{s}) \wedge \theta_{2} ))) \end{array} \] where
\begin{itemize}
\item $\psi(y_{i}, z_{i})$ is a conjunction of linear conditions which isolates the type $\tp^{k}(y_{i},z_{i}/\mathbb{Q} \cup C)$ formulated in the language of the sort $k$, for any instantiation of such $y_{i},z_{i}$.
\item $\theta_{1}$ is the conjunction
\[ \bigwedge_{\pi(v) = \pi(v_{ik})} v = v_{ik} \wedge \bigwedge_{\pi(v) = \pi(w_{ik})} v = w_{ik} \wedge \bigwedge_{\pi(v_{ik}) = \pi(w_{jl})} v_{ik} = w_{jl} \] and $\theta_{2}$ is
\[ \begin{array}{l} \bigwedge_{\pi(v^{(l)}) = \pi(v)} v^{(l)} = v \wedge \bigwedge_{\pi(v^{(l)}) = \pi(v_{ik})} v^{(l)} = v_{ik} \wedge \bigwedge_{\pi(v^{(l)}) = \pi(w_{ik})} v^{(l)} = w_{ik} \wedge \\ \bigwedge_{\pi(v^{(l)}_{st}) = \pi(v_{ik})} v^{(l)}_{st} = v_{ik} \wedge \bigwedge_{\pi(w^{(l)}_{st}) = \pi(w_{ik})} w^{(l)}_{st} = w_{ik} \end{array} \]
\end{itemize}
\end{enumerate}
\end{definition} Axioms 8 and 9 of Definition \ref{definition - theory of an equivariant algebra} together express a significant amount of information. Whereas axiom 8 defines the action of each $\mathbf{U}_{i}$ at a given fiber, axiom 9 is designed to make sure that all of the basis elements in different fibers obtained on repeated application of axiom 8, if they should agree, do agree. For example, given a defining relation of the form $c\mathbf{U}_{\mathbf{j}} - d\mathbf{U}_{\mathbf{k}} = f(h_{1}, \dots, h_{m})$, one expects the basis elements involved in defining the action of $\mathbf{U}_{\mathbf{j}}$ and $\mathbf{U}_{\mathbf{k}}$ to eventually coincide in their respective terminal fibers. And if $f \neq 0$ then there should be more; namely that these terminal basis elements coincide with the basis element we start with. Only with such a stipulation is it possible to make sense of expressions like
\[ c\mathbf{U}_{\mathbf{i}}e - d\mathbf{U}_{j}e = f(x)e \] where $e \in \pi^{-1}(x)$ is a basis element. In this manner, every model of $T_{A}$ is indeed a representation of $A$. The formula $\psi$ is a rigidity condition, ensuring that those roots of polynomials $P_{i}$ which can be related to each other are indeed related to each other. In the initial examples discussed, $\psi$ was implicitly incorporated into the axioms.
\begin{example} Recall Definition \ref{definition - quantum sl2 generic - associated theory} of the theory $T_{q}$ associated to $U_{q}(\mathfrak{sl}_{2}(k))$. The actions of $E$ and $F$ were specified by
\[ \begin{array}{ll} (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow & (\exists v' \in \pi^{-1}(q^{2}x))(\exists y \in k) \\ & (y^{2} = x \wedge Ev = \lambda(y)v' \wedge Fv' = -\lambda(qx)v)) \end{array}\] where $\lambda: L \rightarrow k$ is defined by
\[ \lambda(x) = \frac{y^{-1} + y}{q - q^{-1}} \] Axiom 9 of Definition \ref{definition - theory of an equivariant algebra} would give
\[ \begin{array}{ll} (\forall v \in \pi^{-1}(x))(\mathbf{E}(v,x) \rightarrow & (\forall v_{1} \in \pi^{-1}(q^{2}x))(\forall v_{2} \in \pi^{-1}(x))(\forall w_{1} \in \pi^{-1}(q^{-2}x))(\forall w_{2} \in \pi^{-1}(x)) \\ & (\forall_{i,j \leq 2} y_{ij} \in k)(y_{11}^{2} = y_{21}^{2} = x \wedge y_{12}^{2} = q^{2}x \wedge y_{22}^{2} = q^{-2}x \\
& \wedge Ev = \lambda(y_{11})v_{1} \wedge Fv_{1} = -\lambda(y_{12})v_{2} \wedge Fv = -\lambda(y_{21})w_{1} \\ & \wedge Ew_{1} = \lambda(y_{22})w_{2} \wedge v_{2} = w_{2} = v \wedge \psi \end{array}\] where $\psi$ is chosen to be the formula $y_{12} = qy_{11} \wedge y_{11} = y_{21} \wedge y_{21} = qy_{22}$. With some simplification, this is indeed equivalent to the shorter axiom of Definition \ref{definition - quantum sl2 generic - associated theory} when combined with axiom 8.
\end{example}
\begin{remark} If $A$ is equivariant, we may have some choice of possible $\lambda_{i}, f$ and $P_{i}$. Nevertheless, in defining $T_{A}$ a particular choice of these functions and polynomials is fixed once and for all. If one was being pedantic, the dependence of $T_{A}$ on these functions and polynomials could have been indicated.
\end{remark} A model of $T_{A}$ is therefore a three-sorted structure, which shall be denoted by a tuple $(L,k)$. We have suppressed $V$ from the notation because it is evident that $V$ is in fact definable in $k$. As stated in the introduction, such structures bear a striking resemblance to the $G$-equivariant line bundles (for $G$ a connected algebraic group over $\mathbb{C}$) found in geometric representation theory (see \cite{RTT07}). The proof of the following result takes its inspiration from the construction of the line bundle $L_{\lambda} = G \times_{B} \mathbb{C}_{\lambda}$, where $B$ is a Borel subgroup of $G$ and $\mathbb{C}_{\lambda}$ is a one-dimensional representation of $B$ corresponding to the weight $\lambda$ of the Cartan subgroup $H$ of $G$. The difference below is that we have $(\Gamma \times_{\Gamma} k) \times V$ instead. The resulting structure is then equipped with linear operators $\mathbf{U}_{i}$ between fibers that give us some kind of equivariance, in the sense that $\mathbf{U}_{i}(L_{x}) = L_{\eta_{i}(x)}$ and $\mathbf{U}_{i}: L_{x} \rightarrow L_{\eta_{i}(x)}$ is a linear isomorphism, for each $x \in V$.
\begin{proposition}
$\label{proposition - equivariant algebras - theories are consistent}
T_{A}$ is consistent.
\end{proposition}
\begin{proof} We construct a model of $T_{A}$. Let $k$ be an algebraically closed field of characteristic $0$. Introduce the equivalence relation
\[ (\delta_{1}, a_{1},x_{1}) \sim (\delta_{2}, a_{2}, x_{2}) \Leftrightarrow (\exists \gamma \in \Gamma)(a_{2} = \gamma \cdot a_{1} \wedge \delta_{2} = \gamma^{-1}\delta_{1}) \] on $\Gamma \times k \times V$ and consider the quotient $\Gamma \times k \times V/ \sim$. We shall denote the equivalence class of $(\gamma, a,x)$ by $\overline{(\gamma, a,x)}$. Put
\[ L_{x} = \{\overline{(\gamma, a,x)}: \gamma \in \Gamma, a \in k \} \qquad L = \prod_{x \in V} L_{x} \] Then there is a projection map $\pi: L \rightarrow V$ given by $\pi(\overline{(\gamma, a, x)}) = x$. When $L_{x}$ is understood, we suppress $x$ from the notation and write $\overline{(\gamma, a)}$ for $\overline{(\gamma,a,x)}$.
\\
\\
\textbf{Claim}: Each $L_{x}$ has the structure of a one-dimensional $k$-vector space by
\[ \overline{(\delta_{1}, a_{1})} + \overline{(\delta_{2}, a_{2})} := \overline{(\delta_{2}, \gamma^{-1} a_{1} + a_{2})} \mbox{ where } \delta_{2} = \gamma \delta_{1} \] \[ \lambda \overline{(\delta, a)} := \overline{(\delta, \lambda a)} \]
\begin{proof} Suppose that $(\delta_{1}, a_{1}) \sim (\delta_{1}', a_{1}')$ and $(\delta_{2}, a_{2}) \sim (\delta_{2}', a_{2}')$ and that $\delta_{2} = \gamma \delta_{1}$. There are $\gamma_{1}, \gamma_{2}$ such that $\delta_{1}' = \gamma_{1} \delta_{1}$ and $\delta_{2}' = \gamma_{2} \delta_{2}$. Thus
\[ \delta_{2}' = \gamma_{2}\gamma\gamma_{1}^{-1} \delta_{1}' \] So it remains to prove that
\[ (\delta_{2}, \gamma^{-1}a_{1} + a_{2}) \sim (\delta_{2}', \gamma_{1}\gamma^{-1}\gamma_{2}^{-1}a_{1}' + a_{2}') \] But $\gamma_{1}^{-1}a_{1} = a_{1}'$ and $\gamma_{2}^{-1}a_{2} = a_{2}'$. So
\[ \gamma_{1}\gamma\gamma_{2}^{-1}a_{1}' + a_{2}' = \gamma_{2}^{-1}(\gamma^{-1}a_{1} + a_{2}) \] as required. Scalar multiplication is trivially well-defined and $\overline{(1, 1)}$ is a basis element for $L_{x}$.
\end{proof} Normal basis elements are designated as those of the form $\overline{(\gamma, 1)}$ for $\gamma \in \Gamma$ and it is clear that $\Gamma$ acts faithfully and transitively on the set of normal basis elements of $L_{x}$. We now define maps by
\[ \mathbf{U}_{i} \overline{(1,1,x)} = \lambda_{i}(y_{i})\overline{(1,1,\eta_{i}(x))} \qquad 1 \leq i \leq n\] and extend linearly. By condition $4$ of Definition \ref{definition - equivariant algebras}, the roots $y_{i}$ can be chosen compatibly so that all listed relations of the form
\[ c \prod_{k=0}^{p-1} \lambda_{i_{p} - k}(y_{i_{p-k}}) - d\prod_{k=0}^{q-1} \lambda_{j_{q-k}}(y_{j_{q-k}}) = f(x) \] are satisfied for every $x \in V$. So we use these $y_{i}$ when defining the actions of the $\mathbf{U}_{i}$. It is now clear that our resulting structure satisfies $T_{A}$.
\end{proof} We conclude this subsection with a remark about the types $\Sigma_{c}$. The theory $T_{A}$ will only be adequate insofar as each $\Sigma_{c}$ contains all of the information that is required of $c$. The examples considered above contained at most one constant $q$, and all that was required of $q$ in these cases was that $q$ was generic; namely that it satisfied the type $\Sigma_{q} = \{x^{n} \neq 1: n \in \mathbb{N}, n > 0\}$.
\section{Model Theory of Equivariant Structures: I}
In this section and the next, we build on many of the results of \cite{ZS09}. There it was proved that an uncountably categorical first-order theory can be associated to the Heisenberg algebra and a quantifier elimination result (down to the level of existential formulas) was established. Analogous results are proved here for $T_{A}$ where $A$ is an equivariant algebra with the property that models of $T_{A}$ are, roughly speaking, rather rigid.
\subsection{Categoricity}
We shall fix an equivariant algebra $A$. As part of the definition, we have a certain amount of data (the regular functions $\lambda_{i}$, polynomials $P_{i}$ and $f$, functions $\eta_{i}$). In the language $\mathcal{L}_{A}$, all these entities become definable over $\mathbb{Q}$ and the constants $C$. We recall that each of these constants has its properties fixed by a type $\Sigma_{c}$ over the prime subfield. For ease of reference we recall axiom 9 of Definition \ref{definition - theory of an equivariant algebra}:
\[ \begin{array}{ll} (\forall v \in \pi^{-1}(x))(\forall_{l=1}^{r} v^{(l)} \in \pi^{-1}(\eta_{l}x))( & \\ \quad \mathbf{E}(v,x) \wedge \exists a^{(l)}\varphi(v,v^{(l)},a^{(l)}) \rightarrow & \forall_{i=1}^{r} v_{i} \forall_{i=1}^{r} w_{i} \forall_{i=1}^{r} y_{i} z_{i}(\bigwedge_{i=1}^{r} \phi_{\mathbf{j}_{i}}(v,v_{i}, y_{i}) \\ & \wedge \phi_{\mathbf{k}_{i}}(v,w_{i}, z_{i}) \wedge \theta_{1} \wedge \psi) \wedge \\ & \forall_{s} v^{(l)}_{s} \forall_{s} w^{(l)}_{s} \forall y^{(l)}_{s} z^{(l)}_{s}(\bigwedge_{l=1}^{r} \bigwedge_{s=1}^{r} \phi_{\mathbf{j}_{s}}(v^{(l)},v^{(l)}_{s}, y^{(l)}_{s}) \\ & \wedge \phi_{\mathbf{k}_{s}}(v^{(l)}, w^{(l)}_{s}, z^{(l)}_{s}) \wedge \theta_{2} ))) \end{array} \] where
\begin{itemize}
\item The relations not expressing the adjoint action of $H$ on the $\mathbf{U}_{i}$ are enumerated as
\[ c_{i} \mathbf{U}_{\mathbf{j}_{i}} - d_{i} \mathbf{U}_{\mathbf{k}_{i}} = f_{i}(h_{1}, \dots, h_{m}) \qquad 1 \leq i \leq r\] where $\mathbf{j}_{i}, \mathbf{k}_{i} \in n^{< \omega}$ for each $i$.
\item For any $\mathbf{i} = (i_{1}, \dots, i_{p}) \in n^{< \omega}$ we define the formula $\phi_{\mathbf{i}}(e_{0}, e,y)$ to be
\[ \left( \bigwedge_{k=1}^{p} \mathbf{E}(e^{k}, \eta_{i_{k}} \pi(e^{k-1})) \wedge \mathbf{U}_{i_{k}} e^{k-1} = \lambda_{i_{k}}(y_{k})e^{k} \wedge P_{i_{k}}(y_{k}, \pi(e^{k-1})) = 0 \right) \] where $e^{0} = e_{0}$.
\item $\psi(y_{i}, z_{i})$ is a conjunction of linear conditions which isolates the type $\tp^{k}(y_{i},z_{i}/\mathbb{Q} \cup C)$ formulated in the language of the sort $k$, for any instantiation of such $y_{i},z_{i}$.
\item $\theta_{1}$ and $\theta_{2}$ combined express that those basis elements which should agree (i.e. those which lie in the same fibers), do agree.
\end{itemize} In order for $T_{A}$ to be categorical in uncountable cardinals, given any model $(L,k) \models T_{A}$ where $k$ is an uncountable field, one requires the basis elements in the fibers above the orbit of any $x \in V$ under $\Pi$ to remain rigid under scaling by certain elements of $\Gamma$. Specifically, axioms 8 and 9 provide us with a set of basis elements (one for each fiber) over the orbit $\Pi x$. The requirement is that the truth of axiom 9 should not be affected by shifting these basis elements by certain $\gamma \in \Gamma$. The following technical restriction is designed to achieve this. First some notation. Let $\Xi$ consist of those pairs $(\mathbf{i}, \mathbf{j}) \in (n^{<\omega})^{2}$ selected by $\theta_{1} \wedge \theta_{2}$ in axiom 9 with $\eta_{\mathbf{i}} = \eta_{\mathbf{j}}$, i.e. for any $x \in V$ and basis element $e \in \pi^{-1}(x)$, $\theta_{1} \wedge \theta_{2}$ says that the basis elements used to define $\mathbf{U}_{\mathbf{i}}e$ and $\mathbf{U}_{\mathbf{j}}e$ lying in $\pi^{-1}(\eta_{\mathbf{i}}x)$ and $\pi^{-1}(\eta_{\mathbf{j}}x)$ respectively, agree.
\begin{definition}
\label{definition - equivariant algebras - rigid}
The theory $T_{A}$ is \textbf{$\Gamma$-rigid} if for every model $(L,k) \models T_{A}$ and $(\mathbf{i}, \mathbf{j}) \in \Xi$ with $\mathbf{i} = (i_{1}, \dots, i_{p})$, $\mathbf{j} = (j_{1}, \dots, j_{q})$ we have that
\[ \gamma^{n_{i_{1}}} = \delta^{n_{j_{1}}} = 1 \Rightarrow \gamma^{p} = \delta^{q} \] for every $\gamma, \delta \in \Gamma$.
\end{definition}
\begin{proposition}
\label{proposition - equivariant algebras - rigidity consequences}
Let $T_{A}$ be $\Gamma$-rigid.
\begin{enumerate}
\item For every polynomial $P_{i}(x,y) = y^{n_{i}} - \mu_{i}(x)$, we have $n_{i} \leq 2$.
\item For every relation of the form $c\mathbf{U}_{\mathbf{i}} - d\mathbf{U}_{\mathbf{j}} = f(h_{1}, \dots, h_{m})$, if $\mathbf{i} = (i_{1}, \dots, i_{p})$ and $\mathbf{j} = (j_{1}, \dots, j_{q})$ then one of the following holds:
\begin{enumerate}
\item $p = q$.
\item $p = 2q$.
\item $q = 2p$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item A given $\eta_{i} \in \Pi$ has an inverse $\eta_{j}$ and $(i,j) \in \Xi$. Thus for any $\gamma, \delta \in \Gamma$ such that $\gamma^{n_{i}} = \delta^{n_{j}} = 1$, it follows by $\Gamma$-rigidity (applied with $\eta_{i} \eta_{j} = \eta_{j} \eta_{i}$ and $\eta_{i} \eta_{j} = 1$) that $\gamma^{2} = \delta^{2} = 1$. In particular, this holds if $\gamma$ and $\delta$ are primitive roots of unity. Thus $n_{i}, n_{j} \leq 2$.
\item Suppose that $\gamma^{n_{i_{1}}} = \delta^{n_{j_{1}}} = 1$ where $\gamma$ and $\delta$ are primitive. Then $\gamma^{qn_{i_{1}}} = \delta^{qn_{j_{1}}} = \gamma^{pn_{j_{1}}} = 1$ (by $\gamma^{p} = \delta^{q}$). Because $\gamma$ is primitive, $qn_{i_{1}} \leq pn_{j_{1}}$. The reverse inequality follows by symmetry, hence $qn_{i_{1}} = pn_{j_{1}}$. The result is now immediate by 1.
\end{enumerate}
\end{proof} Proposition \ref{proposition - equivariant algebras - rigidity consequences} gives some indication of the strength of the assumption of $\Gamma$-rigidity. The following results show that $\Gamma$-rigidity is equivalent to uncountable categoricity.
\begin{proposition}
\label{proposition - equivariant algebras - obstruction to categoricity}
Suppose that $(L,k) \models T_{A}$ witnesses the failure of $\Gamma$-rigidity, where $k$ is uncountable. Then there is an automorphism $\sigma$ of $k$ which does not extend to an automorphism of $(L,k)$.
\end{proposition}
\begin{proof} There is $(\mathbf{i}, \mathbf{j}) \in \Xi$ such that for some basis element $e \in \pi^{-1}(x)$, there are $e_{1}, e_{2}, y_{1}, y_{2}$ with
\[ (L,k) \models \phi_{\mathbf{i}}(e,e_{1},y_{1}) \wedge \phi_{\mathbf{j}}(e, e_{2}, y_{2}) \] but there are $\gamma, \delta \in \Gamma$ such that $\gamma^{n_{i_{1}}} = \delta^{n_{j_{1}}} = 1$, and $\gamma^{p} \neq \delta^{q}$ (where $\mathbf{i} = (i_{1}, \dots, i_{p})$, $\mathbf{j} = (j_{1}, \dots, j_{q})$). Let $y_{1}',y_{2}'$ be tuples obtained by transforming $y_{1i} \mapsto \gamma y_{1i}$ and $y_{2i} \mapsto \delta y_{2i}$. Now $\psi$ implies a formula $\psi_{\mathbf{i}, \mathbf{j}}$ which isolates the type $\tp^{k}(y_{1},y_{2}/\mathbb{Q} \cup C)$; indeed $\psi_{\mathbf{i}, \mathbf{j}}$ is just a suitable subformula of $\psi$. In particular it is a conjunction of linear conditions. Because the $y_{1i}$ (respectively $y_{2i}$) are all related to each other via $\psi_{\mathbf{i}, \mathbf{j}}$, $\psi_{\mathbf{i}, \mathbf{j}}$ also holds of $y'_{1}, y'_{2}$. Thus $\tp_{k}(y_{1},y_{2}/\mathbb{Q} \cup C) = \tp_{k}(y'_{1}, y'_{2}/\mathbb{Q} \cup C)$. By saturation of $k$, there is an automorphism $\sigma$ of $k$ such that $\sigma(y_{1}, y_{2}) = (y_{1}',y_{2}')$. Suppose for contradiction that $\sigma$ does extend to an automorphism $\tilde{\sigma}$ of $(L,k)$. Decomposing $\phi_{\mathbf{i}}(e,e_{1},y_{1})$ we obtain
\[ (L,k) \models \mathbf{U}_{\mathbf{i}}e = \prod_{k=1}^{p} \lambda_{i_{k}}(y_{1k})e_{1}^{p} \Rightarrow (L,k) \models \mathbf{U}_{\mathbf{i}}\tilde{\sigma}(e) = \prod_{k=1}^{p} \lambda_{i_{k}}(\sigma(y_{1k}))\tilde{\sigma}(e_{1}^{p}) \] Now $\prod_{k=1}^{p} \lambda_{i_{k}}(\sigma(y_{1k})) = \gamma^{p} \prod_{k=1}^{p} \lambda_{i_{k}}(y_{1k})$. But $y_{11}^{n_{i_{1}}} = \mu_{i_{1}}x$, hence
\[ \mu_{i_{1}}x = y_{11}^{n_{i_{1}}} = \gamma^{n_{i_{1}}} y_{11}^{n_{i_{1}}} = \mu_{i_{1}}\sigma(x) \Rightarrow x = \sigma(x) \] the implication holding because $\Pi$ is a group, hence $\tilde{\sigma}(e)$ and $e$ lie in the same fiber and $\tilde{\sigma}(e) = \mu e$ for some $\mu \in k$. Now we apply the same argument to $\phi_{\mathbf{j}}(e,e_{2}, y_{2})$. By axiom 9, $e^{p}_{1} = e^{p}_{2}$. But then
\[ \tilde{\sigma}(e^{p}_{1}) = \gamma^{-p}\mu e^{p}_{1} = \delta^{-q}\mu e^{p}_{2} = \tilde{\sigma}(e^{p}_{2}) \] hence $\gamma^{p} = \delta^{q}$, resulting in contradiction.
\end{proof}
\begin{theorem}
\label{theorem - equivariant structures - categoricity}
$T_{A}$ is $\Gamma$-rigid if and only if for any models $(L,k), (L',k') \models T_{A}$ with $k,k'$ uncountable, if $\sigma: k \rightarrow k'$ is an isomorphism then $\sigma$ extends to an isomorphism $\tilde{\sigma}: L \rightarrow L'$.
\end{theorem}
\begin{proof} Suppose that $T_{A}$ is $\Gamma$-rigid. Because $V$ is defined over the prime subfield, $\sigma: V \rightarrow V' = \phi(k')$. Similarly $\sigma: \Gamma \rightarrow \Gamma'$ where $\Gamma'$ is the group of $l$-th roots of unity in $k'$. We can assume that $\sigma$ maps constants to constants ($\Sigma_{c}$ are over $\mathbb{Q}$, hence are preserved by $\sigma$. Because all of the information we require of a constant is contained in $\Sigma_{c}$, we may as well reinterpret the constants of $(L',k')$ so that they lie in the image of $C(k)$ under $\sigma$). Partition $V$ up into orbits of the group $\Pi$, thus obtaining
\[ V = \bigcup_{x \in \Lambda} \Pi x \] for some set of representatives $\Lambda$ of each orbit. It is then clear that we have a corresponding partition for $V'$
\[ V' = \bigcup_{\sigma(x) \in \sigma(\Lambda)} \Pi'\sigma(x)\] where $\Pi'$ is the group generated by $\eta_{i}' = \sigma(\eta_{i})$. Define the \textbf{length} of $y$ (with respect to the representative $x$), $l(y)$, to be the length of the smallest sequence $\mathbf{i}$ such that $y = \eta_{\mathbf{i}}(x)$. We extend $\tilde{\sigma}$ to the rest of the orbit $\Pi x$ by induction on length.
\begin{enumerate}
\item $l(y) = 0$, i.e. $y = x$. By axiom 5 of $T_{A}$ there is $e \in \pi^{-1}(x)$ such that $\mathbf{E}(e,x)$ holds in $(L,k)$. Likewise, there is $e' \in \pi^{-1}(\sigma(x))$ such that $(L',k') \models \mathbf{E}(e',\sigma(x))$ and we define $\tilde{\sigma}: L_{x} \rightarrow L'_{\sigma(x)}$ by mapping $e \mapsto e'$ and extending linearly. By repeated application of axiom $8$ we obtain basis elements $e_{1i}, e_{2i}$ and elements $y_{1i}, y_{2i}$ of $k$ such that
\[ (L,k) \models \bigwedge_{i=1}^{r} \phi_{\mathbf{j}_{i}}(e,e_{1i}, y_{1i}) \wedge \phi_{\mathbf{k}_{i}}(e,e_{2i}, y_{2i}) \] where $\mathbf{j}_{i}, \mathbf{k}_{i}$ are such that $c_{i}\mathbf{U}_{\mathbf{j}_{i}} - d_{i}\mathbf{U}_{\mathbf{k}_{i}} = f(h_{1}, \dots, h_{m})$. Similarly we obtain basis elements $e'_{1i}, e'_{2i}$ and $y'_{1i}, y'_{2i} \in k'$ such that
\[ (L',k') \models \bigwedge_{i=1}^{r} \phi_{\mathbf{j}_{i}}(e',e'_{1i}, y'_{1i}) \wedge \phi_{\mathbf{k}_{i}}(e',e'_{2i}, y'_{2i})\] where $\psi'$ is $\psi$ with all parameters from $k$ transformed to their images under $\sigma$. Fix some $1 \leq i \leq r$. Suppose that on decomposing $\phi_{\mathbf{j}_{i}}(e, e_{1i}, y_{1i})$ we obtain
\[ (L,k) \models \bigwedge_{k=1}^{p_{i}} \mathbf{E}(e_{1i}^{k}, \eta_{j_{k}} \pi(e^{k-1}_{1i})) \wedge \mathbf{U}_{j_{k}} e^{k-1}_{1i} = \lambda_{j_{k}}(y_{1ik})e^{k}_{1i} \wedge P_{j_{k}}(y_{1ik}, \pi(e^{k-1}_{1i})) = 0 \] Put $P'_{l} = \sigma(P_{l})$ for each $1 \leq l \leq n$. Note that $\eta_{l}'(\sigma(x)) = \sigma(\eta_{l}(x))$ for every such $l$ and $x \in V$. We also have
\[ (L',k') \models \bigwedge_{k=1}^{p_{i}} \mathbf{E}(e_{1i}^{'k}, \eta'_{j_{k}} \pi(e^{'k-1}_{1i})) \wedge \mathbf{U}_{j_{k}} e^{'k-1}_{1i} = \lambda'_{j_{k}}(y'_{1ik})e^{'k}_{1i} \wedge P'_{j_{k}}(y'_{1ik}, \pi(e^{'k-1}_{1i})) = 0 \] Thus $P'_{j_{k}}(y'_{1ik},\pi(e_{1i}^{'k-1})) = P'_{j_{k}}(\sigma(y_{1ik}),\pi(e_{1i}^{'k-1})) = 0$ for every $k$. Recalling that $P_{j_{k}}(y,x) = y^{n_{j_{k}}} - \mu_{j_{k}}(x)$ and that $n_{j_{k}}$ divides the order of $\Gamma$, it follows that there are $\gamma_{1ik} \in \Gamma$ such that $y'_{1ik} = \gamma_{1ik}\sigma(y_{1ik})$ for every $k$. But then by $\Gamma$-linearity of the $\lambda_{j_{k}}$, it follows that
\[ (L',k') \models \mathbf{U}_{j_{k}} e_{1i}^{'k-1} = \gamma_{1ik}\lambda'_{j_{k}}(\sigma(y_{1ik}))e_{1i}^{'k} \] Thus we define $\tilde{\sigma}$ on the fibers containing the $e_{1i}^{k}$ by mapping $e_{1i}^{k} \mapsto \gamma_{1ik}e_{1i}^{'k}$ and extending linearly. Repeat the above for $\phi_{\mathbf{k}_{i}}(e,e_{2i},y_{2i})$ and every $1 \leq i \leq r$. By axiom 9, $\psi \in \tp^{k}(y_{1i}, y_{2i}/\mathbb{Q} \cup C) \Rightarrow \psi' \in \tp^{k}(\sigma(y_{1i}),\sigma(y_{2i})/\mathbb{Q} \cup C')$. Thus any roots of the various polynomials $P_{l}$ related by $\psi$ transform by the same element of $\Gamma$ under $\sigma$. Now we invoke $\Gamma$-rigidity in $(L',k')$. Let $(\mathbf{i}, \mathbf{j}) \in \Xi$. Recall from the proof of Proposition \ref{proposition - equivariant algebras - obstruction to categoricity} that there is a conjunction of linear conditions $\psi'_{\mathbf{i}, \mathbf{j}}$ implied by $\psi'$ that relates all those roots used to define the action of $\mathbf{U}_{\mathbf{i}}e'$ (respectively $\mathbf{U}_{\mathbf{j}}e'$). Concentrating on $\mathbf{U}_{\mathbf{i}}e'$, for the sake of notational clarity, we assume that these roots are the first $p$ elements of the tuple $y'_{1i}$. Thus there is $\gamma_{1i}$ such that $\gamma_{1i} = \gamma_{lik}$ for $1 \leq k \leq p$ and
\[ (L',k') \models \mathbf{U}_{\mathbf{j}_{i}}e' = \prod_{k=1}^{p} \lambda'_{j_{k}}(y'_{1ik})e_{1i}^{'p} = \gamma_{1i}^{p} \prod_{i=1}\lambda'_{j_{k}}(\sigma(y_{1ik}))e_{1i}^{'p} \] By $\Gamma$-rigidity, $\gamma_{1i}^{p} = 1$. Because the $\gamma_{1i}$ get successively absorbed under $\tilde{\sigma}$ we also have that $\mathbf{U}_{\mathbf{j}_{i}}e' = \prod_{k=1}^{p} \lambda'_{j_{k}}(\sigma(y_{1ik}))\tilde{\sigma}(e_{1i}^{p})$, hence $\tilde{\sigma}(e_{1i}^{p}) = e_{1i}^{'p}$. Exactly the same argument applies to $\mathbf{U}_{\mathbf{j}}e'$, hence applying $\tilde{\sigma}$ as defined still preserves the truth of axiom 9 in $(L',k')$, as required.
\item $l(y) > 0$; thus $\tilde{\sigma}$ has been extended to all those $L_{z}$ for which $l(z) \leq l(y) - 1$. Let $\mathbf{l} = (l_{1}, \dots, l_{s}) \in n^{< \omega}$ witness the length of $y$. Put $z = \eta_{l_{s-1}} \dots \eta_{l_{1}}x$. Then $l(z) \leq l(y) - 1$ and by induction $\tilde{\sigma}$ has already been extended to $\pi^{-1}(z)$. Let $e^{(1)} \in \pi^{-1}(z)$ be the basis element used to define $\mathbf{U}_{l_{s-1}} \dots \mathbf{U}_{l_{1}}e$. Now apply axiom 8 to $\pi^{-1}(z)$ with $e^{(1)}$ to obtain $e^{(2)} \in \pi^{-1}(y)$ in terms of which $\mathbf{U}_{l_{s}}e^{(1)}$ is defined and extend as in the base case. Now repeat the whole argument for the base case with $e^{(2)}$ and the induction step follows.
\end{enumerate}
Repeating the above construction for each orbit completes the construction of $\tilde{\sigma}$. The converse is immediate by Proposition \ref{proposition - equivariant algebras - obstruction to categoricity}.
\end{proof}
\begin{proposition}
\label{proposition - initial examples - rigid}
The theories associated to $A_{1}(k)$, $U_{q}(\mathfrak{sl}_{2}(k))$ and $\mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$ are $\Gamma$-rigid.
\end{proposition}
\begin{proof} We start with $A_{1}(k)$. Recall that for this algebra $\Pi$ is generated by two functions
\[ \eta_{\dagger}(x) = x+1 \qquad \eta(x) = x-1 \] The only relation not expressing the adjoint action of $\mathbf{H}$ is $\mathbf{a}\mathbf{a}^{\dagger} - \mathbf{a}^{\dagger}\mathbf{a} = 1$, hence $\eta_{\dagger}\eta = \eta \eta_{\dagger}$. Now the corresponding polynomials are
\[ P(x,y) = y^{2} - \eta(x) \qquad P_{\dagger}(x,y) = y^{2} - x \] so we must demonstrate that for any $\gamma, \delta \in \Gamma$, $\gamma^{2} = \delta^{2} = 1 \Rightarrow \gamma^{2} = \delta^{2}$ which is trivially true! For $U_{q}(\mathfrak{sl}_{2}(k))$, $\Pi$ is generated by
\[ \eta_{E}(x) = q^{2}x \qquad \eta_{F}(x) = q^{-2}x \] and similarly $\eta_{E}\eta_{F} = \eta_{F}\eta_{E}$. The corresponding polynomials are
\[ P_{E}(x,y) = P_{F}(x,y) = y^{2} - x\] so again, $\Gamma$-rigidity trivially holds. $\Gamma$-rigidity for $\mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$ is immediate because all of the polynomials involved are $P_{i}(x,y) = y - x$, which are linear.
\end{proof}
\subsection{Quantifier elimination}
From now on, we assume that $T_{A}$ is $\Gamma$-rigid. Firstly, we provide some motivation for the definable sets we wish to consider. Fix a model $(L,k) \models T_{A}$ and suppose that $v = (v_{1}, \dots, v_{s})$ is a tuple from the sort $L$. We can re-index the $v_{i}$ according to the fibers of $\pi$ in which they appear. Namely, we fix an enumeration $\{v_{ij}: 1 \leq i \leq t; 1 \leq j \leq s_{i}, \sum_{i} s_{i} = s \}$ so that given $v_{ij}, v_{kl}$, we have $i = k$ if and only if $\pi(v_{ij}) = \pi(v_{kl})$. By the axioms, we can find $m$-tuples $a_{i} \in V \subseteq k^{m}$ such that $\pi(v_{ij}) = a_{i}$ for every $1 \leq i \leq t$. Moreover, because each $\pi^{-1}(a_{i})$ is one-dimensional and we have basis elements $e_{i} \in \mathbf{E}(L, a_{i})$, we can find scalars $\lambda_{ij} \in k$ such that
\[ \bigwedge_{i=1}^{t} \bigwedge_{j=1}^{s_{i}} \lambda_{ij} e_{i} = v_{ij} \] holds in $(L,k)$. Thus one expects all the sentences satisfied by $v$ to be determined by all the inter-relationships between the $e_{i}$. But the relationships between the $e_{i}$ depend on the orbits of $\Pi$ on $V$. We set up some notation to describe these relationships. Suppose that $e_{i}$ and $e_{j}$ lie in the same orbit of $\Pi$. Then there is a `path' in the structure connecting the fiber containing $e_{i}$ to the fiber containing $e_{j}$, i.e. there is $\mathbf{i} \in n^{< \omega}$ such that $\mathbf{U}_{\mathbf{i}}e_{i} \in \pi^{-1}(\pi(e_{j}))$. We wish to construct an existential sentence $\theta_{ij}$ that codes this path. Writing $e_{i}^{0}$ for $e_{i}$, our candidate for $\theta_{ij}$ is the following:
\[\exists \gamma_{ij} \exists_{k=1}^{p} b_{ijk} \exists_{k=1}^{p} e_{i}^{k} \left( \begin{array}{l}\bigwedge_{k = 1}^{p} \mathbf{E}(e_{i}^{k}, \eta_{i_{k}}\pi(e_{i}^{k-1})) \wedge \bigwedge_{k=1}^{p} \mathbf{U}_{i_{k}}e_{i}^{k-1} = \lambda_{i_{k}}(b_{ijk}) e_{i}^{k} \\ \wedge P_{i_{k}}(b_{ijk},\pi(e^{k-1}_{i})) \wedge e_{j} = \gamma_{ij} e^{p}_{i} \end{array} \right) \] This sentence is, of course, satisfied in $(L,k)$. We now have enough information to construct a class of formulas with which to prove quantifier elimination.
\begin{definition}
\label{definition - core formulas}
Let $\{v_{ij}: 1 \leq i \leq t; 1 \leq j \leq s_{i}, \sum_{i} s_{i} = s \}$ and $x = (x_{1}, \dots x_{r})$ be tuples of variables from the sorts $L$ and $k$ respectively. A \textbf{core formula} with variables $(v,x)$ is defined to be a formula of the following shape:
\[ \exists \lambda \exists_{i=1}^{t} e_{i} \exists_{i=1}^{t}y_{i} \exists \gamma \exists b\left( \begin{array}{ll} \bigwedge_{i=1}^{t} \bigwedge_{j =1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij} e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij}(e_{i}, e_{j}, b_{ij}, \gamma_{ij}) \\ \wedge S(\lambda, y, \gamma, b, x) \end{array} \right) \] where
\begin{enumerate}
\item $\Theta$ is a subset of $\{(i,j): 1 \leq i,j \leq t\}$.
\item $S$ defines a Zariski constructible subset of $k^{r_{1}} \times V^{t} \times \Gamma^{r_{2}}$ where
\begin{enumerate}
\item $r_{1} = l(x) + l(b) + s$ (l denotes length)
\item $r_{2} = l(\gamma)$
\end{enumerate}
\item $\phi_{ij}$ is $\theta_{ij}$ with the existential quantification over $\gamma_{ij}, b_{ik}$ removed.
\end{enumerate} A \textbf{core type} is defined to be a consistent collection of core formulas. If $(v,a)$ is a tuple of elements from $L^{s} \times k^{r}$, the \textbf{core type of $(v,a)$} (denoted $\ctp (v,a)$) is defined to be the set of all core formulas satisfied by $(v,a)$.
\end{definition} We now wish to demonstrate that the type of a tuple is determined by its core type.
\begin{proposition}
\label{proposition - core quantifier elimination}
Let $(L,k) \models T_{A}$ be $\aleph_{0}$-saturated. Suppose that $(v,c), (w,d)$ are both tuples from $L^{s} \times k^{r}$ with the property that $\ctp(v,c) = \ctp(w,d)$. Then $\tp(v,c) = \tp(w,d)$.
\end{proposition}
\begin{proof} We shall construct an automorphism $\tilde{\sigma}$ of $(L,k)$ with the property that $\tilde{\sigma}: (v,c) \mapsto (w,d)$. Re-index the tuple $v$ as $\{v_{ij}: 1 \leq i \leq t; 1 \leq j \leq s_{i}, \sum_{i} s_{i} = s \}$ so that given $v_{ij}, v_{kl}$, we have $i = k$ if and only if $\pi(v_{ij}) = \pi(v_{kl})$. By what has already been discussed, the axioms provide us with:
\begin{enumerate}
\item Tuples $a_{i}^{1}$ such that $\pi(v_{ij}) = a_{i}^{1}$ for every $1 \leq i \leq t$.
\item Basis elements $e_{i}^{1} \in \mathbf{E}(L, a_{i})$ and scalars $\lambda_{ij}^{1}$ such that
\[ \bigwedge_{i = 1}^{t} \bigwedge_{j=1}^{s_{i}} \lambda_{ij}^{1} e_{i}^{1} = v_{ij} \] holds.
\end{enumerate} Now we construct the set $\Theta$ so that $(i,j) \in \Theta$ if and only if there is a path from $\pi^{-1}(a_{i}^{1})$ to $\pi^{-1}(a_{j}^{1})$; namely there is a sequence $\mathbf{i} \in n^{< \omega}$ such that $\mathbf{U}_{\mathbf{i}}e_{i}^{1} \in \pi^{-1}(a_{j}^{1})$. Note then that there is a corresponding existential sentence $\theta_{ij}$ that codes this path. Thus the following conjunct holds in $(L,k)$:
\[ \bigwedge_{i=1}^{t} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = a_{i}^{1} \wedge \lambda_{ij}^{1}e_{i}^{1} = v_{ij} \wedge \mathbf{E}(e_{i}^{1}, a_{i}^{1}) \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij}(e^{1}_{i}, e^{1}_{j}, b^{1}_{ij}, \gamma^{1}_{i}) \] We shall denote the above formula by $\phi(v, e^{1}, a^{1}, \lambda^{1}, \gamma^{1}, b^{1})$. Consider the following set of formulas:
\[ \Sigma = \begin{array}{ll} \{\phi(w,e',x', \lambda', \gamma',b') \wedge S(x',\lambda',\gamma',b',d) : \\ \qquad (L,k) \models \phi(v,e^{1},a^{1},\lambda^{1}, \gamma^{1},b^{1}) \wedge S(a^{1}, \lambda^{1},\gamma^{1},b^{1},c) \} \end{array} \] Here the variables have been primed to distinguish them from actual parameters. The $S$ range over all constructible subsets of an appropriate cartesian power of $k$.
\\
\\
\textbf{Claim}: $\Sigma$ is consistent.
\begin{proof} We show that $\Sigma$ is finitely consistent. By definition $\Sigma$ is closed under finite conjuctions, so let $\phi \wedge S \in \Sigma$. Then
\[ (L,k) \models \phi(v,e^{1},a^{1},\lambda^{1},\gamma^{1},b^{1}) \wedge S(a^{1}, \lambda^{1},\gamma^{1},b^{1},c) \] Existentially quantifying out $e^{1},a^{1}, \lambda^{1}, \gamma^{1}$ and $b^{1}$, we obtain a core formula satisfied by $(v,c)$. But $\ctp(v,c) = \ctp(w,d)$, so there are $e^{2}, a^{2}, \lambda^{2}, \gamma^{2}, b^{2}$ such that
\[ (L,k) \models \phi(w,e^{1},a^{1},\lambda^{1},\gamma^{1},b^{1}) \wedge S(a^{1}, \lambda^{1},\gamma^{1},b^{1},d) \] as required.
\end{proof} By saturation, the type $\Sigma$ is satisfied by a tuple $(e^{2}, a^{2}, \lambda^{2}, \gamma^{2}, b^{2})$ say. In particular, we have that
\[ \tp^{k}(a^{1}, \lambda^{1}, \gamma^{1}, b^{1},c) = \tp^{k}(a^{2}, \lambda^{2}, \gamma^{2}, b^{2}, d) \] and by saturation of $k$ we therefore obtain an isomorphism $\sigma$ of $k$ such that
\[ \sigma: (a^{1}, \lambda^{1}, \gamma^{1}, b^{1},c) \mapsto (a^{2}, \lambda^{2}, \gamma^{2}, b^{2}, d) \] It remains to extend $\sigma$ to the whole of $(L,k)$. Partition $V$ up into orbits of $\Pi$:
\[ V = \prod_{x \in \Lambda} \Pi x \] for some set of representatives $\Lambda$. Let $x \in \Lambda$ be such that $\Pi x$ contains $a_{i_{1}} \dots a_{i_{q}}$ and no other $a_{i}$. By re-indexing if necessary, we can assume that the $a_{i_{j}}$ are listed in order of increasing length with respect to $x$. We may also assume that $x = a^{1}_{i_{1}}$ by changing the set of representatives if necessary. We carry out an induction on length which is identical to the proof of Theorem \ref{theorem - equivariant structures - categoricity}. One only has to note that the construction automatically maps $e_{i_{j}}^{1} \mapsto e_{i_{j}}^{2}$ for every $1 \leq j \leq q$. Indeed, suppose that there is a path from $\pi^{-1}(a_{i_{1}}^{1})$ to $\pi^{-1}(a_{i_{j}}^{1})$. Then it is coded in $\theta_{i_{1}, i_{j}}$. But $\phi_{i_{1}, i_{j}}(e^{1}_{i_{1}}, e^{1}_{i_{j}}, b^{1}_{i_{1}, i_{j}}, \gamma^{1}_{i_{1}, i_{j}})$ holds, hence so does $\phi_{i_{1}, i_{j}}(e^{2}_{i_{1}}, e^{2}_{i_{j}}, b^{2}_{i_{1}, i_{j}}, \gamma^{2}_{i_{1}, i_{j}})$ by the fact that $\Sigma$ is realized by $(w, e^{2}, a^{2}, \lambda^{2}, \gamma^{2}, b^{2})$. Thus the following conjunctions hold in $(L,k)$ for $l = 1,2$:
\[ \begin{array}{l} \bigwedge_{k=1}^{n} \mathbf{E}(e^{lk}_{i_{1}}, \eta'_{j_{k}} \pi(e^{l,k-1}_{i_{1}})) \wedge \bigwedge_{k=1}^{n} \mathbf{U}_{j_{k}}e^{l,k-1}_{i_{1}} = \lambda_{j_{k}}(b^{l}_{i_{1}, i_{j}, k}) e^{lk}_{i_{1}} \wedge P_{j_{k}}(b^{l}_{i_{1}, i_{j} ,k}, \pi(e^{l,k-1}_{i_{1}})) \\ \wedge e^{l}_{i_{j}} = \gamma^{l}_{i_{1}, i_{j}}e^{ln}_{i_{1}} \end{array} \] where $e^{l0}_{i_{1}} = e^{l}_{i_{1}}$.
\\
\\
\textbf{Claim}: $\tilde{\sigma}(e^{1k}_{i_{1}}) = e^{2k}_{i_{1}}$ for every $1 \leq k \leq n$.
\begin{proof} This holds by fiat for $k=0$. In constructing $\tilde{\sigma}$, we will have selected $e' \in \pi^{-1}(\eta_{j_{1}}(a^{2}_{i_{1}}))$ such that
\[ \mathbf{U}_{j_{1}}e^{2}_{i_{1}} = \lambda_{j_{1}}(b^{2}_{i_{1},i_{j}, 1})e' \wedge P_{j_{1}}(b^{2}_{i_{1}, i_{j} , 1}, a^{2}_{i_{1}}) \] holds. But $\sigma(b^{1}_{i_{1}, i_{j}, ,1}) = b^{2}_{i_{1}, i_{j}, 1}$, hence we have $\tilde{\sigma}(e^{11}_{i_{1}}) = e'$. But $\mathbf{U}_{j_{1}}e^{2}_{i_{1}} = \lambda_{j_{1}}(b^{2}_{i_{1}, i_{j},1})e^{21}_{i_{1}}$ by the long conjunct, hence $\tilde{\sigma}(e^{11}_{i_{1}}) = e^{21}_{i_{1}}$. Now we repeat the argument till we reach $k=n$.
\end{proof} Because $\sigma(\gamma_{i_{1}, i_{j}}^{1}) = \gamma_{i_{1},i_{j}}^{2}$, it now follows that $\tilde{\sigma}(e_{i_{j}}^{1}) = e_{i_{j}}^{2}$ as required.
\end{proof} It follows by compactness that every $\mathcal{L}_{A}$-formula with parameters from $k$ is equivalent to a boolean combination of core formulas. Some further analysis reveals the structure of subsets of $(L,k)$ defined using parameters from both $L$ and $k$. Intuitively, these should be determined by a class of formulas similar to core formulas, the only difference being that these formulas can also express information about how bases from the fibers containing these parameters from $L$ are connected to other fibers via paths.
\begin{definition}
\label{definition - general core formulas}
Let $e'$ be a tuple of elements from $L$ with length $p$ such that all $e'_{i}$ are basis elements. Let $v = (v_{1}, \dots, v_{m})$, $w = (w_{1}, \dots, w_{n})$ be tuples of variables from $L$. A \textbf{general core formula} with variables $(v,w)$ over $e'$ is defined to be a formula of the following shape:
\[ \exists \lambda \exists \mu \exists_{i=1}^{s} e_{i} \exists_{i=1}^{s} y_{i} \exists \gamma \exists b \left( \bigwedge_{i=1}^{s} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge S(\lambda, \mu, y, \gamma, b, x) \right) \] where
\begin{enumerate}
\item $\{v_{ij}: 1 \leq i \leq s, 1 \leq j \leq s_{i}\}$ is an appropriate enumeration of $v$
\item $\Theta \subseteq \{(i,j): 1 \leq i,j \leq s\}$
\item $S$ is a constructible subset of $k^{r_{1}} \times V^{s} \times \Gamma^{r_{2}}$ where
\begin{enumerate}
\item $r_{1} = l(x) + l(b) + m + n$
\item $r_{2} = l(\gamma)$
\end{enumerate}
\item $\phi$ is defined to be
\[ \bigwedge_{i=1}^{p} \bigwedge_{j=1}^{p_{i}} \mu_{ij} e'_{i} = w_{ij} \wedge \bigwedge_{(i,j) \in \Theta_{1}} \phi_{ij}(e'_{i}, e_{j}, b_{i}, \gamma_{ij}) \wedge \bigwedge_{(i,j) \in \Theta_{2}} \phi_{ij}(e_{i}, e'_{j}, b_{i}, \gamma_{ij}) \] where
\[ \Theta_{1} \subseteq \{(i,j): 1 \leq i \leq p, 1 \leq j \leq s\} \qquad \Theta_{2} \subseteq \{(i,j): 1 \leq i \leq s, 1 \leq j \leq p \} \] and $\{w_{ij}: 1 \leq i \leq p, 1 \leq j \leq p_{i}\}$ is an appropriate enumeration of $w$.
\end{enumerate}
We shall denote such a formula by $\exists e S$ and call $S$ the \textbf{Zariski constructible component} of $\exists e S$.
\end{definition}
\begin{proposition}
\label{proposition - general core quantifier elimination}
Let $(L,k) \models T_{A}$. If $\phi$ is formula with parameters from $L,k$ then it is equivalent to a boolean combination of general core formulas.
\end{proposition}
\begin{proof} Suppose that $\phi(v,x)$ is a formula with free variables $(v,x)$ over a finite set of parameters $w = (w_{1}, \dots, w_{p})$ of $L$ and some unspecified parameters from $k$. Then $\phi(v,x)$ is equivalent to some $\phi_{1}(v,w,x)$ where $\phi_{1}(v,w',x)$ is a formula with free variables $(v,w',x)$ merely over some set of parameters from $k$. Hence $\phi_{1}$ is equivalent to a boolean combination of core formulas over $k$ by Proposition \ref{proposition - core quantifier elimination}. Thus it suffices to prove that a core formula over $k$ with free variables $(v,w',x)$ is equivalent to a boolean combination of general core formulas after substituting $w'$ with the tuple $w$. It transpires that we have the stronger result that every core formula is equivalent to a finite disjunction of general core formulas after substitution, which we now show.
\\
\\
So let $\varphi(v,w',x)$ be a core formula. We can fix an enumeration $\{v_{ij}: 1 \leq i \leq s, 1 \leq j \leq s_{i}, \sum_{i} s_{i} = n\}$ of $(v,w')$ such that
\begin{enumerate}
\item $n$ is the length of $(v,w')$.
\item $\pi(v_{ij}) = \pi(v_{kl})$ if and only if $i = k$.
\item Those $v_{ij}$ for which $v_{ij}$ is not in $w'$ for any $j$ are listed first, i.e. there is a maximum $m \leq s$ such that $v_{ij} \not \in w'$ for all $i \leq m$.
\item For $i > t$, the $w'$ variables are listed last, i.e. there is a minimum $t_{i} \leq s_{i}$ such that $v_{ij} \in w'$ for all $i > t_{i}$.
\end{enumerate} Now $\varphi(v,w',x)$ looks like
\[ \exists \lambda \exists_{i=1}^{t} e_{i} \exists_{i=1}^{t}y_{i} \exists \gamma \exists b\left( \begin{array}{ll} \bigwedge_{i=1}^{t} \bigwedge_{j =1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij} e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij}(e_{i}, e_{j}, b_{ij}, \gamma_{ij}) \\ \wedge S(\lambda, y, \gamma, b, x) \end{array} \right) \] for some $\Theta \subseteq \{(i,j): 1 \leq i,j \leq s\}$ and $S$ over $k$. Substitute $w$ for $w'$. The resulting formula can be simplified by noting that some of the information it expresses is already contained in the theory. If $k > m$, then $y_{k} = \pi(w_{kl})$ is determined, thus $\exists y_{k}$ and such conjuncts can be dropped for $k > m$. Moreover, $\exists e_{k}$ can be dropped by replacing the formula with a finite disjunction, where each disjunct contains $e'_{k}$ for $e_{k}$ and $e'_{k}$ ranges over the finitely many canonical basis elements of $\pi^{-1}(y_{k})$. This allows us to make further deletions from each disjunct, namely $\mathbf{E}(e'_{k}, y_{k})$ (which trivially holds) and $\lambda_{kl}e'_{k} = w_{kl}$ for $l > t_{k}$ (because $\lambda_{kl}$ is determined), and we can therefore drop $\exists \lambda_{kl}$. This leaves us with the formula
\[ \bigvee_{\begin{array}{c} e' = (e_{t+1}, \dots, e_{s}) \\ e'_{k} \in \pi^{-1}(\pi(w_{kl})) \wedge \mathbf{E}(e'_{k}, \pi(w_{kl})) \end{array}} \exists \lambda \exists_{i=1}^{m} e_{i} \exists_{i=1}^{m}y_{i} \exists \gamma \exists b \left(\begin{array}{l} \bigwedge_{i=1}^{m} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \\ \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi' \end{array} \right) \] for appropriate $\phi'$ which we now determine. Clearly in
\[ \bigwedge_{(i,j) \in \Theta} \phi_{ij}(e_{i}, e_{j}, b_{ij}, \gamma_{ij}) \] if we substitute the parameters $e'_{k}$ for $k > m$ then some conjuncts are eliminable; namely those $\phi_{kl}(e'_{k}, e'_{l}, b_{kl}, \gamma_{kl})$ for $k,l > m$ (the theory itself tells us about paths that connect the fibers containing these $e'_{k}, e'_{l}$). Hence the quantifiers $\exists b_{kl}$ and $\exists \gamma_{kl}$ can also be eliminated from each disjunct. Define the sets
\[ \Theta_{1} = \{(i,j) \in \Theta: 1 \leq i \leq m, m < j \leq s \} \qquad \Theta_{2} = \{(i,j) \in \Theta: m < i \leq s, 1 \leq j \leq m \} \] \[ \Phi = \{(i,j) \in \Theta: 1 \leq i,j \leq m \} \] Then we have $\phi'$ as the formula
\[ \bigwedge_{i = m+1}^{s} \bigwedge_{j=1}^{t_{i}} \lambda_{ij}e'_{i} = v_{ij} \wedge \bigwedge_{(i,j) \in \Phi} \phi_{ij} \wedge \bigwedge_{i=1}^{2} \bigwedge_{(i,j) \in \Theta_{i}} \phi_{ij} \wedge S'(\lambda, y, \gamma, b, x) \] where $S'$ is $S$ with the determined parameters $\lambda_{kl}, y_{k}, b_{kl}$ and $\gamma_{kl}$ substituted for the appropriate variables. Now re-label, putting $\mu_{ij} = \lambda_{i+m, j}$. We see that each disjunct is a general core formula as required.
\end{proof}
\subsection{Constructibility}
From now on, we fix an equivariant algebra $A$, a $\Gamma$-rigid theory $T_{A}$ and model $(L,k)$ of $T_{A}$. Proposition \ref{proposition - general core quantifier elimination} suggests taking sets of the form $\exists e C$ (where $C$ defines a closed subset of a cartesian power of $k$) as giving us the closed subsets of a topology on the sorts of $(L,k)$ and their cartesian powers. As expected, it is possible to prove that all definable subsets are constructible after taking some technicalities (adapted suitably from \cite{Zil06}) into account. Namely, given that elements of $\Gamma$ may occur as parameters in $S$ for some general core formula $\exists e S$, we require some formalism dealing with how $S$ transforms under applications of $\Gamma$ to basis elements in the fibers.
\begin{definition}
\label{definition - gamma invariant}
Let $\exists e C$ be a general core formula with $C$ giving a closed subset of $k^{r_{1}} \times V^{s} \times \Gamma^{r_{2}}$. We define the \textbf{action} of $\delta \in \Gamma^{r_{2}}$ on $C$ to be
\[ C^{\delta} = \{(\lambda_{ij}, \mu, y, \gamma, b, a): (\delta_{i}^{-1}\lambda_{ij}, \mu, y, \delta \cdot \gamma, b, a) \in C\} \] where
\[ \delta \cdot \gamma = \left\{ \begin{array}{ll} \delta_{i}^{-1} \gamma_{ij} \delta_{j} & (i,j) \in \Theta \\ \gamma_{ij} \delta_{j} & (i,j) \in \Theta_{1} \\ \delta_{i}^{-1}\gamma_{ij} & (i,j) \in \Theta_{2} \end{array} \right. \] $C$ is defined to be \textbf{$\Gamma$-invariant} if $C^{\delta} = C$ for every $\delta \in \Gamma^{r_{2}}$.
\end{definition} The motivation for this definition is as follows. If a tuple $(v,w,a)$ satisfies $\exists e C$, then there are $\lambda, \mu, e, y, \gamma, b$ such that
\[ \bigwedge_{i=1}^{s} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge C(\lambda, \mu, y, \gamma, b, a)\] holds. Put $e'_{i} = \delta_{i}e_{i}$ for every $i$. Then one can see that $(v,w,a)$ satisfies
\[ \bigwedge_{i=1}^{s} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda'_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge C(\lambda', \mu, y, \delta \cdot \gamma, b, a)\] where $\lambda'_{ij} = \delta_{i}^{-1}\lambda_{ij}$ and $\delta \cdot \gamma$ is defined as above, only if $C^{\delta}(\lambda,\mu,y,\gamma,b,a)$ holds. In particular, if $C$ is $\Gamma$-invariant then for any $\delta \in \Gamma^{r_{2}}$ a base change of this kind can be carried out without affecting validity.
\begin{lemma}
\label{lemma - general core formulas - boolean combinations}
Let $\exists e C_{1}, \exists e C_{2}$ be general core formulas with the same enumeration of variables. Let $C_{1}, C_{2}$ be Zariski closed and suppose that $C_{2}$ $\Gamma$-invariant. Then
\begin{enumerate}
\item $(L,k) \models \exists e(C_{1} \wedge C_{2}) \leftrightarrow \exists e C_{1} \wedge \exists e C_{2}$
\item $(L,k) \models \exists e (\neg C_{2}) \leftrightarrow \neg \exists e C_{2}$
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Left to right is trivial. Conversely, if the right-hand side holds for a tuple $(v,w,a)$, then we may obtain different basis elements $e$ and $e'$ as witnesses to $\exists e C_{1}$ and $\exists e C_{2}$ respectively. But the $\Gamma$-invariance of $C_{2}$ means that we can transform $e'$ to $e$ without affecting validity. So the left-hand side holds.
\item Right to left is easy. Conversely, suppose that $(v,w,a)$ satisfies $\exists e (\neg C_{2})$ and that $e$ is a tuple of basis elements witnessing this. If some basis elements $e'$ witness $\exists e C_{2}$ then we can transform $e'$ to $e$, and using the $\Gamma$-invariance of $C_{2}$ we get a contradiction.
\end{enumerate}
\end{proof}
\begin{lemma}
\label{lemma - constructibility}
If $\exists e S$ is a general core formula then it is equivalent to a disjunction of general core formulas of the type $\exists e (C_{1} \wedge \neg C_{2})$ where $C_{1}, C_{2}$ are Zariski closed and $C_{2}$ is $\Gamma$-invariant.
\end{lemma}
\begin{proof} More or less the same as \cite{Zil06}. Fix a tuple $a \in k$ and recall that $tp^{k}(a)$ denotes the type of $a$ in the language of fields. Put
\[ \Sigma(a) = \{C_{1} \wedge \neg C_{2}: (L,k) \models (C_{1} \wedge \neg C_{2})(a) \mbox{ and $C_{2}$ is $\Gamma$-invariant} \} \] Then it suffices to prove (by Propositions \ref{proposition - core quantifier elimination} and \ref{proposition - general core quantifier elimination}) that $\Sigma(a) \models tp^{k}(a)$. By quantifier-elimination for $k$ and noting that every constructible subset is a disjunction of conjuncts of the kind $C_{1} \wedge \neg C_{2}$, it remains to prove that $C_{1}, C_{2}$ (where $C_{1} \wedge \neg C_{2} \in tp^{k}(a)$) can be replaced with $\tilde{C}_{1}, \tilde{C}_{2}$ (respectively) such that $\tilde{C}_{2}$ is $\Gamma$-invariant and $(L,k) \models (\tilde{C}_{1} \wedge \neg \tilde{C}_{2})\rightarrow (C_{1} \wedge \neg C_{2})$. Put
\[ \tilde{C}_{2} = \bigvee_{\delta \in \Gamma^{r_{2}}} C_{2}^{\delta} \] Evidently $\tilde{C}_{2}$ is closed, $\Gamma$-invariant and $\neg \tilde{C}_{2}$ implies $\neg C_{2}$. If $\tilde{C}_{2} \in p = tp^{k}(a)$ then we are done. Otherwise $\neg C_{2} \wedge \tilde{C}_{2} \in p$. Let $\Delta$ be the maximal (hence proper) subset of $\Gamma^{r_{2}}$ consisting of those $\delta$ such that
\[ \neg D = \bigwedge_{\delta \in \Delta} \neg C_{2}^{\delta} \in p \] $\Delta$ is non-empty because $1 \in \Delta$. Put
\[ \Stab(\Delta) = \{\delta \in \Gamma^{r_{2}}: \delta \Delta = \Delta \} \] If $\delta \not \in \Stab(\Delta)$ then by maximality of $\Delta$ we have $\neg D^{\delta} \wedge \neg D \not \in p$, hence $D^{\delta}\in p$. Thus
\[ \bigwedge_{\delta \in \Gamma^{r_{2}}\setminus \Stab(\Delta)} D^{\delta} \in p \] \textbf{Claim}: We have
\[ (L,k) \models \bigwedge_{\delta \in \Gamma^{r_{2}} \setminus \Stab(\Delta)} D^{\delta} \wedge \bigvee_{\delta \in \Gamma^{r_{2}}} \neg D^{\delta} \rightarrow \bigvee_{\delta \in \Stab(\Delta)} \neg D^{\delta} \]
\begin{proof} Suppose that $b \in k$ is such that $D^{\delta}(b)$ holds for every $\delta \in \Gamma^{r_{2}} \setminus \Stab(\Delta)$ and $\neg D^{\delta_{1}}(b)$ holds for some $\delta_{1} \in \Gamma^{r_{2}}$. Then $\delta_{1} \in \Stab(\Delta)$ and the claim follows.
\end{proof} The latter disjunct is clearly equivalent to $\neg D$ and $\neg D$ implies $\neg C_{2}$. So we take
\[ \tilde{C}_{1} = C_{1} \wedge \bigwedge_{\delta \in \Gamma^{r_{2}}\setminus \Stab(\Delta)} D^{\delta} \] and replace $\tilde{C}_{2}$ with $\bigwedge_{\delta \in \Gamma^{r_{2}}} D^{\delta}$. The result now follows.
\end{proof}
\begin{proposition}
\label{proposition - constructibility}
All definable subsets of $(L,k)$ are constructible, namely every definable subset of $(L,k)$ is a boolean combination of those defined by general core formulas $\exists e C$ where $C$ is Zariski closed and $\Gamma$-invariant.
\end{proposition}
\begin{proof} This is almost immediate by Proposition \ref{proposition - general core quantifier elimination} and Lemmas \ref{lemma - general core formulas - boolean combinations} and \ref{lemma - constructibility}. One has to note, additionally that if we have a general core formula $\exists e C$ where $C$ is just closed, we can replace $C$ with
\[ \tilde{C} = \bigvee_{\delta \in \Gamma^{r_{2}}} C^{\delta} \] to obtain something closed and $\Gamma$-invariant.
\end{proof}
\section{Model Theory of Equivariant Structures: II}
We conclude our analysis of models of $\Gamma$-rigid $T_{A}$ by demonstrating that they are Zariski structures. Intuitively, by inspection of general core formulas one expects all the relevant properties to be verified to reduce predictably to the corresponding properties for algebraic varieties. Vaguely speaking, the Zariski constructible components `dominate' the geometry. We fix an equivariant algebra $A$, $\Gamma$-rigid theory $T_{A}$ and model $(L,k) \models T_{A}$.
\subsection{Topology on $(L,k)$}
We introduce a topology on $L^{n} \times k^{m}$ by taking as a basis of closed subsets those subsets that are defined by general core formulas $\exists e C(v,w,x)$ ($(v,w,x)$ a tuple of variables from $L^{n} \times k^{m}$) where $C$ is Zariski closed and $\Gamma$-invariant. Closed subsets are given by finite unions and arbitrary intersections of basic closed subsets. Note that if $n = 0$, then these formulas reduce to those of the form $C(x)$ where $C$ defines a Zariski closed subset of $k^{m}$. Thus the topology on $(L,k)$ gives us the classical Zariski topology on the sort $k$ and its cartesian powers.
\begin{lemma}
\label{lemma - general core formulas - equivalence}
Let $\exists e C_{1}$, $\exists e C_{2}$ be general core formulas defining basic closed subsets and suppose that both formulas have the same enumeration of $v$ variables. Then
\[ (L,k) \models \exists e C_{1} \leftrightarrow \exists e C_{2} \Rightarrow (L,k) \models C_{1} \leftrightarrow C_{2} \]
\end{lemma}
\begin{proof} By Lemma \ref{lemma - general core formulas - boolean combinations}, $\exists e C_{1} \wedge \neg \exists e C_{2}$ is equivalent to $\exists e (C_{1} \wedge \neg C_{2})$ hence $C_{1} \wedge \neg C_{2}$ must be inconsistent. The rest of the lemma follows by symmetry.
\end{proof} Although a general core formula $\exists e S$ was defined with respect to two tuples of variables $v = (v_{1}, \dots, v_{m})$ and $w = (w_{1}, \dots, w_{n})$, we shall henceforth amalgamate these into one tuple which we enumerate as $\{v_{ij}: 1 \leq i \leq s, 1 \leq j \leq s_{i} \}$ where there is $t \leq s$ for which $v_{ij} \in w$ for all $i > t$.
\begin{proposition}
\label{proposition - Noetherian topology}
The topology defined on $(L,k)$ is Noetherian.
\end{proposition}
\begin{proof} Suppose for contradiction that $(\exists e C_{i}: i \in \mathbb{N})$ defines an infinite descending chain of basic closed subsets, i.e. we have proper inclusions $\exists e C_{i}(L,k) \supset \exists e C_{i+1}(L,k)$ for every $i$. Because there are only finitely many ways of enumerating the variables $v$ as $\{v_{ij}: 1 \leq i \leq s, 1 \leq j \leq s_{i}\}$, there are infinitely many $\exists e C_{i}$ with the same enumeration. Hence we can assume, without loss of generality, that all $\exists e C_{i}$ have the same enumeration of $v$ variables. By Lemma \ref{lemma - general core formulas - boolean combinations},
\[ \exists e C_{i+1}(L,k) = ( \exists e C_{i} \wedge \exists e C_{i+1})(L,k) = \exists e (C_{i} \wedge C_{i+1})(L,k) \] By Lemma \ref{lemma - general core formulas - equivalence} it follows that $C_{i}(k) \supseteq C_{i+1}(k)$. Because $\exists e C_{i}(L,k) \supset \exists e C_{i+1}(L,k)$, Lemma \ref{lemma - general core formulas - boolean combinations} gives that $\exists e(C_{i} \wedge \neg C_{i+1})$ is satisfiable. Thus we have proper inclusions $C_{i}(k) \supset C_{i+1}(k)$ for every $i$, contradicting that the Zariski topology is Noetherian.
\end{proof}
\subsection{Finer Results on Projections and Intersections}
We now work towards the proof that $(L,k)$ is a Zariski structure. A quick glance at the axioms to be verified will indicate that we require more detailed results about projections and intersections of subsets defined by general core formulas. By the results of the previous chapter, it is immediate that a projection of a constructible set is constructible. For a subset defined by a general core formula, we have more.
\begin{proposition}
\label{proposition - general core formulas - projections}
Let $\exists e S$ be a general core formula with the aforementioned convention on enumeration of variables. For a fixed $1 \leq i \leq s$, let $j$ range over a subset $J \subseteq \{1, \dots, s_{i}\}$. Then $\exists v_{ij} \exists e S$ is a general core formula with Zariski constructible component equivalent to one of the following:
\begin{enumerate}
\item $\exists_{j \in J} \lambda_{ij}S$.
\item $\exists_{j \in J} \mu_{i-t,j}S$
\item $\exists \mu_{i-t,1} \exists_{(i-t,j) \in \Theta_{1}} b_{i-t,j} \exists_{(i-t,j) \in \Theta_{1}} \gamma_{i-t,j} \exists_{(k,i-t)\in \Theta_{2}} b_{k,i-t} \exists_{(k,i-t) \in \Theta_{2}} \gamma_{i-t,k} S$
\item $\begin{array}{l} \exists y_{i} \exists_{(i,k) \in \Theta} b_{ik} \exists_{(i,k) \in \Theta} \gamma_{ik} \exists_{(j,i) \in \Theta} b_{ji} \exists_{(j,i) \in \Theta} \gamma_{ji} \\ \exists_{(j-t,i) \in \Theta_{1}} b_{j-t,i} \exists_{(j-t,i) \in \Theta_{1}} \gamma_{j-t,i} \exists_{(i,j-t)\in \Theta_{2}} b_{i,j-t} \exists_{(i,j-t) \in \Theta_{2}} \gamma_{i,j-t} S \end{array}$
\end{enumerate}
\end{proposition}
\begin{proof} The proof divides into four cases:
\begin{enumerate}
\item $1 \leq i \leq t$.
\item $t+1 \leq i \leq s$.
\item $t+1 \leq i \leq s$ and $s_{i} = 1$.
\item $1 \leq i \leq t$ and $s_{i} = 1$.
\end{enumerate} We deal with each of these in turn.
\begin{enumerate}
\item In this case the $v_{ij}$ do not occur in $\phi$ and we can eliminate the conjuncts $\lambda_{ij}e_{i} = v_{ij}$, thus moving the quantifiers $\exists_{j \in J} \lambda_{ij}$ to $S$.
\item In this case $\exists v_{ij} \exists e S$ is equivalent to
\[ \exists \lambda \dots \exists b \left( \bigwedge_{i=1}^{t} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \exists v_{ij} \phi \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge S(\lambda, \mu, y, \gamma, b, a) \right) \] Recall that $\phi$ is
\[ \begin{array}{ll} \bigwedge_{i=t+1}^{p} \bigwedge_{j=1}^{p_{i}} \mu_{i-t,j} e'_{i-t} = v_{ij} \wedge \bigwedge_{(i-t,j) \in \Theta_{1}} \phi_{i-t,j}(e'_{i-t}, e_{j}, b_{i-t,j}, \gamma_{i-t,j}) \wedge \\ \bigwedge_{(i,j-t) \in \Theta_{2}} \phi_{i-t,j}(e_{i}, e'_{j-t}, b_{i,j-t}, \gamma_{i,j-t}) \end{array} \] Thus $\exists v_{ij} \phi$ is equivalent to $\phi'$, where the latter is $\phi$ but with the conjuncts $\mu_{i-t,j} e'_{i-t} = v_{ij}$ removed for $j \in J$. It follows that we can move the quantifiers $\exists_{j \in J} \mu_{i-t,j}$ to $S$ as required.
\item This case is similar to $2$, but more is eliminable from $\phi$ because we can get rid of the parameter $e'_{i}$. Hence we can eliminate $\phi_{i-t,k}(e'_{i-t}, e_{k}, b_{i-t,k}, \gamma_{i-t,k})$ and $\phi_{k,i-t}(e_{k}, e'_{i-t}, b_{k,i-t}, \gamma_{k,i-t})$. The quantifiers
\[ \exists \mu_{i-t,1} \exists_{(i-t,j) \in \Theta_{1}} b_{i-t,j} \exists_{(i-t,j) \in \Theta_{1}} \gamma_{i-t,j} \exists_{(k,i-t)\in \Theta_{2}} b_{k,i-t} \exists_{(k,i-t) \in \Theta_{2}} \gamma_{k, i-t} \] can then be moved to $S$.
\item The most is eliminable in this case. We no longer require $\mathbf{E}(e_{i}, y_{i})$ and those conjuncts $\phi_{jk}$ with $(j,k) \in \Theta$ and $j$ or $k$ equal to $i$. But we can also eliminate conjuncts from $\phi$, namely $\phi_{j-t,i}$ for $(j-t,i) \in \Theta_{1}$ and $\phi_{i, j-t}$ for $(i, j-t) \in \Theta_{2}$. Thus we move the quantifiers
\[ \begin{array}{l} \exists y_{i} \exists_{(i,k) \in \Theta} b_{ik} \exists_{(i,k) \in \Theta} \gamma_{ik} \exists_{(j,i) \in \Theta} b_{ji} \exists_{(j,i) \in \Theta} \gamma_{ji} \\ \exists_{(j-t,i) \in \Theta_{1}} b_{j-t,i} \exists_{(j-t,i) \in \Theta_{1}} \gamma_{j-t,i} \exists_{(i,j-t)\in \Theta_{2}} b_{i,j-t} \exists_{(i,j-t) \in \Theta_{2}} \gamma_{i,j-t} \end{array} \] to $S$.
\end{enumerate}
\end{proof} What if two general core formulas defining basic closed subsets of $L^{n} \times k^{m}$ each have different enumerations of variables, but we wish to determine the intersection of the subsets they define? In this case, we require a common enumeration of both formulas and Lemma \ref{lemma - general core formulas - boolean combinations} should apply, providing that the resulting Zariski constructible components (after re-enumeration) are $\Gamma$-invariant.
\begin{lemma}
\label{lemma - general core formulas - intersections}
Let $\exists e C_{1}$, $\exists e C_{2}$ be general core formulas defining basic closed subsets of $L^{n} \times k^{m}$. Then $\exists e C_{1} \wedge \exists e C_{2}$ is equivalent to a general core formula $\exists e D$ where $D$ is equivalent to
\[ \bigwedge_{p=1}^{2} C_{p}(\lambda,\mu,y,\gamma,b,x) \wedge \bigwedge_{\alpha_{p}(i,1) \sim_{12} \alpha_{p}(j,1)} y_{i} = y_{j} \wedge \bigwedge_{(\alpha_{p}(i,1), \alpha_{p}(j,1)) \sim_{12} (\alpha_{p}(k,1), \alpha_{p}(l,1))} b_{ij} = b_{kl} \wedge \gamma_{ij} = \gamma_{kl} \] for some equivalence relation $\sim_{12}$ on $\{1, \dots, n\}$.
\end{lemma}
\begin{proof} Suppose that $\exists e C_{1}$ and $\exists e C_{2}$ have enumerations
\[ \{v_{ij}: 1 \leq i \leq s_{1}, 1 \leq j \leq s_{1i} \} \qquad \{v_{ij}: 1 \leq i \leq s_{2}, 1 \leq j \leq s_{2i} \} \] respectively. Linearly enumerate the $v_{ij}$ as $v = (v_{1}, \dots, v_{n})$. Thus we obtain bijective maps
\[ \alpha_{p}: \{(i,j): 1 \leq i \leq s_{p}, 1 \leq j \leq s_{p_{i}}\} \rightarrow \{1, \dots, n\} \] for $p = 1,2$. Now introduce the equivalence relations $\sim_{p}$ on $\{1, \dots, n\}$ by
\[ i \sim_{p} j \leftrightarrow \pi_{1}(\alpha_{p}^{-1}(i)) = \pi_{1}(\alpha_{p}^{-1}(j)) \] where $\pi_{1}$ is the projection onto the first coordinate. Let $\sim_{12}$ denote the symmetric closure of the composition $\sim_{2} \circ \sim_{1}$ (hence $\sim_{12}$ is an equivalence relation). It is easy to see that each $\sim_{p}$ refines $\sim_{12}$. Each equivalence class $[i]_{12}$ of $\sim_{12}$ has a canonical representative (take the smallest $i$ in the class) and we let $\Lambda = \{k_{1}, \dots, k_{s}: k_{i} < k_{i+1} \mbox{ for all } i < t\}$ be the set of such representatives. Then one can define a map $\tau: \{1, \dots, n\} \rightarrow \{(i,j): 1 \leq i \leq t, 1 \leq j \leq t_{i}\}$ such that
\begin{enumerate}
\item $\pi_{1}(\tau(j)) = i$ if and only if $j \sim_{12} k_{i}$
\item $\pi_{2}(\tau(j)) < \pi_{2}(\tau(j'))$ if and only if $j < j'$
\end{enumerate} where $\pi_{2}$ denotes the projection onto the second coordinate, and this gives us an new enumeration of $v$.
\\
\\
\textbf{Claim}: For $p=1,2$, $\exists e C_{p}$ is equivalent to
\[ \exists \lambda \exists \mu \exists e \exists \gamma \exists b \left( \bigwedge_{i=1}^{t} \bigwedge_{j=1}^{t_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge C'_{p}(\lambda, \mu, y, \gamma,b,x) \right) \] where $C'_{p}$ is equivalent to
\[ C_{p}(\lambda,\mu,y,\gamma,b,x) \wedge \bigwedge_{\alpha_{p}(i,1) \sim_{12} \alpha_{p}(j,1)} y_{i} = y_{j} \wedge \bigwedge_{(\alpha_{p}(i,1), \alpha_{p}(j,1)) \sim_{12} (\alpha_{p}(k,1), \alpha_{p}(l,1))} b_{ij} = b_{kl} \wedge \gamma_{ij} = \gamma_{kl} \]
\begin{proof} First we obtain a formula that is equivalent to $\exists e C_{p}$ using the linear enumeration of $v$ given by $\alpha_{p}$. Define $C_{\alpha_{p}}(\lambda, \mu, y, \gamma, b, x)$ to be $C_{p}$ with the variables enumerated as follows:
\begin{enumerate}
\item $\lambda_{ij} \mapsto \lambda_{\alpha_{p}(i,j)}$
\item $y_{i} \mapsto y_{\alpha_{p}(i,1)}$
\item $\gamma_{ij} \mapsto \gamma_{\alpha_{p}(i,1), \alpha_{p}(j,1)}$
\item $b_{ijk} \mapsto b_{\alpha_{p}(i,1), \alpha_{p}(j,1),k}$
\end{enumerate} Clearly $\exists e C_{p}$ is then equivalent to
\[ \exists_{i=1}^{n} \lambda \exists \mu \exists_{i=1}^{n} e_{i} \exists \gamma \exists b \left( \begin{array}{l} \bigwedge_{i=1}^{n} \pi(v_{i}) = y_{i} \wedge \lambda_{i}e_{i} = v_{i} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \bigwedge_{i \sim_{p} j} e_{i} = e_{j} \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta'} \phi_{ij} \\ \wedge C_{\alpha_{p}}(\lambda, \mu, y, \gamma, b, x) \wedge \bigwedge_{i \sim_{p} j} y_{i} = y_{j} \wedge \bigwedge_{(i,j) \sim_{p} (k,l)} \gamma_{ij} = \gamma_{kl} \wedge b_{ij} = b_{kl} \end{array} \right) \] where $(i,j) \sim_{p} (k,l)$ is defined to hold if and only if $i \sim_{p} k$ and $j \sim_{p} l$; and $\Theta'$ is an appropriate subset of $\{(i,j): 1 \leq i,j \leq n\}$. Now we define $C_{\tau^{-1} \circ \alpha_{p}}(\lambda, \mu, y, \gamma, b, x)$ to be $C_{\alpha_{p}}$ but with the following enumeration of variables
\begin{enumerate}
\item $\lambda_{i} \mapsto \lambda_{\tau(i)}$
\item $y_{i} \mapsto y_{\pi_{1}(\tau(i))}$
\item $\gamma_{ij} \mapsto \gamma_{\pi_{1}(\tau(i)), \pi_{1}(\tau(j))}$
\item $b_{ijk} \mapsto b_{\pi_{1}(\tau(i)), \pi_{1}(\tau(j)),k}$
\end{enumerate} Because $\sim_{p}$ refines $\sim_{12}$, $i \sim_{p} j$ implies that $\pi_{1}(\tau(i)) = \pi_{1}(\tau(j))$. So using the enumeration of $v$ given by $\tau$, we see that the above formula is equivalent to
\[ \exists \lambda \exists \mu \exists e \exists \gamma \exists b \left( \bigwedge_{i=1}^{t} \bigwedge_{j=1}^{t_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij} e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta''} \phi_{ij} \\ \wedge C_{\tau^{-1} \circ \alpha_{p}}(\lambda, \mu, y, \gamma, b, x) \right) \] where $\Theta''$ is an appropriate subset of $\{(i,j): 1 \leq i,j \leq t\}$. It is easy to see that $C_{\tau^{-1} \circ \alpha_{p}}$ is equivalent to $C_{p}$ with the exception that some of the $y_{i}, b_{ij}, \gamma_{ij}$ become identified according to $\sim_{12}$, and hence the claim follows.
\end{proof} Clearly $C'_{p}$ is also $\Gamma$-invariant, hence by Lemma \ref{lemma - general core formulas - boolean combinations} we obtain the required result.
\end{proof} By definition of the topology on $(L,k)$, it follows by Lemma \ref{lemma - general core formulas - intersections} that all closed subsets are finite unions of basic closed subsets. Thus if a closed subset $\mathbf{C}$ is irreducible, it is basic closed.
\subsection{Zariski Structure}
Let $\exists e C$ be a general core formula defining a basic closed subset. By Lemma \ref{lemma - general core formulas - intersections}, changing the enumeration of the variables can potentially affect $C$ by introducing identifications. Hence if $\exists e C$ is equivalent to $\exists e C'$ where the latter has a different enumeration of variables, it is possible that $\dim C(k) > \dim C'(k)$. Sticking to our philosophy (and corroborative results) that the geometry on $k$ `dominates' the geometry on $(L,k)$, we wish to define the dimension of $\exists e C(L,k)$ to be $\dim C(k)$ for suitable $C$. For this purpose we take the general core formula $\exists e \hat{C}$ defining $\exists e C(L,k)$ with the finest enumeration of $v$ variables possible; namely such that if $\exists e C'$ also defines $\exists e C(L,k)$ then $\sim_{\hat{C}}$ refines $\sim_{C'}$ (with $\sim_{\hat{C}}$ and $\sim_{C'}$ defined as in Lemma \ref{lemma - general core formulas - intersections}). Such an enumeration is clearly possible because there are only finitely many possible enumerations, and we shall call $\exists e \hat{C}$ the \textbf{canonical presentation} of $\exists e C$.
\begin{definition}
\label{definition - dimension}
Let $\exists e C$ define a closed irreducible subset of $L^{n} \times k^{m}$. We define the \textbf{dimension} of $\exists e C(L,k)$ to be
\[ \dim \exists e C(L,k) := \dim \hat{C}(k) \] For $\exists e C(L,k)$ a closed subset, $\dim \exists e C(L,k):=\max\{\mathbf{C}_{i}\}$ where $\mathbf{C}_{i}$ are the irreducible components of $\exists e C(L,k)$. If $\exists e S(L,k)$ is constructible, its dimension is defined to be the dimension of its closure.
\end{definition}
\begin{lemma}
\label{lemma - general core formulas - irreducible sets}
Let $\exists e \hat{C}(L,k)$ be closed and irreducible. Then $\hat{C} = \bigvee D^{\delta}$ where $D$ defines some closed irreducible subset of $\hat{C}(k)$.
\end{lemma}
\begin{proof} Let $D$ be any irreducible component of $\hat{C}$. Then $\bigvee_{\delta \in \Gamma^{r_{2}}} D^{\delta}$ is closed and $\Gamma$-invariant. Hence $\exists e \bigvee D^{\delta}$ defines a basic closed subset. Because $\exists e \hat{C}(L,k)$ is irreducible, it follows that $\exists e \hat{C}(L,k) = \exists e \bigvee D^{\delta}(L,k)$ and by Lemma \ref{lemma - general core formulas - equivalence} the result follows.
\end{proof}
\begin{lemma}
\label{lemma - good dimension}
The notion of dimension defined in Definition \ref{definition - dimension} satifies conditions $1 - 5$ of the definition of a Zariski structure (Definition \ref{definition - Zariski structure - Noetherian}).
\end{lemma}
\begin{proof} For ease of reference, we restate the conditions to be verified:
\begin{enumerate}
\item The dimension of a point is $0$.
\item $\dim(\mathbf{P}_{1} \cup \mathbf{P}_{2}) = \max\{\dim \mathbf{P}_{1}, \dim \mathbf{P}_{2}\}$ for all projective subsets $\mathbf{P}_{1}, \mathbf{P}_{2}$.
\item For $\mathbf{C}$ closed and irreducible in $\mathbf{X}^{n}$ and $\mathbf{C}_{1}$ a closed subset of $\mathbf{C}$, if $\mathbf{C}_{1} \neq \mathbf{C}$ then $\dim \mathbf{C}_{1} < \dim \mathbf{C}$.
\item For $\mathbf{C}$ irreducible and closed in $\mathbf{X}^{n}$, if $\pi: \mathbf{X}^{n} \rightarrow \mathbf{X}^{m}$ is a projection then
\[ \dim \mathbf{C} = \dim \pi(\mathbf{C}) + \min_{a \in \pi(\mathbf{C})} \dim (\pi^{-1}(a) \cap \mathbf{C}) \]
\item For any irreducible closed $\mathbf{C}$ in $\mathbf{X}^{n}$ and projection map $\pi: \mathbf{X}^{n} \rightarrow \mathbf{X}^{m}$, there is a subset $\mathbf{V}$ relatively open in $\pi(\mathbf{C})$ such that
\[ \min_{a \in \pi(\mathbf{C})} \dim(\pi^{-1}(a) \cap \mathbf{C}) = \dim(\pi^{-1}(v) \cap \mathbf{C}) \] for every $v \in \mathbf{V} \cap \pi(\mathbf{C})$.
\end{enumerate}
The conditions on dimensions of points and dimensions of unions are trivial. Let $\exists e \hat{C}$ define an irreducible closed subset. By Lemma \ref{lemma - general core formulas - irreducible sets}, $\hat{C} = \bigvee D^{\delta}$ for some closed irreducible $D(k) \subseteq \hat{C}(k)$. If some $\exists e \hat{C}_{1}$ defines a proper closed subset of $\exists e \hat{C}(L,k)$ then it has the same enumeration of variables because both are canonically presented. Thus $\hat{C}_{1}(k) \subset \hat{C}(k)$ by Lemmas \ref{lemma - general core formulas - boolean combinations} and \ref{lemma - general core formulas - equivalence}. But then $\dim (\hat{C}_{1} \wedge D^{\delta})(k) < \dim D^{\delta}(k) = \dim \hat{C}(k)$ for some $\delta$, hence $\dim \hat{C}_{1}(k) < \dim \hat{C}(k)$, verifying condition $3$.
\\
\\
Now suppose that we have a projection $\pi: L^{n_{1} + n_{2}} \times k^{m_{1} + m_{2}} \rightarrow L^{n_{1}} \times k^{m_{1}}$. Then $\pi(\exists e \hat{C}(L,k))$ is defined by $\exists e \hat{C}'$ where $\hat{C}' = \exists z \hat{C}$ for some appropriate $z \in k$. Thus it remains to prove that
\[ \dim \hat{C}(k) = \dim \exists z \hat{C}(k) + \min \dim (\hat{C}(a,k)) \] where $\hat{C}(a,k) = \pi^{-1}(a) \cap \hat{C}(k)$. But this is known for algebraic varieties, thus giving us $4$. Condition $5$ is proved similarly.
\end{proof}
\begin{theorem}
\label{theorem - Zariski structure}
$(L,k)$ is a Zariski structure.
\end{theorem}
\begin{proof} By Lemma \ref{lemma - good dimension}, it remains to establish semi-properness of projection maps, but this is immediate by constructibility.
\end{proof}
\section{More on Equivariant Zariski Structures}
We conclude our investigation of equivariant algebras and their associated structures by constructing a functor $\nSpec: \mathsf{Equiv}(k)_{\Gamma}^{op} \rightarrow \mathsf{Zar}$, where $\mathsf{Equiv}(k)_{\Gamma}$ is defined to be the full subcategory of $\mathsf{Equiv}(k)$ consisting of those equivariant $k$-algebras whose associated theories are $\Gamma$-rigid. Some additional remarks on equivariant algebras are made.
\subsection{Functorial Correspondence}
We choose a candidate $\nSpec A = (L,\mathbb{K})$ where $\mathbb{K}$ is a large saturated algebraically closed field. We demonstrate that the following diagram commutes:
\[ \xymatrix{ \mathsf{B}^{op} \ar[r] \ar@{^{(}->}[d] & \mathsf{Zar}^{c} \ar@{^{(}->}[d] \\ \mathsf{Equiv}(k)_{\Gamma}^{op} \ar[r] & \mathsf{Zar} } \] where $\mathsf{B}$ is taken to be the category of commutative affine Hopf $k$-algebras, because it is anti-equivalent to the category of affine algebraic groups (Appendix \ref{appendix - Hopf algebras}). The reader may wish to review the contents of Appendix \ref{appendix - Hopf algebras} (in particular Definition \ref{definition - module algebras} and Proposition \ref{proposition - adjoint representation - module algebra}) before embarking on the following results.
\begin{lemma}
\label{lemma - module algebras - eigenvectors}
Let $H$ be a Hopf algebra, $A$ a $H$-module algebra. Suppose that $\mathbf{A}_{1}, \mathbf{A}_{2} \in A$ are eigenvectors of the action of $H$ on $A$, i.e. there are characters $\chi_{i}: H \rightarrow k$ such that
\[ h \cdot \mathbf{A}_{i} = \chi_{i}(h)\mathbf{A}_{i} \qquad h \in H \qquad i = 1,2 \] Then $\mathbf{A}_{1}\mathbf{A}_{2}$ is a $H$-eigenvector with character $\chi(h) = \sum_{(h)} \chi_{1}(h')\chi_{2}(h'')$ for every $h \in H$.
\end{lemma}
\begin{proof} Given $h \in H$ we have
\[ h \cdot (\mathbf{A}_{1}\mathbf{A}_{2}) = \sum_{(h)} (h' \cdot \mathbf{A}_{1})(h'' \cdot \mathbf{A}_{2}) = \sum_{(h)} \chi_{1}(h')\chi_{2}(h'') \mathbf{A}_{1}\mathbf{A}_{2} \] as required.
\end{proof}
\begin{proposition}
\label{proposition - equivariant algebras - functor}
Given a morphism $\varphi: A \rightarrow B$ in $\mathsf{Equiv}(k)_{\Gamma}$, there is a morphism of Zariski structures $\nSpec \varphi: \nSpec B \rightarrow \nSpec A$.
\end{proposition}
\begin{proof} Let $\nSpec A = (L_{A}, V_{A}, \mathbb{K})$, $\nSpec B = (L_{B}, V_{B}, \mathbb{K})$. Suppose that $A$ is equivariant with respect to the Hopf subalgebra $H$ and elements $\mathbf{U}_{11}, \dots \mathbf{U}_{1n_{1}}$ of $A$. Then there is a Hopf subalgebra $H'$ of $B$ such that
\begin{enumerate}
\item $B$ is equivariant with respect to $H'$ and $\mathbf{U}_{21}, \dots, \mathbf{U}_{2n_{2}}$.
\item $\varphi(\mathbf{U}_{1i})$ is a monomial in $\mathbf{U}_{21}, \dots \mathbf{U}_{2n_{2}}$ for each $i$.
\item $\varphi|_{H}: H \rightarrow H'$.
\end{enumerate} Without loss of generality, we can assume that the $\varphi(\mathbf{U}_{1i})$ occur amongst the $\mathbf{U}_{21}, \dots, \mathbf{U}_{2n_{2}}$. Indeed, if some $\varphi(\mathbf{U}_{1i})$ does not occur amongst the $\mathbf{U}_{2j}$ for $1 \leq j \leq n_{2}$, we can add it to this set of generators as a primitive. By Proposition \ref{proposition - adjoint representation - module algebra}, $B$ is a $H'$-module algebra under the adjoint representation. By repeated application of Lemma \ref{lemma - module algebras - eigenvectors}, $\varphi(\mathbf{U}_{1i})$ is a $H'$-eigenvector because it is generated by the $\mathbf{U}_{2j}$. Recall that by the definition of an equivariant algebra (Definition \ref{definition - equivariant algebras}), if $\Pi' = \langle \eta'_{1}, \dots, \eta'_{n_{2}} \rangle$ is the group associated to $B$, then for each $1 \leq j \leq n_{2}$ there must be a $k \leq n_{2}$ such that $\eta_{j}^{-1} = \eta_{k}$. Thus by adding $\varphi(\mathbf{U}_{1i})$ as a generator, we must also add another generator so that this property remains satisfied. This can be done because $\varphi(\mathbf{U}_{1i})$ is a monomial in the $\mathbf{U}_{2j}$.
\\
\\
Now we put $B' = \{\phi(\mathbf{U}_{1i}): 1 \leq i \leq n_{1}\}$. If $\Pi$ is the group associated with $A$, we put $\phi(\Pi)$ as the subgroup of $\Pi'$ generated by those $\eta'_{j}$ for which $\mathbf{U}_{2j} \in B'$. Because $\varphi_{H}: H \rightarrow H'$, we have a corresponding morphism of varieties $f: V_{B} \rightarrow V_{A}$. Partition $V_{B}$ into orbits of $\phi(\Pi)$:
\[ V_{B} = \bigcup_{x \in \Lambda'} \phi(\Pi)x \] for some set of representatives $\Lambda'$. The map $\nSpec \varphi: L_{B} \rightarrow L_{A}$ is then constructed fiberwise on orbits, using an inductive procedure analogous to the proof of Theorem \ref{theorem - equivariant structures - categoricity}. Let $\phi(v,w,x)$ define a basic closed subset of $\nSpec A$; thus $\phi$ is of the form
\[ \exists \lambda \exists \mu \exists e \exists y \exists \gamma \exists b \left( \bigwedge_{i=1}^{s} \bigwedge_{j=1}^{s_{i}} \pi(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \phi \wedge \bigwedge_{(i,j) \in \Theta} \phi_{ij} \wedge C(\lambda, \mu, y, z, \gamma, b, x) \right) \] where $C$ is Zariski closed and $\Gamma$-invariant.
\\
\\
\textbf{Claim}: Let $\pi_{A}: L_{A} \rightarrow V_{A}$ (and $\pi_{B}:L_{B} \rightarrow V_{B}$). The preimage of $\phi(\nSpec A)$ is defined by
\[ \exists \lambda \exists \mu \exists e \exists e'' \exists y \exists \gamma \exists b \left( \begin{array}{c} \bigwedge_{i=1}^{s} \bigwedge_{j=1}^{s_{i}} \pi_{B}(v_{ij}) = y_{i} \wedge \lambda_{ij}e_{i} = v_{ij} \wedge \mathbf{E}(e_{i}, y_{i}) \wedge \\ \bigwedge_{i=1}^{p} \bigwedge_{j=1}^{p_{i}} \pi_{B}(w_{ij}) = z_{i} \wedge \mu_{ij} e''_{i} = w_{ij} \wedge \mathbf{E}(e''_{i}, z'_{i}) \wedge \\ \bigwedge_{(i,j) \in \Theta'} \phi'_{ij} \wedge C'(\lambda, \mu, y, \gamma, b, x) \end{array} \right) \] where
\begin{enumerate}
\item $\Theta' \subseteq \{1 \leq i,j \leq p + s\}$
\item $\phi'_{ij}$ is $\phi_{ij}$ with the $\mathbf{U}_{1i}$ replaced with their images under $\varphi$.
\item $C'(\lambda, \mu,y,z,\gamma,b,x)$ holds if and only if
\begin{itemize}
\item $C(\lambda, \mu, f(y), \gamma, b, x)$ holds in $\nSpec A$.
\item $z' = (z_{i}) \in f^{-1}(z)$ where $z = (\pi_{A}(e'_{i}))$.
\end{itemize}
\end{enumerate}
\begin{proof} Immediate by construction of $\nSpec \varphi$.
\end{proof} Because $C'$ is closed, the formula in the claim defines a closed set in $\nSpec B$, as required.
\end{proof}
\begin{corollary}
\label{corollary - equivariant algebras - functor}
There is a functor $\nSpec: \mathsf{Equiv}(k)_{\Gamma}^{op} \rightarrow \mathsf{Zar}$ extending $\mathsf{B}^{op} \rightarrow \mathsf{Zar}^{c}$.
\end{corollary}
\begin{proof} Immediate by Proposition \ref{proposition - equivariant algebras - functor}. We note that if $H$ is a commutative affine Hopf $k$-algebra then the structure corresponding to $H$ is a line bundle $\pi: L \rightarrow G$ where $G$ is an affine algebraic group. Each fiber $L_{g} = \pi^{-1}(g)$ is the $G$-module given by the character $\chi_{g}: G \rightarrow k$ associated to $g \in G$. It is clear that this structure is definably interpretable in $k$.
\end{proof} A certain amount of algebraic structure is recoverable from an abstract theory $T$ which `looks like' $T_{A}$. If $T$ is formulated in the language
\[ \mathcal{L} = (L,V,k, \pi, \mathbf{E}, \mathbf{U}_{i}, h_{j}, C: 1 \leq j \leq m, 1 \leq i \leq n) \] and satisfies the axioms of Definition \ref{definition - theory of an equivariant algebra}, let $F$ be the free algebra over $k$ on the generators $h_{j}, \mathbf{U}_{i}$. A model $(L,k) \models T$ will be a representation of $A = F/I$ where $I$ is the annihilator of $(L,k)$. By Theorem \ref{theorem - equivariant structures - categoricity}, the algebra $A$ is determined up to the cardinality of the uncountable algebraically closed field $k$. It need not be the case that $T = T_{A}$ for some equivariant $k$-algebra $A$.
\begin{example}
\label{example - theory with no associated equivariant algebra}
Let $A$ be the $k$-algebra with generators $\mathbf{U}, \mathbf{V}^{\pm 1}$ subject to the relation
\[ \mathbf{U}\mathbf{V} = q\mathbf{V}\mathbf{U} \] where $\mathbf{V}$ is invertible and $q$ is a generic parameter. Then $A$ is not equivariant (it is not even semi-equivariant). Yet there is an abstract theory $T$ satisfying the axioms of Definition \ref{definition - theory of an equivariant algebra} from which $A$ can be recovered. $T$ is formulated in the language $\mathcal{L} = (L,V,k,\pi, \mathbf{U}, \mathbf{V}^{\pm 1}, q)$, $V$ is the affine line,
\[ \eta_{\mathbf{V}}(x) = qx \qquad \eta_{\mathbf{V}^{-1}}(x) = q^{-1}x \] $\Gamma = \{1\}$ and $\lambda_{\mathbf{V}}(y,x) = \lambda_{\mathbf{V}^{-1}}(y,x) = y - x$. Clearly $T$ is $\Gamma$-rigid.
\end{example} Thus the full subcategory of $\mathsf{Zar}$ consisting of large saturated models of theories which satisfy Definition \ref{definition - theory of an equivariant algebra} will contain Zariski structures which do not lie in the image of $\nSpec$. An equivalence of categories via $\nSpec$ does not therefore seem possible. The following conjecture remains open.
\begin{conjecture}
\label{conjecture - equivariant algebras - non-classical Zariski}
If $A$ is a non-commutative equivariant algebra then $\nSpec A$ is a non-classical Zariski structure.
\end{conjecture}
\subsection{Quantized Universal Enveloping Algebras}
Despite some important examples falling under the umbrella of equivariant algebras, there are many important algebras which are not equivariant. There is one such collection of algebras that the author firmly had in mind when formulating the mathematics of the previous three chapters; namely the quantized universal enveloping algebras at generic parameter.
\begin{definition} Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra. Let $n$ be the rank of $\mathfrak{g}$, $C = (a_{ij})$ the Cartan matrix of $\mathfrak{g}$ with respect to a choice of Cartan subalgebra $\mathfrak{h}$ and simple roots $\alpha_{1}, \dots, \alpha_{n}$. Let $q \in k$ be generic, $q_{i} = q^{d_{i}}$. The \textbf{quantized enveloping algebra of $\mathfrak{g}$ over $k$} (denoted $U_{q}(\mathfrak{g})$) is the $k$-algebra with generators $E_{i}, F_{i}, K^{\pm 1}_{i}$, $1 \leq i \leq n$ subject to the relations
\begin{enumerate}
\item $K_{i}E_{j}K_{i}^{-1} = q_{i}^{a_{ij}}E_{j} \qquad K_{i}F_{j}K_{i}^{-1} = q_{i}^{-a_{ij}}F_{j}$.
\item $K_{i}K_{j} = K_{j}K_{i}$.
\item $E_{i}F_{j} - F_{j}E_{i} = \delta_{ij} \frac{K_{i} - K_{i}^{-1}}{q_{i} - q_{i}^{1}}$.
\item The quantized Serre relations:
\[ \sum_{l=0}^{1-a_{ij}} (-1)^{l} \left[ \begin{array}{c} 1 - a_{ij} \\ l \end{array} \right] _{q_{i}} E_{i}^{1-a_{ij} - l} E_{j} E^{l}_{i} = 0 \qquad (i \neq j) \] \[ \sum_{l=0}^{1-a_{ij}} (-1)^{l} \left[ \begin{array}{c} 1 - a_{ij} \\ l \end{array} \right] _{q_{i}} F_{i}^{1-a_{ij} - l} F_{j} F^{l}_{i} = 0 \qquad (i \neq j) \]
\end{enumerate}
\end{definition} We refer the reader to Appendix \ref{appendix - Lie theory} for appropriate facts and notation concerning semisimple Lie algebras. The presence of the quantized Serre relations (if non-trivial) prevent $U_{q}(\mathfrak{g})$ from being semi-equivariant, let alone equivariant. Nevertheless, if we discard the quantized Serre relations, we do obtain a semi-equivariant algebra $\tilde{U}_{q}(\mathfrak{g})$: take $H = k[K^{\pm 1}_{i}: 1 \leq i \leq n]$ and equip it with the group Hopf algebra structure, namely that given by
\[ \Delta(K_{i}) = K_{i} \otimes K_{i} \qquad \epsilon(K_{i}) = 1 \qquad S(K_{i}) = K^{-1}_{i} \] for each $i$. Then $E_{i}$ and $F_{i}$ are eigenvectors of the adjoint action of $H$ on $\tilde{U}_{q}(\mathfrak{g})$ by $1$ and the remaining relations satisfied by $E_{i}, F_{i}$ in $3$ are of the required form. By Proposition \ref{proposition - semi-equivariant algebras - epimorphic images} there is an equivariant algebra $U_{q}'(\mathfrak{g})$ of which $\tilde{U}_{q}(\mathfrak{g})$ is an epimorphic image, for which the most likely candidate is the $k$-algebra subject to the relations $1,2$ and
\begin{equation} [E_{i}, F_{i}] = \frac{K_{i} - K_{i}^{-1}}{q - q^{-1}} \qquad 1 \leq i \leq n \end{equation}
\begin{proposition} $U'_{q}(\mathfrak{g})$ is equivariant and its theory is $\Gamma$-rigid.
\end{proposition}
\begin{proof} This is a straightforward generalization of the corresponding result for $U_{q}(\mathfrak{sl}_{2}(k))$. We shall assume for simplicity that $\mathfrak{g}$ is simply laced (i.e. that $d_{i} = 1$ for all $i$). Define the vectors $\mathbf{a}_{j} = (a_{1j}, \dots, a_{nj})$ for $1 \leq j \leq n$ and put $\mathbf{A} = \mathbb{Z}\langle \mathbf{a}_{1}, \dots \mathbf{a}_{n} \rangle$. Then $\Pi = q^{\mathbf{A}}$ acts on $H$ (by multiplication) and the $E_{i}, F_{i}$ move between fibers according to this action on the base. Define
\[ \lambda_{E_{i}}(y) = -\lambda_{F_{i}}(y) = \frac{y_{i} + y_{i}^{-1}}{q - q^{-1}} \qquad y = (y_{1}, \dots y_{n}) \qquad 1 \leq i \leq n \] \[ P_{E_{i}}(x,y) = P_{F_{i}}(x,y) = y^{2} - x \] Then these maps and polynomials satisfy the required conditions. As with the $U_{q}(\mathfrak{sl}_{2}(k))$ case, we have to be careful about picking the roots and for this purpose, we partition $(k^{*})^{n}$ into cosets of $q^{\mathbf{A}}$:
\[ (k^{*})^{n} = \bigcup_{x \in \Lambda} q^{\mathbf{A}}x \] where $\Lambda$ is a set of representatives. For $x \in \Lambda$, choose any $y$ such that $y^{2} = x$ (i.e. $y = (y_{1}, \dots, y_{n})$ with $y_{i}^{2} = x_{i}$ for every $i$). If $z \in q^{\mathbf{A}}x$ then there is $\mathbf{a} \in \mathbf{A}$ such that $z = q^{\mathbf{a}}x$ and we choose the square root $q^{\mathbf{a}/2}y$ of $z$. The associated theory is trivially $\Gamma$-rigid (see the proof of Proposition \ref{proposition - initial examples - rigid}).
\end{proof} It is unlikely that $\tilde{U}_{q}(\mathfrak{g})$ itself is equivariant, although the author has been unable to prove this. A calculation using the above $\lambda_{E_{i}}$, $\lambda_{F_{i}}$, $P_{E_{i}}$ demonstrates that they do not satisfy $E_{i}F_{j} - F_{j}E_{i} = 0$ for $i \neq j$. It also does not seem possible that a more sophisticated selection of functions and polynomials can rectify this without violating the corresponding relations for $(7.1)$. Nevertheless, that there is an epimorphism $U_{q}'(\mathfrak{g}) \rightarrow \tilde{U}_{q}(\mathfrak{g})$ suggests (by a possible generalization of Corollary \ref{corollary - equivariant algebras - functor}) that a putative geometric structure corresponding to $\tilde{U}(\mathfrak{g})$ could map to the Zariski structure $\nSpec U'(\mathfrak{g})$.
\subsection{Total Equivariance}
We isolate a particularly nice class of equivariant algebras with the following definition.
\begin{definition}
\label{definition - totally equivariant algebras}
An equivariant $k$-algebra $A$ is \textbf{totally equivariant} if any maximal commutative subalgebra has the structure of a Hopf algebra with respect to which $A$ is equivariant.
\end{definition}
\begin{example}
\label{example - quantum torus generic - totally equivariant}
Let $\mathcal{O}_{} = \mathcal{O}_{q}((k^{\times})^{2})$ be the quantum 2-torus, i.e. the $k$-algebra with generators $\mathbf{U}^{\pm 1}, \mathbf{V}^{\pm 1}$ subject to the relation
\[ \mathbf{U}\mathbf{V}= q \mathbf{V}\mathbf{U} \qquad \] where $q$ is generic. Then $\mathcal{O}_{q}$ is totally equivariant.
\end{example}
\begin{proof} The algebra $\mathcal{O}_{q}$ is equivariant because $\mathcal{O}_{\mathbf{q}}((k^{\times})^{n})$ is (see Subsection \ref{subsection - initial examples are equivariant}). Now any maximal commutative subalgebra $H$ must be generated by some $c\mathbf{U}^{p}\mathbf{V}^{q}$ and its inverse, where $p,q \in \mathbb{Z}$ and $c \in k^{\times}$. Thus taking the additional generators $\mathbf{V}^{-q}\mathbf{U}^{1-p}$, $\mathbf{V}^{1-q}\mathbf{U}^{-p}$ and their inverses gives the whole of $\mathcal{O}_{q}$. It is easy to see that these generators are eigenvectors for $H$ under the adjoint action, either directly or by use of Lemma \ref{lemma - module algebras - eigenvectors}.
\end{proof} A totally equivariant algebra $A$ has many associated Zariski structures, each depending on the particular Hopf subalgebra chosen. For those maximal commutative subalgebras which are conjugated by an automorphism of $A$, there is a corresponding isomorphism of the associated Zariski structures by Corollary \ref{corollary - equivariant algebras - functor}. In Example \ref{example - quantum torus generic - totally equivariant}, one could consider the maximal commutative subalgebras $k[\mathbf{U}^{\pm 1}]$ and $k[\mathbf{V}^{\pm 1}]$. By total equivariance, there are two corresponding Zariski structures $\nSpec \mathcal{O}_{q}$ and $\nSpec \mathcal{O}_{q}'$ respectively. Let $\varphi$ be the $k$-algebra automorphism of $\mathcal{O}_{q}$ given by
\[ \varphi(\mathbf{U}) = \mathbf{V} \qquad \varphi(\mathbf{V}) = \mathbf{U}^{-1} \] Then $\varphi$ is an equivariant automorphism and it follows that there is a corresponding Zariski isomorphism $\nSpec \mathcal{O}_{\epsilon} \simeq \nSpec \mathcal{O}_{\epsilon}'$.
|
1,108,101,565,478 | arxiv | \section{Introduction}
Reinforcement learning (RL) is a framework that allows an agent to complete tasks in an environment, even when a model of the environment is not known.
The agent `learns' to complete a task by maximizing its expected long-term reward, where the reward signal is supplied by the environment.
RL algorithms have been successfully implemented in many fields, including robotics \cite{hafner2011reinforcement, lillicrap2016continuous}, and games \cite{mnih2015human, silver2016mastering} .
However, it remains difficult for an RL agent to master new tasks in unseen environments.
This is especially true when the reward given by the environment is sparse/ significantly delayed.
It may be possible to guide an RL agent towards more promising solutions faster, if it is equipped with some form of \emph{prior knowledge} about the environment.
This can be encoded by modifying the reward signal received by the agent during training.
However, the modification must be carried out in a principled manner, since providing an additional reward at each step might distract the agent from the true goal \cite{randlov1998learning}.
Potential-based reward shaping (PBRS) is one such method that augments the reward in an environment specified by a Markov Decision Process (MDP) with a term that is a difference of \emph{potentials} \cite{Ng1999policy}.
This method is attractive since it easily allows for the recovery of optimal policies, while enabling the agent to learn these policies faster.
Potential functions are typically functions of states.
This could be a limitation, since in some cases, such a function may not be able to encode all information available in the environment.
To allow for imparting more information to the agent, a potential-based advice (PBA) scheme was proposed in \cite{Wiewiora2003principled}.
The potential functions in PBA include both states and actions as their arguments.
To the best of our knowledge, PBRS and PBA schemes in the literature \cite{Ng1999policy, Wiewiora2003principled, Devlin2012dynamic} assume that an optimal policy is deterministic.
This will not always be the case, since an optimal policy might be a stochastic policy.
This is especially true when there are states in the environment that are partially observable or indistinguishable from each other.
Moreover, the aforementioned papers limit their focus to discrete state and action spaces.
In this paper, we study the addition of PBRS and PBA schemes to the reward, in settings where: \emph{i)} the optimal policy will be stochastic, and \emph{ii)} state and action spaces may be continuous.
We additionally provide guarantees on the convergence of an advantage actor-critic architecture that is augmented with a PBA scheme.
We make the following contributions:
\begin{itemize}
\item We prove that the ability of an agent to learn an optimal stochastic policy remains unaffected when augmenting PBRS to soft Q-learning
\item We propose a technique for adapting PBA in policy-based methods, in order to use these schemes in environments with continuous state and action spaces.
\item We present an Algorithm, \textbf{AC-PBA}, describing an advantage actor-critic architecture augmented with PBA, and provide guarantees on its convergence.
\item We evaluate our approach on two experimental domains: a discrete-state, discrete-action \emph{Puddle-jump Gridworld} that has indistinguishable states, and a continuous-state, continuous-action \emph{Mountain Car}.
\end{itemize}
The remainder of this paper is organized as follows: Section \ref{RelWork} presents related work in reward shaping. Required preliminaries to RL, PBRS and PBA is presented in Section \ref{PrelimSection}.
Section \ref{PBRSSection} presents our results on using PBRS for stochastic policy learning.
We present a method to augment PBA to policy gradient frameworks and an algorithm detailing this
in Section \ref{PBASection}.
Experiments validating our approach are reported in Section \ref{ExptSection}, and we conclude the paper in Section \ref{ConclusionSection}.
\section{Related Work}\label{RelWork}
Shaping or augmenting the reward received by an RL agent in order to enable it to learn optimal policies faster is an active area of research.
Reward modification via human feedback was used in \cite{thomaz2006reinforcement, knox2010combining} to interactively shape an agent's response so that it learned a desired behavior.
However, frequent human supervision is usually costly and may not possible in every situation.
A curiosity-based RL algorithm for sparse reward environments was presented in \cite{pathak2017curiosity}, where an intrinsic reward signal characterized the prediction error of the agent as a curiosity reward.
The reward received by the agent was augmented with a function that represented the number of times the agent had visited a state in \cite{tang2017exploration}.
Entropy regularization as a way to encourage exploration of policies during the early stages of learning was studied in \cite{williams1991function} and \cite{mnih2016asynchronous}.
This was used to lead a policy towards states with a high reward in \cite {levine2013guided} and \cite{levine2016end}.
Static potential-based functions were shown to preserve the optimality of deterministic policies in \cite{Ng1999policy}.
This property was extended to dynamic potential-based functions in \cite{Devlin2012dynamic}.
The authors of \cite{Wiewiora2003init} showed that when an agent learned a policy using
Q-learning,
applying PBRS at each training step was equivalent to initializing the Q-function with the potentials.
They studied value-based methods, but restricted their focus to learning deterministic policies.
The authors of \cite{Devlin2012express} demonstrated a method to transform a reward function into a potential-based function during training.
The potential function in PBA was obtained using an `experience filter' in \cite{li2018introspective}.
The use of PBRS in model-based RL was studied in \cite{asmuth2008potential}, and for episodic RL in \cite{grzes2017reward}.
PBRS was extended to planning in partially observable domains in \cite{eck2016potential}.
However, these papers only considered the finite-horizon case.
In comparison, we consider the infinite horizon, discounted cost setting in this paper.
In control theoretic settings, RL algorithms have been used to establish guarantees on convergence to an optimal controller for the Linear Quadratic Regulator, when a model of the underlying system was not known in \cite{bradtke1993reinforcement, fazel2018global}.
A survey of using RL for control is presented in \cite{bucsoniu2018reinforcement}.
OpenAI Gym \cite{brockman2016openai} enables the solving of several problems in classical control using RL algorithms.
\section{Preliminaries}\label{PrelimSection}
\subsection{Reinforcement Learning}
An MDP \cite{puterman2014markov} is a tuple $(S,A,\mathbb{T},\rho_0, R)$.
$S$ is the set of states, $A$ the set of actions,
$\mathbb{T}:S \times A \times S \rightarrow [0,1]$ encodes $\mathbb{P}(s_{t+1}|s_t,a_t)$, the probability of transition to $s_{t+1}$, given current state $s_t$ and action $a_t$.
$\rho_0$ is a probability distribution over the initial states.
$R : S \times A \rightarrow \mathbb{R}$ denotes the reward that the agent receives when transitioning from $s_t$ while taking action $a_t$.
In this paper, $R < \infty$.
The goal for an RL agent \cite{sutton2018reinforcement} is to learn a \emph{policy} $\pi$,
in order to maximize
$J:=\mathbb{E}_{\tau \sim \pi}[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)]$.
Here, $\gamma$ is a discounting factor, and the expectation is taken over the trajectory $\tau=(s_0,a_0,r_0,s_1,\dots)$ induced by policy $\pi$.
If $\pi: S \rightarrow A$, the policy is \emph{deterministic}.
On the other hand, a randomized policy returns a probability distribution over the set of actions, and is denoted $\pi: S \times A \rightarrow [0,1]$.
The value of a state-action pair $(s,a)$ following policy $\pi$ is represented by the \emph{Q-function}, written $Q^{\pi}(s,a) = \mathbb{E}_{\tau \sim \pi}[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)|s_0=s,a_0=a]$.
The Q-function allows us to calculate the state value $V^{\pi}(s) = \mathbb{E}_{a \sim \pi}[Q^{\pi}(s,a)]$.
The advantage of a particular action $a$, over other actions at a state $s$ is defined by $A^{\pi}(s,a) := Q^{\pi}(s,a)-V^{\pi}(s)$.
\subsection{Value-based and Policy-based Methods}
The RL problem has two general solution techniques.
\emph{Value-based} methods determine an optimal policy by maintaining a set of reward estimates when following a particular policy.
At each state, an action that achieves the highest (expected) reward is taken.
Typical value-based methods to learn greedy (determininistic) policies include Q-learning and Sarsa-learning \cite{sutton2018reinforcement}.
Recently, the authors of \cite{haarnoja2017reinforcement} proposed \emph{soft Q-learning}, which is a value-based method that is able to learn stochastic policies.
In comparison, \emph{policy-based} methods directly search over the policy space \cite{sutton2018reinforcement}.
Starting from an initial policy, specified by a set of parameters, these methods compute the expected reward for this policy, and update the parameter set according to certain rules to improve the policy.
Policy gradient \cite{sutton2000policy} is one way to achieve policy improvement.
This method repeatedly computes (an estimate of) the gradient of the expected reward with respect to the policy parameters.
Policy-based approaches usually exhibit better convergence properties, and can be used in continuous action spaces \cite{haarnoja2018soft}.
They can also be used to learn stochastic policies.
REINFORCE and actor-critic are examples of policy gradient algorithms \cite{sutton2018reinforcement}.
\subsection{PBRS and PBA}
Reward shaping methods augment the environment reward $R$ with an additional reward $F \in \mathbb{R}$, $F< \infty$.
This changes
the structure of the original MDP $M(=(S,A,\mathbb{T},\rho_0, R))$ to $M'=(S,A,\mathbb{T},\rho_0, R+F)$.
The goal is to choose $F$ so that an optimal policy for $M'$, $\pi^{*}_{M'}$, is also optimal for the original MDP $M$.
\emph{Potential-based reward shaping} (PBRS) schemes were shown to be able to preserve the optimality of deterministic policies in \cite{Ng1999policy}.
In PBRS, the function $F$ is defined as a difference of \emph{potentials}, $\phi(\cdot)$.
Specifically, $F(s_t,a_t,s_{t+1}) := \gamma \phi(s_{t+1}) - \phi(s_t)$.
Then, the Q-function, $Q^{*}_{M}(s,a)$, of the optimal greedy policy for $M$ and the optimal Q-function $Q^{*}_{M'}(s,a)$ for $M'$ are related by:
$Q^{*}_{M'}(s,a)= Q^{*}_{M}(s,a) - \phi(s)$.
Therefore, the optimal greedy policy is not changed \cite{Ng1999policy, Devlin2012dynamic}, since:
\begin{align*}
&\pi^{*}_{M'}(s) \in \argmax_{a\in A}~Q^{*}_{M'}(s,a) \\
&\qquad = \argmax_{a\in A}~\big(Q^{*}_{M}(s,a) - \phi(s)\big) = \argmax_{a\in A}~Q^{*}_{M}(s,a).
\end{align*}
The authors of \cite{Wiewiora2003principled} augmented $\phi(s)$ to include action $a$ as an argument.
They termed this \emph{potential-based advice} (PBA).
There are two forms-- \emph{look-ahead PBA} and \emph{look-back PBA}-- respectively defined by:
\begin{align}
F(s_{t},a_{t},s_{t+1},a_{t+1}) &= \gamma \phi(s_{t+1},a_{t+1}) - \phi(s_{t},a_{t})\label{lookaheadPBA}\\
F(s_{t},a_{t},s_{t-1},a_{t-1}) &= \phi(s_{t},a_{t}) - {\gamma}^{-1}\phi(s_{t-1},a_{t-1}).\label{lookbackPBA}
\end{align}
For the look-ahead PBA scheme, the state-action value function for $M$ following policy $\pi$ is given by:
\begin{align}\label{PBAq}
Q^{\pi}_{M}(s,a) = Q^{\pi}_{M'}(s,a)+\phi(s,a).
\end{align}
The optimal greedy policy for $M$ can be recovered from the optimal state-action value function for $M'$ from:
\begin{align}\label{PBAp}
\pi^*_{M}(s_t) &\in \argmax_{a \in A} \big(Q^{*}_{M'}(s_t,a)+\phi(s_t,a)\big).
\end{align}
The optimal greedy policy for $M$ using look-back PBA can be recovered similarly.
\section{PBRS for Stochastic Policy Learning}\label{PBRSSection}
The existing literature on PBRS has focused on augmenting value-based methods to learn optimal deterministic policies.
In this section, we first show that PBRS preserves optimality, when the optimal policy is stochastic.
Then, we show that the \emph{learnability} will not be changed when using PBRS in soft Q-learning
\begin{prop}\label{PBRSResult}
Assume that the optimal policy is stochastic.
Then, with $F:=\gamma\phi(s_{t+1})-\phi(s_t)$, PBRS preserves the optimality of stochastic policies.
\end{prop}
\begin{proof}
The goal in the original MDP $M$ was to find a policy $\pi$ in order to maximize:
\begin{align}\label{PBRS_M}
{\pi}_M^* &= \argmax_{\pi} \mathbb{E}_{\tau \sim \pi}\left[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)\right].
\end{align}
In PBRS, the goal is to determine a policy so that:
\begin{align}
&{\pi}_{M'}^* = \argmax_{\pi} \mathbb{E}_{\tau \sim \pi}\big[\sum_{t=0}^{\infty}\gamma^t \big(R(s_t,a_t)+ F(s_t,a_t,s_{t+1},a_{t+1})\big)\big] \nonumber\\
&= \argmax_{\pi} \mathbb{E}_{\tau \sim \pi}\big[\sum_{t=0}^{\infty}\gamma^t \big(R(s_t,a_t)+\gamma\phi(s_{t+1})-\phi(s_t)\big)\big] \nonumber\\
&= \argmax_{\pi} \bigg[\mathbb{E}_{\tau \sim \pi}\big[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)\big]-\mathbb{E}_{\tau \sim \pi}\big[\phi(s_0)\big]\bigg]\nonumber\\
&=\argmax_{\pi}~ \mathbb{E}_{\tau \sim \pi}\big[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)\big]-\int_{s}\rho_0(s)\phi(s)\text{d}s.\label{PBRS_M'}
\end{align}
The last term in Equation (\ref{PBRS_M'}) is constant, and doesn't affect the identity of the maximizing policy of (\ref{PBRS_M}).
\end{proof}
%
Next, we examine the effect on learnability when using PBRS with soft Q-learning.
Soft Q-learning is a value-based method for stochastic policy learning that was proposed in \cite{haarnoja2017reinforcement}.
Different from Equation (\ref{PBRS_M}), the goal is to maximize both, the accumulated reward, and the policy entropy at each visited state:
\begin{align}\label{SoftQf}
{\pi}_{\text{soft}}^* &= \argmax_{\pi} \mathbb{E}_{\tau \sim \pi}\big[\sum_{t=0}^{\infty}\gamma^t \big(R(s_t,a_t)+\alpha\mathcal{H}(\pi(\cdot|s_t))\big)\big].
\end{align}
The entropy term $\mathcal{H}(\pi(\cdot|s_t))$ encourages exploration of the state space, and the parameter $\alpha$ is a trade-off between exploitation and exploration.
Before stating our result, we summarize the soft Q-learning update procedure.
From \cite{haarnoja2017reinforcement}, the optimal value-function, $V^*_{\text{soft}}(s_t) $, is given by:
\begin{align}\label{SoftV}
V^*_{\text{soft}}(s_t) = \alpha \log \int_{A}\exp\big(\frac{1}{\alpha}Q^*_{\text{soft}}(s_t,a)\big)\text{d} a.
\end{align}
The optimal soft Q-function is determined by solving the soft Bellman equation:
\begin{align}\label{SoftQ}
Q^*_{\text{soft}}\big(s_t,a_t\big) = r_t+\gamma \mathbb{E}_{s_{t+1}}\big[V^*_{\text{soft}}(s_{t+1})\big].
\end{align}
The optimal policy can be obtained from Equation (\ref{SoftQ}) as:
\begin{align}\label{Softp}
{\pi}_{\text{soft}}^*(a_t|s_t) = \exp\big(\frac{1}{\alpha}\big(Q^*_{\text{soft}}(s_t,a_t)-V^*_{\text{soft}}(s_t)\big)\big),
\end{align}
In the rest of this Section, we assume both, states and actions are discrete and no function approximator is used.
We also omit subscripts for $Q_{\text{soft}}$ and $V_{\text{soft}}$, and set $\alpha=1$ for simplicity.
From Equation (\ref{SoftQ}), and as in Q-learning, soft Q-learning updates the soft Q-function by minimizing the soft Bellman error:
\begin{align}\label{Bellman_error}
\begin{split}
\delta Q_k(s_k,a_k) = r(s_k,a_k) + \gamma V_k(s_{k+1})-Q_k(s_k,a_k),
\end{split}
\end{align}
where $V_k(s_{t+1}) = \log \sum_{a \in A}\exp\big(Q_k(s_{t+1},a)\big)$.
During training, $\pi_k(a_t|s_t) = \exp\big(Q_k(s_t,a_t)-V_k(s_t)\big)$.
With $\lambda$ denoting the learning rate, the Q-function update is given by:
\begin{align}\label{softupdate}
\begin{split}
Q_{k+1}(s_k,a_k) = Q_k(s_k,a_k) + \lambda\delta Q_k(s_k,a_k).
\end{split}
\end{align}
The main result of this section shows that the ability of an agent to learn an optimal policy is unaffected when using soft Q-learning augmented with PBRS.
We define a notion of \emph{learnability}, and use this to establish our claim.
During training, an agent encounters a sequence of states, actions, and rewards that serves as `raw-data' which is fed to the RL algorithm.
Let $L$ and $L'$ denote two RL agents.
Let $\mathcal{D}_k = (s_k,a_k,r_k,s_{k+1})$ and $\mathcal{D}'_k = (s'_k,a'_k,r'_k,s'_{k+1})$ denote the experience tuple at learning step $k$ from a trajectory used by $L$ and $L'$, respectively.
\begin{df}[Learnability]\label{learnability}
Denote the accumulated difference in the Q-functions of $L$ and $L'$ after learning for $k$ steps by $\Delta Q_k(s,a)$ and $\Delta Q'_k(s,a)$, respectively.
Then, given identical sample experiences, (that is, $\mathcal{D}_{k'}=\mathcal{D}'_{k'}$ $\forall k' \leq k$), $L$ and $L'$ are said to have the same learnability if $\Delta Q_{k'}(s,a)=\Delta Q'_{k'}(s,a)$ $\forall k' \leq k ~ \forall s \forall a$.
\end{df}
\begin{prop}\label{learnability_prop}
Soft Q-learning, with initial soft Q-values $Q(s,a)=Q_0(s,a)$ and augmented with PBRS where state potential is $\phi(s)$, has the same learnability as soft Q-learning without PBRS but with its soft Q-values initialized to $Q(s,a)=Q_0(s,a)+\phi(s)$.
\end{prop}
\begin{proof}
Consider an agent $L$ that uses a PBRS scheme during learning and an agent $L'$ that does not use PBRS, but has its soft Q-values initialized as $Q'_0(s,a):=Q_0(s,a)+\phi(s)$, where $Q_0(s,a)$ is the initial Q-value of $L$.
We further assume that $L$ and $L'$ adopt the same learning rate.
From Definition \ref{learnability}, to show that $L$ and $L'$ have the same learnability, we need to show that the soft Bellman errors $\delta Q_{k}(s_t,a_t)$ and $\delta Q'_{k}(s_k,a_k)$ are equal at each training step $k$, given the same experience sets $\mathcal{D}_k$ and $\mathcal{D}_k'$.
From Equation (\ref{Bellman_error}), the soft Bellman errors for $L$ and $L'$ can be respectively written as:
\begin{align*
\delta Q_k(s_k,a_k)& = r(s_k,a_k) + \gamma\phi(s_{k+1})-\phi(s_k)+ \\
&\qquad \gamma V_k(s_{k+1})-Q_k(s_k,a_k)\\
\delta Q'_k(s'_k,a'_k) &= r(s'_k,a'_k) + \gamma V'_k(s'_{k+1})-Q'_k(s'_k,a'_k).
\end{align*}
Since $\mathcal{D}_{k'}=\mathcal{D}_{k'}'$ for each $k' \leq k$, comparing $\delta Q'_k(s_k,a_k)$ and $\delta Q_k(s'_k,a'_k)$ is reduced to comparing $\delta Q'_k(s_k,a_k)$ and $\delta Q_k(s_k,a_k)$.
We show this by induction.
At training step $k=0$ there is no update. Thus, $\delta Q_0(s_0,a_0)=\delta Q'_0(s_0,a_0)$.
Assume that the Bellman errors are identical up to a step $k=K$.
That is, $\delta Q_k(s_k,a_k)=\delta Q'_k(s_k,a_k)$ $\forall k\leq K$.
Then, the accumulated errors for the two agents until this step are also identical.
That is, $\Delta Q_K(s,a)=\Delta Q'_K(s,a) ~ \forall s \forall a$.
Consider training step $k=K+1$.
The state values at this step are:
$V_K(s_{K+1}) = \log \sum_{a \in A}\exp\big[Q_0(s_{K+1},a)+\Delta Q_K(s_{K+1},a)\big]$ and
$V'_K(s_{K+1})=\log \sum_{a \in A}\exp\big[Q_0(s_{K+1},a)+\phi(s_{K+1})+\Delta Q'_K(s_{K+1},a)\big]$ respectively.
The Bellman errors at $k=K+1$ are:
\begin{align*}
\delta Q_{K+1}(s_K,a_K) = r(s_K,a_K) + &\gamma\phi(s_{K+1})-\phi(s_K)\\
&+\gamma V_K(s_{K+1})-Q_K(s_K,a_K)\\
= r(s_K,a_K) + \gamma\phi(s_{K+1})&-\phi(s_K)+\gamma V_K(s_{K+1})\\
&-Q_0(s_K,a_K)-\Delta Q_K(s_K,a_K)
\end{align*}
\begin{align*}
&\delta Q'_{K+1}(s_K,a_K) = r(s_K,a_K) + \gamma V'_K(s_{K+1})-Q'_K(s_K,a_K)\\
&=r(s_K,a_K) + \gamma V'_K(s_{K+1}) -Q_0(s_K,a_K)-\phi(s_K)-\Delta Q'_K(s_K,a_K)\\
&=\delta Q_{K+1}(s_K,a_K)-\gamma\phi(s_{K+1})+\gamma (V'_K(s_{K+1})-V_K(s_{K+1}))\\
&=\delta Q_{K+1}(s_K,a_K)-\gamma\phi(s_{K+1})+\gamma\phi(s_{K+1})\\
&=\delta Q_{K+1}(s_K,a_K).
\end{align*}
It follows that $\Delta Q_{K+1}(s,a)=\Delta Q'_{K+1}(s,a) ~ \forall s \forall a$.
\end{proof}
\begin{rem}
If the Q-function is represented by a function approximator (as is typical for continuous action spaces), then Proposition \ref{learnability_prop} may not hold.
This is because the Q-function in this scenario is updated using gradient descent, instead of Equation (\ref{softupdate}).
Gradient descent is sensitive to initialization.
Thus, different initial values will result in different updates of the Q-function.
\end{rem}
\section{PBA for Stochastic Policy Learning}\label{PBASection}
Although PBRS can preserve the optimality of policies in several settings, it suffers from the drawback of being unable to encode richer information, such as desired relations between states and actions.
The authors of \cite{Wiewiora2003principled} proposed \emph{potential-based advice} (PBA), a scheme that augments the potential function by including actions as an argument together with states.
In this section, we show that while using PBA, recovering the optimal policy can be difficult if the optimal policy is stochastic.
Then, we propose a novel way to impart prior information in order to learn a stochastic policy with PBA.
\subsection{Stochastic policy learning with PBA}
Assume that we can compute $Q^{*}_{M}(s,a)$, the optimal value for state-action pair $(s,a)$ in MDP $M$.
The optimal stochastic policy for $M$ is $\pi^*_M = \argmax_{\tau \sim \pi}\mathbb{E}_{\pi}\big[Q^{*}_{M}(s,a)\big]$.
From Equation (\ref{PBAq}), the optimal stochastic policy for the modified MDP $M'$ that has its reward augmented with PBA is given by $\pi^*_{M'} = \argmax_{\pi}\mathbb{E}_{\tau \sim \pi}\big[Q^{*}_{M}(s,a)-\phi(s,a)\big]$.
Without loss of generality, $\pi^*_M \neq \pi^*_{M'}$.
If the optimal policy is deterministic, then the policy for $M$ can be recovered easily from that for $M'$ using Equation (\ref{PBAp}).
However, when it is stochastic, we need to average over trajectories in the MDP, which makes it difficult to recover the optimal policy for $M$ from that of $M'$.
In the sequel, we will propose a novel way to take advantage of PBA in the policy gradient framework in order to directly learn a stochastic policy.
\subsection{Imparting PBA in policy gradient}
Let $J_M(\theta)$ denote the value of a parameterized policy $\pi_{\theta}$ in MDP $M$.
That is, $J_M(\theta) = \mathbb{E}_{\tau \sim \pi_{\theta}}\left[\sum_{t=0}^{\infty}\gamma^t R(s_t,a_t)\right]$.
Following the policy gradient theorem \cite{sutton2018reinforcement}, and defining $G(s_t,a_t):=\sum_{i=t}^{i=\infty}\gamma^{i-t}r_i$, the gradient of $J(\theta)$ with respect to the parameter $\theta$ is given by:
\begin{align}\label{REINFORCE}
\nabla_{\theta}J_M(\theta) = \mathbb{E}_{\tau \sim \pi_{\theta}}\big[G(s_t,a_t)\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big].
\end{align}
Then, $\mathbb{E}_{\tau \sim\pi_{\theta}}\big[G(s_t,a_t)\big]=Q^{\pi_{\theta}}(s_t,a_t)$.
REINFORCE \cite{sutton2018reinforcement} is a policy gradient method that uses Monte Carlo simulation to learn $\theta$, where the parameter update is performed only at the end of an episode (a trajectory of length $T$).
If we apply a look-ahead PBA scheme as in Equation (\ref{lookaheadPBA}) along with REINFORCE, then the total return from time $t$ is given by:
\begin{align}
\begin{split}
G^{a}(s_t,a_t)&=\sum_{i=t}^{i=T}\gamma^{i-t}r_i+\gamma^{T-t}\phi(s_T,a_T)-\phi(s_t,a_t) \\
&=G(s_t,a_t)+\gamma^{T-t}\phi(s_T,a_T)-\phi(s_t,a_t).
\end{split}
\end{align}
Notice that if $G^{a}(s_t,a_t)$ is used in Equation (\ref{REINFORCE}) instead of $G(s_t,a_t)$, then the policy gradient is biased.
One way to resolve the problem is to add the difference $-\gamma^{T-t}\phi(s_T,a_T)+\phi(s_t,a_t)$ to $G^{a}(s_t,a_t)$.
However, this makes the learning process identical to the original REINFORCE and PBA is not used.
While using PBA in a policy gradient setup, it it important to add the term $\phi(s,a)$ so that the policy gradient is unbiased, and also leverage the advantage that PBA offers during learning.
To apply PBA in policy gradient, we turn to temporal difference (TD) methods.
TD methods update estimates of the accumulated return based in part on other learned estimates, before the end of an episode.
A popular TD-based policy gradient method is the actor-critic framework \cite{sutton2018reinforcement}.
In this setup, after performing action $a_t$ at step $t$, the accumulated return $G(s_t,a_t)$ is estimated by $Q_M(s_t,a_t)$ which, in turn, is estimated by $r_t+\gamma V_M(s_{t+1})$.
It should be noted that the estimates are unbiased.
When the reward is augmented with look-ahead PBA, the accumulated return is changed to $Q_{M'}(s_t,a_t)$, which is estimated by $r_t+\gamma\phi(s_{t+1},a_{t+1})-\phi(s_t,a_t)+\gamma V_{M'}(s_{t+1})$.
From Equation (\ref{PBAq}), at steady state, $Q_M(s_t,a_t)-Q_{M'}(s_t,a_t)=\phi(s_t,a_t)$.
Intuitively, to keep policy gradient unbiased when augmented with look-ahead PBA, we can add $\phi(s_t,a_t)$ at each training step.
In other words, we can use $r_t+\gamma\phi(s_{t+1},a_{t+1})+\gamma V_{M'}(s_{t+1})$ as the estimated return.
It should be noted that before the policy reaches steady state, adding $\phi(s_t,a_t)$ at each time step will not cancel out the effect of PBA.
This is unlike in REINFORCE, where the addition of this term negates the effect of using PBA.
In the advantage actor-critic, an advantage term is used instead of the Q-function in order to reduce the variance of the estimated policy gradient.
In this case also, the potential term $\phi(s_t,a_t)$ can be added in order to keep the policy gradient unbiased.
\begin{varalgorithm}{AC-PBA}
\caption{: Actor-critic augmented with PBA}
\begin{algorithmic} \label{Algo1}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\REQUIRE Differentiable policy function $\pi_{\theta}(a|s)
\hspace{8.8mm}Differentiable value function $V^{\omega}(s)
\hspace{8.8mm}Potential-based advice $\phi(s,a)$
\hspace{8.8mm}Maximum episode $T_{max}$
\textbf{Initialization}: \\
policy parameter $\theta$, value parameter $\omega$, learning rate $\alpha^{\theta}$ and $\alpha^{\omega}$, discount factor $\gamma$, episode counter $T \leftarrow 0$
\REPEAT
\STATE initialize state $s_0$, $t \leftarrow 0$
\REPEAT
\STATE Sample action $a_t \sim \pi_{\theta}(\cdot|s_t)$
\STATE Take action $a_t$, observe reward $r_t$, next state $s_{t+1}$
\STATE $R= \begin{cases}
0, & \text{if }
\begin{aligned}[t]
s_{t+1} \text{ is a terminal state },
\end{aligned}
\\
V^{\omega}(s_{t+1}), & \text{otherwise.}
\end{cases}$
\IF{use look-ahead advice}
\STATE $\delta_t=r_t + \gamma\phi(s_{t+1},a_{t+1})-\phi(s_t,a_t)+\gamma R - V^{\omega}(s_t)$
\STATE Update $\theta \leftarrow \theta + \alpha^{\theta} \big(\delta_t+\phi(s_t,a_t)\big)\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)$
\ELSE
\STATE $\delta_t = r_t + \phi(s_{t},a_{t})-\gamma^{-1}\phi(s_{t-1},a_{t-1})+\gamma R - V^{\omega}(s_t)$
\STATE Update $\theta \leftarrow \theta + \alpha^{\theta} \delta_t\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)$
\ENDIF
\STATE Update $\omega \leftarrow \omega - \alpha^{\omega} \delta_t\nabla_{\omega}V^{\omega}(s_t)$
\UNTIL{$s_{t+1}$ is a terminal state}
\STATE $T \leftarrow T+1$
\UNTIL{$T>T_{max}$}
\end{algorithmic}
\end{varalgorithm}
A procedure for augmenting the advantage actor-critic with PBA is presented in Algorithm \ref{Algo1}.
$\alpha^{\theta}$ and $\alpha^{\omega}$ denote learning rates for the actor and critic respectively.
When applying look-ahead PBA, at training step $t$, parameter $\omega$ of the critic $V^{\omega}(s)$ is updated as follows:
\begin{align}
\delta^a_t &= r_t + \gamma\phi(s_{t+1},a_{t+1})-\phi(s_t,a_t)+\gamma V^{\omega}(s_{t+1}) - V^{\omega}(s_t)\nonumber \\
\omega &= \omega - \alpha^{\omega} \delta^a_t\nabla_{\omega}V^{\omega}(s_t),\nonumber
\end{align}
where $\delta^a_t$ is the estimation error of the state value after receiving new reward $[r_t + \gamma\phi(s_{t+1},a_{t+1})-\phi(s_t,a_t)]$ at step $t$.
To ensure an unbiased estimate of the policy gradient, the potential term $\phi(s_t,a_t)$ is added while updating $\theta$ as:
\begin{align}
\theta = \theta + \alpha^{\theta} \big(\delta^a_t+\phi(s_t,a_t)\big)\nabla_{\theta}\log\pi_{\theta}(a_t|s_t).\nonumber
\end{align}
A similar method can be used when learning with look-back PBA.
In this case, the critic and the policy parameter are updated as follows:
\begin{align}
\delta^b_t &= r_t + \phi(s_{t},a_{t})-\gamma^{-1}\phi(s_{t-1},a_{t-1})+\gamma V^{\omega}(s_{t+1}) - V^{\omega}(s_t)\nonumber\\
\omega &= \omega - \alpha^{\omega} \delta^b_t\nabla_{\omega}V^{\omega}(s_t),\nonumber\\
\theta &= \theta + \alpha \big(\delta^b_t+\gamma^{-1}\mathbb{E}\big[\phi(s_{t-1},a_{t-1})|s_t\big]\big)\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\label{addback}
\end{align}
In fact, the potential term need not be added to ensure an unbiased estimate in this case.
Then, the policy parameter update becomes:
\begin{align}\label{noaddback}
\theta = \theta + \alpha \delta^b_t\nabla_{\theta}\log\pi_{\theta}(a_t|s_t),
\end{align}
which is exactly the policy update of the advantage actor-critic.
This is formally stated in Proposition \ref{PBAprop}
\begin{prop} \label{PBAprop}
When the actor-critic is augmented with look-back PBA, Equations (\ref{addback}) and (\ref{noaddback}) are equal in the sense of expectation. That is
\begin{align}
\mathbb{E}_{(s_t,a_t) \sim \rho^{\pi_\theta}}\big[\big(\delta^b_t+&\gamma^{-1}\mathbb{E}\big[\phi(s_{t-1},a_{t-1})|s_t\big]\big)\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big] \nonumber\\
= \quad&\mathbb{E}_{(s_t,a_t) \sim \rho^{\pi_\theta}}\big[\delta^b_t\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big],
\end{align}
where $\rho^{\pi_\theta}$ is the distribution induced by the policy $\pi_\theta$.
\end{prop}
\begin{proof}
It is equivalent to show that:
\begin{align}
\mathbb{E}_{(s_t,a_t) \sim \rho^{\pi_\theta}}\big[\mathbb{E}\big[\phi(s_{t-1},a_{t-1})|s_t\big]\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big] = 0.
\end{align}
The inner expectation $\mathbb{E}\big[\phi(s_{t-1},a_{t-1})|s_t\big]$ is a function of $s_t$, policy $\pi_{\theta}$, and transition probability $\mathbb{T}$.
Denoting this expectation by $f(s_t,\pi_{\theta},\mathbb{T})$, we obtain:
\begin{align}
&\mathbb{E}_{(s_t,a_t) \sim \rho^{\pi_\theta}}\big[f(s_t,\pi_{\theta},\mathbb{T})\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big]\nonumber \\=
&\mathbb{E}_{s_t \sim
\rho^{\pi_\theta}}\bigg[\mathbb{E}_{a_t\sim \pi_\theta}\big[f(s_t,\pi_{\theta},\mathbb{T})\nabla_{\theta}\log\pi_{\theta}(a_t|s_t)\big]\bigg]\nonumber \\=
&\mathbb{E}_{s_t \sim
\rho^{\pi_\theta}}\bigg[\int_{A}\pi_{\theta}(a_t|s_t)f(s_t,\pi_{\theta},\mathbb{T})\frac{\nabla_{\theta}\pi_{\theta}(a_t|s_t)}{\pi_{\theta}(a_t|s_t)}\text{d}a\bigg] \nonumber\\=
&\mathbb{E}_{s_t \sim
\rho^{\pi_\theta}}\bigg[f(s_t,\pi_{\theta},\mathbb{T})\nabla_{\theta}\int_{A}\pi_{\theta}(a_t|s_t)\text{d}a\bigg] = 0.
\end{align}
The last equality follows from the fact that the integral evaluates to $1$, and its gradient is $0$.
\end{proof}
The main result of this paper presents guarantees on the convergence of Algorithm \ref{Algo1} using the theory of `two time-scale stochastic analysis' \cite{Borkar2000ODE}. Assume that:
\begin{itemize}
\item \textbf{A1}: The value function $V^{\omega}(s)$ belongs to a linear family. That is, $V^{\omega}=\Phi \omega$, where $\Phi \in \mathbb{R}^{|S|\times k}, k <S$ is a known full-rank feature matrix, and $\omega \in \Omega \subseteq \mathbb{R}^{k}$
\item \textbf{A2}: For the set of policies $\{\pi_{\theta}, \theta \in \Theta\subseteq \mathbb{R}^{d}\}$, there exists a constant $C_{\Theta}$ such that $\norm{\nabla_{\theta}\log\pi_{\theta}}_2\leq C_{\Theta}$.
\item \textbf{A3}: Learning rates of the actor and critic satisfy: $\sum_t \alpha_t^{\theta}=\sum_t \alpha_t^{\omega}=\infty$, $\sum_t [(\alpha_t^{\theta})^2+(\alpha_t^{\omega})^2]<\infty$,
$\lim\limits_{t\rightarrow \infty}\frac{\alpha_t^{\theta}}{\alpha_t^{\omega}}=0$.
\end{itemize}
For any probability measure $\mu$ on a finite set $\mathcal{M}$, the $\ell_2$-norm of $f$ with respect to $\mu$ is given by $\norm{f}_{\mu}:=\big[\int_{\mathcal{M}}|f(x)|^2\text{d}\mu(x)\big]^{\frac{1}{2}}$.
Theorem \ref{convergence} gives a bound on the error introduced as a result of approximating the value function $V_{M'}$ with $V_{M'}^{\omega}$ as in assumption \textbf{A1}.
This error term is small if the family $\Omega$ is rich.
In fact, if the critic is updated in batches, a tighter bound can be achieved, as shown in Proposition 1 of \cite{Yang2018finite}.
Extending the result to the case of online updates is a subject of future work.
\begin{thm}\label{convergence}
Let $\mathcal{E}(\theta):=\norm{V^{\omega(\theta)}_{M'}(s)-V^{\pi_{\theta}}_{M'}(s)}_{\rho^{\pi_{\theta}}}$.
Then, for any limit point $(\theta^*,\omega^*) :=\lim\limits_{T_{max} \to \infty}(\theta_{T_{max}},\omega_{T_{max}})\}$ of Algorithm \ref{Algo1},
$\norm{\nabla_{\theta}J_M(\theta^*)}_2\leq C\mathcal{E}(\theta^*)$.
\end{thm}
\begin{proof}
We consider only look-ahead PBA.
The proof for look-back PBA follows similarly.
Define $F:=F(s,a,s',a')$.
From assumption \textbf{A3}, the actor is updated at a slower rate than the critic.
This allows us to fix the actor to study the asymptotic behavior of the critic \cite{Bhatnagar2009}.
The update dynamics of the critic can be represented by
\begin{align}\label{ODE_critic}
\dot{\omega}=\mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\delta_{\omega}\nabla_{\omega}V^{\omega}_{M'}(s)\big],
\end{align}
where $\delta_{\omega}=r(s,a) + \gamma\phi(s',a')-\phi(s,a)+\gamma V^{{\omega}}(s') - V^{\omega}(s)$ if look-ahead PBA is applied.
When the critic is
approximated by a linear function (assumption \textbf{A1}), $\omega$ will converge to $\omega(\theta)$, an asymptotically stable equilibrium of Equation (\ref{ODE_critic}).
The update of the actor is then
\begin{align}\label{ODE_actor}
\dot{\theta}=\mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\nabla_{\theta}\log \pi_{\theta}(a|s)\big(r(s,a)+F+\gamma V_{M'}^{\omega(\theta)}(s')+\phi(s,a)\big)\big].
\end{align}
Let $\Theta_s$ denote the set of asymptotic stable equilibria in Equation (\ref{ODE_actor}).
Any $\theta \in \Theta_s$ will satisfy $\dot{\theta}=0$ in Equation (\ref{ODE_actor}).
Then, $\{(\theta_t,\omega_t)\}_{t>0}$ will converge to $\{(\theta,\omega(\theta)):\theta \in \Theta_s\}$.
Now, consider the evaluation of $\pi_{\theta}$, $\theta \in \Theta_s$, in the original MDP $M$.
We obtain the following equations:
\begin{align}\label{evaluation}
&\nabla_{\theta}J_M(\theta) =
\mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\nabla_{\theta}\log \pi_{\theta}(a|s)Q^{\pi_{\theta}}_{M}(s,a)\big] \nonumber\\
&= \mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\nabla_{\theta}\log \pi_{\theta}(a|s)\big(Q^{\pi_{\theta}}_{M'}(s,a)+\phi(s,a)\big)\big] \nonumber\\
& = \mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\nabla_{\theta}\log \pi_{\theta}(a|s)\big(r(s,a)+F+\gamma V^{\pi_{\theta}}_{M'}(s')+\phi(s,a)\big)\big].
\end{align}
Subtracting Equation (\ref{ODE_actor}) from Equation (\ref{evaluation}), and applying the Cauchy-Schwarz inequality to the result yields:
\begin{align*
\nabla_{\theta}J_M(\theta)&=\gamma \mathbb{E}_{ \rho^{\pi_{\theta}}}\big[\nabla_{\theta}\log \pi_{\theta}(a|s)\big( V_{M'}^{\omega(\theta)}(s')-V^{\pi_{\theta}}_{M'}(s')\big)\big]\\
\therefore \norm{\nabla_{\theta}J_M(\theta)}_2&\leq \gamma\norm{\nabla_{\theta}\log \pi_{\theta}(a|s)}_{\rho^{\pi_{\theta}}}\norm{V_{M'}^{\omega(\theta)}(s)-V^{\pi_{\theta}}_{M'}(s)}_{\rho^{\pi_{\theta}}}.
\end{align*}
The result follows by applying assumption \textbf{A2}.
\end{proof}
\begin{rem}
Look-back PBA could result in better performance compared to look-ahead PBA since look-back PBA does not involve estimating a future action.
\end{rem}
\section{Experiments}\label{ExptSection}
Our experiments seek to compare the performance of an actor-critic architecture augmented with PBA and with PBRS with the `vanilla' advantage actor-critic (A2C).
We consider two setups.
The first is a \emph{Puddle-Jump Gridworld} \cite{marom2018belief}, where the state and action spaces are discrete.
The second environment we study is a continuous state and action space \emph{mountain car} \cite{brockman2016openai}.
In each experiment, we compare the rewards received by the agent when it uses the following schemes: \emph{i)}: `vanilla' (A2C); \emph{ii)}: A2C augmented with PBRS; \emph{iii)}: A2C with look-ahead PBA; \emph{iv)}: A2C with look-back PBA.
\subsection{Puddle-Jump Gridworld}
\begin{figure}
\centering
\includegraphics[width=1.5 in]{gridworld.pdf}
\caption{Schematic of the puddle-jump gridworld. The state of the agent is its position $(x,y)$. The shaded row (row $2$) represents the puddle the agent should jump over. The two blue grids denote states that are indistinguishable to the agent. The agent can choose an action from the set $\{up, down, left, right, jump\}$ at each step.}\label{GridWorld}
\end{figure}
Figure \ref{GridWorld} depicts the \emph{Puddle-jump gridworld} environment as a 10x10 grid.
The state space is $s=(x,y)$ denoting the position of the agent in the grid, where $x,y \in \{0,1,\dots,9\}$.
The goal of the agent is to navigate from the start state $S= (0,0)$ to the goal $G=(9,9)$.
At each step, the agent can choose from actions in the set $A = \{up, down, left, right, jump\}$.
There is a \emph{puddle} along row $2$ which the agent should jump over. Further, the states $(9,8)$ and $(8,9)$ (blue squares in Figure \ref{GridWorld}) are indistinguishable to the agent.
As a result, any optimal policy for the agent is a stochastic policy.
If the $jump$ action is chosen in rows $3$ or $1$, the agent will land on the other side of the puddle with probability $p_j$, and remain in the same state otherwise.
This action chosen in other rows will keep the agent in its current state.
Any action that will move the agent off the grid will keep its state unchanged.
The agent receives a reward of $-0.05$ for each action, and $+1000$ for reaching $G$.
When using PBRS, we set $\phi^{PBRS}(s):=u_0$ for states in rows $0$ and $1$, and $\phi^{PBRS}(s):=u_1$ for all other states.
We need $u_1>u_0$ to encourage the agent to jump over the puddle.
Unlike in PBRS, PBA can provide the agent with more information about the actions it can take.
We set $\phi^{PBA}(s,a)$ to a `large' value if action $a$ at state $s$ results in the agent moving closer to the goal according to the $\ell_1$ norm, $\big(|G-x|+|G-y|
\big)$.
We additionally stipulate that $\frac{1}{|A|}\sum_{a\in A}\phi^{PBA}(s,a) = \phi^{PBRS}(s)$. That is, the state potential of PBA is the same as the state potential of PBRS under a uniform distribution over the actions.
This is to ensure a fair comparison between PBRS and PBA.
In our experiment, we set the discount factor $\gamma=1$.
Since the dimensions of the state and action spaces is not large, we do not use a function approximator for the policy $\pi$.
A parameter $\theta_{s,a}$ is associated to each state-action pair, and the policy is computed as:
$\pi_{\theta}(a|s)=\frac{\exp(\theta_{s,a})}{\sum_{a\in A}\exp(\theta_{s,a})}$.
We fix $\alpha^{\omega}= 0.001$, and $\alpha^{\theta}= 0.2$ for all cases.
From Figure \ref{cliff_results}, we observe that the look-back PBA scheme performs the best, in that the agent converges to the goal in \textbf{five times} fewer episodes ($25$ vs. $125$ episodes) than A2C without advice.
When A2C is augmented with PBRS, convergence to the goal is slightly faster than without any reward shaping.
When augmented with look-ahead PBA, in the first few episodes, the reward increases faster than in the case of A2C augmented with PBRS.
However, this slows down after the early training stages and the policy converges to the goal in about the same number of episodes as a policy trained without advice.
A reason for this could be that during later stages of training, a look-ahead PBA scheme might advise an agent with `bad' actions, leading to bad policies, thereby impeding the progress of learning.
For example, an action $a_t$ might be a good choice at state $s_t$, but the look-ahead PBA scheme might indicate that $a_t$ is bad, due to a poor estimate of the future action $a_{t+1}$.
\begin{figure}
\centering
\includegraphics[width=2.8 in]{cliff_jump.png}
\caption{Average rewards in puddle-jump gridworld when jump success probability $p_j=0.2$. The baseline is the advantage actor-critic without advice.}\label{cliff_results}
\end{figure}
A smaller jump success probability $p_j$ is an indication that it is more difficult for the agent to reach the goal state $G$.
Figure \ref{cliff_results2} shows that look-back PBA results in the highest reward for a more difficult task (lower $p_j$), when compared with the other reward shaping schemes.
\begin{figure}
\centering
\includegraphics[width=2.8 in]{cliff_jump2.png}
\caption{Average reward for the first 100 episodes with respect to the jump success probability $p_j$.}\label{cliff_results2}
\end{figure}
\subsection{Continuous Mountain Car}
In the mountain car (MC) environment, an under powered car in a valley has to drive up a steep hill to reach the goal. In order to achieve this, the car should learn how to accumulate momentum.
A schematic for this environment is shown in Figure \ref{MountainCarFig}.
\begin{figure}
\centering
\includegraphics[width=3 in]{MountainCar.png}
\caption{Schematic of the mountain-car environment. The agent's state is represented by its position $p_t$ (along the $x-$coordinate) and velocity $v_t$. The action $a_t$ is a force applied to the car. The goal is marked as a flag.}\label{MountainCarFig}
\end{figure}
This MC environment has continuous state and action spaces.
The state $s=(p,v)$ denotes position $p \in [-1.2,0.6]$ and velocity $v \in [-0.07,0.07]$.
The action $a \in [-1,+1]$.
The continuous action space makes it difficult to use classic value-based methods, such as Q-learning and Sarsa-learning.
The reward provided by the environment depends on the action and whether the car reaches the goal.
Specifically, once the car reaches the goal it receives $+100$, and before that, the reward at time $t$ is $-|a_t|^2$.
This reward structure therefore discourages the waste of energy.
This acts as a barrier for learning, because there appears to be a sub-optimal solution where the agent remains at the bottom of the valley.
Moreover, the reward for reaching the goal is significantly delayed, which makes it difficult for the conventional actor-critic algorithm to learn a good policy.
One choice of a potential function while using PBRS in this environment is $\phi^{PBRS}(s_t):=p_t+2$, where the offset is so that the potential is positive.
An interpretation of this scheme is: \emph{`state value is larger when the car is horizontally closer to the goal.'}
The PBA scheme we use for this environment encourages the accumulation of momentum by the car-- the direction of the action is encouraged to be the same as the current direction of the car's velocity.
In the meanwhile, we discourage inaction.
Mathematically, the potential advice function has a larger value if \emph{ $a_t\neq 0$}.
We let $\phi^{PBA}(s_t,a_t)=1$, if $a_tv_t >0$, and $\phi^{PBA}(s_t,a_t)=0$, otherwise.
In our experiments, we set $\gamma= 0.99$.
To deal with the continuous state space, we use a neural network (NN) as a function approximator.
The policy distribution $\pi_{\theta}(a|s)$ is approximated by a normal distribution, the mean and variance of which are the outputs of the NN.
The value function is also represented by an NN.
We set $\alpha^{\theta}=1\times 10^{-5}$ and $\alpha^{\omega}=5.6\times 10^{-4}$, and use Adam \cite{Adam} to update the NN parameters.
The results we report are averaged over 10 different environment seeds
\begin{figure}
\centering
\includegraphics[width=2.8 in]{MC.png}
\caption{Average rewards for continuous mountain car problem (averaged over 10 different environment random seeds). The baseline is the A2C without advice.}\label{MountainCar_results}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{|c | c | c | c |}
\hline
No advice & PBRS & Look-ahead PBA & Look-back PBA \\ [0.5ex]
\hline\hline
10\% & 20\% & 40\% & 100\% \\
\hline
\end{tabular}
\caption{Percentage of trials where policy converges correctly in continuous mountain car problem.}
\label{table:1}
\end{table}
Our experiments indicate that the policy makes the agent converge to one of two points: the goal, or remain stationary at the bottom of the valley.
The percentage of solutions that converge to the goal is shown in Table \ref{table:1}.
From Figure \ref{MountainCar_results} and Table \ref{table:1}, when learning with the vanilla A2C, the agent is able to reach the goal only in $10\%$ of the trials (out of 10 trials), and was stuck at the sub-optimal solution for the remaining trials.
With PBRS, the agent could converge correctly in only $20\%$ of the trials.
This is because the agent might have to take an action that moves it away from the goal in order to accumulate momentum.
However, the potential function $\phi^{PBRS}(\cdot)$ discourages such actions.
In comparison, the average reward when using look-ahead PBA is slightly higher, but the agent is able to reach the goal in only $40\%$ of the trials. Similar to the gridworld setup, look-back PBA performs the best, where the agent is able to reach the goal in $100\%$ of the trials.
\section{Conclusion}\label{ConclusionSection}
This paper presented a framework for augmenting the reward received by an RL agent with PBRS and with PBA.
Different from prior work, we demonstrated that our approach can be used in environments with continuous states and actions, and when the optimal policy is stochastic.
We presented guarantees on the convergence of an algorithm that augments an A2C architecture with these schemes.
Our experiments indicated that these schemes allowed the agent to achieve higher average rewards, and learn an optimal policy faster.
Future work will focus on establishing tighter bounds for Theorem \ref{convergence}, and extending our approach to the average reward case.
\bibliographystyle{IEEEtran}
|
1,108,101,565,479 | arxiv | \section{Extension to Low Rank Phase Retrieval (LRPR)} \label{algo_thm_proof_lrpr}
In LRPR, recall that, we measure ${\y_{(mag)}}_k = |\bm{A}_k \x^*_k|$.
This problem commonly occurs in dynamic phaseless imaging applications such as Fourier ptychography.
Because of the magnitude-only measurements, we can recover each column only up to a global phase uncertainty. We use $\mathrm{dist}(\x^*,\hat\x): = \min_{\theta \in [-\pi, \pi]} \|\x^* - e^{-j \theta} \hat\x\|$ to quantify this phase invariant distance \cite{pr_altmin,rwf}. Also, for a complex number, $z$, we use $\bar{z}$ to denote its conjugate and we use $phase(z):= z /|z|$.%
\subsection{AltGD-Min-LRPR algorithm}
With three simple changes that we explain next, the AltGD-Min approach also solves LRPR and provides the fastest existing solution for it.
First, observe that because of the magnitude-only measurements, we cannot use $\hat\X_0$ with $\bm{y}_{ki}$ replaced by ${\y_{(mag)}}_{ki}$ for initialization. The reason is $\mathbb{E}[\a_{ki} {\y_{(mag)}}_{ki}] = 0$ and so $\mathbb{E}[ \a_{ki} {\y_{(mag)}}_{ki} \mathbbm{1}_{{\y_{(mag)}}_{ki} \le \sqrt{\alpha}}] = 0$ too. In fact, because of this, it is not even possible to define a different matrix $\hat\X$ whose expected value can be shown to be close to $\X^*$. Instead, we have to use the initialization approach of \cite{lrpr_it}. This is given in line 5 of Algorithm \ref{AltGD-Min_lrpr}. The matrix $\bm{Y}_U$ is such that its expected value is close to $\X^* \X^*{}^\top + c \bm{I}$. This fact is used to argue that its top $r$ singular vectors span a subspace that is close to that spanned by columns of $\U^*{}$.
Next, consider the GDmin iterations. We use the following idea to deal with the magnitude-only measurements: ${\y_{(mag)}}_{ki}:=|\bm{y}_{ki}|$. Let $\bm{c}_{ki}:= \mathrm{phase}(\a_{ki}{}^\top \x^*_k)$. Then, clearly,
\[
\bm{y}_{ki} = \bm{c}_{ki} {\y_{(mag)}}_{ki}
\]
and ${\y_{(mag)}}_{ki} = \bar{\bm{c}}_{ki} \bm{y}_{ki}$.
We do not observe $\bm{c}_{ki}$, but we can estimate it using $\hat\x_k$ which is an estimate of $\x^*_k$. Using the estimated phase, we can get an estimate $\hat\bm{y}_{ki}$ of $\bm{y}_{ki}$.
We replace $\nabla_{\bm{U}} f({\bm{U}},\bm{B})$ by its estimate which uses $\hat\bm{y}_{ki}= {\y_{(mag)}}_{ki} \hat\bm{c}_{ki}$, with $\hat\bm{c}_{ki} = phase(\a_{ki}{}^\top \hat\x_k)$, to replace $\bm{y}_{ki}$. See line 10 of Algorithm \ref{AltGD-Min_lrpr}
Lastly, because of the magnitude-only measurements, the update step for updating $\hat\b_k$s is no longer an LS problem. We now need to solve an $r$-dimensional standard PR problem: $\min_{\hat\b} \|{\y_{(mag)}}_k - |\bm{A}_k {\bm{U}} \b| \|^2$.
This can be solved using any of the order-optimal algorithms for standard PR, e.g., Truncated Wirtinger Flow (TWF) \cite{twf} or Reshaped WF (RWF) \cite{rwf}. For concreteness, we assume that RWF is used. We should point out here that we only need to run $T_{RWF,t}$ iterations of RWF at outer loop iteration $t$, with $T_{RWF,t}$ set below in our theorem (we set this to ensure that the error level of this step is of order $\delta_t$).
The entire algorithm, AltGD-Min-LRPR, is summarized in Algorithm \ref{AltGD-Min_lrpr}.
\subsection{Main Result}
We can prove the following result with simple changes to the proof of Theorem \ref{gdmin_thm}.
\begin{theorem}
Consider Algorithm \ref{AltGD-Min_lrpr}. Set $\eta = c / {\sigma_{\max}^*}^2$, $ \tilde{C} = 9 \kappa^2 \mu^2$, $T = C \kappa^2 \log(1/\epsilon)$, and $T_{RWF,t} = C(t + c \log r)$.
Assume that Assumption \ref{right_incoh} holds.
If
\[
mq \ge C \kappa^6 \mu^2 (n + q) r^2 (r + \log(1/\epsilon) \log \kappa )
\]
and $m \ge C \max(\log q, \log n) \log(1/\epsilon)$,
then,
w.p. $1- n^{-10}$,
$ \SE}%{\SE_F(\U^*{},{\bm{U}}_T) \le \epsilon$, $\mathrm{dist}( (\hat\x_k)_T,\x^*_k) \le \epsilon \|\x^*_k\|$ for all $k \in [q]$, and $\sum_k \mathrm{dist}^2( (\hat\x_k)_T,\x^*_k) \le \epsilon^2 {\sigma_{\max}^*}^2.$%
\label{gdmin_lrpr_thm}
\end{theorem}
We prove this result in Sec. \ref{lrpr_proof}.
Notice the $\log(1/\epsilon)$ in the sample complexity of Theorem \ref{gdmin_thm} is now replaced by $(r+\log(1/\epsilon))$. The reason is because of the different initialization approach which needs $nr^3$ samples instead of $nr^2$.
This is needed because PR is a more difficult problem: we cannot define a matrix $\hat\X_0$ for it for which $\mathbb{E}[\hat\X_0]$ is close to $\X^*$.
Observe that AltGD-Min-LRPR has the same sample complexity as that for the AltMin solution from \cite{lrpr_best}. But its time complexity is better by a factor of $\log(1/\epsilon)$ making it the fastest solution for LRPR. Also, we should mention here that, for solutions to the two related problems -- sparse PR (phaseless but global measurements) and LRMC (linear but non-global measurements) -- that have been extensively studied for nearly a decade, the best sample complexity guarantees for iterative (and hence fast) algorithms are sub-optimal. The best sparse PR guarantee \cite{cai} requires $m$ to be of order $s^2$ for the initialization step. Here $s$ is the sparsity level. LRPR has both phaseless and non-global measurements. This is why its initialization step needs two extra factors of $r$ compared to the optimal. Once initialized close enough to the true solution, it is well known that a PR problem behaves like a linear one. This is true for AltGD-Min-LRPR too.
}% {\color{red}
Consider a comparison with use of a standard PR approach to recover each column of $\X^*$ individually. If TWF \cite{twf} or RWF \cite{rwf} were used for this, this would require $m \gtrsim n$. In comparison, ignoring log factors, our solution for LRPR needs $m \gtrsim (n/q) r^3$. Thus, the use of altGD-min is a better idea when the rank, $r$, of the matrix $\X^*$ is small enough so that $q \gtrsim r^3$.
\color{black}
\subsection{Proof of Theorem \ref{gdmin_lrpr_thm}} \label{lrpr_proof}
For the initialization, we use the bound from \cite{lrpr_it}.
\begin{lemma}[\cite{lrpr_it}]
\label{bounding_U_lrpr}
Let $\text{{SubsDist}}_2({\bm{U}}_0,\U^*{}) = \|(\bm{I} - \U^*{} \U^*{}^\top) {\bm{U}}_0\|$. Pick a $\delta_\init < 0.1$. Then, w.p. at least
$ 1 - 2\exp\left( n (\log 17) -c \frac{\delta_\init^2 mq}{ \kappa^4 r^2} \right) - 2 \exp\left(-c \frac{\delta_\init^2 mq}{ \kappa^4 \mu^2 r^2} \right) $,
\[
\text{{SubsDist}}_2({\bm{U}}_0,\U^*{}) \leq \delta_\init \text{ and so } \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \le \sqrt{r} \delta_\init.
\]
\end{lemma}
For the iterations, without loss of generality, as also done in past works on PR, e.g., \cite{pr_altmin,twf,rwf,lrpr_best}, to make things simpler, we assume that, for each $k$, $\x^*_k$ is replaced by $\bar{z} \x^*_k$ where $z = \mathrm{phase}(\langle \x^*_k, \hat\x_k \rangle)$.
With this, $\mathrm{dist}(\x^*_k, \hat\x_k) = \|\x^*_k - \hat\x_k\|$.
We modify Lemma \ref{algebra} using the following idea. Let ${\bm{U}} = {\bm{U}}_t$ and $\bm{B} = \bm{B}_t$. For LRPR, the GD step uses an approximate gradient w.r.t. the old cost function $f({\bm{U}},\bm{B})$. Let
\begin{align*}
\mathrm{Err}: = \widehat{\mathrm{GradU}} - \mathrm{GradU}.
\end{align*}
Here $\widehat{\mathrm{GradU}} = \sum_{ki} (\hat\bm{y}_{ki} - \a_{ki}{}^\top \hat\x_k) ) \a_{ki} \hat\b_k{}^\top$ and $\mathrm{GradU} = \nabla_{\bm{U}} f({\bm{U}},\bm{B}) = \sum_{ki} (\bm{y}_{ki} - \a_{ki}{}^\top \hat\x_k) ) \a_{ki} \hat\b_k{}^\top$ is the same as earlier. Thus,%
\begin{align*}
\mathrm{Err}
& = \sum_{ki} (\hat\bm{y}_{ki} - \bm{y}_{ki}) \a_{ki} \hat\b_k{}^\top \\
& = \sum_{ki} (\hat\bm{c}_{ki} - \bm{c}_{ki} ) |\a_{ki}^\top \x^*_k| \a_{ki} \hat\b_k{}^\top \\
& = \sum_{ki} (\hat\bm{c}_{ki} \bar{\bm{c}}_{ki} - 1 ) (\a_{ki}^\top \x^*_k) \a_{ki} \hat\b_k{}^\top
\end{align*}
Proceeding as in the proof of Lemma \ref{algebra}, and using $\| (\bm{I} - \U^*{} \U^*{}^\top) \mathrm{Err}\|_F \le \|\mathrm{Err}\|_F$ and $\|\mathrm{Err}\| \le \|\mathrm{Err}\|_F$, we can conclude the following
{\footnotesize
\begin{align*}
& \text{{SubsDist}}(\U^*{}, {\bm{U}}^+) \le \\
& \frac{ \|(I - (\eta/m) \mathrm{Hess}\| \cdot \text{{SubsDist}}(\U^*{},{\bm{U}}) + (\eta/m) \|\mathrm{Term2}\|_F + (\eta/m)\|\mathrm{Err}\|_F}{1 - (\eta/m) \|\mathrm{GradU}\| - (\eta/m) \|\mathrm{Err}\|_F}
\end{align*}
}
where the expressions for $\mathrm{GradU}, \mathrm{Term2}, \mathrm{Hess}$ are the same as before with one change: $\hat\b_k$ is now obtained by solving a noisy $r$-dimensional PR problem (instead of a LS problem) using RWF \cite{rwf}. Thus, to complete the proof, (i) we need to bound
\[
\|\mathrm{Err}\|_F = \max_{{\bm{W}} \in \S_{nr} } \sum_{ki} (\hat\bm{c}_{ki} \bar{\bm{c}}_{ki} - 1 ) (\a_{ki}^\top \x^*_k) (\a_{ki}^\top {\bm{W}} \hat\b_k)
\]
and (ii) we need bounds on the three other terms that were also bounded earlier for the linear case.
The term $\|\mathrm{Err}\|_F$, is bounded in Lemma 4 of \cite{lrpr_best} . We repeat the lemma below.
\newcommand{\delta_{t,F}}{\delta_t}
\begin{lemma}
Assume that $\SE}%{\SE_F({\bm{U}}_t, \U^*{}) \le \delta_{t,F}$ with $\delta_{t,F} < c/\kappa^2$.
Then,
w.p. at least $1- 2 \exp\left( nr \log(17) - c \frac{m q\epsilon_2^2}{ \mu^2 \kappa r} \right) - \exp(\log q + r - c m )$,
\[
\|\mathrm{Err}\|_F \le C m ( \epsilon_2 + \sqrt{\delta_{t,F}} ) \delta_{t,F} {\sigma_{\max}^*}^2
\]
\label{Err_bnd}
\end{lemma}
Consider the other three terms: $\mathrm{GradU}, \mathrm{Term2}, \mathrm{Hess}$. These were bounded in Lemma \ref{terms_bnds} for the linear case. The statement and proof of this lemma remain the same as earlier because of the following reason. Its proof uses the bounds on $\hat\b_k$, $\hat\x_k$ from Lemma \ref{B_lemma}. The statement of this lemma also remains the same with one change: we replace $\|\x^* - \bm{x}\|$ by $\mathrm{dist}(\x^*, \bm{x})$ and $\|{\bm{X}}^* - {\bm{X}}\|_F^2$ by $\sum_{k=1}^q \mathrm{dist}^2(\x^*_k,\bm{x}_k)$, and the same for $\b^*} %{\tilde\b^*_k, \bm{g}_k$. The first part of Lemma \ref{B_lemma} now follows by the first part of \cite[Lemma 3.3]{lrpr_best}.
All the subparts of the second part of Lemma \ref{B_lemma} follow exactly as given in its proof in Sec. \ref{B_lemma_proof}.
\section{The Proposed AltGD-Min Algorithm and Guarantee
\label{algo_thm}
\subsection{The AltGD-Min algorithm} \label{algo_explain}
We would like to design a fast GD algorithm to find the matrix ${\bm{X}}$ that minimizes the squared-loss cost function
$
\tilde{f}({\bm{X}}): = \sum_{k=1}^q \|\bm{y}_k - \bm{A}_k \hat\x_k\|^2.
$
For reasons described earlier, we decompose ${\bm{X}} = {\bm{U}} \bm{B}$ and develop an alternating GD-min (AltGD-Min) approach for the squared loss function,
\[
f({\bm{U}},\bm{B}) := \tilde{f}({\bm{U}}\bm{B}) = \sum_k \|\bm{y}_k - \bm{A}_k {\bm{U}} \b_k\|^2.
\]
Starting with a careful initialization for ${\bm{U}}$ explained below, AltGD-Min proceeds as follows. At each new iteration,
\begin{itemize}} \newcommand{\ei}{\end{itemize}
\item {\em Min-B: } update $\bm{B}$ by solving $\bm{B} \leftarrow \arg\min_{\tilde\bm{B}} f({\bm{U}},\tilde\bm{B})$. Since $\b_k$ only occurs in the $k$-th summand of $f({\bm{U}},\bm{B})$, this decouples to a much simpler column-wise least squares (LS) problem: $\b_k \leftarrow \arg\min_{\tilde\b_k} \|\bm{y}_k - \bm{A}_k {\bm{U}} \tilde\b_k\|^2$. This is solved in closed form as $\hat\b_k = (\bm{A}_k {\bm{U}})^\dag \bm{y}_k$ for each $k$; here $\bm{M}^\dag:=(\bm{M}^\top \bm{M})^{-1}\bm{M}^\top$.
\item {\em ProjGD-U: } update ${\bm{U}}$ by one GD step for it, $\hat\U^+ \leftarrow {\bm{U}} - \eta \nabla_U f({\bm{U}},\bm{B})$, followed by projecting $\hat\U^+$ onto the space of matrices with orthonormal columns to get the updated ${\bm{U}}^+$.
We get ${\bm{U}}^+$ by QR decomposition: $\hat\U^+ \overset{\mathrm{QR}}=} %{\stackrel{EVD}{=} {\bm{U}}^+ {\bm{R}}^+$.
\ei
Notice that, because of the decoupling for $\bm{B}$, the min step only involves solving $q$ $r$-dimensional Least Squares (LS) problems, in addition to also first computing the matrices, $\bm{A}_k {\bm{U}}$.
Computing the matrices needs time of order $mnr$, and solving one LS problem needs time of order $mr^2$. Thus, the LS step needs time $O(q\max(mnr, m r^2)) = O(mqnr)$ since $r \le n$. This is equal to the time needed to compute the gradient w.r.t. ${\bm{U}}$; and thus, the per-iteration cost of AltGD-Min is only $O(mqnr)$. The QR decomposition of an $n \times r$ matrix takes time only $nr^2$.
Since $f({\bm{U}},\bm{B})$ is not a convex function of the unknowns $\{{\bm{U}}, \bm{B}\}$, a careful initialization is needed.
Borrowing the spectral initialization idea from LRMC and LRMS solutions, we should initialize ${\bm{U}}_0$ by computing the top $r$ singular vectors o
\[
{\bm{X}}_{0,full} = \frac{1}{m} [ (\bm{A}_1^\top \bm{y}_1), (\bm{A}_2^\top \bm{y}_2), \dots, (\bm{A}_k^\top \bm{y}_k), \dots (\bm{A}_q^\top \bm{y}_q)]
\]
Clearly the expected value of the $k$-th column of this matrix equals $\x^*_k$ and thus $\mathbb{E}[{\bm{X}}_{0,full}] = \X^*$. But, as we explain next, it is not clear how to prove that this matrix concentrates around $\X^*$.
Observe that it can also be written as
\begin{align*}
{\bm{X}}_{0,full} := \frac{1}{m}\sum_{k=1}^q \sum_{i=1}^m \a_{ki} \bm{y}_{ki} \bm{e}_k{}^\top
\end{align*}
Its summands are independent sub-exponential r.v.s with maximum sub-exponential norm $\max_k \|\x^*_k\| \le \mu \sqrt{r/q} {\sigma_{\max}^*}$. This is too large and does not allow us to bound $\|{\bm{X}}_{0,full} - \X^*\|$ under the desired sample complexity; see Appendix \ref{algo_understand}.
To resolve this issue, we borrow the truncation idea from earlier work on PR \cite{twf,lrpr_it} and initialize ${\bm{U}}_0$ as the top $r$ left singular vectors of
\begin{eqnarray}
\hat\X_{0} & := & \frac{1}{m}\sum_{k=1}^q \sum_{i=1}^m \a_{ki} \bm{y}_{ki} \bm{e}_k{}^\top \mathbbm{1}_{\left\{ \bm{y}_{ki}^2 \le \alpha \right\} } \nonumber \\
& = & \frac{1}{m}\sum_{k=1}^q \bm{A}_k^\top \bm{y}_{k,trunc}(\alpha) \bm{e}_k^\top
\label{newinit}
\end{eqnarray}
where $\alpha := \tilde{C} \frac{\sum_{ki} (\bm{y}_{ki})^2}{mq}$ and $\bm{y}_{k,trunc}(\alpha) := \bm{y}_k \circ \mathbbm{1}(|\bm{y}_k| \le \sqrt\alpha )$. We set $\tilde{C}$ in our main result.
Observe that we are summing over only those $i,k$ for which $\bm{y}_{ki}^2$ is not too large (is not much larger than its empirically computed average value). This {truncation} filters out the too large (outlier-like) measurements and sums over the rest. Theoretically, this converts the summands into sub-Gaussian r.v.s which have lighter tails than the un-truncated ones. This allows us to prove the desired concentration bound.
Different from the above setting, in \cite{twf,lrpr_it}, truncation was applied to symmetric positive definite matrices and was used to convert summands that were heavier-tailed than sub-exponential to sub-exponential.%
We summarize the complete algorithm in Algorithm \ref{gdmin}. This uses sample-splitting which is a commonly used approach in the LR recovery literature \cite{lowrank_altmin,fastmc,rmc_gd} as well as in other compressive sensing settings. It helps ensure that the measurement matrices in each iteration for updating ${\bm{U}}$ and $\bm{B}$ are independent of all previous iterates. This allows one to use concentration bounds for sums of independent r.v.s. We provide a detailed discussion in Sec. \ref{samplesplit}.
\subsubsection{Practical algorithm and setting algorithm parameters}
First, when we implement the algorithm, we use Algorithm \ref{gdmin} with using the full set of measurements for all the steps (no sample-splitting).
The algorithm has 4 parameters: $\eta$, $T$, $\tilde{C}$ and the rank $r$. According to the theorem below, we should set $\eta = c / {\sigma_{\max}^*}^2$ with $c<0.5$. But ${\sigma_{\max}^*}$ is not known. The initialization matrix $\hat\X_0$ provides an approximation to $\X^*$ and hence we can set $\eta = c/\|\hat\X_0\|^2$.
Consider $\tilde{C}$. The theorem requires setting $\tilde{C} = 9 \kappa^2 \mu^2$, however $\kappa,\mu$ are functions of $\X^*$ which is unknown. Using the definition of $\mu$ from Assumption \ref{right_incoh}, we can replace $\kappa^2 \mu^2$ by an estimate of its lower bound: $q \cdot \max_k \widehat{\|\x^*_k\|^2} / \widehat{\|\X^*\|_F^2}$ with $ \widehat{\|\x^*_k\|^2} = (1/m) \sum_i \bm{y}_{ki}^2$ and $\widehat{\|\X^*\|_F^2} = (1/m) \sum_k \sum_i \bm{y}_{ki}^2$.
To set the total number of algorithm iterations $T$, we can use a large maximum value along with breaking the loop if a stopping criterion is satisfied. A common stopping criterion for GD is to stop when the iterates do not change much. One way to do this is to stop when $\text{{SubsDist}}({\bm{U}}_t, {\bm{U}}_{t-1}) \le 0.01 \sqrt{r}$ for last few iterations.
As explained in \cite{lrpr_gdmin_mri}, we can use the following constraints to set the rank. We need our choice of rank, $\hat{r}$, to be sufficiently small compared to $\min(n,q)$ for the algorithm to take advantage of the LR assumption. Moreover, for the LS step for updating $\hat\b_k$'s (which are $r$-length vectors) to work well (for its error to be small), we also need it to also be small compared with $m$.
One approach that is used often is to use the ``$b\%$ energy threshold'' on singular values.
Thus, one good heuristic that respects the above constraints is to compute the ``$b\%$ energy threshold'' of the first $\min(n,q,m)/10$ singular values, i.e. compute $\hat{r}$ as the smallest value of $r$ for which
\[
\sum_{j=1}^{{r}} \sigma_j({\bm{X}}_0)^2 \ge (b/100) \cdot \sum_{j=1}^{\min(n,q,m)/10} \sigma_j({\bm{X}}_0)^2
\]
for a $b \le 100$. In our MRI experiments in \cite{lrpr_gdmin_mri}, we used $b=85$. We also realized from the experiments that the algorithm is not very sensitive to this value as long as $\hat{r} \ll \min(n,q,m)$.
\subsubsection{Federating the algorithm}
Suppose that our sketches $\bm{y}_k$ are geographically distributed across a set of $L$ nodes. Each node ${\ell}$ stores a subset, denoted $\mathcal{S}_{\ell}$, of the $\bm{y}_k$s with $|\mathcal{S}_{\ell}|=q_{\ell}$. These subsets are mutually disjoint so that $\sum_{\ell} q_{\ell} = q$.
Typically $L \ll q$.
Privacy constraints dictate that we cannot share the $\bm{y}_k$s with the central server; although summaries computed using the $\bm{y}_k$s can be shared at each algorithm iteration.
This will be done as follows. Consider the GDmin steps of Algorithm \ref{gdmin} first. Line 13 (Update $\b_k$s, $\bm{x}_k$s) is done locally at the node that stores the corresponding $\bm{y}_k$. For line 14 (Gradient w.r.t ${\bm{U}}$), the partial sums over $k \in \mathcal{S}_{\ell}$ are computed at node ${\ell}$ and transmitted to the center which adds all the partial sums to obtain $\nabla_{\bm{U}} f({\bm{U}},\bm{B})$. Line 15 (GD step) and line 16 (projection via QR) are done at the center. The updated ${\bm{U}}$ is then broadcast to all the nodes for use in the next iteration.
The per node time complexity of this algorithm is thus $mnr q_{\ell}$ at each iteration. The center only performs additions and a QR decomposition (an order $nr^2$ operation) in each iteration. Thus, the time complexity of the federated solution is only $mnr(\max_{\ell} q_{\ell})T$ per node.
The initialization step can be federated by using the Power Method (PM) \cite{golub89,npm_hardt} to compute the top $r$ eigenvectors of $\hat\X_0 \hat\X_0{}^\top$. Any PM guarantee helps ensure that its output is close in subspace distance to the span of the top $r$ eigenvectors of $\hat\X_0 \hat\X_0{}^\top$ after a sufficient number of iterations.
The communication complexity of the federated implementation is thus just $nr$ per node per iteration (need to share the partial gradient sums).
Observe also that the information shared with the center is not sufficient to recover $\X^*$ centrally. It is only sufficient to recover $\mathrm{span}(\U^*{})$.
The recovery of the columns of $\bm{B}$, $\b^*} %{\tilde\b^*_k$, is entirely done locally at the node where the corresponding $\bm{y}_k$ is stored, thus ensuring privacy.
\begin{algorithm}[t]
\caption{\small{The AltGD-Min algorithm. Let $\bm{M}^\dagger:= (\bm{M}^\top\bm{M})^{-1} \bm{M}^\top$.}}
\label{gdmin}
\begin{algorithmic}[1]
\State {\bfseries Input:} $\bm{y}_k, \bm{A}_k, k \in [q]$
\State {\bfseries Parameters:} Multiplier in specifying $\alpha$ for init step, $\tilde{C}$; GD step size, $\eta$; Number of iterations, $T$
\State {\bfseries Sample-split:} Partition the measurements and measurement matrices into $2T+1$ equal-sized disjoint sets: one set for initialization and $2T$ sets for the iterations. Denote these by $\bm{y}_k^{(\tau)}, \bm{A}_k^{(\tau)}, \tau=0,1,\dots 2T$.
\State {\bfseries Initialization:}
\State Using $\bm{y}_k \equiv \bm{y}_k^{(0)}, \bm{A}_k \equiv \bm{A}_k^{(0)}$,
set
\\ $\alpha = \tilde{C} \frac{1}{mq}\sum_{ki}\big|\bm{y}_{ki}\big|^2$,
\\
$\bm{y}_{k,trunc}(\alpha):= \bm{y}_k \circ \mathbbm{1}\{|\bm{y}_k| \le \sqrt{\alpha} \}$
\\
$\displaystyle \hat\X_{0}:= (1/m) \sum_{k \in [q]} \bm{A}_k^\top \bm{y}_{k,trunc}(\alpha) \bm{e}_k^\top$
\State Set ${\bm{U}}_0 \leftarrow $ top-$r$-singular-vectors of $\hat\X_0$
\State {\bfseries GDmin iterations:}
\For{$t=1$ {\bfseries to} $T$}
\State Let ${\bm{U}} \leftarrow {\bm{U}}_{t-1}$.
\State {\bfseries Update $\hat\b_k, \hat\x_k$: } For each $k \in [q]$, set $(\hat\b_k)_{t} \leftarrow (\bm{A}_k^{(t)} {\bm{U}})^\dagger \bm{y}_k^{(t)}$ and set $(\hat\x_k)_{t} \leftarrow {\bm{U}} (\hat\b_k)_{t}$
\State {\bfseries Gradient w.r.t. ${\bm{U}}$: } With $\bm{y}_k \equiv \bm{y}_k^{(T+t)}, \bm{A}_k \equiv \bm{A}_k^{(T+t)}$, compute $\nabla_{\bm{U}} f({\bm{U}}, \bm{B}_t) = \sum_k \bm{A}_k^\top (\bm{A}_k {\bm{U}} (\hat\b_k)_t - \bm{y}_k) (\hat\b_k)_t^\top$
\State {\bfseries GD step: } Set $\displaystyle \hat\U^+ \leftarrow {\bm{U}} - (\eta/m) \nabla_{\bm{U}} f({\bm{U}}, \bm{B}_t)$.
\State {\bfseries Projection step: } Compute $\hat\U^+ \overset{\mathrm{QR}}=} %{\stackrel{EVD}{=} {\bm{U}}^+ {\bm{R}}^+$.
\State Set ${\bm{U}}_t \leftarrow {\bm{U}}^+$.
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Main Result}\label{main_res}
We can prove the following result.
\begin{theorem}
Consider Algorithm \ref{gdmin}. Let $m_t$ denote the number of samples used in iteration $t$.
Set $\tilde{C} = 9\kappa^2 \mu^2$, $\eta = c / {\sigma_{\max}^*}^2$ with a $c \le 0.5$, and $T = C \kappa^2 \log(1/\epsilon)$. Assume that Assumption \ref{right_incoh} holds and that the $\bm{A}_k$s are i.i.d. and each contains i.i.d. standard Gaussian entries.
If
\[
m_0 q \ge C \kappa^6 \mu^2 (n + q) r^2,
\]
and $m_t$ for $t \ge 1$ satisfie
\[
m_t q \ge C \kappa^4 \mu^2 (n + q) r^2 \log \kappa \text{ and } m_t \ge C \max(r,\log q, \log n)
\]
then, with probability (w.p.) at least $1- t n^{-10}$, for all $t \ge 0$,
\[
\SE}%{\SE_F({\bm{U}}_t, \U^*{}) \le \left(1 - \frac{(\eta {\sigma_{\max}^*}^2) 0.4}{\kappa^2} \right)^t \delta_0
\]
with $\delta_0 = 0.09/\kappa^2.$
Thus, with $T= C \kappa^2 \log(1/\epsilon)$ and $\eta = 0.5/{\sigma_{\max}^*}^2$, w.p. at least $1 - (T+1) n^{-10}$,
\begin{align*}
& \SE}%{\SE_F({\bm{U}}_T,\U^*{}) \le \epsilon, \ {\|(\hat\x_{k})_T-\x^*_{k}\|} \le \epsilon {\|\x^*_k\|}, \text{ for all $k \in [q]$, } \\
& \|\hat\X_T - \X^*\|_F \le 1.4 \epsilon \|\X^*\|
\end{align*}
\label{gdmin_thm}
\end{theorem}
{\em Sample complexity}
The sample complexity (total number of samples needed to achieve $\epsilon$-accurate recovery) is $m_\mathrm{tot} = \sum_{\tau=0}^T m_\tau \ge m_0 + T \min_{t\ge 1} m_t$. From the above result, this needs to satisfy $m_{tot} q \ge C \kappa^6 \mu^2 (n+q) r^2 \log(1/\epsilon) \log (\kappa)$ and $m_{tot} > C \kappa^2 \max(r,\log q, \log n ) \log(1/\epsilon)$.
{\em Time complexity}
Let $m \equiv m_t$. The initialization step needs time $mq n $ for computing $\hat\X_0$; and time of order $nqr$ times the number of iterations used in the $r$-SVD step. Since we only need a $\delta_0$-accurate initial estimate of $\mathrm{span}(\U^*{})$, with $\delta_0 = c/\kappa^2$, order $\log(\kappa)$ number of iterations suffice for this SVD step. Thus the complexity is $O(nq(m+r) \cdot \log \kappa) = O(mqn \cdot \log \kappa)$ since $m \ge r$. One gradient computation needs time $O(mq nr)$. The QR decomposition needs time of order $nr^2$. The update of columns of $\bm{B}$ by LS also needs time $O(mqnr)$ (explained earlier).
As we prove above, we need to repeat these steps $T = O(\kappa^2 \log (1/\epsilon))$ times.
Thus the total time complexity is $O(mqn\log\kappa + \max(mqnr, nr^2, mqnr) \cdot T ) = O( \kappa^2 mqnr \log(1/\epsilon) \log\kappa)$.
{\em Communication complexity} The communication complexity per node per iteration for a federated implementation is just order $nr$. Thus, the total is $O( nr \cdot \kappa^2 \log(1/\epsilon))$.
Thus, we have the following corollary.
\begin{corollary}[AltGD-Min]
In the setting of Theorem \ref{gdmin_thm}, if Assumption \ref{right_incoh} holds, and if
\[
m_{tot} q \ge C \kappa^6 \mu^2 (n+q) r^2 \log(1/\epsilon) \log (\kappa)
\]
and $m_{tot} > C \kappa^2 \max(r,\log q, \log n ) \log(1/\epsilon)$,
then, w.p. at least $1 - (C\kappa^2\log(1/\epsilon)) n^{-10}$,
$ \|\hat\X - \X^*\|_F \le 1.4 \epsilon \|\X^*\|$ and ${\|\hat\x_{k}-\x^*_{k}\|} \le \epsilon { \|\x^*_k\|}$ for all $k \in [q]$.
The time complexity is $C \kappa^2 mqnr \log (1/\epsilon) \log\kappa$ and the communication complexity is $O( nr \cdot \kappa^2 \log(1/\epsilon))$.
\end{corollary}
Observe that the above results show that after $T=C \kappa^2 \log(1/\epsilon)$ iterations, $\SE}%{\SE_F({\bm{U}}_T,\U^*{}) \le \epsilon$, ${\|\hat\x_{k}-\x^*_{k}\|} \le \epsilon { \|\x^*_k\|}$, and $\|\hat\X_T - \X^*\|_F \le 1.4 \epsilon \|\X^*\|$. The RHS in the third bound does indeed contain $\|\X^*\|$ (the induced 2-norm). This is correct because, $\text{{SubsDist}}(.,.)$ is a Frobenius norm subspace distance. We explain this in Sec. \ref{outline_iters}
\subsection{Discussion and comparison with the best LRMC results}
An algorithm is called linear time if its time complexity is the same order as the time needed to load all input data. In our case, this is $O(mqn)$. Treating $\kappa$ as a constant, the AltGD-Min complexity is worse than linear-time by a factor of only $r \log(1/\epsilon)$. As can be seen from Table \ref{compare_lrccs}, the same is also true for the fastest LRMC solution, projGD-X \cite{rmc_gd}. For LRMC, linear time is $O(mq)$.
To our best knowledge, this is the case for the fastest algorithms for all LR problems.%
Consider the sample complexity.
The degrees of freedom (number of unknowns) of a rank-$r$ $n \times q$ matrix are $(n+q)r$. A sample complexity of $\Omega ( (n+q) r)$ samples (or, sometimes this times log factors) is called ``optimal''.
Thus, ignoring the log factors, our sample complexity of $m_{tot} q \gtrsim (n + q) r^2$ is sub-optimal only by a factor of $r$.
As can also be seen from Table \ref{compare_lrccs}, this suboptimality matches that of the best results for LRMC solutions that are not convex relaxation based \cite{rmc_gd,rpca_gd,lafferty_lrmc}.
The need for exploiting incoherence while obtaining the high probability bounds on the recovery error terms is what introduces the extra factor of $r$ for both LRMC and LRcCS. LRMC has been extensively studied for over a decade and there does not seem to be a way to obtain an (order-) optimal sample complexity guarantee for it except when studying convex relaxation solutions (which are much slower).
In addition, we also need $m \gtrsim \max(r,\log q, \log n)$. This is redundant except for very large $q,n$. This is needed because, the recovery of each column of $\B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*$ is a decoupled $r$-dimensional LS problem. We analyze this step in Lemma \ref{B_lemma}; notice that the bound on the recovery error of column $k$ holds w.p. at least $1 - \exp(r - cm)$. By union bound, it holds for all $q$ columns w.p. at least at least $1 - q \exp(r - cm) = 1 - \exp(\log q + r - c m)$. This probability is at least $1 - n^{-10} = 1 - \exp(-10 \log n)$ if $m \gtrsim \max(r,\log q, \log n)$.
\subsection{Detailed comparison with existing LRcCS results}\label{detailed_compare}
There are two existing solutions for LRcCS -- AltMin \cite{lrpr_icml,lrpr_it,lrpr_best} and the convex relaxation (mixed norm minimization) \cite{lee2019neurips}.
Mixed norm is defined as $\|{\bm{X}}\|_{mixed}:= \inf_{ \{{\bm{U}}, {\bm{V}}: {\bm{U}} {\bm{V}} = {\bm{X}}\} } \|{\bm{U}}\|_F \max_{k \in [q]} \|\v_k\|$, where ${\bm{U}}$ is $n \times r$ and ${\bm{V}}:=[\v_1, \v_2, \dots \v_q]$ is an $r \times q$ matrix.
In our notation, for the noise-free case ($\sigma=0$), their main result states the following.%
\begin{prop}[Convex relaxation (mixed norm min) in the $\sigma=0$ (noise-free) setting \cite{lee2019neurips}]\label{convex_sol}
Consider a matrix $\X^* \in \{ \X^*: \max_k \|\x^*_k\|^2 \le \alpha^2, \|\X^*\|_{mixed} \le R \le \alpha \sqrt{r} \}$. Then, w.p. $ 1- \exp( - c_2 n R^2 / \alpha^2)$,
$
\frac{\|\hat\X - \X^*\|_F^2}{\|\X^*\|_F^2} \le c_1 \frac{\alpha^2}{\|\X^*\|_F^2/ q } \sqrt{ \frac{(n+q) r \log^6 n}{m_{tot} q} }
$
Under our Assumption \ref{right_incoh}, $\max_k \|\x^*_k\|^2 \le \mu^2 (r/q) {\sigma_{\max}^*}^2 = (\mu^2 \kappa^2)(r/q) {\sigma_{\min}^*}^2 \le (\kappa^2 \mu^2 ) \|\X^*\|_F^2/q$, i.e. $\frac{\alpha^2}{\|\X^*\|_F^2/ q } = (\kappa^2 \mu^2) $.
Thus, the above result can also be stated as:
For all matrices $\X^*$ that satisfy Assumption \ref{right_incoh} and for which $\|\X^*\|_{mixed} \le \sqrt{r} \cdot \kappa \mu \|\X^*\|_F/\sqrt{q} $, if
$$m_{tot} q \ge C_1 \kappa^4 \mu^4 (n+q) r \log^6 n \cdot \frac{1}{\epsilon^4},$$ then, w.p. at least $1 - \exp(-c_2 n)$, ${\|\hat\X - \X^*\|_F} \le \epsilon {\|\X^*\|_F}$.
The time complexity is $C mq nr \min( \frac{1}{\sqrt\epsilon}, n^3 r^3)$ (explained earlier in Sec. \ref{relwork}).
\end{prop}
Notice that both the sample and the time complexity of the convex solution depend on powers of $1/\sqrt{\epsilon}$: the sample complexity grows as $1/\epsilon^4$ while the time complexity grows as $1/\sqrt{\epsilon}$. However, its sample complexity has an order-optimal dependence on $r$. For AltGD-Min, both sample and time complexities depend only logarithmically on $\epsilon$ only as $\log(1/\epsilon)$. But its sample complexity depends sub-optimally on $r$, it grows as $r^2$. In summary, the time complexity of the convex solution is always much worse, its sample complexity is worse when a solution with accuracy level $\epsilon < 1/{r}^{1/4}$ is needed.
A second point to mention is that our result for AltGD-Min provides a column-wise error bound (bounds $\|\x^*_k - \hat\x_k\|/\x^*_k\|$). The convex result only provides a bound on the Frobenius norm of the entire matrix. Thus it is possible that some columns have much larger recovery error than others. This can be problematic in applications such as dynamic MRI where each column corresponds to one signal/image of a time sequence and where the goal is to ensure accurate-enough recovery of all columns.
On the other hand, the advantage of the convex guarantee is that it holds w.h.p. for all matrices $\X^*$ in the specified set, where as our result only holds w.h.p. for {\em a} matrix $\X^*$ satisfying Assumption \ref{right_incoh}.
The reason for these last two points and the reason that we cannot avoid using sample-splitting is the same: the update of $\bm{B}$ is a column-wise LS problem. We explain the reasoning carefully in Sec. \ref{samplesplit} where we discuss the limitations of our approach.
A second advantage of the convex result is that it directly studies the noisy version of the LRcCS problem. This should be possible for AltGD-Min too, we postpone it to future work.
The best result for AltMin is from \cite{lrpr_best}, it states the following.%
\begin{prop}[AltMin \cite{lrpr_best}]
Under Assumption \ref{right_incoh}, if $$m_{tot} q \ge C \kappa^8 \mu^2 n r^2 (r + \log(1/\epsilon)) \text{ and } m_{tot} > \max(r,\log q, \log n ),$$ then, w.p. at least $1 - (\log(1/\epsilon)) n^{-10}$, $ \|\hat\X - \X^*\| \le \epsilon \|\X^*\|$ and ${\|\hat\x_{k}-\x^*_{k}\|} \le \epsilon { \|\x^*_k\|}$ for all $k \in [q]$.
The time complexity is $C mqnr \log^2 (1/\epsilon)$.
\label{altmin_best}
\end{prop}
Treating $\kappa$ as a numerical constant, compared with the above result for AltMin, the sample complexity of AltGD-Min is either better by a factor of $ r$ or is as good. It is better when $r > \log(1/\epsilon)$. Also, the time complexity is always better by a factor $\log(1/\epsilon)$. As a function of $\kappa$, the AltGD-Min sample complexity is better by a factor of $\kappa^2$, but its time is worse by a factor of $\kappa^2$ compared to that of AltMin. The reason is that its error decays as $(1-c/\kappa^2)^t$. For AltMin the error decays as $c^t$.
Experimentally, GD is usually much faster than AltMin because the constants in its time complexity are also lower
\section{Numerical Experiments}\label{sims}
Our first experiment compares AltGD-Min with the mixed norm minimization solution from \cite{lee2019neurips} (mixed-norm-min) and with
the AltMin algorithm \cite{lrpr_icml,lrpr_it,lrpr_best} modified for the linear LRcCS problem (replace the PR step for updating $\b_k$'s by a simple LS step). We implement this with using two possible initializations: the initialization developed in \cite{lrpr_icml,lrpr_it,lrpr_best} for LRPR (AltMinLin-LRPRinit), and with the initialization approach developed in this work (AltMinLin-LRCSinit).
For mixed norm min, we used the code downloaded from \url{https://www.dropbox.com/sh/lywtzc0y9awpvgz/AABbjuiuLWPy_8y7C3GQKo8pa?dl=0}, which is provided by the authors. For AltMin, we used the code from \url{https://github.com/praneethmurthy/}. We implemented AltGD-Min with $\eta = 0.4/\|\hat\X_0\|^2$ and $\tilde{C} = 9$. Also, we used {\em one} set of measurements for all its iterations.
For chosen values of $n,q,r$ and $m$, we simulated the data as follows. We simulated $\U^*{}$ by orthogonalizing an $n \times r$ standard Gaussian matrix; and $\b^*} %{\tilde\b^*_k$s were generated i.i.d. from ${\cal{N}}(0,\bm{I}_r)$. These were generated once. For each of 100 Monte Carlo runs, the measurement matrices $\bm{A}_k$ contained i.i.d. standard Gaussian entries. We obtained $\bm{y}_k = \bm{A}_k \U^*{} \b^*} %{\tilde\b^*_k$, $k \in [q]$. For the LRPR experiment, we used ${\y_{(mag)}}_k = |\bm{y}_k|$ as the measurements.
We plot the empirical average of $\|\hat\X - \X^*\|_F/\|\X^*\|_F$ at each iteration $t$ on the y-axis (labeled ``Error-X'' in the plots) and the time taken by the algorithm until iteration $t$ on the x-axis.
For our first experiment, shown in Fig. \ref{m80_gdmin}, we used $n=600,q=600,r=4$ and $m=80$. In this case, mixed-norm-min error decays to about 2-5\% but does not reduce any further. But, for our algorithm, AltGD-Min, and for both versions of AltMin, the error decays to $10^{-15}$. Notice also that AltGD-Min is much faster than all the other approaches.
Fig. \ref{m50_gdmin} reduced $m$ to $m=50$. Here a similar trend is observed, except that the error decays to only around $10^{-13}$ for AltGD-Min and $10^{-11}$ for the two AltMin approaches. Finally, for Fig. \ref{m30_gdmin}, we reduced $m$ to $m=30$. In this case, only AltGDmin and AltMin-LRCSinit work, while mixed-norm-min and AltMin-LRPRinit errors do not decrease at all. The reason is both these need a higher sample complexity (see Table \ref{compare_lrccs}).
}% {\color{red}
Finally, we also tried an experiment with very large $m$: $n=100,q=120,r=2$ and $m=0.9n = 90$, see Fig. \ref{m90_n100_gdmin}. Even for such a large value of $m$ (compared to $n$), observe that the mixed-norm-min error saturates at around 1-2\%. The likely reason for this that, in the guarantee for mixed-norm-min \cite{lee2019neurips} (summarized for the noiseless case in Proposition \ref{convex_sol} given earlier), even for $m = n$, the error is bounded by a multiplier (more than 1) times $\sqrt{r/q}$.%
\color{black}
For the comparisons for the LRPR problem shown in Fig. \ref{m250_lrpr}, we need a much larger $q$ and $m$ since LRPR requires $mq$ to scale as $nr^3$ both for initialization and for the GDmin iterations and the multiplying constants are also much larger for LRPR. We used $n=600,q=1000,r=4$ and $m=250$.
Notice that altGD-Min-LRPR is faster than AltMin-LRPR.
We implemented altGD-Min-LRPR with $\eta = 0.9/\|\hat\X_0\|^2$, $\tilde{C} = 9$, and $T_{RWF,t}= \max(5+t,40)$ in the RWF code (code for \cite{rwf}, downloaded from the specified site). Also, here again, we used {\em one} set of measurements for all its iterations
\subsubsection{#1} \vspace{-0.12in} }
\newcommand{\item} %\newcommand{\Item}{ \vspace{-0.05in} \item \vspace{-0.1in} }{\item}
\newcommand{\SE}%{\SE_F}{\text{{SubsDist}}
\renewcommand{\P}{\bm{P}}
\newcommand{\otimes}{\otimes}
\newcommand{{\U_{vec}}}{{{\bm{U}}_{vec}}}
\newcommand{{\Z_{vec}}}{{{\bm{{Z}}}_{vec}}}
\newcommand{{\y_{(mag)}}}{{\bm{y}_{(mag)}}}
\renewcommand{\forall}{\text{ for all }}
\newcommand{\epsilon}{\epsilon}
\newcommand{\Section}[1]{\section{#1}}
\newcommand{\Subsection}[1]{\subsection{#1}}
\renewcommand{\hat\b}{\b} \renewcommand{\hat\B}{\bm{B}}
\renewcommand{\hat\x}{\bm{x}} \renewcommand{\hat\X}{{\bm{X}}}
\input{Revise2_GDmin_intro}
\input{Revise2_GDmin_algo_thm}
\input{Revise2_GDmin_proofoutline_proof}
\input{Revise2_GDmin_lemmas_proof}
\input{Revise2_GDmin_PR}
\input{Revise2_GDmin_PR_proof}
\input{Revise2_GDmin_limitations}
\input{Revise2_GDmin_expts}
\section{Conclusions}\label{conclude}
This work developed a sample-efficient and fast gradient descent (GD) solution, called AltGD-Min, for provably recovering a low-rank (LR) matrix from mutually independent column-wise linear projections. This problem, which we refer to as ``Low Rank column-wise Compressive Sensing (LRcCS)'', frequently occurs in LR-based accelerated low rank dynamic MRI and in federated sketching. If used in a federated setting, AltGD-Min is also communication-efficient. The LRcCS problem has not received little attention in the theoretical literature unlike the other well-studied LR recovery problems (matrix completion, sensing, or multivariate regression).
\appendices \renewcommand\thetheorem{\Alph{section}.\arabic{theorem}}
\input{Revise2_GDmin_why_not_lrmc_ideas}
\input{Revise2_GDmin_init_proof_no_resamp}
\bibliographystyle{IEEEtran}
\section{Proof of Initialization Theorem \ref{init_thm} without sample-splitting}
\label{init_reuse_proof}
Consider the initialization using $\hat\X_0$ defined in \eqref{newinit}. We we want to bound the initialization error without sample-splitting. This means that the threshold $\alpha$ is not independent of the $\a_{ki}, \bm{y}_{ki}$ used in the expression for $\hat\X_0$ and thus, it is not clear how to compute its expected value even if we condition on $\alpha$. However, the following slightly more complicated approach can be used.
Using Fact \ref{sumyik_bnd} and Assumption \ref{right_incoh}, it is possible to show that $\hat\X_0$ is close to a matrix, ${\bm{X}}_+(\epsilon_1)$ given next for which $\mathbb{E}[{\bm{X}}_+]$ is easily computed:
Let
\[
\alpha_+:= \tilde{C} (1+ \epsilon_1) \frac{\|\X^*\|_F^2}{q}
\]
and define
\begin{align}
{\bm{X}}_+ (\epsilon_1) & := \frac{1}{m} \sum_{ki} \a_{ki} \bm{y}_{ki} \bm{e}_k{}^\top \mathbbm{1}_{ \small \{ \bm{y}_{ki}^2 \le \alpha_+ \} }. \text{ Then, } \nonumber \\
\mathbb{E}[{\bm{X}}_+] & = \X^* {\bm{D}}(\epsilon_1), \nonumber\\ {\bm{D}} &:= diagonal(\beta_k(\epsilon_1)), \nonumber\\ \beta_k(\epsilon_1)&:= \mathbb{E}\left[\zeta^2 \mathbbm{1}_{\small \left\{ \zeta^2 \le \frac{\alpha_+}{\|\x^*_k\|^2} \right\} } \right]
\label{Xplus}
\end{align}
with $\zeta$ being a scalar standard Gaussian.
Thus $\hat\X_+$ is $\hat\X_0$ with the threshold $\alpha$ replaced by $\alpha_+$ which is deterministic. Consequently $ \mathbb{E}[{\bm{X}}_+]$ has a similar form too and is obtained as explained in the proof of Lemma \ref{Wedinlemma} given in Sec. \ref{Wedinlemma_proof}.
Next, recall that $\X^* \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} {\bm\Sigma^*} {\V^*}} %{{\B^*}$ and $\tilde{C} = 9 \kappa^2 \mu^2$. Let $\tilde{c} = c/\tilde{C}$ for a $c<1$. Clearly, the span of the top $r$ singular vectors of $\mathbb{E}[{\bm{X}}_+] = \X^* {\bm{D}}$ equals $\mathrm{span}(\U^*{})$ and it is rank $r$ matrix. Let,
\[
\mathbb{E}[{\bm{X}}_+]= \X^* {\bm{D}} \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} \check{\bm\Sigma^*} \check{\V}}%{\check{\B}
\]
be its $r$-SVD (here $\check{\V}}%{\check{\B}$ is an $r \times q$ matrix with its rows containing the $r$ right singular vectors). We thus have
\begin{align*}
\sigma_r(\mathbb{E}[{\bm{X}}_+]) &= \sigma_{\min}(\check{\bm\Sigma^*}) =\sigma_{\min}({\bm\Sigma^*} {\V^*}} %{{\B^*} {\bm{D}} \check{\V}}%{\check{\B}{}^\top) \\&\ge \sigma_{\min}({\bm\Sigma^*})\sigma_{\min}({\V^*}} %{{\B^*})\sigma_{\min}({\bm{D}})\sigma_{\min}(\check{\V}}%{\check{\B}{}^\top) \\&= {\sigma_{\min}^*} \cdot 1 \cdot (\min_k \beta_k) \cdot 1
\end{align*}
Fact \ref{betak_bnd} given earlier shows that $(\min_k \beta_k) \ge 0.9$ and thus,
\[
\sigma_r(\mathbb{E}[{\bm{X}}_+]) \ge 0.9 {\sigma_{\min}^*}
\]
Also, $\sigma_{r+1}(\mathbb{E}[{\bm{X}}_+]) = 0$ since it is a rank $r$ matrix. Thus, using Wedin's $\sin \Theta$ theorem for $\text{{SubsDist}}$ (summarized in Theorem \ref{Wedin_sintheta}) applied with $\bm{M} \equiv \hat\X_0$, $\bm{M}^* \equiv \mathbb{E}[{\bm{X}}_+]$ gives
\begin{align}\label{Wedin}
&\SE}%{\SE_F({\bm{U}}_0,\U^*{})\nonumber \\& \le \dfrac{\sqrt{2} \max\left( \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+])^\top \U^*{} \|_F , \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+]) \check{\V}}%{\check{\B}{}^\top \|_F \right)}{0.9 {\sigma_{\min}^*} - \|\hat\X_0 - \mathbb{E}[{\bm{X}}_+]\|}
\end{align}
In the next three subsections, we prove a set of six lemmas that help bound the three terms in the expression above. {\em The main new ideas over the proof given earlier in Sec \ref{init_proof}, are in the proof of the first lemma, Lemma \ref{Xhat0_2} given below, and in the proof of Claim \ref{claim:expect_init} that is used in this proof.}
\begin{claim}
\label{claim:expect_init}
\label{EXhat0_Xplus}
Let $\x^* \in \Re^n$, $\bm{z} \in \Re^n$ be two deterministic vectors and let $\alpha$ be a deterministic scalar. Let $\a \sim {\cal{N}}(0,\bm{I}_n)$ be a standard Gaussian vector and define $\bm{y} := \a^\top\x^*$.
For an $0 < \epsilon < 1$,
\[
\mathbb{E}\left[ |\bm{y} (\a{}^\top\bm{z}) | \mathbbm{1}_{\{ \bm{y}^2 \in [1\pm\epsilon ]\alpha \}} \right] \leq C \epsilon \|\bm{z}\| \sqrt{\alpha}.
\]
\end{claim}
Combining Lemmas \ref{Xhat0_1} and \ref{Xhat0_2} and using Fact \ref{sumyik_bnd}, and setting $\epsilon_1 = c\delta_0 / \sqrt{r} \kappa $, we conclude that, w.p. at least
\\ $1-2\exp( (n+q)-\tilde{c}\epsilon_1^2 mq ) - \exp(-\tilde{c} mq \epsilon_1^2 ) \ge 1-2\exp( (n+q)- \tilde{c} mq \delta_0^2 / r \kappa^2 ) - \exp(-\tilde{c} mq \delta_0^2 /r \kappa^2)$,
\[
\|\hat\X_0 - \mathbb{E}[{\bm{X}}_+]\| \lesssim \epsilon_1 \|\X^*\|_F \lesssim c \delta_0 {\sigma_{\min}^*}
\]
By combining Lemmas \ref{Xhat0_Bstar_2}, \ref{Xhat0_Bstar_1}, \ref{Xhat0_Ustar_2}, and \ref{Xhat0_Ustar_1} and using Fact \ref{sumyik_bnd}, and setting $\epsilon_1 = c \delta_0 / \sqrt{r} \kappa $, we conclude that, w.p. at least
\\ $1-2\exp( n r - \tilde{c} mq \delta_0^2 / r \kappa^2 ) - 2\exp( q r - \tilde{c} mq \delta_0^2 / r \kappa^2 ) - \exp(- \tilde{c} mq \delta_0^2 /r \kappa^2 )$,
\[
\max\left( \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+])^\top \U^*{} \|_F , \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+]) \check{\V}}%{\check{\B}^\top \|_F \right) \lesssim c \delta_0 {\sigma_{\min}^*}
\]
Plugging these into \eqref{Wedin} proves Theorem \ref{init_thm}
\subsection{Bounding the denominator term}\label{denom_bnd}
By triangle inequality, $\|\hat\X_0 - \mathbb{E}[{\bm{X}}_+]\| \le \|{\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\| + \|\hat\X_0 - {\bm{X}}_+\|.$ The next two lemmas bound these two terms. The lemmas assume the claim of Fact \ref{sumyik_bnd} holds, i.e., that $\frac{1}{mq}\sum_{ki} \bm{y}_{ki}^2 \in [1\pm \epsilon_1] \tilde{C} \|\X^*\|_F^2/q$ where $\tilde{C} = 9\mu^2\kappa^2$.
\begin{lemma}
\label{lem:init_denom_term}
\label{Xhat0_2}
Assume that $\frac{1}{mq}\sum_{ki} \bm{y}_{ki}^2 \in [1\pm \epsilon_1] \tilde{C} \|\X^*\|_F^2/q$ (claim of Fact \ref{sumyik_bnd} holds). Then, w.p. $1-\exp(C(n+q)-\epsilon_1^2mq/\mu^2\kappa^2)$,
\[
\|\hat\X_0 - {\bm{X}}_+\| \leq C\epsilon_1 \mu\kappa\|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_denom_term}]
We have
\begin{align*}
\|{\bm{X}}_+ - \hat\X_0 \| &= \max_{\bm{z}\in \S^n,~\bm{w}\in \S^q} \bm{z}{}^\top\left({\bm{X}}_+ - \hat\X_0 \right)\bm{w} \\&= \max_{\bm{z}\in \S^n,~\bm{w}\in \S^q} \frac{1}{m} \sum_{ki} \bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\\&\qquad\times\mathbbm{1}_{\left\{ \frac{\tilde{C}}{mq}\sum_{ki} \bm{y}_{ki}^2\leq \bm{y}_{ki}^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} }.
\end{align*}
For the last expression above, we have used the assumption $\sum_{ki} \bm{y}_{ki}^2/m \le \tilde{C}(1+\epsilon_1)\|\X^*\|_F^2$.
Consider the RHS for a fixed unit norm $\bm{z}$ and $\bm{w}$. The lower threshold of the indicator function is itself a r.v.. To convert it into a deterministic bound, we need the following sequence of bounding steps: To use our assumption that $\sum_{ki} \bm{y}_{ki}^2 / m \geq (1-\epsilon_1)\tilde{C} \|\X^*\|_F^2$, we first need to bound the summands by their absolute values. This is done as follows:
\begin{align*}
|\bm{z}{}^\top\left({\bm{X}}_+ - \hat\X_0 \right)\bm{w} |
&\leq \frac{1}{m} \sum_{ki} \big|\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\big|\\&\qquad\times\mathbbm{1}_{\left\{ \frac{\tilde{C}}{mq}\sum_{ki} \bm{y}_{ki}^2\leq |\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} },\\
&\leq \frac{1}{m} \sum_{ki} \big|\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\big|\\&\qquad\times\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} },
\end{align*}
where in the last line we used our assumption that $\sum_{ki} \bm{y}_{ki}^2 / m \geq (1-\epsilon_1)\tilde{C} \|\X^*\|_F^2$.
This final expression is a sum of mutually independent sub-Gaussian r.v.s with subGaussian norm $K_{ki} \leq C |\bm{w}(k)| \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F/\sqrt{q} \le \sqrt{\tilde{C}}|\bm{w}(k)| \|\X^*\|_F/\sqrt{q}$. Thus, by applying the sub-Gaussian Hoeffding inequality, Theorem 2.6.2 of \cite{versh_book},
\begin{align*}
&\Pr\left\{ \Big| \sum_{ki} \big|\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} } \right.\\&- \left.\mathbb{E}\left[\sum_{ki} \big|\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] \Big|\geq t \right\}\\
& \qquad\leq 2\exp\left[-c\frac{t^2}{\sum_{ki} K_{ki}^2}\right].
\end{align*}
By setting $t = \epsilon_1 m \|\X^*\|_F$,
\[
\frac{t^2}{\sum_{ki} K_{ki}^2} \geq \frac{m^2 q \epsilon_1^2 \|\X^*\|_F^2}{\sum_{ki} \tilde{C} \|\X^*\|_F^2 |\bm{w}(k)|^2} = \frac{\epsilon_1^2mq}{\tilde{C}}.
\]
Since $\tilde{C} = 9\mu^2\kappa^2$, thus, w.p. $1-\exp(-c\epsilon_1^2mq/\mu^2\kappa^2)$, for a fixed $\bm{z}$ and $\bm{w}$,
\[
\bm{z}{}^\top\left(\hat\X_0 - {\bm{X}}_+\right)\bm{w} \leq \epsilon_1 \|\X^*\|_F + \mathbb{E}\left[\frac{1}{m}\sum_{ki} \big|\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] .
\]
By using Claim \ref{claim:expect_init} and $|\bm{w}(k)| \|\bm{z}\| = |\bm{w}(k)|$ we have
\begin{align*}
&\mathbb{E}\left[\frac{1}{m}\sum_{ki} \big|\bm{y}_{ki}(\a_{ki}{}^\top\bm{z}) \bm{w}(k) \big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] \\&\leq \sqrt{\tilde{C}(1+\epsilon_1)}\epsilon_1\|\X^*\|_F \sum_{k} \big|\bm{w}(k)\big|/\sqrt{q} \leq C\epsilon_1\mu\kappa \|\X^*\|_F,
\end{align*}
where in the last inequality we used Cauchy-Schwarz to show that $\sum_{k} \big|\bm{w}(k)\big|/\sqrt{q} \le \sqrt{ \sum_k \big|\bm{w}(k)\big|^2 \sum_k (1/q) } = 1$. Or this also follows by $\|\bm{w}\|_1/\sqrt{q} \le \|\bm{w}\| = 1$. Also, we used $\sqrt{\tilde{C}} = C \kappa \mu$.
Thus, w.p. $1-\exp(-c\epsilon_1^2mq/\mu^2\kappa^2)$, for a fixed $\bm{z}$ and $\bm{w}$, $\bm{z}{}^\top\left(\hat\X_0 - {\bm{X}}_+\right)\bm{w} \leq C\epsilon_1\mu\kappa \|\X^*\|_F$.
By Proposition \ref{epsnet_MW}, $\max_{\bm{z}\in \S^n,~\bm{w}\in \S^q} \bm{z}{}^\top\left(\hat\X_0 - {\bm{X}}_+\right)\bm{w} \le 1.4 C \epsilon_1 \mu \kappa \|\X^*\|_F$ w.p. at least $1-\exp( (n+q)\log(17)-c\epsilon_1^2mq/\mu^2\kappa^2)$.
\end{proof}
\begin{lemma}
\label{lem:init_denom_term2}
\label{Xhat0_1}
Consider ${\bm{X}}_+$. Fix $1 < \epsilon_1 < 1$. Then, w.p. $1-\exp\left[C(n+q)-c\epsilon_1^2mq/\mu^2\kappa^2\right]$
\[
\|{\bm{X}}_+ -\mathbb{E}[{\bm{X}}_+]\| \leq C\epsilon_1 \|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_denom_term2}]
The proof involves an application of the sub-Gaussian Hoeffding inequality followed by an epsilon-net argument, both almost the same as those used in the proof of Lemma \ref{lem:init_denom_term} given above. We have,
\[
\|{\bm{X}}_+ -\mathbb{E}[{\bm{X}}_+]\| = \max_{\bm{z}\in\mathcal{S}_n, \bm{w}\in\mathcal{S}_q} \langle {\bm{X}}_+ -\mathbb{E}[{\bm{X}}_+], ~\bm{z}\bm{w}{}^\top\rangle.
\]
For a fixed $\bm{z}\in\mathcal{S}_n, \bm{w}\in\mathcal{S}_q$, we have
\begin{align*}
&\langle {\bm{X}}_+ -\mathbb{E}[{\bm{X}}_+], ~\bm{z}\bm{w}{}^\top\rangle \\&= \frac{1}{m} \sum_{ki} \left(\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\mathbbm{1}_{\left\{|\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} } \right.\\&\qquad-\left. \mathbb{E}\left[\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\mathbbm{1}_{\left\{|\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} } \right]\right) .
\end{align*}
The summands are mutually independent, zero mean sub-Gaussian r.v.s with norm $K_{ki}\leq C |\bm{w}(k)| \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F/\sqrt{q}$. We will again apply the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book}.
Let $t=\epsilon_1m\|\X^*\|_F$. Then
\[
\frac{t^2}{\sum_{ki} K_{ki}^2} \geq \frac{\epsilon_1^2m^2\|\X^*\|_F^2}{\sum_{ki} \tilde{C}(1+\epsilon_1)\|\X^*\|_F^2/q} \geq \frac{\epsilon_1^2mq}{C\mu^2\kappa^2}
\]
Thus, for a fixed $\bm{z}\in\mathcal{S}_n, \bm{w}\in\mathcal{S}_q$, by sub-Gaussian Hoeffding, we conclude that, w.p. at least $1-\exp\left[-c\epsilon_1^2mq/\mu^2\kappa^2\right]$,
\[
\langle {\bm{X}}_+ -\mathbb{E}[{\bm{X}}_+], ~\bm{z}\bm{w}{}^\top\rangle \leq C \epsilon_1 \|\X^*\|_F.
\]
By Proposition \ref{epsnet_Mwz}, the above bound holds w.p. at least $1-\exp\left[ (n+q) -c\epsilon_1^2mq/\mu^2\kappa^2\right]$.
\end{proof}
\subsection{Bounding the $\check{\V}}%{\check{\B}$ numerator term}
We bound $ \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+]) \check{\V}}%{\check{\B}^\top \|_F $ in this section. By triangle inequality. it is bounded by $\|\left(\hat\X_0 - {\bm{X}}_+\right)\check{\V}}%{\check{\B}{}^\top\|_F + \|\left(\hat\X_+ -\mathbb{E}[ {\bm{X}}_+]\right)\check{\V}}%{\check{\B}{}^\top\|_F$.
\begin{lemma}
\label{lem:init_nom_B_term1}
\label{Xhat0_Bstar_2}
Assume that $\frac{1}{m}\sum_{ki} \bm{y}_{ki}^2\in[1\pm \epsilon_1]\|\X^*\|_F^2$. Then, w.p. $1-\exp\left[ nr-c\epsilon_1^2mq/\mu^2\kappa^2\right]$,
\[
\|\left(\hat\X_0 - {\bm{X}}_+\right)\check{\V}}%{\check{\B}{}^\top\|_F \leq C\epsilon_1\mu\kappa\|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_nom_B_term1}]
The initial part of the proof is very similar to the that of the proof of Lemma \ref{Xhat0_2}.
We have,
$
\|\left(\hat\X_0 - {\bm{X}}_+\right)\check{\V}}%{\check{\B}{}^\top\|_F = \max_{{\bm{W}}\in \S_{nr} } \langle {\bm{W}},~ \left(\hat\X - {\bm{X}}_+\right)\check{\V}}%{\check{\B}{}^\top\rangle.
$
For a fixed ${\bm{W}}\in \S_{nr}$,
\begin{align*}
&\langle {\bm{W}},~ \left(\hat\X_0 - {\bm{X}}_+\right)\check{\V}}%{\check{\B}{}^\top\rangle \\&= \frac{1}{m} \sum_{ki} \bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k}) \mathbbm{1}_{\left\{ \frac{\tilde{C}}{mq}\sum_{ki} \bm{y}_{ki}^2\leq |\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} }
\end{align*}
Proceeding as in the proof of Lemma \ref{Xhat0_2},
\begin{align*}
&\frac{1}{m} \sum_{ki} \bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\mathbbm{1}_{\left\{ \frac{\tilde{C}}{mq}\sum_{ki} \bm{y}_{ki}^2\leq |\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} }\\
&\leq \frac{1}{m} \sum_{ki} \big|\bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\big|\mathbbm{1}_{\left\{ \frac{\tilde{C}}{mq}\sum_{ki} \bm{y}_{ki}^2\leq |\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} } ,\\
&\leq \frac{1}{m} \sum_{ki} |\bm{y}_{ki}| |(\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})| \mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm \epsilon_1] \frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} } .
\end{align*}
The summands are mutually independent sub-Gaussian r.v.s with norm $K_{ki} \leq C \sqrt{\tilde{C}(1+\epsilon_1)} \|{\bm{W}}\check{\v}}%{\check{\b}_{k}\| \|\X^*\|_F/\sqrt{q}$.
Thus, we can apply the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book}.
Set $t=\epsilon_1m\|\X^*\|_F$. Then we have
\[
\frac{t^2}{\sum_{ki} K^2_{ki}} \geq \frac{\epsilon_1^2m^2\|\X^*\|_F^2}{ (\sum_{ki} \|{\bm{W}}\check{\v}}%{\check{\b}_{k}\|^2 ) \tilde{C}(1+\epsilon_1)\|\X^*\|_F^2/q} \geq \frac{\epsilon_1^2mq}{C\mu^2\kappa^2},
\]
where we used the fact that $\check{\V}}%{\check{\B}\Bcheck{}{}^\top=\bm{I}$ ($\check{\V}}%{\check{\B}^\top$ contains right singular vectors of a matrix) and thus $\|{\bm{W}}\check{\V}}%{\check{\B}\|_F = 1$. Applying sub-Gaussian Hoeffding, we can conclude that, w.p., $1-\exp\left[-c\epsilon_1^2mq/\mu^2\kappa^2\right]$
\begin{align*}
&\frac{1}{m} \sum_{ki} \big|\bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm \epsilon_1] \frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} } \\&\leq \epsilon_1 \|\X^*\|_F \\&\qquad+ \frac{1}{m}\sum_{ki} \mathbb{E}\left[\big|\bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm \epsilon_1] \frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] .
\end{align*}
We use Claim \ref{claim:expect_init} to bound the expectation term. Using this lemma with $\alpha^2 \equiv \tilde{C} (1 +\epsilon_1) \|\X^*\|_F^2/q$, $\bm{z} \equiv {\bm{W}} \check{\v}}%{\check{\b}_k$
\begin{align*}
&\frac{1}{m}\sum_{ki} \mathbb{E}\left[\big|\bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\big|\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \in [1\pm \epsilon_1] \frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] \\&\leq \frac{1}{m}\sum_{ki} \sqrt{\tilde{C}(1+\epsilon_1)}\epsilon_1 \|\X^*\|_F \|{\bm{W}}\check{\v}}%{\check{\b}_{k}\|/\sqrt{q} \leq C\epsilon_1\mu\kappa\|\X^*\|_F.
\end{align*}
where the last inequality used Cauchy-Schwarz on $\sum_k \|{\bm{W}} \check{\v}}%{\check{\b}_k\|/\sqrt{q}$ to conclude that $\sum_k \|{\bm{W}} \check{\v}}%{\check{\b}_k\| (1/\sqrt{q}) \le \sqrt{ \sum_k \|{\bm{W}} \check{\v}}%{\check{\b}_k\|^2 \sum_k (1/q) } =\sqrt{ \|{\bm{W}} \check{\V}}%{\check{\B}\|_F^2 \cdot 1} = 1$ since $ \|{\bm{W}} \check{\V}}%{\check{\B}\|_F=1$.
By Proposition \ref{epsnet_MW}, the above bound holds for all ${\bm{W}} \in \S_{nr}$, w.p. at least $1-\exp\left[nr\log(1+2/\epsilon_{net})-c\epsilon_1^2mq/\mu^2\kappa^2\right]$.
\end{proof}
\begin{lemma}
\label{lem:init_nom_B_term2}
\label{Xhat0_Bstar_1}
Consider $0 < \epsilon_1 < 1$. Then, w.p. $1-\exp\left[nr -\epsilon_1^2mq/\mu^2\kappa^2\right]$
\[
\|\left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right)\check{\V}}%{\check{\B}{}^\top\|_F \leq C\epsilon_1\|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_nom_B_term2}]
The proof is quite similar to the previous one.
For a fixed ${\bm{W}}\in\S_{nr}$ we have,
\begin{align*}
&\langle \left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle \\& = \frac{1}{m}\sum_{ki} \left( \bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} } - \mathbb{E}[.]\right)
\end{align*}
where $\mathbb{E}[.]$ is the expected value of the first term.
The summands are independent, zero mean, sub-Gaussian r.v.s with subGaussian norm less than $K_{ki} \leq C \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F\|{\bm{W}}\b_k\|/\sqrt{q}$. Thus, by applying the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book},
with $t=\epsilon_1 m \|\X^*\|_F$, and using $\|{\bm{W}}\check{\V}}%{\check{\B}\|_F = 1$, we can conclude that,
w.p. $1-\exp\left[-\epsilon_1^2mq/(C\mu^2\kappa^2)\right]$,
$$
\langle \left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle\leq C \epsilon_1 \|\X^*\|_F.
$$
By Proposition \ref{epsnet_MW}, the above bound holds for all ${\bm{W}} \in \S_{nr}$ w.p. $1-\exp\left[nr -\epsilon_1^2mq/(C\mu^2\kappa^2)\right]$.
\end{proof}
\subsection{Bounding the U* numerator term}
We bound $ \| (\hat\X_0 - \mathbb{E}[{\bm{X}}_+])^\top \U^*{} \|_F $ here. By triangle inequality, it is bounded by $\|\left(\hat\X_0 - {\bm{X}}_+ \right){}^\top\U^*{} \|_F + \|\left(\hat\X_+ - \mathbb{E}[{\bm{X}}_+] \right){}^\top\U^*{} \|_F$.
\begin{lemma}
\label{lem:init_term1}
\label{Xhat0_Ustar_2}
Assume that $\frac{1}{mq}\sum_{ki} \bm{y}_{ki}^2 \in [1\pm \epsilon_1]\|\X^*\|_F^2/q$. Then, w.p. $1-\exp\left[qr-c\epsilon_1^2mq/\mu^2\kappa^2\right]$
\[
\|\left(\hat\X_0 - {\bm{X}}_+ \right){}^\top\U^*{} \|_F \leq C\epsilon_1 \mu\kappa\|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_term1}]
The proof is similar to that of Lemmas \ref{Xhat0_2} and \ref{Xhat0_Bstar_2}.
We have,
$
\|\left(\hat\X_0 - {\bm{X}}_+ \right){}^\top\U^*{}\|_F = \max_{{\bm{W}}\in\S_{qr}} \langle {\bm{W}},~ \left(\hat\X - {\bm{X}}_+ \right){}^\top\U^*{} \rangle.
$
For a fixed ${\bm{W}} \in \S_{qr}$, using the same approach as in Lemma \ref{Xhat0_2}, and letting $\bm{w}_k$ be the $k$-th column of the $r \times q$ matrix ${\bm{W}}$,
\begin{align*}
&\langle {\bm{W}},~ \left(\hat\X_0 - {\bm{X}}_+ \right){}^\top\U^*{} \rangle \\&\qquad\leq \frac{1}{m}\sum_{ki} \big|\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k) \big|\mathbbm{1}_{ \left\{ \frac{\tilde{C}}{mq}\sum_{ki} |\bm{y}_{ki}|^2 \leq|\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} },\\
&\qquad\leq \frac{1}{m}\sum_{ki} \big|\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k) \big|\mathbbm{1}_{ \left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }.
\end{align*}
The summands are now mutually independent sub-Gaussian r.v.s with norm $K_{ki} \leq \sqrt{\tilde{C}(1+\epsilon_1)}\|\bm{w}_k\| \|\X^*\|_F/\sqrt{q}$. Thus, we can apply the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book}, to conclude that,
for a fixed ${\bm{W}} \in \S_{qr}$, w.p. $1-\exp\left[-c\epsilon_1^2mq/\mu^2\kappa^2\right]$,
\begin{align*}
&\frac{1}{m}\sum_{ki} \big|\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k) \big|\mathbbm{1}_{ \left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} } \\&\leq \epsilon_1 \|\X^*\|_F + \frac{1}{m}\sum_{k}\mathbb{E}\left[\big|\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k) \big|\mathbbm{1}_{ \left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right]
\end{align*}
By Claim \ref{claim:expect_init}, and using $\sum_{k} \|\bm{w}_k \|/\sqrt{q} \leq \sqrt{\sum_{k}\|\bm{w}_k \|^2}~\sqrt{\sum_k 1/q} = 1$,
\begin{align*}
&\frac{1}{m}\sum_{k}\mathbb{E}\left[\big|\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k) \big|\mathbbm{1}_{ \left\{ |\bm{y}_{ki}|^2 \in [1\pm\epsilon_1]\frac{\tilde{C}}{q}\|\X^*\|_F^2 \right\} }\right] \\&\leq \frac{1}{m}\sum_{ki} \epsilon_1 \|\bm{w}_k\| \sqrt{\tilde{C}(1+\epsilon_1)/q}\|\X^*\|_F,\\
&\leq C\epsilon_1\mu\kappa \|\X^*\|_F,
\end{align*}
By Proposition \ref{epsnet_MW} (epsilon net argument), the bound holds for all unit norm ${\bm{W}}$ w.p. $1-\exp\left[ qr-c\epsilon_1^2mq/\mu^2\kappa^2\right]$.
\end{proof}
\begin{lemma}
\label{lem:init_term1_2}
\label{Xhat0_Ustar_1}
Consider $0 < \epsilon_1 < 1$. Then, w.p. $1-\exp\left[ qr -\epsilon_1^2/mq\mu^2\kappa^2\right]$
\[
\|\left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right){}^\top\U^*{}\|_F \leq C\epsilon_1\|\X^*\|_F.
\]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:init_term1_2}]
For fixed ${\bm{W}} \in \mathcal{S}_{qr}$,
\begin{align*}
&\mathrm{trace}\left({\bm{W}}{}^\top \left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right){}^\top\U^*{}\right) \\
&\qquad= \frac{1}{m}\sum_{ki} \left( \bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k)\mathbbm{1}_{\left\{|\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} } \right.\\&\qquad-\left. \mathbb{E}\left[\bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k)\mathbbm{1}_{\left\{|\bm{y}_{ki}|^2 \leq \frac{\tilde{C}(1+\epsilon_1)}{q}\|\X^*\|_F^2 \right\} }\right] \right)
\end{align*}
The summands are independent zero mean sub-Gaussian r.v.s with norm less than $K_{ki} \leq \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F\|\bm{w}_k\|/\sqrt{q}$. Thus, by applying the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book},
with $t=\epsilon_1 m \|\X^*\|_F$, we can conclude that,
for a fixed ${\bm{W}} \in \mathcal{S}_{qr}$, w.p. $1-\exp\left[-\epsilon_1^2mq/C\mu^2\kappa^2\right]$,
\[
\mathrm{trace}\left({\bm{W}}{}^\top\left({\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\right){}^\top\U^*{} \right) \leq \epsilon_1 \|\X^*\|_F.
\]
By Proposition \ref{epsnet_MW} (epsilon net argument), the bound holds for all unit norm ${\bm{W}}$ w.p. $1-\exp\left[qr -\epsilon_1^2mq/C\mu^2\kappa^2\right]$.
\end{proof}
\subsection{Proof of Claim \ref{EXhat0_Xplus}}
\begin{proof
We can write $\x^* = \|\x^*\| {\bm{Q}} \bm{e}_1$ where ${\bm{Q}}$ is a unitary matrix with first column proportional to $\x^*_k$. We need to bound
\begin{align*}
&\mathbb{E}[ \|\x^*\| \cdot | (\a^\top {\bm{Q}} \bm{e}_1) (\a^\top{\bm{Q}} {\bm{Q}}^\top \bm{z}) | \mathbbm{1}_{\{ \|\x^*\|^2 | \a^\top {\bm{Q}} e_1|^2 \in [1\pm \epsilon] \alpha } \} ]
\\&= \|\x^*\| \cdot \|\bm{z}\| \cdot \mathbb{E}[ | \tilde\a(1) \tilde\a^\top \bar\bm{z}_Q | \mathbbm{1}_{ \{ |\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 } \} ]
\end{align*}
where $\bar\bm{z}_Q:= {\bm{Q}}^\top \bm{z} / \|\bm{z}\|$, $\tilde\a:= {\bm{Q}}^\top\a$ and $\beta: = \sqrt{\alpha}/\|\x^*\|$. Since ${\bm{Q}}$ is unitary and $\a$ Gaussian, thus $\tilde\a$ has the same distribution as $\a$.
Let $\tilde\a(1)$ be its first entry and $\tilde\a(\mathrm{rest})$ be the $(n-1)$-length vector with the rest of the $n-1$ entries and similarly for $\bar\bm{z}_Q$. Then, $ \tilde\a^\top \bar\bm{z}_Q =\tilde\a(1)\cdot\bar\bm{z}_Q(1) + \tilde\a(\mathrm{rest})^\top \bar\bm{z}_Q(\mathrm{rest})$.
Since $\tilde\a(1)$ and $\tilde\a(\mathrm{rest})$ are independent,
\begin{align*}
&\mathbb{E}[ | \tilde\a(1) \tilde\a^\top \bar\bm{z}_Q | \mathbbm{1}_{|\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 } ]
\\&\le |\bar\bm{z}_Q(1)| \mathbb{E}[ | \tilde\a(1)^2 | \mathbbm{1}_{|\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 } ] \\&\qquad+ \mathbb{E}[ | \tilde\a (\mathrm{rest})^\top \bar\bm{z}_Q(\mathrm{rest}) | ] \ \mathbb{E}[ | \tilde\a(1)| \mathbbm{1}_{|\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 }] \\
\le & \mathbb{E}[ | \tilde\a(1)^2 | \mathbbm{1}_{|\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 } ] + 2 \mathbb{E}[ | \tilde\a(1)| \mathbbm{1}_{|\tilde\a(1)|^2 \in [1\pm \epsilon] \beta^2 }] \\
\le & \epsilon \beta + 2 \epsilon \beta = 3\epsilon \beta = C \epsilon \frac{\sqrt{\alpha}}{ \|\x^*\|}.
\end{align*}
The second inequality used the facts that (i) $|\bar\bm{z}_Q(1)| \le \|\bar\bm{z}_Q\|=1$ by definition and (ii) $\zeta:=\tilde\a (\mathrm{rest})^\top \bar\bm{z}_Q(\mathrm{rest})$ is a scalar standard Gaussian r.v. and so $\mathbb{E}[|\zeta|] \le 2$.
The third one relies on the following two bounds:
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item
\begin{align*}
&\mathbb{E}\left[ |\a(1)|^2\mathbbm{1}_{ \left\{ |\a(1) |^2 \in [1\pm\epsilon]\beta^2 \right\}} \right] \\&= \frac{2}{\sqrt{2\pi}}\int_{\sqrt{1-\epsilon}\beta}^{\sqrt{1+\epsilon}\beta} z^2\exp(-z^2/2)dz,\\
&\leq \frac{2e^{-1/2}}{\sqrt{2\pi}}\int_{\sqrt{1-\epsilon}\beta}^{\sqrt{1+\epsilon}\beta} dz \leq \frac{2e^{-1/2}}{\sqrt{2\pi}} \epsilon\beta\leq \epsilon\beta/3
\end{align*}
where we used the facts that $z^2\exp(-z^2/2) \leq \exp(-1/2)$ for all $z \in\Re$; $\sqrt{1-\epsilon} \geq 1-\epsilon/2$ and $\sqrt{1+\epsilon} \leq 1+\epsilon/2$ for $0 < \epsilon <1$.
\item Similarly, we can show that
\begin{align*}
&\mathbb{E}\left[ |\a(1)|\mathbbm{1}_{\{ |\a(1)|^2 \in [1\pm\epsilon]\beta^2 \}} \right] \\&= \frac{2}{\sqrt{2\pi}}\int_{\sqrt{1-\epsilon}\beta}^{\sqrt{1+\epsilon}\beta} z\exp(-z^2/2)dz,\\
&\leq \frac{2e^{-1/2}}{\sqrt{2\pi}}\int_{\sqrt{1-\epsilon}\beta}^{\sqrt{1+\epsilon}\beta} dz = \frac{2e^{-1/2}}{\sqrt{2\pi}} \epsilon\beta\leq \epsilon\beta/3
\end{align*}
\een
The claim follows by combining the two equations given above.
\end{proof}
\section{Introduction
This work develops a sample-efficient, fast, and communication-efficient gradient descent (GD) solution, called AltGD-Min, for provably recovering a low-rank (LR) matrix from a set of mutually independent linear projections of each of its columns. The communication-efficiency considers a federated setting.
This problem, which we henceforth refer to as ``Low Rank column-wise Compressive Sensing (LRcCS)'', is precisely defined below.
Unlike the other well-studied LR problems -- multivariate regression (MVR) \cite{wainwright_linear_columnwise}, LR matrix sensing \cite{lowrank_altmin} and LR matrix completion (LRMC) \cite{matcomp_candes,lowrank_altmin} -- LRcCS has received little attention so far in terms of approaches with provable guarantees. There are only two existing provably correct solutions.
(1) Its generalization {\em LR phase retrieval (LRPR)}, was studied in our recent work \cite{lrpr_icml,lrpr_it,lrpr_best} where we developed a provably correct alternating minimization (AltMin) solution. Since LRPR is a generalization, the algorithm also solves LRcCS. (2) In parallel work, \cite{lee2019neurips} developed and analyzed a convex relaxation (mixed-norm minimization) for LRcCS. Both solutions are much slower than GD-based methods, and, in most practical settings, also have worse sample complexity.
LRcCS occurs in accelerated LR dynamic MRI \cite{st_imaging,dyn_mri1,dyn_mri2}, and in distributed/federated sketching \cite{hughes_icml_2014,aarti_singh_subs_learn,lee2019neurips}. We explain these in Sec. \ref{apps}.
We show the speed and performance advantage of AltGD-Min for dynamic MRI in \cite{lrpr_gdmin_mri}.
\subsection{Problem Setting, Notation, and Assumption}\label{probdef}
\bfpara{Problem definition}
The goal is to recover an $n \times q$ rank-$r$ matrix $\X^* =[\x^*_1, \x^*_2, \dots, \x^*_q]$ from $m$ linear projections (sketches) of each of its $q$ columns, i.e. fro
\begin{eqnarray}
\bm{y}_k := \bm{A}_k \x^*_k, \ k \in [q]
\label{ykvec}
\end{eqnarray}
where each $\bm{y}_k$ is an $m$-length vector, $[q]:=\{1,2,\dots, q\}$, and the measurement/sketching matrices $\bm{A}_k$ are mutually independent and known. The setting of interest is low-rank (LR), $r \ll \min(n,q)$, and undersampled measurements, $m < n$. Our guarantees assume that each $\bm{A}_k$ is random-Gaussian: each entry of it is independent and identically distributed (i.i.d.) standard Gaussian.
We also study the magnitude-only measurements' setting, LRPR \cite{lrpr_icml,lrpr_it,lrpr_best}. This involves recovering $\X^*$ from
$${\y_{(mag)}}_k := |\bm{A}_k \x^*_k|, \ k \in [q].$$
Here $|\bm{z}|$ takes the entry-wise absolute value of entries of the vector $\bm{z}$.
\bfpara{Notation}
Everywhere, $\|.\|_F$ denotes the Frobenius norm, $\|.\|$ without a subscript denotes the (induced) $l_2$ norm (often called the operator norm or spectral norm), $\|\bm{M}\|_{\max}$ is the maximum magnitude entry of the matrix $\bm{M}$, $^\top$ denotes matrix or vector transpose, and $|\bm{z}|$ for a vector $\bm{z}$ denotes element-wise absolute values. $\bm{I}_n$ (or sometimes just $\bm{I}$) denotes the $n \times n$ identity matrix. We use $\bm{e}_k$ to denote the $k$-th canonical basis vector, i.e., the $k$-th column of $\bm{I}$.
For any matrix ${\bm{{Z}}}$, $\bm{z}_k$ denotes its $k$-th column.
We say ${\bm{U}}$ is a {\em basis matrix} if it contains orthonormal columns. For basis matrices ${\bm{U}}_1, {\bm{U}}_2$, we use $$\SE}%{\SE_F({\bm{U}}_1, {\bm{U}}_2): = \|(\bm{I} - {\bm{U}}_1 {\bm{U}}_1{}^\top){\bm{U}}_2\|_F$$ as the Subspace Distance (SD) measure. For two $r$-dimensional subspaces, this is the $l_2$ norm of the sines of the $r$ principal angles between $\mathrm{span}({\bm{U}}_1)$ and $\mathrm{span}({\bm{U}}_2)$. $\SE}%{\SE_F({\bm{U}}_1,{\bm{U}}_2)$ is symmetric when ${\bm{U}}_1,{\bm{U}}_2$ are both $n \times r$ basis matrices. Notice here we are using the Frobenius SD, unlike many recent works including our older work \cite{lrpr_it} that use the induced 2-norm one. This is done because it enables us to prove the desired guarantees easily.
We reuse the letters $c,C$ to denote different numerical constants in each use with the convention that $c < 1$ and $C \ge 1$.
The notation $a \in \Omega(b)$ means $a \ge C b$ while $a \in O(b)$ means $a \le Cb$. We use $\mathbbm{1}_{\text{statement}}$ to denote an indicator function that takes the value 1 if $\text{statement}$ is true and zero otherwise.
For a vector $\bm{w}$, we sometimes use $\bm{w}(k)$ to denote the $k$-th entry of $\bm{w}$. For a vector $\bm{w}$ and a scalar $\alpha$, $\mathbbm{1}(\bm{w} \le \alpha)$ returns a vector of 1s and 0s of the same length as $\bm{w}$, with 1s where $(\bm{w}(k) \le \alpha)$ and zero everywhere else. We use $\circ$ to denote the Hadamard product. Thus $\bm{z}:=\bm{w} \circ \mathbbm{1}(\bm{w} \le \alpha)$ zeroes out entries of $\bm{w}$ larger than $\alpha$, while keeping the smaller ones as is.
For $\X^*$ which is a rank-$r$ matrix, we let
\[
\X^* \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} \underbrace{{\bm\Sigma^*} {\V^*}} %{{\B^*}{}}_{\B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*} := \U^*{} \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*
\]
denote its reduced (rank $r$) SVD, i.e., $\U^*{}$ and ${\V^*}} %{{\B^*}^\top$ are matrices with orthonormal columns {\em (basis matrices)}, $\U^*{}$ is $n \times r$ and ${\V^*}} %{{\B^*}$ is $r \times q$, and ${\bm\Sigma^*}$ is an $r \times r$ diagonal matrix with non-negative entries. We use $\kappa:= {\sigma_{\max}^*}/{\sigma_{\min}^*}$ to denote the condition number of ${\bm\Sigma^*}$. This is not the condition number of $\X^*$ (whose minimum singular value is zero).
We let $\B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*:= {\bm\Sigma^*} \V^*{}{}$ and we use $\b^*} %{\tilde\b^*_k$ to denote its $k$-th column.
We use the phrase {\em $\epsilon$-accurate recovery} to refer to $\text{{SubsDist}}({\bm{U}}, \U^*{}) \le \epsilon$ or $\|\hat\X - \X^*\|_F \le \epsilon \|\X^*\|_F$ or both
\bfpara{Assumption}
Another way to understand \eqref{ykvec} is as follows: each scalar measurement $\bm{y}_{ki}$ ($i$-th entry of $\bm{y}_k$) satisfies
\[
\bm{y}_{ki} : = \langle \a_{ki}, \x^*_k \rangle , \ i \in [m], \ k \in [q]
\]
with $\a_{ki}{}^\top$ being the $i$-th row of $\bm{A}_k$. Observe that the measurements are not global, i.e., no $\bm{y}_{ki}$ is a function of the entire matrix $\X^*$. They are global for each column ($\bm{y}_{ki}$ is a function of column $\x^*_k$) but not across the different columns. We thus need an assumption that enables correct interpolation across the different columns. The following assumption, which is a slightly weaker version of incoherence (w.r.t. the canonical basis) of right singular vectors suffices for this purpose.
\begin{assu}[(Weakened) Right Singular Vectors' Incoherence]
Assume that
\[
\max_k \|\b^*} %{\tilde\b^*_k\| \le {\sigma_{\max}^*} \mu \sqrt{r/q}.
\]
for a constant $\mu \ge 1 $ ($\mu$ does not grow with $n,q,r$).
Since $\|\x^*_k\| =\|\b^*} %{\tilde\b^*_k\|$, this implies that $\max_k \|\x^*_k\| \le {\sigma_{\max}^*} \mu \sqrt{r/q}$. Also, since ${\sigma_{\min}^*} \sqrt{r} \le \|\X^*\|_F$, this also implies that $\max_k \|\x^*_k\| \le \kappa \mu {\|\X^*\|_F}/{\sqrt{q}}$.
\label{right_incoh}
\end{assu}
Right singular vectors incoherence is the assumption $\max_k \|\v^*} %{\b^*_k\| \le \mu \sqrt{r/q}$. Since $\b^*} %{\tilde\b^*_k = {\bm\Sigma^*} \v^*} %{\b^*_k$, this implies that the above holds. Incoherence of both left and right singular vectors was introduced for guaranteeing correct ``interpolation'' for the LRMC problem \cite{matcomp_candes,lowrank_altmin}.
\subsection{Existing Work}\label{relwork}
\bfpara{Existing solutions for LRcCS and LRPR}
Since it is always possible to obtain magnitude-only measurements ${\y_{(mag)}}_k$ from linear ones $\bm{y}_k$ as ${\y_{(mag)}}_k = |\bm{y}_k|$, a solution to LRPR also automatically solves LRcCS under the same assumptions. Hence the AltMin algorithm for LRPR from \cite{lrpr_icml,lrpr_it} is the first provably correct solution for LRcCS. Of course, since LRcCS is an easier problem than LRPR, we expect a direct solution to LRcCS to need weaker assumptions. As we show in this paper, this is indeed true. A more recent work \cite{lee2019neurips} studied the noisy version of LRcCS and developed a convex relaxation (mixed norm minimization) to provably solve it. Its time complexity is not discussed in the paper, however, it is well known that solvers for convex programs are much slower when compared to direct iterative algorithms: they either require number of iterations proportional to $1/\sqrt{\epsilon}$ or the per-iteration cost has cubic dependence on the problem size (here $((n+q)r)^3$) \cite{lowrank_altmin}. Thus, if $q \le n$, its time complexity $O(mqnr \cdot \min(1/\sqrt{\epsilon}, n^3 r^3))$.
In \cite{lrpr_best}, we provided the best possible guarantee for the AltMin algorithm for solving LRPR, and hence LRcCS.
We discuss these results in detail in Sec. \ref{detailed_compare} and summarize them in Table \ref{compare_lrccs}.
\bfpara{Other well-studied LR recovery problems} The multivariate regression (MVR) problem, studied in \cite{wainwright_linear_columnwise}, is our problem with $\bm{A}_k = \bm{A}$. However this is a very different setting than ours because, with $\bm{A}_k = \bm{A}$, the different $\bm{y}_k$'s are no longer mutually independent. As a result, one cannot exploit law of large numbers' arguments over all $mq$ scalar measurements $\bm{y}_{ki}$. Consequently, the required value of $m$ can never be less than $n$. The result of \cite{wainwright_linear_columnwise} shows that $m$ of order $(n+q)r$ is both necessary and sufficient.
LRMS involves recovering $\X^*$ from $\bm{y}_i = \langle \bm{A}_i, \X^* \rangle, \ i=1,2,\dots,mq$ with $\bm{A}_i$ being dense matrices, typically i.i.d. Gaussian \cite{lowrank_altmin}. Thus all measurements are i.i.d. and {\em global}: each contains information about the entire quantity-of-interest, here $\X^*$. Because of this, for LRMS, one can prove a LR Restricted Isometry Property (RIP) that simplifies the rest of the analysis. This is what makes it very different from, and easier than, our problem.
{\em LRMC}, which involves recovering $\X^*$ from a subset of its observed entries, {\em is the most closely related problem to ours} since it also involves recovery from non-global measurements. The typical model assumed is that each matrix entry is observed with probability $p$ independent of others \cite{matcomp_candes,lowrank_altmin}. Setting unobserved entries to zero, this can be written as $\bm{y}_{jk} = \delta_{jk} \X^*_{jk}$ with $\delta_{jk} \stackrel{\mathrm{iid}}{\thicksim } Bernoulli(p)$. LRMC measurements are both row-wise and column-wise local. To allow correct interpolation across both rows and columns, it needs the incoherence assumption on both its left and right singular vectors.
For our problem, the measurements are global for each column, but not across the different columns. For this reason, only right singular vectors' incoherence is needed. In fact, because of the nature of our measurements, even if left incoherence were assumed, it would not help.
This {\em asymmetry in our measurement model and the fact that our measurements are unbounded (each $\bm{y}_{ki}$ is a Gaussian r.v) are two key differences} between LRMC and LRcCS {\em that prevent us from borrowing LRMC proof techniques for our work}. Here {\em symmetric} means: if we replace $\X^*$ by its transpose, the probability distribution of the set of measurements does not change. {\em Bounded} means that the measurements' magnitude has a {\em uniform} bound. This bound is $\|\X^*\|_{\max}$ for LRMC measurements.
Non-convex (iterative, not convex relaxation based) LRMC algorithms with the best sample complexity are GD-based. There are two common approaches for designing GD algorithms in the LR recovery literature, and in particular for LRMC. The first is to use standard projected GD on ${\bm{X}}$ ({\em projGD-X}), also referred to as Iterative Hard Thresholding: at each iteration, perform one step of GD for minimizing the squared loss cost function, $\tilde{f}({\bm{X}})$, w.r.t. ${\bm{X}}$, followed by projecting the resulting matrix onto the space of rank $r$ matrices (by SVD). This was studied in \cite{fastmc,rmc_gd} for solving LRMC.
This is shown to converge geometrically with a constant GD step size, while needing only $ \Omega ( (n+q)r^2 \log^2 n \log^2 (1/\epsilon))$ samples on average.
The second is to let ${\bm{X}} = {\bm{U}} \bm{B}$ where ${\bm{U}}$ is $n \times r$ and $\bm{B}$ is $r \times q$ and perform alternating GD for the cost function $f({\bm{U}},\bm{B}) := \tilde{f}({\bm{U}}\bm{B})$, i.e., update $\bm{B}$ with one step of GD for minimizing $f({\bm{U}},\bm{B})$ while keeping ${\bm{U}}$ fixed at its previous value, and then do the same for ${\bm{U}}$ with $\bm{B}$ fixed, and repeat. Since the ${\bm{X}} = {\bm{U}} \bm{B}$ factorization is not unique, i.e., ${\bm{X}} = {\bm{U}} {\bm{R}}^{-1} {\bm{R}} \bm{B}$ for any invertible $r \times r$ matrix ${\bm{R}}$, this approach can result in the norm of one of ${\bm{U}}$ or $\bm{B}$ growing in an unbounded fashion, while that of the other decreases at the same rate, causing numerical problems. A typical approach to resolve this issue, and one that was used for LRMC \cite{rpca_gd,lafferty_lrmc}, is to change the cost function to minimize to $f({\bm{U}}, \bm{B}) + \lambda f_2({\bm{U}},\bm{B})$ where $f_2({\bm{U}},\bm{B}):=\|{\bm{U}}^\top {\bm{U}} - \bm{B} \bm{B}^\top\|_F$ is the ``norm-balancing term'' (helps ensure that norms of ${\bm{U}}$ and $\bm{B}$ remain similar). We henceforth refer to this approach as {\em altGDnormbal}.
The sample complexity bound for this approach is similar to that for projGD-X. But, it needs a GD step size of order $1/r$ or smaller \cite{rpca_gd,lafferty_lrmc}; making it $r$-times slower than projGD-X
\begin{table*}
\begin{center}
\resizebox{0.95\linewidth}{!}
{
\begin{tabular}{llllll} \toprule
& Sample Comp. & Time Comp. & Communic. Comp. & Holds for & Column-wise \\
& $mq \gtrsim$ & & per node (predicted) & all $\X^*$? & error bound? \\
\hline \midrule
Convex \cite{lee2019neurips} & $n r \frac{1}{\epsilon^4}$ & $\text{\scriptsize{linear-time}} \cdot \min\left( \frac{1}{\sqrt\epsilon}, n^3 r^3\right)$ & not clear & yes & no \\
&&&&& \\
AltMin \cite{lrpr_icml,lrpr_it} & $n r^4 \log(\frac{1}{\epsilon}) $ & $\text{\scriptsize{linear-time}} \cdot r\log^2(\frac{1}{\epsilon})$ & $nr\log(\frac{1}{\epsilon}) \cdot r\log^2(\frac{1}{\epsilon})$ & no \\
&&&&& \\
AltMin \cite{lrpr_best} & $n r^2 (r+\log(\frac{1}{\epsilon}) )$ & $\text{\scriptsize{linear-time}} \cdot r\log^2(\frac{1}{\epsilon})$ & $nr\log(\frac{1}{\epsilon}) \cdot r\log^2(\frac{1}{\epsilon})$ & no & yes \\
&&&&& \\
{\color{blue} altGD-Min} & \color{blue} $\mathbf{n r^2 \log(\frac{1}{\epsilon})}$ & \color{blue} $\mathbf{\text{\scriptsize{linear-time}} \cdot r\log(\frac{1}{\epsilon})}$ & \color{blue} $\mathbf{nr \cdot r\log(\frac{1}{\epsilon})}$ & \color{blue} no & \color{blue} yes \\
{\color{blue} (proposed)} && &&& \\
&&&&& \\
\bottomrule
\multicolumn{6}{l}{Best sample LRMC algorithms among those that do not solve a convex relaxation} \\
\toprule
ProjGD-X & $\max(n,q) r^2 \log^2 n \log^2(\frac{1}{\epsilon})$ & $\bf{\text{\scriptsize{linear-time}} \cdot r \log(\frac{1}{\epsilon})}$ & $nq$ ** && \\
\cite{rmc_gd} &&&&& \\
&&&&& \\
AltGDnormbal & $\mathbf{\max(n,q) r^2 \log n} $ & $\text{\scriptsize{linear-time}} \cdot r^2 \log(\frac{1}{\epsilon})$ & $\mathbf{\max(n,q)r}$ & & \\
\cite{rpca_gd} &&&&& \\
&&&&& \\
\bottomrule
\end{tabular}
}
\end{center}
**The communication complexity of ProjGD-X would be $nq$ because the gradient w.r.t. ${\bm{X}}$ computed at each node will need to be transmitted by the nodes to the center. The gradient w.r.t. ${\bm{X}}$ is not low rank (LR), and hence one cannot transmit just its rank $r$ SVD.
\vspace{-0.05in}
\caption{\small{Existing work versus our work.
For brevity, this table assumes $q \le n$ and treats $\kappa,\mu$ as numerical constants.
All approaches also need $m \ge \max(r,\log q, \log n)$.
Column-wise error bound exists means $\max_k \|\x^*_k - \hat\x_k\|/\|\x^*_k\| \le \epsilon$ holds in addition to a similar bound on matrix Frobenius norm error.
Linear-time is the time needed to read all algorithm inputs. For LRcCS, this is $\bm{y}_k,\bm{A}_k$ for all $k \in [q]$ and thus linear-time is order $mnq$. For LRMC, this is the set of observed entries and their locations and thus linear-time is order $mq$.
}% {\color{red}
None of the other algorithms have been studied in the federated context and hence the communication complexity (Comm. Comp.) listed in the fourth column is based on our understanding of how one would federate the algorithm.
\color{black}
Notice that AltGD-min has the best time and communication complexities; and for $\epsilon^4 < r$, it also has the best sample complexity.
}}
\label{compare_lrccs}
\vspace{-0.1in}
\end{table*}
\subsection{Contributions and Novelty}\label{contrib}
\noindent \bfpara{Contribution to solving LRcCS and LRPR}
(1) This work develops a novel GD-based solution to LRcCS, called AltGD-Min, that is fast and communication-efficient. We show that, with high probability (w.h.p.), AltGD-Min obtains an $\epsilon$-accurate estimate in order $\kappa^2 \log(1/\epsilon)$ iterations, as long as Assumption \ref{right_incoh} holds, the matrices $\bm{A}_k$ are i.i.d., with each containing i.i.d. standard Gaussian entries, $mq \in \Omega ( \kappa^6 \mu^2 (n + q) r^2 \log(1/\epsilon) )$, and $m \in \Omega(\max(\log q, \log n)\log(1/\epsilon))$. Its time complexity is $O( mqnr \cdot \kappa^2 \log(1/\epsilon))$ and its communication complexity per node is $O(nr\cdot \kappa^2 \log(1/\epsilon))$.
We provide a comparison of our guarantee with those of other works in Table \ref{compare_lrccs}.
This table also summarizes the guarantees for the two most sample-efficient LRMC solutions: projGD-X and altGDnormbal. The former is also the fastest LRMC solution, while the latter is the most communication-efficient.
As mentioned earlier, LRMC is the most similar problem to ours that has been extensively studied. Notice that, our sample complexity matches that of the best results for LRMC algorithms that do solve a convex relaxation.
(2) We show that a simple extension of AltGD-Min also provides the fastest provable solution to LRPR, as long as the above assumptions hold and $mq \in \Omega(\kappa^6 \mu^2 nr^2(r + \log(1/\epsilon))$. Its time complexity is the same too.
\noindent \bfpara{Contributions / Novelty of algorithm design and proof techniques}\label{novelty}
As explained earlier, there are three commonly used provably correct iterative algorithms for LR recovery problems -- altMin, projGD-X, and altGD (altGDnormbal to be precise). AltMin is slower than GD-based methods because, for updating both ${\bm{U}}$ and $\bm{B}$, it requires solving a minimization problem keeping the other variable fixed. For our specific asymmetric problem, the min step for ${\bm{U}}$ is the slow one. ProjGD-X and altGDnormbal are faster, but it is not clear how to analyze them for LRcCS under the desired sample complexity\footnote{}% {\color{red}
In order to show that a GD-based algorithm converges, one needs to be able to bound the norm of the gradient and show that it goes to zero with iterations. When studying both projGD-X and altGDnormbal, for different reasons, the estimates of the different columns are coupled. Consequently, it is not possible to get a tight enough bound on $\max_k\|\x^*_k - \hat\x_k\|$. But, due to the form of the LRcCS measurement model, such a bound is needed to get a tight enough bound on the 2-norm of the gradient of the cost function, and show that it decreases sufficiently at each iteration, under the desired sample complexity. Moreover, in case of projGD-X, even if one could somehow get the desired bound, it would not suffice because the summands will still be too heavy tailed. This point is explained in detail in Appendix \ref{algo_understand}.}. Our novel altGD-min approach however resolves both issues: it is fast as projGD-X and it can be analyzed.
Moreover, its communication complexity for a federated implementation (and its memory complexity) is only $nr$ per node per iteration, instead of $nq$ for projGD-X.
As can be seen from Table \ref{compare_lrccs}, treating $\kappa,\mu$ as numerical constants, it has the best sample-, time-, and communication/memory- complexity among all approaches for LRcCS and all fast (iterative) approaches for LRMC as well. Because of this, an AltGD-Min type algorithm may also be of interest for solving LRMC in a fast, sample-efficient and communication-efficient fashion. In fact, it can be also be useful for other bilinear inverse problems such as blind deconvolution.
\bfpara{AltGDmin algorithm} The main idea is as follows. Express ${\bm{X}}$ as ${\bm{X}} = {\bm{U}} \bm{B}$ and alternatively update ${\bm{U}}$ and $\bm{B}$ as follows: (a) keeping $\bm{B}$ fixed at its previous value, update ${\bm{U}}$ by a GD step for it for the cost function $f({\bm{U}},\bm{B})$ followed by projecting the output onto the space of matrices with orthonormal columns; and (b) keeping ${\bm{U}}$ fixed at its previous value, update $\bm{B}$ by minimizing $f({\bm{U}}, \bm{B})$ over it. Because of the column-wise decoupled form of our measurement model, step (b) is as fast as the GD step and thus the per-iteration time complexity of AltGD-Min is equal to that of any other GD method such as projGD-X or altGDnormbal. This decoupling (which means that, given ${\bm{U}}$, $\b_k$ only depends on $\x^*_k$, and not on the other columns of $\X^*$) also allows us to get the desired tight-enough bound on $\max_k\|\hat\b_k- {\bm{U}}^\top \x^*_k\|$ and hence on $\max_k\|\hat\x_k - \x^*_k\|$. This, and the fact that we use the gradient w.r.t. ${\bm{U}}$ in our algorithm, means that the summands in the gradient, and in other error bound terms, are {\em nice-enough sub-exponential random variables (r.v.s)}: sub-exponential r.v.s whose maximum sub-exponential norm is small enough (is proportional to $(r/q)$), so that the summation can be bounded w.h.p. under the desired sample complexity.
\bfpara{AltGDmin analysis} When we analyzed the AltMin approach for LRPR \cite{lrpr_it,lrpr_best}, we could directly modify proof techniques from AltMin for LRMC \cite{lowrank_altmin} for getting a bound on $\text{{SubsDist}}({\bm{U}},\U^*{})$ in terms of the bound on this distance from the previous iteration.
We cannot do this for AltGD-Min because the algorithm itself is different from the two GD approaches studied for solving LRMC. We instead analyze AltGD-Min by a novel use of the fundamental theorem of calculus \cite{lan93} that, along with other linear algebra tricks, helps us get a bound on $\text{{SubsDist}}({\bm{U}},\U^*{})$ which has the desired property: the terms in it are sums of {\em nice-enough sub-exponentials}. See Lemma \ref{algebra} and its proof.
}% {\color{red}
The use of this result is motivated by its use in \cite{pr_mc_reuse_meas}, and many earlier works, where it is used in a standard way: to bound the Euclidean distance, $\|\bm{x} - \x^*\|$, for standard GD to solve the PR problem for recovering a single vector $\x^*$. Thus, at the true solution $\bm{x}=\x^*$, the gradient of the cost function was zero. In our case, there are two differences: (i) we need to bound the subspace distance error, and (ii) our algorithm is not standard GD, and this means that $\nabla_U f(\U^*{} \U^*{}^\top {\bm{U}}, \bm{B}) \neq 0$. We explain our approach in Sec. \ref{outline_iters}.
\color{black}
\bfpara{AltGDmin initialization} The standard LR spectral initialization approach cannot be used because its summands are sub-exponential r.v.s that are not {\em nice-enough}.
We give a detailed explanation in Appendix \ref{algo_understand}.
We address this issue by borrowing the truncation idea from the PR literature \cite{twf,rwf,lrpr_it}. But, in our case, truncation is applied to a non-symmetric matrix. Thus the sandwiching arguments developed for symmetric matrices in \cite{twf}, and modified in \cite{rwf,lrpr_it}, cannot be borrowed. We need a different argument which is used for proving Lemma \ref{lem:init_denom_term} and is briefly explained in Sec. \ref{outline_init}.
\subsection{Applications}\label{apps}
The LRcCS and LRPR problems occur in projection imaging applications involving sets of images, e.g., dynamic MRI \cite{st_imaging,dyn_mri1,dyn_mri2}, federated LR sketching \cite{hughes_icml_2014,lee2019neurips}, and dynamic Fourier ptychography (LRPR) \cite{TCIgauri}.
In MRI, Fourier projections of the region of interest, e.g., a cross-section of the brain or the heart, are acquired one coefficient at a time, making the scanning (data acquisition) quite slow. Hence, reduced sample complexity enables accelerated scanning.
Since medical image sequences are usually slow changing, the LR model is a valid assumption for a time sequence \cite{st_imaging,dyn_mri1,dyn_mri2}.
In our notation, $\x^*_k$ is the vectorized version of the $k$-th image of the sequence and there are a total of $q$ images. The matrices $\bm{A}_k$ are random Fourier, i.e., $\bm{A}_k = \H_k {\bm{F}}$ where ${\bm{F}}$ is the $n \times n$ matrix that models computation of the 2D discrete Fourier transform as a matrix-vector operation, and $\H_k$ is an $m \times n$ random sampling ``mask'' matrix that models the frequency selection.
In \cite{lrpr_gdmin_mri}, we have shown the power of AltGD-Min for fast undersampled dynamic MRI of medical image sequences. It is both much faster, and in most cases, also provides better reconstructions, than many existing solutions from the MRI literature.
Large scale usage of smartphones results in large amounts of geographically distributed data, e.g., images. There is a need to compress/sketch this data before storing it. Sketch refers to a compression approach where the compression end is low complexity, usually simple linear projections \cite{hughes_icml_2014,lee2019neurips}. Consider the setting where different subsets of columns of $\X^*$ (each column corresponds to one vectorized image) are available at each of the $\rho \le q$ nodes. The goal is to sketch them so that they can be correctly recovered using a federated algorithm.
We can store the sketches $\bm{y}_k := \bm{A}_k \x^*_k$ with $\bm{A}_k$'s being i.i.d. Gaussian. This way we store a total of only $mq$ scalars, with $mq$ of order roughly just $(n+q)r^2$.
Traditional LR sketching approaches, e.g., \cite{cov_sketch}, are designed for centralized settings and will not be efficient in a distributed setting.
\subsection{Organization}
In Sec. \ref{algo_thm}, we develop AltGD-Min, give its guarantee for solving LRcCS, and compare it with existing results. We state and prove the two theorems that help prove our main result in Sec. \ref{proving_mainres}. This section also contains brief proof outlines before the actual proofs. The lemmas used in these proofs are proved in Sec. \ref{proof_lemmas}. The extension for solving LRPR is developed, and its guarantee is stated and proved, in Sec. \ref{algo_thm_proof_lrpr}. We discuss the limitations of our results in Sec. \ref{limitations}. Simulation experiments are provided in Sec. \ref{sims}. We conclude in Sec. \ref{conclude}.%
\section{Proofs of all the lemmas
\label{proof_lemmas}
\subsection{Basic tools used}
Our proofs use the following results and definitions:
\begin{theorem}[Wedin $\sin \Theta$ theorem for Frobenius norm subspace distance \cite{wedin,spectral_init_review}[Theorem 2.3.1]]
For two $n_1 \times n_2$ matrices $\bm{M}^*$, $\bm{M}$, let $\U^*{}, {\bm{U}}$ denote the matrices containing their top $r$ singular vectors and let $\V^*{}^\top, {\bm{V}}^\top$ be the matrices of their right singular vectors (recall from problem definition that we defined SVD with the right matrix transposed). Let $\sigma^*_r, \sigma^*_{r+1}$ denote the $r$-th and $(r+1)$-th singular values of $\bm{M}^*$.
If $\|\bm{M} - \bm{M}^*\| \le \sigma^*_r - \sigma^*_{r+1}$, then
\begin{align*}
&\text{{SubsDist}}({\bm{U}}, \U^*{}) \\
&\qquad \le \frac{\sqrt{2} \max(\|(\bm{M} - \bm{M}^*)^\top \U^*{}\|_F, \|(\bm{M} - \bm{M}^*)^\top \V^*{}^\top\|_F )}{\sigma^*_r - \sigma^*_{r+1} - \|\bm{M} - \bm{M}^*\|}
\end{align*}
\label{Wedin_sintheta}
\end{theorem}
\begin{theorem}[Fundamental theorem of calculus \cite{lan93}[Chapter XIII, Theorem 4.2], \cite{pr_mc_reuse_meas}]
For two vectors $\bm{z}_0, \z^* \in \Re^d$, and a differentiable vector function $g(\bm{z}) \in \Re^{d_2}$,
\[
g(\bm{z}_0) - g(\z^*) = \left( \int_{\tau=0}^1 \nabla g (\bm{z}(\tau) ) d\tau \right) (\bm{z}_0 - \z^*),
\]
\text{ where }
\[
\bm{z}(\tau) = \z^* + \tau (\bm{z}_0 - \z^*).
\]
Observe that $\nabla_{\bm{z}} g(\bm{z})$ is a $d_2 \times d$ matrix.
\label{funda_calc}
\end{theorem}
\begin{definition}
For any $n\times r$ matrix ${\bm{{Z}}}$, let ${\Z_{vec}}$ denote the $nr$ length vector formed by arranging all $r$ columns of ${\bm{{Z}}}$ one below the other.
Thus, for $n$-length and $r$-length vectors $\a$ and $\b$,
\begin{itemize}} \newcommand{\ei}{\end{itemize}
\item $(\a \b^\top)_{vec} = \a \otimes \b$ with $\otimes$ being the Kronecker product;
\item $\a^\top {\bm{U}} \b = \mathrm{trace}(\a^\top {\bm{U}} \b) = \mathrm{trace}(\b \a^\top {\bm{U}}) = \langle (\a \b^\top), {\bm{U}} \rangle =\langle \a \otimes \b, {\bm{U}}_{vec} \rangle $;
\ei
$f({\U_{vec}}, \bm{B}) = \sum_{ki} ((\a_{ki} \otimes \b_k)^\top {\U_{vec}} - \bm{y}_{ki})^2$ and
\begin{eqnarray}
(\nabla_{{\bm{U}}}f({\bm{U}}, \bm{B}) )_{vec} = \nabla_{{\bm{U}}_{vec}}f({\bm{U}}_{vec}, \bm{B})
\label{gradU_vecmat}
\end{eqnarray}
\end{definition}
\begin{definition}
At various places, $\nabla f({\bm{U}}, \hat\B)$ is short for $\nabla_{\bm{U}} f({\bm{U}}, \hat\B) = \sum_{ki} \a_{ki} \hat\b_k{}^\top(\a_{ki}{}^\top {\bm{U}} \hat\b_k - \bm{y}_{ki})$ and similarly $\nabla f({\U_{vec}},\hat\B)$ is short for $\nabla_{\U_{vec}} f({\U_{vec}},\hat\B) = \sum_{ki} (\a_{ki} \otimes \b_k) ((\a_{ki} \otimes \b_k)^\top {\U_{vec}} - \bm{y}_{ki})$.
\end{definition}
\begin{definition}
For any vector $\bm{w}$, we use $\bm{w}(k)$ to denote its $k$-th entry.
\end{definition}
\begin{definition}
Everywhere we use $\S_{nr}$ to denote both the set of matrices $\{{\bm{W}} \in \Re^{n \times r}: \|{\bm{W}}\|_F = 1 \}$ and the set of these matrices vectorized $\{\bm{w} \in \Re^{nr}: \|\bm{w}\|=1 \}$. We also switch between the two sometimes. In the entire writing below, $\bm{w} = {\bm{W}}_{vec}$.
\end{definition}
All the high probability bounds for initialization use subGaussian Hoeffding inequality, while those for GD lemmas use the sub-exponential Bernstein inequality, both are from \cite{versh_book}.
In addition, these lemmas also use the following results to ``epsilon-net'' extend a bound holding for a fixed unit norm ${\bm{W}}$ (or $\bm{w}$) to all unit norm ${\bm{W}}$s (or $\bm{w}$s)
\begin{prop}[Epsilon-netting for bounding $\max_{\bm{w} \in \S_n, \bm{z} \in \S_r} |\bm{w}^\top \bm{M} \bm{z} |$]
For an $n \times r$ matrix $\bm{M}$ and fixed vectors $\bm{w}, \bm{z}$ with, $\bm{w} \in \S_n$ and $\bm{z} \in \S_r$, suppose that $|\bm{w}^\top \bm{M} \bm{z} | \le b_0$ w.p. at least $1-p_0$.
Consider an $\epsilon_{net}$ net covering $\S_n $ and $\S_r$, $\bar\S_n$, $\bar\S_r$
Then w.p. at least $1 - (1+2/\epsilon_{net})^{n+r} p_0$,
\begin{itemize}} \newcommand{\ei}{\end{itemize}
\item $\max_{\bm{w} \in \bar\S_n, \bm{z} \in \bar\S_r} |\bm{w}^\top \bm{M} \bm{z} | \le b_0$ and
\item $\max_{\bm{w} \in \S_n, \bm{z} \in \S_r} |\bm{w}^\top \bm{M} \bm{z} | \le \frac{1}{1 - 2\epsilon_{net} - \epsilon_{net}^2} b_0$.
\ei
Using $\epsilon_{net}=1/8$, this implies the following simpler conclusion: \\
W.p. at least $1 - 17^{n+r} p_0= 1- \exp( (\log 17) (n+r) ) \cdot p_0$, $\max_{\bm{w} \in \S_n, \bm{z} \in \S_r} |\bm{w}^\top \bm{M} \bm{z} | \le 1.4 b_0$.
\label{epsnet_Mwz}
\end{prop}
\begin{proof} The proof follows that of Lemma 4.4.1 of \cite{versh_book} \end{proof}
\begin{prop}[Epsilon-netting for bounding $\max_{{\bm{W}} \in \S_{nr}} \langle \bm{M}, {\bm{W}}$]
For an $n \times r$ matrix $\bm{M}$ and a fixed $n \times r$ matrix ${\bm{W}} \in \S_{nr}$ (unit Frobenius norm matrix), suppose that $\langle \bm{M}, {\bm{W}} \rangle \le b_0$ w.p. at least $1-p_0$.
Consider an $\epsilon_{net}$ net covering $\S_{nr}$, $\bar\S_{nr}$.
Then w.p. at least $1 - (1+2/\epsilon_{net})^{nr} p_0$,
\begin{itemize}} \newcommand{\ei}{\end{itemize}
\item $\max_{{\bm{W}} \in \bar\S_{nr}} \langle \bm{M}, {\bm{W}} \rangle \le b_0$ and
\item $\max_{{\bm{W}} \in \S_{nr}} \langle \bm{M}, {\bm{W}} \rangle \le \frac{1}{1 - \epsilon_{net}} b_0$.
\ei
Using $\epsilon_{net}=1/8$, this implies the following simpler conclusion: \\
w.p. at least $1 - 17^{nr} p_0 = 1- \exp( (\log 17) (nr) ) \cdot p_0$, $\max_{{\bm{W}} \in \S_{nr}} \langle \bm{M}, {\bm{W}} \rangle \le 1.2 b_0$.
\label{epsnet_MW}
\end{prop}
\begin{proof} The proof follows exactly as that of Exercise 4.4.3 of \cite{versh_book} \end{proof}
\begin{prop}[Epsilon-netting for upper and lower bounding $\sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2$ over all ${\bm{W}} \in \S_{nr}$]
For an $n \times r$ matrices $\bm{M}_{ki}$ and a fixed ${\bm{W}} \in \S_{nr}$, suppose that, w.p. at least $1-p_0$,
\[
b_1 \le \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 \le b_2
\]
Consider an $\epsilon_{net}$ net covering $\S_{nr}$, $\bar\S_{nr}$.
Then, w.p. at least $1 - (1+2/\epsilon_{net})^{nr} p_0$,
\[
\max_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 \le \frac{1}{1 - \epsilon_{net}^2 - 2 \epsilon_{net}} b_2
\]
and
\[
\min_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 \ge b_1 - 2 \epsilon_{net} \cdot \frac{1}{1 - \epsilon_{net}^2 - 2 \epsilon_{net}} b_2
\]
Picking $\epsilon_{net} = b_1/ (8 b_2)$ guarantees that the above lower bound is non-negative.
In particular, it implies the following:
\\
w.p. at least $1 - (24b_2/b_1)^{nr} p_0 = 1 - \exp(C nr \log(b_2/b_1) ) \cdot p_0$,
$
0.8 b_1 \le \min_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 \le \max_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 \le 1.4 b_2
$
\label{epsnet_MWsquared}
\end{prop}
\begin{proof}
By union bound, for all $\bar{\bm{W}} \in \bar\S_{nr}$,
$b_1 \le \sum_{ki} \langle \bm{M}_{ki}, \bar{\bm{W}} \rangle^2 \le b_2$ holds w.p. at least $1 - (1+2/\epsilon_{net})^{nr} p_0$.
Proof for the upper bound:
Let $\gamma^* = \max_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2$.
Writing ${\bm{W}} = \bar{\bm{W}} + ({\bm{W}} - \bar{\bm{W}})$ where $\bar{\bm{W}}$ is the closest point to ${\bm{W}}$ on $\bar\S_{nr}$, we have
$
\sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2 = \sum_{ki} \langle \bm{M}_{ki}, \bar{\bm{W}} \rangle^2 + \sum_{ki} \langle \bm{M}_{ki}, ({\bm{W}} - \bar{\bm{W}}) \rangle^2 + 2 \sum_{ki} \langle \bm{M}_{ki}, \bar{\bm{W}} \rangle \cdot \sum_{ki} \langle \bm{M}_{ki}, ({\bm{W}} - \bar{\bm{W}}) \rangle$ and
$\|({\bm{W}} - \bar{\bm{W}})\|_F \le \epsilon_{net}$.
Rewriting $({\bm{W}} - \bar{\bm{W}}) = ({\bm{W}} - \bar{\bm{W}}) \cdot ({\bm{W}} - \bar{\bm{W}}) / \|({\bm{W}} - \bar{\bm{W}})\|_F$ and using the fact that $({\bm{W}} - \bar{\bm{W}})/\|({\bm{W}} - \bar{\bm{W}})\|_F \in \S_{nr}$ and $\|({\bm{W}} - \bar{\bm{W}})\|_F \le \epsilon_{net}$ and using Cauchy-Schwarz for the third term in the above expression, we have
\[
\gamma^* \le b_2 + \epsilon_{net}^2 \gamma^* + 2 \sqrt{\gamma^* \cdot \epsilon_{net}^2 \gamma^*} = b_2 + \epsilon_{net}^2 \gamma^* + 2\epsilon_{net} \gamma^*
\]
Thus, $\gamma^* \le 1/(1 - \epsilon_{net}^2 - 2 \epsilon_{net}) \cdot b_2$.
Proof for the lower bound:
Let $\beta^* = \min_{{\bm{W}} \in \S_{nr}} \sum_{ki} \langle \bm{M}_{ki}, {\bm{W}} \rangle^2$. Proceeding as above, we have
\[
\beta^* \ge b_1 - 2 \sqrt{\gamma^* \cdot \epsilon_{net}^2 \gamma^*} = b_1 - 2\epsilon_{net} \gamma^*
\]
\end{proof}
\subsection{Proving GD iterations' lemmas: Proof of Lemma \ref{algebra} (algebra lemma)}
\label{algebra_proof}
Recall that ${\U_{vec}}$ denotes the vectorized ${\bm{U}}$.
We use this so that we can apply the simple vector version of the fundamental theorem of calculus \cite[Chapter XIII, Theorem 4.2]{lan93},\cite[Lemma 2 proof]{pr_mc_reuse_meas} (given in Theorem \ref{funda_calc}) on the $nr$ length vector $\nabla f({\U_{vec}},\hat\B)$, and so that the Hessian can be expressed as an $nr \times nr$ matrix.
We apply Theorem \ref{funda_calc} with $\bm{z}_0 \equiv {\U_{vec}}$, $\z^* \equiv (\U^*{} \U^*{}^\top {\bm{U}})_{vec}$, and $g(\bm{z}) = \nabla f(\bm{z},\hat\B)$. Thus $d = d_2 = nr$ and $\nabla g(\bm{z})$ is the Hessian of $f(\bm{z},\hat\B)$ computed at $\bm{z}$.
Let ${\bm{U}}(\tau) := \U^*{} \U^*{}{}^\top {\bm{U}} + \tau ({\bm{U}} - \U^*{} \U^*{}{}^\top {\bm{U}})$.
Applying the theorem,
\begin{align}
&\nabla f({\bm{U}}_{vec},\hat\B) - \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B) \nonumber\\
&= ( \int_{\tau=0}^1 \nabla_{{\U_{vec}}}^2 f({\bm{U}}(\tau)_{vec}, \hat\B) d\tau ) ( {\bm{U}}_{vec} - (\U^*{}\Ustar{}^\top {\bm{U}})_{vec})
\label{funda_thm_app_0}
\end{align}
where
\begin{align}\label{Hess_compute}
\nabla_{{\U_{vec}}}^2 f({\bm{U}}(\tau)_{vec}, \hat\B) = \sum_{ki} (\a_{ki} \otimes \hat\b_k) (\a_{ki} \otimes \hat\b_k)^\top : = \ \mathrm{Hess} \
\end{align}
This is an $nr \times nr$ matrix. Because the cost function is quadratic, the Hessian is constant w.r.t. $\tau$. Henceforth, we refer to it as $ \ \mathrm{Hess} \ $. With this, the above simplifies to
\begin{align}
&\nabla f({\bm{U}}_{vec},\hat\B) - \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B) \nonumber\\
&= \ \mathrm{Hess} \ ( {\bm{U}}_{vec} - (\U^*{}\Ustar{}^\top {\bm{U}})_{vec}) = \mathrm{Hess} \ (\P {\bm{U}})_{vec}
\label{funda_thm_app}
\end{align}
with
\[
\P : = \bm{I} - \U^*{} \U^*{}^\top
\]
denoting the $n \times n$ projection matrix to project orthogonal to $\U^*{}$.
This proof is motivated by a similar approach used in \cite[Lemma 2 proof]{pr_mc_reuse_meas} to analyze GD for standard PR. However, there the application was much simpler because $f(.)$ was a function of one variable and at the true solution the gradient was zero, i.e., $\nabla f(\x^*) = \bm{0}$. In our case $\nabla f(\U^*{} \U^*{}^\top {\bm{U}}, \hat\B) \neq \bm{0}$ because $\hat\B \neq \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*$. But we can show that $\mathbb{E}[ (\bm{I} - \U^*{} \U^*{}^\top) \nabla f(\U^*{} \U^*{}^\top {\bm{U}}, \hat\B)] = \bm{0}$ and this helps us get the final desired result.
From Algorithm \ref{gdmin}, recall that $\hat\U^+ = {\bm{U}} - (\eta/m) \nabla f({\bm{U}},\hat\B)$. Vectorizing this equation,
and using \eqref{funda_thm_app}, we get
\begin{align}
(\hat\U^+)_{vec}
& = {\bm{U}}_{vec} - (\eta/m) \nabla f({\bm{U}}_{vec}, \hat\B) \nonumber \\
& = {\bm{U}}_{vec} - (\eta/m) \ \mathrm{Hess} \ ( \P {\bm{U}})_{vec} \nonumber\\
&\qquad- (\eta/m) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B) )
\label{Uplusvec_eq}
\end{align}
We can prove our final result by using \eqref{gradU_vecmat} and the following simple facts:
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item For an $n \times n$ matrix $\bm{M}$, let
$
\mathrm{big}(\bm{M}) := \bm{I}_r \otimes \bm{M}.
$
be an $nr \times nr$ block diagonal matrix with $\bm{M}$ in the diagonal blocks.
For any $n \times r$ matrix ${\bm{{Z}}}$,
\begin{align}
\label{MZ}
& \mathrm{big}(\bm{M}) {\Z_{vec}} = (\bm{M} {\bm{{Z}}})_{vec}
\end{align}
\item Since $\P$ is idempotent, $\P = \P^2$. Also, because of its block diagonal structure, $\mathrm{big}(\bm{M}^2)= (\mathrm{big}(\bm{M}))^2$. Thus,
\begin{align}
\mathrm{big}(\P)& = \mathrm{big}(\P^2) = (\mathrm{big}(\P))^2 = \mathrm{big}(\P)) \bm{I}_{nr} (\mathrm{big}(\P)
\label{P_props}
\end{align}
\een
Left multiplying both sides of \eqref{Uplusvec_eq} by $\mathrm{big}(\P)$, and using \eqref{MZ}, \eqref{P_props}, and \eqref{gradU_vecmat},
\begin{align*}
& \mathrm{big}(\P) (\hat\U^+)_{vec}
= \mathrm{big}(\P) {\bm{U}}_{vec} - (\eta/m) \mathrm{big}(\P) \ \mathrm{Hess} \ (\P {\bm{U}})_{vec} \\
&\qquad- (\eta/m) \mathrm{big}(\P) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B) \\
& = \mathrm{big}(\P)\bm{I}_{nr} \mathrm{big}(\P) {\bm{U}}_{vec} - (\eta/m) \mathrm{big}(\P) \ \mathrm{Hess} \ \mathrm{big}(\P) {\bm{U}}_{vec} \\
&\qquad - (\eta/m) \mathrm{big}(\P) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B) \\
& = \mathrm{big}(\P) (\bm{I}_{nr} - (\eta/m) \ \mathrm{Hess}) \mathrm{big}(\P) {\bm{U}}_{vec} \\
&\qquad - (\eta/m) \mathrm{big}(\P) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}})_{vec},\hat\B).
\end{align*}
Thus, using $\|\mathrm{big}(\P)\| = \|\P\| = 1$, \eqref{MZ}, and \eqref{gradU_vecmat},
\begin{align}
\| ( \P \hat\U^+)_{vec}\|
& \le \|\bm{I}_{nr} - (\eta/m) \ \mathrm{Hess} \ \| \ \| (\P {\bm{U}})_{vec} \| \nonumber\\
&+ (\eta/m) \| ( \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B))_{vec} \|
\end{align}
Converting the vectors to matrices, using $||\bm{M}_{vec}|| = ||\bm{M}||_F$, and substituting for $\P$,
\begin{align*}
&\|(\bm{I} - \U^*{} \U^*{}{}^\top)\hat\U^+\|_F \\
&\le \|\bm{I}_{nr} - (\eta/m) \ \mathrm{Hess} \ \| \ \|(\bm{I} - \U^*{} \U^*{}{}^\top){\bm{U}}\|_F\\
&\qquad + (\eta/m) \|(\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B) \|_F
\end{align*}
Since $\hat\U^+ \overset{\mathrm{QR}}=} %{\stackrel{EVD}{=} {\bm{U}}^+ {\bm{R}}^+$ and since $\|\bm{M}_1 \bm{M}_2\|_F \le \|\bm{M}_1\|_F \|\bm{M}_2\|$, this means that
$$\SE}%{\SE_F(\U^*{}, {\bm{U}}^+) \le \|(\bm{I} - \U^*{} \U^*{}{}^\top)\hat\U^+\|_F \| ({\bm{R}}^+)^{-1}\|.$$
Since $\| ({\bm{R}}^+)^{-1}\| = 1/\sigma_{\min}({\bm{R}}^+) = 1/\sigma_{\min}(\hat\U^+)$, using $\hat\U^+ = {\bm{U}} - (\eta/m) \nabla f({\bm{U}}, \hat\B)$,
\begin{align*}
\| ({\bm{R}}^+)^{-1}\| &= \frac{1}{\sigma_{\min}({\bm{U}} - (\eta/m) \nabla f({\bm{U}}, \hat\B))} \\
&\le \frac{1}{1 - (\eta/m) \|\nabla f({\bm{U}}, \hat\B)\|}
\end{align*}
where we used $\sigma_{\min}({\bm{U}} - (\eta/m) \nabla f({\bm{U}}, \hat\B)) \ge \sigma_{\min}({\bm{U}}) - (\eta/m) \|\nabla f({\bm{U}}, \hat\B)\| = 1- (\eta/m) \|\nabla f({\bm{U}}, \hat\B)\|$ for the last inequality.
Combining the last three equations above proves our lemma.
\subsection{Proof of GD iterations' lemmas: Proof of Lemma \ref{terms_bnds}} \label{terms_bnds_proof}
\subsubsection{Upper and Lower bounding the Hessian eigenvalues and hence HessTerm}
\label{subsec:hessian}
First assume the event that implies that the conclusions of Lemma \ref{B_lemma} hold.
Recall from \eqref{Hess_compute} that
$
\ \mathrm{Hess} \ := \nabla_{\tilde{\bm{U}}_{vec}}^2 f(\tilde{\bm{U}}_{vec}; \hat\B) = \sum_{ki} (\a_{ki} \otimes \hat\b_{k}) (\a_{ki} \otimes \hat\b_k){}^\top.
$
Since $ \ \mathrm{Hess} \ $ is a positive semi-definite matrix,
$
\lambda_{\min}\left( \ \mathrm{Hess} \ \right) = \min_{\bm{w}\in\S_{nr}} \bm{w}{}^\top \ \mathrm{Hess} \ \ \bm{w} $ and $\lambda_{\max}\left( \ \mathrm{Hess} \ \right) = \max_{\bm{w}\in\S_{nr}} \bm{w}{}^\top \ \mathrm{Hess} \ \ \bm{w}.
$
For a fixed $\bm{w}\in\S_{nr}$,
\[
\bm{w}{}^\top \ \mathrm{Hess} \ \ \bm{w} = \sum_{ki} (\a_{ki}{}^\top {\bm{W}} \hat\b_k)^2
\]
where ${\bm{W}}$ is an $n \times r$ matrix with $\|{\bm{W}}\|_F = 1$.
Clearly $(\a_{ki}{}^\top {\bm{W}} \hat\b_k)^2 $ are mutually independent sub-exponential random variables (r.v.) with sub-exponential norm $K_{ki}\leq \|{\bm{W}}\hat\b_{k}\|^2$. Also, $\mathbb{E}[ (\a_{ki}{}^\top {\bm{W}} \hat\b_k)^2] = \|{\bm{W}} \hat\b_k\|^2$ and thus $\mathbb{E}[ \sum_{ki} (\a_{ki}{}^\top {\bm{W}} \hat\b_k)^2]= m\|{\bm{W}}\hat\B\|_F^2$.
Applying the sub-exponential Bernstein inequality, Theorem 2.8.1 of \cite{versh_book}, for a fixed ${\bm{W}}\in\S_{nr}$ yields
\begin{align*}
&\Pr\left\{ \Big|\sum_{ki} \big|\a_{ki}{}^\top{\bm{W}}\hat\b_{k}\big|^2 - m\|{\bm{W}}\hat\B\|_F^2 \Big| \geq t \right\}\\
&\qquad \leq \exp\left[-c\min\left( \frac{t^2}{\sum_{ki} K_{ki}^2},~\frac{t}{\max_{ki} K_{ki}} \right)\right].
\end{align*}
We set $t = \epsilon_3 m {\sigma_{\min}^*}^2$. By Lemma \ref{B_lemma}, $\|\hat\b_{k}\|^2 \leq 1.1 \mu^2 {\sigma_{\max}^*}^2 (r/q) = 1.1 \kappa^2 \mu^2 {\sigma_{\min}^*}^2 (r/q)$. Thus,
\begin{align*}
\frac{t^2}{\sum_{ki} K_{ki}^2} &\geq \frac{\epsilon_3^2m^2 {\sigma_{\min}^*}^4}{\sum_{ki} \|{\bm{W}}\hat\b_{k}\|^4 } \geq \frac{\epsilon_3^2 m {\sigma_{\min}^*}^4}{ \max_k \|\hat\b_{k}\|^2 \sum_{k}\|{\bm{W}}\hat\b_{k}\|^2 } \\
&\geq \frac{\epsilon_3^2m {\sigma_{\min}^*}^4}{ \mu^2 {\sigma_{\max}^*}^2 (r/q) 1.1. {\sigma_{\max}^*}^2 } = c\epsilon_3^2mq/ r \mu^2\kappa^4
\end{align*}
Here we used $\sum_{k}\|{\bm{W}}\hat\b_{k}\|^2 = \|{\bm{W}} \hat\B\|_F^2 \le \|{\bm{W}}\|_F \|\hat\B\|_2 \le 1.1. {\sigma_{\max}^*}$ using the bound on $\|\hat\B\|_2$ from Lemma \ref{B_lemma}.
Also,
\begin{align*}
\frac{t}{\max_{ki} K_{ki}} &\geq \frac{\epsilon_3 m {\sigma_{\min}^*}^2}{\max_{ki} \|{\bm{W}}\hat\b_{k}\|^2} \geq \frac{\epsilon_3 m {\sigma_{\min}^*}^2}{1.1 \mu^2 {\sigma_{\max}^*}^2 (r/q)} \\
&= c\epsilon_3 mq/r\mu^2\kappa^2.
\end{align*}
Therefore, for a fixed ${\bm{W}} \in \S_{nr}$, w.p. $1-\exp\left[-c\epsilon_3^2 mq/r \mu^2\kappa^4 \right]$ we have
\begin{align}\label{bnd_term}
\Big|\sum_{ki} \big|\a_{ki}{}^\top{\bm{W}}\hat\b_{k}\big|^2 - m\|{\bm{W}}\hat\B\|_F^2 \Big| \leq \epsilon_3 m {\sigma_{\min}^*}^2.
\end{align}
and hence, by Lemma \ref{B_lemma}, w.p. $1-\exp\left[-c\epsilon_3^2 mq/r \mu^2\kappa^4 \right]$,
\begin{align}
\label{fixedW_upperbnd}
&\sum_{ki} \big|\a_{ki}{}^\top{\bm{W}}\hat\b_{k}\big|^2 \leq m\|{\bm{W}}\hat\B\|_F^2 + \epsilon_3m{\sigma_{\min}^*}^2 \nonumber \\
&\qquad\leq m\|\hat\B\|^2 + \epsilon_3 m {\sigma_{\min}^*}^2
\leq m(1.1 + \epsilon_3/\kappa^2) {\sigma_{\max}^*}^2.
\end{align}
and
\begin{align}
\label{fixedW_lowerbnd}
&\sum_{ki} \big|\a_{ki}{}^\top{\bm{W}}\hat\b_{k}\big|^2 \ge m\|{\bm{W}}\hat\B\|_F^2 - \epsilon_3m{\sigma_{\min}^*}^2 \nonumber \\
&\qquad \ge 0.9 m {\sigma_{\min}^*}^2 + \epsilon_3 m{\sigma_{\min}^*}^2
\geq m(0.9 - \epsilon_3) {\sigma_{\min}^*}^2.
\end{align}
To extend these bounds to all ${\bm{W}} \in \S_{nr}$ we apply Proposition \ref{epsnet_MWsquared} with $b_1 \equiv m(0.9 - \epsilon_3) {\sigma_{\min}^*}^2$ and $b_2 \equiv m(1.1 + \epsilon_3/\kappa^2) {\sigma_{\max}^*}^2$. Applying it we can conclude that, given the event that the claims of Lemma \ref{B_lemma} holds,
w.p. at least $1 - \exp( nr \log \kappa - c mq \epsilon_3^2 / r \mu^2 \kappa^4 )$,
\begin{align*}
m (0.7 - 1.2 \epsilon_3) {\sigma_{\min}^*}^2 &\le \lambda_{\min}( \ \mathrm{Hess} \ ) \\&\le \lambda_{\max}( \ \mathrm{Hess} \ ) \le m (1.1 + \epsilon_3) {\sigma_{\max}^*}^2
\end{align*}
Using the probability from Lemma \ref{B_lemma},
the above bound holds w.p. at least $1 - \exp( nr \log \kappa - c mq \epsilon_3^2 / r \mu^2 \kappa^4 ) - \exp(\log q + r - c m)$.
\subsubsection{Bounding the GradU Term}
\label{subsec:sigmin_R_U}
We have
$
\|\nabla f({\bm{U}},\hat\B)\| = \max_{\bm{z}\in\S_n, \bm{w}\in\S_r} \bm{z}{}^\top \nabla f({\bm{U}},\hat\B)\bm{w}.
$
For a fixed $\bm{z}\in\S_n, \bm{w}\in\S_r$ we have
\begin{align*}
&\bm{z}{}^\top \left( \nabla f({\bm{U}},\hat\B) - \mathbb{E}[\nabla f({\bm{U}},\hat\B)] \right) \bm{w} \\
&\qquad= \sum_{ki} \left[ \left(\a_{ki}{}^\top{\bm{U}}\hat\b_{k}-\bm{y}_{ki} \right)\left(\a_{ki}{}^\top\bm{z}\right)\left(\bm{w}{}^\top\hat\b_{k}\right) - \mathbb{E}[.] \right]
\end{align*}
where $\mathbb{E}[.]$ is the expected value of the first term.
Clearly, the summands are independent sub-exponential r.v.s with norm $K_{ki} \leq C \|\hat\x_{k}-\x^*_{k}\|\|\hat\b_{k}\|$. We apply the sub-exponential Bernstein inequality, Theorem 2.8.1 of \cite{versh_book}, with $t = \epsilon_1 \delta_t m {\sigma_{\max}^*}^2$.
To apply this, we use bounds on $\|\hat\b_{k}\|$, $\|\X^* - \hat\X\|_F$ and $\|\hat\x_{k}-\x^*_{k}\|$ from Lemma \ref{B_lemma} to show that
\begin{align*}
\frac{t^2}{\sum_{ki} K_{ki}^2} &\geq c\frac{\epsilon_1^2\delta_t^2 m^2 {\sigma_{\max}^*}^4}{ m \max_k \|\hat\b_{k}\|^2 \sum_k\|\hat\x_{k}-\x^*_{k}\|^2 } \\&\geq c\frac{\epsilon_1^2 \delta_t^2 m {\sigma_{\max}^*}^4}{C \mu^2 {\sigma_{\max}^*}^2 (r/q) \|\hat\X - \X^*\|_F^2 } \\&\geq c \frac{\epsilon_1^2 \delta_t^2 mq {\sigma_{\max}^*}^4}{C \mu^2 {\sigma_{\max}^*}^2 r \delta_t^2 {\sigma_{\max}^*}^2 } = c \epsilon_1^2 \frac{mq }{ r \mu^2}.
\end{align*}
and
\[
\frac{t}{\max_{ki} K_{ki}} \geq c \frac{\epsilon_1\delta_tm{\sigma_{\max}^*}^2}{C\delta_t {\sigma_{\max}^*}^2 \mu^2 (r/q) } \ge c \epsilon_1 \frac{mq }{ r \mu^2 }.
\]
Therefore, for a fixed $\bm{z}\in\S_n, \bm{w}\in\S_{r}$ w.p. $1-\exp (-c\epsilon_1^2mq/r \mu^2 )$,
\begin{align*}
\bm{z}{}^\top\left( \nabla f({\bm{U}},\hat\B) - \mathbb{E}[\nabla f({\bm{U}},\hat\B)] \right)\bm{w} &\leq \epsilon_1 \delta_t m {\sigma_{\max}^*}^2
\end{align*}
Since $\nabla f({\bm{U}},\hat\B) = \sum_{ki} \a_{ki}\a_{ki}{}^\top(\hat\x_{k}-\x^*_{k})\hat\b_{k}{}^\top$,
\[
\mathbb{E} [\nabla f({\bm{U}},\hat\B) ] = m\sum_k (\hat\x_{k}-\x^*_{k})\hat\b_{k}{}^\top = m\left(\hat\X -\X^* \right)\hat\B{}^\top.
\]
Using the bounds on $\|\X^* - \hat\X\|_F$ and $\|\hat\B\|$ from Lemma \ref{B_lemma},
\begin{align*}
\|\mathbb{E} [\nabla f({\bm{U}},\hat\B) ] \| &= m \| (\hat\X -\X^* )\hat\B{}^\top\| \\&\leq m \|\hat\X -\X^*\|~\|\hat\B\| \\& \le m \|\hat\X -\X^*\|_F ~\|\hat\B\| \\& \leq 1.1 m\delta_t{\sigma_{\max}^*}^2
\end{align*}
Hence, for a fixed $\bm{z}\in\S_n, \bm{w}\in\S_{r}$ w.p. $1-\exp\left[-c \epsilon_1^2 mq/r \mu^2 \right]$ we have
\[
| \bm{z}^\top \nabla f({\bm{U}},\hat\B) \bm{w} | \leq (1.1 + \epsilon_1) m\delta_t {\sigma_{\max}^*}^2.
\]
Applying Proposition \ref{epsnet_Mwz}, this implies that, w.p. $1-\exp ((n+r) (\log 17) -c\epsilon_1^2mq/r \mu^2 )$, $ \max_{\bm{z}\in\S_n, \bm{w}\in\S_r} \bm{z}{}^\top \nabla f({\bm{U}},\hat\B)\bm{w} \le 1.4 (1.1 + \epsilon_1) m\delta_t {\sigma_{\max}^*}^2.$
\subsubsection{Bounding Term2}
First, since $\mathrm{Term2} = (\bm{I} - \U^*{} \U^*{}{}^\top) \sum_{ki} \a_{ki} (\a_{ki}{}^\top\U^*{} (\U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k) ) \hat\b_{k}{}{}^\top$, and $\mathbb{E}[\a_{ki} \a_{ki}{}^\top] = \bm{I}$,
\[
\mathbb{E}[\mathrm{Term2}] = 0
\]
We have
\begin{align*}
&\|(\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B) \|_F \\&\qquad=\max_{{\bm{W}}\in\S_{nr}} \langle (\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B) ,~{\bm{W}}\rangle
\end{align*}
For a fixed $n \times r$ matrix ${\bm{W}}$ with unit Frobenius norm,
\begin{align*}
&\langle (\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B) ,~{\bm{W}}\rangle \\&\qquad= \sum_{ki} \left(\a_{ki}{}^\top\U^*{} (\U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k) \right)\left(\a_{ki}{}^\top(\bm{I} - \U^*{} \U^*{}{}^\top){\bm{W}}\hat\b_{k}\right)
\end{align*}
Observe that the summands are independent, zero mean, sub-exponential r.v.s with sub-exponential norm $K_{ki}\leq C \| \U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k \| \|(\bm{I} - \U^*{} \U^*{}{}^\top){\bm{W}}\hat\b_{k}\| \le
\| \U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k \| \| {\bm{W}}\hat\b_{k}\|$.
We can now apply the sub-exponential Bernstein inequality Theorem 2.8.1 of \cite{versh_book}.
Let $t=\epsilon_2\delta_t m {\sigma_{\max}^*}^2$.
Using the bound on $\| \U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k \|$ from Lemma \ref{B_lemma} followed by Assumption \ref{right_incoh} (right incoherence), and also the bound on $\|\hat\B\|$ from Lemma \ref{B_lemma},
\begin{align*}
\frac{t^2}{\sum_{ki} K^2_{ki}} &\geq \frac{\epsilon_2^2\delta_t^2m^2{\sigma_{\max}^*}^4}{ \delta_t^2 {\sigma_{\max}^*}^2 \mu^2 (r/q) \sum_{ki} \|{\bm{W}}\hat\b_{k}\|^2 } \\&\geq \frac{\epsilon_2^2 m^2 {\sigma_{\max}^*}^2}{C \mu^2 (r/q) m \|{\bm{W}}\hat\B\|_F^2 }\geq \frac{\epsilon_2^2m^2 {\sigma_{\max}^*}^2}{ \mu^2 (r/q) m {\sigma_{\max}^*}^2 } \\&\geq c \epsilon_2^2mq/ r\mu^2 ,
\end{align*}
and
\[
\frac{t}{\max_{ki} K_{ki}}\geq \frac{\epsilon_2\delta_tm{\sigma_{\max}^*}^2}{C\delta_t \kappa^2 \mu^2 {\sigma_{\max}^*}^2 (r/q)}\geq c \epsilon_2 mq/ (r \kappa^2 \mu^2) .
\]
Thus, by the sub-exponential Bernstein inequality,
for a fixed ${\bm{W}}\in\S_{nr}$, w.p. $1-\exp (-c\epsilon_2^2mq/r \kappa^2 \mu^2 )$,
\[
\langle (\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B) ,~{\bm{W}}\rangle \leq \epsilon_2 \delta_t m {\sigma_{\max}^*}^2.
\]
Applying Proposition \ref{epsnet_MW}, w.p. at least $1-\exp ( nr-c\epsilon_2^2mq/r \kappa^2 \mu^2 )$, $\max_{{\bm{W}}\in\S_{nr}} \langle (\bm{I} - \U^*{} \U^*{}{}^\top) \nabla f((\U^*{}\Ustar{}^\top{\bm{U}}),\hat\B), {\bm{W}} \rangle \le 1.2 \epsilon_2 \delta_t m {\sigma_{\max}^*}^2.$
\newcommand{\A_k}{\bm{A}_k}
\newcommand{\A_k{}^\top}{\bm{A}_k{}^\top}
\subsection{Proof of GD iterations' lemmas: Proof of Lemma \ref{B_lemma}, all parts other than the first part} \label{B_lemma_proof}
Recall that $\bm{g}_k = {\bm{U}}^\top \x^*_k = {\bm{U}}^\top \U^*{} \b^*} %{\tilde\b^*_k$, and $\bm{G} = {\bm{U}}^\top \U^*{} \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*$.
Using the $\text{{SubsDist}}$ bound and the first part, $\|\bm{g}_k - \hat\b_k\| \le 0.4 \delta_t \|\b^*} %{\tilde\b^*_k\|$.
Since $\x^*_k - \hat\x_k = {\bm{U}} \bm{g}_k + (\bm{I} - {\bm{U}} {\bm{U}}^\top) \x^*_k - {\bm{U}} \hat\b_k = {\bm{U}} (\bm{g}_k - \hat\b_k) + (\bm{I} - {\bm{U}} {\bm{U}}^\top) \x^*_k $, using \eqref{bhatk_bnd},
\begin{align*}
\|\x^*_k - \hat\x_k\| & \le \|\bm{g}_k - \hat\b_k\| + \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\|
\le 1.4 \delta_t \|\b^*} %{\tilde\b^*_k\|.
\end{align*}
$\| \U^*{}{}^\top{\bm{U}}\hat\b_{k} - \b^*} %{\tilde\b^*_k \| = \| \U^*{}\Ustar{}^\top{\bm{U}}\hat\b_{k} - \U^*{}\b^*} %{\tilde\b^*_k \| = \|{\bm{U}} \hat\b_k - (\bm{I} - \U^*{}\Ustar{}^\top){\bm{U}}\hat\b_{k} - \U^*{}\b^*} %{\tilde\b^*_k \| = \|\hat\x_k - (\bm{I} - \U^*{}\Ustar{}^\top){\bm{U}}\hat\b_{k} - \x^*_k \|
\le \|\hat\x_{k}-\x^*_{k}\| + \|(\bm{I} - \U^*{} \U^*{}{}^\top) {\bm{U}} \hat\b_k\| \le 2.4 \delta_t \|\b^*} %{\tilde\b^*_k\| $
Bounding $\|\bm{G} - \hat\B\|_F$ and $\|\X^* - \hat\X\|_F$:
Since $\sum_k \|\bm{M} \b^*} %{\tilde\b^*_k\|^2 = \|\bm{M} \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*\|_F^2 \le \|\bm{M}\|_F^2 \|\B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*\|^2 =\|\bm{M}\|_F^2 {\sigma_{\max}^*}^2$, we can use the first bound from \eqref{bhatk_bnd} to conclude that
\begin{align*}
\|\bm{G} - \hat\B\|_F^2 & = \sum_k \|\bm{g}_k - \hat\b_k\|^2 \\
& \le 0.4^2 \sum_k \| (\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\|^2
\\&= 0.4^2 \| (\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^* \|_F^2
\le 0.4^2 \delta_t^2 {\sigma_{\max}^*}^2
\end{align*}
and, similarly,
\begin{align*}
\|\X^* - \hat\X\|_F^2 &\le \sum_k \|\bm{g}_k - \hat\b_k\|^2 + \sum_k \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\|^2 \\&\le (0.4^2 +1^2) \delta_t^2 {\sigma_{\max}^*}^2
\end{align*}
Incoherence of $\hat\b_k$'s: Using the bound on $\|\hat\b_k - \bm{g}_k\|$, and using $\|\bm{g}_k\| \le \|\b^*} %{\tilde\b^*_k\|$ and the right incoherence assumption,
\begin{align*}
\|\hat\b_k\| & = \| (\hat\b_k - \bm{g}_k + \bm{g}_k) \| \le (1 + 0.4 \delta_t) \|\b^*} %{\tilde\b^*_k\| \le 1.04 {\sigma_{\max}^*} \sqrt{r/q}.
\end{align*}
Lower and Upper Bounds on $\sigma_i(\hat\B)$): Using the bound on $\|\bm{G} - \hat\B\|_F$ and using $\text{{SubsDist}}({\bm{U}},\U^*{}) \le \delta_t < c/\kappa$,
\begin{align*}
\sigma_{\min}(\hat\B) &\geq \sigma_{\min}(\bm{G}) - \|\bm{G} - \hat\B\| \\
& \geq \sigma_{\min}({\bm{U}}^\top\U^*{}) \sigma_{\min}( \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^* ) - \|\bm{G} - \hat\B\|_F \\
& \ge \sqrt{1- \|\U^*{}_\perp{}^\top {\bm{U}}\|^2} {\sigma_{\min}^*} - 0.4 \delta_t {\sigma_{\max}^*} \\
& \ge \sqrt{1- \delta_t^2} {\sigma_{\min}^*} -0.4 \delta_t {\sigma_{\max}^*} \ge 0.9 {\sigma_{\min}^*}
\end{align*}
since we assumed $\delta_t \le \delta_0 < 0.1/ \kappa$.
Similarly,
\begin{align*}
\|\hat\B\|= \sigma_{\max}(\hat\B) & \leq \sigma_{\max}({\bm{U}}^\top\U^*{}) \sigma_{\max}( \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^* ) + \|\bm{G} - \hat\B\|_F \\
& \le {\sigma_{\max}^*} + 0.4 \delta_t {\sigma_{\max}^*} \le 1.1 {\sigma_{\max}^*}
\end{align*}
\subsection{Proof of GD iterations' lemmas: Proof of Lemma \ref{B_lemma}, first part} \label{proof_B_lemma_part1}
We bound $\|\bm{g}_k - \hat\b_{k}\|$ here.
Recall that $\bm{g}_k = {\bm{U}}^\top \x^*_k$.
Since $\bm{y}_k = \A_k\x^*_{k} = \A_k{\bm{U}}\U{}^\top\x^*_{k} + \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}$, therefore
\begin{align*}
\hat\b_{k} &= \left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}({\bm{U}}{}^\top\A_k{}^\top)\A_k{\bm{U}}\U{}^\top\x^*_{k} \\&\qquad+ \left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}({\bm{U}}{}^\top\A_k{}^\top) \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k},\\
&=\left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}\left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right){\bm{U}}{}^\top\x^*_{k} \\&\qquad+ \left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}({\bm{U}}{}^\top\A_k{}^\top) \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k},\\
&=\bm{g}_k + \left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}({\bm{U}}{}^\top\A_k{}^\top) \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}.
\end{align*}
Thus,
\begin{align}
\|\hat\b_{k} - \bm{g}_k\| &\leq \|\left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}\| \nonumber\\& \qquad\times ~\|{\bm{U}}{}^\top\A_k{}^\top \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|.
\label{gk_bhatk_bnd}
\end{align}
Using standard results from \cite{versh_book}, one can show the following:
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item
W.p. $\ge 1-q\exp\left(r-cm\right)$, for all $k\in[q]$, $\min_{\bm{w} \in \S_r} \sum_i \big|\a_{ki}{}^\top{\bm{U}}\bm{w}\big|^2 \ge 0.7 m$ and so
\begin{align*}
\|\left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}}\right)^{-1}\| &= \frac{1}{\sigma_{\min}\left({\bm{U}}{}^\top\A_k{}^\top\A_k{\bm{U}} \right)} \\&= \frac{1}{\min_{\bm{w}\in\S_{r}} \sum_i \langle {\bm{U}}^\top \a_{ki} , \bm{w} \rangle^2 }\\& \le \frac{1}{0.7 m}
\end{align*}
\item W.p. at least $1-q\exp(r-cm)$, $ \forall k\in[q]$,
\[
\|{\bm{U}}{}^\top\A_k{}^\top \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\| \leq 0.15 m \| (\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|
\]
\een
Combining the above two bounds and \eqref{gk_bhatk_bnd}, w.p. at least $1 - 2\exp(\log q + r - c m)$, $\forall k\in[q]$,
\[
\|\bm{g}_k - \hat\b_{k}\| \leq 0.4 \|\left(\bm{I}_n-{\bm{U}}\U^\top\right)\U^*{}\b^*} %{\tilde\b^*_k\|.
\]
This completes the proof. We explain next how to get the above two bounds.
The first bound above follows by a restatement of Theorem 4.6.1 of \cite{versh_book}. Or, it follows more directly by using $\mathbb{E}[\sum_i \big|\a_{ki}{}^\top{\bm{U}}\bm{w}\big|^2] = m$, applying the sub-exponential Bernstein inequality \cite[Theorem 2.8.1]{vershynin} to bound the deviation from this mean, and then applying Proposition \ref{epsnet_MWsquared} with $n \equiv 1, r \equiv r$ (epsilon net argument).
The second bound is obtained as follows. Notice that
\begin{align*}
&\|{\bm{U}}{}^\top\A_k{}^\top \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\| \\
&\qquad= \max_{\bm{w}\in\S_{r}} \bm{w}{}^\top{\bm{U}}{}^\top\A_k{}^\top \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k} \\&\qquad= \max_{\bm{w}\in\S_{r}} \sum_i (\a_{ki}{}^\top{\bm{U}}\bm{w})(\a_{ki}{}^\top(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k} )
\end{align*}
Clearly $\mathbb{E}\left[ {\bm{U}}{}^\top\A_k{}^\top \A_k(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\right] = {\bm{U}}{}^\top (\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k} = 0$.
Moreover, the summands are products of sub-Gaussian r.v.s and are thus sub-exponential. Also, the different summands are mutually independent and zero mean.
Applying sub-exponential Bernstein with $t = \epsilon_0 m \|(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|$ for a fixed $\bm{w}\in\S_{r}$,
$$ | \sum_i (\a_{ki}{}^\top{\bm{U}}\bm{w})(\a_{ki}{}^\top(\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k} ) | \leq \epsilon_0 m \| (\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|$$
w.p. at least $1-\exp (-c\epsilon_0^2m )$.
Setting $\epsilon_0 = 0.1$, this implies that the above is bounded by $0.1 m \| (\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|$ w.p. at least $1-\exp (-c m)$. By Proposition \ref{epsnet_MW} with $n \equiv 1, r \equiv r$, the above is bounded by $0.12 m \| (\bm{I}-{\bm{U}}\U{}^\top)\x^*_{k}\|$ for all $\bm{w} \in \S_r$ w.p. at least $1-\exp(r -c m)$. Using a union bound over all $q$ columns, the bound holds for all $q$ columns w.p. at least $1- q \exp(r -c m)$.
\subsection{Proof of Initialization lemmas/facts: Proof of Lemma \ref{Wedinlemma}} \label{Wedinlemma_proof}
To see why \eqref{X0} holds, it suffices to show that $\mathbb{E}[(\hat\X_0)_k |\alpha] = \x^*_k \beta_k(\alpha)$ for each $k$. The easiest way to see this is to express $\x^*_k = \|\x^*_k\| {\bm{Q}}_k \bm{e}_1$ where ${\bm{Q}}_k$ is an $n \times n$ unitary matrix with first column $\x^*_k/\|\x^*_k\|$; and to use the fact that $\tilde\a_{ki}:={\bm{Q}}_k^\top \a_{ki}$ has the same distribution as $\a_{ki}$, both are ${\cal{N}}(0,\bm{I}_n)$.
Using ${\bm{Q}}_k {\bm{Q}}_k^\top = \bm{I}$, $(\hat\X_0)_k = (1/m) \sum_i {\bm{Q}}_k {\bm{Q}}_k^\top \a_{ki} \a_{ki}^\top \|\x^*_k\| {\bm{Q}}_k \bm{e}_1 \mathbbm{1}_{ \|\x^*_k\| |\a_{ki}^\top {\bm{Q}}_k \bm{e}_1 | \le \sqrt{\alpha} } = (1/m) \sum_i {\bm{Q}}_k \|\x^*_k\| \tilde\a_{ki} \tilde\a_{ki} (1) \mathbbm{1}_{ |\tilde\a_{ki}(1)| \le \sqrt{\alpha}/\|\x^*_k\\|}$. Thus $\mathbb{E}[((\hat\X_0)_k] =(1/m) m {\bm{Q}}_k \|\x^*_k\| \bm{e}_1 \mathbb{E}[ \zeta^2 \mathbbm{1}_{|\zeta| < \sqrt{\alpha}/\|\x^*_k\\|} ]$.
This follows because $\mathbb{E}[\a \a(1) \mathbbm{1}_{ |\a(1)| < \beta } = \bm{e}_1 \mathbb{E}[\a(1)^2 \mathbbm{1}_{ |\a(1)| < \beta }]$.
Recall that $\tilde{C} = 9 \kappa^2 \mu^2$ and $\tilde{c} = c/\tilde{C}$ for a $c<1$.
Recall also that $\X^* \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} {\bm\Sigma^*} {\V^*}} %{{\B^*}$ and $\mathbb{E}[\hat\X_{0}|\alpha] \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} \check{\bm\Sigma^*} \check{\V}}%{\check{\B}$. Thus, using \eqref{X0}, $\check{\bm\Sigma^*} = {\bm\Sigma^*} {\V^*}} %{{\B^*} {\bm{D}} \check{\V}}%{\check{\B}{}^\top$. Hence,
\begin{align*}
\sigma_r(\mathbb{E}[\hat\X_{0}|\alpha]) &= \sigma_{\min}(\check{\bm\Sigma^*}) \\&=\sigma_{\min}({\bm\Sigma^*} {\V^*}} %{{\B^*} {\bm{D}} \check{\V}}%{\check{\B}{}^\top) \\&\ge \sigma_{\min}({\bm\Sigma^*})\sigma_{\min}({\V^*}} %{{\B^*})\sigma_{\min}({\bm{D}})\sigma_{\min}(\check{\V}}%{\check{\B}{}^\top) \\&= {\sigma_{\min}^*} \cdot 1 \cdot (\min_k \beta_k (\alpha)) \cdot 1
\end{align*}
Also, $\sigma_{r+1}(\mathbb{E}[\hat\X_{0}]) = 0$ since it is a rank $r$ matrix. Thus, using Wedin's $\sin \Theta$ theorem for the Frobenius norm subspace distance $\text{{SubsDist}}$ \cite{wedin,spectral_init_review}[Theorem 2.3.1, second row] (specified in Theorem \ref{Wedin_sintheta} above) applied with $\bm{M} \equiv \hat\X_0$, $\bm{M}^* \equiv \mathbb{E}[\hat\X_{0}]$ we get \eqref{Wedin_main}.
\subsection{Proof of Initialization lemmas and facts: Proof of Lemma \ref{init_terms_bnd}} \label{init_terms_bnd_proof}
\begin{proof}[Proof of first part of Lemma \ref{init_terms_bnd}]
The proof involves an application of the sub-Gaussian Hoeffding inequality, Theorem 2.6.2 of \cite{versh_book}, followed by an epsilon-net argument. The application of sub-Gaussian Hoeffding uses conditioning on $\alpha$ for $\alpha \in \mathcal{E}$.
For $\alpha \in \mathcal{E}$, $\alpha \leq \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F/\sqrt{q}$ and this helps get a simple probability bound. Since $\alpha$ is independent of all $\a_{ki}, \bm{y}_{ki}$'s used in defining $\hat\X_0$, the conditioning does not change anything else in our proof. For example, the different summands are mutually independent even conditioned on it.
We have,
\[
\|\hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha]\| = \max_{\bm{z}\in\S_n, \bm{w}\in\S_q} \langle \hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha], ~\bm{z}\bm{w}{}^\top\rangle.
\]
For a fixed $\bm{z}\in\S_n, \bm{w}\in\S_q$, we have
\begin{align*}
&\langle \hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha], ~\bm{z}\bm{w}{}^\top\rangle \\&\qquad= \frac{1}{m} \sum_{ki} \bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\mathbbm{1}_{ \{|\bm{y}_{ki}|^2 \leq \alpha \} } \\&\qquad - \mathbb{E}\left[\bm{w}(k)\bm{y}_{ki}(\a_{ki}{}^\top\bm{z})\mathbbm{1}_{ \{|\bm{y}_{ki}|^2 \leq \alpha \} } \right] .
\end{align*}
The summands are mutually independent, zero mean sub-Gaussian r.v.s with sub-Gaussian norm $K_{ki} \le C |\bm{w}(k)| \sqrt{\alpha} / m$. For $\alpha \in \mathcal{E}$, $\alpha \leq \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F/m \sqrt{q}$.
Let $t=\epsilon_1 \|\X^*\|_F$. Then, for any $\alpha \in \mathcal{E}$,
\[
\frac{t^2}{\sum_{ki} K_{ki}^2} \geq \frac{\epsilon_1^2\|\X^*\|_F^2}{\sum_{ki} \tilde{C}(1+\epsilon_1) \bm{w}(k)^2 \|\X^*\|_F^2/m^2 q} \geq \frac{\epsilon_1^2mq}{C\mu^2\kappa^2}
\]
since $\sum_k \bm{w}(k)^2 = \|\bm{w}\|^2 = 1$.
Thus, for a fixed $\bm{z}\in \S_n, \bm{w}\in \S_q$, by sub-Gaussian Hoeffding, we conclude that, conditioned on $\alpha$, for any $\alpha \in \mathcal{E}$, w.p. at least $1-\exp\left[-c\epsilon_1^2mq/\mu^2\kappa^2\right]$,
\[
\langle \hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha], ~\bm{z}\bm{w}{}^\top\rangle \leq C \epsilon_1 \|\X^*\|_F.
\]
The rest of the proof follows by a standard epsilon net argument summarized in Proposition \ref{epsnet_Mwz}. Applying it, conditioned on $\alpha$, for any $\alpha \in \mathcal{E}$,
w.p. at least $1-\exp\left[ (n+q) -c\epsilon_1^2mq/\mu^2\kappa^2\right]$, $\max_{\bm{z}\in\S_n, \bm{w}\in\S_q} \langle \hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha], ~\bm{z}\bm{w}{}^\top\rangle \leq 1.4 C \epsilon_1 \|\X^*\|_F.$
\end{proof}
\begin{proof}[Proof of second part of Lemma \ref{init_terms_bnd}]
We have
\[
\|\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{}\|_F =
\max_{{\bm{W}}\in\S_{qr}} \langle {\bm{W}}, \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{} \rangle
\]
For a fixed ${\bm{W}} \in \S_{qr}$,
\begin{align*}
&\langle {\bm{W}}, \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{} \rangle \\&\qquad= \mathrm{trace}\left({\bm{W}}{}^\top \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{}\right)
\\&\qquad= \frac{1}{m}\sum_{ki} \left( \bm{y}_{ki}(\a_{ki}{}^\top\U^*{}\bm{w}_k)\mathbbm{1}_{\left\{|\bm{y}_{ki}|^2 \leq \alpha \right\} } - \mathbb{E}[.] \right)
\end{align*}
Conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$, the summands are independent zero mean sub-Gaussian r.v.s with subGaussian norm $K_{ki} \leq \sqrt{\alpha} \|\bm{w}_k\|/m \le \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F\|\bm{w}_k\|/m\sqrt{q}$. Thus,
\[
\sum_{ki} K_{ki}^2 \le m \tilde{C}(1+ \epsilon_1) \|{\bm{W}}\|_F^2 \|\X^*\|_F^2 / m^2q = \tilde{C} \|\X^*\|_F^2 / mq
\]
Applying the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book}, for a fixed ${\bm{W}} \in \S_{qr}$, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$, w.p. $1-\exp\left[-\epsilon_1^2mq/C\mu^2\kappa^2\right]$,
\[
\mathrm{trace}\left({\bm{W}}{}^\top\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{} \right) \leq \epsilon_1 \|\X^*\|_F.
\]
The rest of the proof follows by a standard epsilon net argument summarized in Proposition \ref{epsnet_MW}.
Applying Proposition \ref{epsnet_MW}, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$, w.p. at least $1-\exp\left[qr -c \epsilon_1^2mq/ \mu^2\kappa^2\right]$, $\max_{{\bm{W}} \in \S_{qr}} \mathrm{trace}\left({\bm{W}}{}^\top\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{} \right) < 1.2 \epsilon_1 \|\X^*\|_F$.
\end{proof}
\begin{proof}[Proof of third part of Lemma \ref{init_terms_bnd}]
We have
\[
\|\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top\|_F = \max_{{\bm{W}}\in\S_{nr}} \langle \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle.
\]
For a fixed ${\bm{W}}\in\S_{nr}$ we have,
\begin{align*}
&\langle \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle \\&\qquad= \frac{1}{m}\sum_{ki} \left( \bm{y}_{ki} (\a_{ki}{}^\top{\bm{W}}\check{\v}}%{\check{\b}_{k})\mathbbm{1}_{\left\{ |\bm{y}_{ki}|^2 \leq \alpha \right\} } - \mathbb{E}[.]\right)
\end{align*}
where $\mathbb{E}[.]$ is the expected value of the first term.
Conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$, the summands are independent, zero mean, sub-Gaussian r.v.s with subGaussian norm $K_{ki} \leq C \sqrt\alpha \|{\bm{W}}\check{\v}}%{\check{\b}_k\| \leq C \sqrt{\tilde{C}(1+\epsilon_1)}\|\X^*\|_F\|{\bm{W}}\check{\v}}%{\check{\b}_k\|/m\sqrt{q}$. Thus, by applying the sub-Gaussian Hoeffding inequality Theorem 2.6.2 of \cite{versh_book},
with $t=\epsilon_1 \|\X^*\|_F$, and using $\|{\bm{W}}\check{\V}}%{\check{\B}\|_F = 1$ (holds since $\check{\V}}%{\check{\B}$ contains orthormal rows which are right singular vectors of $\mathbb{E}[{\bm{X}}_0|\alpha]$), conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$,
we will get that,
\[
\frac{t^2}{\sum_{ki} K_{ki}^2} \geq \frac{m^2\epsilon_1^2 \|\X^*\|_F^2}{\sum_{ki} \tilde{C}(1+\epsilon_1)\|\X^*\|_F^2\|{\bm{W}}\check{\v}}%{\check{\b}_{k}\|^2/q } = \frac{mq\epsilon_1^2}{C\mu^2\kappa^2},
\]
w.p. $1-\exp\left[- c\epsilon_1^2mq/(\mu^2\kappa^2)\right]$. Here we used the fact that $\check{\V}}%{\check{\B}\Bcheck{}{}^\top=\bm{I}$ and thus $\|{\bm{W}}\check{\V}}%{\check{\B}\|_F^2 = 1$.
$$
\langle \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle \leq C \epsilon_1 \|\X^*\|_F.
$$
Applying Proposition \ref{epsnet_MW}, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$, w.p. at least $1-\exp\left[nr -c\epsilon_1^2mq/(\mu^2\kappa^2)\right]$, $\max_{{\bm{W}} \in \S_{nr}} \langle \left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top,~{\bm{W}}\rangle \le 1.2 C \epsilon_1 \|\X^*\|_F.$
\end{proof}
\subsection{Proof of Initialization lemmas and facts: Proof of Facts}\label{facts_proof}
\begin{proof}[Proof of Fact \ref{sumyik_bnd}] Apply sub-exponential Bernstein. \end{proof}
\begin{proof}[Proof of Fact \ref{betak_bnd}]
Let $\gamma_k = \frac{\sqrt{\tilde{C}(1-\epsilon_1)}\|\X^*\|_F}{\sqrt{q}\|\x^*_{k}\|}$.
Since $\tilde{C} = 9\mu^2\kappa^2$ and $\|\x^*_{k}\|^2\leq \mu^2\kappa^2\|\X^*\|_F^2/q$ (Assumption \ref{right_incoh}) thus
\[
\gamma_k \geq 3.
\]
Now,
\begin{align*}
\mathbb{E}\left[ \zeta^2\mathbbm{1}_{\left\{ |\zeta| \leq \gamma_k \right\}} \right]
=& 1 - \mathbb{E}\left[ \zeta^2\mathbbm{1}_{\left\{ |\zeta| \geq \gamma_k \right\}} \right]\\
\ge & 1 - \frac{2}{\sqrt{2\pi}}\int_{3}^{\infty}z^2\exp(-z^2/2)dz \\
\geq & 1 - \frac{2e^{-1/2}}{\sqrt{\pi}}\int_{3}^{\infty}z\exp(-z^2/4)dz
\\&= 1 - \frac{2e^{-11/4}}{\sqrt{\pi}} \geq 0.92.
\end{align*}
The first inequality used $\gamma_k \ge 3$. The second used the fact that $z\exp(-z^2/4) \leq \sqrt{2e}$ for all $z\in \Re$.
\end{proof}
In all the proofs above, notice that the only thing we used about $\check{\V}}%{\check{\B}$ is the fact that its rows contain singular vectors and thus $\check{\V}}%{\check{\B} \check{\V}}%{\check{\B}^\top = \bm{I}$ and so $\sigma_r(\check{\V}}%{\check{\B}) = \sigma_1(\check{\V}}%{\check{\B}) = 1$. We never required incoherence for it
\section{Limitations of our results} \label{limitations}
Our results have three limitations: (i) the algorithm that is analyzed needs sample-splitting, even though, in numerical experiments this is not needed; (ii) our bound holds w.h.p. for a single matrix $\X^*$ satisfying Assumption \ref{right_incoh} (and not for all such matrices); and (iii) for obtaining exactly zero error, we need an infinite number of samples. We explain here the reasons why we are unable to address these issues.
We should mention here that, since all computers are finite precision, (iii) is entirely a theoretical curiosity. Also, many other results in the LR recovery literature, e.g., \cite{lowrank_altmin,fastmc,rmc_gd}, also have all these limitations.
\subsection{Need for sample-splitting} \label{samplesplit}
In Algorithm \ref{gdmin}, sample-splitting (line 3) helps ensure that the measurement matrices in each iteration for updating each of ${\bm{U}}$ and $\bm{B}$ are independent of all previous iterates: we split our sample set into $2T+1$ subsets, we use one subset for initialization of ${\bm{U}}$ and one subset each for $T$ iterations of updating $\bm{B}$ and updating ${\bm{U}}$.
This helps prove the desired error decay bound by applying the sub-exponential Bernstein inequality \cite{versh_book} which requires the summands to be mutually independent. This becomes true in our case because, conditioned on past measurement matrices, the current set of $\a_{ki}$'s are independent of the last updated values of ${\bm{U}},\bm{B}$; and the $\a_{ki}$s for different $(i,k)$ are mutually independent by definition. Thus, under the conditioning, the summands are mutually independent.
Since we prove convergence in order $\log(1/\epsilon)$ iterations, this only adds a multiplicative factor of $\log(1/\epsilon)$ in the sample complexity.
Sample-splitting and the above overall idea is a standard approach used in many older works; in fact it is assumed for most of the LRMC guarantees for solutions that do not solve a convex relaxation (are iterative algorithms) \cite{lowrank_altmin, fastmc, rmc_gd}. An exception is \cite{rpca_gd}.
There are a few commonly used approaches to avoid sample splitting. (1) One is using the leave-one-out strategy as done in \cite{pr_mc_reuse_meas}. But this means that the sample complexity dependence on $r$ worsens: the LRMC sample complexity with this approach is $(n+q)r^3$ times log factors. Also, it is not clear how to develop this approach for alternating ${\bm{U}}, \bm{B}$ updates.
(2) The second is to try to prove error decay for all matrices that are close enough to the true $\X^*$ and that satisfy the other assumptions of the guarantee. There are at least two different approaches to doing this. (2a) The first, which was used in \cite{rpca_gd}, works for LRMC since its measurements are bounded and symmetric: the authors are able to utilize i.i.d. Bernoulli sampling and left and right singular vectors' incoherence to prove key probabilistic bounds for all matrices of the form ${\bm{U}} {\bm{V}}$ with ${\bm{U}},{\bm{V}}$ both being incoherent. This does not work in our case because our measurements are asymmetric and unbounded (which means for example that $\bm{y}_{ki}$ times its estimate is heavier-tailed than $\bm{y}_{ki}$).
(2b) An alternative approach is the following overall idea, which has been successfully used for analyzing standard PR algorithms, e.g., see \cite{twf,rwf}, but does not always work for other problems. In our setting, this means the following: At iteration $t+1$, suppose that the previous estimate ${\bm{U}}_t$ satisfies $\text{{SubsDist}}({\bm{U}}_t,\U^*{}) \le \delta_t$.
We need to try to show that, for all ${\bm{U}}$ that are a subspace distance $\delta_t$ away from the true subspace, the next iterate (which is a function of ${\bm{U}}$ and of the current $\bm{A}_k,\bm{y}_k$ for all $k$) is a distance $c \delta_t$ away with a $c<1$. To be precise, for all ${\bm{U}} \in \mathcal{T}:=\{{\bm{U}}: {\bm{U}}^\top {\bm{U}} = \bm{I} \text{ and } \text{{SubsDist}}({\bm{U}},\U^*{}) \le \delta_t\}$, we need ${\bm{U}}^+({\bm{U}}) = orth({\bm{U}} - \eta \nabla_U f({\bm{U}},\bm{B}))$ to satisfy $\text{{SubsDist}}({\bm{U}}^+, \U^*{}) \le c \delta_t$ for a $c<1$. Here $orth(\bm{M})$ is a matrix with orthonormal columns spanning the same subspace as those of $\bm{M}$. Also recall that the columns of $\bm{B}$ are $\b_k:= (\bm{A}_k {\bm{U}})^\dag \bm{y}_k$ for all $k \in [q]$.
One can show this for all ${\bm{U}} \in \mathcal{T}$ by covering $\mathcal{T}$ by a net containing a finite number of points that are such that any point in $\mathcal{T}$ is with a subspace distance $0.25\delta_t$ of some point in the net, and first proving that this bound holds for all ${\bm{U}}$ in the net. The first step for proving such a bound is to bound the error in the estimates $\b_k$ for all ${\bm{U}}$ in this net. Because of the decoupled column-wise recovery of the $\b_k$'s, for {\em one} ${\bm{U}}$ in this net, the bound on $\|\b_k({\bm{U}}) - {\bm{U}}^\top \x^*_k\|$ holds w.p. $\ge 1 - q\exp(r - c m)$. This is proved in Lemma \ref{B_lemma}. If we want this bound to hold for all ${\bm{U}}$'s in the net covering $\mathcal{T}$, we will need a union bound over all points in the net. The smallest sized net to cover $\mathcal{T}$ with accuracy $\epsilon_{net}=0.25 \delta_t$ has size upper bounded by $C^{nr}$ \cite{versh_book}.
With using this, the probability lower bound becomes $1- \exp(nr + \log q + r - c m)$. For this to even just be non-negative, we need $m > C nr$ which is too large and makes our guarantee useless.
\subsection{Why we cannot prove our result for all $X^*$}
The inability to obtain a useful union bound over a net of size $C^{nr}$ explained above is also why we cannot do this.
\subsection{Why sample complexity depends on the desired final accuracy $\epsilon$}
Observe from our result that the number of samples required to achieve a certain accuracy $\epsilon$ grows as $\log(1/\epsilon)$. This means that, for the algorithm to achieve zero error, we need an infinite number of samples. We should mention that this problem is not unique to our result. It is often seen for results that use sample-splitting, e.g., \cite{lowrank_altmin, rmc_gd}. An exception is \cite{fastmc} for LRMC, where the following basic idea is used: one tries to show that after enough iterations, e.g., when the recovery error is $\epsilon_0 = 1/n$ or smaller, one can start reusing the same samples and still prove error decay. This is also the idea used in \cite{pr_mc_reuse_meas}.
Briefly, the reason we are unable to circumvent this problem using a similar idea to that of \cite{fastmc} is that our algorithm is not a regular GD or projected GD method.
To use a similar idea in our setting, we would need to proceed as follows. We use independent samples until the error is below an $\epsilon_0$ that is small enough. Pick $\epsilon_0 = 1/(\kappa^2n^2)$.
This happens after $T(\epsilon_0) = C\kappa^2 \log(n)\log(\kappa)$ iterations. Consider $t=T+1$. At this time, $\delta_{t} = \epsilon_0 = 1/(\kappa^2n^2)$. Thus, by Lemma \ref{B_lemma}, $\|\hat\b_k - {\bm{U}}^\top \x^*_k\| \lesssim (1/(\kappa^2n^2) ) \|\x^*_k\|$ and all the other bounds also hold with $\delta_t$ replaced by $\epsilon_0$.
We try to show error decay by applying Lemma \ref{algebra}. For this to work, we need to be able to show all of the following without using independence between ${\bm{U}},\bm{B}$ and the $\bm{A}_k$s: (i) upper and lower bound the eigenvalues of $\mathrm{Hess} = \sum_{ki} (\a_{ki} \otimes \b_k) (.)^\top$ as those proved earlier, (ii) bound $\|\nabla_{\bm{U}} f({\bm{U}},\bm{B})\|/m$ by $c_0 {\sigma_{\min}^*}^2$ for a small constant $c_0< 1$ (in fact even in our main proof, such a bound is sufficient since this term only appears in the denominator), and (iii) bound $\|\mathrm{Term2}\|_F/m$ by $(c_2/\kappa^2) \delta_t {\sigma_{\max}^*}^2$ with a $c_2$ sufficiently less than one.
As we explain next, (i) and (ii) can be obtained easily, but (iii) cannot. We can obtain (i) by showing that $\mathrm{Hess}$ is close to $\mathrm{Hess}^* = \sum_{ki} (\a_{ki} \otimes ({\bm{U}}^\top \U^*{} \b^*} %{\tilde\b^*_k)) (.)^\top$; and $\mathrm{Hess}^*$ can be bounded almost exactly as done in our proof earlier since $\bm{A}_k$s are independent of $\x^*_k$s. The ${\bm{U}}$ in the expression for $\mathrm{Hess}^*$ does not matter because ${\bm{U}}^\top \U^*{}$ is an $r \times r$ rotation matrix and one can take a maximum over all rotation matrices. Using the loose bounds $\|\a_{ki}\|\le 5 \sqrt{n}$ w.h.p., one can show that $\|\mathrm{Hess}^* - \mathrm{Hess}\| \le mq \max_{ki} [ \max_{{\bm{W}} \in \S_{nr}} |\a_{ki}^\top {\bm{W}} \bm{g}_k| \cdot \max_{{\bm{W}} \in \S_{nr}} |\a_{ki}^\top {\bm{W}} (\bm{g}_k - \b_k)| ] \lesssim mq \sqrt{n} \mu \sqrt{r/q} {\sigma_{\max}^*} \cdot \sqrt{n} \epsilon_0 \mu \sqrt{r/q} {\sigma_{\max}^*} \le m \mu^2 (r/n) {\sigma_{\min}^*}^2 $. Similarly, for (ii), $\sum_{ki} \|\a_{ki} \a_{ki}^\top (\x^*_k - \bm{x}_k) \b_k^\top\| \lesssim mq \cdot \sqrt{n} \cdot \sqrt{n} \cdot \epsilon_0 \cdot (\mu^2 r/q){\sigma_{\max}^*}^2 = m (\mu^2 r/n) {\sigma_{\min}^*}^2$. Using $(\mu^2 r/n) \ll 1$, claims (i) and (ii) follow.
However, proving (iii) seems to be impossible without using the fact that $\mathbb{E}[\mathrm{Term2}] = 0$. But this expected value is zero only when $\bm{A}_k$s are independent of ${\bm{U}},\bm{B}$.
\bfpara{Possible ways to prove (iii)}
For bounding $\mathrm{Term2}$ for times $t > T(\epsilon_0)$, we can try one of the following ideas. (1) Try to use Cauchy-Schwarz in a way that the projection orthogonal to $\U^*{}$ is used. There does not seem to be a way to make this work. (2) Try to use the leave-one-out strategy of \cite{pr_mc_reuse_meas} only for $t > T(\epsilon_0)$.
\section{Proving Theorem \ref{gdmin_thm}} \label{proving_mainres}
\subsection{Two key results for proving Theorem \ref{gdmin_thm} and its proof}\label{gdmin_thm_proof}
Theorem \ref{gdmin_thm} is an almost immediate consequence of the following two results.
\begin{theorem}[Initialization]
\label{init_thm}
Pick a $\delta_0 < 0.1$.
If $mq \ge C \kappa^4 \mu^2 (n + q)r^2 / \delta_0^2$ , then w.p. at least $1 - \exp(-c (n+q))$,
\[
\SE}%{\SE_F(\U^*{}, {\bm{U}}_0) \le \delta_0.
\]
\end{theorem}
\begin{proof} See Sec. \ref{init_proof} (simpler proof with sample-splitting for $\alpha$) or Appendix \ref{init_reuse_proof} (proof without sample-splitting). Proof outline is given in Sec. \ref{outline_init}. \end{proof}
\begin{theorem}[GD Descent]
\label{iters_thm}
If, at each iteration $t$, $m q \ge C \kappa^4 \mu^2 (n+q)r^2 \log \kappa$ and $m > C \max(\log q, \log n)$;
if $\SE}%{\SE_F(\U^*{}, {\bm{U}}_0) \le \delta_0 = c/\kappa^2$ for a $c \le 0.1/1.1$;
and if $\eta \le 0.5 / {\sigma_{\max}^*}^2$,
then w.p. at least $1 - (t+1) n^{-10}$,
$$
\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t+1}) \le \delta_{t+1}:= \left(1 - (\eta {\sigma_{\max}^*}^2) \tfrac{0.4}{\kappa^2}\right)^{t+1} \delta_0.
$$
If $\eta = 0.5 {\sigma_{\max}^*}^2$, this simplifies to $\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t+1}) \le (1 -0.2/\kappa^2)^{t+1} \delta_0$.
Also, with the above probability,
\[
\|(1/m)\nabla_U f({\bm{U}}_t,\bm{B}_{t+1})\| \le 1.6 \delta_t {\sigma_{\max}^*}^2.
\]
with $\delta_t$ defined in the $\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t+1})$ bound above.
\end{theorem}
Since $\delta_t$ decays exponentially with $t$, the same is also true for the gradient norm at iteration $t$, $\|(1/m)\nabla_U f({\bm{U}}_t,\bm{B}_{t+1})\|$.
\begin{proof} See Sec. \ref{iters_proof}. Proof outline is given in Sec. \ref{outline_iters}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{gdmin_thm}]
The $\text{{SubsDist}}(.)$ bound is an immediate consequence of Theorems \ref{init_thm} and \ref{iters_thm}. To apply Theorem \ref{iters_thm}, we need $\delta_0 = c / \kappa^2$. By Theorem \ref{init_thm}, if $mq \ge C \kappa^6 \mu^2 (n+q)r^2$, then, w.p. at least $1-n^{-10}$, $\SE}%{\SE_F(\U^*{}, {\bm{U}}_0) \le \delta_0 = c / \kappa^2$.
With this, if, at each iteration, $mq \ge C \kappa^4 \mu^2 (n+q)r^2 \log \kappa$ and $m \ge C \max(\log q, \log n)$, then by Theorem \ref{iters_thm}, w.p. at least $1-(t+1) n^{-10}$, the stated bound on $\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t+1})$ holds.
By setting $T = C \kappa^2 \log(1/\epsilon)$ in this, we can guarantee $\left(1 - \tfrac{c_1}{\kappa^2}\right)^T \le \epsilon$. This proves the $\text{{SubsDist}}({\bm{U}}_T, \U^*{})$ bound.
The bounds on $\|\hat\x_{k}-\x^*_{k}\|$ and $\|\hat\X - \X^*\|_F$ follow by Lemma \ref{B_lemma} given in Sec. \ref{iters_proof}.%
\end{proof}
\subsection{Proof outline (and novelty) for Theorem \ref{iters_thm}}\label{outline_iters}
For proving exponential error decay, we need to show this: at iteration $t$, if $\text{{SubsDist}}({\bm{U}},\U^*{}) \le \delta_t$ with $\delta_t < \delta_0 = c/\kappa^2$. Then, $\text{{SubsDist}}({\bm{U}}^+,\U^*{}) \le c \delta_t$ for a $c<1$. We explain how to do this next.
Suppose that, at iteration $t$, $\text{{SubsDist}}({\bm{U}},\U^*{}) \le \delta_t < \delta_0 = 0.1/\kappa^2$.
\bfpara{Analyzing the minimization step for updating $\bm{B}$ (Lemma \ref{B_lemma})}
Recall from Algorithm \ref{gdmin} that $\b_k = (\bm{A}_k {\bm{U}})^\dag \bm{y}_k$, $\bm{x}_k = {\bm{U}} \b_k$, and $\x^*_k = \U^*{} \b^*} %{\tilde\b^*_k$. Using standard results from \cite{versh_book}, we can show that the estimates $\hat\b_k$ satisfy $\|\hat\b_k - {\bm{U}}^\top \x^*_k\| \le 0.4 \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\|$. This then implies that (i) $\hat\b_k$'s are incoherent, i.e., $\|\hat\b_k\| \le 1.1\mu {\sigma_{\max}^*} \sqrt{r/q}$; and (ii) $\|\hat\x_k - \x^*_k\| \le 1.4 \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\| \le 1.4\delta_t \max_k \|\x^*_k\|$, i.e., we can get the desired column-wise error bound.
Also (iii) $\|\hat\X - \X^*\|_F \le 1.4 \delta_t {\sigma_{\max}^*}$ (notice this bound does not contain $r$). We get this as follows:
\begin{align*}
\|\hat\X - \X^*\|_F
& = \sqrt{\sum_k \|\hat\x_k - \x^*_k\|^2} \\
& \le \sqrt{1.4^2\sum_k \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \b^*} %{\tilde\b^*_k\|^2} \\
& = 1.4 \|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^* \|_F \\
& \le 1.4\|(\bm{I} - {\bm{U}} {\bm{U}}^\top) \U^*{} \|_F {\sigma_{\max}^*}
\end{align*}
Similarly, $\|\hat\B - {\bm{U}}^\top \X^*\|_F \le 0.4 \delta_t {\sigma_{\max}^*}.$
(iv) Using Weyl's inequality and $\delta_t <0.1/\kappa^2$, this then implies that $\sigma_{\max}(\bm{B}) \le 1.1 {\sigma_{\max}^*}$ and $\sigma_{\min}(\bm{B}) \ge 0.9 {\sigma_{\min}^*}$.
\bfpara{Bounding $\text{{SubsDist}}({\bm{U}}^+,\U^*{})$ by a novel use of fundamental theorem of calculus (Lemma \ref{algebra})}
Recall from Algorithm \ref{gdmin} that $\hat\U^+ = \hat\U - (\eta/m) \nabla_U f({\bm{U}},\bm{B})$ and $\hat\U^+ \overset{\mathrm{QR}}=} %{\stackrel{EVD}{=} {\bm{U}}^+ {\bm{R}}^+$.
We bound $\text{{SubsDist}}({\bm{U}}^+, \U^*{})$ using the fundamental theorem of calculus \cite[Chapter XIII, Theorem 4.2]{lan93},\cite{pr_mc_reuse_meas}, summarized in Theorem \ref{funda_calc}. The use of this result is motivated by its use in \cite{pr_mc_reuse_meas}, and many earlier works, where it is used in a standard way: to bound the Euclidean norm error $\|\bm{x} - \x^*\|$ for standard GD to solve the PR problem for recovering a single vector $\x^*$. Thus, at the true solution $\bm{x}=\x^*$, the gradient of the cost function was zero. In our case, there are two differences: (i) we need to bound the subspace distance error, and (ii) our algorithm is not standard GD; in particular, this means that $\nabla_U f(\U^*{} \U^*{}^\top {\bm{U}}, \bm{B}) \neq 0$.
To deal with (i) and (ii), we proceed as follows.
We first bound $\|(\bm{I} - \U^*{} \U^*{}^\top) \hat\U^+\|_F$. To do this, we apply Theorem \ref{funda_calc} on vectorized $\nabla_U f({\bm{U}},\bm{B})$ with the pivot being vectorized $\nabla_U f(\U^*{} \U^*{}^\top {\bm{U}}, \bm{B})$, and use this in the equation for $\hat\U^+$. Next, we project both sides of this expression orthogonal to $\U^*{}$ followed by some careful linear algebra. Notice here that $\nabla_U f(\U^*{} \U^*{}^\top {\bm{U}}, \bm{B}) \neq 0$, because $\bm{B} \neq \B^*} \newcommand{\b^*} %{\tilde\b^*}{\b^*$. Because of this, we get an extra term, $\mathrm{Term2}:=(\bm{I} - \U^*{} \U^*{}^\top) \nabla_U f(\U^*{} \U^*{}^\top {\bm{U}}, \bm{B})$, in our bound other than the usual term containing the Hessian. We are able to bound it by $\epsilon \delta_t {\sigma_{\max}^*}^2$ for any constant small enough $\epsilon$, by realizing that $\mathbb{E}[\mathrm{Term2}]=0$ (conditioned on past measurements), and that its summands are {\em nice-enough} subexponentials.
Next, we bound $\text{{SubsDist}}(\U^*{}, {\bm{U}}^+)$ by using
\begin{align*}
\text{{SubsDist}}(\U^*{}, {\bm{U}}^+)
& \le \|(\bm{I} - \U^*{} \U^*{}^\top) \hat\U^+\|_F \|({\bm{R}}^+)^{-1}\| \\
& = \frac{\|(\bm{I} - \U^*{} \U^*{}^\top) \hat\U^+\|_F}{ \sigma_{\min}(\hat\U^+) }
\end{align*}
and
$\sigma_{\min}(\hat\U^+)= \sigma_{\min}({\bm{U}} - (\eta/m) \nabla_{\bm{U}} f({\bm{U}},\bm{B})) \ge 1 - (\eta/m) \|\nabla_{\bm{U}} f({\bm{U}},\bm{B})\|$.
\bfpara{Bounding the terms in the $\text{{SubsDist}}(\U^*{},{\bm{U}}^+)$ bound (Lemma \ref{terms_bnds})}
Consider $\|\nabla_{\bm{U}} f({\bm{U}},\bm{B})\|$.
Using Lemma \ref{B_lemma}, it can be shown that, for unit vectors $\bm{w},\bm{z}$, the maximum sub-exponential norm of any summand of $\bm{w}^\top \nabla_{\bm{U}} f({\bm{U}},\bm{B}) \bm{z}$
is bounded by $\|\hat\x_k - \x^*_k\| \cdot \|\b_k\| \le 1.1 \mu^2 {\sigma_{\max}^*}^2 \delta_t (r/q)$. Observe that we get this (sufficiently small) bound because of the extra $\b_k^\top$ term in the summands of $\nabla_{\bm{U}} f({\bm{U}},\bm{B})$ compared to those in $\nabla_{\bm{X}} \tilde{f}({\bm{X}})$.
This, along with using the sub-exponential Bernstein inequality \cite{versh_book} followed by a standard epsilon-net argument, and bounding $\|\mathbb{E}[\nabla_U f]\|$ using $\|\mathbb{E}[\nabla_U f]\|= \|m(\hat\X - \X^*) \bm{B}^\top\| \le m \delta_t {\sigma_{\max}^*}^2$ (by Lemma \ref{B_lemma}), helps guarantee that $\|\nabla_U f\| \lesssim 2 m \delta_t {\sigma_{\max}^*}^2$ w.h.p. as long as $mq \gtrsim (n+q)r^2$.
We bound $\|\mathrm{Term2}\|_F$ using similar ideas and the key fact that $\mathbb{E}[\mathrm{Term2}]=0$. This is true because of sample-splitting. We upper and lower bound the eigenvalues of the Hessian, $\mathrm{Hess}$, using similar ideas and the following: for a unit vector $\bm{w}$ of length $nr$ and its rearranged unit Frobenius norm matrix ${\bm{W}}$ of size $n \times r$, $\mathbb{E}[\bm{w}^\top \mathrm{Hess} \ \bm{w} ] = \mathbb{E}[ \sum_{ki} ( \a_{ki}{}^\top {\bm{W}} \b_k)^2] = m \|{\bm{W}} \bm{B}\|_F^2$. Using the bounds on $\sigma_i(\bm{B})$ from Lemma \ref{B_lemma}, this can be upper and lower bounded.
\subsection{Lemmas for proving GD descent Theorem \ref{iters_thm} and its proof} \label{iters_proof}
Let ${\bm{U}} \equiv {\bm{U}}_{t}$, $\hat\B \equiv \hat\B_{t+1}$. The proof follows using the following 3 lemmas.
\begin{lemma}[Error bound on $\bm{B}$ and its implications]
Let $U \equiv {\bm{U}}_t$, $\bm{B} \equiv \bm{B}_{t+1}$, and
$$\bm{g}_k: = {\bm{U}}^\top \x^*_k.$$
Assume that $\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t}) \le \delta_t$ with $\delta_t < \delta_0 = c/\kappa^2$ (this bound on $\delta_t$ is needed for the second part of this lemma).
Then, w.p. $\ge 1 - q \exp(r - c m)$,
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item
\begin{align}
\|\bm{g}_k - \hat\b_k \| & \leq 0.4 \|\left(\bm{I}_n-{\bm{U}}\U^\top \right)\U^*{}\b^*} %{\tilde\b^*_k\|
\label{bhatk_bnd}
\end{align}
\item This in turn implies all of the following.
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item $ \|\hat\x_{k}-\x^*_{k}\| \leq 1.4 \|\left(\bm{I}-{\bm{U}}\U^\top \right)\U^*{}\b^*} %{\tilde\b^*_k\|$
\item $\|\bm{G}-\hat\B\|_F \leq 0.4 \delta_t {\sigma_{\max}^*}$ and $\|\X^*-\hat\X\|_F \le \sqrt{1.16} \delta_t {\sigma_{\max}^*}$,
\item $\|\bm{g}_k - \hat\b_k \| \leq 0.4 \delta_t \|\b^*} %{\tilde\b^*_k\|$ and $ \|\hat\x_{k}-\x^*_{k}\| \le 1.4 \delta_t \|\x^*_k\|$,
\item $\| \U^*{}{}^\top{\bm{U}} \hat\b_{k} - \b^*} %{\tilde\b^*_k \| \le 2.4 \delta_t \|\b^*} %{\tilde\b^*_k\| $,
\item $\|\hat\b_k\| \le 1.1 \mu {\sigma_{\max}^*} \sqrt{r/q}$.
\item $\sigma_{\min}(\hat\B) \ge 0.9 {\sigma_{\min}^*}$ and $\sigma_{\max}(\hat\B) \le 1.1 {\sigma_{\max}^*}$,
\een
\een
\label{B_lemma}
\end{lemma}
\begin{proof} See Sec. \ref{B_lemma_proof}. \end{proof}
\begin{lemma}\label{algebra}
Let ${\bm{U}} \equiv {\bm{U}}_{t}$, $\hat\B \equiv \hat\B_{t+1}$. Let $\otimes$ denote the Kronecker product. We have
\begin{align*}
&\SE}%{\SE_F({\bm{U}}_{t+1},\U^*{}) \\
&\qquad \le \dfrac{ \|\bm{I}_{nr} - (\eta/m) \mathrm{Hess}\| \cdot \SE}%{\SE_F(\U^*{},{\bm{U}}) + (\eta/m) \|\mathrm{Term2}\|_F }{ 1 - (\eta/m) \|\mathrm{GradU}\|},
\end{align*}
where,
\begin{align*}
\mathrm{GradU} & := \nabla_{\bm{U}} f({\bm{U}},\hat\B) = \sum_{ki} (\bm{y}_{ki} - \a_{ki}{}^\top {\bm{U}} \hat\b_k) \a_{ki} \hat\b_k{}^\top
\\
\mathrm{Term2} & := (\bm{I} - \U^*{} \U^*{}^\top) \nabla_{\bm{U}} f((\U^*{}\Ustar^\top {\bm{U}}),\hat\B) \\
& = (\bm{I} - \U^*{} \U^*{}^\top) \sum_{ki} (\bm{y}_{ki} - \a_{ki}{}^\top \U^*{}\Ustar^\top{\bm{U}} \hat\b_k) \a_{ki} \hat\b_k{}^\top
\\
\mathrm{Hess} & := \sum_{ki} (\a_{ki} \otimes \hat\b_k) (\a_{ki} \otimes \hat\b_k)^\top
\end{align*}
\end{lemma}
\begin{proof} See Sec. \ref{algebra_proof} \end{proof}
\begin{lemma}
\label{terms_bnds}
Assume $\SE}%{\SE_F(\U^*{}, {\bm{U}}) \le \delta_t < \delta_0 =c/\kappa^2$. Then,%
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item w.p. at least $1 - \exp( (n+r) - c mq \epsilon_1^2 / r \mu^2 ) - \exp(\log q + r - c m)$,
\[
\|\mathrm{GradU}\| \le 1.5(1.1+ \epsilon_1) m \delta_t {\sigma_{\max}^*}^2 ;
\]
\item w.p. at least $1 - \exp( nr - c mq \epsilon_2^2 / r \mu^2 ) - \exp(\log q + r - c m)$,
\[
\|\mathrm{Term2}\|_F \le 1.1 m \epsilon_2 \delta_t {\sigma_{\max}^*}^2 ;
\]
\item w.p. at least $1 - \exp( nr \log \kappa - c mq \epsilon_3^2 / r \kappa^4 \mu^2 ) - \exp(\log q + r - c m)$,%
{\small
\begin{align*}
m (0.65- 1.2 \epsilon_3) {\sigma_{\min}^*}^2 &\le \lambda_{\min}( \mathrm{Hess} ) \\
&\le \lambda_{\max}( \mathrm{Hess} ) \le m (1.1 + \epsilon_3) {\sigma_{\max}^*}^2.
\end{align*}
}
\een
\end{lemma}
\begin{proof} See Sec. \ref{terms_bnds_proof}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{iters_thm}]
The proof follows by induction. Base case for $t=0$ is true by assumption. Induction assumption: Assume that, w.p. at least $1-t n^{-10}$, $\SE}%{\SE_F(\U^*{}, {\bm{U}}_{t}) \le \delta_t$ with $\delta_t \le \delta_0 =c_0/\kappa^2$.
Set $ \epsilon_1 = 0.1$, $ \epsilon_3 = 0.01$, $\epsilon_2 = 0.01 / 1.1 \kappa^2$ and, $c_0 =0.1 / 1.5(1.1+0.1) $.
The upper bound on $\lambda_{\max}(\mathrm{Hess})$ and using $\eta \le 0.5 /{\sigma_{\max}^*}^2$ implies that
$
\lambda_{\min}(\bm{I}_{nr} - (\eta/m) \mathrm{Hess}) = 1 - (\eta/m) \lambda_{\max}(\mathrm{Hess}) \ge 1 - \frac{0.5 (1.1 + 0.01) m{\sigma_{\max}^*}^2}{m {\sigma_{\max}^*}^2} > 1-0.555 > 0
$
i.e. $\bm{I}_{nr} - (\eta/m) \mathrm{Hess}$ is positive definite.
Thus,
$
\| \bm{I}_{nr} - (\eta/m) \mathrm{Hess} \| = \lambda_{\max}(\bm{I}_{nr} - (\eta/m) \mathrm{Hess}) = 1 - (\eta/m) \lambda_{\min}(\mathrm{Hess}) \le 1 - (\eta/m) m (0.65- 1.2 \epsilon_3) {\sigma_{\min}^*}^2 \le 1 - (\eta {\sigma_{\max}^*}^2) 0.63 / \kappa^2 .
$
By Lemma \ref{algebra}, Lemma \ref{terms_bnds}, and the above,
w.p. at least $1 -t n^{-10} - \exp( (n+q) - c mq / r \mu^2 ) - \exp( nr - c mq / r \kappa^4 \mu^2 ) - \exp( nr \log \kappa - c mq / r \kappa^4 \mu^2) - \exp(\log q + r - c m)$,
\begin{align*}
& \SE}%{\SE_F(\U^*{}, {\bm{U}}_{t+1}) \\
& \le \dfrac{ (1 - (\eta {\sigma_{\max}^*}^2) 0.63 / \kappa^2)\cdot \delta_t + (\eta/m) 1.1 m \epsilon_2 {\sigma_{\max}^*}^2 \delta_t }{1 - (\eta/m) 1.5(1.1+\epsilon_1) m \delta_t {\sigma_{\max}^*}^2} \\
& \le \left( \frac{ 1 - (\eta {\sigma_{\max}^*}^2) 0.63/\kappa^2 + (\eta {\sigma_{\max}^*}^2) 0.01/\kappa^2 }{1 - (\eta {\sigma_{\max}^*}^2) 0.1/\kappa^2} \right) \delta_t \\
& \le \left( 1 - (\eta {\sigma_{\max}^*}^2) \frac{0.42}{ \kappa^2} \right) \delta_t
\end{align*}
The second inequality substituted the values of $\epsilon_j$'s and used $\delta_t < \delta_0= 0.1 / ( 1.5(1.1+0.1) \kappa^2)$ for its denominator term. The third inequality used $(1 - (\eta {\sigma_{\max}^*}^2) 0.1/\kappa^2)^{-1} \le (1 + (\eta {\sigma_{\max}^*}^2) 0.2/\kappa^2)$ (for $0 < x < 1$, $1/(1-x) \le 1+2x$).
By plugging in the epsilon values in the probability, the above holds w.p.
$\ge 1 - tn^{-10} - 0.2\exp( (n+q) - c mq / r \mu^2 ) - 0.2\exp( nr - c mq / r \mu^2 \kappa^4 ) - 0.2\exp( nr \log \kappa - c mq / r \mu^2 \kappa^4 ) - \exp(\log q + r - c m)$ .
If $mq \ge C \kappa^4 (n+q)r^2 \log \kappa$ and $m \ge C\max(r, \log q, \log n)$ for a $C$ large enough, then, this probability is $\ge 1- tn^{-10} - 0.2\exp( - c (n+q)) - 0.4 \exp( - c n r) - n^{-10} > 1 - (t+1) n^{-10}$.
\end{proof}
\subsection{Proof outline (and novelty) for Initialization Theorem \ref{init_thm}}\label{outline_init}
Recall that we compute ${\bm{U}}_0$ as the top $r$ left singular vectors of $\hat\X_0$ defined in \eqref{newinit} and that this is a truncated version of $\hat\X_{0,full}$. As noted there, we cannot use ${\bm{X}}_{0,full}$ because its summands are not {\em nice-enough sub-exponentials}. Truncation converts the summands into sub-Gaussian r.v.s.
For these, we can use the sub-Gaussian Hoeffding inequality \cite[Chap 2]{versh_book} which needs a small enough bound on only the squared sum of the sub-Gaussian norms of the $mq$ summands, and not on their maximum value (as needed by the sub-exponential Bernstein inequality). This is an easier requirement that gets satisfied for our problem.
Of course, truncation also means that the summands of $\hat\X_0$ are not mutually independent (each summand depends on the truncation threshold $\alpha$ which is computed using all measurements $\bm{y}_{ki}$) and that $\mathbb{E}[\hat\X_0] \neq \X^*$.
There are two ways to resolve this issue. The first and simpler approach, but one that assumes more sample-splitting is given below in Sec \ref{init_proof}. This assumes that $\alpha$ is a computed using a different independent set of measurements than those used to define the rest of $\hat\X_0$. With this, $\mathbb{E}[\hat\X_0|\alpha] = \X^* {\bm{D}}(\alpha)$, where ${\bm{D}}$ is a diagonal matrix defined below in Lemma \ref{Wedinlemma} and the summands are independent conditioned on $\alpha$. Thus, we can apply Wedin's $\sin \Theta$ theorem \cite{wedin,spectral_init_review} (given in Proposition \ref{Wedin_sintheta}) on $\hat\X_0$ and $\mathbb{E}[\hat\X_0|\alpha]$ to bound $\text{{SubsDist}}({\bm{U}}_0, \U^*{})$, followed by subGaussian Hoeffding and a standard epsilon-net argument, to bound the terms in this bound.
To avoid sample-splitting for $\alpha$, we need to significantly modify the sandwiching arguments from \cite{twf,lrpr_it} for our setting. This is done in Appendix \ref{init_reuse_proof}. In the previous works, sandwiching was used for a symmetric positive definite (p.d.) matrix. Here we need such an argument for a non-symmetric matrix. Briefly, we do this as follows. We define a matrix ${\bm{X}}_+$ that is such that the span of top $r$ left singular vectors of its expected value equals that of $\U^*{}$ and that can be shown to be close to $\hat\X_0$. ${\bm{X}}_+$ is $\hat\X_0$ with $\alpha$ replaced by $\tilde{C}(1+\epsilon) \|\X^*\|_F^2/q$.
We bound $\|\hat\X_0 - \mathbb{E}[{\bm{X}}_+]\|$ by bounding $\|{\bm{X}}_+ - \hat\X_0\|$ and $\|{\bm{X}}_+ - \mathbb{E}[{\bm{X}}_+]\|$. Bounding the latter is simple.
Bounding $\|{\bm{X}}_+ - \hat\X_0\|$ requires bounding $\bm{w}^\top ({\bm{X}}_+ - \hat\X_0) \bm{z}$ for unit vectors $\bm{w}, \bm{z}$ and this is not straightforward because its summands are not mutually independent. To deal with this, we first bound each summand by its absolute value, and then bound the indicator function term to get a new one that is non-random so that the summands of this new term are mutually independent.
But, its summands are no longer zero mean (because of taking the absolute values), and hence more work is needed to get the desired small enough bound on the expected value of this term
\subsection{Simpler proof of Theorem \ref{init_thm} that assumes independent measurements used for computing $\alpha$}\label{init_proof}
For the simpler proof given here, assume that we use a different independent set of measurements for computing $\alpha$ than those used for the rest of $\hat\X_{0}$, i.e., let
\[
\alpha = \tilde{C} \frac{\sum_{ki} (\bm{y}_{ki}^{nrmX})^2}{mq}
\]
with $\bm{y}_{ki}^{nrmX}$ independent of $\{\bm{A}_k^{(0)}, \bm{y}_k^{(0)} \}$.
With this change, it is possible to compute $\mathbb{E}[\hat\X_0|\alpha]$ easily. But, it does not affect the sample complexity order and so it does not change our theorem statement.
The proof follows by combining the two lemmas and facts given next.
\begin{lemma} \label{Wedinlemma}
Conditioned on $\alpha$, we have the following conclusions.
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item Let $\zeta$ be a scalar standard Gaussian r.v.. Define
\[
\beta_{k}(\alpha) := \mathbb{E}[\zeta^2 \mathbbm{1}_{\{\|\x^*_{k}\|^2\zeta^2 \leq \alpha\}}].
\]
Then,
\begin{align}
&\mathbb{E}[\hat\X_0|\alpha] = \X^* {\bm{D}}(\alpha), \nonumber\\
&\text{ where } {\bm{D}}(\alpha):=diagonal(\beta_k(\alpha),k \in [q])
\label{X0}
\end{align}
i.e. ${\bm{D}}(\alpha)$ is a diagonal matrix of size $q\times q$ with diagonal entries $\beta_{k}$ defined above
\item Let $\mathbb{E}[\hat\X_{0}|\alpha] = \X^* {\bm{D}}(\alpha) \overset{\mathrm{SVD}}=} %{\stackrel{EVD}{=} \U^*{} \check{\bm\Sigma^*} \check{\V}}%{\check{\B}$ be its $r$-SVD. Then,
\begin{align}\label{Wedin_main}
& \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \le \nonumber \\
& \dfrac{\sqrt{2} \max\left( \| (\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha] )^\top \U^*{} \|_F , \| (\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha] ) \check{\V}}%{\check{\B}{}^\top \|_F \right)}{{\sigma_{\min}^*} \min_k \beta_k(\alpha) - \|\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha] \|}
\end{align}
as long as the denominator is non-negative.
\een
\end{lemma}
\begin{proof} See Sec. \ref{Wedinlemma_proof} \end{proof}
\newcommand{\mathcal{E}}{\mathcal{E}}
}% {\color{red}
Define the set $\mathcal{E}$ as follows
\begin{eqnarray}
\mathcal{E}:= \left\{ \tilde{C}(1 - \epsilon_1) \frac{\|\X^*\|_F^2}{q} \le \alpha \le \tilde{C} (1 + \epsilon_1) \frac{\|\X^*\|_F^2}{q} \right\}.
\label{def_ev}
\end{eqnarray}
The following fact is an immediate consequence of sub-exponential Bernstein inequality for bounding $|\alpha - \|\X^*\|_F^2/q|$.
\begin{fact}\label{sumyik_bnd}
$\Pr(\alpha \in \mathcal{E}) \ge 1 - \exp(- \tilde{c} mq \epsilon_1^2):= 1 - p_\alpha$. Here $\tilde{c} = c/\tilde{C} = c / \kappa^2 \mu^2.$
\end{fact}
\color{black}
The next lemma bounds the terms of Lemma \ref{Wedinlemma}.
\begin{lemma} \label{init_terms_bnd}
Fix $0 < \epsilon_1 < 1$. Then,
\begin{enumerate}} \newcommand{\een}{\end{enumerate}
\item \label{Xhat0_1}
w.p. at least $1-\exp\left[(n+q)-c\epsilon_1^2mq/\mu^2\kappa^2\right]$, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$,
\[
\|\hat\X_{0} -\mathbb{E}[\hat\X_{0}|\alpha]\| \leq 1.1 \epsilon_1 \|\X^*\|_F
\]
\item
\label{Xhat0_Ustar_1}
w.p. at least $1-\exp\left[ qr - c \epsilon_1^2 mq / \mu^2\kappa^2\right]$, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$,
\[
\|\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right){}^\top\U^*{}\|_F \leq 1.1 \epsilon_1\|\X^*\|_F
\]
\item
\label{lem:init_nom_B_term2}
\label{Xhat0_Bstar_1}
w.p. at least $1-\exp\left[nr - c \epsilon_1^2mq/\mu^2\kappa^2\right]$, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$,
\[
\|\left(\hat\X_{0} - \mathbb{E}[\hat\X_{0}|\alpha]\right)\check{\V}}%{\check{\B}{}^\top\|_F \leq 1.1 \epsilon_1\|\X^*\|_F.
\]
\een
\end{lemma}
\begin{proof} See Sec. \ref{init_terms_bnd_proof} \end{proof}
We also need to the following fact.
\begin{fact}
\label{betak_bnd}
For any $\epsilon_1 \leq 0.1$,
$
\min_k \mathbb{E}\left[\zeta^2 \mathbbm{1}_{ \left\{ |\zeta| \leq \tilde{C} \frac{ \sqrt{1- \epsilon_1} \|\X^*\|_F }{ \sqrt{q}\|\x^*_{k}\| } \right\} } \right] \geq 0.92.
$
\end{fact}
\begin{proof}[Proof of Theorem \ref{init_thm}]
Set $\epsilon_1 = 0.4 \delta_0 / \sqrt{r} \kappa $. Define
$
p_0 = 2\exp( (n+q)- c mq \delta_0^2 / r \kappa^2 ) + 2\exp( n r - c mq \delta_0^2 / r \kappa^2 ) + 2\exp( q r - c mq \delta_0^2 / r \kappa^2 ).
$
Recall that $\Pr(\alpha \in \mathcal{E}) \ge 1 - p_\alpha$ with
$
p_\alpha = \exp(- \tilde{c} mq \epsilon_1^2) = \exp(- c mq \delta_0^2 /r \mu^2 \kappa^2 ).
$
Using Lemma \ref{init_terms_bnd}, conditioned on $\alpha$, for an $\alpha \in \mathcal{E}$,
\begin{itemize}} \newcommand{\ei}{\end{itemize}
\item w.p. at least $ 1-p_0$,
$
\|\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha]\| \le 1.1 \epsilon_1 \|\X^*\|_F = 0.44 \delta_0 {\sigma_{\min}^*},
$ and %
{\small
$
\max\left( \| (\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha])^\top \U^*{} \|_F , \| (\hat\X_0 - \mathbb{E}[\hat\X_{0}|\alpha]) \check{\V}}%{\check{\B}^\top \|_F \right) \le 0.44 \delta_0 {\sigma_{\min}^*}
$
}
\item
$\min_k \beta_k(\alpha) \ge \min_k \mathbb{E}\left[\zeta^2 \mathbbm{1}_{\{ |\zeta| \leq \tilde{C} \frac{ \sqrt{1- \epsilon_1} \|\X^*\|_F }{ \sqrt{q}\|\x^*_{k}\| } \} } \right] \ge 0.9
$
The first inequality is an immediate consequence of $\alpha \in \mathcal{E}$ and the second follows by Fact \ref{betak_bnd}.
\ei
Plugging the above bounds into \eqref{Wedin_main} of Lemma \ref{Wedinlemma}, conditioned on $\alpha$, for any $\alpha \in \mathcal{E}$, w.p. at least $ 1 - p_0$,
$\SE}%{\SE_F({\bm{U}}_0,\U^*{}) \le \frac{0.44 \delta_0}{0.9 - 0.44 \delta_0} < \delta_0$ since $\delta_0 < 0.1$. In other words,
\begin{align}
& \Pr\left( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \ge \delta_0 | \alpha \right) \le p_0 \ \text{for any $\alpha \in \mathcal{E}$}.
\label{SD_given_alpha_bnd}
\end{align}
Since (i)
$
\Pr( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \ge \delta_0 )
\le \Pr( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \ge \delta_0 \text{ and } \alpha \in \mathcal{E}) + \Pr (\alpha \notin \mathcal{E}),
$ and
(ii)
$
\Pr( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \ge \delta_0 \text{ and } \alpha \in \mathcal{E} ) \le \Pr(\alpha \in \mathcal{E}) \max_{\alpha \in \mathcal{E}}\Pr( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \le \delta_0 |\alpha ),
$
thus, using Fact \ref{sumyik_bnd} and \eqref{SD_given_alpha_bnd}, we can conclude that
\[
\Pr \left( \SE}%{\SE_F({\bm{U}}_0,\U^*{}) \ge \delta_0 \right) \le p_0 (1 - p_\alpha) + p_\alpha \le p_0 + p_\alpha
\]
Thus, for a $\delta_0 < 0.1$, $\SE}%{\SE_F({\bm{U}}_0,\U^*{}) < \delta_0$ w.p. at least $ 1- p_0 - p_\alpha = 1 - 2\exp( (n+q)- c mq \delta_0^2 / r \kappa^2 ) - 2\exp( n r - c mq \delta_0^2 / r \kappa^2 ) - 2\exp( q r - c mq \delta_0^2 / r \kappa^2 ) - \exp(- c mq \delta_0^2 /r \mu^2 \kappa^4 )$.
This is $\ge 1 - 5 \exp(-c(n+q))$ if $mq > C \kappa^2 \mu^2 (n+q)r^2 / \delta_0^2$. This finishes our proof.
\end{proof}
\section{Understanding why LRMC-style GD approaches cannot be easily analyzed for LRcCS} \label{algo_understand}
\begin{table*}[t]
\caption{\small{
Understanding why LRMC style projected-GD on ${\bm{X}}$ does not work in our case.
}}
\label{LRMC_diff}
\vspace{-0.1in}
\begin{center}
\renewcommand*{\arraystretch}{1.01}
{
\begin{tabular}{l l l} \toprule
& LRMC & Our Problem, LRcCS \\ \midrule
$\tilde{f}(\hat\X)$ & $\displaystyle \sum_{k=1}^q \sum_{j=1}^n (\bm{y}_{jk} - \delta_{jk} \hat\X_{jk})^2 $ & $\displaystyle \sum_{k=1}^q \sum_{i=1}^m (\bm{y}_{ki} - \a_{ki}^\top \hat\x_k)^2 $ \\
& $\delta_{jk} \stackrel{\mathrm{iid}}{\thicksim } Bernoulli(p)$ & $\a_{ki} \stackrel{\mathrm{iid}}{\thicksim } {\cal{N}}(0, \bm{I}_n)$ \\
\hline
$\nabla_X \tilde{f}(\hat\X)$ & $\displaystyle \sum_{k=1}^q \sum_{j=1}^n \delta_{jk}(\bm{y}_{jk} - \delta_{jk} \hat\X_{jk}) \bm{e}_j \bm{e}_k^\top$ & $\displaystyle \sum_{k=1}^q \sum_{i=1}^m (\bm{y}_{ki} - \a_{ki}^\top \hat\x_k) \a_{ki} \bm{e}_k^\top $ \\
& $\displaystyle = \sum_{k=1}^q \sum_{j=1}^n \delta_{jk}( \X^*_{jk} - \hat\X_{jk}) \bm{e}_j \bm{e}_k^\top$ & $\displaystyle = \sum_{k=1}^q \sum_{i=1}^m \a_{ki}^\top (\x^*_k - \hat\x_k) \a_{ki} \bm{e}_k^\top $ \\
$\tilde\H:= \H - \eta \nabla f(\hat\X)$ & $\displaystyle \sum_{k=1}^q \sum_{j=1}^n (1 - \frac{\delta_{jk}}{p}) \H_{jk} \bm{e}_j \bm{e}_k{}^\top$ & $\displaystyle \frac{1}{m} \sum_{k=1}^q \sum_{i=1}^m (\bm{I} - \a_{ki} \a_{ki}{}^\top ) \bm{h}_k \bm{e}_k{}^\top$ \\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-0.15in}
\end{table*}
\subsection{Gradient Descent} \label{lrmc_compare_iters}
}% {\color{red}
The iterates of a gradient descent (GD) algorithm converge when the gradient approaches zero.
Thus, in order to show its convergence, one needs to be able to bound the norm of the gradient and show that it goes to zero with iterations. In order to show fast enough convergence (reach $\epsilon$ error in order $\log(1/\epsilon)$ iterations), one further needs to show that this bound on the gradient norm decreases sufficiently with each iteration.
\color{black}
Consider projGD-X which was studied in \cite{rmc_gd} for solving LRMC. ProjGD-X iterations involve computing $\hat\X^+ \leftarrow \mathcal{P}_r( \hat\X - \nabla_{\bm{X}} \tilde{f}(\hat\X) )$, here $\mathcal{P}_r(\bm{M})$ projects its argument onto the space of rank-$r$ matrices.
To bound $\|\nabla_{\bm{X}} \tilde{f}({\bm{X}})\|$, we need to bound $|\bm{w}^\top \nabla_{\bm{X}} \tilde{f}({\bm{X}}) \bm{z}|$ for any unit norm vectors $\bm{w},\bm{z}$.
We show the cost function $\tilde{f}(\hat\X)$ and its gradient for both LRMC and LRcCS in Table \ref{LRMC_diff}. Observe that, for LRcCS, $\bm{w}^\top \nabla_{\bm{X}} \tilde{f}({\bm{X}}) \bm{z}$ is a sum of sub-exponential r.v.s with sub-exponential norms bounded by $K_e = \max_k \|\bm{w}\| \cdot \|\x^*_k - \hat\x_k\| \cdot |\bm{z}_k| \le \max_k \|\x^*_k - \hat\x_k\|$. Thus, in order to get a small enough bound on $|\bm{w}^\top \nabla_{\bm{X}} \tilde{f}({\bm{X}}) \bm{z}|$ by applying the sub-exponential Bernstein inequality \cite{versh_book}, we need a small enough bound on $\max_k \|\x^*_k - \hat\x_k\|$ (column-wise error bound).
It is not clear how to get this because the projection step introduces coupling between the different columns of the estimated matrix ${\bm{X}}$ \footnote{
Let $\H := {\bm{X}} - \X^*$, $\tilde\H:= (\hat\X - \eta \nabla f(\hat\X)) - \X^* = \H - \eta \nabla f(\hat\X)$, and $\H^+ = \hat\X^+ - \X^* = \mathcal{P}_r(\hat\X - \nabla f(\hat\X)) - \X^* = \mathcal{P}_r(\X^* + \tilde\H) - \X^* $.
To bound the LRMC projGD-X errors, one needs an entry-wise bound of the form $\|\H^+\|_{\max} \le \delta_t \|\X^*\|_{\max}$ with $\delta_t$ decaying exponentially.
We show the expressions for $\tilde\H$ in the table. For LRMC, notice that different summands of $\tilde\H$ are mutually independent and each depends on only one entry of $\H$. This fact is carefully exploited in \cite[Lemma 1]{rmc_gd} and \cite[Lemma 1]{fastmc}. By borrowing ideas from the literature on spectral statistics of Erdos-Renyi graphs \cite{ekyy}, the authors are able to obtain expressions for higher powers of $(\tilde\H \tilde\H^\top)$. These expressions help them get the desired bound under the desired sample complexity.
For LRcCS, using the gradient expression, we need a bound on $\max_k \|\bm{h}^+_k\|$ in terms of $\|\bm{h}_k\|$ in order to show its exponential decay. Since the different entries of $\tilde\H$ are not mutually independent and not bounded, the LRMC proof approach cannot be borrowed.
}.
Moreover, even if we could somehow get such a bound, in the best case, it would be proportional to $\delta_t \max_k \|\x^*_k\|$ with $\delta_t<1$ and decaying exponentially with $t$. Using Assumption \ref{right_incoh},
this would then imply that $K_e \le \delta_t \max_k \|\x^*_k\| \le \delta_t \mu \sqrt{r/q} {\sigma_{\max}^*}$. But, this is not small enough. We need it to be proportional to $\delta_t (r/q)$ in order to be able to bound the gradient norm under the desired sample complexity.
Consider altGDnormbal studied in \cite{lafferty_lrmc,rpca_gd} for LRMC. In this case again, the desired column-wise error bound cannot be obtained because the update step for $\bm{B}$ involves GD w.r.t. $f({\bm{U}},\bm{B}) + f_2({\bm{U}},\bm{B})$. The gradient w.r.t $f_2$ (norm-balancing term) introduces coupling between the different columns of $\bm{B}$, and hence, also between columns of ${\bm{X}} = {\bm{U}} \bm{B}$.
Thus, once again, it is not clear how to get a tight bound on $\max_k \|\x^*_k - \hat\x_k\|$.
For AltGD-Min, because the min step for updating $\bm{B}$ is a decoupled LS problem, it is possible to get the desired column-wise error bound. Secondly, because we use GD w.r.t ${\bm{U}}$, there is an extra $\b_k^\top$ term in the gradient summands. This makes the gradient (and its deviation from its expected value), a sum of {\em nice-enough} sub-exponential r.v.s as explained in Sec. \ref{outline_iters}.%
\begin{table*}[t]
\caption{\small{
Why the LRMC initialization approach cannot be directly borrowed?
}}
\label{LRMC_init_diff}
\vspace{-0.1in}
\begin{center}
\renewcommand*{\arraystretch}{1.01}
{
\begin{tabular}{lll} \toprule
& LRMC & Our Problem, LRcCS \\ \midrule
$\hat\X_{0,full}=$ & $\displaystyle \sum_k \sum_j \frac{\delta_{jk}}{p} \bm{y}_{jk} \bm{e}_j \bm{e}_k{}^\top$ & $\displaystyle \frac{1}{m} \sum_k \sum_i \a_{ki} \bm{y}_{ki} \bm{e}_k{}^\top$ \\
& $\delta_{jk} \stackrel{\mathrm{iid}}{\thicksim } Bernoulli(p)$ & $\a_{ki} \stackrel{\mathrm{iid}}{\thicksim } {\cal{N}}(0, \bm{I}_n)$ \\
$\H_0 =\hat\X_{0,full} - \X^*$ & $\displaystyle \sum_{k=1}^q \sum_{j=1}^n (1 - \frac{\delta_{jk}}{p}) \X^*_{jk} \bm{e}_j \bm{e}_k{}^\top$ & $\displaystyle \frac{1}{m} \sum_{k=1}^q \sum_{i=1}^m (\bm{I} - \a_{ki} \a_{ki}{}^\top ) \x^*_k \bm{e}_k{}^\top$ \\
Each summand is & nicely bounded by & unbounded \& sub-expo. norm$^{**}$ is \\
& $\mu^2 {\sigma_{\max}^*} (r/\sqrt{nq}) $ & $\mu {\sigma_{\max}^*} \sqrt{r/q}$ (too large, need $r/q$) \\
Concen. ineq. & Matrix Bernstein \cite{tail_bound} & Sub-expo Bernstein \cite{versh_book} \\
& gives desired sample comp. & does not give desired sample comp. \\
\bottomrule
\end{tabular}
}
\end{center}
{\footnotesize ${**}$: ``max sub-expo. norm": max sub-exponential norm of $(\a_{ki}{}^\top\bm{w}) (\a_{ki}{}^\top \x^*_k) (\bm{e}_k^\top \bm{z})$ for any unit vectors $\bm{w},\bm{z}$.}
\vspace{-0.15in}
\end{table*}
\subsection{Initialization} \label{lrmc_compare_init}
The standard approach used for initializing iterative algorithms for LRMC (as well as other linear LRR problems) is to compute the top $r$ left singular vectors of the matrix ${\bm{X}}_{0,full}$ that satisfies $({\bm{X}}_{0,full})_{vec}= \mathcal{A}^\top (\bm{y}_{all})$, where $\bm{y}_{all}$ is the $mq$-length vector of all measurements and $\mathcal{A}$ denotes the linear mapping from $(\X^*)_{vec}$ to $\bm{y}_{all}$. In case of LRMC and LRcCS, this is computed is as given in Table \ref{LRMC_init_diff}. It is not hard to see that, in both cases, $\mathbb{E}[\hat\X_{0,full}]= \X^*$.
To show that this approach works, one typically uses a $\sin \Theta$ theorem, e.g., Davis-Kahan or Wedin, to bound $\text{{SubsDist}}(\U^*{}, {\bm{U}}_0)$ as a function of terms that depend on $\H_0:= \hat\X_{0,full} - \X^*$. Thus a first requirement is to bound $\|\H_0\|$. For LRMC, this can be done easily since $\H_0$ is a sum of the independent one-sparse random matrices shown in the table with each matrix containing an i.i.d. Bernoulli r.v. times $\X^*_{jk}$ ($jk$-th entry of $\X^*$) as its nonzero entry. Using the left and right singular vectors' incoherence (assumed in all LRMC guarantees), and $\X^*_{jk} = \bm{e}_j^\top \X^* \bm{e}_k$, one can argue that, for unit vectors $\bm{w},\bm{z}$, each summand of $|\bm{w}^\top \H_0 \bm{z}|$ is of order at most $(1/p) {\sigma_{\max}^*} r/\sqrt{nq}$. This bound, along with a bound on the ``variance parameter" needed for applying matrix Bernstein \cite{tail_bound},\cite[Chap 5]{versh_book} helps show that $\|\H_0\| \le c {\sigma_{\max}^*}$ w.h.p., under the desired sample complexity bound.
For LRcCS, the summands of $\hat\X_{0,full}$, and hence of $\H_0$, are sub-exponential r.v.s. These can be bounded using the sub-exponential Bernstein inequality \cite[Chap 2]{versh_book}. This requires a bound on the maximum sub-exponential norm of any summand. Denote this bound by $K_e$. In order to show that $\|\H_0\| \le c {\sigma_{\max}^*}$ w.h.p, under the desired sample complexity, we need $K_e$ to be of order $(r/q)$ or smaller. However, for our summands, we can only guarantee $K_e \le (1/m) \max_k \|\x^*_k\| \le (1/m) \mu \sqrt{r/q} {\sigma_{\max}^*}$. This is not small enough, i.e., the summands are not {\em nice-enough} subexponentials. It will require $mq \gtrsim (n+q) r \cdot \sqrt{q}$ which is too large.%
\section*{Author Biographies}
{\bf Seyedehsara Nayer (Email: [email protected])} recently completed her Ph.D. in ECE at Iowa State University. She has an M.S. from Sharif University in Iran. She works as a Senior Engineer at ASML in Santa Clara, CA.
Her research interests are around various aspects of information science and focuses on Signal Processing, and Statistical Machine Learning.
{\bf Namrata Vaswani (Email: [email protected])} received a B.Tech from IIT-Delhi in India in 1999 and a Ph.D. from the University of Maryland, College Park in 2004, both in Electrical Engineering. Since Fall 2005, she has been with the Iowa State University where she is currently the Anderlik Professor of Electrical and Computer Engineering. Her research interests lie in a data science, with a particular focus on Statistical Machine Learning and Signal Processing. She has served two terms as an Associate Editor for the IEEE Transactions on Signal Processing; as a lead guest-editor for a 2018 Proceedings of the IEEE Special Issue (Rethinking PCA for modern datasets); and as an Area Editor for the IEEE Signal Processing Magazine (2018-2020). Vaswani is a recipient of the Iowa State Early Career Engineering Faculty Research Award (2014), the Iowa State University Mid-Career Achievement in Research Award (2019) and University of Maryland’s ECE Distinguished Alumni Award (2019). She also received the 2014 IEEE Signal Processing Society Best Paper Award for her 2010 IEEE Transactions on Signal Processing paper co-authored with her student Wei Lu on “Modified-CS: Modifying compressive sensing for problems with partially known support”. She is a Fellow of the IEEE Fellow (class of 2019).
|
1,108,101,565,480 | arxiv | \section{Introduction}~
Coherence underlies quantum phenomena. Familiar from waves, coherence gives rise to interference effects that power quantum systems to be different from and more useful than everyday objects. Quantum coherence enables computation \cite{Shor1999}, measurement \cite{Dowling1998}, teleportation \cite{Bouwmeesteretal1997}, and more, making it an important resource to quantify \cite{Aberg2006arxiv,Baumgratzetal2014,LeviMintert2014,WinterYang2016}. Our ability to generate and transform coherence is thus vital to the success of these ventures.
Here, we show how to ideally transfer arbitrary amounts of coherence from light to atomic systems.
Previous work found the ideal states of light for transferring maximal coherence to a single atom: \textit{transcoherent} states do this job and can be approximated by easier-to-generate squeezed light in the appropriate limits \cite{GoldbergSteinberg2020}. These are important to the plethora of applications requiring maximally coherent atomic states, such as quantum engines \cite{Korzekwaetal2016} and quantum state preparation with quantum logic gates \cite{Mulleretal2009}. In other applications, arbitrary superpositions of a two-level atom's ground and excited states $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ may be desired, with the most general state being
\eq{
\ket{\theta,\phi}\equiv \cos\frac{\theta}{2}\ket{\mathrm{g}}+\sin\frac{\theta}{2}\text{e}^{\text{i}\phi}\ket{\mathrm{e}};
\label{eq:theta phi atomic state}
} the atom may be a physical atom or any other physical system with two energy levels, known as a qubit. We find the ideal states for all of these applications and demonstrate how they avoid residual light-atom entanglement and other deleterious effects that ruin the quality of the atomic coherence.
Light is routinely used for controlling atomic states. Strong, classical light with a frequency close to the transition frequency of a two-level atom induces ``Rabi flopping'' that coherently drives the atom between $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ at the Rabi frequency $\Omega_0\sqrt{\bar{n}}$, with $\bar{n}$ equal to the intensity of the field in the appropriate units that amount to the single-photon intensity when the field is quantized \cite{AllenEberly1987}. Waiting an appropriate time $\Omega_0\sqrt{\bar{n}}t=\theta$, for example, will lead to an atom in state $\ket{\mathrm{g}}$ rotating to state $\ket{\theta,0}$. However, even quasiclassical light in a coherent state is fundamentally made from a superposition of different numbers of photons \cite{Sudarshan1963,Glauber1963}, which each drive oscillations in the atom at a different Rabi frequency; these give rise to famous effects such as the collapses and revivals of Rabi oscillations that help demonstrate the existence of quantized photons underlying quasiclassical light \cite{Eberlyetal1980,Rempeetal1987}.
When considering the quantized version of light's interaction with a single atom, the Jaynes-Cummings model (JCM) dictates that the light will generally become entangled with the atom \cite{GeaBanacloche1990,GeaBanacloche1991,PhoenixKnight1991a,GeaBanacloche2002,vanEnkKimble2002,SilberfarbDeutsch}. This prevents the atom from being in any pure state $\ket{\theta,\phi}$ and always tends to degrade the quality of the atomic state thus created. Such is the problem that transcoherent states surmount for $\theta=\tfrac{\pi}{2}$ and that we generalize here.
A natural, further generalization of these results is to field states interacting with a collection of atoms. While the dynamics between a single atom and a mode of light are straightforward to solve through the JCM, the same with a collection of atoms, known as the Tavis-Cummings model (TCM), cannot usually be done in closed form. This enriches our problem and allows us to incorporate strategies from semiclassical quantization in our investigations.
The above interactions are linear in the electromagnetic field operators. We lastly extend these results for arbitrary atomic control to interactions that involve nonlinear contributions from the electromagnetic field, such as $m$-photon absorption processes. These showcase the reach of our transcoherence idea well beyond the initial goal of transferring coherence from light to atoms.
\subsection{Jaynes-Cummings model}~
The Jaynes-Cummings Hamiltonian governs the resonant interaction between a single bosonic mode annihilated by ${a}$ and a two-level atom with ground and excited states $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$, respectively:
\eq{
H=\omega\left({a}^\dagger\vphantom{a}{a}+\ket{\mathrm{e}}\bra{\mathrm{e}}\right)+\frac{\Omega_0}{2}\left({a}\sigma_+ + {a}^\dagger\vphantom{a}\sigma_-\right),
} where $\omega$ is the resonance frequency, $\Omega_0$ is the single-excitation Rabi frequency (sometimes known as the vacuum Rabi frequency), and $\sigma_+=\sigma_-^\dagger=\ket{\mathrm{e}}\bra{\mathrm{g}}$ is the atomic raising operator. The JCM characterizes light-matter interactions in a variety of physical systems including circuit quantum electrodynamics (QED) \cite{Raimondetal2001}, cavity QED \cite{Finketal2008}, and parametric amplification \cite{GutierrezJaureguiAgarwal2021}. This interaction conserves total energy and total excitation number, as can be seen from its eigenstates
\eq{
\ket{\pm,n}=\frac{\ket{n}\otimes\ket{\mathrm{e}}\pm\ket{n+1}\otimes\ket{\mathrm{g}}}{\sqrt{2}},
\label{eq:JCM eigenstates}
} with eigenenergies $\pm\tfrac{\Omega_n}{2}$ for the quantized Rabi frequencies
\eq{
\Omega_n=\Omega_0\sqrt{n+1}.
} These are responsible for the field and the atom periodically exchanging an excitation with frequency $\tfrac{\Omega_n}{2}$ when the initial state is either $\ket{n}\otimes\ket{\mathrm{e}}$ or $\ket{n+1}\otimes\ket{\mathrm{g}}$.
We will work in the interaction picture with Hamiltonian
\eq{
H_{\mathrm{I}}=
\frac{\Omega_0}{2}\left({a}\sigma_+ + {a}^\dagger\vphantom{a}\sigma_-\right);
} the Schr\"odinger-picture results can thence be obtained with the substitutions $\ket{n}\to\text{e}^{-\text{i}\omega nt}\ket{n}$ and $\ket{\mathrm{e}}\to\text{e}^{-\text{i} \omega t}\ket{\mathrm{e}}$.
When the atom is initially in its ground state and the field in state $\sum_n \psi_n\ket{n}$, the evolved state takes the form
\eq{
\ket{\Psi(t)}=&\psi_0\ket{0}\otimes\ket{\mathrm{g}}+\sum_{n=0}^\infty \psi_{n+1}\left(\cos\frac{\Omega_n t}{2}\ket{n+1}\otimes\ket{\mathrm{g}}
\right.\\&\qquad\qquad\qquad\qquad\left.
-\text{i}\sin\frac{\Omega_n t}{2}\ket{n}\otimes \ket{\mathrm{e}}\right)\\
=&\sum_{n=0}^\infty \ket{n}\otimes\left(\psi_n\cos\frac{\Omega_{n-1}t}{2}\ket{\mathrm{g}}-\text{i} \psi_{n+1}\sin\frac{\Omega_n t}{2}\ket{\mathrm{e}}\right).
\label{eq:JCM from ground}
} Similarly,
when the atom is initially in its excited state and the field in state $\sum_n \psi_n\ket{n}$, the evolved state takes the form
\eq{
\ket{\Psi(t)}=&\sum_{n=0}^\infty \ket{n}\otimes \left(\psi_n\cos\frac{\Omega_n t}{2}\ket{\mathrm{e}}
-\text{i} \psi_{n-1}\sin\frac{\Omega_{n-1} t}{2}\ket{\mathrm{g}}\right).
\label{eq:JCM from excited}
}
To achieve an arbitrary final atomic state, the most intuitive procedure is to begin with the atom in its ground state and the field in the target atomic state $\cos\frac{\theta}{2}\ket{0}+\text{i}\sin\frac{\theta}{2}\text{e}^{\text{i}\phi}\ket{1}$ and wait for the duration of a ``single-excitation $\pi$ pulse'' $\Omega_0 t=\pi$ to enact the transformation
\eq{
&\left(\cos\frac{\theta}{2}\ket{0}+\text{i}\sin\frac{\theta}{2}\text{e}^{\text{i}\phi}\ket{1}\right)\otimes\ket{\mathrm{g}}
\\
&\qquad\qquad\to\ket{0}
\otimes \left(\cos\frac{\theta}{2}\ket{\mathrm{g}}+\sin\frac{\theta}{2}\text{e}^{\text{i}\phi}\ket{\mathrm{e}}\right)= \ket{0}\otimes\ket{\theta,\phi}.
} We exhaustively show in Section \ref{sec:perfectly generating arbitrary coherence} how to achieve this transformation with other field states, with no residual atom-field entanglement, at faster rates, and with more feasible pulses of light.
Since the free atomic evolution enacts $\phi\to\phi-\omega t$, we can generate the states $\ket{\theta,\phi}$ with any value of $\phi$ and simply allow free evolution to generate the same state with any other value of $\phi$, so in the following we set $\phi=0$ (alternatively, direct solutions with $\phi\neq 0$ can readily be obtained by adjusting the relative phases between the photon-number states).
\section{Optimal field states for generating arbitrary amounts of atomic coherence}~
\label{sec:perfectly generating arbitrary coherence}
What are the optimal field states that can generate arbitrary pulse areas? That is, which field states
\eq{
\ket{\psi}=\sum_n \psi_n\ket{n}
} can achieve the transformations
\eq{
\ket{\psi}\otimes\ket{\mathrm{g}}\to\ket{\psi^\prime}\otimes \ket{\theta}
} or
\eq{
\ket{\psi}\otimes\ket{\mathrm{e}}\to \ket{\psi^\prime}\otimes \ket{\theta},
} where the former correspond to ``$\theta$ pulses,'' the latter to ``$\theta+\pi$ pulses,'' and we have defined the atomic state $\ket{\theta}\equiv\ket{\theta,0}$ by allowing $\theta$ to extend to $2\pi$.
We specifically seek transformations for which the final state has zero residual entanglement between the atom and the light, such that the atomic state can be used in arbitrary quantum information protocols without degradation.
\subsection{Transcoherent states}~
In Ref. \cite{GoldbergSteinberg2020} we defined \textit{transcoherent states} as those enabling $\tfrac{\pi}{2}$ pulses. For atoms initially in their ground states, perfect $\tfrac{\pi}{2}$ pulses can be achieved by the transcoherent states whose coefficients in the photon-number basis satisfy the recurrence relation
\eq{
\psi_{n+1}=\text{i}\frac{\cos\frac{\Omega_{n-1} t}{2}}{\sin\frac{\Omega_n t}{2}}\psi_n
\label{eq:recurrence relation transcoherent ground}
} to ensure that the amplitudes of $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ in the evolved state in Eq. \eqref{eq:JCM from ground} are equal. This can be satisfied by field states with $\psi_n=0$ for $n> n_{\mathrm{max}}$ for \textit{any} chosen maximum photon number $n_{\mathrm{max}}\geq 1$, so long as the total interaction time satisfies
\eq{
\Omega_{n_{\mathrm{max}}-1}t=\pi,
\label{eq:interaction time transcoherent ground}
} which ensures that the highest-excitation subspace spanned by $\ket{\pm,n_{\mathrm{max}}-1}$ undergoes a $\pi$ pulse such that it transfers all of its excitation probability from $\ket{n_{\mathrm{max}}}\otimes\ket{\mathrm{g}}$ to $\ket{n_{\mathrm{max}}-1}\otimes\ket{\mathrm{e}}$.
In the large-$n_{\mathrm{max}}$ limit, these states strongly approximate Gaussian states with an average photon number $\bar{n}$ whose photon-number distributions are squeezed from that of a canonical coherent state, $\sigma^2_{\mathrm{coh}}=\bar{n}$, by a factor of $\tfrac{\pi}{2}$.
Similarly, another set of transcoherent states has its photon-number distribution satisfy the same recurrence relation as Eq. \eqref{eq:recurrence relation transcoherent ground}, with the lowest-excitation manifold undergoing a $(2k)\pi$ pulse and the highest a $(2k+1)\pi$ pulse for any $k\in\mathds{N}_0$. This pulse, in the large-$\bar{n}$ limit, corresponds to a $\frac{4k+1}{2}\pi$ pulse produced by a coherent state with its photon-number distribution squeezed by a factor of $\frac{4k+1}{2}\pi$.
Superpositions of such states with nonzero coefficients all satisfying $(2k)^2 n_{\mathrm{max}}\leq n\leq (2k+1)^2 n_{\mathrm{max}}$ will also enact perfect $\tfrac{\pi}{2}$ pulses in a time $\Omega_{n_{\mathrm{max}}-1}t=\pi$.
Another set of transcoherent states is found when the atom is initially in its excited state. This setup requires the initial field state's coefficients to satisfy a different recurrence relation to ensure equal coefficients of $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ in Eq. \eqref{eq:JCM from excited}:
\eq{
\psi_{n+1}=-\text{i}\frac{\sin\frac{\Omega_{n} t}{2}}{\cos\frac{\Omega_{n+1} t}{2}}\psi_{n}.
\label{eq:recurrence relation transcoherent excited}
} As well, the lowest-excitation sector must now undergo a $(2k+1)\pi$ pulse while the highest undergoes a $2(2k+1)\pi$ pulse. This happens in a time \eq{
\Omega_{n_{\mathrm{min}}}t=(2k+1)\pi,
\label{eq:interaction time transcoherent excited}
} corresponding in the large-$\bar{n}$ limit to a $\tfrac{4k+3}{2}\pi$ pulse generated by a coherent state that has been photon-number squeezed by $\tfrac{4k+3}{2}\pi$. Superpositions of commensurate states will again achieve perfect coherence transfer.
\subsection{Beyond transcoherent states}~
We can generalize the recurrence relations of Eqs. \eqref{eq:recurrence relation transcoherent ground} and \eqref{eq:recurrence relation transcoherent excited} to generate arbitrary states of the form of Eq. \eqref{eq:theta phi atomic state}.
When the atom is initially in its ground state, it will evolve to a state of the form of Eq. \eqref{eq:theta phi atomic state} if and only if the initial field state's coefficients obey the recurrence relation [again, c.f. Eq. \eqref{eq:JCM from ground}]
\eq{
\psi_{n+1}=\text{i}\tan{\frac{\theta}{2}}\frac{\cos\frac{\Omega_{n-1} t}{2}}{\sin\frac{\Omega_n t}{2}}\psi_n.
\label{eq:recurrence relation beyond transcoherent ground}
} The same boundary conditions as for transcoherent states hold, meaning that we require interaction times of the form of Eq. \eqref{eq:interaction time transcoherent ground} such that the lowest-excitation manifold undergoes a $0\pi$ pulse and the highest a $\pi$ pulse; extensions to other excitation manifolds are similarly permissible. We plot a number of such states in Fig. \ref{fig:beyond transcoherent from g}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{beyond_states_nmax_200_various_pulses_v2}
\caption{Photon-number probability distributions for field states that exactly generate arbitrary rotations $\theta$ on atoms initially in their ground states (various shapes correspond to different values of $\theta$). The field states are calculated using the recursion relation Eq. \eqref{eq:recurrence relation beyond transcoherent ground} with $n_{\mathrm{max}}=200$. Also plotted are the photon-number distributions for coherent states with the same average energies (solid curves). For the same $n_{\mathrm{max}}$ and thus the same value of $\Omega_0 t$, a higher-energy pulse generates a larger rotation angle $\theta$, with more photon-number squeezing being necessary for larger rotation angles.}
\label{fig:beyond transcoherent from g}
\end{figure}
When the atom is initially in its excited state, it will evolve to a state of the form of Eq. \eqref{eq:theta phi atomic state} if and only if the initial field state's coefficients obey the recurrence relation [again, c.f. Eq. \eqref{eq:JCM from excited}]
\eq{
\psi_{n+1}=-\text{i}\tan{\frac{\theta}{2}}\frac{\sin\frac{\Omega_{n} t}{2}}{\cos\frac{\Omega_{n+1} t}{2}}\psi_{n}
.
\label{eq:recurrence relation beyond transcoherent excited}
} The same boundary conditions as for transcoherent states hold, meaning that we require interaction times of the form of Eq. \eqref{eq:interaction time transcoherent excited} such that the lowest-excitation manifold undergoes a $(2k+1)\pi$ pulse and the highest a $(4k+2)\pi$ pulse; extensions to superpositions of excitation manifolds are again permissible.
What are the properties of these extended transcoherent states whose coefficients are given respectively by Eqs. \eqref{eq:recurrence relation beyond transcoherent ground} and \eqref{eq:recurrence relation beyond transcoherent excited}? In Ref. \cite{GoldbergSteinberg2020}, we discussed how transcoherent states maximize the coherence \eq{
\mathcal{C}(t)&=\left|\bra{\Psi(t)}\sigma_+\ket{\Psi(t)}\right|+\left|\bra{\Psi(t)}\sigma_-\ket{\Psi(t)}\right|.
\label{eq:coherence definition}
} This is achieved through a careful balance between the narrowness of the distribution $|\psi_n|^2$, which favours nearly equivalent Rabi frequencies, and its broadness, which leads to larger overlap terms $|\psi_{n+1}\psi_n|$ in Eq.
\eqref{eq:coherence definition}.
By creating the atomic states of Eq. \eqref{eq:theta phi atomic state}, we are attaining arbitrary values of $\mathcal{C}(t)$. How do the conditions of Eqs. \eqref{eq:recurrence relation beyond transcoherent ground} and \eqref{eq:recurrence relation beyond transcoherent excited} achieve the optimal balance for the distribution $|\psi_n|^2$?
We begin with the case of atoms initially in their ground states: Eq. \eqref{eq:recurrence relation beyond transcoherent ground}. By choosing states that satisfy the condition of Eq. \eqref{eq:interaction time transcoherent ground}, we ensure that the recursion relation truncates. Then, since the ratio $|\psi_{n+1}/\psi_n|$ monotonically decreases until it reaches zero, where the series truncates, the photon-number distribution approaches a smooth, singly peaked distribution;
for sufficiently large $\bar{n}$, this distribution is Gaussian.
The recursion relation of Eq. \eqref{eq:recurrence relation beyond transcoherent ground} is stationary when \eq{
\cos\frac{\theta}{2}\sin\frac{\Omega_n t}{2}-\sin\frac{\theta}{2}\cos\frac{\Omega_{n-1}t}{2}=0.
} For large $\bar{n}$, this occurs at a sufficiently large value of $n$ such that $\Omega_n\approx \Omega_{n-1}$; we will refer to this value as $\tilde{n}$, which we will later see to be on the order of the average number of photons $\tilde{n}=\mathcal{O}(\bar{n})$. This leads to the condition
\eq{
\sin\left(\frac{\theta}{2}-\frac{\Omega_{\tilde{n}} t}{2}\right)\approx 0\qquad
\Rightarrow\qquad \Omega_{\tilde{n}} t\approx \theta ,
\label{eq:stationary point ground}
}
corresponding with the classical scenario in which a pulse area of $\theta$ is applied when the Rabi frequency and total interaction time satisfy $\Omega_{\tilde{n}} t=\theta$.
Substituting this condition into Eq. \eqref{eq:recurrence relation beyond transcoherent ground} leads to the expansion about large $\tilde{n}$:
\eq{
\tan{\frac{\theta}{2}}\frac{\cos\frac{\Omega_{{n}-1} t}{2}}{\sin\frac{\Omega_{{n}} t}{2}}=&\tan{\frac{\theta}{2}}\frac{\cos\frac{\theta\sqrt{n}}{2\sqrt{\tilde{n}+1}}}{\sin\frac{\theta\sqrt{n+1}}{2\sqrt{\tilde{n}+1}}}=\tan{\frac{\theta}{2}}\frac{\cos\frac{\theta\sqrt{\tilde{n}+\delta}}{2\sqrt{\tilde{n}+1}}}{\sin\frac{\theta\sqrt{\tilde{n}+\delta+1}}{2\sqrt{\tilde{n}+1}}}\\
&
\approx 1- \frac{\theta}{2\tilde{n}\sin\theta}\left(\delta-\sin^2\frac{\theta}{2}\right)
.} From this we find the approximate difference relation:
\eq{
\frac{\psi_{\tilde{n}+\delta}-\psi_{\tilde{n}}}{\delta}\approx -\frac{\theta}{2\tilde{n}\sin\theta}\left(\delta-\sin^2\frac{\theta}{2}\right)\psi_{\tilde{n}}.
} This describes a Gaussian distribution
\eq{
|\psi_{\tilde{n}+\delta}|^2\approx |\psi_{\tilde{n}}|^2 \exp\left[
-\frac{\theta}{2\tilde{n}\sin\theta}\left(\delta-\sin^2\frac{\theta}{2}\right)^2
\right]
\label{eq:Gaussian fround ground}
} with photon-number variance
\eq{
\sigma^2=\tilde{n}\sinc\theta.
\label{eq:sinc variance}
} The mean is slightly shifted to $\bar{n}=\tilde{n}+\sin^2\frac{\theta}{2}$, due to the discreteness of $n$; had we instead set the stationary point of the recursion relation to be at $\tilde{n}-1$, we would have accordingly found the argument of the Gaussian to be $\left(\delta+\cos^2\frac{\theta}{2}\right)^2$.
The same calculation can be performed for an atom initially in its excited state. Looking at the stationary point of the recursion relation of Eq. \eqref{eq:recurrence relation beyond transcoherent excited}, namely,
\eq{
\cos\frac{\theta}{2}\cos\frac{\Omega_{n+1}t}{2}-\sin\frac{\theta}{2}\sin\frac{\Omega_{n}t}{2}=0,
} we now arrive at the condition
\eq{
\cos\left(\frac{\theta}{2}+\frac{\Omega_{\tilde{n}t}}{2}\right)=0\qquad \Rightarrow \qquad \Omega_{\tilde{n}}t\approx \theta+\pi.
\label{eq:stationary point excited}
} This similarly corresponds to the classical scenario in which a pulse area of $\theta+\pi$ (i.e., rotating from $\ket{\mathrm{e}}$ to $\ket{\mathrm{g}}$ to $\ket{\theta}$) is achieved when the Rabi frequency and total interaction time satisfy $\Omega_{\bar{n}}t=\theta+\pi$.
Substituting this new condition into Eq. \eqref{eq:recurrence relation beyond transcoherent excited} leads to the expansion:
\eq{
-\tan\frac{\theta}{2}\frac{\sin\frac{\Omega_{n}t}{2}}{ \cos\frac{\Omega_{n+1}t}{2} }
\approx 1-\frac{\theta+\pi}{2\tilde{n}\sin\theta}\left(\delta+\cos^2\frac{\theta}{2}\right).
} Using this for the approximate difference equation leads to the Gaussian distribution
\eq{
|\psi_{\tilde{n}+\delta}|^2\approx |\psi_{\tilde{n}}|^2 \exp\left[
-\frac{\theta+\pi}{2\tilde{n}\sin\theta}\left(\delta+\cos^2\frac{\theta}{2}\right)^2
\right].
} This differs from Eq. \eqref{eq:Gaussian fround ground} by an innocuous-looking addition of $\pi$ that is responsible for a number of important properties (as well as the stationary point being shifted from $tilde{n}$ by 1).
\subsection{Discussion}~
The states defined by Eqs. \eqref{eq:recurrence relation beyond transcoherent ground} and \eqref{eq:recurrence relation beyond transcoherent excited} directly generalize the transcoherent states of Ref. \cite{GoldbergSteinberg2020}. Transcoherent states, in the large-$\bar{n}$ limit, enact $\tfrac{4k+1}{2}\pi$ pulses on state $\ket{\mathrm{g}}$ because their photon-number distributions are squeezed by $\tfrac{4k+1}{2}\pi$ and $\tfrac{4k+3}{2}\pi$ pulses on state $\ket{\mathrm{e}}$ because their photon-number distributions are squeezed by $\tfrac{4k+3}{2}\pi$.
We can now understand from where these factors truly arise: number squeezing by $\sinc \theta$ leads to pulse areas of $\theta$ on states $\ket{\mathrm{g}}$ and by $\left(\theta+\pi\right)^{-1}\sin\theta=-\sinc\left(\theta+\pi\right)$ leads to pulse areas of $\theta+\pi$ on states $\ket{\mathrm{e}}$. The properties of the $\sinc$ function are responsible for the forms of the viable solutions to optimally delivering arbitrary pulse areas.
Variances cannot be negative. The $\sinc$ function, however, flips its sign periodically with period $\pi$. This means that the only pulses that can be delivered to state $\ket{\mathrm{g}}$ are those with
\eq{
(2k)\pi \leq \theta\leq (2k+1)\pi\quad \Leftarrow\quad \sigma^2=\bar{n}\sinc\theta
\label{eq:theta constraints from ground}
} and to $\ket{\mathrm{e}}$ are those with
\eq{
(2k+1)\pi \leq \theta+\pi\leq (2k+2)\pi\quad \Leftarrow\quad \sigma^2=-\bar{n}\sinc\left(\theta+\pi\right),
\label{eq:theta constraints from excited}
} where $k\in\mathds{N}_0$. These ranges, together with Eqs. \eqref{eq:stationary point ground} and \eqref{eq:stationary point excited}, cover all of the classically allowed possibilities for pulse areas, now in a fully quantized regime. The periodic maxima of these functions correspond to the transcoherent states, the negative regions explain why certain pulse areas are only accessible to atoms initially in their ground or excited states, and the property $\left|\sinc \theta\right|\leq 1$ implies that only photon-number squeezing, not photon-number broadening, is useful for gaining quantum advantages in generating atomic states with arbitrary coherence properties.
This also explains why we chose to retain the $-$ sign from Eq. \eqref{eq:recurrence relation beyond transcoherent excited} in Eq. \eqref{eq:stationary point excited} and similarly how we chose the signs of the terms in Eq. \eqref{eq:stationary point ground}: the solutions found from the alternate choices of signs lead to minima in the recurrence relations instead of maxima, corresponding to regions where the $\sinc$ functions are negative, which do not lead to valid solutions.
We can inspect a number of limits to ensure that these states behave sensibly. In the limit of a pulse area of $0$, where we desire no change in the state of the system, the best states have no number squeezing: $\sigma^2=\bar{n}$. Moreover, these states require an interaction time satisfying $\Omega_{\tilde{n}}t=0$, so $\bar{n}=0$, and the optimal solution is that the field is in its vacuum state, which is the trivial case of a coherent state with no photon-number squeezing. This solution works for atoms initially in either state $\ket{\mathrm{g}}$ or $\ket{\mathrm{e}}$; for the latter, the pulse area has $\theta+\pi=0$, which seems like it imparts a negative photon-number variance because $-\sinc 0=-1$, but this is not a problem because the product of this negative squeezing factor and $\bar{n}=0$ still vanishes.
To achieve a pulse area $l\pi$ for $l\in\mathds{N}$, a variance of zero is required, corresponding to the zeroes of the $\sinc$ function. Equivalently, the only field states that exactly generate $\pi$ pulses, $2\pi$ pulse, and so on are those that are ``infinitely'' photon-number-squeezed coherent states: number states. This directly accords with the eigenstates of the JCM Hamiltonian $\ket{\pm,n}$ found in Eq. \eqref{eq:JCM eigenstates}: when the joint system begins in either $\ket{n}\otimes\ket{\mathrm{e}}$ or $\ket{n+1}\otimes\ket{\mathrm{g}}$, an interaction time of $\Omega_n t=(2l+1)\pi$ swaps the excitation between the field and the atom, while an interaction time of $\Omega_n t=(2l)\pi$ returns the excitations to their original starting points.
Complementing the properties of the $\sinc$ function, there is another reason why the ideal field states only exist for the pulse areas described by Eqs. \eqref{eq:theta constraints from ground} and \eqref{eq:theta constraints from excited}. An ideal pulse acting on $\ket{\mathrm{g}}$ must undergo a $(2k)\pi$ pulse in the lowest-excitation manifold and $(2k+1)\pi$ pulse in the highest. The pulse area for the \textit{average} photon number $\bar{n}$ must therefore always be between $(2k)\pi$ and $(2k+1)\pi$, never between $(2k+1)\pi$ and $(2k+2)\pi$; the converse holds for atoms initially in state $\ket{\mathrm{e}}$. This is why, for example, a perfect $\tfrac{\pi}{2}$ pulse can never be applied to state $\ket{\mathrm{e}}$, which must instead experience a perfect $\tfrac{3}{2}\pi$ pulse. Controlling the pulse areas of the lowest- and highest-excitation sections is paramount for ideal coherence transfer.
The idea of tailoring pulses such that the highest-excitation manifold undergoes a $\pi$ pulse was recently explored in a different context Ref. \cite{Liuetal2021constructing}.
There, this paradigm was used to create a universal set of quantum operations that could be used for quantum computation. We thus stress the importance of using the fully quantized JCM to surmount information leakage in light-matter-interaction protocols.
A useful property of transcoherent states and beyond is that the field states experience less backaction from the interaction with the atom than standard coherent states. This allows them to be repeatedly used as ``catalysts'' for transferring coherence to the atoms before eventually running out of energy and coherence to impart and degrading through repeated interactions \cite{GoldbergSteinberg2020}.\footnote{C.f. quantum catalysis as studied in the JCM in Ref. \cite{Messingeretal2020}.} Therefore, the cost associated with producing a transcoherent state should, in some sense, be reduced by a factor of the number of times that state can be used for practical coherence transfer.
\section{Optimal field states for generating $\Theta$ pulses on arbitrary atomic states}~
Transcoherent states and their generalizations in Section \ref{sec:perfectly generating arbitrary coherence} are the unique optimal field states that generate arbitrary atomic states $\ket{\theta}$ in arbitrarily short times from atoms initially in state $\ket{\mathrm{g}}$ or $\ket{\mathrm{e}}$. For quantum information protocols, one often seeks a transformation that transforms arbitrary initial atomic states in the same way. The transcoherent states offer a method for enacting the ideal rotation by $\tfrac{\pi}{2}$ about the $y$ axis on the Bloch sphere:
\eq{
\begin{pmatrix}
\ket{\mathrm{g}}\\\ket{\mathrm{e}}
\end{pmatrix}\to\frac{1}{\sqrt{2}}\begin{pmatrix}
1&1\\-1&1
\end{pmatrix}\begin{pmatrix}
\ket{\mathrm{g}}\\\ket{\mathrm{e}}
\end{pmatrix},
\label{eq:general pi/2 pulse}
} equivalent to a $\tfrac{\pi}{2}$ pulse, on either of the initial states $\ket{\mathrm{g}}$ or $\ket{\mathrm{e}}$. This is similar to the Hadamard transformation, which is also useful for generating coherence.
What is the optimal field state for enacting a $\tfrac{\pi}{2}$ transformation on arbitrary initial states $\ket{\theta,\phi}$ to the transformed states:
\eq{
\ket{\theta,\phi}_{\frac{\pi}{2}}\equiv
\frac{\cos\frac{\theta}{2}-\text{e}^{\text{i}\phi}\sin\frac{\theta}{2}}{\sqrt{2}}\ket{\mathrm{g}}+\frac{\cos\frac{\theta}{2}+\text{e}^{\text{i}\phi}\sin\frac{\theta}{2}}{\sqrt{2}}\ket{\mathrm{e}}?
\label{eq:theta phi state after Hadamard}
} It is impossible to do this perfectly, in contrast to transcoherent states and their generalizations in Sec. \ref{sec:perfectly generating arbitrary coherence} that can do this perfectly, because the highest-excitation manifold must undergo a $(2k)\pi$ pulse for most initial states $\ket{\theta,\phi}$ but a $(2k+1)\pi$ pulse for initial state $\ket{\mathrm{e}}$. We thus seek states that perform the best on average.
A straightforward way of determining the success of creating the state depicted in Eq. \eqref{eq:theta phi state after Hadamard} is as follows:
we begin with some state $\ket{\theta,\phi}$ and some initial field state, evolve the joint system using the JCM, measure the overlap of the evolved state $\ket{\Psi(t)}$ with $\ket{\theta,\phi}_{\frac{\pi}{2}}$, and average the result over all initial atomic state angles $\theta$ and $\phi$. The result should depend on the initial field state, so we can ask what field state maximizes the resulting averaged fidelity, equivalent to the averaged success probability.
By combining Eqs. \eqref{eq:JCM from ground} and \eqref{eq:JCM from excited}, we learn that a state $\sum_n \psi_n\ket{n}\otimes\ket{\theta,\phi}$ evolves to
\eq{
\ket{\Psi(t)}=\sum_{n=0}^\infty\ket{n}\otimes\left(\cos\frac{\theta}{2}\ket{G_n}+\sin\frac{\theta}{2}\text{e}^{\text{i}\phi}\ket{E_n}\right),
\label{eq:transformed joint state from arbitrary initial}
} where we have defined the atomic states
\eq{
\ket{G_n}&=\psi_n\cos\frac{\Omega_{n-1}t}{2}\ket{\mathrm{g}}-\text{i} \psi_{n+1}\sin\frac{\Omega_n t}{2}\ket{\mathrm{e}} ,\\
\ket{E_n}&=\psi_n\cos\frac{\Omega_n t}{2}\ket{\mathrm{e}}
-\text{i} \psi_{n-1}\sin\frac{\Omega_{n-1} t}{2}\ket{\mathrm{g}}.
}
Comparing this result to the desired state $\ket{\theta,\phi}_{\frac{\pi}{2}}$, we observe that a strongly peaked photon-number distribution around $\bar{n}$ satisfying \eq{
\psi_{\bar{n}}\cos\frac{\Omega_{\bar{n}-1}t}{2}\approx \psi_{\bar{n}}\cos\frac{\Omega_{\bar{n}}t}{2}
&
\approx \text{i} \psi_{{\bar{n}}-1}\sin\frac{\Omega_{\bar{n}-1}t}{2}\\&\approx
-\text{i} \psi_{{\bar{n}}+1}\sin\frac{\Omega_{\bar{n}}t}{2}
\label{eq:desired distribution properties}
} would be highly beneficial for performing the transformation of Eq. \eqref{eq:general pi/2 pulse}. This seems to suggest an optimal interaction time conforming to the classical relationship $\Omega_{\bar{n}}t=\tfrac{\pi}{2}$.
There are a few considerations to calculating the averaged fidelity. To be viable for arbitrary initial states, all angles of the Bloch sphere should be equally weighted in the average.
If the azimuthal angle of the initial atomic state were known, on the other hand, one could integrate only over the polar coordinate, after waiting an appropriate free evolution such that $\phi\to 0$.
It is not obvious that there will be a single unique solution: some points on the Bloch sphere are hardly rotated by a $\tfrac{\pi}{2}$ pulse because they are close to the $\pm y$-axis thereof.
Indeed, these processes yield different ideal states and will be considered in turn.
\subsection{Averaging fidelity over all initial atomic states}~
We calculate the squared overlap averaged over the entire surface of the Bloch sphere for $\tfrac{\pi}{2}$-pulses in Appendix \ref{app:averaging fidelity} to find, for an optimal phase relationship $\psi_n \psi_{n+1}^*=-\text{i}\left|\psi_n \psi_{n+1}\right|$,
\eq{
\mathcal{F}=&\frac{1}{2}+\frac{1}{6}\sum_n\left|\psi_n\right|^2 \cos\frac{\Omega_n t}{2}\cos\frac{\Omega_{n-1}t}{2}\\
&+\left|
\psi_{n-1}\psi_{n+1}\right|\sin\frac{\Omega_n t}{2}\sin\frac{\Omega_{n-1}t}{2}
\\
&+ \left|\psi_n \psi_{n+1}\right|\sin\frac{\Omega_{n}t}{2}\left(
2\cos\frac{\Omega_n t}{2}+
\cos\frac{\Omega_{n-1}t}{2}+
\cos\frac{\Omega_{n+1}t}{2}
\right).
\label{eq:averaged fidelity general}
} What photon-number distributions and what times optimize this averaged fidelity?
It is clear from counting the terms in Eq. \eqref{eq:averaged fidelity general} that achieving distributions resembling Eq. \eqref{eq:desired distribution properties} would allow for averaged fidelities approaching unity. But the relationships in Eq. \eqref{eq:desired distribution properties} compete with each other: a narrow photon-number distribution ensures that the Rabi frequencies coincide, while a broad distribution increases the overlap between adjacent photon-number coefficients. We are thus faced with the same problem of optimizing the width of the photon-number distribution that we faced in finding the transcoherent states.
Writing the photon-number distribution as
\eq{
|\psi_n|^2\propto\exp\left[-\frac{\left(n-\bar{n}\right)^2}{2\sigma^2}\right]
\label{eq:Gaussian photon number distribution}
} up until some manual cutoff $n_{\mathrm{max}}\gg \bar{n}+\sigma$, we can optimize over the variances $\sigma^2$ for various values of the average photon number $\bar{n}$. Intriguingly, we find that, for sufficiently large $\bar{n}$, the optimal variance is always \textit{slightly} number squeezed, approaching
\eq{
\sigma_{\mathrm{optimal}}^2\approx 0.9\bar{n}.
\label{eq:optimal variance for average pi over 2}
} This is depicted in Fig. \ref{fig:nbar20}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{nmax_400_nbar_20_ave_fids}
\caption{Average fidelity $\mathcal{F}$ for rotating atoms in arbitrary initial states by $\tfrac{\pi}{2}$ calculated from Eq. \eqref{eq:averaged fidelity general} using states of the form of Eq. \eqref{eq:Gaussian photon number distribution} with various variances. The optimal photon-number variances are scaled by that of coherent light with $\sigma^2=\bar{n}$. The cutoff point was chosen to be $n_{\mathrm{max}}=400$; here, $\bar{n}=20$.}
\label{fig:nbar20}
\end{figure}
We can repeat this calculation to find the optimal squeezed to achieve any $\Theta$ pulse when averaged over all initial atomic states. The calculation yields a more cumbersome version of Eq. \eqref{eq:averaged fidelity general} and is given in Appendix \ref{app:averaging fidelity any pulse}. Optimizing this expression numerically over Gaussian field states, we find the intriguing relationship (Fig. \ref{fig:optimal pulses all situations})
\eq{
\sigma^2_{\mathrm{optimal}}= \bar{n}\sinc\frac{\Theta}{2},
\label{eq:sinc variance any atom}
} which explains the $0.9$ found in Eq. \eqref{eq:optimal variance for average pi over 2}. The maximum achievable fidelity decreases with $\Theta$ for a given $\bar{n}$ and is notably less than the perfect fidelities achievable when the atom is initially in its ground state. In fact, comparing this expression with Eq. \eqref{eq:sinc variance}, we see that the optimal field state for delivering a $\Theta$ pulse to an unknown atomic state has the same amount of squeezing as the optimal field state for delivering a $\Theta/2$ pulse to an atom in its ground state. While we only speculate on the origin of this conclusion, we are confident that it arises from some averaging of the distance that an atom must traverse on the Bloch sphere, which ranges from $0$ to $\Theta$; i.e., the average atom must rotate half as far as a ground-state atom during a $\Theta$ pulse, so the average atom requires a $\Theta/2$ pulse.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fidelity_and_variance_all_situations_nbar500_v2}
\caption{Optimal fidelity and squeezing for field states imparting $\Theta$ pulses on atoms with known (blue dots), unknown (orange triangles), and partially known (green $\times$s) initial states. Each point represents a different value of $\Theta$ for which the average fidelities were optimized over all possible variances and a given, fixed $\bar{n}=500$. \textbf{(a)} The average fidelities achieved are all large because we have used field states with large intensities. Known-state atoms can be always be perfectly transformed, while it is easier to achieve shorter pulses with smaller $\Theta$ for atoms in unknown states. \textbf{(b)} The optimal variances are plotted in units of $\bar{n}$. Plotted on top are the curves $\sinc\Theta$ (blue), $\sinc\tfrac{\Theta}{2}$ (orange), and $\sqrt{1/2}\sinc\tfrac{\Theta}{2}$ (green). The blue and green curves intersect at $\Theta=\pi/2$ when $\sigma^2=2\bar{n}/\pi$ (red, dashed).}
\label{fig:optimal pulses all situations}
\end{figure}
\subsection{Averaging fidelity over initial states with known azimuth}~
We next take the case where the initial azimuthal angle of the atom is known; this is equivalent to taking $\phi=0$ instead of averaging over that coordinate. As calculated in Appendix \ref{app:averaging fidelity any pulse fixed phi}, the fidelity for a $\tfrac{\pi}{2}$ pulse, averaged over all initial values of the atom's polar coordinate $\theta$, is
\eq{
\mathcal{F}=&\frac{1}{2}
+\frac{1}{4}\sum_n \left|\psi_n \psi_{n+1}\right|\sin\frac{\Omega_n t}{2}\\
&\times\left(2\cos\frac{\Omega_n t}{2}+
\cos\frac{\Omega_{n-1} t}{2}+
\cos\frac{\Omega_{n+1} t}{2}\right)
.
}
By numerically maximizing this quantity over all states with Gaussian photon-number distributions, we find that the optimal field state is a coherent state that is number squeezed by $\tfrac{\pi}{2}$. This is exactly the same result as for transcoherent states, even though the corresponding quantity in Ref. \cite{GoldbergSteinberg2020} [Eq. (13) there] has a single cosine term instead of all three, so we proceed with the same method to justify our optimal solution. To maximize the fidelity, we need to maximize the inner product between vectors with components $\psi_{n+1} \sin\tfrac{\Omega_n t}{2}$ and $\psi_{n}\left(2\cos\frac{\Omega_n t}{2}+
\cos\frac{\Omega_{n-1} t}{2}+
\cos\frac{\Omega_{n+1} t}{2}\right)/4$. By the Cauchy-Schwarz inequality, this is achieved when the vectors are parallel, satisfying
\eq{
\psi_{n+1} \sin\tfrac{\Omega_n t}{2}=\frac{\psi_{n}}{4}\left(2\cos\frac{\Omega_n t}{2}+
\cos\frac{\Omega_{n-1} t}{2}+
\cos\frac{\Omega_{n+1} t}{2}\right).
} This generates a recursion relation for the ideal state coefficients that can be expanded about their peak at $\bar{n}$ for an evolution time $\Omega_{\bar{n}}t=\tfrac{\pi}{2}$:
\eq{
\frac{2\cos\frac{\Omega_{\bar{n}+\delta} t}{2}+
\cos\frac{\Omega_{\bar{n}+\delta-1} t}{2}+
\cos\frac{\Omega_{\bar{n}+\delta+1} t}{2}}{4\sin\tfrac{\Omega_{\bar{n}+\delta} t}{2}}\approx 1-\frac{\pi\delta}{4\bar{n}}.
}
We select a probability distribution satisfying the approximate difference equation
\eq{
\frac{\psi_{\bar{n}+\delta}-\psi_{\bar{n}}}{\delta}\approx -\frac{\pi\delta}{4\bar{n}}\psi_{\bar{n}}
,
} whose solution is the photon-number-squeezed Gaussian distribution
\eq{
\psi_{\bar{n}+\delta}\approx \psi_{\bar{n}}\exp\left(-\frac{\delta^2}{4\sigma^2}\right),\quad \sigma^2=\frac{2\bar{n}}{\pi}.
}
We thus observe that the best states for \textit{exactly} producing a $\tfrac{\pi}{2}$ pulse and for \textit{on average} producing a $\tfrac{\pi}{2}$ pulse are the same, so long as the azimuthal coordinate of the initial atomic state is known.
The same calculation can be done for arbitrary pulse areas $\Theta$. Averaging the success probability for acting on atoms with $\phi=0$ and arbitrary $\theta$ (Appendix \ref{app:averaging fidelity any pulse fixed phi}), we find that the optimal variances obey (Fig. \ref{fig:optimal pulses all situations})
\eq{
\sigma^2=\frac{2\bar{n}}{\pi}\frac{\sinc \frac{\Theta}{2}}{\sinc \frac{\pi}{4}}=\bar{n}\frac{\sinc \frac{\Theta}{2}}{\sqrt{2}}.
\label{eq:sinc variance known phi}
} As usual, larger pulse areas require more photon-number squeezing, and we find that more squeezing is required than when averaging over the entire Bloch sphere.
\subsection{Discussion}~
The optimal field state for enacting a pulse area of $\Theta$ on an atomic state depends on the atomic state. We can collect some of our results: when the atom is initially in its ground state, the optimal photon-number variance is [Eq. \eqref{eq:sinc variance}] $\bar{n}\sinc\Theta$; when the atom is initially in a state with some known $\phi$ but unknown polar angle, the optimal variance is [Eq. \eqref{eq:sinc variance known phi}] $\bar{n}\sinc\tfrac{\Theta}{2}/\sqrt{2}$; and, when the atom is initially in an unknown state, the optimal variance is [Eq. \eqref{eq:sinc variance any atom}] $\bar{n}\sinc\tfrac{\Theta}{2}$. How do all of these compare with each other?
In terms of fidelity, not knowing the initial state leads to poorer performance. Surprisingly, averaging over a known azimuth leads to slightly smaller fidelities than averaging over the entire sphere. This discrepancy arises from the different Jacobian factors when integrating over a circle versus a sphere, implying that the ratio of the performances of states initially near the equator to states initially at the poles is what controls the overall success on average.
All of the scenarios require more photon-number squeezing for larger pulse areas. When the pulse area is $\tfrac{\pi}{2}$ or greater, an atom in its ground state requires the most squeezing because it has to travel the furthest, an atom oriented along a known meridian requires less squeezing on average, and an atom oriented in an unknown direction requires the least squeezing on average. That the completely unknown orientation requires the least squeezing makes sense: on average, such atoms need to traverse an angular distance of $\Theta/2$ for a rotation about some fixed axis on the Bloch sphere. That the polar-angle-unknown orientation requires less squeezing than ground-state atoms is more surprising: this implies that it requires more ``effort,'' in terms of greater squeezing, to travel from the poles of the Bloch sphere than from any other point. The cause of this discrepancy for angles other than $\Theta=\tfrac{\pi}{2}$ remains an open question for further study.
When the pulse area is less than $\tfrac{\pi}{2}$, the variances quoted above are more squeezed for the case with known $\phi$ than for atoms initially in their ground states. While this may be an empirical phenomenon, there is a fly in the ointment: it is not clear for $\Theta<\tfrac{\pi}{2}$ what the optimal relationship \eq{\varphi\equiv \arg \psi_{n+1}-\arg \psi_n} in Appendix \ref{app:averaging fidelity any pulse fixed phi}
should be for the initial field states. However, performing a multiparameter optimization over $\varphi$ and $\sigma^2$, we always find the optimal value to have $\varphi\approx\tfrac{\pi}{2}$ and thus maintain the variance relationship of Eq. \eqref{eq:sinc variance known phi}. We must then conclude that, somehow, ground-state atoms require less squeezing than others for rotations by $\Theta<\tfrac{\pi}{2}$ about a great circle and more squeezing than others for rotations by $\Theta>\tfrac{\pi}{2}$ about the same great circle. This is an intriguing phenomenon that surely deserves further research.
Typically, quantum computing algorithms require the same operation to be performed on arbitrary initial states. Given the optimal field state for this purpose that is squeezed by $\sinc\tfrac{\theta}{2}$, what advantage can one acquire relative to standard quantum computing protocols that use coherent states to perform logic gates on atoms? We plot in Fig. \ref{fig:fidelity improvement} the improvement in the average fidelity [Eq. \eqref{eq:averaged fidelity general}] that one can attain using the optimally squeezed states relative to coherent states with no squeezing, comparing how this improvement changes with the energy of the field state. The fidelity improvement of squeezed states over coherent states increases quickly with the rotation angle and the improvement lessens with increasing $\bar{n}$, while the relative error decreases with rotation angle and is independent from $\bar{n}$. These imply that quantum computing applications that are limited in average photon number, that are using many $\pi$ gates, or that possess any significant error rates may benefit the most from using squeezed light to improve their logic gates.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fidelity_improvement_unknown_atom}
\caption{\textbf{(a)} Additive increase in the fidelity $\mathcal{F}$ of performing a $\Theta$-rotation on an atom averaged over all initial atomic states using light whose photon-number distribution is squeezed by an optimal amount $\sinc\tfrac{\Theta}{2}$ relative to coherent light with the same strength. The improvement is significant for larger rotation angles and smaller field-state strengths. \textbf{(b)} Multiplicative decrease in the error $1-\mathcal{F}$ of performing a $\Theta$-rotation on an atom averaged over all initial atomic states using light whose photon-number distribution is squeezed by an optimal amount $\sinc\tfrac{\Theta}{2}$ relative to coherent light with the same strength. The improvement is significant for larger rotation angles and is largely independent of the field-state strength.
}
\label{fig:fidelity improvement}
\end{figure}
\section{Generating $\tfrac{\pi}{2}$ pulses for collections of atoms}~
Can the transcoherent states be generalized to field states that impart optimal pulses on collections of atoms?
A set of atoms all in the maximally coherent state $\ket{\tfrac{\pi}{2}}$ is useful for applications such as creating lasers with noise-free amplification \cite{ScullyZubairy1988}.
We investigate the collective interaction governed by the Tavis-Cummings interaction Hamiltonian \cite{TavisCummings1968}
\eq{
H_{\mathrm{TC}}=\frac{\Omega_0}{2}( {a} J_+ + {a}^\dagger\vphantom{a} J_-),
} where we now employ the collective excitation operators from SU(2):
\eq{
J_i=\sum_{k=1}^{2J}\mathds{I}^{(1)}\otimes\cdots\otimes\mathds{I}^{(k-1)}\otimes\sigma_i^{(k)}\otimes\mathds{I}^{(k+1)}\otimes\cdots\otimes\mathds{I}^{(2J)}.
} Here, $2J$ is the total number of atoms, where the $k$th atom has its own Pauli operators $\sigma_i^{(k)}$ and the permutation-symmetric states of the $2J$ atoms are equivalent to a single spin-$J$ particle. The transcoherent state problem begins with all of the atomic states in their collective ground state $\ket{J,-J}=\ket{\mathrm{g}}^{\otimes 2J}$ that is annihilated by $J_-$ and is an eigenstate of $J_z$ with minimal eigenvalue $-J$. A $\tfrac{\pi}{2}$ pulse acting on all atoms simultaneously would enact the transformation to the spin-coherent state $\ket{\mathrm{g}}^{\otimes 2J}\to 2^{-J} (\ket{\mathrm{g}}+\ket{\mathrm{e}})^{\otimes 2J}$ that is an eigenstate of $J_x$ with maximal eigenvalue $J$, which can be expressed in the basis of $J_z$ eigenstates as
\eq{
\ket{J,-J}\to\frac{1}{2^J}\sum_{m=-J}^J \sqrt{\binom{2J}{m+J}}\ket{J,m}.
\label{eq:pi/2 pulse on 2J atoms}
} How can this best be performed?
\subsection{Optimal pulses for maximum coherence generation}~
We can investigate a series of field states to find which ones best impart a $\tfrac{\pi}{2}$ pulse on a collection of $2J$ atoms. Unlike in the case of the JCM, it is not convenient to write a closed-form expression for the fidelity as a function of the field-state coefficients. Instead, we choose a variety of representative field states, from which we evolve the TCM numerically using \texttt{QuTiP} \cite{Johanssonetal2012,Johanssonetal2013}. These can then be compared to the optimal final state from Eq. \eqref{eq:pi/2 pulse on 2J atoms} and optimized accordingly.
To make the optimization tractable, we choose to optimize over field states with Gaussian photon-number distributions with varying widths. This is motivated in part by the optimal states for the JCM always having Gaussian photon-number distributions, in part because Gaussian states are among the easiest to prepare experimentally, and in part because field states with sufficiently large average photon number and sufficiently localized photon-number distributions will convert $H_{\mathrm{TC}}$ into a rotation of the form
\eq{
\exp\left(-\text{i} H_{\mathrm{TC}}t\right)\underset{\bar{n}\gg 1}{\approx}\exp\left[-i\Omega_0 \sqrt{\bar{n}}t\left(J_x\cos\varphi+J_y\sin\varphi\right)\right],
\label{eq:strong field approx TCM rotation}
} where again $\varphi$ encodes the relative phases of the field-state coefficients. For a given fixed $\bar{n}$, we thus expect the fidelity to be optimized by an interaction time $\Omega_0 t\sqrt{\bar{n}}\approx \pi/2$.
Figure \ref{fig:TCM optimals} plots the optimal field-state variances and interaction times for achieving $\tfrac{\pi}{2}$ pulses for various values of $J$ and $\bar{n}$. These parameters are the best ones found using the Nelder–Mead method implemented in \texttt{SciPy} with a variety of random seeds. As expected, the overall fidelities increase with increasing $\bar{n}$. It is perhaps unsurprising that they decrease with increasing $J$, as $J=1/2$ is the only situation in which perfect $\tfrac{\pi}{2}$ pulses can be implemented, and that the increase is quadratic in $J$ from this minimum. The optimal squeezing and time parameters follow opposite trends such that the optimal photon-number variance and interaction time obey the following relationship for a given average photon number: \eq{
\sigma_{\mathrm{optimal}}^2&\approx \frac{2\bar{n}}{\pi} ,\\
\Omega_0 t_{\mathrm{optimal}}\sqrt{\bar{n}}&\approx \frac{\pi}{2} ,\\
\Rightarrow\quad
\sigma_{\mathrm{optimal}}^2\Omega_0 t_{\mathrm{optimal}}&\approx \sqrt{\bar{n}}.
\label{eq:optimal params TCM}
} There is also some residual dependence on $J$ that may warrant the replacement $\bar{n}\to\bar{n}-J/2+1/2$ in Eq. \eqref{eq:optimal params TCM}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{infidelity_variance_time_nbar100_200_v2}
\caption{Optimal $\tfrac{\pi}{2}$ pulses on a collection of $2J$ atoms or another spin-$J$ system, for various average energies (blue dots and orange triangles have $\bar{n}=100$ and $\bar{n}=200$, respectively). \textbf{(a)} Error probability, i.e., unity minus the fidelity, of enacting a perfect $\tfrac{\pi}{2}$ pulse to achieve the transformation of Eq. \eqref{eq:pi/2 pulse on 2J atoms}. All of the fidelities are excellent, with larger $\bar{n}$ being more favourable and larger $J$ being less favourable. \textbf{(b)} Optimal variances for the initial field states. These all have their photon-number distributions squeezed by approximately $\tfrac{\pi}{2}$ relative to coherent states, with increasing squeezing required for increasing $J$. The scatter in the plot implies that not all of the results have converged to their optimal values, which may only be achievable with larger $\bar{n}$ and longer optimization times. \textbf{(c)} Optimal interaction times to achieve the desired transformation. These are all approximately the classical times for a $\tfrac{\pi}{2}$ pulse, with a slight increase in optimal time with increasing $J$. Of note, the products of the optimal variances and times are approximately unity in these units, implying that $\sigma^2\Omega_0 t/\sqrt{\bar{n}}\approx 1$.
}
\label{fig:TCM optimals}
\end{figure}
\subsection{Semiclassical limit}~
The semiclassical limit of the TCM with a highly energetic field has been studied in Refs. \cite{DrobnyJex1993,Chumakovetal1994,KlimovChumakov1995,Retamaletal1997}. Those showed that, like in the JCM \cite{GeaBanacloche1990,PhoenixKnight1991a,PhoenixKnight1991b,GeaBanacloche1991,GeaBanacloche1992,GeaBanacloche2002,vanEnkKimble2002,SilberfarbDeutsch}, one cannot simply employ the replacement ${a}\to\alpha$ in $H_{\mathrm{TC}}$ in the strong-field limit as per Eq. \eqref{eq:strong field approx TCM rotation}, as this neglects possible atom-field entanglement for any finite $\bar{n}$ and wrongly predicts the final atomic state to be pure. In Ref. \cite{Retamaletal1997}, for example, we find that the TCM Hamiltonian can be approximated in the presence of a strong field with the appropriate relative phase relationships as
\eq{
\tilde{H}_{\mathrm{TC}}=-\Omega_0\sqrt{{a}^\dagger\vphantom{a}{a}-J+1/2}{J}_y.
\label{eq:TCM approx H}
} This approximation is valid for $\bar{n}-J+1/2\gg 1$ and all of the interaction times we consider here $(\Omega_0 t\sim \bar{n}^{-1/2}\ll \bar{n})$). It serves to rotate the collective atomic state at a Rabi frequency
\eq{
\Omega(J,n)=\Omega_0\sqrt{n-J+1/2}
} for a given field-state energy level $\ket{n}$, which is smaller than $\Omega_n$ above and decreases with $J$, explaining why slightly longer interaction times are required with increasing $J$ to achieve the same $\tfrac{\pi}{2}$ pulses [Fig. \ref{fig:TCM optimals}\textbf{(c)}]. However, the actual functional dependence in Fig. \ref{fig:TCM optimals}\textbf{(c)} looks like it follows $\Omega(J/2,n)$ instead of $\Omega(J,n)$, so we will investigate further to elucidate whether this is simply a numerical artifact. That one cannot simply replace $\bar{n}\to\bar{n}-J+1/2$ in Eqs. \eqref{eq:optimal params TCM} to match the replacement $\Omega_n\to\Omega(J,n)$ may be justified by the competition between Rabi frequencies with a variety of values of $n$.
We can approximate the full evolution of our initial state using Eq. \eqref{eq:TCM approx H} and the unitary evolution
\eq{
U_{\mathrm{TC}}=Q\exp\left(-\text{i}\tilde{H}_{\mathrm{TC}}t\right)Q^\dagger.
} Here,
\eq{
Q=\sum_{m=-J}^J\text{e}^{\text{i}\hat{\phi}(J+m)}\otimes\ket{J,m}\bra{J,m}
} is an almost-unitary operator ($\left[Q,Q^\dagger\right]=|0\rangle\langle 0|$) and \eq{
\exp(\text{i} \hat{\phi})=\sum_{n=0}^\infty \ket{n}\bra{n+1}
} is a phase operator for the field (we have reserved the caret for the operator $\hat{\phi}$ in deference to the intricacies of phase operators \cite{BarnettVaccaro2007}). The first transformation leaves the state unchanged:
\eq{
Q^\dagger \sum_n \psi_n\ket{n}\otimes \ket{J,-J}=\sum_n \psi_n\ket{n}\otimes\ket{J,-J}.
}
Next, the effective Hamiltonian enacts a rotation of the atomic states by an angle that depends on the field's energy level in a manner reminiscent of Eq. \eqref{eq:pi/2 pulse on 2J atoms}:
\eq{
&\text{e}^{-\text{i}\tilde{H}_{\mathrm{TC}}t}\sum_n \psi_n\ket{n}\otimes\ket{J,-J}=\sum_n \psi_n\ket{n}\\
&\otimes \sum_{m=-J}^J\sqrt{\binom{2J}{m+J}}\cos^{J-m}\frac{\Omega(J,n)t}{2}\sin^{J+m}\frac{\Omega(J,n)t}{2}\ket{J,m}.
\label{eq:midway approximate psi TCM}
}
Note that the atomic states in the superposition are SU(2)-coherent states that are eigenvalues of the spin operator pointing at different angles for different field energy levels $J_x \sin\left[\Omega(J,n)t\right]-J_z\cos\left[\Omega(J,n)t\right]$; if this was the end of the evolution, an initial field state with definite photon number $n$ would suffice to perfectly enact a rotation by $\Omega(J,n)$ on all of the atoms.
Finally, using
\eq{
\exp(\text{i} \hat{\phi}k)=\sum_{n=0}^\infty \ket{n}\bra{n+k},
} the transformed state becomes
\eq{
\ket{\Psi(t)}=\sum_{n=2J+1}^\infty\sum_{m=-J}^J
\sqrt{\binom{2J}{m+J}}\psi_{n+J+m}\ket{n}\otimes\ket{J,m}\\
\times \cos^{J-m}\frac{\Omega(J,n+J+m)t}{2}\sin^{J+m}\frac{\Omega(J,n+J+m)t}{2},
\label{eq:approx psi(t) TCM}
} where we have restricted our attention to states with $\psi_n= 0$ for $n\leq 2J$ such that $Q$ is unitary.
We can then inspect the properties of this state to see how to optimally achieve the $\tfrac{\pi}{2}$ pulse of Eq. \eqref{eq:pi/2 pulse on 2J atoms}.
For Eq. \eqref{eq:approx psi(t) TCM} to best approximate a $\tfrac{\pi}{2}$ pulse, we would like
\eq{
\psi_{n+k} \cos^{2J-k}\frac{\Omega(J,n+k)t}{2}\sin^{k}\frac{\Omega(J,n+k)t}{2}\approx \frac{1}{2^J}
} for all values of $n$ and $k$ such that the final state is most separable and the atomic state is closest to that of Eq. \eqref{eq:pi/2 pulse on 2J atoms}. The width of the trigonometric terms' distribution changes with $n$ and $k$, so it is not obvious how to choose the appropriate optimal width for the photon-number distribution, although we note that all Gaussian distributions centred at $\bar{n}$ with interaction times $\Omega(J,\bar{n})t\approx\pi/2$ will converge to the proper limit with large $\bar{n}$. Instead, we can look at the overlap between $\ket{\Psi(t)}$ and a state rotated by $\Theta$:
\eq{
\left|\braket{J,\Theta}{\Psi(t)}\right|^2=
\sum_{n=2J+1}^\infty\left|\sum_{m=-J}^{J}
{\binom{2J}{m+J}}\psi_{n+J+m}\right.\\
\times \cos^{J-m}\frac{\Omega(J,n+J+m)t}{2}\sin^{J+m}\frac{\Omega(J,n+J+m)t}{2}\\
\left.\times \cos^{J-m}\frac{\Theta}{2}\sin^{J+m}\frac{\Theta}{2}\right|^2.
\label{eq:approx fidelity TCM}
} By selecting $\Theta=\pi/2$ and Gaussian photon-number distributions centred at $\bar{n}$, we can optimize this result over all interaction times $\Omega_0 t$ and photon-number variances $\sigma^2$. Exemplary results are plotted in
Fig. \ref{fig:approx infidelities}, with the data from Fig. \ref{fig:TCM optimals} overlain.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{approx_infidelity_variance_time_nbar100_200_500_v2}
\caption{Optimal $\tfrac{\pi}{2}$ pulses on a collection of $2J$ atoms as in Fig. \ref{fig:TCM optimals}, but now optimized using the approximate fidelity of Eq. \eqref{eq:approx fidelity TCM} to permit larger $\bar{n}$ and $J$ ($\bar{n}=100$, $200$, and $500$ are the blue dots, orange triangles, and green diamonds, respectively). The approximation breaks down when $\sqrt{\bar{n}}\gg J$ is not achieved. \textbf{(a)} The error probabilities $1-\mathcal{F}$ with this method are again quite low but are nonnegligible when the approximation fails. \textbf{(b)} The optimal variances best match the curve $\sigma^2=2(\bar{n}-1.2J+0.5)/\pi$. \textbf{(c)} The optimal interaction times best match the curve $\Omega_0 t\sqrt{\bar{n}-1.2J+0.5}=\pi/2$. In all cases, the solutions found using the exact evolution (red $\times$s and purple $+$s for $\bar{n}=100$ and $200$, respectively) best match the $\bar{n}=500$ solution (i.e., the largest energy considered) for all $J$.
}
\label{fig:approx infidelities}
\end{figure}
Comparing the results between the two optimization methods in Fig. \ref{fig:approx infidelities}, we see that they match in the case of large $\bar{n}$. Intriguingly, the replacement $\bar{n}\to \bar{n}-qJ+1/2$ in Eq. \eqref{eq:optimal params TCM} seems to always hold, but the optimal order-unity parameter $q$ does not seem consistent between $\sigma^2_{\mathrm{optimal}}$ and $t_{\mathrm{optimal}}$.
This may be partially explained by setting $|\psi_{\bar{n}+\delta}|\propto \exp(-\delta^2/4\sigma^2)$ for small $\delta$ in Eq. \eqref{eq:approx fidelity TCM}, expanding around $n=\bar{n}$, expanding the sinusoidal terms around large $\bar{n}$, and asking what photon-number variance $\sigma^2$ will cancel all of the $\mathcal{O}(\delta)$ terms; the result is
\eq{
\sigma^2
=\frac{2J+m+m^\prime}{m+m^\prime}\left[\frac{2(\bar{n}-J+1/2)}{\pi}+\frac{J (m+m^\prime)+m^2+m^{\prime 2}}{2}\right],
} which resembles Eq. \eqref{eq:optimal params TCM} with the replacement $\bar{n}\to\bar{n}-J+1/2$ but changes with different values of $m$ and makes no sense (goes negative) when $m+m^\prime\leq 0$.
It is thus always a good approximation to use Eq. \eqref{eq:optimal params TCM} and then update $\bar{n}$ as a function of $J$ for the problem at hand.
A complementary strategy for optimizing the initial field states is to look at the evolution of the expectation values of $J_z$ and $J_x$. For a perfect $\tfrac{\pi}{2}$ pulse, the expectation value of $J_z$ should go from its minimal value $-J$ to $0$, while that of $J_x$ should go from $0$ to its maximal value $J$. In fact, given a fixed total spin, any of the collective operators $J_i$ attaining its maximal eigenvalue is a sufficient condition for the spin state to be in a pure state and thus for there to be no residual entanglement with the light. These goals can then give us constraints on our initial field states in the spirit of transcoherence.
The $J_z$ operator evolves in the Heisenberg picture as \cite{KlimovChumakov1995}
\eq{
U_{\mathrm{TC}}J_z U_{\mathrm{TC}}^\dagger=J_z \cos\hat{\tau}-J_y\sin\hat{\tau}
} for the field operator $\hat{\tau}=\Omega_0 t\sqrt{{a}^\dagger\vphantom{a}{a}-J+1/2}$. Since the initial atomic state has $\expct{(J_x,J_y,J_z)}=(0,0,-J)$, the final state has
\eq{
\langle \Psi(t)|J_z|\Psi(t)\rangle=-J\expct{\cos\hat{\tau}},
} where the final expectation value is taken with respect to the initial field state. For this quantity
\eq{
\expct{\cos\hat{\tau}}=\sum_n |\psi_n|^2\cos\left(\Omega_0 t\sqrt{n-J+1/2}\right)
}
to vanish, the photon-number distribution should be strongly peaked around $\expct{\tau}=\pi/2$, again corresponding to a classical $\tfrac{\pi}{2}$ pulse with average field strength $\bar{n}$ and interaction time $\Omega(J,\bar{n})t=\pi/2$.
We then look at the evolution of $J_x$. Since we already have an expression for the evolved state, we can directly calculate
\eq{
\langle \Psi(t)|J_x|\Psi(t)\rangle=\sum_{n=2J+1}^\infty\sum_{m=-J}^J\binom{2J}{J+m}(J-m)\psi_{n+J+m}\psi_{n+J+m+1}^*\\
\times \cos^{J-m}\frac{\Omega(J,n+J+m)t}{2}\sin^{J+m}\frac{\Omega(J,n+J+m)t}{2}\\
\times \cos^{J-m-1}\frac{\Omega(J,n+J+m+1)t}{2}\sin^{J+m+1}\frac{\Omega(J,n+J+m+1)t}{2}.
} Expanding around $n+J=\bar{n}+\delta$ for small $\delta$ and again using $|\psi_{\bar{n}+\delta}|\propto\exp(-\delta^2/4\sigma^2)$, we find
\eq{
\frac{\psi_{n+J+m}\psi_{n+J+m+1}^*}{|\psi_{\bar{n}}|^2}\approx 1-\frac{(\delta+m)^2+(\delta+m+1)^2}{4 \sigma^2}+\mathcal{O}(\sigma^{-4}).
} The sinusoidal terms expand to leading order in $\bar{n}-J+1/2$ as
\eq{
4^{-J}\left\{1+\frac{\pi \left[2 \delta m+\delta+2 m (m+1)+1\right]}{4\left(\bar{n}-J+1/2\right)}\right\}
\nonumber
} Multiplying these terms by the coefficients $\binom{2J}{J+m}(J-m)$ and summing from $m=-J$ to $m=J$ yields, to leading order in $\sigma^2$ and $\bar{n}-J+1/2$,
\eq{
\langle \Psi(t)|J_x|\Psi(t)\rangle\approx J\sum_{\delta} |\psi_{\bar{n}}|^2\left[ 1-\frac{ 2 \delta^2+J}{4 \sigma^2}+\frac{\pi J}{4 (\bar{n}-J+1/2)}\right].
} Performing the sum from $\delta=-l$ to $\delta=l$ for some $l$, the leading-order terms cancel each other when
\eq{
\sigma^2=\frac{\bar{n}-J+1/2}{\pi }\left(1+\frac{2l(l+1)}{3J}\right).
} This is exactly the result of Eq. \eqref{eq:optimal params TCM} with the replacement $\bar{n}\to\bar{n}-J+1/2$ and the number-squeezing factor being replaced as $\tfrac{2}{\pi}\to \left(1+\tfrac{2l(l+1)}{3J}\right)/\pi$; if the normalization is given by $|\psi_{\bar{n}}|^2\approx 1/\sqrt{2J+1}$, we find $l=(\sqrt{2J+1}-1)/2$ and the number-squeezing factor being exactly $\tfrac{2}{\pi}$ together yield the best result for $\langle \Psi(t)|J_x|\Psi(t)\rangle\approx J$. The overall dependence of the optimal variance $\sigma^2$ on $\bar{n}$ and $J$ is now apparent and considerations of different ranges of the sum over $\delta$ change the dependence of the optimal variance on $J$.
\subsection{Perfect pulses cannot be generated for the Tavis-Cummings interaction}~
The evolution of the TCM can always be solved exactly \cite{TavisCummings1968,Scharf1970,HeppLieb1973} but requires the intricate solution of the Bethe ansatz equations \cite{Bogoliubovetal1996,Luetal2021arxiv}. We solve this system of equations for the case of $N=2$ atoms, given the initial state $\sum_n \psi_n\ket{n}\otimes \ket{J,-J}$, to show that perfect $\tfrac{\pi}{2}$ pulses are not generally possible. Since total excitation number is conserved and the atomic subspace is spanned by only three states $\ket{1,\pm 1}$ and $\ket{1,0}$, we can analytically solve this model.
The evolved state at any time $t$ is (using $\ket{m,n}\equiv\ket{m}\otimes\ket{J,n}$ for brevity in this subsection alone):
\eq{
\ket{\Psi(t)}&=\psi_0\ket{0,-1}+\psi_1\left(\cos\frac{\Omega_0 t}{\sqrt{2}}\ket{1,-1}-\text{i}\sin\frac{\Omega_0 t}{\sqrt{2}}\ket{0,0}\right)\\
&+\sum_{n\geq 2} \psi_n\ket{n,-1}\left(\frac{n-1}{2n-1}+\frac{n}{2n-1}\cos\frac{\Omega_0 t\sqrt{2n-1}}{\sqrt{2}}\right)\\
&+ \psi_n\ket{n-2,1}\frac{\sqrt{n(n-1)}}{2n-1}\left(-1+\cos\frac{\Omega_0 t\sqrt{2n-1}}{\sqrt{2}}\right)\\
&-\text{i} \psi_n\ket{n-1,0}\sqrt{\frac{n}{2n-1}}\sin\frac{\Omega_0 t\sqrt{2n-1}}{\sqrt{2}}.
}
We found this using the eigenstates
\eq{
\ket{n\pm}=\frac{1}{\sqrt{2}}\left(\sqrt{\frac{n}{2n-1}}\ket{n,-1}\pm\ket{n-1,0}+\sqrt{\frac{n-1}{2n-1}}\ket{n-2,1}\right)
} with eigenvalues $\pm\sqrt{2n-1}\Omega_0/\sqrt{2}$ and the null eigenstate
\eq{
\ket{n0}=-\sqrt{\frac{n-1}{2n-1}}\ket{n,-1}+\sqrt{\frac{n-1}{2n-1}}\ket{n-2,1}.
} Can this ever perfectly create the desired pulse?
Projecting onto $\ket{0}$ for the field state, the first requirement for the atoms to be in the correct rotated state is that
\eq{
\psi_0\ket{-1}-\text{i} \psi_1\sin\frac{\Omega_0 t}{\sqrt{2}}\ket{0}+\psi_2\frac{\sqrt{2}}{3}\left(\cos\frac{\Omega_0 t\sqrt{3}}{\sqrt{2}}-1\right)\ket{1},\\
\propto \ket{-1}+\sqrt{2}\ket{0}+\ket{1}.
} This immediately yields two constraints for the three free parameters $\psi_1$, $\psi_2$, and $t$:
\eq{
\psi_0&=\psi_2\frac{\sqrt{2}}{3}\left(\cos\frac{\Omega_0 t \sqrt{3}}{\sqrt{2}}-1\right)\\
\psi_0\sqrt{2}&=-\text{i} \psi_1\sin\frac{\Omega_0 t}{\sqrt{2}}.
} If any of these three coefficients vanishes then they all must, unless the timing spares $\psi_1$ or $\psi_2$ from needing to vanish. The next requirement found by projecting the field onto state $\ket{1}$ is that
\eq{
&\psi_1\cos\frac{\Omega_0 t}{\sqrt{2}}\ket{-1}-\text{i} \psi_2\sqrt{\frac{2}{3}}\sin\frac{\Omega_0 t\sqrt{3}}{\sqrt{2}}\ket{0}
\\
&+\psi_3\frac{\sqrt{6}}{5}\left(\cos\frac{\Omega_0 t\sqrt{5}}{\sqrt{2}}-1\right)\ket{1}
\propto \ket{-1}+\sqrt{2}\ket{0}+\ket{1}.
} Again, if any of the coefficients vanishes then they all must, unless rescued by an exact timing prescription. This introduces another required relationship between $\psi_1$ and $\psi_2$,
\eq{
\psi_1\sqrt{2}\cos\frac{\Omega_0 t}{\sqrt{2}}=-\text{i} \psi_2\sqrt{\frac{2}{3}}\sin\frac{\Omega_0 t\sqrt{3}}{\sqrt{2}},
}
which together impose the timing requirement:
\eq{
\frac{\frac{2}{3}\left(\cos\frac{\Omega_0 t \sqrt{3}}{\sqrt{2}}-1\right)}{-\text{i}\sin\frac{\Omega_0 t}{\sqrt{2}}}=
-\text{i} \frac{\frac{1}{\sqrt{3}}\sin\frac{\Omega_0 t \sqrt{3}}{\sqrt{2}}}{\cos\frac{\Omega_0 t}{\sqrt{2}}}\\
\frac{2}{\sqrt{3}}\left(\cos\frac{\Omega_0 t \sqrt{3}}{\sqrt{2}}-1\right)=-\sin\frac{\Omega_0 t \sqrt{3}}{\sqrt{2}}
\tan\frac{\Omega_0 t}{\sqrt{2}}.
} This only holds when $\Omega_0 t \sqrt{3/2}=2k\pi,\,k\in\mathds{N}$. But then we find that $\psi_0=0$, $\psi_1=0$, and so on, and we have no solution. So it is impossible to perfectly solve this problem for two atoms.
Even considering the general case where the transformed state approximately follows Eq. \eqref{eq:approx psi(t) TCM}, the parameters cannot be chosen precisely enough such that no excitation escapes. Consider that there will always be some probability of finding the atom in any state $\ket{J,m}$ for rotations with $0<\Theta<\pi$. Every field state $\ket{n}$ must be in a tensor product with the same superposition of atomic states, so, whenever any coefficient $\psi_{n+J+m}$ is nonzero, every other coefficient $\psi_{n+J+m^\prime}$ must also be nonzero for all $-J\leq m^\prime\leq J$. Since $n$ can vary by $1$ and the range of values of $m$ must vary by at least $2$ for any $J>1/2$ (i.e., for anything but the JCM case of a single atom), there can be no maximal coefficient beyond which all of the coefficients vanish. Some excitation will always leak out of the subspaces in which the atoms are in the proper rotated state. The sole alternative to setting coefficients to zero is that one of the sinusoidal terms vanishes for a particular $n$ and $m$ and $t$. This suffices in the case of the JCM because there is only a single maximal coefficient that must be constrained, but is insufficient in the $J>1/2$ case where entire ranges of coefficients must vanish in the field state's maximal photon number. This is why, although squeezing is beneficial regardless of $J$, the true transcoherent states that perfectly enact $\Theta$ rotations only exist for the case of $J=1/2$. We note incidentally that a true transcoherent state for arbitrary $J$ can be achieved in the trivial case of a rotation by $0$ (i.e., the field should be in the vacuum state) and for $\Theta=\pi$ with a Fock state. This is exact for the JCM and holds precisely within the approximations of the TCM that lead to Eq. \eqref{eq:approx psi(t) TCM}, because then state in Eq. \eqref{eq:midway approximate psi TCM} is separable and remains separable following the application of the almost-unitary operator $Q^\dagger$ on the state.
\subsection{Discussion}~
Photon-number squeezing increases the probability of successfully imparting a $\tfrac{\pi}{2}$ pulse on a collection of ground-state atoms. Given the ubiquity of this result for other initial atomic states in the JCM, we expect photon-number squeezing to similarly enhance arbitrary pulse areas in the TCM for unknown initial atomic states. This problem is slightly more numerically cumbersome due to the lack of a closed-form expression for the fidelities averaged over all initial atomic states, so we leave this expectation as a conjecture that could be evidenced by numerical investigations of Eq. \eqref{eq:approx fidelity TCM} with various $\Theta$.
Earlier studies showed that a collection of partially excited atoms in the final state of Eq. \eqref{eq:pi/2 pulse on 2J atoms} will interact with a coherent field state to number squeeze the latter \cite{Retamaletal1997}. It thus comes as no surprise that photon-number-squeezed states are useful for enacting, in some sense, the reverse of this process. This idea of matching the squeezing to the interplay between different Rabi frequencies should be useful for a variety of light-matter-interaction protocols.
\section{Arbitrary coherence cannot be generated in the presence of nonzero detuning}~
All of the models thus far have dealt with resonant interactions between the field mode and the atoms. We show here that the same can \textit{never} be achieved \textit{exactly} for nonzero values of detuning.
One reason to consider nonzero detuning is to establish transformations like the Hadamard transform. This sends an atom in its ground state to an even superposition of its ground and excited state, but does so by a $\pi$ rotation of the spin vector about the $\tfrac{x+z}{\sqrt{2}}$-axis of the Bloch sphere, instead of the $\tfrac{\pi}{2}$ pulses we have been discussing here. For large field states, the interaction parts of the JCM and TCM Hamiltonian effectively rotate the average spin vector about some axis in the $xy$-plane, so the only method for truly imparting a Hadamard gate is by the introduction of a nonzero detuning that allows for effective rotations about other spin axes.
The Jaynes-Cummings interaction Hamiltonian at nonzero detuning $\delta$ takes the form
\eq{
H_{\mathrm{II}}=\frac{\delta}{2}\sigma_z+\frac{\Omega_0}{2}\left({a}\sigma_-+{a}^\dagger\vphantom{a}\sigma_+\right).
} The new eigenstates are
\eq{
\ket{+,n}_\delta&=\cos\frac{\alpha_n}{2}\ket{n}\otimes\ket{\mathrm{e}}+\sin\frac{\alpha_n}{2}\ket{n+1}\otimes\ket{\mathrm{g}},\\
\ket{-,n}_\delta&=\sin\frac{\alpha_n}{2}\ket{n}\otimes\ket{\mathrm{e}}-\cos\frac{\alpha_n}{2}\ket{n+1}\otimes\ket{\mathrm{g}}
} and have interaction-picture energies
\eq{
E_\pm(n)=\pm\frac{\Omega(n)}{2},
}
where \eq{
\alpha_n=\tan^{-1}\frac{\Omega_0\sqrt{n+1}}{\delta}
} and we have defined the detuned Rabi frequencies
\eq{
\Omega(n)=\sqrt{\Omega_n^2+\delta^2}=\sqrt{\Omega_0^2(n+1)+\delta^2}.
} An atom initially in its ground state and the field initially in the general state $\sum_n \psi_n\ket{n}$ together evolve to
\eq{
\ket{\Psi(t)}=&\sum_{n=-1}^\infty \psi_{n+1}\left[
-\text{i} \sin\frac{\Omega(n)t}{2}\sin\frac{\alpha_n}{2}\cos\frac{\alpha_n}{2}\ket{n}\otimes\ket{\mathrm{e}}\right.\\
&\left.
\left(\text{e}^{-\text{i}\Omega(n)t/2}\sin^2\frac{\alpha_n}{2}+\text{e}^{\text{i}\Omega(n)t/2}\cos^2\frac{\alpha_n}{2}\right)\ket{n+1}\otimes\ket{\mathrm{g}}\right].
}
We can proceed as usual to find initial field states and interaction times that create separable states with the atom completely coherent. Here, there are more complicated conditions that need to be solved, but there is the extra degree of freedom in the detuning that could account for them.
Projecting the evolved state onto states with definite photon number and requiring the result to be proportional to $\cos\frac{\theta}{2}\ket{\mathrm{g}}+\sin\frac{\theta}{2}\ket{\mathrm{e}}$ leads to the recurrence relation:
\eq{
\frac{\psi_{n+1}}{\psi_n}=-\text{i}\tan\frac{\theta}{2
\frac{\text{e}^{-\text{i}\Omega(n-1)t/2}\sin^2\frac{\alpha_{n-1}}{2}+\text{e}^{\text{i}\Omega(n-1)t/2}\cos^2\frac{\alpha_{n-1}}{2}}{\sin\frac{\Omega(n) t}{2}\sin\frac{\alpha_n}{2}\cos\frac{\alpha_n}{2}}.
} For a given $t$, $\Omega_0$, and $\delta$, this series is uniquely determined. Can we force this series to truncate on both sides?
To not have any probability leak out of the lowest-excitation subspace with the smallest nonzero coefficient $\psi_{n_{\mathrm{min}}}$, we require
\eq{
\sin\frac{\Omega({n_{\mathrm{min}}-1}) t}{2}\sin\frac{\alpha_{n_{\mathrm{min}}-1}}{2}\cos\frac{\alpha_{n_{\mathrm{min}}-1}}{2}=0.
}
This constraint is readily satisfied when $n_{\mathrm{min}}=0$, because $\alpha_{-1}=0$.
For larger $n_{\mathrm{min}}$, this amounts to the requirement
\eq{
\Omega({n_{\mathrm{min}}-1}) t=\Omega_0 t\sqrt{ n_{\mathrm{min}}+\left(\frac{\delta}{\Omega_0}\right)^2}=2m\pi,\quad m\in\mathds{N},
}
which can readily be satisfied by an appropriate interaction time $\Omega_0 t$ and detuning $\delta$.
However, it is impossible to not have any probability leak out of the highest-excitation subspace with the largest nonzero coefficient $\psi_{n_{\mathrm{max}}}$. To do so, we would require both
\eq{
\cos\frac{\Omega({n_{\mathrm{max}}}-1)t}{2}\left(\sin^2\frac{\alpha_{{n_{\mathrm{max}}}-1}}{2}+\cos^2\frac{\alpha_{{n_{\mathrm{max}}}-1}}{2}\right)=0
} and
\eq{
\sin\frac{\Omega({n_{\mathrm{max}}}-1)t}{2}\left(\sin^2\frac{\alpha_{{n_{\mathrm{max}}}-1}}{2}-\cos^2\frac{\alpha_{{n_{\mathrm{max}}}-1}}{2}\right)=0.
} While the former is readily converted into the satisfiable condition
\eq{
\sqrt{\Omega_0^2 n_{\mathrm{max}}+\delta^2}t&=(2l+1)\pi,\quad l\in\mathds{N}
,
}
the latter can \textit{only} be satisfied by \eq{
\alpha_{n_{\mathrm{max}}-1}=\frac{\pi}{2} \quad\Rightarrow\quad \delta=0.
} Any presence of nonzero detuning allows excitations to leak beyond the highest-excitation subspace, so there can never be perfect transfer of coherence from a field state to an atom that leaves no residual entanglement when the interaction is nonresonant. This accords with complete transfer of probability between $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ being impossible with nonzero detuning in the Rabi model with a classical field.
\section{$m$-photon processes}~
\subsection{Beyond the Jaynes-Cummings model}~
What happens when it takes more than one photon to excite an atom? We can consider the nonlinear interaction that requires $m$-photon absorption to transform $\ket{\mathrm{g}}$ to $\ket{\mathrm{e}}$:
\eq{
H=\omega\left(\frac{1}{m}{a}^\dagger\vphantom{a}{a}+\ket{\mathrm{e}}\bra{\mathrm{e}}\right)+\frac{\Omega_0^{(m)}}{2}\left({a}^m\sigma_+ + {a}^\dagger\vphantom{a}^m\sigma_-\right),
} where $\omega$ is the resonance frequency but now each individual photon provides energy $\tfrac{\omega}{m}$ and $\Omega_0^{(m)}$ is the coupling strength that depends on the $m$th-order nonlinearity. This interaction conserves total energy and a form of the total excitation number, as can be seen from its eigenstates
\eq{
\ket{\pm,n}=\frac{\ket{n}\otimes\ket{\mathrm{e}}\pm\ket{n+m}\otimes\ket{\mathrm{g}}}{\sqrt{2}},
\label{eq:JCM eigenstates}
} which now have the quantized-Rabi-like frequencies
\eq{
\Omega_n^{(m)}=\Omega_0^{(m)}\sqrt{(n+m)(n+m-1)\cdots(n+1)}.
} We will work in the interaction picture with Hamiltonian
\eq{
H_{\mathrm{I}}=
\frac{\Omega_0^{(m)}}{2}\left({a}^m\sigma_+ + {a}^\dagger\vphantom{a}^m\sigma_-\right);
} the Schr\"odinger-picture results can thence be obtained with the substitutions $\ket{n}\to\text{e}^{-\text{i}\omega nt/m}\ket{n}$ and $\ket{\mathrm{e}}\to\text{e}^{-\text{i} \omega t}\ket{\mathrm{e}}$.
When the atom is initially in its ground state and the field in state $\sum_n \psi_n\ket{n}$, the evolved state takes the form [c.f. Eq. \eqref{eq:JCM from ground}]
\eq{
\ket{\Psi(t)}
=&\sum_{n=0}^\infty \ket{n}\otimes\left(\psi_n\cos\frac{\Omega_{n-m}^{(m)}t}{2}\ket{\mathrm{g}}-\text{i} \psi_{n+m}\sin\frac{\Omega_n^{(m)} t}{2}\ket{\mathrm{e}}\right).
} Similarly,
when the atom is initially in its excited state and the field in state $\sum_n \psi_n\ket{n}$, the evolved state takes the form [c.f. Eq. \eqref{eq:JCM from excited}]
\eq{
\ket{\Psi(t)}=&\sum_{n=0}^\infty \ket{n}\otimes \left(\psi_n\cos\frac{\Omega_n^{(m)} t}{2}\ket{\mathrm{e}}
-\text{i} \psi_{n-m}\sin\frac{\Omega_{n-m}^{(m)} t}{2}\ket{\mathrm{g}}\right).
}
\subsection{Transcoherent states and beyond}~
What are the optimal field states that can generate arbitrary pulse areas for this nonlinear interaction?
We again seek transformations for which the final state has zero residual entanglement between the atom and the light, such that the atomic state can be used in arbitrary quantum information protocols without degradation.
From the ground state, arbitrary transformations can be performed by field states whose photon-number coefficients satisfy
\eq{
\psi_{n+m}=\text{i}\tan\frac{\theta}{2}\frac{\cos\frac{\Omega_{n-m}^{(m)} t}{2}}{\sin\frac{\Omega_n^{(m)} t}{2}}\psi_n
} to ensure that the amplitudes of $\ket{\mathrm{g}}$ and $\ket{\mathrm{e}}$ in the evolved state in Eq. \eqref{eq:JCM from ground} are equal. This can be satisfied by field states with $\psi_n=0$ for $n> n_{\mathrm{max}}$ for some chosen $n_{\mathrm{max}}\geq 1$, so long as the total interaction time satisfies
\eq{
\Omega_{n_{\mathrm{max}}-1}^{(m)}t=\pi,
} which ensures that the highest-excitation subspace spanned by $\ket{\pm,n_{\mathrm{max}}-1}$ undergoes a $\pi$ pulse. Now, in contrast to the $m=1$ scenario, there are $m$ independent recursion relations that must all truncate at the same time $t$, which cannot occur because the oscillation frequencies cannot have an integer ratio $\Omega_{n_k}^{(m)}/\Omega_{n}^{(m)}$ for any integer $k$. Therefore, in order to exactly produce the desired atomic state, one must use a state of light that sets to zero $m-1$ of the coefficients from $\psi_0$ to $\psi_{m-1}$ and thus that only has population in photon numbers spaced $m$ apart. The alternative, which can still outperform coherent states, is to use a state with a large average number of photons that will approximate the recursion relation by a squeezed state.
The same can be said for the atom initially in its excited state, where now the optimal recursion relation takes the form
\eq{
\psi_{n+m}=-\text{i}\tan\frac{\theta}{2}\frac{\sin\frac{\Omega_{n}^{(m)} t}{2}}{\cos\frac{\Omega_{n+m}^{(m)} t}{2}}\psi_{n}.
}
What are the properties of the field states that approximate these recursion relations in the limit of large numbers of photons? As usual, the optimal interaction time matches the semiclassical one:
\eq{
\Omega_0^{(m)} t&=\frac{\theta}{\sqrt{(\bar{n}+m)(\bar{n}+m-1)\cdots(\bar{n}+1)}}\\
&\approx \frac{\theta}{\sqrt{\bar{n}^m+\frac{m(m+1)}{2}\bar{n}^{m-1}}}
} from the ground state and
\eq{
\Omega_0^{(m)} t\approx\frac{\theta+\pi}{\sqrt{\bar{n}^m+\frac{m(m+1)}{2}\bar{n}^{m-1}}}
} from the excited state. The ratios in the recursion relations obey
\eq{
\tan\frac{\theta}{2}\frac{\cos\frac{\Omega_{\bar{n}+\delta-m}^{(m)} t}{2}}{\sin\frac{\Omega_{\bar{n}+\delta}^{(m)} t}{2}}\approx 1-\frac{m }{2 \bar{n}\sinc\theta}\left(\delta-m\sin^2\frac{\theta}{2}\right)
} and (recall that starting in $\ket{\mathrm{e}}$ requires $\sinc(\theta+\pi)$ to be negative in order to rotate by $\theta+\pi$ to $\ket{\theta}$)
\eq{
-\tan\frac{\theta}{2}\frac{\sin\frac{\Omega_{\bar{n}+\delta}^{(m)} t}{2}}{\cos\frac{\Omega_{\bar{n}+\delta+m}^{(m)} t}{2}}\approx 1+\frac{m}{2\bar{n}\sinc(\theta+\pi)}\left(\delta+m\sin^2\frac{\theta+\pi}{2}\right).
} For comparison, an exponential distribution for coefficients separated by $m$ obeys
\eq{
\left|\frac{\psi_{\bar{n}+\delta+m}}{\psi_{\bar{n}+\delta}}\right|=\exp\left[-\frac{(\delta+m)^2-\delta^2}{4\sigma^2}\right]\approx1-\frac{m}{2\sigma^2}\left(\delta+\frac{m}{2}\right).
} We see that \textit{no different number squeezing is needed} for $m$-photon processes relative to the JCM, with the mean photon numbers being shifted from the ideal classical ones by a factor of $\pm m\sin^2\frac{\theta}{2}$.
\color{black}
\section{Conclusions}~
We have performed a detailed investigation of the optimal field states for transferring arbitrary amounts of coherence to individual and collections of atoms. When a single atom is initially its ground or excited state, there exists a field state to rotate it by an arbitrary amount in an arbitrarily short amount of time that generates no residual entanglement with the field. Since these unitary operations can be reversed and, therefore, composed, we have thus found field states for perfectly performing arbitrary rotations on arbitrary atomic states without resorting to any semiclassical approximations.
The perfect field states and interaction times depend on knowing the initial state of the atom. When this initial state is unknown, field states with their photon-number distributions squeezed relative to coherent states retain an advantage in their ability to perform arbitrary operations on some average atomic state. These squeezed field states can then be useful for tasks like creating logic gates for quantum computers.
We showed that squeezed light is also useful for transferring coherence to a collection of atoms or to any spin system. This cements squeezed light as a resource beyond traditional realms such as metrology \cite{LIGO2011} and computation \cite{Madsenetal2022}. More squeezing is required to perform larger rotations on atomic states, so the continuing improvements in squeezing capabilities motivated by said traditional applications provides increasing benefit to our light-matter-interaction scenarios.
Finally, we found that generalizations of the JCM to nonlinear processes responsible for high-harmonic generation and $m$-photon absorption can also have transcoherent states and beyond. The optimal field states for rotation atoms by $\theta$ through nonlinear interactions are \textit{also} squeezed in their photon-number variances by a factor of $\sinc\theta$. All of the results from linear interactions thus extend \textit{mutatis mutandis} to nonlinear ones.
Transcoherent states and beyond stimulate many questions that may be explored in future work. Are these states easiest to generate in a cavity; if so, does the number of atoms in the cavity affect how the field states may enter the cavity? Can they be generated in an optomechanical system using phonons instead of photons as the bosonic mode? If one instead uses a beam of light travelling through free space, for which the JCM is no longer the exact model \cite{KiilerichMolmer2019,KiilerichMolmer2020}, how does squeezing affect coherence transfer to atoms? Does transcoherence lead to better design of field states in the presence of nonzero detuning between the atomic energy gap and the field frequency; can there still be an advantage in coherence transfer due to squeezing? Are there other interactions beyond the JCM and the generalizations considered here for which squeezing can confer additional advantages relative to coherent light? These exciting questions are but a fraction of what can now be studied.
Our previous work explored quantum catalysis as a particularly useful application of transcoherent states that generate perfect $\tfrac{\pi}{2}$ pulses to individual atoms. Now that the toolbox has been expanded to arbitrary rotations and arbitrary numbers of atoms, we strongly believe that our transcoherent states and beyond will be important to application in which light is used to precisely control atoms in any desired fashion.
\begin{acknowledgments}
AZG and KH acknowledge that the NRC headquarters is located on the traditional unceded territory of the Algonquin Anishinaabe and Mohawk people. This work was supported by NSERC. AMS acknowledges support as a CIFAR Fellow. AZG thanks Andrei Klimov for useful discussions.
\end{acknowledgments}
|
1,108,101,565,481 | arxiv | \section{Alanine dipeptide in vacuum}
The conformational transition between conformers $\alpha$ and $\beta$ of the alanine dipeptide molecule
has been extensively studied as an example of rare event~\cite{SM-2005-JCP-Ren-Weinan, SM-2013-PRL-Tiwary-Parrinello,
SM-2014-JCTC-Salvalaglio-Parrinello,SM-2017-JMM-Cuny-Mineva,SM-2018-arXiv-Gimondi-Salvalaglio}.
The two stable states are differentiated by the values
of the backbone dihedral angles $\Phi$ and $\Psi$, as defined in the inset of Fig.~\ref{figSM1} (left panel),
and are separated by a activation free energy (FE) barrier of $\approx 8$ kcal/mol.
We used a Langevin thermostat to enforce the temperature~\cite{SM-2007-JCP-Bussi-Parrinello}, a time step of $0.2$ fs,
AMBER03 forcefield~\cite{SM-2005-JCC-Case-Woods} and GROMACS~5.1 molecular dynamics code~\cite{SM-2001-JMM-Lindahl-VanDerSpoel}
patched with PLUMED~2.3~\cite{SM-2014-CPC-Tribello-Bussi}.
To reconstruct the FE surface, we performed well-tempered metaD (WT-metaD) atomistic
simulations~\cite{SM-2008-PRL-Barducci-Parrinello,SM-2014-PRL-Dama-Voth} using both torsional angles $\Phi$ and $\Psi$
as collective variables (CVs), a bias factor of $15$ at $300$ K.
The initial Gaussian height was $1.25$ kJ/mol, the width was $0.25$ rad, and the deposition stride was $0.12$ ps.
A single alanine dipeptide molecule was kept in a periodic cubic box of edge $\approx 3$ nm.
The LINCS algorithm~\cite{SM-1997-JCC-Hess-Fraaije} handled bond constraints while
the particle-mesh Ewald scheme~\cite{SM-1993-JCP-Darden-Pedersen} was used to treat long-range
electrostatic interactions. The non-bonded van der Waals cutoff radius was $0.8$ nm.
\begin{figure}[b]
\includegraphics[width=1.0 \textwidth, angle=-0]{figSM1.eps}
\caption{\textbf{Left panel:} FE surface associated with the conformational transition between conformers
$\alpha$ and $\beta$ of alanine dipeptide in vacuum as a function of the two dihedral angles $\Phi$ and $\Psi$ (see inset).
The contour lines are every half $k_B T$. The typical MFEP obtained within the steepest descent
framework~\cite{SM-2013-JCP-Chen-Xiao} is shown in red color along with the locations of the transition states, TS1 and TS2.
%
\textbf{Middle panel:} FE profile of the alanine dipeptide in vacuum as a function of the progression along the typical MFEP
(normalized to unity) obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao}.
The nonlinear least-squares Marquardt-Levenberg algorithm was implemented to fit the parameters $\omega_0$ and $\omega_m$,
measured in the equilibrium and metastable states, respectively.
%
\textbf{Right panel:} FE profile of alanine dipeptide in vacuum reconstructed along the dihedral angle $\Phi$ obtained within
the WT-metaD framework. The nonlinear least-squares Marquardt-Levenberg algorithm was implemented to fit the parameters
$\omega_0$ and $\omega_m$, measured in the equilibrium and metastable states, respectively.
}
\label{figSM1}
\end{figure}
\\
Fig.~\ref{figSM1} (left panel) shows the FE surface for this molecule, along with the rough locations of the stable states.
The two minima $C_{7eq}$ and $C'_{7eq}$ are combined in the $\beta$ basin
as in Refs.~\cite{SM-2000-Bolhuis-Chandler,SM-2013-PRL-Tiwary-Parrinello}.
The location of the metastable basins and the heigh of the FE barrier are in agreement with the ones found
in the literature~\cite{SM-2005-JCP-Ren-Weinan,SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello}.
We determined the value of the FE of formation, $\Delta F^0_{\alpha \beta} = F(\beta) - F(\alpha) = 3.6 \pm 0.4~k_B T$
and the activation energies, $\Delta F_{\alpha \to \beta} = 9.1 \pm 0.1~k_B T$
and $\Delta F_{\beta \to \alpha} = 12.6 \pm 0.1~k_B T$ along the MFEP obtained within the steepest descent
framework~\cite{SM-2013-JCP-Chen-Xiao}, as shown in Fig.~\ref{figSM1} (middle panel). The FE of formation,
$\Delta F^*_{\alpha \beta} = 4.7 \pm 0.1~k_B T$, defined in term of the probability distribution of $\Phi$ and $\Psi$,
was computed considering the successive isosurfaces in the FE basins depicted in Fig.~\ref{figSM1} (left panel)
as integration domains (cf. Eq.~(13) in the main text).
In Fig.~\ref{figSM1} (middle panel), we show the FE of the peptide as a function of the progression along
the typical MFEP (normalized to unity). The nonlinear least-squares Marquardt-Levenberg algorithm was implemented
to fit the parameters $\omega_0$ and $\omega_{m}$ with Gaussian distribution. We obtained
$\omega_{m} = 5.0 \pm 0.1$ and $\omega_0 = 3.3 \pm 0.1$ for the metastable ($\alpha$) and equilibrium ($\beta$) basins, respectively.
Assuming that the effective friction coefficient, $\gamma$,
remains unchanged in the transitions $\alpha \leftrightarrow \beta$, one obtains the transition rate ratio,
$k_{\beta \to \alpha}/k_{\alpha \to \beta} = (5.6 \pm 2.0) \times 10^{-2}$.
In Fig.~\ref{figSM1} (right panel), we show the FE profile of the peptide along the dihedral angle $\Phi$ reconstructed within
the WT-metaD framework. We determined the value of the FE of formation, $\Delta F^0_{\alpha \beta} = 3.9 \pm 0.1~k_B T$
and the activation energies, $\Delta F_{\alpha \to \beta} = 7.7 \pm 0.1~k_B T$
and $\Delta F_{\beta \to \alpha} = 11.6 \pm 0.1~k_B T$. The nonlinear least-squares Marquardt-Levenberg algorithm was implemented
to fit the parameters $\omega_0$ and $\omega_{m}$ with Gaussian distribution. We obtained
$\omega_{m} = 6.8 \pm 0.1$ and $\omega_0 = 4.7 \pm 0.2$ for the metastable ($\alpha$) and equilibrium ($\beta$) basins, respectively.
The \textit{standard} KT yields $k^{(st)}_{\beta \to \alpha}/k^{(st)}_{\alpha \to \beta} = (1.4 \pm 0.2) \times 10^{-2}$.
We extended the Metadynamics scope~\cite{SM-2010-JCP-Xin-Hamelberg,SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello}
to estimate the mean transition times between the metastable ($\alpha$) and the equilibrium ($\beta$) states
of the peptide. WT-metaD was performed using both torsional angles $\Phi$ and $\Psi$ as CV.
We denote by $\tau$, the mean transition time over the barrier from the states,
and by $\tau_M$, the mean transition time for the metadynamics run. The latter changes as the simulation
progresses and is linked to the former through the acceleration factor
$\alpha(t) \equiv \langle e^{\beta V(s,t)} \rangle_M = \tau/\tau_M(t)$, where the angular brackets
$\langle \dots \rangle_M$ denote an average over a metadynamics run confined to the metastable basin,
and $V(s,t)$ is the metadynamics time-dependent bias. To avoid depositing bias in the transition state region,
we increase the time lag between two successive Gaussian depositions in the WT-metaD
framework~\cite{SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello} to $20$ ps and decrease the bias
factor to $5$. The statistics for $\tau_{\alpha \to \beta}^{(\textrm{num})}$ and $\tau_{\beta \to \alpha}^{(\textrm{num})}$
conformed to a Poisson distribution with means $\mu_{\alpha \to \beta}= 5 \pm 2$~ns and $\mu_{\beta \to \alpha} = 125 \pm 37$~ns
and variance $\lambda_{\alpha \to \beta}=6$~ns and $\lambda_{\beta \to \alpha} = 116$~ns, respectively .
The statistics obey a two-sample Kolmogorov-Smirnov test~\cite{SM-2014-JCTC-Salvalaglio-Parrinello} with $p$-value
equal to $0.81$ and $0.76$, respectively.
This yields the numerical ratio $k^{(num)}_{\beta \to \alpha}/k^{(num)}_{\alpha \to \beta} = (4.0 \pm 1.5) \times 10^{-2}$.
\section{Linear DNA denaturation bubble}
The cooperative opening and closure of a sequence of DNA consecutive base-pairs (bps) is central in biological mechanisms.
The associated characteristic times measured experimentally by Altan-Bonnet \textit{et al.}~\cite{SM-2003-PRL-Altan-Krichevsky}
showed large bubble lifetimes of $20 - 100~\mu$s and nucleation time of several $m$s.
We use the DNA model of Refs.~\onlinecite{SM-2013-PRE-Dasanna-Manghi,SM-2015-JCP-Sicard-Manghi},
where the mesoscopic DNA model consists in two interacting bead-spring chains each made of $N = 50$ beads
(of diameter $a = 0.34$ nm) at position $\textbf{r}_i$, with a AT-rich region of $30$ bps in the middle,
and a GC region of $10$ bps at each extremity. The Hamiltonian is $\mathcal{H} =\mathcal{H}_{el}^{(1)}
+ \mathcal{H}_{el}^{(2)} + \mathcal{H}_{tor} + \mathcal{H}_{int}$, where the first two contributions are elastic
energies of the strands $j=1,2$ which include both stretching and bending energies
\begin{equation}
\mathcal{H}_{el}^{(j)} = \sum_{i=0}^{N-1} \frac{\kappa_s}{2}(r_{i,i+1}-a_\textrm{ref})^2
+ \sum_{i=0}^{N-1}\frac{\kappa_\theta}{2}(\theta_i-\theta_\textrm{ref})^2.
\end{equation}
The stretching modulus, $a^2\beta_0 \kappa_s = 100$, is a compromise between numerical efficiency
and experimental values~\cite{SM-Hugel-PRL2005}, where $\beta_0^{-1} = k_B T_0$ is the thermal energy,
$T_0 = 300$ K is the room temperature, and $a_\textrm{ref}=0.357$ nm. The bending modulus is large, $\beta_0 \kappa_\theta = 600$,
to maintain the angle between two consecutive tangent vectors along each strand $\theta_i$ to the
fixed value $\theta_\textrm{ref} = 0.41$ rad. Each strand is thus modeled as a freely rotating chain (FRC)~\cite{SM-Grosberg-AIP1994}.
The third and fourth terms of $\mathcal{H}$ are the torsional energy and hydrogen-bonding interactions, respectively.
The torsional energy is modeled by a harmonic potential
\begin{equation}
\mathcal{H}_{tor} = \sum_{i=0}^{N-1} \frac{\kappa_{\phi,i}}{2}(\phi_i-\phi_\textrm{ref})^2 ,
\end{equation}
where $\phi_i$ is defined as the angle between two consecutive base-pair vectors
$\brho_i \equiv \textbf{r}_i^{(1)}-\textbf{r}_i^{(2)}$ and $\brho_{i+1}$ ($\phi_\textrm{ref} = 0.62$ rad).
The stacking interaction between base pairs is modeled through a $\kappa_{\phi,i}$ that depends on
the value of the \textit{bare} dsDNA torsional modulus $\kappa_\phi$, and the distances between complementary bases,
$\kappa_{\phi,i} = \kappa_\phi [1-f(\rho_i)f(\rho_{i+1})]$, where
\begin{equation}
f(\rho_i) = \frac{1}{2}\Big[1+\erf\Big(\frac{\rho_i -\rho_b}{\lambda'}\Big)\Big],
\label{stacking}
\end{equation}
and $\rho_i =|\brho_i|$. Hence, $\kappa_{\phi,i} = \kappa_\phi$ in the dsDNA state and $\kappa_{\phi,i} = 0$
in the ssDNA one. The actual values in the dsDNA state after equilibration, $\kappa^*_{\phi,\rm ds}$,
are however different from the prescribed values, $\kappa_{\phi}$, due to thermal fluctuations and non-linear potentials
entering the Hamiltonian.
The hydrogen-bonding interaction is modeled by a Morse potential
\begin{equation}
\mathcal{H}_{int} = \sum_{i=0}^{N-1} A (e^{-2\frac{\rho_i-\rho_\textrm{ref}}{\lambda}} -2e^{-\frac{\rho_i-\rho_\textrm{ref}}{\lambda}}) ,
\end{equation}
where $\rho_\textrm{ref}=1$ nm, $\lambda=0.2$ nm, and $\beta_0 A=8$ and $12$ for AT and GC bonding, respectively,
as in Refs.~\onlinecite{SM-Dasanna-EPL2012, SM-2013-PRE-Dasanna-Manghi,SM-2015-JCP-Sicard-Manghi}.
The fitted values for the dsDNA persistence length and the pitch are $\ell_{\rm ds}\simeq160$~bps
and $p = 12$~bps for the relevant range of $\beta_0\kappa_\phi$ we are interested in, which are comparable to
the actual dsDNA values ($\ell_{\rm ds}\simeq150$~bps and $p= 10.4$~bps). The ssDNA persistence length
is $\ell_{\rm ss} = 3.7$~nm, compatible with experimental measurement~\cite{SM-Tinland-Macro1997},
even though in the upper range of measured values.
The evolution of $\textbf{r}_i(t)$ is governed by the overdamped Langevin equation,
integrated using a Euler's scheme,
\begin{equation}
\zeta \frac{d\textbf{r}_i}{dt} = -\nabla_{\textbf{r}_i}\mathcal{H}({\textbf{r}_j}) + \mathbf{\xi}(t) ,
\end{equation}
where $\zeta=3\pi\eta a$ is the friction coefficient for each bead of diameter $a$ with
$\eta=10^{-3}$ Pa.s the water viscosity.
The diffusion coefficient, $D_\textrm{diff} \equiv k_BT_0/3\pi\eta a$, thus takes into account
the level of coarse-graining of the mesoscopic model involved in the kinetics associated
to the smoothed free-energy landscape~\cite{SM-Murtola-PCCP2009}.
The random force of zero mean $\mathbf{\xi}_i(t)$ obeys
the fluctuation-dissipation relation $\langle \mathbf{\xi}_i(t).\mathbf{\xi}_i(t')\rangle =6k_BT\zeta\delta_{ij}\delta(t-t')$.
Lengths and energies are made dimensionless in the units of $a=0.34$ nm and $k_BT_0$, respectively.
The dimensionless time step is $\delta\tau = \delta t k_B T_0/(a^2\zeta)$, set to $5 \times 10^{-4}$
($\delta t=0.045$ ps) for sufficient accuracy~\cite{SM-Dasanna-EPL2012,SM-2013-PRE-Dasanna-Manghi,SM-2015-JCP-Sicard-Manghi}.
This set of parameters induces zipping velocities $v \approx 0.2-2$ bp/ns, compatible with experimental
measurements~\cite{SM-Bustamante-COSB2000}.\\
\begin{figure}[b]
\includegraphics[width=1.0 \textwidth, angle=-0]{figSM2.eps}
\caption{\textbf{Left panel} FE surface associated with the DNA bubble closure/nucleation mechanism projected
along the maximal distance between paired bases $\rho_{\textrm{max}}$ and the minimal twist angle between successive bps,
$\phi_{\textrm{min}}$ (see inset). The contour lines are every two $k_B T$. The two stables basins associated with the opened (op) and closed (cl) states of the DNA bubble and the typical MFEP obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao}
are shown (red).
\textbf{Middle panel} FE of the DNA bubble as a function of the progression along the typical MFEP (normalized to unity)
obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao}. The nonlinear least-squares Marquardt-Levenberg algorithm was implemented to fit the parameters $\omega_0$ and $\omega_m$, measured in the equilibrium and metastable states, respectively.
\textbf{Right panel} FE profile of the system along $\rho_{\textrm{max}}$ reconstructed within the WT-metaD framework.
The nonlinear least-squares Marquardt-Levenberg algorithm was implemented to fit the parameters
$\omega_0$ and $\omega_m$, measured in the equilibrium and metastable states, respectively.
}
\label{figSM2}
\end{figure}
To reconstruct the FE surface, we performed WT-metaD coarse-grained simulations with the width of the DNA bubble,
$\rho_{\max}(t)$, as CV using the version 2.3 of the plugin for free-energy calculation, named PLUMED \cite{SM-2014-CPC-Tribello-Bussi}
According to the algorithm introduced by Barducci \textit{et al.}~\cite{SM-2008-PRL-Barducci-Parrinello,SM-2009-JCC-Bonomi-Parrinello}
a Gaussian is deposited every $25$ ps with intial height of $0.1\,k_B T$ and a bias factor of
$5$ at $T=300$ K. The resolution of the recovered free-energy landscape is determined by the width
of the Gaussians $\sigma = 0.1$ in units of the CV.
As described in previous work~\cite{SM-2015-JCP-Sicard-Manghi}, we put a wall
at $\rho_{\max} \approx 4$ nm to prevent the system to escape from the metastable state
(and therefore entering in the zipping regime, \textit{i.e.} a far from equilibrium
process~\cite{SM-Dasanna-EPL2012, SM-2013-PRE-Dasanna-Manghi}).
We checked that a slight change in the position of the wall ($\rho_{\max}=3.5,4,4.5,5.5,7$ nm) does not change significantly
the results, particularly the positions of the local minimum and the saddle, as well as the barrier height.
To explore the \textit{slow} entropic contribution associated to the DNA bubble metastable basin we chose to follow the
evolution of the minimal twist angle $\Phi_{\textrm{min}}$ inside the bubble~\cite{SM-2015-JCP-Sicard-Manghi}, as shown in the inset
in Fig.~\ref{figSM2} (left panel), reconstructed afterwards using the \textit{reweighting technique}
of Bonomi et al.~\cite{SM-2009-JCC-Bonomi-Parrinello}.\\
The analysis of the FE surface associated with the bubble closure and opening mechanisms,
as shown in Fig.~\ref{figSM2} (left panel), allowed us to determine the value the FE of formation,
$\Delta F^0 = F(op) - F(cl) = 9.0 \pm 0.1~k_B T$ and the activation energies,
$\Delta F_{cl \to op} = 21.8 \pm 0.1~k_B T$
and $\Delta F_{op \to cl} = 12.9 \pm 0.1~k_B T$ along the MFEP obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao},
as shown in Fig.~\ref{figSM2} (middle panel), The FE of formation, $\Delta F^* = 6.7 \pm 0.1~k_B T$,
defined in term of the probability distribution of $\rho_{\textrm{max}}$ and $\Phi_{\textrm{min}}$,
was computed considering the successive isosurfaces in the FE basins depicted in Fig.~\ref{figSM2} (left panel)
as integration domains (cf. Eq.~(13) in the main text).
In Fig.~\ref{figSM2} (middle panel), we show the FE of the system as a function of the progression along
the typical MFEP (normalized to unity). The nonlinear least-squares Marquardt-Levenberg algorithm was implemented
to fit the parameters $\omega_0$ and $\omega_{m}$ with Gaussian or skew-Gaussian distributions depending
on the symmetric or asymmetric nature of the FE profile, respectively. We obtained
$\omega_{m} = 5.3 \pm 0.2$ and $\omega_0 = 64.2 \pm 2.1$ for the metastable ($cl$) and equilibrium ($op$) basins, respectively.
Considering the Rouse model~\cite{SM-2005-JPCM-Ambjornsson-Metzler} valid for flexible polymer chain,
the effective friction coefficient, $\gamma$, in Eq.~$16$ in the main text depends on the number of opened bps,
$N_{\textrm{bub}}$, in the DNA bubble. The typical size observed in the simulations, $N_{bub}\approx 10$ bps,
yields the relation $\gamma_{op}/\gamma_{cl} \approx N_{\textrm{bub}}$ between the effective frictions.
We obtain the transition rate ratio, $k_{cl \to op}/k_{op \to cl} = (1.5 \pm 0.6) \times 10^{-3}$.
In Fig.~\ref{figSM2} (right panel), we show the FE profile of the system along $\rho_{\textrm{max}}$ reconstructed within
the WT-metaD framework. We determined the value of the FE of formation, $\Delta F^0 = 10.3 \pm 0.1~k_B T$
and the activation energies, $\Delta F_{cl \to op} = 22.6 \pm 0.1~k_B T$
and $\Delta F_{op \to cl} = 12.3 \pm 0.1~k_B T$. The nonlinear least-squares Marquardt-Levenberg algorithm was implemented
to fit the parameters $\omega_0$ and $\omega_{m}$ with Gaussian distribution. We obtained
$\omega_{m} = 5.4 \pm 0.4$ and $\omega_0 = 64.3 \pm 1.9$ for the metastable ($\alpha$) and equilibrium ($\beta$) basins, respectively.
The \textit{standard} KT yields $k^{(st)}_{cl \to op}/k^{(st)}_{op \to cl} = (4.0 \pm 0.7) \times 10^{-3}$.
We extended the Metadynamics scope~\cite{SM-2010-JCP-Xin-Hamelberg,SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello}
to estimate the mean transition times between the metastable ($op$) and the equilibrium ($cl$) states
of the DNA bubble. WT-metaD was performed using the width $\rho_{\textrm{max}}$ as CV.
Unlike in the FE surface reconstruction, no wall was added along the CV $\rho_{\textrm{max}}$ in that case.
We denote by $\tau$, the mean transition time over the barrier from the states,
and by $\tau_M$, the mean transition time for the metadynamics run. To avoid depositing bias in the transition state region,
we increase the time lag between two successive Gaussian depositions in the WT-metaD
framework~\cite{SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello} to $700$ ps and decrease the bias
factor to $3$. The statistics for $\tau_{op \to cl}^{(\textrm{num})}$ and $\tau_{cl \to op}^{(\textrm{num})}$
conformed to a Poisson distribution with means $\mu_{op \to cl}= 121 \pm 12~\mu$s and $\mu_{cl \to op} = 67 \pm 8$~ms
and variance $\lambda_{op \to cl}=110~\mu$s and $\lambda_{cl \to op}=67$~ms, respectively .
The statistics obey a two-sample Kolmogorov-Smirnov test~\cite{SM-2014-JCTC-Salvalaglio-Parrinello} with $p$-value
equal to $0.86$ and $0.65$, respectively.
This yields the numerical ratio $k^{(num)}_{cl \to op}/k^{(num)}_{op \to cl} = (1.8 \pm 0.4) \times 10^{-3}$.
\section{Circular DNA denaturation bubble}
The circular DNA (cDNA) is described with the same DNA model used for the linear DNA, where the two single strands
are modeled as freely rotating chains of $N=246$ beads of diameter $a=0.34$ nm with a AT-rich region of $30$ bps
clamped by a closed circular GC region of $(N-30)$ bps. The size of these AT-rich regions was chosen
so that it is larger than the size of the representative \textit{long-lived} denaturation bubbles studied in this work.
The dsDNA minicircle is described by a circular helix where a helical line of radius $\alpha$ coils around
a torus of radius $R$ in the $x-y$ plane. The centers of the beads on each strand initially coincide with
the surface of this torus in Cartesian space according to the equations
\begin{equation}
\left\{
\begin{aligned}
x_n^{(j)} &= \Big( \alpha \sin\Big(n\frac{2\pi}{p} + \psi^{(j)}\Big) + R \Big) \times \cos(n\theta) \\
y_n^{(j)} &= \Big( \alpha \sin\Big(n\frac{2\pi}{p} + \psi^{(j)}\Big) + R \Big) \times \sin(n\theta) \\
z_n^{(j)} &= \alpha \cos\Big(n\frac{2\pi}{p} + \psi^{(j)}\Big)
\end{aligned}
\right.
\end{equation}
with $x_n^{(j)}$, $y_n^{(j)}$ and $z_n^{(j)}$ the Cartesian coordinates of bead $n$ on strand $j$.
The parameter $\psi^{(1)}=0$ for the first strand and $\psi^{(1)}=\pi$ for the second strand. The cross-sectional
radius $\alpha$ is set equal to half the equilibrium base-pair distance, $\rho_{\textrm{ref}} = 1$~nm, considered
in previous work~\cite{SM-2013-PRE-Dasanna-Manghi,SM-2015-JCP-Sicard-Manghi}.
The twist angle between two base-pairs is defined as $\phi =2\pi/p $, where $p=12.3$ is the DNA pitch,
\textit{i.e.} the number of bps corresponding to one complete helix turn. For purposes of generating the initial
conformations, the bending angle per axis segment between the centers of two consecutive bps
is set initially at $\theta = 2\pi/N$.
We constrained a sequence of 10 GC bps on each extremity of the AT-rich region to be aligned arbitrarily
along the Z-axis, as depicted in Fig.~\ref{figSM3} (left panel).
The superhelical densities $\sigma = \frac{Lk - Lk^0}{Lk^0} = \frac{\Delta Lk}{Lk^0}$ along with the sizes $N$ of the
minicircles was specifically chosen to tune the value of the excess of linking number $\Delta L_k < 1$.
The parameter $Lk = 20$ represents the linking numbers of the cDNA molecule
and $Lk^0$ is defined as $Lk^0 = N/p_0$, with $p_0 = 12.0$ the equilibrium pitch
measured in the \textit{open linear} states.\\
The analysis of the FE surface associated with the bubble closure and opening mechanisms,
as shown in Fig.~\ref{figSM3} (left panel), allowed us to determine the value the FE of formation,
$\Delta F^0 = F(op) - F(cl) = -4.4 \pm 0.5~k_B T$ and the activation energies,
$\Delta F_{cl \to op} = 17.8 \pm 0.5~k_B T$
and $\Delta F_{op \to cl} = 23.5 \pm 0.4~k_B T$ along the MFEP obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao},
as shown in Fig.~\ref{figSM3} (middle panel). The FE of formation, $\Delta F^* = -8.6 \pm 0.4~k_B T$,
defined in term of the probability distribution of $\rho_{\textrm{max}}$ and $\Phi_{\textrm{min}}$,
was computed considering the successive isosurfaces in the FE basins depicted in Fig.~\ref{figSM3} (left panel)
as integration domains.
In Fig.~\ref{figSM3} (middle panel), we show the FE of the system as a function of the progression along
the typical MFEP (normalized to unity). The nonlinear least-squares Marquardt-Levenberg algorithm was implemented
to fit the parameters $\omega_0$ and $\omega_{m}$ with skew-Gaussian distributions due to the asymmetric nature of the FE shape.
We obtained $\omega_{m} = 69.5 \pm 3.1$ and $\omega_0 = 3.7 \pm 0.2$ for the metastable ($op$) and equilibrium ($cl$) basins, respectively.
Considering the Rouse model~\cite{SM-2005-JPCM-Ambjornsson-Metzler} valid for flexible polymer chain,
the effective friction coefficient, $\gamma$, in Eq.~$16$ in the main text depends on the number of opened bps,
$N_{\textrm{bub}}$, in the DNA bubble. The typical size observed in the simulations, $N_{bub}\approx 12$ bps,
yields the relation $\gamma_{op}/\gamma_{cl} \approx N_{\textrm{bub}}$ between the effective frictions.
We obtain the transition rate ratio, $k_{cl \to op}/k_{op \to cl} = (1.0 \pm 0.4) \times 10^{6}$.
In Fig.~\ref{figSM3} (right panel), we show the temporal evolution of the FE profile of the system
along $\rho_{\textrm{max}}$ reconstructed in the WT-metaD simulation. In such case, the
convergence of the FE profile could not be achieved due to large entropic fluctuations.
However, the analysis of the converged FE surface obtained in Fig.~\ref{figSM3} (left panel) was achievable
with the appropriate use of the auxiliary variable $\Phi_{\textrm{min}}$.
We extended the Metadynamics scope~\cite{SM-2010-JCP-Xin-Hamelberg,SM-2013-PRL-Tiwary-Parrinello,SM-2014-JCTC-Salvalaglio-Parrinello}
to estimate the mean transition times between the metastable ($op$) and the equilibrium ($cl$) states
of the DNA bubble. WT-metaD was performed using the width $\rho_{\textrm{max}}$ as CV.
The statistics for $\tau_{cl \to op}^{(\textrm{num})}$ conformed to a Poisson distribution with means
$\mu_{op \to cl}= 4.9 \pm 0.6$ ms and variance $\lambda_{op \to cl}=6.0$ ms. The statistics obeys a two-sample
Kolmogorov-Smirnov test~\cite{SM-2014-JCTC-Salvalaglio-Parrinello} with $p$-value equal to $0.71$.
However, the numerical estimation of the transition time $\tau_{cl \to op}^{(\textrm{num})}$ was not achievable
within the metaD framework, as the shape of the original FE surface could not be evenly maintained after
the addition of the bias potential due to \textit{large} entropic fluctuations.
Nevertheless, our approach allowed us to asses the transition rate ration and to estimate $\tau_{cl \to op} = 80 \pm 40$ min.
\begin{figure}[h]
\includegraphics[width=1.0 \textwidth, angle=-0]{figSM3.eps}
\caption{\textbf{Left panel} FE surface associated with the circular DNA bubble closure/nucleation mechanism projected
along the maximal distance between paired bases $\rho_{\textrm{max}}$ and the minimal twist angle between successive bps,
$\phi_{\textrm{min}}$. The contour lines are every two $k_B T$. The two stables basins associated with the opened (op) and closed (cl) states of the DNA bubble and the typical MFEP obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao}
are shown (red).
\textbf{Middle panel} FE of the circular DNA bubble as a function of the progression along the typical MFEP (normalized to unity)
obtained within the steepest descent framework~\cite{SM-2013-JCP-Chen-Xiao}. The nonlinear least-squares Marquardt-Levenberg algorithm was implemented to fit the parameters $\omega_0$ and $\omega_m$, measured in the equilibrium and metastable states, respectively.
\textbf{Right panel} Temporal evolution of the FE profile of the system along $\rho_{\textrm{max}}$ reconstructed
in the WT-metaD simulation. The convergence of the FE profile could not be achieved due to large entropic fluctuations.}
\label{figSM3}
\end{figure}
|
1,108,101,565,482 | arxiv | \section{\@startsection {section}{1}{\z@}%
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\renewcommand{\textfraction}{0.15}
\defq^2_\mathrm{max}{q^2_\mathrm{max}}
\defs_\mathrm{th}{s_\mathrm{th}}
\defm_{B^*}{m_{B^*}}
\def|V_{ub}|{|V_{ub}|}
\def\n#1e#2n{#1\times10^{#2}}
\def\mathrm{i}{\mathrm{i}}
\def\mathrm{e}{\mathrm{e}}
\def\,\mathrm{GeV}{\,\mathrm{GeV}}
\def\,\mathrm{MeV}{\,\mathrm{MeV}}
\def\d{\mathrm{d}}
\def\mathcal{F}{\mathcal{F}}
\begin{document}
\begin{center}\Large\bfseries\sffamily
$|V_{ub}|$ from Exclusive Semileptonic $B\to\rho$ Decays
\end{center}
\begin{center}
\textbf{\textsf{Jonathan M Flynn${}^\mathrm{a}$, Yoshiyuki
Nakagawa${}^\mathrm{b}$, Juan
Nieves${}^\mathrm{c}$ and Hiroshi Toki${}^\mathrm{b}$}}\\[2ex]
${}^\mathrm{a}$School of Physics and Astronomy, University of
Southampton, Southampton SO17~1BJ, UK\\
${}^\mathrm{b}$Research Center for Nuclear Physics, Osaka
University,
Ibaraki, Osaka 567-0047, Japan \\
${}^\mathrm{c}$Instituto de F{\'\i}sica Corpuscular (centro mixto
CSIC-UV),
Institutos de Investigaci\'on de Paterna,
46071, Valencia, Spain
\end{center}
\medskip
\begin{quote}
\begin{center}\textbf{\textsf{Abstract}}\end{center}
We use Omn\`es representations of the form factors $V$, $A_1$ and
$A_2$ for exclusive semileptonic $B\to\rho$ decays, and apply them to
combine experimental partial branching fraction information with
theoretical calculations of the three form factors to extract
$|V_{ub}|$. We find a slightly lower result,
$|V_{ub}|=(2.8\pm0.2)\times10^{-3}$ , than the values extracted from
exclusive semileptonic $B\to\pi$ decays,
$(3.47\pm0.29\pm0.03)\times10^{-3}$~\cite{Flynn:2007ii},
$(3.36\pm0.23)\times10^{-3}$~\cite{Bourrely:2008za},
$(3.38\pm0.35)\times10^{-3}$~\cite{Bailey:2008wp}, and using all other
inputs in CKM fits, $(3.55\pm0.15)\times10^{-3}$~\cite{Bona:2006ah,
UTfit:web}. The disagreement is greater when we compare to the result
extracted from inclusive $B \to X_u l \nu$ decays, $|V_{ub}| =
(4.10\pm0.30_\mathrm{exp}\pm0.29_\mathrm{th})
\times10^{-3}$~\cite{Neubert:FPCP2007}.
\end{quote}
\section{Introduction}
The magnitude of the element $V_{ub}$ of the Cabibbo-Kobayashi-Maskawa
(CKM) quark mixing matrix plays a critical role in testing the
consistency of the Standard Model of particle physics and, in
particular, the description of CP violation. Any inconsistency could
be a sign of new physics beyond the standard model. $V_{ub}$ is
currently the least well-known element of the CKM matrix and
improvement in the precision of its determination is highly desirable
and topical.
$|V_{ub}|$ can be determined using inclusive or exclusive charmless
semileptonic $B$ decays. The inclusive method has historically
provided a more precise result, but recent
experimental~\cite{Athar:2003yg,Aubert:2005cd,Hokuue:2006nr,Aubert:2006ry,Gray:2007pw,
Adam:2007pv} and theoretical
developments~\cite{Arnesen:2005ez,Becher:2005bg,LCSR_04_BZ,
Flynn:2000gd, Flynn:2006vr,Ball:2006jz, Flynn:2007ki,
Flynn:2007qd,Bourrely:2008za, Duplancic:2008ix, Duplancic:2008zz} are
allowing the exclusive semileptonic $B\to \pi$ method to approach the
same level of precision.
Recently~\cite{Flynn:2007ii} we extracted $|V_{ub}|$ from combined
experimental partial branching fraction information and theoretical
[lattice QCD (LQCD) and Light cone sum rules (LCSR)] information on
exclusive semileptonic $B\to\pi$ decays. The Omn\`es representation
was employed to provide parametrisations of the form factors. The
extracted value turned out to be in striking agreement with that
extracted using all other inputs in CKM fits and in some disagreement
with $|V_{ub}|$ extracted from inclusive semileptonic decays.
The aim of this letter is to extend the above formalism to study the
exclusive semileptonic $B\to \rho$ decay and independently extract
$|V_{ub}|$ from the recent measurements of the partially integrated
branching fraction by BABAR~\cite{Aubert:2005cd},
Belle~\cite{Hokuue:2006nr} and CLEO~\cite{Gray:2007pw,Adam:2007pv}. We
will make use of quenched LQCD form factor
results~\cite{BowlerKC:JHEP05:2004, Abada:2002ie} for the high $q^2$
region, and LCSR values~\cite{Ball:2004rg} at $q^2=0$. Thanks to the
Omn\`es representation of the form-factors, we are able to combine all
these inputs, as we previously showed for $B\to\pi$ decays.
\section{Fit Procedure}
\subsection{Form-factors and differential decay width}
The semileptonic decay $B^0 \to \rho^- \ell^+ \nu_l$ is determined by
the matrix element of the $V-A$ weak current between a $B$ meson and a
$\rho$ meson. The matrix element is
\begin{equation}
\langle \rho (k, \eta) | \bar b \gamma^\mu (1-\gamma_5) u | B
(p) \rangle = \eta^*_\beta T^{\mu\beta},
\end{equation}
with form factor decomposition
\begin{eqnarray}
T_{\mu\beta} &=&
\frac{2V(q^2)}{m_B+m_\rho}\epsilon_{\mu\gamma\delta\beta}p^\gamma
k^\delta - \mathrm{i} (m_B+m_\rho)A_1(q^2)
g_{\mu\beta}\nonumber\\ &&\mbox{} + \mathrm{i}
\frac{A_2(q^2)}{m_B+m_\rho} (p+k)_\mu q_\beta - \mathrm{i}
\frac{2A(q^2)}{q^2} m_\rho q_\mu (p+k)_\beta,
\end{eqnarray}
where $q=p-k$ is the four-momentum transfer and $\eta$ is the $\rho$
polarisation vector. The meson masses are $m_B=5279.5$ MeV and
$m_\rho=775.5$ MeV for $B^0$ and $\rho^-$, respectively. In the
helicity basis each of the form factors corresponds to a transition
amplitude with definite spin-parity quantum numbers in the center
of mass frame of the lepton pair. This relates the form factors $V$,
$A_1$ and $A_2$ to the total angular momentum and parity quantum
numbers of the $B\rho$ meson pair, $J^P=1^-, 1^+$ and $1^+$,
respectively~\cite{Wirbel:1985ji}. The physical region for the squared
four-momentum transfer is $0\le q^2 \le q^2_\mathrm{max} \equiv
(m_B-m_\rho)^2$. If the lepton mass can be ignored ($l=e$ or $\mu$),
the total decay rate is given by
\begin{equation}
\Gamma\left( B^0 \to \rho^- \ell^+ \nu_l \right) =
\frac{G_F^2|V_{ub}|^2}{192\pi^3m^3_B} \int_0^{q^2_{\rm max}}
dq^2 q^2\left[\lambda (q^2)\right]^\frac12 \left
(|H^+(q^2)|^2+|H^-(q^2)|^2+|H^0(q^2)|^2 \right)
\label{eq:gamma}
\end{equation}
where $G_F= 1.16637\times 10^{-5}$ GeV$^{-2}$ is the Fermi constant
and $\lambda(q^2)=(m^2_B+m^2_\rho-q^2)^2-4m^2_Bm^2_\rho$. $H^0$ comes
from the contribution of the longitudinally polarised $\rho$ and is
given by
\begin{equation}
H^0(q^2) = - \frac{1}{2m_\rho\sqrt {q^2}}\left \{ \left(
m_B^2-m^2_\rho-q^2\right)\left( m_B+m_\rho\right)A_1(q^2)-
\frac{4m_B^2|\vec{k}\,|^2}{m_B+m_\rho}A_2(q^2) \right \}
\end{equation}
where $\vec{k}$ is the momentum of the $\rho$ in the $B$-meson rest
frame. $H^{\pm}$ correspond to the contribution of the transverse
polarisations of the vector meson and are given by~\footnote{Note a
typo in Eq.~(1.7) of Ref.~\cite{BowlerKC:JHEP05:2004}, the $\pm$
sign should be $\mp$, as used in previous papers of the UKQCD
Collaboration~\cite{Bowler:1994zr, Flynn:1995dc}.}
\begin{equation}
H^{\pm} = - \left \{ (m_B+m_\rho) A_1(q^2) \mp
\frac{2m_B|\vec{k}\,|}{m_B+m_\rho} V(q^2) \right \}
\end{equation}
The CLEO Collaboration has also measured partial branching fractions
of the differential distribution~\cite{Gray:2007pw, Adam:2007pv}
\begin{eqnarray}
\frac{d\Gamma(B^0\to \rho^- \ell^+\nu)}{dq^2\,d\cos\theta_{W\ell}} & =
&
\frac{G_F^2 |V_{ub}|^2}{512\pi^3 m_B^3} q^2 \left [ \lambda
(q^2)\right]^\frac12 \Big\{ 2\sin^2\theta_{W\ell} |H^0(q^2)|^2
\nonumber \\
& & \mbox{} +
(1-\cos\theta_{W\ell})^2|H^+(q^2)|^2+
(1+\cos\theta_{W\ell})^2|H^-(q^2)|^2
\Big\} \label{eq:cos}
\end{eqnarray}
with $\theta_{W\ell}$ the angle between the charged lepton direction in
the virtual $W-$gauge boson rest frame and the virtual $W$ in the
$B$-meson rest frame.
\subsection{Omn\`es parametrisations}
We have
previously~\cite{Flynn:2007ii,Flynn:2000gd,Flynn:2006vr,Flynn:2007ki,
Flynn:2007qd } used a multiply subtracted Omn\`es dispersion
relation~\cite{omnes,mushkelishvili}, based on unitarity and
analyticity properties, to describe $B \to \pi$ semileptonic decays.
Here, we apply these ideas to $B \to \rho$ decays and use for $(n+1)$
subtractions~\cite{Flynn:2007qd}
\begin{equation}
F(q^2) = \frac{1}{s_0-q^2} \prod_{i=0}^n \left[F
(s_i)(s_0-s_i)\right]^{\alpha_i(q^2)}, \quad \alpha_i(s)\equiv
\prod_{j=0, j\ne i} \frac{s-s_j}{s_i-s_j}, \qquad F=V,A_1,A_2
\label{eq:omn-param}
\end{equation}
where $s_0$ corresponds to a pole of the form factor $F$. We fix
$s_0=m_{B^*}^2$ and $s_0 = s_{\rm th}=(m_B+m_\rho)^2$ for $V$ and
$A_1$ and $A_2$ form factors, respectively. In principle, for the
axial form factors one should use the square of the $1^+$ $B$-meson
mass. The mass of this latter hadron is not well established yet, but
it appears to be heavier than the $1^-$ $B^*$ resonance. Thus and for
the purposes of this exploratory work, since it would be reasonably
far from $\sqrt{q^2_{\rm max}}$, it is sufficient to employ $s_{\rm
th}$. The parametrisation of Eq.~(\ref{eq:omn-param}) amounts to
finding an interpolating polynomial for $\ln[(s_0-q^2)F(q^2)]$ passing
through the points $(s_0^2-s_i)F(s_i)$. While one could always propose
a parametrisation using an interpolating polynomial for
$\ln[g(q^2)F(q^2)]$ for a suitable function $g(q^2)$, the derivation
using the Omn\`es representation shows that taking $g(q^2)=s_0^2-q^2$
is physically motivated~\cite{Flynn:2007qd}.
\subsection{Theoretical and experimental inputs}
We have used experimental partial branching fraction data from
CLEO~\cite{Gray:2007pw,Adam:2007pv}, Belle~\cite{Hokuue:2006nr} and
BABAR~\cite{Aubert:2005cd}. CLEO and BABAR combine results for neutral
and charged $B$-meson decays using isospin symmetry, while Belle give
separate values for $B^0 \to \rho^- \ell^+ \nu_l$ and $B^+ \to \rho^0
\ell^+ \nu_l$ decays. Belle use three $q^2$ intervals, and we have
added in quadrature the two different systematic errors quoted for
each $q^2$ bin, and combined charged and neutral $B$-meson results. We
take the resulting systematic errors to be fully correlated. BABAR's
untagged analysis also uses three $q^2$ bins and we have assumed that
the quoted percentage systematic errors for the partial branching
fractions divided by total branching fraction are representative for
the partial branching fractions alone and, following BABAR, took them
to be fully correlated. CLEO determines partial branching fractions as
a function of both $q^2$ and of $\cos\theta_{W\ell}$ (see
Eq.~(\ref{eq:cos})) and complete correlation matrices are given
in~\cite{Gray:2007pw} for both statistical uncertainties and
systematic errors that we have used in our fits.
When computing partial branching fractions, we have used $\tau_{B^0}=
1/\Gamma_\mathrm{Tot} = \n(1.527\pm
0.008)e-12n\,\mathrm{s}$~\cite{Barberio:2007cr} for the $B^0$
lifetime. All the branching fraction inputs are listed in
Table~\ref{tab:in_exp}.
\begin{table}
\begin{center}
\begin{tabular}{ @{} l >{$} c <{$} >{$} c <{$} >{$} c <{$} >{$} c <{$} @{} }
\hline
& q^2 \mbox{\ range } [\,\mathrm{GeV}^2] & \cos\theta_{W\ell} \mbox{\ range }
& 10^4{B}^{\textrm{in}}_k & 10^4{B}^{\textrm{Omn\`es}}_k \\
\hline
BELLE~\cite{Hokuue:2006nr}
& 0 - 8 & [-1,1] & 0.62 \pm 0.14 \pm 0.06 & 0.69 \pm 0.12\\
& 8 - 16 & [-1,1] & 1.20 \pm 0.23 \pm 0.11 & 1.12 \pm 0.15\\
& > 16 & [-1,1] & 0.53 \pm 0.20 \pm 0.12 & 0.53 \pm 0.08\\
\hline
BABAR~\cite{Aubert:2005cd}
& 0 - 10 & [-1,1] & 0.73 \pm 0.17 \pm 0.21 & 0.96 \pm 0.15\\
& 10 - 15 & [-1,1] & 0.82 \pm 0.10 \pm 0.13 & 0.71 \pm 0.10\\
& > 15 & [-1,1] & 0.59 \pm 0.07 \pm 0.16 & 0.68 \pm 0.10\\
\hline
CLEO ~\cite{Gray:2007pw}
& 0 - 2 & [-1,1] & 0.45 \pm 0.20 \pm 0.15 & 0.08 \pm 0.03\\
& 2 - 8 & [-1,1] & 0.96 \pm 0.20 \pm 0.29 & 0.61 \pm 0.10\\
& 8 - 16 & \phantom{-}[0,1] & 0.75 \pm 0.16 \pm 0.14 & 0.74 \pm 0.10\\
& > 16 & \phantom{-}[0,1] & 0.35 \pm 0.07 \pm 0.05 & 0.39 \pm 0.06\\
& > 8 & [-1,0] & 0.42 \pm 0.18 \pm 0.31 & 0.51 \pm 0.07\\
\hline
\end{tabular}
\end{center}
\caption{Experimental branching fraction inputs for the $\chi^2$
function defined in Eq.~(\ref{eq:chi2}). Statistical and systematic
errors are shown. We also give branching fractions calculated using
our fitted form factors and $|V_{ub}|$.}
\label{tab:in_exp}
\end{table}
For theoretical form-factor inputs (listed in Table~\ref{tab:in_th}),
we use the lightcone sumrule (LCSR) results at $q^2=0$ of
Ref.~\cite{Ball:2004rg} and lattice QCD results from the
UKQCD~\cite{BowlerKC:JHEP05:2004} and SPQcdR~\cite{Abada:2002ie}
Collaborations, near $q^2_\mathrm{max}$. LQCD inputs have been obtained in the
quenched approximation. There is therefore an uncontrolled systematic
error, which is not fully included in the errors given in
Table~\ref{tab:in_th}.
\begin{table}
\begin{center}
\begin{tabular}{@{}l >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} @{}}
\hline
& q^2 \mbox{ [GeV$^2$]} & V & A_1 & A_2 \\
\hline
LCSR~\cite{Ball:2004rg} & 0 & 0.323 \pm 0.029 & 0.242 \pm 0.024 & 0.221 \pm 0.023 \\
\hline \tstrut
UKQCD~\cite{BowlerKC:JHEP05:2004}
& 12.67 & 0.684 \pm 0.162 \, ^{+ 0.00} _{- 0.56 }& 0.439 \pm 0.067 \,^{+ 0.000} _{-0.080 } & 0.70 \pm 0.49\,^{+ 0.08 } _{- 0.03 }\\\tstrut
& 13.01 & 0.714 \pm 0.162 \, ^{+ 0.00} _{- 0.50} & 0.448 \pm 0.065 \,^{+ 0.000} _{-0.079 }& 0.71 \pm 0.46\, ^{+ 0.08 } _{- 0.03 }\\\tstrut
& 13.51 & 0.763 \pm 0.155 \,^{+ 0.00} _{- 0.40 } & 0.460 \pm 0.063 \,^{+ 0.000} _{-0.075 }& 0.72 \pm 0.43\, ^{+ 0.10 } _{- 0.02 }\\\tstrut
& 14.02 & 0.818 \pm 0.147 \, ^{+ 0.00} _{- 0.31 }& 0.472 \pm 0.059 \,^{+ 0.000} _{-0.073 }& 0.73 \pm 0.42\, ^{+ 0.12 } _{- 0.01 } \\\tstrut
& 14.52 & 0.883 \pm 0.141 \, ^{+ 0.00} _{- 0.24 }& 0.485 \pm 0.055 \, ^{+ 0.000} _{-0.070 }& 0.76 \pm 0.42\,^{+ 0.14 } _{- 0.03 } \\\tstrut
& 15.03 & 0.967 \pm 0.137 \, ^{+ 0.00} _{- 0.20} & 0.498 \pm 0.051 \,^{+ 0.000} _{-0.068 }& 0.78 \pm 0.46\,^{+ 0.16 } _{- 0.05 } \\\tstrut
& 15.53 & 1.057 \pm 0.134 \, ^{+ 0.00} _{- 0.19} & 0.513 \pm 0.049 \, ^{+ 0.000} _{-0.067 }& 0.81 \pm 0.54\,^{+ 0.18} _{- 0.06 } \\\tstrut
& 16.04 & 1.164 \pm 0.150 \,^{+ 0.10 } _{- 0.21} & 0.529 \pm 0.047 \,^{+ 0.000} _{-0.066 }& 0.84 \pm 0.71\,^{+ 0.20 } _{- 0.07 } \\\tstrut
& 16.54 & 1.296 \pm 0.184 \,^{+ 0.21 } _{- 0.25} & 0.544 \pm 0.043 \,^{+ 0.000} _{-0.062 }& 0.87 \pm 0.97\,^{+ 0.23 } _{- 0.08 } \\\tstrut
& 17.05 & 1.46 \pm 0.26 \,^{+ 0.34 } _{- 0.30} & 0.560 \pm 0.043 \,^{+ 0.000} _{-0.059 }& 0.90 \pm 1.35\,^{+ 0.27 } _{- 0.07 } \\\tstrut
& 17.55 & 1.67 \pm 0.40 \, ^{+ 0.49 } _{- 0.36}& 0.577 \pm 0.043 \,^{+ 0.000} _{-0.058 }& 0.90 \pm 1.89\,^{+ 0.33 } _{- 0.03 } \\\tstrut
& 18.17 & 2.02 \pm 0.68 \, ^{+ 0.73 } _{- 0.48} & 0.599 \pm
0.052 \,^{+ 0.000 } _{-0.058 } & 0.9 \pm 2.9\,^{+ 0.4 } _{- 0.1 } \\\\
\hline\tstrut
SPQcdR~\cite{Abada:2002ie}
& 10.69 & 0.51 \pm 0.26 & 0.354 \pm 0.085 & 0.38 \pm 0.26\\
& 12.02 & 0.61 \pm 0.28 & 0.384 \pm 0.087 & 0.49 \pm 0.30\\
& 13.35 & 0.74 \pm 0.30 & 0.421 \pm 0.089 & 0.65 \pm 0.35\\
& 14.68 & 0.93 \pm 0.31 & 0.465 \pm 0.092 & 0.93 \pm 0.41\\
& 16.01 & 1.20 \pm 0.32& 0.519 \pm 0.097 & 1.41 \pm 0.56\\
& 17.34 & 1.61 \pm 0.33& 0.588 \pm 0.108 & 2.39 \pm 1.23\\
& 18.67 & 2.26 \pm 0.55& 0.678 \pm 0.134 & 4.7 \pm 4.1\\
\hline
\end{tabular}
\end{center}
\caption{Form factor inputs for the $\chi^2$ function defined in
Eq.~(\ref{eq:chi2}). For UKQCD we show both statistical (symmetrized)
and systematical errors, while SPQcdR errors include both
systematic and statistical uncertainties (we are indebted with
C.M. Maynard and F. Mescia for providing us with these form factors).}
\label{tab:in_th}
\end{table}
\subsection{Definition of $\chi^2$}
We implement the following fitting procedure. Choose a set of
subtraction points spanning the physical range to use in the Omn\`es
formula of equation~(\ref{eq:omn-param}). Now find the best-fit value
of $|V_{ub}|$ and the form factors at the subtraction points to match
both theoretical input form factor values and the experimental partial
branching fraction inputs. The $\chi^2$ function for the fit is :
\begin{eqnarray}
\chi^2 &=& \sum_{i,j=1}^{60}
\left[F^\mathrm{in}_i-\fom{i}\right]
C^{-1}_{ij}\left[F^\mathrm{in}_j-\fom{j}\right]\nonumber\\
& & \mbox{} +
\sum_{k,l=1}^{11}
\left[B_k^\mathrm{in} -
B_k^{\mbox{\scriptsize Omn\`es}}(|V_{ub}|,F_0,F_1,F_2)\right]
C^{-1}_{B\,kl}
\left[B_l^\mathrm{in} -
B_l^{\mbox{\scriptsize Omn\`es}}(|V_{ub}|,F_0,F_1,F_2)\right],
\label{eq:chi2}
\end{eqnarray}
where $F^\mathrm{in}_i$ are input LCSR or lattice QCD values for
$V(q^2_i), \,A_1(q^2_i)$ and $A_2(q^2_i)$, and $B^\mathrm{in}_k$ are
input experimental partial branching fractions. Moreover, $\fom i$
stands for each of the form factors $F=V,A_1, A_2$ at $q^2=q^2_i$, and
it is given by equation~(\ref{eq:omn-param}) with three subtractions
$(s_l, F(s_l))$ at $(0,F_0)$, $(2q^2_\mathrm{max}/3,F_1)$ and
$(q^2_\mathrm{max},F_2)$. The branching fractions $B^{\mbox{\scriptsize
Omn\`es}}$ are calculated using $F^{\mbox{\scriptsize Omn\`es}}$, for
$V, A_1$ and $A_2$ form factors . There are in total 10 fit
parameters: $V(0)$, $V(2q^2_\mathrm{max}/3)$, $V(q^2_\mathrm{max})$, $A_1(0)$,
$A_1(2q^2_\mathrm{max}/3)$, $A_1(q^2_\mathrm{max})$, $A_2(0)$, $A_2(2q^2_\mathrm{max}/3)$,
$A_2(q^2_\mathrm{max})$ and $|V_{ub}|$. The latter parameter is used when
computing $B^{\mbox{\scriptsize Omn\`es}}$.
We have assumed that the LCSR and LQCD form factor values have
independent statistical uncertainties and treated the errors listed in
Table~\ref{tab:in_th} for the SPQcdR inputs as purely statistical. For
the UKQCD data we have put the form factor values in the centre of
their systematic range and use half that range as the systematic
error. We have built a covariance matrix where the statistical
uncertainties ($\sigma_i$) are uncorrelated and the systematic errors
($\epsilon_i$) are fully correlated, leading to a $60\times 60$
covariance matrix with three diagonal blocks. The first $3\times 3$
and second $21\times 21$ blocks are for the LCSR and SPQcdR results
and have the form $C_{ij}=\sigma_i^2 \delta_{ij}$. The third block is
for the UKQCD data and has the form $C_{ij}=\sigma_i^2
\delta_{ij}+\epsilon_i\epsilon_j$. We will further discuss the effect
of the UKQCD systematic errors on $|V_{ub}|$ below.
The covariance matrix, $C_B$, for the partial branching
fraction inputs is constructed as follows. For Belle and BABAR input
data, we have assumed independent statistical uncertainties and
fully-correlated systematic errors leading to an
$6\times 6$ covariance matrix with two diagonal blocks of the form
$C_{B\,ij}=\sigma_i^2 \delta_{ij} + \epsilon_i\epsilon_j$. For the
CLEO input, we use an $5\times 5$ covariance matrix $C_{B\,ij}^{\rm
CLEO}= \sigma_i \sigma_j {\cal C}_{B\,ij}^{\,\rm CLEO-stat}+
\epsilon_i \epsilon_j {\cal C}_{B\,ij}^{\,\rm CLEO-sys} $, where we
have read off the statistical and systematic correlation matrices
(${\cal C}_{B\,ij}^{\,\rm CLEO-stat/sys}$) from
tables X and XI, respectively, of Ref.~\cite{Gray:2007pw}.
We do not consider any correlation between measurements from different
experiments, or between different sources of theoretical inputs. Nor
do we consider correlations between experimental and theoretical
inputs.
\section{Results and discussion}
The best fit parameters are
\begin{equation}
\begin{array}{rcl}
|V_{ub}| & = & \left (2.76 \pm 0.21\right) \times 10^{-3} \\
V(0) & = & 0.322 \pm 0.030 \\
V(2q^2_\mathrm{max}/3) & = & 0.681 \pm 0.073 \\
V(q^2_\mathrm{max}) & = & 4.21 \pm 0.76 \\
A_1(0) & = & 0.223 \pm 0.021 \\
A_1(2q^2_\mathrm{max}/3) & = & 0.449 \pm 0.020 \\
A_1(q^2_\mathrm{max}) & = & 0.657 \pm 0.055 \\
A_2(0) & = & 0.231 \pm 0.022 \\
A_2(2q^2_\mathrm{max}/3) & = & 0.679 \pm 0.098 \\
A_2(q^2_\mathrm{max}) & = & 2.76 \pm 1.38 \\
\end{array}
\label{eq:besfit}
\end{equation}
The fit has $\chi^2/\mathrm{d.o.f.}=0.21$ for $61$ degrees of freedom,
while the Gaussian correlation matrix can be found in the
appendix~\ref{sec:app}. In figure~\ref{fig:results} we show the
fitted form factors and the differential decay rate calculated from
our fit. Partial branching fractions calculated for the same bins as
used experimentally are given in the last column of
Table~\ref{tab:in_exp}. Our calculated total branching ratio turns out
to be $(2.30^{+0.24}_{-0.26}) \times 10^{-4}$, in reasonable agreement
with $(2.80\pm 0.18 \pm 0.16)\times 10^{-4}$ quoted by the Heavy
Flavours Averaging Group (HFAG)~\cite{Barberio:2007cr}.
We have further investigated the effect of the highly asymmetric UKQCD
systematic errors on $|V_{ub}|$. First, we have completely dropped them
and used only the statistical uncertainties on the UKQCD points. We
find $|V_{ub}|= \left (2.68 \pm 0.19\right) \times 10^{-3}$. Second, we
have performed a Monte Carlo where we randomly choose each UKQCD form
factor value within its systematic error range, with complete
correlation between all systematic shifts. For each trial we perform a
fit like our original one, but setting to zero the systematic errors
on the UKQCD inputs. In this case, we find $|V_{ub}|= \left (2.85 \pm
0.10\right) \times 10^{-3}$. Note that this last result is the mean
and the standard deviation of the fit result for $|V_{ub}|$ over all
the trials, whereas the result above and that quoted in
Eq.~(\ref{eq:besfit}) are the fit result and error from a single fit.
Thus, the result from this second procedure should be understood as a
shift of $+0.09 \pm 0.10$ in the value of $|V_{ub}|$ in
Eq.~(\ref{eq:besfit}). Finally, we have repeated the latter procedure,
but taking the $A_2$ systematic error to be anticorrelated with those
of the $V$ and $A_1$. This results in $|V_{ub}|= \left (2.86 \pm
0.15\right) \times 10^{-3}$.
From the above discussion, we estimate
\begin{equation}
|V_{ub}| = 2.8 \pm 0.2
\label{eq:vubresult}
\end{equation}
which constitutes our main result. Quenched approximation systematic
effects from LQCD are not accounted for by the $0.2$ error quoted
above. These are difficult to quantify and are a limitation here.
However, unquenched lattice simulations are now standard and future
lattice QCD results will address this limitation (although they will
also face the problem of an unstable $\rho$ meson for light enough
simulated up and down quark masses).
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig-fits.eps}
\end{center}
\caption{Results obtained from the fit to experimental partial
branching fraction data and theoretical form factor calculations.
The top and the left bottom plots show the three form factors with
their 68\% CL bands (shaded) together with
the lattice and LCSR input points (green square LCSR, red dots
UKQCD, blue triangles
SPQcdR). The bottom right plot shows the differential decay rate with
68\% CL band (shaded) together
with the experimental partial branching fractions divided by the
appropriate bin-width (histograms and points). Green squares, red
dots and blue triangles
denote BABAR, Belle and CLEO results, respectively.}
\label{fig:results}
\end{figure}
Nevertherless, we see that the Omn\`es framework used here provides a
fair description of all available experimental and theoretical results
for semileptonic $B\to\rho$ decays, leading to a further independent
determination of $|V_{ub}|$. The result is lower than the values
obtained in the most recent studies of the exclusive semileptonic
$B\to\pi$ decay,
$(3.47\pm0.29\pm0.03)\times10^{-3}$~\cite{Flynn:2007ii},
$(3.36\pm0.23)\times10^{-3}$~\cite{Bourrely:2008za},
$(3.38\pm0.35)\times10^{-3}$~\cite{Bailey:2008wp}, $(3.5 \pm
0.4_\mathrm{th} \pm 0.2_\mathrm{shape}\pm
0.1_\mathrm{BR})\times10^{-3}$~\cite{Duplancic:2008ix}, and using all
other inputs in CKM fits,
$(3.55\pm0.15)\times10^{-3}$~\cite{Bona:2006ah, UTfit:web}. The
disagreement is greater when we compare to the most precise result
extracted from inclusive $B \to X_u l \nu$ decays, $|V_{ub}| =
(4.10\pm0.30_\mathrm{exp}\pm0.29_\mathrm{th})
\times10^{-3}$~\cite{Neubert:FPCP2007}. Thus, the hints of
disagreement between inclusive and exclusive/global-CKM-fit
determinations are strengthened.
\subsubsection*{Acknowledgments}
JMF and JN acknowledge support from the EU Human Resources and
Mobility Activity, FLAVIAnet, contract number MRTN--CT--2006--035482,
PPARC grant PP/D000211/1 and MEC grant FIS2005--00810.
|
1,108,101,565,483 | arxiv | \section{Introduction}
A number of ways of introducing inverses of graphs have been proposed, all based on inverting adjacency matrices. For a graph with a non-singular adjacency matrix a first thought might be to hope that the inverse matrix defines a graph again. It turns out, however, that this happens to be the case only for unions of isolated edges \cite{Har}. A successful approach was initiated by Godsil \cite{Godsil1985} who defined a graph to be invertible if the inverse of its (non-singular) adjacency matrix is diagonally similar (c.f. \cite{Zas}) to a nonnegative integral matrix representing the adjacency matrix of the inverse graph in which positive labels determine edge multiplicities. This way of introducing invertibility has the appealing property that inverting an inverse graph gives back the original graph. For a survey of results and other approaches to graph inverses we recommend \cite{McMc}.
Inverse graphs are of interest in estimating the least positive eigenvalue in families of graphs, a task for which there appears to be lack of suitable bounds. However, if the graphs are invertible, one can apply one of the (many) known upper bounds on largest eigenvalues of the inverse graphs instead (cf. \cite{Pavlikova1990, Pavlikova2015}). Properties of spectra of inverse graphs can also be used to estimate the difference between the minimal positive and maximal negative eigenvalue (the so-called HOMO-LUMO gap) for structural models of chemical molecules, as it was done e.g. for graphene in \cite{YeKlein}.
Godsil's ideas have been further developed in several ways. Akbari and Kirkland \cite{KirklandAkb2007} and Bapat and Ghorbani \cite{Bapat} studied inverses of edge-labeled graphs with labels in a ring, Ye \emph{et al.} \cite{Ye} considered connections of graph inverses with median eigenvalues, and Pavl\'{\i}kov\'a \cite{Pavlikova2015} developed constructive methods for generating invertible graphs by edge overlapping. A large number of related results, including a unifying approach to inverting graphs, were proposed in a recent survey paper by McLeman and McNicholas \cite{McMc}, with emphasis on inverses of bipartite graphs and diagonal similarity to nonnegative matrices.
Less attention has been given to the study of invertibility of non-bipartite graphs and their spectral properties which is the goal of this paper. After introducing basic concepts, in Section 2 we present an example of a non-bipartite graph representing an important chemical molecule of fulvene. Its adjacency matrix has the remarkable additional property that its inverse is integral and diagonally similar to a nonpositive rather than a nonnegative matrix. This motivated us to introduce negative invertibility as a natural counterpart of Godsil \cite{Godsil1985} concept: A negatively invertible graph is one with a non-singular adjacency matrix whose inverse is diagonally similar to a nonpositive matrix. The negative of this matrix is then the adjacency matrix of the inverse graph. Positively and negatively invertible graphs are subfamilies of integrally invertible graphs, whose adjacency matrices have an integral inverse. The corresponding inverse graphs, however, would have to be interpreted as labeled graphs with both positive and negative (integral) edge labels.
Results of the paper are organized as follows. In Section 3 we develop constructions of new integrally invertible graphs from old ones by `bridging' two such graphs over subsets of their vertices. This yields a wide range of new families of integrally invertible graphs. We derive sufficient conditions for their positive and negative invertibility. In contrast to purely graph-theoretical approach we use methods of matrix analysis and in particular results on inverting block matrices such as the Schur complement theorem and the Woodbury and Morrison-Sherman formulae. Using this approach enables us to derive useful bounds on the spectra of graphs arising from bridging construction in Section 4. We then illustrate our results in Section 5 on a recursively defined family of fulvene-like graphs. In the final Section 6 we discuss arbitrariness in the bridging construction and give a census of all invertible graphs on at most 6 vertices with a unique 1-factor.
\section{Invertible graphs}\label{sec-intro}
In this section we recall a classical concept of an invertible graph due to Godsil \cite{Godsil1985}. Let $G$ be an undirected graph, possibly with multiple edges, and with a (symmetric) adjacency matrix $A_G$. Conversely, if $A$ is a nonnegative integral symmetric matrix, we will use the symbol $G_A$ to denote the graph with the adjacency matrix $A$.
The spectrum $\sigma(G)$ of $G$ consists of eigenvalues (i.e., including multiplicities) of $A_G$ (cf. \cite{Cvetkovic1978, Cvetkovic1988}). If the spectrum does not contain zero then the adjacency matrix $A$ is invertible. We begin with a definition of an integrally invertible graph.
\begin{definition}\label{def-integerinv}
A graph $G=G_A$ is said to be integrally invertible if the inverse $A^{-1}$ of its adjacency matrix exists and is integral.
\end{definition}
It follows that a graph $G_A$ is integrally invertible if and only if $det(A)=\pm1$ (cf. \cite{KirklandAkb2007}). Note that, in such a case the inverse matrix $A^{-1}$ need not represent a graph as it may contain negative entries.
Following the idea due to Godsil, the concept of the inverse graph $G_A^{-1}$ is based on the inverse matrix $A^{-1}$ for which we require signability to a nonnegative or nonpositive matrix. We say that a symmetric matrix $B$ is positively (negatively) {signable} if there exists a diagonal $\pm 1$ matrix $D$ such that $D B D$ is nonnegative (nonpositive). We also say that $D$ is a signature matrix.
\begin{definition}\label{def:inv}
A graph $G_A$ is called positively (negatively) invertible if $A^{-1}$ exists and is signable to a nonnegative (nonpositive) integral matrix. If $D$ is the corresponding signature matrix, the positive (negative) inverse graph $H=G_A^{-1}$ is defined by the adjacency matrix $A_H=D A^{-1} D$ ($A_H= - DA^{-1} D$).
\end{definition}
The concept of positive invertibility coincides with the original notion of invertibility introduced by Godsil \cite{Godsil1985}. Definition~\ref{def:inv} extends Godsil's original concept to a larger class of integrally invertible graphs with inverses of adjacency matrices signable to nonpositive matrices.
Notice that for a diagonal matrix $D^A$ containing $\pm1$ elements only, we have $D^A D^A = I$. It means that $(D^A)^{-1} = D^A$.
\begin{remark}
The idea behind the definition of an inverse graph consists of the following useful property. If $G$ is a positively (negatively) invertible graph then $G^{-1}$ is again a positively (negatively) invertible graph and $G=(G^{-1})^{-1}$. As far as the spectral properties are concerned, we have
\[
\sigma(G^{-1}) = 1/\sigma(G) = \{ 1/\lambda, \ \lambda\in\sigma(G) \},
\]
for any positively invertible graph $G$. On the other hand, if $G$ is negatively invertible then
\[
\sigma(G^{-1}) = -1/\sigma(G) = \{ - 1/\lambda, \ \lambda\in\sigma(G) \}.
\]
\end{remark}
\begin{figure}
\begin{center}
\includegraphics[width=3.5truecm]{figures/fulvene}
\ \
\includegraphics[width=3.5truecm]{figures/fulvene-inverse}
\ \
\includegraphics[width=2.5truecm]{figures/fulvene-chem}
\end{center}
\caption{
An example of a negatively invertible non-bipartite graph $F_0$ (left) and its inverse graph $(F_0)^{-1}$ (middle) of the fulvene chemical organic molecule (right).}
\label{fig-fulvene}
\end{figure}
Fig.~\ref{fig-fulvene} (left) shows the graph $F_0$ on 6 vertices representing the organic molecule of the fulvene hydrocarbon (5-methylidenecyclopenta-1,3-diene) (right). The graph $F_0$ is negatively (but not positively) invertible with the inverse graph $(F_0)^{-1}$ depicted in Fig.~\ref{fig-fulvene} (middle). The spectrum consists of the following eigenvalues:
\[
\sigma(F_0)= \{ -1.8608, -q, -0.2541, 1/q, 1, 2.1149 \},
\]
where $q=(\sqrt{5}+1)/2$ is the golden ratio with the least positive eigenvalue $\lambda_1^+(F_0)=1/q$. The inverse adjacency matrix $A_{F_0}^{-1}$ is signable to a nonpositive integral matrix by the signature matrix $D^{A_{F_0}} = diag(-1,-1,1,1,1-1)$.
\subsection{Bipartite graphs and their invertibility}
A graph $G_B$ is called bipartite if the set of vertices can be partitioned into two disjoint subsets such that no two vertices within the same subset are adjacent. The adjacency matrix $B$ of a bipartite graph $G_B$ can be given in a block form:
\[
B =
\left(
\begin{array}{cc}
0 & K \\
K^T & 0\\
\end{array}
\right),
\]
where $K$ is a matrix with nonnegative integer entries. Clearly, the adjacency matrix $B$ of a bipartite graph $G_B$ is invertible if and only if the number of its vertices is even and the matrix $K$ is invertible. If we consider the labeled graph corresponding to the adjacency matrix $B^{-1}$ and the product of edge labels on every cycle in this graph is positive, then $B^{-1}$ is signable to a nonnegative matrix and so $G_B$ is a positively invertible graph (see \cite{KirklandTif2009}).
Recall that a 1-factor (a perfect matching) of a graph is a spanning 1-regular subgraph with all vertices of degree 1. If $G$ is a bipartite graph with a 1-factor $M$ such that the graph $G/M$ obtained from $G$ by contracting edges of $M$ is bipartite then $G$ is a positively invertible graph (c.f. \cite{Godsil1985, Pavlikova1990}).
Bipartiteness and invertibility are related as follows.
\begin{theorem}\label{theo-bipartite}
Let $G$ be an integrally invertible graph. Then $G$ is bipartite if and only if $G$ is simultaneously positively and negatively invertible.
\end{theorem}
\noindent P r o o f.
Let $G=G_B$ be an integrally invertible bipartite graph. Assume that $G_B$ is positively invertible, i.e., there exists a signature matrix $D^+ = diag(D_1, D_2)$ such that the matrix
\begin{eqnarray*}
D^+ B^{-1} D^+ &=&
\left(
\begin{array}{cc}
D_1 & 0 \\
0 & D_2\\
\end{array}
\right)
\left(
\begin{array}{cc}
0 & (K^{-1})^T \\
K^{-1} & 0\\
\end{array}
\right)
\left(
\begin{array}{cc}
D_1 & 0 \\
0 & D_2\\
\end{array}
\right)
\\
&=&
\left(
\begin{array}{cc}
0 & D_1 (K^{-1})^T D_2 \\
D_2 K^{-1} D_1 & 0\\
\end{array}
\right)
\end{eqnarray*}
contains nonnegative integer entries only. Then for the $\{\pm1\}$ diagonal matrix $D^- = diag(D_1, - D_2)$ the matrix
\[
D^- B^{-1} D^- =
\left(
\begin{array}{cc}
0 & -D_1 (K^{-1})^T D_2 \\
-D_2 K^{-1} D_1 & 0\\
\end{array}
\right)
\]
contains nonpositive integers only. Hence $G_B$ is negatively invertible, and vice versa.
On the other hand, suppose that $G$ is simultaneously positively and negatively invertible. We will prove that $G$ is a bipartite graph with even number of vertices. Let $n$ be the number of vertices of the graph $G$. Let $D^\pm$ be diagonal $\{\pm 1\}$-matrices such that $D^+ A^{-1} D^+$ contains nonnegative integers and $D^- A^{-1} D^-$ contains nonpositive integers only. Since $(D^\pm A^{-1} D^\pm)_{ij} = D^\pm_{ii} (A^{-1})_{ij} D^\pm_{jj}$ we conclude that $(D^\pm A^{-1} D^\pm)_{ij}\not=0$ if and only if $(A^{-1})_{ij}\not=0$. Hence
\begin{equation}
D^+ A^{-1} D^+ = - D^- A^{-1} D^- .
\label{Dpm}
\end{equation}
As $det(A^{-1}) = det(D^+ A^{-1} D^+) = (-1)^n det(D^- A^{-1} D^-)= (-1)^n det(A^{-1})$ we conclude that $n$ is even, i.~e. $n=2m$.
Recall that for the trace operator $tr(Z)=\sum_{i=1}^n Z_{ii}$ of an $n\times n$ matrix $Z$ we have $tr(X Y) = tr(Y X)$ where $X,Y$ are $n\times n$ matrices. With respect to (\ref{Dpm}) we obtain:
\begin{eqnarray*}
tr(D^+D^-) &=& tr(A D^+ D^- A^{-1}) = tr(D^- A D^+ D^- A^{-1} D^-)
\\
&=& - tr(D^- A D^+ D^+ A^{-1} D^+) = - tr(D^-D^+)= - tr(D^+D^-).
\end{eqnarray*}
Thus $ tr(D^+D^-)=0$. Since $D^\pm$ are diagonal $\{\pm 1\}$-matrices, there exists an $n\times n$ permutation matrix $P$ such that
\[
P^T D^+ D^- P =
\left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right),\ \ i.e.
\quad
D^+ D^- = D^- D^+ =
P \left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right) P^T ,
\]
where $I$ is the $m\times m$ identity matrix. It follows from (\ref{Dpm}) that
\[
A^{-1} = - D^+ D^- A^{-1} D^- D^+ =
- P \left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
P^T A^{-1} P
\left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
P^T.
\]
Since $P^T P = P P^T =I$ we have
\[
P^T A^{-1} P
=
- \left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
P^T A^{-1} P
\left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
\]
If we write $P^T A^{-1} P$ as a block matrix we obtain
\begin{eqnarray*}
P^T A^{-1} P \equiv
\left(
\begin{array}{cc}
V & H \\
H^T & W\\
\end{array}
\right)
&=&
- \left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
\left(
\begin{array}{cc}
V & H \\
H^T & W\\
\end{array}
\right)
\left(
\begin{array}{cc}
I & 0 \\
0 & -I\\
\end{array}
\right)
\\
&=&
\left(
\begin{array}{cc}
-V & H \\
H^T & -W\\
\end{array}
\right).
\end{eqnarray*}
Therefore $V=W=0$ and
\[
P^T A^{-1} P =
\left(
\begin{array}{cc}
0 & H \\
H^T & 0\\
\end{array}
\right)
\ \Longrightarrow \
P^T A P =
\left(
\begin{array}{cc}
0 & (H^T)^{-1} \\
H^{-1} & 0\\
\end{array}
\right).
\]
This means that the adjacency matrix $A$ represents a bipartite graph $G=G_A$ after a permutation of its vertices given by the matrix $P$.
\hfill $\diamondsuit$
\section{Integrally invertible graphs arising by bridging}
Let $G_A$ and $G_B$ be undirected graphs on $n$ and $m$ vertices, respectively. By ${\mathcal B}_k(G_A,G_B)$ we shall denote the graph $G_C$ on $n+m$ vertices which is obtained by bridging the last $k$ vertices of the graph $G_A$ to the first $k$ vertices of $G_B$. The adjacency matrix $C$ of the graph $G_C$ has the form:
\[
C = \left(
\begin{array}{cc}
A & H\\
H^T & B
\end{array}
\right),
\]
where the $n\times m$ matrix $H$ has the block structure:
\[
H= \left(
\begin{array}{cc}
0 & 0\\
I & 0
\end{array}
\right)
= F E^T,\quad\hbox{where}\quad
F=\left(
\begin{array}{l}
0\\
I
\end{array}
\right),\ \
E=\left(
\begin{array}{l}
I\\
0
\end{array}
\right),
\]
and $I$ is the $k\times k$ identity matrix.
Assume that $A$ and $B$ are symmetric $n\times n$ and $m\times m$ invertible matrices, respectively. With regard to the Schur complement theorem we obtain
\begin{equation}
C^{-1}=\left(
\begin{array}{cc}
A & H\\
H^T & B
\end{array}
\right)^{-1}
=
\left(
\begin{array}{cc}
S^{-1} & - S^{-1} H B^{-1} \\
- B^{-1} H^T S^{-1} & B^{-1} + B^{-1} H^T S^{-1} H B^{-1}
\end{array}
\right),
\label{invC}
\end{equation}
where $S= A- H B^{-1} H^T$ is the Schur complement (see e.~g. \cite[Theorem A.6]{Maja2013}). To facilitate further notation let us introduce the following matrices:
\[
P = F^T A^{-1} F,\qquad R = E^T B^{-1} E.
\]
In order to compute the inverse of the Schur complement $S$ we follow derivation of the Woodbury and Morrison-Sherman formulae (c.f. \cite[Corollary A.6, A.7]{Maja2013}). More precisely, equation $S x =y$ can be rewritten as follows:
\[
y=(A- H B^{-1} H^T)x = A x - F E^T B^{-1} E F^T x = A x -F R F^T x
\]
and thus $x= A^{-1} y + A^{-1} F R F^T x$. Hence
\[
F^T x = F^T A^{-1} y + F^T A^{-1} F R F^T x = F^T A^{-1} y + P R F^T x .
\]
If we assume that the matrix $I - P R$ is invertible then $F^T x = (I - P R)^{-1}F^T A^{-1} y$, and
\begin{equation}
S^{-1}=(A- H B^{-1} H^T)^{-1} = A^{-1} + A^{-1} F R (I - P R)^{-1}F^T A^{-1}.
\label{invS}
\end{equation}
Note that the matrix $I - P R$ is integrally invertible provided that either $P R=0$ or $P R = 2 I$.
Clearly, an $m\times m$ matrix with a zero principal $k\times k$ diagonal block is invertible only for $k\le m$. Consequently, there are no connected invertible graphs with $E^T B^{-1} E=0$ for $k > m/2$.
\begin{theorem}\label{theo-1}
Let $G_A$ and $G_B$ be integrally invertible graphs on $n$ and $m$ vertices, and let $R$ and $P$ be the upper left and lower right $k\times k$ principal submatrices of $B^{-1}$ and $A^{-1}$, respectively.
Let $G_C={\mathcal B}_k(G_A,G_B)$ be the graph obtained by bridging $G_A$ and $G_B$ over the last $k$ vertices of $G_A$ and the first $k$ vertices of $G_B$. If $P R=0$ or $P R = 2 I$, then the graph $G_C$ is integrally invertible.
\end{theorem}
\noindent P r o o f. Since $P R=0$ or $P R = 2 I$, then the inverse $(I - P R)^{-1}$ exists and is equal to $\pm I$. Hence the inverse $S^{-1}$ of the Schur complement is an integral matrix. Therefore the block matrix $C$ given by
\[
C = \left(
\begin{array}{cc}
A & H\\
H^T & B
\end{array}
\right)
\]
is invertible, and hence so is the bridged graph $G_C$. Moreover, $C^{-1}$ is an integral matrix because $A^{-1}, B^{-1}, S^{-1}$ are integral.
\hfill $\diamondsuit$
\begin{definition}\label{def-arbitrarily}
Let $G_B$ be a graph on $m$ vertices with an invertible adjacency matrix $B$. We say that $G_B$ is arbitrarily bridgeable over a subset of $k\le m/2$ vertices if the $k\times k$ upper principal submatrix $R\equiv E^T B^{-1} E$ of the inverse matrix $B^{-1}$ is a null matrix, that is $R = 0$.
\end{definition}
In view of Definition~\ref{def-arbitrarily}, the bridged graph $G_C={\mathcal B}_k(G_A,G_B)$ is integrally invertible provided that $G_B$ is arbitrarily bridgeable over the set of its 'first' $k$ vertices.
In the next theorem we address the question of invertibility of the bridged graph $G_C={\mathcal B}_k(G_A,G_B)$ under the assumption that $G_A$ and $G_B$ are positive (negative) invertible graphs.
\begin{theorem}\label{theo-si}
Let $G_A$ and $G_B$ be graphs on $n$ and $m$ vertices, respectively. Assume that they are either both positively invertible or both negatively invertible graphs with signature matrices $D^A$ and $D^B$. Then the graph $G_C={\mathcal B}_k(G_A,G_B)$ is positively (negatively) invertible if we have $P R =0$ and either the matrix $D^A H D^B$ or $-D^A H D^B$ contains nonnegative integers only.
\end{theorem}
\noindent P r o o f. Let $C$ be the adjacency matrix to the graph $G_C={\mathcal B}_k(G_A,G_B)$. If $P R =0$ then for the inverse of the Schur complement (see (\ref{invS})) we have
\[
S^{-1} = A^{-1} + A^{-1} F R F^T A^{-1} = A^{-1} + A^{-1} H B^{-1} H^T A^{-1}
\]
because $F R F^T = F E^T B^{-1} E F^T = H B^{-1} H^T$. Therefore
\begin{eqnarray*}
D^A S^{-1} D^A &=& D^A A^{-1} D^A + D^A A^{-1} H B^{-1} H^T A^{-1} D^A
\\
&=& D^A A^{-1} D^A
\\
&& + (D^A A^{-1} D^A)( D^A H D^B)( D^B B^{-1} D^B)( D^B H^T D^A)( D^A A^{-1} D^A)
\end{eqnarray*}
and so $D^A S^{-1} D^A$ is a nonnegative (nonpositive) integer matrix because the matrices $D^A A^{-1} D^A$ and $D^B B^{-1} D^B$ are simultaneously nonnegative (nonpositive) and $D^A H D^B$ or $-D^A H D^B$ contains nonnegative integers only.
In the case when $D^A H D^B$ is nonnegative we will prove that $C^{-1}$ is diagonally similar to a nonnegative (nonpositive) integer matrix with $D^C=diag(D^A,-D^B)$\ ($D^C=diag(D^A,D^B)$). With regard to (\ref{invC}) we have
\begin{eqnarray*}
D^C C^{-1} D^C &=&
\left(
\begin{array}{cc}
D^A & 0\\
0 & -D^B
\end{array}
\right)
\left(
\begin{array}{cc}
S^{-1} & - S^{-1} H B^{-1} \\
- B^{-1} H^T S^{-1} & B^{-1} + B^{-1} H^T S^{-1} H B^{-1}
\end{array}
\right)
\\
&& \qquad\qquad \times \left(
\begin{array}{cc}
D^A & 0\\
0 & -D^B
\end{array}
\right)
\end{eqnarray*}
\begin{equation*}
=
\left(
\begin{array}{cc}
D^A S^{-1} D^A & D^A S^{-1} D^A D^A H D^B D^B B^{-1} D^B \\
D^B B^{-1} D^B D^B H^T D^A D^A S^{-1} D^A & D^B B^{-1} D^B + W,
\end{array}
\right)
\end{equation*}
where
\[
W = (D^B B^{-1} D^B)( D^B H^T D^A)( D^A S^{-1} D^A)( D^A H D^B)( D^B B^{-1} D^B) .
\]
In the expression for the matrix $W$ we have intentionally used the matrices $D^A D^A=D^B D^B = I$ instead of the identity matrix $I$. Since the matrices $D^A A^{-1} D^A$, $D^B B^{-1} D^B$, and $D^A S^{-1} D^A$ contain nonnegative (nonpositive) integers only and $D^A H D^B$ is nonnegative, we conclude that $C^{-1}$ is diagonally similar to a nonnegative (nonpositive) integral matrix.
In the case when $-D^A H D^B$ is nonnegative we can proceed similarly as before and conclude that $C^{-1}$ is diagonally similar to a nonnegative (nonpositive) integral matrix.
Hence the graph $G_C$ is positively (negatively) invertible, as claimed.
\hfill $\diamondsuit$
\bigskip
As a consequence we obtain the following:
\begin{corollary}
Let $G_A, G_B$ be two positively (negatively) invertible graphs such that $(B^{-1})_{11}=0$. Then the graph $G_C={\mathcal B}_1(G_A,G_B)$ bridged over the first vertex is again positively (negatively) invertible.
\end{corollary}
\noindent P r o o f. For $k=1$ the condition $(B^{-1})_{11}=0$ implies $R\equiv E^T B^{-1} E=0$, i.e. $G_B$ is arbitrarily bridgeable. The matrix $D^A H D^B$ contains only one nonzero element, equal to $\pm H$. Hence the assumptions of Theorem~\ref{theo-si} are fulfilled and so $G_C$ is positively (negatively) invertible.
\hfill $\diamondsuit$
\medskip
With regard to Theorem~\ref{theo-bipartite} and Theorem~\ref{theo-si} we obtain the following result:
\begin{corollary}
Let $G_A, G_B$ be two bipartite positively and negatively invertible graphs such that $G_B$ is arbitrarily bridgeable over the first $k$ vertices. Let $D^A_+$ and $D^B_+$ ($D^A_-$ and $D^B_-$) be diagonal $\{\pm1\}$-matrices such that $D^A_+ A^{-1} D^A_+$ and $D^B_+ B^{-1} D^B_+$ ($D^A_- A^{-1} D^A_-$ and $D^B_- B^{-1} D^B_-$) are nonnegative (nonpositive) matrices. If $D^A_\pm H D^B_\pm$ are either both nonnegative or both nonpositive then the bridged graph $G_C={\mathcal B}_k(G_A,G_B)$ is again bipartite positively and negatively invertible.
\end{corollary}
\begin{figure}
\begin{center}
\includegraphics[width=7truecm]{figures/fig-bridged}
\end{center}
\caption{
An example of bridging of two bipartite positively and negatively invertible graphs $G_A$ and $G_B$ through vertices $3\leftrightarrow 1', 4\leftrightarrow 2'$. The resulting graph $G_C={\mathcal B}_2(G_A, G_B)$.
}
\label{fig-bridged}
\end{figure}
\begin{example}\rm
In what follows, we present an example showing that the assumption made on nonnegativity or nonpositivity of matrices $D^A_+ H D^B_+$ and $D^A_- H D^B_-$ cannot be relaxed. To do so, we will construct a bridged graph $G_C$ from two integrally invertible bipartite graphs such that $G_C$ is only positively but not negatively invertible graph and, as a consequence of Theorem~\ref{theo-bipartite}, the graph $G_C$ is not bipartite.
Let $G_A$ and $G_B$ be two simultaneously positively and negatively invertible bipartite graphs shown in Fig.~\ref{fig-bridged}. We will bridge them over a set of $k=2$ vertices to obtain the graph $G_C$ with inverses given by
\[
\scriptsize
A^{-1}=\left(
\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & -1\\
1 & 0 & -1 & 0\\
\end{array}
\right),
\quad
B^{-1}=\left(
\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 1 & -1\\
0 & 1 & 0 & 0\\
1 & -1 & 0 & 0\\
\end{array}
\right),
\]
\[\scriptsize
C^{-1}=\left(
\begin{array}{cccccccc}
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & -1\\
0 & 1 & 0 & -1 & 0 & 0 & 1 & -1\\
1 & 0 & -1 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\
-1 & 0 & 1 & 0 & 0 & 1 & 0 & -1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 2\\
\end{array}
\right).
\]
The graphs $G_A, G_B$ are isomorphic with eigenvalues: $\{\pm 1.6180, \pm 0.6180\}$.
The upper left $2\times 2$ principal submatrix $R$ of $B^{-1}$ is zero, so that the graph $G_B$ can be arbitrarily bridged to an integrally invertible graph $G_A$.
It is easy to verify that the inverse matrices $A^{-1}$ and $B^{-1}$ can be signed to nonnegative matrices by signature matrices $D^A_+=D^B_+ = diag(-1,1,1,-1)$. At the same time they can be signed to a nonpositive matrix by $D^A_-=diag(-1,-1,1,1)$ and $D^B_-=diag(-1,1,-1,1)$. Furthermore, $D^A_+ H D^B_+ = -H$ is a nonpositive matrix. By Theorem~\ref{theo-si}, the graph $G_C$ is positively invertible. On the other hand,
\[\scriptsize
D^A_- H D^B_- = \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
\end{array}
\right)
\]
is neither nonnegative nor nonpositive. Indeed, the graph $G_C$ is not bipartite and it is just positively (and not negatively) invertible, with spectrum
\[
\sigma(G_C)= \{-1.9738, -1.8019, -0.7764, -0.445, 0.2163, 1.247, 1.4427, 2.0912\}.
\]
\end{example}
\begin{remark}
If the graph $G_B$ is arbitrarily bridgeable over the first $k$ vertices then $R=0$, and, consequently the assumption $P R =0$ appearing in Theorem~\ref{theo-si} is satisfied. On the other hand, if we consider the graph $G_C$ with the vertex set $\{1,2,3,4,1',2',3',4'\}$ shown in Figure~\ref{fig-bridged} then the inverse matrix $C^{-1}$ contains the principal submatrix
\[\scriptsize
P = \left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
\end{array}
\right)
\]
corresponding to vertices $4,1',2',3'$. Consider the same graph $\tilde{G}_{\tilde{C}}$ on the vertex set $\{\tilde{1},\tilde{2},\tilde{3},\tilde{4},\tilde{1}',\tilde{2}',\tilde{3}',\tilde{4}'\}$. Then, after permuting vertices, the inverse matrix $\tilde{C}^{-1}$ has the upper left principal $4\times 4$ submatrix
\[\scriptsize
R = \left(
\begin{array}{cccc}
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{array}
\right).
\]
Hence $P R =0$ but neither $P$ nor $R$ is an all-zero $4\times4$ matrix. By Theorem~\ref{theo-si} the bridged graph ${\mathcal B}_4(G_C, \tilde{G}_{\tilde{C}})$ over the set of vertices $4\leftrightarrow\tilde{2}', 1'\leftrightarrow \tilde{3}', 2'\leftrightarrow \tilde{4}, 3'\leftrightarrow \tilde{1}'$ is integrally invertible.
\end{remark}
\section{Spectral bounds for graphs arising by bridging}
In this section we derive a lower bound for the least positive eigenvalue of bridged graphs ${\mathcal B}_k(G_A,G_B)$ in terms of the least positive eigenvalues of graphs $G_A$ and $G_B$. Throughout this section we assume that the adjacency matrices $A,B$ are invertible but we do not require their integral invertibility.
Before stating and proving our spectral estimate we need the following auxiliary Lemma.
\begin{lemma}\label{lemmaMax}
Assume that $D$ is an $n\times m$ matrix and $\alpha,\beta>0$ are positive constants. Then, for the optimal value $\lambda^*$ of the following constrained optimization problem:
\begin{equation}\label{minD}
\begin{array}{rl}
\lambda^*= \max & \alpha \Vert x- D y\Vert^2 +\beta \Vert y\Vert^2 \\
{\rm s. t.} & \Vert x\Vert^2 + \Vert y\Vert^2 = 1,\ \ x\in\mathbb{R}^n, y\in\mathbb{R}^m,
\end{array}
\end{equation}
we have an explicit expression of the form:
\[
\lambda^* = \max\left\{ \lambda,\ \frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} \in\sigma(D^T D)\right\}
\]
\[
= \frac{\alpha\mu^* + \alpha+\beta + \sqrt{(\alpha\mu^* + \alpha+\beta)^2-4\alpha\beta}}{2},
\]
where $\mu^*=\max\{\sigma(D^T D)\}$ is the maximal eigenvalue of the matrix $D^T D$.
\end{lemma}
\noindent P r o o f.
The proof is straightforward and is based on standard application of the Lagrange multiplier method (see e.g. \cite{Maja2013}). We give details for the reader's convenience.
Let us introduce the Lagrange function:
\begin{eqnarray*}
L(x,y,\lambda) &=& \alpha \Vert x- D y\Vert^2 +\beta \Vert y\Vert^2 - \lambda( \Vert x\Vert^2 + \Vert y\Vert^2)
\\
&=& \alpha x^T x -2\alpha x^T D y +\alpha y^T D^T D y +\beta y^T y -\lambda x^T x -\lambda y^T y.
\end{eqnarray*}
Now, it follows from the first order necessary conditions for constrained maximum $(x,y)$ (see e.~g. \cite{Maja2013}) that there exists a Lagrange multiplier $\lambda\in\mathbb{R}$ such that
\begin{eqnarray}
\label{A1}
0 &=& L^\prime_x \equiv 2 \alpha x^T -2\alpha y^T D^T -2 \lambda x^T,
\\
0 &=& L^\prime_y \equiv -2\alpha x^T D +2 \alpha y^T D^T D + 2 \beta y^T -2 \lambda y^T.
\end{eqnarray}
In the case $\lambda\not=\alpha$ we obtain
\[
x = \frac{\alpha}{\alpha-\lambda} D y,
\qquad
\left[ \alpha D^T D -(\lambda-\beta)\right] y = \alpha D^T x = \frac{\alpha^2}{\alpha-\lambda} D^T Dy.
\]
Therefore,
\[
D^T D y = \frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} y \ \Rightarrow \
\frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} \in\sigma(D^T D).
\]
Now, from the constraint $x^T x + y^T y =1$ we deduce that
\[
1 = x^T x + y^T y = \frac{\alpha^2}{(\alpha-\lambda)^2} y^T D^T D y + y^T y =
\left( \frac{\alpha^2}{(\alpha-\lambda)^2} \frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} +1 \right) \Vert y\Vert^2.
\]
Hence
\[
\Vert y\Vert^2 = \frac{(\lambda-\alpha)\lambda}{\lambda^2 -\alpha\beta}, \qquad
\Vert x\Vert^2 = 1- \Vert y\Vert^2 = \frac{(\lambda-\beta)\alpha}{\lambda^2 -\alpha\beta}.
\]
Finally, for the value function $f(x,y) = \alpha \Vert x- D y\Vert^2 +\beta \Vert y\Vert^2 $ of the constrained optimization problem (\ref{minD}) we obtain
\begin{eqnarray*}
f(x,y) &=& \alpha x^T x -2\alpha x^T D y +\alpha y^T D^T D y +\beta y^T y
\\
&=& \alpha x^T x -2\frac{\alpha^2}{\alpha-\lambda} y^T D^T D y +\alpha y^T D^T D y +\beta y^T y
\\
&=& \frac{\alpha^2(\lambda-\beta)}{\lambda^2-\alpha\beta}
+ \left(\alpha - \frac{2\alpha^2}{\alpha-\lambda}\right)\frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} \frac{(\lambda-\alpha)\lambda}{\lambda^2 -\alpha\beta}
+
\beta \frac{(\lambda-\alpha)\lambda}{\lambda^2 -\alpha\beta}
\\
&=& \lambda.
\end{eqnarray*}
In the case $\lambda=\alpha$, one sees from (\ref{A1}) that $D y=0$ and so $f(x,y)= \alpha \Vert x\Vert^2 +\beta \Vert y\Vert^2 \le \max\{\alpha,\beta\} \le \lambda^*$. In summary,
\[\lambda^* = \max\left\{ \lambda,\ \frac{(\lambda-\alpha)(\lambda-\beta)}{\alpha\lambda} \in\sigma(D^T D)\right\},
\]
as claimed. \hfill$\diamondsuit$
We are in a position to present our spectral bound.
\begin{theorem}\label{theo-2}
Let $G_A$ and $G_B$ be graphs on $n$ and $m$ vertices with invertible adjacency matrices. Assume $G_B$ is arbitrarily bridgeable over the first $k$ vertices. Then the least positive eigenvalue $\lambda_1^+(G_C)$ of its adjacency matrix $C$ of the bridged graph $G_C={\mathcal B}_k(G_A,G_B)$ satisfies
\[
\lambda_1^+(G_C) \ge \lambda_{lb}(G_A,G_B,k) :=\frac{2}{\alpha\mu^* + \alpha+\beta + \sqrt{(\alpha\mu^* + \alpha+\beta)^2-4\alpha\beta}},
\]
where $\mu^*=\max\{\sigma(B^{-1} H^T H B^{-1})\}$ is the maximal eigenvalue of the positive semidefinite $m\times m$ matrix $B^{-1} H^T H B^{-1}, \alpha = 1/\lambda_1^+(G_A)$ and $\beta = 1/\lambda_1^+(G_B)$.
\end{theorem}
\noindent P r o o f.
The idea of the proof is based on estimation of the numerical range of the matrix $C^{-1}$. Since
$\lambda_1^+(G_C)=\lambda_1^+(C)=1/\lambda_{max}(C^{-1})$ where $\lambda_{max}(C^{-1})$ is the maximal eigenvalue of the inverse matrix $C^{-1}$, the lower bound for $\lambda_1^+(C)$ can be derived from the upper bound for $\lambda_{max}(C^{-1})$. As stated in Definition~\ref{def-arbitrarily}, the assumption that $G_B$ is an arbitrarily bridgeable graph implies $S^{-1}=A^{-1}$. Thus formula (\ref{invC}) for the inverse matrix $C^{-1}$ becomes:
\begin{eqnarray*}
C^{-1} &=&
\left(
\begin{array}{cc}
A^{-1} & - A^{-1} H B^{-1} \\
- B^{-1} H^T A^{-1} & B^{-1} + B^{-1} H^T A^{-1} H B^{-1}
\end{array}
\right)
\\
&=&
\left(
\begin{array}{cc}
I & 0 \\
- B^{-1} H^T & I
\end{array}
\right)
\left(
\begin{array}{cc}
A^{-1} & 0 \\
0 & B^{-1}
\end{array}
\right)
\left(
\begin{array}{cc}
I & - H B^{-1} \\
0 & I
\end{array}
\right).
\end{eqnarray*}
Let $z=(x,y)^T\in \mathbb{R}^{n+m}$ where $x\in\mathbb{R}^n, y\in\mathbb{R}^m$.
For the Euclidean inner product $\langle C^{-1} z, z \rangle $ in $\mathbb{R}^{n+m}$ we obtain
\begin{eqnarray*}
\langle C^{-1} z, z\rangle &=& \langle A^{-1} (x-H B^{-1} y), x-H B^{-1} y \rangle
+ \langle B^{-1} y, y \rangle
\\
&\le& \lambda_{max}(A^{-1}) \Vert x-H B^{-1} y\Vert^2 + \lambda_{max}(B^{-1}) \Vert y\Vert^2 .
\end{eqnarray*}
Letting $\alpha=\lambda_{max}(A^{-1}), \beta=\lambda_{max}(B^{-1})$ and $D=H B^{-1}$, by Lemma~\ref{lemmaMax} we obtain
\[
\langle C^{-1} z, z\rangle \le \frac{1}{\lambda_{lb}(G_A,G_B,k)} \Vert z\Vert^2,
\]
for any $z\in\mathbb{R}^{n+m}$. Since
\[
\lambda_{max}(C^{-1}) = \max_{z\not=0} \frac{ \langle C^{-1} z, z\rangle}{\Vert z\Vert^2}
\le \frac{1}{\lambda_{lb}(G_A,G_B,k)}
\]
our Theorem follows because $\alpha = \lambda_{max}(A^{-1})= 1/\lambda_1^+(A)= 1/\lambda_1^+(G_A)$ and $\beta = \lambda_{max}(B^{-1})= 1/\lambda_1^+(B)= 1/\lambda_1^+(G_B)$.
\hfill$\diamondsuit$
\bigskip
To illustrate this on an example, for the graph $G_C = {\mathcal B}_k(G_A,G_B)$ shown in Fig.~\ref{fig-bridged} we have $\lambda_1^+(G_C) = 0.2163$ and the lower bound derived above gives $\lambda_{lb}(G_A,G_B,k) = 0.1408$.
\section{A ``fulvene'' family of integrally invertible graphs}
The aim of this section is to present construction of a family of integrally invertible graphs grown from the ``fulvene'' graph of Fig.~\ref{fig-fulvene} (left), which is the same as the $H_{10}$ in Fig.~\ref{fig-hexafamily}. With regard to Table~\ref{tab-2} (see Section~6), the graph $F_0\equiv H_{10}$ can be arbitrarily bridged over the pair of vertices labeled by $1,2$ (see the left part of Fig.~\ref{fig-fulvene}) to any integrally invertible graph.
Our construction begins with the fulvene graph $F_0$. The next iteration $F_1$ is obtained by bridging $F_0$ to another copy of $F_0$ over the vertex set $\{1,2\}$ in both copies (see Fig.~\ref{fig-fulvenefamily1}).
We now describe a recursive construction of graphs $F_n$ from $F_{n-1}$. For $n\ge 2$, the graph $F_n$ will be obtained from $F_{n-1}$ by bridging a certain number $f_n$ (to be described below) copies of the graph $F_0$ over the vertex set $\{1,2\}$ to vertices of $F_{n-1}$ of degree $1$ or $2$. By definition, we set $f_1=f_2:=2$. Let $|V^{(i)}(F_n)|,\ i=1,2,3,$ denote the number of vertices of $F_n$ with degree $i$.
The order of bridging is as follows:
\begin{itemize}
\item two copies of $F_0$ are bridged to every vertex of degree $1$ which belonged to $F_{n-2}$ and remained in $F_{n-1}$ with degree 1. The other vertex of $F_0$ is bridged to the shortest path distance vertex of degree $2$ belonging to $F_{n-1}$. The number $f^{(1)}_n$ of graphs $F_0$ added to $F_{n-1}$ is given by: $f^{(1)}_n = 2 f_{n-2}$. This way one uses $|V^{(2)}(F_{n-1})| - 2 f_{n-2}$ of vertices of degree $2$ from $F_{n-1}$.
\item The remaining $f^{(2)}_n = f_n -f^{(1)}_n$ copies of $F_0$ are bridged to $F_{n-1}$ through $|V^{(2)}(F_{n-1})| - 2 f_{n-2}$ vertices of degree $2$ in such a way that the graph is bridged to the pair vertices of degree $2$ with the shortest distance. By construction we have $|V^{(2)}(F_n)|= 2 f_n$. Hence, the number $|V^{(2)}(F_{n-1})| - 2 f_{n-2} = 2( f_{n-1} - f_{n-2})$ is even and so $f^{(2)}_n = f_{n-1} - f_{n-2}$. Moreover, the number $|V^{(1)}(F_n)|$ of vertices of degree $1$ is given by: $|V^{(1)}(F_n)| = f_n + f_{n-1}$ as the vertices of degree $1$ from $F_{n-1}\setminus F_{n-2}$ have not been bridged.
\end{itemize}
Since $f^{(1)}_n = 2 f_{n-2}$ and $f^{(2)}_n = f_{n-1} - f_{n-2}$ the total number $f_n=f^{(1)}_n + f^{(2)}_n$ of newly added graphs $F_0$ satisfies the Fibonacci recurrence
\[
f_n=f_{n-1} + f_{n-2}, \ f_1=f_2=2.
\]
It can be explicitly expressed as:
\[
f_n = \frac{2}{\sqrt{5}} \left( q^n - q^{-n}\right),
\]
where $q=(1+\sqrt{5})/2$ is the golden ratio.
Properties of the fulvene family of graphs $F_n$, $n\ge 0$, (for $F_0,F_1,F_2,F_3$ see Fig.~\ref{fig-fulvenefamily1}) can be summarized as follows.
\begin{theorem}\label{theo-fulvene}
Let $F_n, n\ge 0$, be a graph from the fulvene family of graphs. Then
\begin{enumerate}
\item $F_n$ is integrally invertible;
\item $F_n$ is a planar graph of maximum degree 3, with
\begin{eqnarray*}
|V^{(1)}(F_n)| &=& f_n + f_{n-1}, \\
|V^{(2)}(F_n)| &=& 2 f_n, \\
|V^{(3)}(F_n)| &=& |V(F_n)| - |V^{(1)}(F_n)| - |V^{(2)}(F_n)| = 6 \sum_{k=1}^n f_k - 3 f_n - f_{n-1},
\end{eqnarray*}
where $|V(F_n)| = 6 \sum_{k=1}^n f_k$ is the number of vertices of $F_n$;
\item $F_n$ is asymptotically cubic in the sense that
\[
\lim_{n\to\infty} \frac{|V^{(3)}(F_n)|}{|V(F_n)|} =1;
\]
\item the least positive eigenvalue $\lambda_1^+(F_n)$ satisfies the estimate:
\[
\lambda_1^+(F_n)\ge \frac{1}{q} \frac{5}{6^{n+1}-1}.
\]
\end{enumerate}
\end{theorem}
\noindent P r o o f. The number of vertices and integral invertibility of $F_n$ have been derived during construction of $F_n$.
To prove the lower bound for the least positive eigenvalue $\lambda_1^+(F_n)$ of the integrally invertible graph $F_n$ constructed in Section 5. Recall that the next generation $F_n$ is constructed from $F_{n-1}$ by bridging $f_n$ basic fulvene graphs $F_0$ to vertices of degree 1 and 2 of $F_{n-1}$, which can be described as
\[
F_n = {\mathcal B}_{2f_n} (F_{n-1}, G_{B_n}),
\]
where the graph $G_{B_n}$ has an $M\times M$ adjacency matrix $B_n$ of the block diagonal form:
\[
B_n =diag( \underbrace{B,\cdots,B}_{f_n\ times}).
\]
Here $M=6f_n$ and $B=A_{F_0}$ is the adjacency matrix to the graph $F_0$. Therefore
\[
A_{F_n} =
\left(\begin{array}{cc}
A_{F_{n-1}} & H_n\\
H_n^T & B_n
\end{array}
\right)
\]
where $H_n = (H_n^1, \cdots, H_n^{f_n})$ is an $N\times M$ block matrix with $N=|V(F_{n-1})|$. Each $H_n^r$ is an $N\times 6$ $\{0,1\}$-matrix of the form $H_n^r=(u^r, v^r, 0, 0, 0, 0)$ where $u^r_i=1$ ($v^r_i=1$) if and only if the vertex 1 (2) of the $r$-th fulvene graph $F_0$ is bridged to the $i$-th vertex of $F_{n-1}$.
In order to apply the spectral estimate from Theorem~\ref{theo-2} we will derive an upper bound on the optimum value of $\mu^* = \max\sigma(B_n^{-1} H_n^T H_n B_n^{-1})$. Clearly, the matrix $B_n^{-1} H_n^T H_n B_n^{-1}$ satisfies
\[
(B_n^{-1} H_n^T H_n B_n^{-1})_{rs} = B^{-1} (H_n^r)^T H_n^s B^{-1}.
\]
Now,
\[
(H_n^r)^T H_n^s =
\left\{
\begin{array}{ll}
diag(1,1,0,0,0,0), & \hbox{if } r=s,\\
\\
diag(1,0,0,0,0,0), & \hbox{if $r\not=s$ and the $r$-th and $s$-th graph $F_0$} \\
& \hbox{are bridged to the same vertex of $F_{n-1}$,} \\
\\
0, & \hbox{otherwise.}
\end{array}
\right.
\]
Since
\[\scriptsize
B^{-1}
=\left(
\begin{array}{rrrrrr}
0 & 0 & 0 & 0 & 1 & -1\\
0 & 0 & 1 & 0 & 0 & -1\\
0 & 1 & 0 & 0 & -1 & 1\\
0 & 0 & 0 & 0 & 0 & 1\\
1 & 0 & -1 & 0 & 0 & 1\\
-1 & -1 & 1 & 1 & 1 & -2
\end{array}
\right)
\]
it can be verified by an easy calculation that
\[
\max\sigma(B^{-1} (H_n^r)^T H_n^s B^{-1}) =
\left\{
\begin{array}{ll}
3, & \hbox{if } r=s,\\
\\
2, & \hbox{if $r\not=s$ and the $r$-th and $s$-th graph $F_0$} \\
& \hbox{are bridged to the same vertex of $F_{n-1}$,} \\
\\
0, & \hbox{otherwise.}
\end{array}
\right.
\]
Thus, for any vector $z=(z^1, \cdots, z^{f_n}) \in \mathbb{R}^M, z^i\in\mathbb{R}^6$, we have
\begin{eqnarray*}
z^T B_n^{-1} H_n^T H_n B_n^{-1} z
&=& \sum_{r,s=1}^{f_n} (z^r)^T B^{-1} (H_n^r)^T H_n^s B^{-1} z^s
\\
&\le& 3\Vert z\Vert^2 + \sum_{r\not=s} (z^r)^T B^{-1} (H_n^r)^T H_n^s B^{-1} z^s
\\
&\le& 3\Vert z\Vert^2 + 2 \sum_{r\not=s} \frac12 (\Vert z^r\Vert^2 + \Vert z^s\Vert^2)\le 5\Vert z\Vert^2,
\end{eqnarray*}
because for the symmetric matrix $W=B^{-1} (H_n^r)^T H_n^s B^{-1}$ it holds:
$|u^T W v| \le \max |\sigma(W)| \frac12 (\Vert u\Vert^2 + \Vert v\Vert^2)$. Hence,
\[
\mu^* = \max\sigma(B_n^{-1} H_n^T H_n B_n^{-1}) \le 5.
\]
Finally, we establish the lower bound for the least positive eigenvalue $\lambda_1^+(F_n)$. With regard to Theorem~\ref{theo-2} we have
\[
\lambda_1^+(F_n)\ge
\frac{2}{\alpha(\mu^* + 1)+\beta + \sqrt{(\alpha(\mu^* +1)+\beta)^2-4\alpha\beta}},
\]
where $\alpha=1/\lambda_1^+(F_{n-1}), \beta=1/\lambda_1^+(G_{B_n}) = 1/\lambda_1^+(F_0)=q$. If we denote $y_n=1/\lambda_1^+(F_n)$ we obtain
\begin{eqnarray*}
y_n &\le& \frac12 \left( (\mu^*+1) y_{n-1} + q + \sqrt{ ((\mu^*+1) y_{n-1} +q)^2 -4 q y_{n-1} }\right)\\
&\le& (\mu^*+1) y_{n-1} + q \le 6 y_{n-1} + q.
\end{eqnarray*}
Solving the above difference inequality yields $y_n\le \frac{q}{5}( 6^{n+1}-1)$ and so
\[
\lambda_1^+(F_n)\ge \frac{1}{q} \frac{5}{6^{n+1}-1},
\]
as claimed. \hfill $\diamondsuit$
\medskip
\begin{remark}
The asymptotic behavior of $\lambda_1^+(F_n)\to0$ as $n\to\infty$ is not surprising. For example, if we consider a cycle $C_N$ on $N$ vertices then, we have $\lambda_1^+(C_N)= 2\cos\frac\pi2 \frac{N-1}{N}$ and so $\lambda_1^+(C_N)= O(N^{-1})=O(|V(C_N)|^{-a})$ with the polynomial decay rate $a=1$. In the case of the graph $F_n$ the number of its vertices growths exponentially $N=O(q^{n+1})$, and so the lower bound $\lambda_1^+(F_n) \ge O(6^{-n-1}) = O(|V(F_n)|^{-a})$ with the polynomial decay rate $a= \ln 6/\ln q \dot{=} 3.7234$ as $|V(F_n)|\to\infty$ can be expected.
\end{remark}
\begin{figure}
\begin{center}
\includegraphics[angle=90,width=1.2truecm]{figures/fulvene1}
\qquad
\includegraphics[angle=90,width=1truecm]{figures/fulvene2}
\qquad
\includegraphics[angle=90,width=5truecm]{figures/fulvene4}
\\
\ \hglue-2truecm $F_0$ \hskip 2truecm $F_1$ \hskip 3truecm $F_2$
\includegraphics[width=6.5truecm]{figures/fulvene8}
\\
$F_3$
\end{center}
\caption{
The graphs $F_0,F_1,F_2, F_3$ of the ``fulvene'' family of integrally invertible graphs.
}
\label{fig-fulvenefamily1}
\end{figure}
\section{Arbitrarily bridgeable connected graphs with a unique 1-factor}
In this section we present a census of invertible graphs on $m\le 6$ vertices with a unique 1-factor, such that they can be arbitrarily bridged to an invertible graph through a set of $k\le m/2$ vertices. Recall that a graph $G$ has a unique 1-factor if $G$ contains a unique 1-regular spanning subgraph (i.e., a perfect matching). Note that any graph having a 1-factor should have even number of vertices.
For $m=2$ the graph $K_2$ is the unique connected graph with a unique 1-factor. It is a positively invertible bipartite graph with the spectrum $\sigma(K_2)=\{-1,1\}$.
For $m=4$ there are two connected graphs $Q_1, Q_2$ with a unique 1-factor shown in Fig.~\ref{fig-quarticfamily}. Both graphs are positively invertible with the spectra
\[
\sigma(Q_1) = \{\pm 1.6180, \pm 0.6180 \},\qquad \sigma(Q_2)=\{-1.4812, -1, 0.3111, 2.1701\}.
\]
The graph $Q_1$ can be arbitrarily bridged over the singleton sets $\{1\}$, $\{2\}$, $\{3\}$, $\{4\}$ and over pairs of vertices: $\{2,3\}, \{1,3\}, \{2,4\}$. The graph $Q_1$ can be arbitrarily bridged over the singletons $\{2\}, \{3\}, \{4\}$ and over the pair $\{2,3\}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{figures/quartic}
\\
$Q_1$ \hskip 0.3\textwidth $Q_2$
\end{center}
\caption{
The family of graphs on $4$ vertices with a unique 1-factor.
}
\label{fig-quarticfamily}
\end{figure}
The situation is more interesting and, at the same time, more complicated, for connected graphs on $m=6$ vertices with a unique $1$-factor. To this end, we recall the well-known Kotzig's theorem stating that a graph with a unique 1-factor has a bridge that belongs to the 1-factor sub-graph. Splitting of $6$ vertices into two subsets of 3 vertices connected by a bridge leads to graphs $H_1, H_4, H_{19}$ shown in Fig.~\ref{fig-hexafamily}. Splitting into subsets of 2 and 4 vertices is impossible because the bridge should belong to the 1-factor and so the hanging leaf vertex of a 2-vertices sub-graph is not contained in the 1-factor. Splitting into a 1 vertex graph and 5-vertices graph lead to the remaining 17 graphs shown in Fig.~\ref{fig-hexafamily}. One can construct these 17 graphs from the set of all 10 graphs on four vertices (including disconnected graphs) by bridging to $K_2$ using up to $4$ edges.
In summary, there exist 20 undirected connected graphs on $m=6$ vertices with a unique 1-factor shown in Fig.~\ref{fig-hexafamily}. All of them have invertible adjacency matrix. Except of the graph $H_{19}$ they are integrally invertible.
In this census, there are three bipartite graphs $H_1, H_2, H_6$ which are simultaneously positively and negatively invertible. There are twelve graphs
\[
H_3, H_4, H_7, H_8, H_9, H_{13}, H_{14}, H_{15}, H_{16}, H_{17}, H_{18}, H_{20},
\]
which are positively invertible. The three graphs $H_5, H_{10}, H_{12}$ are negatively invertible. The integrally invertible graph $H_{11}$ is neither positively nor negatively invertible. The graphs $H_8$ and $H_{18}$ are iso-spectral but not isomorphic.
\begin{figure}
\begin{center}
\includegraphics[width=0.3\textwidth]{figures/hexa-01}
\quad
\includegraphics[width=0.25\textwidth]{figures/hexa-02}
\quad
\includegraphics[width=0.25\textwidth]{figures/hexa-03}
\\
$H_1$ \hskip 0.3\textwidth $H_2$ \hskip 0.3\textwidth $H_3$
\smallskip
\includegraphics[width=0.24\textwidth]{figures/hexa-04}
\hskip 0.08\textwidth
\includegraphics[width=0.18\textwidth]{figures/hexa-05}
\hskip 0.15\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-06}
\\
$H_4$ \hskip 0.3\textwidth $H_5$ \hskip 0.3\textwidth $H_6$
\medskip
\includegraphics[width=0.12\textwidth]{figures/hexa-07}
\hskip 0.19\textwidth
\includegraphics[width=0.18\textwidth]{figures/hexa-08}
\hskip 0.19\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-09}
\\
$H_7$ \hskip 0.3\textwidth $H_8$ \hskip 0.3\textwidth $H_9$
\medskip
\includegraphics[width=0.16\textwidth]{figures/hexa-10}
\hskip 0.19\textwidth
\includegraphics[width=0.16\textwidth]{figures/hexa-11}
\hskip 0.19\textwidth
\includegraphics[width=0.16\textwidth]{figures/hexa-12}
\\
$H_{10}$ \hskip 0.3\textwidth $H_{11}$ \hskip 0.3\textwidth $H_{12}$
\medskip
\includegraphics[width=0.18\textwidth]{figures/hexa-13}
\hskip 0.19\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-14}
\hskip 0.19\textwidth
\includegraphics[width=0.18\textwidth]{figures/hexa-15}
\\
$H_{13}$ \hskip 0.3\textwidth $H_{14}$ \hskip 0.3\textwidth $H_{15}$
\medskip
\includegraphics[width=0.18\textwidth]{figures/hexa-16}
\hskip 0.19\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-17}
\hskip 0.19\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-18}
\\
$H_{16}$ \hskip 0.3\textwidth $H_{17}$ \hskip 0.3\textwidth $H_{18}$
\medskip
\includegraphics[width=0.18\textwidth]{figures/hexa-19}
\hskip 0.19\textwidth
\includegraphics[width=0.12\textwidth]{figures/hexa-20}
\\
$H_{19}$\hskip 0.3\textwidth $H_{20}$
\medskip
\end{center}
\caption{
The family of graphs on $6$ vertices with a unique 1-factor.
}
\label{fig-hexafamily}
\end{figure}
\begin{table}
\small
\caption{The family of graphs on 6 vertices with a unique 1-factor, their signability and spectrum. Graphs $H_8$ and $H_{18}$ are iso-spectral but not isomorphic.}
\begin{center}
\begin{tabular}{c||c|c}
Graph & invertibility & spectrum \\
\hline\hline
$H_1$ & pos, neg & $\{ -1.8019, -1.2470, -0.4450, 0.4450, 1.2470, 1.8019 \}$ \\
\hline
$H_2$ & pos, neg & $\{ -1.9319, -1.0000, -0.5176, 0.5176, 1.0000, 1.9319 \}$ \\
\hline
$H_3$ & pos & $\{ -1.7397, -1.3738, -0.5945, 0.2742, 1.0996, 2.3342 \}$ \\
\hline
$H_4$ & pos & $\{ -1.7746, -1.0000, -1.0000, 0.1859, 1.3604, 2.2283 \}$ \\
\hline
$H_5$ & neg & $\{ -1.6180, -1.6180, -0.4142, 0.6180, 0.6180, 2.4142 \}$ \\
\hline
$H_6$ & pos, neg & $\{ -2.2470, -0.8019, -0.5550, 0.5550, 0.8019, 2.2470 \}$ \\
\hline
$H_7$ & pos & $\{ -1.8942, -1.3293, -0.6093, 0.3064, 0.7727, 2.7537 \}$ \\
\hline
$H_8$ & pos & $\{ -1.9032, -1.0000, -1.0000, 0.1939, 1.0000, 2.7093 \}$ \\
\hline
$H_9$ & pos & $\{ -1.6180, -1.3914, -1.0000, 0.2271, 0.6180, 3.1642 \}$ \\
\hline
$H_{10}$ & neg & $\{ -1.8608, -1.6180, -0.2541, 0.6180, 1.0000, 2.1149 \}$ \\
\hline
$H_{11}$ & int inv & $\{ -1.8241, -1.6180, -0.5482, 0.3285, 0.6180, 3.0437 \}$ \\
\hline
$H_{12}$ & neg & $\{ -2.1420, -1.3053, -0.3848, 0.4669, 0.7661, 2.5991 \}$ \\
\hline
$H_{13}$ & pos & $\{ -1.8563, -1.4780, -0.7248, 0.1967, 0.8481, 3.0143 \}$ \\
\hline
$H_{14}$ & pos & $\{ -1.9202, -1.0000, -0.7510, 0.2914, 1.0000, 2.3799 \}$ \\
\hline
$H_{15}$ & pos & $\{ -1.6783, -1.3198, -1.0000, 0.1397, 1.2297, 2.6287 \}$ \\
\hline
$H_{16}$ & pos & $\{ -2.1364, -1.2061, -0.5406, 0.2611, 1.0825, 2.5395 \}$ \\
\hline
$H_{17}$ & pos & $\{ -1.8619, -1.2827, -1.0000, 0.2512, 0.4897, 3.4037 \}$ \\
\hline
$H_{18}$ & pos & $\{ -1.9032, -1.0000, -1.0000, 0.1939, 1.0000, 2.7093 \}$ \\
\hline
$H_{19}$ & nonint inv & $\{ -1.7321, -1.0000, -1.0000, -0.4142, 1.7321, 2.4142 \}$ \\
\hline
$H_{20}$ & pos & $\{ -2.3117, -1.0000, -0.6570, 0.3088, 0.7272, 2.9327 \}$ \\
\hline
\hline\hline
\end{tabular}
\end{center}
\noindent Legend: 'pos'/'neg' stands for a positively/negatively invertible graph, 'int inv' means an integrally invertible graph which is neither positively nor negatively invertible, 'nonint inv' stands for a graph with an adjacency matrix which is invertible but it is not integral.
\end{table}
\begin{table}
\label{tab-2}
\small
\caption{The family of graphs on 6 vertices with a unique 1-factor which can be arbitrarily bridged through $k=1,2,3$ vertices.}
\begin{center}
\rotatebox{90}{
\begin{tabular}{c||c|c|c}
\tiny Graph & $k=1$ & $k=2$ & $k=3$ \\
\hline\hline
\tiny $H_1$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{4,6\}, \{2,6\}, \{4,5\}, \{3,5\}, \{2,5\}, \{1,5\}, \{2,4\}, \{2,3\}, \{1,3\}$ & \tiny $\{2,4,6\}, \{2,4,5\}, \{2,3,5\}, \{1,3,5\}$ \\
\hline
\tiny $H_2$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{4,6\}, \{2,6\}, \{3,5\}, \{2,5\}, \{1,5\}, \{3,4\}, \{2,4\}, \{1,4\}, \{2,3\}, \{1,3\}$ & \tiny $\{2,4,6\}, \{2,3,5\}, \{1,3,5\}, \{2,3,4\}, \{1,3,4\}$ \\
\hline
\tiny $H_3$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}$ & \tiny $\{4,6\}, \{2,6\}, \{3,5\}, \{2,5\}, \{3,4\}, \{2,4\}, \{2,3\}$ & \tiny $\{2,4,6\}, \{2,3,5\}, \{2,3,4\}$ \\
\hline
\tiny $H_4$ & \tiny $\{6\}, \{5\}, \{4\}, \{2\}$ & \tiny $\{4,6\}, \{2,6\}, \{4,5\}, \{2,5\}, \{2,4\}$ & \tiny $\{2,4,6\}, \{2,4,5\}$ \\
\hline
\tiny $H_5$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{3,6\}, \{2,6\}, \{4,5\}, \{3,5\}, \{2,5\}, \{1,5\}, \{2,4\}, \{2,3\}, \{1,3\}$ & \tiny $\{2,3,6\}, \{2,4,5\}, \{2,3,5\}, \{1,3,5\}$ \\
\hline
\tiny $H_6$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{5,6\}, \{4,6\}, \{2,6\}, \{3,5\}, \{1,5\}, \{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,4,6\}, \{1,3,5\}, \{2,3,4\}, \{1,2,3\},$ \\
\hline
\tiny $H_7$ & \tiny $\{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{3,5\}, \{1,5\}, \{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{1,3,5\}, \{2,3,4\}, \{1,2,3\}$ \\
\hline
\tiny $H_8$ & \tiny $\{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{3,4\}, \{2,4\}, \{1,4\}, \{2,3\}, \{1,2\}$ & \tiny $\{2,3,4\}, \{1,2,4\}$ \\
\hline
\tiny $H_9$ & \tiny $\{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,3,4\}, \{1,2,3\}$ \\
\hline
\tiny $H_{10}$ & \tiny $\{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $ \{4,5\}, \{2,5\}, \{3,4\}, \{2,4\}, \{1,4\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,4,5\}, \{1,3,4\}, \{1,2,4\}$ \\
\hline
\tiny $H_{11}$ & \tiny $\{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $ \{4,5\}, \{2,5\}, \{3,4\}, \{2,4\}, \{1,4\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,4,5\}, \{1,3,4\}, \{1,2,4\}$ \\
\hline
\tiny $H_{12}$ & \tiny $\{6\}, \{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $ \{5,6\}, \{4,5\}, \{2,5\}, \{3,4\}, \{2,4\}, \{1,4\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,4,5\}, \{1,3,4\}, \{1,2,4\}$ \\
\hline
\tiny $H_{13}$ & \tiny $\{5\}, \{4\}, \{2\}, \{1\}$ & \tiny $\{4,5\}, \{2,5\}, \{1,5\}, \{2,4\}, \{1,2\}$ & \tiny $\{2,4,5\}, \{1,2,5\}$\\
\hline
\tiny $H_{14}$ & \tiny $\{6\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{4,6\}, \{2,6\}, \{1,6\}, \{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,4,6\}, \{1,2,6\}, \{2,3,4\}, \{1,2,3\}$ \\
\hline
\tiny $H_{15}$ & \tiny $ \{5\}, \{4\}, \{2\}, \{1\}$ & \tiny $\{4,5\}, \{2,5\}, \{1,5\}, \{2,4\}, \{1,2\}$ & \tiny $\{2,4,5\}, \{1,2,5\}$ \\
\hline
\tiny $H_{16}$ & \tiny $\{6\}, \{5\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{5,6\}, \{3,5\}, \{1,5\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{1,3,5\}, \{1,2,3\}$ \\
\hline
\tiny $H_{17}$ & \tiny $\{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{2,3,4\}, \{1,2,3\}$\\
\hline
\tiny $H_{18}$ & \tiny $\{5\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{4,5\}, \{3,5\}, \{1,5\}, \{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $ \{3,4,5\}, \{1,3,5\}, \{2,3,4\}, \{1,2,3\} $ \\
\hline
\tiny $H_{19}$ & --- & --- & --- \\
\hline
\tiny $H_{20}$ & \tiny $\{6\}, \{4\}, \{3\}, \{2\}, \{1\}$ & \tiny $\{4,6\}, \{1,6\}, \{3,4\}, \{2,4\}, \{2,3\}, \{1,3\}, \{1,2\}$ & \tiny $\{1,2,3\}, \{2,3,4\}$\\
\hline\hline
\end{tabular}
}
\end{center}
\end{table}
\section*{Acknowledgements} The authors are thankful to Prof.~Jozef~\v{S}ir\'a\v{n} and the anonymous referee for many helpful comments and suggestions that helped us to improve the paper. The research was supported by the Research Grants APVV 0136-12 and 15-0220, VEGA 1/0026/16 and 1/0142/17 (SP), and by the VEGA Research Grant 1/0780/15 (D\v{S}).
|
1,108,101,565,484 | arxiv | \section{Introduction}
\label{sec:introduction}
Confinement is one of the peculiar features of Quantum Chromodynamics,
the theory of strong interactions. Thanks to 25 years of intensive
research in the field of lattice gauge theory, a few mechanisms for
confinement have been identified. These mechanisms are associated
either with monopoles or vortices and seem to be closely related to
each other~\cite{Greensite:2003bk}. In this context, confinement was
and is stated mostly by a non-vanishing string tension
$\sigma_{\mathrm{Wilson}}$ which expresses the minimal energy of the gluon field
between a pair of static quarks. The string tension is defined by
Wilson loops and can be extracted in the limit of large Euclidean time
from the Wilson loop's area-law decay. This definition, however, is
not completely satisfying, because not only quarks but also gluons are
confined, and there is no area law in the more realistic case of light
dynamical quarks present in the vacuum.
There are two other, though less popular, approaches that might help
to shed additional light on the phenomenon of confinement. One is given
by the Hamiltonian approach which promises to present an understanding
not only of bound states but also of the vacuum structure in terms of
wave functionals. The other is a more field-theoretically inspired
approach that focuses on the QCD Green's functions and their infrared
behavior. The QCD Green's functions may serve as input to a hadron
phenomenology based on the Bethe-Salpeter and Faddeev equations.
There, the ultimate goal is to attain a coherent description
of hadronic states and processes based on the dynamics of confined
gluons, ghosts and quarks (see, e.g., Ref.~\cite{Alkofer:2000wg}).
Both the Hamiltonian approach and investigations of QCD Green's functions
require to fix a gauge. This introduces the well-known Gribov
ambiguity present in the Coulomb as well as in covariant gauges. One
should keep in mind that the confinement mechanisms associated with
monopoles and vortices, that received credit by reproducing the
Wilson string tension, also mostly require a gauge condition.
In the Coulomb gauge, the Gribov ambiguity represents a severe
source of uncertainty and its effect on the results must be faithfully
checked. On the other hand, the Coulomb gauge yields a particularly
interesting confinement picture called the Gribov-Zwanziger
scenario~\cite{Gribov:1977wm,Zwanziger:1998ez}.
This scenario might provide an understanding of confinement
even in the presence of dynamical quarks when the Wilson-loop
criterion fails.
A central element of the Gribov-Zwanziger confinement scenario in
Coulomb gauge is the instantaneous color-Coulomb potential involving
the Faddeev-Popov operator $M$ (in Coulomb gauge) and the infrared
spectral properties of the latter~\cite{Greensite:2004ur,Nakagawa:2007fa}.
The expression
\begin{equation}
V_{\mathrm{Coul}} (x-y)\delta^{ab} = \left\langle
g^2\left[M^{-1}(-\triangle)M^{-1} \right]^{ab}(x,y) \right\rangle \;
\label{eq:coulomb_potential}
\end{equation}
is defined through the vacuum expectation value of the potential part of the
Hamilton operator
\begin{equation}
H = \frac{1}{2} \int d^{3}x \, \left( {\vec \Pi}_{tr}^{2}({\vec x})
+ {\vec B}^{2}({\vec x}) \right) + H_{\mathrm{Coul}}
\label{eq:Hamiltonian}
\end{equation}
resulting from the elimination of longitudinal degrees of freedom.
Here the potential term $H_{\mathrm{Coul}}$ is expressed in terms of the color charge
density (including external sources and the charge density of the gluon field
itself) by means of the color-Coulomb potential,
\begin{equation}
H_{\mathrm{Coul}} = \frac{1}{2} \int d^{3}x d^{3}y \, \rho^{a}({\vec x})
V_{\mathrm{Coul}} (x-y)\delta^{ab}
\rho^{b}({\vec y}) \; .
\label{eq:Coulomb_energy}
\end{equation}
As Zwanziger has shown~\cite{Zwanziger:1998ez}, the Coulomb potential does
not equal the Wilson potential $V_{\mathrm{Wilson}}$ used to extract the string tension
$\sigma_{\mathrm{Wilson}}$ as an order parameter for confinement. Instead, for large
spatial distances $r$ the Coulomb potential represents an upper bound for the
rise of the Wilson potential,
\begin{equation}
V_{\mathrm{Wilson}}(r) \leq - \frac{4}{3} V_{\mathrm{Coul}}(r) \; .
\label{eq:Zwanziger_inequality}
\end{equation}
In other words, there is no confinement without Coulomb confinement
since the Coulomb string tension is an upper bound for the Wilson string
tension~\cite{Zwanziger:2002sh},
\begin{equation}
\sigma_{\mathrm{Wilson}} \leq \frac{4}{3} \sigma_{\mathrm{Coul}} \; .
\label{eq:Zwanziger_inequality_2}
\end{equation}
Zwanziger has continuously developed the confinement scenario originally
proposed by Gribov~\cite{Gribov:1977wm}. He has put forward the Coulomb
potential as a new order parameter for
confinement~\cite{Zwanziger:1998ez,Zwanziger:2002sh,Zwanziger:2003de}.
In fact, the Coulomb potential is expected to linearly rise at large~$r$ even in
the presence of dynamical quarks when the Wilson-loop criterion fails. Recent
lattice studies have shown, however, that the relation
(\ref{eq:Zwanziger_inequality}) is only a {\it necessary}~\cite{Greensite:2003nf}
condition for confinement, and that the Coulomb potential
can be linearly rising with spatial distance even in the deconfinement
phase~\cite{Nakagawa:2007fa,Nakagawa:2006fk}.
Using lattice techniques, a linearly rising Coulomb
potential~\cite{Greensite:2003nf,Nakamura:2005ux,Nakagawa:2006fk}
and a connection between the center-vortex mechanism and the Gribov-Zwanziger
scenario~\cite{Greensite:2004cz,Greensite:2004ke,Greensite:2004ur}
have been observed. Furthermore, Greensite et al.\ proposed~\cite{Greensite:2003xf}
to use correlators of partial Polyakov loops to measure the Coulomb potential.
Corresponding $SU(2)$ as well as $SU(3)$ studies revealed that the Coulomb
string tension $\sigma_{\mathrm{Coul}}$ could well be 2-3 times larger than
the Wilson string tension
$\sigma_{\mathrm{Wilson}}$~\cite{Greensite:2003xf,Greensite:2004cz,Nakamura:2005ux,Nakagawa:2006fk}.
This is in contrast to results of $SU(2)$ studies where the Coulomb potential
was measured by means of its very definition via \Eq{eq:coulomb_potential} suggesting
$\sigma_{\mathrm{Coul}} = \sigma_{\mathrm{Wilson}}$ \cite{Cucchieri:2002su,Langfeld:2004qs}.
In the present study we provide a (yet missing) thorough measurement
of the Coulomb potential in $SU(3)$ gauge theory based on its very
definition in \Eq{eq:coulomb_potential}. We investigate the relation
between $\sigma_{\mathrm{Wilson}}$ and $\sigma_{\mathrm{Coul}}$ and find, though
hedged with large uncertainty, $\sigma_{\mathrm{Coul}}$ to be 1.6 times larger than
$\sigma_{\mathrm{Wilson}}$. The origin of the systematic uncertainty will be discussed.
The paper is organized as follows. In Sect.~\ref{sec:detailsofsimulation}
we describe the details of our numerical simulation and define the
lattice observables measured. We investigate finite-volume effects,
lattice-spacing effects and the effects due to the Gribov ambiguity
in Sect.~\ref{sec:systematics}. In Sect.~\ref{sec:infrared}
we analyze the infrared behavior of the effective Coulomb potential.
A summary concludes this paper.
\section{Details of the numerical simulation}
\label{sec:detailsofsimulation}
\subsection{Lattice samples and gauge-fixing algorithms}
\label{sec:gaugefixing}
For our study we use the standard lattice formulation of $SU(3)$
Yang-Mills theory in Coulomb gauge where we always start from
non-gauge-fixed $SU(3)$ gauge configurations and apply the Coulomb gauge
condition subsequently. Our sets of gauge configurations were generated
with Wilson's one-plaquette action at three values of the inverse
coupling, $\beta = 5.8$, $6.0$ and $6.2$, for a couple of lattice sizes
$L_{s}^{3} \times L_{t}$ where $L_{t}$ and $L_{s}$ denote the
spatial and temporal lattice extension, respectively. We have only
considered hyper-cubic lattices with $L_{s}=L_{t}=L=12$, 16, 24, 32 and 48.
Those ensembles were then gauge-fixed to the Coulomb gauge by
minimizing the gauge functional
\begin{equation}
F_{U}[g] = \frac{1}{3} \sum_{x}
\sum_{i=1}^{3} \Re\operatorname{Tr}\left(1 -g_x U_{x,i} g^{\dagger}_{x+\hat{i}} \right) \; ,
\label{eq:func_coulomb_gauge}
\end{equation}
that involves all space-like links on the lattice. This was
accomplished by adjusting the gauge transformations $g_x \in SU(3)$ while
keeping the original gauge configuration $U$ fixed. Due to the
particular form of $F_{U}[g]$ no condition is
imposed on time-like links. Consequently, the different time-slices can
be minimized independently. We considered gauge-fixing within a given
time-slice successful as soon as the stopping criterion
\begin{equation}
\max_{\vec{x},\ t\,\text{fix}} \operatorname{Tr} \left[ \left(\partial_{i} \,^{g}\!
A_{x,i} \right) \left(\partial_{i} \,^{g}\! A_{x,i} \right)^{\dagger}
\right] < 10^{-13}
\label{eq:local_gauge_violation}
\end{equation}
was satisfied. Here the lattice gauge-potential is defined in the
usual way as
\begin{equation}
^g\!A_{x+\hat{i}/2,i} = \frac{1}{2iag_{0}} \Big( \,^gU_{x,i} - \,
^gU^{\dagger}_{x,i} \Big)\Big|_{traceless} \; ,
\label{eq:transversal_potential}
\end{equation}
where $^gU_{x,i} \equiv g_x U_{x,i} g^{\dagger}_{x+\hat{i}}\;$, $a$ is
the lattice spacing and $g_0$ the bare coupling constant which
is related to $\beta$ through $\beta=6/g^2_0$.
To minimize the gauge functional we used an over-relaxation (OR) algorithm
preceded by an optimally-tuned simulated annealing (SA) algorithm.
In what follows, we call this particular combination of simulated annealing and
over-relaxation steps the SA-OR algorithm. To assess the influence of
Gribov copies, we also generated some gauge copies with the pure OR
algorithm without preconditioning. In all cases, the over-relaxation
parameter was tuned to $\omega=1.70$ on the small and
$\omega=1.60$ on the large lattices. More details are given below.
The SA algorithm has been proven to be very useful in handling various
optimization problems. For this algorithm the gauge functional $F_{U}[g]$ is
regarded as a ``spin Hamiltonian'' where the gauge transformation
fields $g_x$ take the role of ``spin variables'' coupled through the links
$U_{x,i}$ (kept fixed). Minimizing $F_{U}[g]$ is achieved by
adiabatically lowering the auxiliary temperature $T$ of a statistical
spin glass system characterized by the Gibbs weight
\begin{equation}
W[g] \propto \exp\left(-F_U[g]/T\right) \; .
\label{eq:spin-model}
\end{equation}
The minimization process always starts with equilibrating this spin
system at some initial temperature $T=T_i$ which is then slowly decreased.
Formally, in the limit of (adiabatically) lowering $T \to 0$ this system
approaches the ground state and hence the gauge functional reaches its
absolute minimum for a given gauge configuration.
For the practical purpose considered here such an adiabatic cooling-down process
is not feasible as it would require an enormous amount of computing time.
Nevertheless, we find that much lower minima for $F_{U}[g]$ can be reached,
compared to applying only over-relaxation (OR), if we combine the SA with
the OR algorithm as follows: We start from an initial temperature of
$T_{i}=0.45$ and linearly decrease the temperature down to $T_{f}=0.01$
within 1500 ``compound sweeps''. Each such sweep consists of one
heatbath and three microcanonical update sweeps. After this, we use the OR
algorithm until the Coulomb gauge is reached, i.e.\ the
stopping criterion (\ref{eq:local_gauge_violation}) is satisfied.
The advantage of using the SA-OR instead of the OR algorithm alone
becomes more pronounced as the lattice becomes larger. Furthermore,
the number of necessary iterations in the subsequent OR algorithm is
drastically reduced by a preceding SA algorithm, the more the lower the
final $T_f$ is chosen. Note that instead
of adding subsequent OR steps, we could also have used SA on its own
extending its use to a much lower temperature $T_{f}$ to fix to Coulomb gauge.
This, however, is much more CPU-time intensive and we find no benefit
in doing this, because after gauge-fixing the
transversality condition (\ref{eq:local_gauge_violation}) must at any case
be guaranteed with high precision, which can be achieved only by the finalizing OR.
We observe that the time-slices of a given
configuration may behave very differently during the iterative
gauge-fixing process. In fact, we find the number of necessary
iterations may differ by a factor of 10 to 20 between the individual
time-slices of a given configuration. In the majority of cases,
time-slices did not show any recalcitrancy during gauge fixing,
although in some cases time-slices could not be fixed within a
certain (predefined) number of iterations. In the latter case we simply repeated
the entire gauge-fixing process for {\it these} time-slices, using the same
algorithm but starting from a different randomly chosen gauge
transformation. The ``well-behaved'' and hence already gauge-fixed
time-slices were not touched again.
After all individual time-slices had been minimized,
the original configuration $U$ was gauge-transformed, i.e.,
$U_{x,\mu}\to{}^gU_{x,\mu}$. To simplify the notation we drop the label
$g$ in what follows and assume that a gauge configuration $U$
satisfies the Coulomb gauge condition already. Since our observables,
namely the effective Coulomb potential and the ghost propagator, are genuine
three-dimensional, instantaneous observables defined by space-like links only,
we did not have to fix the residual gauge freedom.
The latter, after the Coulomb gauge has been fixed, resides in spatially constant
but time-dependent gauge transformations (for a continuum view at this problem
see \cite{Watson:2007fm}).
\subsection{Observables of interest}
The Coulomb energy is a complicated functional of the
transverse gauge potential $A_i({\vec x})$ and the total color charge
density. Nevertheless, it is instructive to characterize its gross
features through the infrared and ultraviolet behavior of the
expectation value of the color-diagonal part of the kernel
$M^{-1} (-\triangle) M^{-1}$ in momentum space alone. On the
lattice this is defined as the MC average
\begin{equation}
V^{L}_{\mathrm{Coul}}({\vec k}) = \frac{1}{8 L_s^3} \left\langle
\sum_{a,{\vec x},{\vec y}} e^{i{\vec k} \cdot ({\vec x}-{\vec y})}
\Big[M^{-1}(-\triangle)M^{-1}\Big]^{aa}_{\vec{x}\vec{y}} \right\rangle \; ,
\label{eq:potential_in_momentumspace}
\end{equation}
where we use a shorthand notation for the scalar product
${\vec{k}\cdot\vec{x}} = 2\pi \sum_{i=1}^3 k_{i}x_{i}/L_i$ with
integer-valued lattice momenta $k_i$ and lattice coordinates $x_i$.
$M$ is the lattice Faddeev-Popov operator for the Coulomb gauge
\begin{align}
M^{ab}_{xy} = \delta_{x_4,y_4} &\sum_{i=1}^{3}
\Re\operatorname{Tr}\Big[\left\{T^a,T^b\right\} \left(U_{x,i} +
U_{x-\hat{i},i}\right) \delta_{{\vec x},{\vec y}} \nonumber
\\ &\hspace{-3em}-2\,T^bT^a\, U_{x,i}\, \delta_{\vec{x}+\hat{i},\vec{y}}
- 2\,T^aT^b\, U_{x-\hat{i},i}\,\delta_{\vec{x}-\hat{i},\vec{y}}
\Big] \; .
\label{eq:FP_operator}
\end{align}
Note that the Faddeev-Popov operator is a direct sum of operators
acting within individual time-slices. In coordinate space these
three-dimensional operators define the Coulomb energy of a given
dynamical (gluonic) color-charge density plus an external one
(cf.~\Eq{eq:Coulomb_energy}). Given the tree-level form of the
Coulomb potential on a three-dimensional lattice we relate
integer-valued lattice momenta $k_i\in (-L_i/2,L_i/2]$ to physical
ones by
\begin{equation}
q_{i}(k_{i}) = \frac{2}{a} \sin\left(\frac{\pi k_{i}}{L_i}\right) \; .
\label{eq:physical_momenta}
\end{equation}
Physical units are assigned by using the interpolation formula for
$r_0/a$ as given in \cite{Necco:2001xg} setting $r_0=0.5~\text{fm}$. To
simplify the writing we introduce $q$ as abbreviation for
$|\vec{q}\,|$ whenever appropriate.
In Ref.~\cite{Zwanziger:2003de} an analytic calculation of the Coulomb
potential is presented which reads, upon Fourier transformation,
\begin{equation}
V_{\mathrm{Coul}}(q) = q^2 G^2(q) + V^{\text{c}}(q) \; .
\label{eq:cp_factorisation}
\end{equation}
Here $G$ denotes the ghost propagator (entering the disconnected part)
and $V^{\text{c}}$ denotes the connected part of the potential. Under the
assumption that the (yet unknown) connected part can be neglected, an infrared
asymptotic limit for $V_{\mathrm{Coul}}$ has been given in \cite{Zwanziger:2003de}.
It will be analyzed below at what momenta the factorization
$V_{\mathrm{Coul}}(q) \simeq q^2 G^2(q)$ is justified from our data concerning both
the effective Coulomb potential and the ghost propagator. The latter
can be estimated in momentum space as the MC average
\begin{equation}
G^{L}\big(\vec{k}\big) = \frac{1}{8 L_s^3} \left\langle
\sum_{a,\vec{x},\vec{y}} e^{i\vec{k} \cdot (\vec{x}-\vec{y})}
\big[M^{-1}\big]^{aa}_{\vec{x}\vec{y}} \right\rangle
\label{eq:ghostprop_in_momentumspace}
\end{equation}
at non-zero lattice momenta $k$. As for the Coulomb potential we use
\Eq{eq:physical_momenta} to assign physical momenta to $G$. To invert the
Faddeev-Popov operator we adapted the techniques
developed in Landau-gauge studies of the ghost propagator
(see, e.g., \cite{Sternbeck:2005tk}).
The data for the ghost propagator used to test the factorization hypothesis
will not be presented in the present publication. They have been presented
at Lattice 2007~\cite{Voigt:2007wd} and will be discussed more in depth
in a forthcoming paper~\cite{BerlinOsaka:2008}.
Note that both the evaluation of the effective Coulomb potential and
of the ghost propagator involve CPU-time intensive operations. As a
consequence, we have restricted our measurements to lattice momenta
$k$ that survive a cylinder cut. Our cylinder cut is the obvious
adaptation of the Landau-gauge cylinder cut~\cite{Leinweber:1998uu}.
To minimize finite-volumes effects, we also cone cut our data
\cite{Leinweber:1998uu} if they refer to lattices smaller than $(2.5~\text{fm})^4$.
\subsection{Running coupling and physical scale}
\label{sec:running_coupling}
The Coulomb potential is a renormalization-group
invariant which can be written as (here for pure SU(3) gauge
theory)~\cite{Cucchieri:2000hv}
\begin{equation}
q^2 V_{\mathrm{Coul}}(q) = \frac{12}{11}~g_{\mathrm{Coul}}^2(q/\Lambda_{\mathrm{Coul}}) \; ,
\end{equation}
where $\Lambda_{\mathrm{Coul}}$ is a special QCD scale parameter characteristic of
the Coulomb gauge, that defines a running coupling constant $g_{\mathrm{Coul}}$.
The latter has to satisfy the renormalization-group equation
\begin{equation}
\label{eq:rge}
q\frac{\partial g_{\mathrm{Coul}}}{\partial q}=\beta_{\mathrm{Coul}}(g_{\mathrm{Coul}}) \; ,
\end{equation}
where the beta function, $\beta_{\mathrm{Coul}}$, has the usual weak-coupling
expansion starting with the two standard scheme-independent coefficients
\begin{equation}
\label{eq:b0_b1}
b_0 \equiv \frac{11}{16\pi^2} \quad\textrm{and}\quad b_1 \equiv
\frac{51}{128\pi^4}
\end{equation}
(see \cite{Cucchieri:2000hv} for higher terms).
For sufficiently large
$q$, the product $11q^2V_{\mathrm{Coul}}(q)/12$
is expected to be described through the two-loop
expression of the running coupling
\begin{equation}
\label{eq:alpha_s_twoloop}
g_{\mathrm{Coul}}^2(q) =
\frac{1}{b_0\ln(q^2/\Lambda_{\mathrm{Coul}}^2)}\Bigg[
1 - \frac{b_1}{b_0^2}
\frac{\ln[\ln(q^2/\Lambda_{\mathrm{Coul}}^2)]}{\ln(q^2/\Lambda_{\mathrm{Coul}}^2)}
\Bigg] \; .
\end{equation}
Lattice data describing $q^2V_{\mathrm{Coul}}(q)$ do not depend on the lattice
spacing~$a$ in the asymptotic scaling region. At larger~$a$, scaling
violations should be expected though, and they will be discussed below
for the lattice spacings used in this study.
In a previous analysis of the data~\cite{Voigt:2007wd} we used an
ultraviolet fit to the one-loop expression
(cf.\ the first term of \Eq{eq:alpha_s_twoloop}) to fix the unknown
physical scale of the effective Coulomb potential (see
Ref.~\cite{Voigt:2007wd} for details). For the present study, we
scrutinized if the highest momenta accessible in our simulations
really permit a feasible fit to the one-loop or the two-loop expression
given in \Eq{eq:alpha_s_twoloop}. We find that this is not the case and
that the ultraviolet fit described in~\cite{Voigt:2007wd} has artificially
up-scaled our data by a free factor bigger than one.
In the present study, we therefore do not rely anymore on this ultraviolet fit.
Indeed, the physical scale is fixed by
simply multiplying the bare lattice data for the effective Coulomb
potential with $6/(\beta a^{2})$
\begin{equation}
V_{\mathrm{Coul}}(q) = \frac{6}{\beta} a^{2} V^{L}_{\mathrm{Coul}}(k,\beta) \; ,
\end{equation}
where $a$ denotes the lattice spacing in $\mbox{GeV}^{-1}$.
Again, we use the interpolation formula in \cite{Necco:2001xg}
to set $a$ assuming $r_0=0.5\ \text{fm}$. For all figures in the present
paper, the physical scale of the effective Coulomb potential is fixed in
this way.
\section{Studying systematic effects}
\label{sec:systematics}
\begin{figure*}
\includegraphics[width=1.0\textwidth]{cp_finitevolume}
\caption{(Color online) The Coulomb potential multiplied by $q^4$
shown as a function of $q^2$ in physical units. We show data for
different lattice sizes at $\beta=5.8$ (left) and $\beta=6.0$ (right)
to illustrate finite-volume effects. These seem to be under control
for data on lattices larger than $16^4$ because those fall roughly
on the same curve. For both $\beta$ values we only used data from
first SA-OR copies (fc).}
\label{fig:cp_finitevolume}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.0\textwidth]{cp_errorinbeta}
\caption{(Color online) The Coulomb potential multiplied by $q^4$
shown versus $q^2$ measured on comparable physical volumes.
Data for $\beta=5.8$ and $6.0$ are shown on the left,
and for $\beta=6.0$ and $6.2$ on the right hand side.
Only data from first SA-OR copies is shown.}
\label{fig:cp_errorinbeta}
\end{figure*}
In this section we discuss the effects of finite lattice volumes
and lattice spacings as well as the influence of the Gribov ambiguity on the
effective Coulomb potential.
\subsection{Lattice artifacts}
As we are primarily interested
in the product $q^4~V_{\mathrm{Coul}}(q)$, we directly discuss this product
instead of the effective Coulomb potential itself.
Note that we investigate effects of finite lattice spacings and
volumes by considering Coulomb-potential data collected for first
SA-OR copies only.
Finite-volume effects are studied by varying the lattice sizes
from $12^4$ to $48^4$ but keeping $\beta$, and hence the lattice
spacing $a$, fixed. We find that only data obtained on the smaller
lattices, $12^4$ and $16^4$, at $\beta=6.0$ show visible finite-volume
effects at lower momenta. For larger lattices, namely $24^4$,
$32^4$ and $48^4$, effects seem to be mild
(see the magnified view in the right panel of \Fig{fig:cp_finitevolume}).
At $\beta=5.8$ (reaching even lower momenta) only the $12^4$ data
clearly deviate from the other data (see \Fig{fig:cp_finitevolume},
left panel).
Lattice-spacing effects are investigated by comparing
data from lattices of equal physical volume for different values of
$\beta$. Within our choice of $\beta$ values and lattice sizes $L^4$,
we can find only a few combinations of $\beta$ and $L$ where
this is approximately possible. For those we can compare data at
approximately equal physical momenta and disentangle by eye the effect
of varying $a$. As demonstrated in \Fig{fig:cp_errorinbeta} these
discretization effects are small and of the order of 10 to 15\% at
largest. The difference between data for $\beta=6.2$ and 6.0 is
smaller (right panel) than the difference between data for $\beta=6.0$
and 5.8 (left panel).
\begin{figure*}
\includegraphics[width=0.9\textwidth]{cpandfunc_convergence}
\caption{(Color online) The upper row shows the convergence of $V_{\mathrm{Coul}}$
with increasing number $N_{\mathrm{cp}}$ of copies obtained with either the
SA-OR (circles) or the OR algorithm (crosses). The Coulomb potential is
shown for the smallest (on-axis) lattice momentum available for each of
the three lattice sizes, i.e. $\vec{k}=([1,0,0])$ with square brackets
indicating that the average over all permutations is taken.
The lower row shows the corresponding convergence of the (average) deviation
of the gauge functional from its best value known for each configuration.
As a function of $N_{\mathrm{cp}} < N_{\mathrm{cp}}^{\mathrm{max}}$, the values of $V_{\mathrm{Coul}}$ and
$F$ are to be understood as those for the ``currently best'' gauge copy
among the $N_{\mathrm{cp}}$ copies inspected for each configuration after $N_{\mathrm{cp}}$
repetitions of gauge fixing.}
\label{fig:cpandfunc_convergence}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.6\linewidth]{cp_bcvsfc}
\caption{(Color online) The ratio of the Coulomb potential (as function of $q^2$)
if evaluated either on arbitrary (first) copies from simple OR or on best gauge
copies from repeated SA-OR. Note that the enhancement for the first copy
tremendously grows with $q^2 \to 0$ compared to what is known in the case of the
ghost propagator. In contrast to the upper panel of \Fig{fig:cpandfunc_convergence}
only momenta are included that are allowed by the cuts mentioned above.}
\label{fig:cp_bcvsfc}
\end{figure*}
\subsection{Effects due to the Gribov ambiguity}
In comparison to lattice artifacts, the Gribov ambiguity turns out to
have a much larger impact on the Coulomb potential data. In order to
assess the influence of this ambiguity we follow the ``first copy --
best copy'' (fc-bc) strategy applied before in Landau-gauge studies of
gluon and ghost propagators \cite{Bakeev:2003rr,Sternbeck:2005tk}.
Here, we use this strategy in two different ways.
First, we estimate the number $N_{\mathrm{cp}}$ of gauge-fixed copies per configuration
necessary to achieve ``quasi convergence'' of the Coulomb potential, considered
as a function of $N_{\mathrm{cp}}$. Second, we quantify the systematic error of the
Coulomb potential that is admitted if {\it an arbitrary} (first) gauge-fixed copy
is chosen instead of {\it the best} copy among $N_{\mathrm{cp}}$ copies for each gauge
configuration. A copy is considered to be the best among all $N_{\mathrm{cp}}$ gauge-fixed
copies of a given configuration if its gauge functional is lower than
those of all the other $N_{\mathrm{cp}}-1$ copies after the gauge-fixing has been attempted
$N_{\mathrm{cp}}$-times.
Let us first compare the convergence of the bare data describing the Coulomb
potential upon increasing $N_{\mathrm{cp}} \to N_{\mathrm{cp}}^{\mathrm{max}}$ for the two gauge-fixing
algorithms OR and SA-OR. As an example, in \Fig{fig:cpandfunc_convergence} we
show for each lattice size data describing $V_{\mathrm{Coul}}$ for the lowest (on-axis)
lattice momentum available, i.e. $\vec{k}=([1,0,0])$ with square brackets
indicating that the average over all permutations is taken.
Note that in contrast to the rest of this paper we did not apply neither the
cylinder nor the cone cut here. The obtained deviations from the best-copy value
can be considered as an upper bound for all other momenta.
The data were obtained at $\beta=6.0$ on lattice sizes $L^4=12^4,16^4$ and $24^4$.
For each gauge-field configuration a number of $N_{\mathrm{cp}}^{\mathrm{max}}=20$, 30 and 40 independent
gauge-fixed copies was generated separately with the OR and the SA-OR algorithm.
From the figure we see that upon increasing $N_{\mathrm{cp}}$ the effective Coulomb potential
decreases and becomes (more or less) independent of $N_{\mathrm{cp}}$ for $N_{\mathrm{cp}}$ coming
closer to $N_{\mathrm{cp}}^{\mathrm{max}}$. What $N_{\mathrm{cp}}$ is sufficient to achieve ``quasi convergence''
depends, of course, on the gauge-fixing algorithm and on the lattice size. In fact,
it is clearly visible in \Fig{fig:cpandfunc_convergence} that the number of
gauge copies necessary to achieve convergence is substantially lower
for the SA-OR algorithm than for the OR algorithm. With both algorithms one needs
to consider more gauge copies with growing $L$.
For example, if we use the SA-OR algorithm, a number of copies $N_{\mathrm{cp}}=5$
on a $12^4$ lattice and $N_{\mathrm{cp}}=15$ on a $24^4$ lattice is sufficient. In contrast,
if we were using the OR algorithm, $N_{\mathrm{cp}}=40$ or more copies are necessary for a
lattice like $24^4$. Note that the observed increase with $L$ at fixed $\beta$ is
partly due to the smaller physical value associated with the lowest on-axis lattice
momentum that needs to be considered with increasing $L$.
On the larger lattices $32^4$ and $48^4$ we could not afford to gauge-fix more
than a single gauge copy per configuration with the SA-OR algorithm. This was
simply due to a drastic increase of the necessary number of iterations, but also
due to a growing number of ``trouble-making'' time-slices encountered on those
larger lattices. Thus, we did not apply the OR algorithm for the purpose of
comparison, and therefore we are not in the position to assess the influence of Gribov
copies on these lattices at the present stage.
As mentioned above, we also used the fc-bc strategy to estimate the impact of
Gribov-copy effects on the Coulomb potential data at different physical momenta.
For this purpose we gauge-fixed our field configurations at $\beta=5.8$, $6.0$ and $6.2$
to Coulomb gauge only once with the OR algorithm on one hand and $N_{\mathrm{cp}}$-times
with the SA-OR algorithm on the other. In order to obtain the results shown in
\Fig{fig:cp_bcvsfc} we have chosen, refering to \Fig{fig:cpandfunc_convergence},
$N_{\mathrm{cp}}=10$, 15 and 20 as sufficient numbers of gauge-fixed copies per configuration
for the lattice sizes $12^4$, $16^4$ and $24^4$, respectively. Then the Coulomb
potential was measured separately on the set of single copies obtained with the
OR algorithm on one hand and on the set of best copies obtained with the repeated
SA-OR algorithm on the other. For brevity we refer below to these two sets as the
first OR and the best SA-OR copies.
The ratio of the effective Coulomb potential measured for first OR copies and
best SA-OR copies is depicted in \Fig{fig:cp_bcvsfc} as a function of momentum
squared. The Gribov ambiguity has a dramatic impact on the effective Coulomb
potential at $q^2 < 10~\text{GeV}^2$. Even for the rather
small lattices considered here, and hence for rather high physical
momenta, the measurement of the effective Coulomb potential on first OR-copies
gives results larger by up to 100\% than the results on best SA-OR copies.
Note that this effect is much stronger than what has been observed for the ghost
propagator using the same method \cite{Voigt:2007wd,BerlinOsaka:2008}.
There an enhancement of about 5 to 10\% was typical for the presently
accessible lowest momenta. Note also that here we are comparing the
standard with one of the best presently known methods of gauge fixing.
In order to assess next the difference between $V_{\mathrm{Coul}}$ obtained from an arbitrary,
first SA-OR copy (fc) and the best among a sufficient number of SA-OR copies (bc),
restricted, however, to the smallest lattice momentum for each lattice size, i.e.
${\vec k} = ([1,0,0])$, we have to look back to \Fig{fig:cpandfunc_convergence}.
Considering the ratio $R$ of the effective Coulomb potential measured either for
first or best SA-OR copies as a function of the lattice size $L$, we find that this
is well described by $R \approx c - d/L$. If such an ansatz was used to extrapolate
the ratio $R$ at $\beta=6.0$ to $L=48$, a ratio $R=1.6$ would be obtained. For
the first OR copy an overestimation factor $R=2.7$ would be expected.
Both are reasonable upper bounds for the overestimation of the Coulomb potential
at any fixed physical momentum for first SA-OR copies and - even worse - first
OR copies. This estimate will be needed in the next Sect.~\ref{sec:infrared}.
We conclude that the effective Coulomb potential is less affected by Gribov-copy
effects if we use the SA-OR algorithm instead of the OR algorithm.
This conclusion rests on the observation that the ``quasi convergence''
of the Coulomb potential is faster for the SA-OR than for the OR algorithm.
Second, the results obtained on arbitrary, first SA-OR copies (fc) are less affected
by the Gribov ambiguity than those obtained on arbitrary, first OR-copies.
Therefore, we have used the SA-OR algorithm as our method of choice for the results
to be presented in the following. Recall, however, that if only first SA-OR copies
are available for analysis, measurements of the Coulomb potential in the infrared
region will be accompanied with an increased uncertainty. For instance, an overestimation
of about $60 \%$ for the smallest lattice momentum on a $48^4$ lattice at $\beta=6.0$
must be expected.
\section{Infrared behavior}
\label{sec:infrared}
Despite the Gribov ambiguity being that large, we now try to summarize what we know
about the momentum dependence of the effective Coulomb potential, globally and in
particular in the low-momentum region. As mentioned above, we were not in the position
to generate more than a single SA-OR gauge copy per configuration on the larger
lattices $32^4$ and $48^4$.
Therefore we present here a full set of data concerning the Coulomb potential for
a single SA-OR copy (fc) per configuration for all $\beta$ values and lattices sizes,
ensuring in this way an equal treatment of Gribov copy effects on both small and large
lattices. As is well known, this choice is equivalent to an averaging over all local
minima of all configurations, i.e. all over the {\it Gribov Region}. Best-copy data,
that we have available only up to lattices $24^4$ (after inspecting a sufficient number
$N_{\mathrm{cp}}$ of copies) would come close to a prescription that requires an average over
only the absolute minimum per configuration, i.e. restricted to the {\it Fundamental
Modular Region}. Zwanziger has argued that these averages should approach each other
in the limit of large volumes.
As long as they did not converge, we are admitting a strong systematic effect when
we restrict the analysis to the first SA-OR copy.
This can be clearly seen in our data from smaller lattices, $12^4$, $16^4$ and $24^4$.
In order not to overload the \Fig{fig:cp_runningcoupling} we show here
(and lateron in \Fig{fig:cp_stringtension}) only {\it selected}
results from SA-OR best-copies. The data are from the $24^4$ lattice where we had
the choice between 40 copies at $\beta=6.0$ and between 20 copies at $\beta=5.8$.
We try here (and later for the Coulomb potential) our best to estimate the systematic
error emerging from the ignorance of further Gribov copies at larger lattices.
A thorough study of the Gribov ambiguity for larger lattices remains highly desirable.
In the infrared momentum region, the running coupling given through the effective
Coulomb potential diverges stronger than $1/q^2$. This is shown in the left panel of
\Fig{fig:cp_runningcoupling}. The very fact of the infrared enhancement will not need to
be revised if once a systematic account for the Gribov effect would be undertaken,
although the divergence would be less pronounced. A rough indication of the size of
the effect is given by the filled symbols in that figure. These are best-copy results
for the lattice $24^4$.
In Ref.~\cite{Zwanziger:2003de}, Zwanziger presented an analytic calculation of the
Coulomb potential. By only considering the disconnected part of the expectation value
of the effective Coulomb potential (cf.~\Eq{eq:cp_factorisation}) Zwanziger predicted
an almost linearly rising effective Coulomb potential in the infrared limit.
Using our data for the ghost propagator~\cite{Voigt:2007wd} we are now in the position
to test the validity of his factorization hypothesis. If Zwanziger's assumption were valid,
the ratio
\begin{equation}
F_{\mathrm{Coul}}(q) = \frac{V^L_{\mathrm{Coul}}(q)}{(a~q~G^L(q))^{2}} \; .
\end{equation}
should be constant as function of the momenta. Note that $V^{L}$ and $G^L$ denote the bare
lattice Coulomb potential and the lattice ghost propagator taken at the
{\it physical} momentum $q$. The resulting plot shown in the right panel of
\Fig{fig:cp_runningcoupling} demonstrates that the assumption of factorization is valid
only for $q^2 > 10 \mbox{~GeV}^2$, but it is not correct in the momentum range
$q^2 \leq 10 \mbox{~GeV}^2$. This is in agreement with the results
of Langfeld and Moyaerts for $SU(2)$ pure gauge theory~\cite{Langfeld:2004qs}.
Much alike the enhancement of the running coupling, our conclusion that
the factorization hypothesis is violated would also not be invalidated if the effect of Gribov
copies was taken into account properly.
Similar to the left panel, the anticipated Gribov effect on the violation of factorization
is shown by the filled symbols in the right panel, which are the best-copy results for the
lattice $24^4$.
\begin{figure*}
\includegraphics[width=1.0\textwidth]{cp_runningcouplingfactorisation_withbc_v2}
\caption{(Color online) Left: The running coupling $g^2_{\mathrm{Coul}}(q) \propto q^2 V_{\mathrm{Coul}}(q)$
diverges in the infrared region and tends to zero in the asymptotic limit
$q^2 \rightarrow \infty$.
Right: The factorization of the effective Coulomb potential is violated for momenta
$0.04 \mbox{ GeV}^2 \leq q^2 \leq 10 \mbox{ GeV}^2$. For both the running coupling
and the test of the factorization hypothesis, the open symbols (including stars)
represent measurements on the first SA-OR copies per configuration. For comparison,
the filled triangles and filled squares show selected results for the best SA-OR
copy (bc) per configuration for two $\beta$ values on the largest lattice $24^4$
where the Gribov problem was fully under control.}
\label{fig:cp_runningcoupling}
\end{figure*}
We discuss now the momentum dependence of the effective Coulomb potential
in three clearly emerging momentum ranges, the {\it high-momentum range}, the
{\it intermediate momentum range} and the {\it low-momentum range}, and
describe the influence of the Gribov ambiguity in each range separatly.
The left panel of \Fig{fig:cp_stringtension} shows the presently known picture
concerning the momentum dependence of $q^4~V_{\mathrm{Coul}}(q)$. A logarithmic momentum
scale has been chosen in order to give a global view including the ultraviolet
and infrared behavior.
For the largest momenta in the {\it high-momentum range} $q^2 \geq 10 \mbox{~GeV}^2$,
the Coulomb potential shows roughly the expected $1/q^2$ behavior leading to an increase
of $q^4~V_{\mathrm{Coul}}(q)$ linear in $q^2$. From \Fig{fig:cp_bcvsfc} it is clear that the
high-momentum region is robust with respect to the Gribov ambiguity.
Although the inspection by eye suggests that we are seeing the tree-level form of the
Coulomb potential, we could not find reasonable fits of our data by the one-loop or the two-loop
expressions given in \Eq{eq:alpha_s_twoloop}.
We conclude that much higher momenta must be considered to get an estimate of the Coulomb
scale parameter $\Lambda_{\mathrm{Coul}}$ from such a fit.
With decreasing physical momenta, the first-copy data for $q^4~V_{\mathrm{Coul}}(q)$ reach
an almost flat region in the {\it intermediate momentum range}
$ 0.2 \mbox{~GeV}^2 \leq q^2 \leq 6 \mbox{~GeV}^2$, although a little bulge is visible
in the left panel of \Fig{fig:cp_stringtension}.
If the function $q^4~V_{\mathrm{Coul}}(q)$ stayed constant on the level of $\approx 20 \mbox{~GeV}^2$
in the limit $q^2 \to 0$, this would imply a perfect linearly confining potential
corresponding to an estimate of $\sigma_{\mathrm{Coul}} \approx (890 \mbox{~MeV})^2$.
This figure is more likely an upper bound.
Indeed, if for large spatial distances $r$ we assume the simple ansatz~\cite{Cucchieri:2002su}
\begin{equation}
V_{\mathrm{Coul}}(r) = - \sigma_{\mathrm{Coul}}\,r + C/r \; ,
\label{eq:cp_infraredpot}
\end{equation}
this suggests a momentum behavior
\begin{equation}
V_{\mathrm{Coul}}(q) = \frac{8\pi \sigma_{\mathrm{Coul}}}{q^{4}} + \frac{4\pi C}{q^{2}} \; ,
\label{eq:cp_infraredfit}
\end{equation}
with the intercept of $q^4~V_{\mathrm{Coul}}(q)$ at $q^2=0$ defining the Coulomb string tension $\sigma_{\mathrm{Coul}}$.
In the {\it intermediate momentum range} $ 0.2 \mbox{~GeV}^2 \leq q^2 \leq 6 \mbox{~GeV}^2$
the Gribov effect sets in and becomes apparently more severe with decreasing momentum.
For instance, for the smallest (on-axis) lattice momenta on lattices of sizes $12^4$,
$16^4$ and $24^4$ we have seen in \Fig{fig:cpandfunc_convergence} that the Coulomb potential
$V_{\mathrm{Coul}}(q)$ is overestimated by the first SA-OR copies compared with the best SA-OR copies
(among 40 copies). For the $24^4$ lattice at $\beta=6.0$ the overestimation amounts to
$\approx 40\%$. This can be extrapolated to the $48^4$ lattice where the effect amounts to
$\approx 60 \%$. This is an upper bound for the Gribov effect experienced by $V_{\mathrm{Coul}}(q)$
at {\it physical} momenta that are allowed by the cylinder and cone cuts.
In agreement with these estimates it can be seen in the left panel of
\Fig{fig:cp_stringtension} that in the {\it intermediate momentum range}
$ 0.2 \mbox{~GeV}^2 \leq q^2 \leq 6 \mbox{~GeV}^2$ the best-copy data from
the $24^4$ lattice (shown as filled symbols) provide us with another, independent
early indication of a plateau. The somewhat lower level of $\approx 10 \mbox{~GeV}^2$
would correspond to $\sigma_{\mathrm{Coul}} \approx (630 \mbox{~MeV})^2$. In view of this the
bulge must be understood as an artefact of insufficient gauge fixing.
With the simulation reported here, on our largest lattices the {\it low-momentum range}
with $q^2 < 0.2$ has become accessible for the first time.
Rather unexpectedly in this region the first-copy data for $q^4~V_{\mathrm{Coul}}(q)$ drop
with decreasing momentum as seen in the left panel. The right panel of
\Fig{fig:cp_stringtension} shows the infrared region magnified and in a linear scale
in $q^2$. This picture shows that a fit ansatz linear in $q^2$ describes the drop of
the first-copy data very well. We do not know whether a similar effect, namely the
onset of an apparently new infrared regime in the {\it low-momentum range} will happen
for the best-copy data as well. For the time being we assume that the bulge and the
new infrared regime is only a matter of measurements on insufficiently gauge-fixed
configurations. We have fitted the behavior according to \Eq{eq:cp_infraredfit}.
The Coulomb string tension is estimated as
\begin{equation}
\sigma_{\mathrm{Coul}} = (552 \pm 35 \mbox{~MeV})^2 \; .
\end{equation}
With some caution we may consider this as the {\it common limit} for $q^2 \to 0$
and the {\it common lower bound} for the Coulomb string tension (common to both
standards of gauge-fixing).
The other fit parameter, the ``Coulombic'' coefficient $C$ in front of the $1/r$ term
in \Eq{eq:cp_infraredpot}, is obtained as
\begin{equation}
C = 6.0 \pm 1.0 \; .
\end{equation}
This parameter has no relation to the ``Coulombic'' part $1/r$
in \Eq{eq:cp_infraredpot}. It rather describes the narrow momentum interval where the
single-copy data probably converge to the best-copy results for $q^4~V_{\mathrm{Coul}}(q)$.
The small number of data points is another reason why we give not much significance
to the fit. Still, details of the least $\chi^2$ fit of the first-copy data are
shown in \Tab{tab:infraredfit}.
\begin{table}
\caption{Results of $\chi^2$ fits to the single-copy data at momenta $q^2\le q_{\mathrm{max}}^2$.}
\label{tab:infraredfit}
\begin{tabular}{ l@{\quad} c@{\quad} c@{\quad} c@{\quad} r }
\hline\hline
$q_{\mathrm{max}}^2 \mbox{ [GeV}^2\mbox{]}$ & \# data & $\sqrt{\sigma_{\mathrm{Coul}}} \mbox{ [MeV]} $
& $C$ & $\chi^2/\mbox{ndf}$ \\
& points & & & \\ \hline
0.11 & 5 & $534(16)$ & 6.6(3) & 2.9 \\
0.16 & 6 & $526(18)$ & 6.8(4) & 1.8 \\
0.17 & 7 & $558(20)$ & 5.8(2) & 2.5 \\
0.18 & 8 & $587(28)$ & 4.9(2) & 3.8 \\ \hline\hline
\end{tabular}
\end{table}
We remark that the choice of the upper momentum cutoff for the fitting range, $q_{\mathrm{max}}$,
has only weak influence on the fit results.
In units of the Wilson string tension the fit result is
\begin{equation}
\sigma_{\mathrm{Coul}} = (1.6 \pm 0.2 ) ~\sigma_{Wilson} \; .
\end{equation}
This is the tentative lower bound for the Coulomb string tension.
Our estimate for the Coulomb string tension is in agreement with Zwanziger's inequality.
The relevance of this estimate is, however, faced with three sources of uncertainty.
\begin{itemize}
\item First, it relies only on first-copy data in a rather small number of data points,
and the obtained $\chi^2/\mbox{ndf}$ values are rather large (see \Tab{tab:infraredfit}).
The latter might be interpreted as a probable inadequacy of the assumed infrared ansatz
\Eq{eq:cp_infraredfit}. On the other hand, the right panel of \Fig{fig:cp_stringtension}
supports such a behavior.
\item Second, the weak but visible scaling violation of the effective Coulomb potential has the
effect that our estimate of the Coulomb string tension would be higher if we considered
higher inverse coupling constants $\beta$. The effective Coulomb potential in general
slightly increases with increasing $\beta$.
\item Third, the strong Gribov effect is neglected
in this estimate for the Coulomb string tension. If we had consequently looked for the best
SA-OR copies and had measured $q^4~V_{\mathrm{Coul}}(q)$ for these, the amount of overestimation
by the first-copy data in the bulge region is of the estimated order. One possibility
is that by the drop described by the fit given above the (yet unknown) level of the best-copy
results is reached. However, we cannot exclude the possibility that the best-copy data
in the {\it low-momentum range} will also enter a new infrared regime with a similar
decrease, such that the overestimation by the first-copy data remains. In this case
the final estimate of the Coulomb string tension would be close to the Wilson string
tension.
\end{itemize}
In the light of these uncertainties, we find it difficult to draw a conclusion on the
exact value of the Coulomb string tension.
Our value is larger than the values reported in previous $SU(2)$ investigations starting
also from the definition \Eq{eq:coulomb_potential}. These authors arrived at an estimate
close to the Wilson string tension~\cite{Cucchieri:2002su,Langfeld:2004qs}.
However, the Gribov copy problem for the effective Coulomb potential was ignored in these
studies. Furthermore, in Ref.~\cite{Cucchieri:2002su} the estimate of the Coulomb string
tension actually relies on data in the perturbative region, while the first plateau of
$q^4~V_{\mathrm{Coul}}(q)$ has been considered as a finite-volume effect. Such a plateau could
be observed in Ref.~\cite{Langfeld:2004qs} but the further decrease of $q^4~V_{\mathrm{Coul}}$
for even lower momenta was beyond the possibilities of this investigation. In contrast,
$SU(3)$ studies using incomplete (partial-length) Polyakov lines, made in order to
interpolate between the Coulomb string tension and the Wilson string tension, gave
$\sigma_{\mathrm{Coul}} = (2-3) \sigma_{Wilson}$ \cite{Nakamura:2005ux,Nakagawa:2006fk}.
These studies also have neglected the problem of Gribov copies that might have affected
the measured correlators.
\begin{figure*}
\includegraphics[width=1.0\linewidth]{cp_stringtension_withbc_v3}
\caption{(Color online) The effective Coulomb potential multiplied by $q^4$ as a
function of the physical momentum squared. Left: with a logarithmic momentum scale
in order to overlook both the IR and UV behavior. Right: the infrared momentum
region is shown in a linear scale in $q^2$ in order to judge the adequacy of the
linear fit of the first-copy data in the extremely IR region. The infrared fit
used to extract the corresponding Coulomb string tension is also shown in the
left panel. Open symbols (including stars) represent measurements on the first
SA-OR copies per configuration. For comparison, the filled triangles and filled
squares in both panels show results for the best SA-OR gauge copies (bc) for two
$\beta$ values on the lattice $24^4$, the largest lattice where the Gribov problem
was under scrutiny.}
\label{fig:cp_stringtension}
\end{figure*}
\section{Conclusions}
\label{sec:conclusion}
In this study we have attempted a thorough measurement of the effective
Coulomb potential in $SU(3)$ lattice gauge theory. We used a broad range of
lattice sizes, $12^4 - 48^4$, to perform Monte-Carlo simulations at
the three values $\beta=5.8$, $6.0$ and $6.2$. This has allowed us to show that
finite-volume effects are hardly visible on the larger lattices and
discretization effects are modest. Additionally, the use of
the fc-bc strategy has revealed a dramatic dependence of the Coulomb
potential on the choice of Gribov copy.
Unfortunately, by computer resources we were forced to restrict this ``Gribov analysis''
to the smaller lattices $12^4$, $16^4$ and $24^4$.
Thus, performing a full Gribov study up to the largest lattices still remains a highly
desirable goal. We note that the necessity of choosing best copies versus first (and
hence arbitrary) gauge-fixed copies is a matter of current debate (see also
\cite{Bogolubsky:2005wf,Bogolubsky:2007bw}). As another example, in a
BRST formulation, an average over all Gribov copies is taken which, on
the lattice, usually leads to the well-known Neuberger $0/0$ problem
\cite{Neuberger:1986vv, Neuberger:1986xz}. For a recent lattice BRST
formulation without this complication see \cite{vonSmekal:2007ns}.
What can be said here with confidence is that for the effective Coulomb
potential we find an extraordinarily strong Gribov-copy effect which has
never been observed before for other observables (say, the gluon and ghost
propagators).
We see a strong violation of the factorization hypothesis for the effective
Coulomb potential in momentum space below $q^2 \sim 5 \mbox{~GeV}^2$. For
smaller momenta the ``connected part'' of the corresponding expectation value
in \Eq{eq:cp_factorisation} is not negligible anymore.
This spoils any simple relation between the momentum dependence of the effective Coulomb
potential and the behavior of the ghost propagator.
Using only one SA-OR gauge copy per configuration and hence
allowing a strong systematic Gribov effect to be included in the bargain,
we found a new infrared regime of the effective Coulomb potential.
The first plateau of $q^4~V_{\mathrm{Coul}}(q)$, encountered with decreasing
momenta, turns out {\it not} to represent the asymptotic behavior,
because there is a further step-like decline for even smaller momenta.
The size of the step is of the same order as the extrapolated difference
between single-copy and many-copy SA-OR results. Therefore, we adopted the
point of view that the ``breakthrough'' to some new infrared regime at large
enough volume is a feature only of the single-copy data, and that some kind of
convergence (between averaging over the Gribov Region and the Fundamental Modular
Region) is behind this observation.
Future studies shall scrutinize whether the presumed common infrared limit
of these two schemes really exists or, alternatively, the Gribov ambiguity
persists at lower momentum for larger volumes. We estimated the Coulomb
string tension by fitting the data at the lowest momenta and found it
approximately 1.6 times larger than the Wilson string tension.
If the Gribov ambiguity persists, it is not excluded that in the -- further
delayed -- infrared limit finally $\sigma_{\mathrm{Coul}}=\sigma_{\mathrm{Wilson}}$ will be found.
\section{Acknowledgements}
This work is supported by the DFG under contract FOR 465
(Research Group {\it Lattice Hadron Phenomenology}), and by the
Australian Research Council. A major part of the simulations were
done on the IBM pSeries 690 at HLRN, Germany. We thank H.~St\"uben
for contributing parts of the code. We are grateful to Y.~Nagakwa,
A.~Nakamura, T.~Saito, L.~von Smekal and H.~Toki for inspiring discussions.
\bibliographystyle{apsrev}
|
1,108,101,565,485 | arxiv | \section{Introduction}
The Casimir effect which results in a force acting between two
parallel electrically neutral material plates separated with a gap
of width $a$ finds many prospective applications ranging from
fundamental physics to nanotechnology.\cite{1}
For real dissimilar material plates described by the dielectric
permittivities $\varepsilon^{(1,2)}(\omega)$ and magnetic
permeabilities $\mu^{(1,2)}(\omega)$ at temperature $T$
in thermal equilibrium the
generalized Lifshitz formula for the Casimir (van der Waals)
pressure takes the form\cite{2}\cdash\cite{5}
\begin{eqnarray}
P(a,T)&=&-\frac{k_BT}{\pi}
\sum_{l=0}^{\infty}{\vphantom{\sum}}^{\prime}
\int_{0}^{\infty}\!\! q_lk_{\bot}dk_{\bot}\left\{
\left[\frac{e^{2aq_l}}{r_{\rm TM}^{(1)}(i\xi_l,k_{\bot})
r_{\rm TM}^{(2)}(i\xi_l,k_{\bot})}-1\right]^{-1}\right.
\nonumber \\
&&\left.
+\left[\frac{e^{2aq_l}}{r_{\rm TE}^{(1)}(i\xi_l,k_{\bot})
r_{\rm TE}^{(2)}(i\xi_l,k_{\bot})}-1\right]^{-1}\right\}.
\label{eq1}
\end{eqnarray}
\noindent
Here, $k_B$ is the Boltzmann constant, prime adds a multiple
one half to the term with $l=0$, $k_{\bot}$ is the projection
of the wave vector on the plane of the plates, and
$\xi_l=2\pi k_BTl/\hbar$ with $l=0,\,1,\,2,\,\ldots$ are
the Matsubara frequencies. The reflection coefficients for
the transverse magnetic and transverse electric polarizatons
of the electromagnetic field are given by
\begin{equation}
r_{\rm TM}^{(n)}(i\xi_l,k_{\bot})=
\frac{\varepsilon_l^{(n)}q_l-k_l^{(n)}}{\varepsilon_l^{(n)}q_l
+k_l^{(n)}}, \qquad
r_{\rm TM}^{(n)}(i\xi_l,k_{\bot})=
\frac{\mu_l^{(n)}q_l-k_l^{(n)}}{\mu_l^{(n)}q_l+k_l^{(n)}},
\label{eq2}
\end{equation}
\noindent
where $\varepsilon_l^{(n)}\equiv\varepsilon^{(n)}(i\xi_l)$,
$\mu_l^{(n)}\equiv\mu^{(n)}(i\xi_l)$, the index $n=1,\,2$
numerates the plates and
\begin{equation}
q_l^2=k_{\bot}^2+\frac{\xi_l^2}{c^2}, \qquad
{k_l^{(n)}}^2=k_{\bot}^2+
\varepsilon_l^{(n)}\mu_l^{(n)}\frac{\xi_l^2}{c^2}.
\label{eq3}
\end{equation}
As was noticed long ago,\cite{6} ``In the majority of cases,
the contribution to the van der Waals interaction due to the
magnetic properties of real materials is extremely small.''
A large contribution (including the Casimir repulsion through
a vacuum gap for some range of parameters) was found\cite{7}
using the approximation of frequency-independent
$\varepsilon$ and $\mu$. Later, however, it was shown\cite{8}
that for real materials $\mu$ is nearly equal to unity in the
range of frequencies which gives a major contribution to the
Casimir pressure. This problem was reconsidered\cite{4}
at both zero and nonzero temperature for one metallic and
one magnetodielectric plate using the description of a metal by
means of the Drude model and of magnetodielectric by a
simplified model of the Drude-Lorentz type. At $T\neq 0$ the
Casimir force was found to be always attractive.
In this paper we investigate the thermal Casimir pressure
between plates made of ferromagnetic metal, ferromagnetic
dielectric and nonmagnetic metal taking into account
realistic dependences of $\varepsilon$ and $\mu$ on the
frequency and using different approaches to the theory of
the thermal Casimir force suggested in the literature.
We demonstrate how the use of different approaches
influences the Casimir pressure and find when the
Casimir repulsion through a vacuum gap is feasible.
In Sec.~2 we provide a brief review of magnetic properties.
Sec.~3 deals with ferromagnetic metals and Sec.~4 with
ferromagnetic dielectrics. In Sec.~5 we consider the
behavior of the Casimir pressure in the vicinity of Curie
temperature. Sec.~6 contains our conclusions.
\section{Review of magnetic properties}
The magnetic permeability along the imaginary frequency axis
is represented in the form
\begin{equation}
\mu(i\xi)=1+4\pi\chi(i\xi),
\label{eq4}
\end{equation}
\noindent
where $\chi(i\xi)$ is the magnetic susceptibility.
The magnitude of $\chi(i\xi)$ decreases monotonously when
$\xi$ increases. All materials possess diamagnetic polarization
for which\cite{9}\cdash\cite{11} $\chi(0)<0$, $\mu(0)<1$
and $|\mu(0)-1|\sim 10^{-5}$. Diamagnets (such materials as,
for instance, Au, Si, Cu and Ag) do not possess any other
type of magnetic polarization. For them one can put
$\mu_l=1$, $l=0,\,1,\,2,\,\ldots$ in computations using
(\ref{eq1}) so that magnetic properties of diamagnets do not
influence the Casimir force.
Some materials also possess paramagnetic polarization (in a
broad sense) which is larger in magnitude than the diamagnetic
one and leads to\cite{11}$\chi(0)>0$, $\mu(0)>1$.
Paramagnets (in a narrow sense) are materials with $\mu(0)>1$
if the interaction of magnetic moments of their constituent
particles is neglibibly small. Paramagnets may consist of
microparticles which are paramagnetic magnetizable but have no
intrinsic magnetic moment (the Van Vleck polarization
paramagnetism\cite{12}) and of microparticles possessing
a permanent magnetic moment (the orientational
paramagnetism\cite{9}\cdash\cite{12}). For all paramagnets in a
narrow sense it is true that
$\chi(0)<10^{-4}$ and one can put $\mu_l=1$ for all $l$.
This conclusion is unchanged for all paramagnets in a broad sense
(with the single exception of ferromagnets) because $\chi(0)$
remains as small as mentioned above and takes only a
slightly larger
values in the vicinity of $T=0$
even at temperatures below
the critical temperature $T_{\rm cr}$ of the magnetic phase
transitions \cite{9}\cdash\cite{11,13}\cdash\cite{15}
(for different materials $T_{\rm cr}$ varies from a few K to more
than thousand K).
For the subdivision of paramagnetic materials called ferromagnets
it is true that
$\mu(0)\gg 1$ at $T<T_{\rm cr}$ (in this case $T_{\rm cr}$
is referred to as the Curie temperature, $T_{\rm cr}\equiv T_C$).
There is a lot of ferromagnetic materials, both metals and
dielectrics.\cite{16} The rate of decrease of $\mu(i\xi)$ for
ferromagnets depends on their electric resistance. Thus, for
ferromagnetic metals and dielectrics $\mu(i\xi)$ becomes
approximately equal to unity for $\xi$ above $10^4$
and $10^9\,$Hz, respectively. Keeping in mind that the first
Matsubara frequency $\xi_1\sim 10^{14}\,$Hz at $T=300\,$K
we arrive at the conclusion that ferromagnets can affect the
Casimir force between macroscopic bodies only through the
contribution of the zero-frequency term of the Lifshitz formula
(\ref{eq1}). In all terms of this formula with $l\geq 1$ one can
put $\mu_l=1$. Note that below we do not consider so-called
hard ferromagnetic materials possessing a spontaneous
magnetization because the magnetic interaction between
the plates made of such materials far exceeds any conceivable
Casimir force. The subject of our interset is the soft
ferromagnetic materials which do not possess a spontaneous
magnetization. It is well known that the magnetic permeability
of ferromagnets depends on the applied magnetic
field.\cite{9}\cdash\cite{11} Since in the Casimir interaction
the mean applied field is equal to zero, below we consider
the so-called {\it initial} (zero field) permeability, i.e.,
$\mu=\mu(\mbox{\boldmath$H$}=0)$.
\section{Ferromagnetic metals}
First we consider the case when both Casimir plates are made
of common ferromagnetic metal Co, Cd, Fe or Ni.
The dielectric properties of a metal are described by the
Drude\cite{17,18} or the plasma\cite{19,20} model approaches,
i.e., using the dielectric functions of the form
\begin{equation}
\varepsilon_D(i\xi)=1+\frac{\omega_p^2}{\xi(\xi+\gamma)},
\qquad
\varepsilon_D(i\xi)=1+\frac{\omega_p^2}{\xi^2},
\label{eq5}
\end{equation}
\noindent
where $\omega_p$ is the plasma frequency, $\gamma$ is the
relaxation parameter. By considering different models
proposed in the literature we aim to determine whether or not
the magnetic properties influence the magnitude of the
Casimir pressure and is it possible to experimentally
distinguish between alternative theoretical predictions.
For all $l\geq 1$ we put $\mu_l=1$.
For two similar plates Eq.~(\ref{eq2}) leads to the following
reflection coefficients at $\xi=0$ if the Drude and plasma
models are used
\begin{eqnarray}
&&
r_{{\rm TM},D}(0,k_{\bot})=r_{{\rm TM},p}(0,k_{\bot})=1,
\quad
r_{{\rm TE},D}(0,k_{\bot})=
\frac{\mu(0)-1}{\mu(0)+1}\equiv r_{\mu},
\nonumber \\
&&
r_{{\rm TE},p}(0,k_{\bot})=\frac{\mu(0)ck_{\bot}-
[c^2k_{\bot}^2+\mu(0)\omega_p^2]^{1/2}}{\mu(0)ck_{\bot}+
[c^2k_{\bot}^2+\mu(0)\omega_p^2]^{1/2}}.
\label{eq6}
\end{eqnarray}
\noindent
In the limiting case of large separations (high $T$) only the
zero-frequency term in Eq.~(\ref{eq1}) contributes
to the Casimir pressure and all calculations can be performed
analytically. When the Drude model is used, the result is
\begin{equation}
P_{D}(a,T)=-\frac{k_BT}{8\pi a^3}\left[\zeta(3)+
\mbox{Li}_3(r_{\mu}^2)\right],
\label{eq7}
\end{equation}
\noindent
where $\zeta(z)$ is the Riemann zeta function and
$\mbox{Li}_n(z)$ is polylogarithm function.
Under the conditions $\mu(0)\gg 1$ (valid for ferromagnetic
metals) and $\mu(0)=1$ (valid for nonmagnetic
metals) Eq.~(\ref{eq7}) leads to
\begin{equation}
P_{D,{\rm fm}}(a,T)=-\frac{k_BT}{4\pi a^3}\zeta(3),
\qquad
P_{D,{\rm nm}}(a,T)=-\frac{k_BT}{8\pi a^3}\zeta(3),
\label{eq8}
\end{equation}
\noindent
respectively. As can be seen from Eq.~(\ref{eq8}), if the Drude
model is used, the account of magnetic properties of ferromagnetic
metals doubles the magnitude of the Casimir pressure at large
separations. If, however, the plasma model is used at large
separations under the condition
$\sqrt{\mu(0)}\delta_0/a\ll 1$, Eq.~(\ref{eq1}) results in
\begin{equation}
P_{p,{\rm fm}}(a,T)=-\frac{k_BT}{4\pi a^3}\zeta(3)
\left[1-3\sqrt{\mu(0)}\frac{\delta_0}{a}\right],
\label{eq9}
\end{equation}
\noindent
where $\delta_0=c/\omega_p$ is the skin depth.
The same expression, but with $\mu(0)=1$, is obtained for
nonmagnetic metals described by the plasma model.
At $T=300\,$K Eqs.(\ref{eq7})--(\ref{eq9}) are
applicable for $a>6\,\mu$m.
\begin{figure}[b]
\vspace*{-13.7cm}
\hspace*{-5.1cm}
\psfig{file=figGK-1.ps,width=8.5in}
\vspace*{-13.5cm}
\caption{The relative Casimir pressure as a function of separation
in the configuration of two parallel Co plates with
inclusion of
magnetic properties (the solid lines) and with magnetic
properties neglected (the dashed lines). Computations are
performed with the dielectric permittivity (a) of the Drude model and
(b) of the plasma model.}
\label{aba:fig1}
\end{figure}
{}From an experimental point of view the most interesting region
is from $a=0.5\,\mu$m to $a=1\,\mu$m (for $a<0.5\,\mu$m the
contribution of the zero-frequency term and, thus, of magnetic
properties is not large enough).
We have performed numerical computations of the Casimir pressure,
Eq.~(\ref{eq1}), in the region from 0.5 to $6\,\mu$m for Co
with parameters\cite{21,22}
$\omega_{p,{\rm Co}}=3.97\,$eV, $\gamma_{\rm Co}=0.036\,$eV
and $\mu_{\rm Co}(0)=70$. In Fig.~1 we plot the ratio of the
Casimir pressure $P$ between two Co plates at $T=300\,$K
to $P_0=-\pi^2\hbar c/(240a^4)$ computed using (a) the Drude
model and (b) the plasma model. The solid lines take into account
the magnetic properties and the dashed lines are computed with
magnetic properties neglected. Note that the solid line
in Fig.~1(a) is almost coincident with the dashed line in
Fig.~1(b). At small separations $a<1\,\mu$m, the difference
between the dashed line in Fig.~1(a) and the solid line
in Fig.~1(b) is also not observable in the limits of the
experimental precision. Thus the experiments on an indirect
measurement of the Casimir pressure by means of a micromechanical
ocsillator\cite{23} at separations of about 500--600\,nm
can allow us to choose one of the following situations.
1. The experimental data are in favour of the solid line
in Fig.~1(a) and the dashed line in Fig.~1(b). This means
that either the magnetic properties affect the Casimir
pressure and metals should be described by the Drude model
or the magnetic properties do not affect the Casimir
pressure and metals should be described by the plasma model.
2. The experimental data are in favour of the dashed line
in Fig.~1(a) and the solid line in Fig.~1(b). In this
case either the magnetic properties affect the Casimir
pressure and metals should be described by the plasma model
or the magnetic properties do not affect the Casimir
pressure and metals should be described by the Drude model.
Now let one plate be made of a ferromagnetic metal ($n=1$)
and the other of a nonmagnetic metal ($n=2$). Here, in the
limit of large separations one obtains
\begin{equation}
P_D(a,T)=-\frac{k_BT}{8\pi a^3}\zeta(3)
\label{eq10}
\end{equation}
\noindent
if the Drude model is used. For the plasma model under
conditions
$\sqrt{\mu(0)}\delta_{01}/a\ll 1$ and
$\delta_{02}/a\ll 1$ it follows
\begin{equation}
P_{p}(a,T)=-\frac{k_BT}{4\pi a^3}\zeta(3)
\left[1-
\frac{3(\sqrt{\mu(0)}\delta_{01}+\delta_{02})}{2a}\right].
\label{eq11}
\end{equation}
\noindent
Note than when the Drude model is used
$r_{\rm TE}^{(2)}(0,k_{\bot})=0$ and, thus,
the magnetic properties of a ferromagnetic plate entering only
through $r_{\rm TE}^{(1)}(0,k_{\bot})$ do not influence
the result.
The results of the numerical computations for the Co plate interacting
with the Au plate
($\omega_{p,{\rm Au}}=9.0\,$eV, $\gamma_{{\rm Au}}=0.035\,$eV)
in the case when the plasma model is used are shown in Fig.~2(a).
\begin{figure}[t]
\vspace*{-13.7cm}
\hspace*{-5.1cm}
\psfig{file=figGK-2.ps,width=8.5in}
\vspace*{-13.5cm}
\caption{The relative Casimir pressure as a function of separation
in the configuration of two parallel plates, one made of Au and the other
of (a) Co and (b) ferromagnetic dielectric with inclusion of
magnetic properties (the solid lines) and with magnetic
properties neglected (the dashed lines). Computations are
performed using the plasma model for the dielectric permittivity
of metals.}
\label{aba:fig2}
\end{figure}
It is seen that here the inclusion of the magnetic properties
(the solid line) decreases the magnitude of the Casimir
pressure. The influence of the magnetic properties is, however,
very moderate and can be observed only in the experiment
on measuring the difference Casimir pressure above a
patterned plate\cite{24} one section of which is made of
Co and the other of Au. Such an experiment allows one to
choose between the two alternatives in each of the situations
described above. This will provide a complete
experimental answer to questions whether the magnetic
properties influence the Casimir force and what
dielectric model should be used in the Lifshitz theory
to describe real metals.
\section{Ferromagnetic dielectrics}
Ferromagnetic dielectrics are materials that, while
displaying physical properties characteristic of dielectrics,
show ferromagnetic behavior under the influence of an external
magnetic field.\cite{25} Such materials are widely used in
different magneto-optical devices. As an example, we consider
a composite material of polystyrene with a
volume fraction $f$ of ferromagnetic metal nanoparticles in
the mixture. The permittivity of such a material can be
presented in the form\cite{26}
\begin{equation}
\varepsilon_{\rm fd}(i\xi)=\varepsilon_{\rm d}(i\xi)
\left(1+\frac{3f}{1-f}\right),
\label{eq12}
\end{equation}
\noindent
where $\varepsilon_{\rm d}$ is the permittivity of
polystyrene.\cite{27}
We have performed computations of the Casimir pressure for
two parallel plates one of which is made of ferromagnetic
dielectric [$f=0.25$, $\varepsilon_{\rm fd}(0)=5.12$,
$\mu(0)=25$] and the other of Au described by the plasma model.
Recall that if Au is described by the Drude model the
magnetic properties do not influence the Casimir pressure
as explained in Sec.~3. The computational results for
$P/P_0$ as a function of $a$ are presented in Fig.~2(b)
where the solid line takes the magnetic properties into
account and the dashed line neglects them.
As can be seen in Fig.~2(b), magnetic properties
have an important influence
on the Casimir pressure and even lead to the
change of sign of the force (from attraction to repulsion).
This important conclusion can be confirmed analytically in the
limiting case of large $a$. If the metallic properties of Au
plate are described by the Drude model, one obtains
\begin{equation}
P_D(a,T)=-\frac{k_BT}{8\pi a^3}\mbox{Li}_3(r_{\varepsilon}),
\qquad
r_{\varepsilon}\equiv
\frac{\varepsilon_{\rm fd}-1}{\varepsilon_{\rm fd}+1}.
\label{eq13}
\end{equation}
\noindent
This does not depend on the magnetic properties. If, however,
the plasma model is used, then, under the condition
$\delta_{02}/a\ll 1$, one arrives at
\begin{equation}
P_p(a,T)=-\frac{k_BT}{8\pi a^3}\left[\mbox{Li}_3(r_{\varepsilon})
+\mbox{Li}_3(-r_{\mu})\left(1-
3\frac{\delta_{02}}{a}\right)\right].
\label{eq14}
\end{equation}
\noindent
The expression on the right-hand side of Eq.~(\ref{eq14}) is
positive and the respective Casimir force is repulsive if the
following condition is satisfied:
\begin{equation}
\mbox{Li}_3(r_{\varepsilon})<
\left|\mbox{Li}_3(-r_{\mu})\left(1-
3\frac{\delta_{02}}{a}\right)\right|.
\label{eq15}
\end{equation}
\noindent
This condition is easily satisfied for real materials.
\section{Vicinity of the Curie temperature}
At the Curie temperature $T_C$ specific for each material,
ferromagnets undergo a magnetic phase transition.\cite{14,16}
At higher temperature they lose ferromagnetic properties and
become paramagnets in the narrow sense. Thus, for Fe, Co, Ni
and Gd the Curie temperature is equal to 1043\,K, 1388\,K, 627\,K
and 293\,K, respectively.\cite{28} Here, we consider the
behavior of the Casimir pressure under the magnetic phase
thansition which occurs with the increase of $T$ in the
configuration of two similar plates made of Gd. The Drude
parameters of Gd are equal to\cite{29}
$\omega_{p,{\rm Gd}}=9.1\,$eV, $\gamma_{{\rm Gd}}=0.58\,$eV.
Computations of the Casimir pressure between two parallel plates
made of Gd in the vicinity of the Curie temperature require respective
values of $\mu(0)$ for Gd at $T<T_C$ [at $T>T_C$,
$\mu_{\rm Gd}(0)=1$ to high accuracy].
In Fig.~3(a) the magnetic permeability of Gd is shown as a
function of temperature in the region from 280\,K to
300\,K on the basis of the experimental data.\cite{30}
\begin{figure}[t]
\vspace*{-13.7cm}
\hspace*{-5.1cm}
\psfig{file=figGK-3.ps,width=8.5in}
\vspace*{-13.5cm}
\caption{(a) The static magnetic permeability of Gd
at the magnetic phase transition as a function of temperature.
(b)
The relative Casimir pressure as a function of temperature
in the configuration of two parallel Gd plates at the separation
$a=0.5\,\mu$m. The solid and dashed lines include and
neglect the magnetic properties, respectively. The pairs
of lines marked 1 and 2 indicate the respective computational
results obtained using the Drude and the plasma models.
}
\label{aba:fig3}
\end{figure}
The Casimir pressure as a function of temperature was computed
at the separation $a=500\,$nm between the plates using
Eq.~(\ref{eq1}). The computational results obtained using
the Drude and the plasma models are shown in Fig.~3(b) by
the pairs of
lines 1 and 2, respectively. In each pair the solid line
takes into account the magnetic properties and the dashed line is
computed with these properties disregarded.
As can be seen from Fig.~3{b},
experiments on the magnetic phase transition can also
be used to determine the influence of magnetic properties on the
Casimir force and as a test for different models of the dielectric
properties of metals.
\section{Conclusions}
The investigation of the influence of magnetic properties on the
Casimir force performed above leads
to the following conclusions.
\begin{enumerate}
\item[1.]
Of all the real materials, only ferromagnets might affect
the Casimir force.
\item[2.]
At all feasible temperatures the possible influence of
ferromagnets on the Casimir force occurs solely through
the contribution of the zero-frequency term in the
Lifshitz formula.
\item[3.]
In the framework of the Lifshitz theory the Casimir
repulsion of two macroscopic bodies separated by a vacuum
gap arises for only the case when one body is made of
ferromagnetic dielectric and the other is metallic. In doing
so the metal is described by the plasma model.
\item[4.]
Modern experimental techniques present good
opportunities to check
whether the magnetic properties of the plate material influence
the Casimir force. Experiments with magnetic bodies allow
independent test of the plasma and Drude model approaches
to the description of the dielectric properties of metals.
\end{enumerate}
\section*{Acknowledgments}
The authors are grateful to the
Deutsche Forschungsgemeinschaft
Grant No.~GE\,696/9--1
for partial financial support.
|
1,108,101,565,486 | arxiv | \section{Introduction}
The searches for supersymmetry (SUSY) are motivated by the
solutions of the most important problems: the hierarchy problem,
gauge coupling unification and dark matter problem \cite{1.}.
Experimental searches
for SUSY in the most probable channels for the superparticle
production at the LHC did not lead to the desired results and
set new lower limits in the mass range about 2 TeV for gluino
and squarks \cite{2.}. This fact led to the need for
SUSY searches in other sectors, for example, in the electroweak
sector. As highlighted in CERN Courier \cite{3.}:
"Based on data recorded in 2016, CMS has
covered models of electroweak production of "wino"-like
charginos and neutralinos with searches in different final states.
More results are expected soon, and the sensitivity of the searches
will largely profit from the extension of the data set in the
remaining two years of LHC Run 2". Another important sector
for SUSY searches in low mass range of 1 TeV are the searches
for extended Higgs boson sector predicted by Minimal Supersymmetric
Standard Model (MSSM) \cite{4.}, that consists of five Higgs bosons:
CP even Higgs bosons, h and H, CP odd Higgs boson, A, charged
Higgs bosons, H$^{\pm}$.
The purpose of our paper is to calculate the production
cross section of such particles at the energy of 14 TeV
at the LHC in the most optimal space of parameters of the MSSM model.
\section{ Optimal parameter space for studying of the properties
of MSSM Higgs bosons}
The masses of five Higgs bosons of MSSM model
at tree level are calculated through the
masses of gauge boson, M$_W$, M$_Z$,
and two additional parameters such as
the pseudoscalar mass, M$_A$ and the
ratio of vacuum expectation values of
two Higgs doublets, tan$\beta\equiv\upsilon_u/\upsilon_d$
\cite{5.}:
\[M^2_{H^{\pm}}=M^2_A+M^2_W\ ,\]
\[M^2_{h,H}=\frac{1}{2}\Biggl(M^2_A+M^2_Z\mp\sqrt{(M^2_A+M^2_Z)^2
-4M^2_AM^2_Z\mbox{cos}^22\beta}\Biggr)\ .\]
In the paper \cite{6.} the theoretical
predictions of the MSSM Higgs particles in
the low tan$\beta$ regime, $1 \leq$tan$\beta\leq 3$
are reviewed, with the assumption that SUSY
should be in the range of 1 TeV. It was
showed that the heavier MSSM neutral H/A
and charged H$^{\pm}$ states can decay
into gauge bosons, lighter Higgs bosons
and top quarks, presented in Fig.1
\begin{center}
{\includegraphics[width=0.85\textwidth]{1}}\\
\emph{{Fig.1.}} {\emph{The branching ratios as functions of
masses of MSSM Higgs bosons (A left, H center, H$^{\pm}$ right)
for tan$\beta$=2.5, from \cite{6.}.}}\\
\end{center}
In the Handbook of LHC Higgs cross sections,
2017 \cite{5.} are given examples of
sensitivity on the [tan$\beta$, M$_A$]
parameter space for the "model independent"
hMSSM approach \cite{6.}, compared
to the second approach \cite{7.}
so called "low-tb-high" approach in the MSSM,
that is orthogonal to the one previous.
Relative differences in
BR(H$\rightarrow$ WW) between the predictions of
the "low-tb-high" scenario and the
corresponding predictions obtained
with the hMSSM+HDECAY combination are presented in Fig.2.
\begin{center}
{\includegraphics[width=0.51\textwidth]{2}}\\
\emph{{Fig.2.}} {\emph{
Relative differences in BR(H$\rightarrow$WW) between the hMSSM+HDECAY
scenario and the "low-tb-high" scenario, from \cite{5.}.}}\\
\end{center}
The results of ATLAS \cite{8.}
and CMS \cite{9.} Collaborations
excluded at the 95$\%$ confidence level (CL) a
significant part of the [tan$\beta$, M$_A$] plane.
We'll use the benchmark scenarios of the model
independent approach for the Higgs sector,
the hMSSM with M$_h$ = 125 GeV for the
experimental limits on the cross sections
times branching ratios in the context
of the MSSM \cite{10.}.
The results for the branching fractions
received with the program HDECAY
\cite{11.} for the Higgs decays in the [tan$\beta$, M$_A$]
plane are displayed in Fig. 3 with red area
for the large decay rates and blue area for the small one.
\begin{center}
{\includegraphics[width=0.75\textwidth]{3}}\\
\emph{{Fig.3.}} {\emph{The branching ratios of the neutral
Higgs bosons in the [tan$\beta$; M$_A$] parameter
space of the hMSSM model, from \cite{12.}. }}\\
\end{center}
The production cross sections for
A and H bosons are displayed in Fig. 4
in the [tan$\beta$, M$_A$]
hMSSM parameter space for 14 TeV at the LHC
\begin{center}
{\includegraphics[width=0.85\textwidth]{4}}\\
\emph{{Fig.4.}} {\emph{The production cross sections
of the Higgs bosons A (left)
and H (right) at the LHC with $\sqrt{s}$=14 TeV in the
[tan$\beta$; M$_A$] hMSSM plane,
from \cite{12.}.}}\\
\end{center}
\section{Calculations of the production
cross sections times branching fractions for Higgs bosons}
1) CP-even Higgs boson, H\\
Searches for heavy Higgs bosons by Run-2 ATLAS Collaboration
at the LHC in the $H\rightarrow ZZ$ and $H\rightarrow WW$ decay channels
are relevant due to the possibility of evidence for
new particles beyond the Standard Model. The limits
on $\sigma(pp\rightarrow H)\times BR(H\rightarrow ZZ)$ and
$\sigma(pp\rightarrow H)\times BR(H\rightarrow WW)$ at 95$\%$ CL from \cite{13.}
and \cite{14.} correspondingly are presented in Fig. 5
\begin{center}
{\includegraphics[width=0.85\textwidth]{5}}\\
\emph{{Fig.5.}} {\emph{
Limits on $\sigma(pp\rightarrow H)\times BR(H\rightarrow ZZ)$ (a) and
$\sigma(pp\rightarrow H)\times BR(H\rightarrow WW)$ (b) via
gluon-gluon fusion at 95$\%$ CL.}}\\
\end{center}
Using the restricted parameter set for [tan$\beta$; M$_A$] plane,
presented in the previous section and computer programs SusHi
\cite{15.}
and SOFTSUSY4.0 \cite{16.},
we calculated $\sigma(pp\rightarrow H)\times BR(H\rightarrow ZZ)$ and
$\sigma(pp\rightarrow H)\times BR(H\rightarrow WW)$ for $\sqrt{s}$=14 TeV
at the LHC, presented in Fig. 6
\begin{center}
{\includegraphics[width=0.81\textwidth]{6}}\\
\emph{{Fig.6.}} {\emph{
$\sigma(pp\rightarrow H)\times BR(H\rightarrow ZZ)$ (left) and
$\sigma(pp\rightarrow H)\times BR(H\rightarrow WW)$ (right)
for $\sqrt{s}$=14 TeV at the LHC. }}\\
\end{center}
From Fig. 6 we can see the increase in value $\sigma\times Br$
for ggh fusion process compared with bbh fusion
process of heavy Higgs boson, H production.
Since the branching ratios
for the decays $H\rightarrow bb$ and $H\rightarrow tt$
are significant values according to our calculations with
SOFTSUSY4.0 program, we have performed
calculations of $\sigma(pp\rightarrow H)\times BR(H\rightarrow tt)$
and $\sigma(pp\rightarrow H)\times BR(H\rightarrow bb)$ for the planned
at the LHC energy of 14 TeV, presented in Fig. 7
\begin{center}
{\includegraphics[width=0.81\textwidth]{7}}\\
\emph{{Fig.7.}} {\emph{$\sigma(pp\rightarrow H)\times BR(H\rightarrow bb)$
(left) and $\sigma(pp\rightarrow H)\times BR(H\rightarrow tt)$ (right)
for $\sqrt{s}$=14 TeV at the LHC.}}\\
\end{center}
From the comparison of our calculations, presented
above, we can see significant predominance of
the values $\sigma\times Br$ for the second variant (Fig. 7)
compared to the first one (Fig. 6). It is
also important to stress the necessity of
N3LO calculations for essential enlargement
of the $\sigma\times Br$ value. \\
2) CP-odd Higgs boson, A\\
In this section we have considered the following
decay processes of A boson: $A\rightarrow bb$ and $A \rightarrow tt$.
The consideration of these processes of A boson
decay is connected with the large value of
branching ratio, that is represented in Fig.1.
As we have calculated the process $ A\rightarrow Zh$ in
\cite{17.} and currently
there are no other experimental data, for future
experimental searches it was
of interest to perform calculations for the two other decay
channels from the three maximal. Using the
computer programs SOFTSUSY4.0 and SusHi, we
have performed the calculations of $\sigma\times Br$ for CP-odd
Higgs boson, A. As the branching ratio for A
boson is maximal for the decays $A\rightarrow bb$ and
$A\rightarrow tt$ in the selected set of parameters,
it was interesting to calculate $\sigma\times Br$ for this
both processes over a wide range of boson
masses, from 500 GeV to 3450 GeV. The
results of our calculations are presented in Fig. 8
\begin{center}
{\includegraphics[width=0.81\textwidth]{8}}\\
\emph{{Fig.8.}} {\emph{
$\sigma(pp\rightarrow A)\times BR(A\rightarrow bb)$ in the mass
range 500-2200 GeV (left) and 1800-3450 GeV (right) (a)
and $\sigma(pp\rightarrow A)\times BR(A\rightarrow tt)$
in the mass
range 500-2200 GeV (left) and 1800-3450 GeV (right) (b).
}}\\
\end{center}
From Fig. 8 we can see the predominance of
the ggh process of A boson formation over
the bbh one except for the (b) case of
$A\rightarrow tt$ process in the mass range of 500-2200 GeV
with interesting intersection points between
bbh and ggh processes. It is also necessary
to stress the largest value of $\sigma\times Br$ for the
smallest masses, $m_A$, what is easily explained
in connection with the lower mass of the Higgs boson A. \\
3) charged Higgs bosons, H$\pm$\\
As is known \cite{18.},
the production of charged Higgs boson
depends on its mass and for m$_{H^{+}}$ $>$ m$_t$,
H$^+$ production mode is associated
with a top quark, as illustrated in Fig. 9
\begin{center}
{\includegraphics[width=0.31\textwidth]{9}}\\
\emph{{Fig.9.}} {\emph{
Leading-order Feynman diagram for the production of H$^+$
in association with a top quark in five flavor scheme.}}\\
\end{center}
In Fig. 10 are shown the expected and
observed limits for the production of
$H^+\rightarrow tb$ in association with a top quark,
bands for 68$\%$ (in green) and 95$\%$ (in yellow)
confidence intervals and the signal
prediction in the m$^{mod-}_h$ benchmark scenario
of the MSSM \cite{19.}.
\begin{center}
{\includegraphics[width=0.55\textwidth]{10}}\\
\emph{{Fig.10.}} {\emph{
Expected and observed limits for the production o
f H$^+\rightarrow tb$
in association with a top quark, from \cite{18.}}.}\\
\end{center}
As model points with $0.5\leq$tan$\beta\leq 0.6$, tan$\beta\approx 0.5$,
tan$\beta$=0.7 and tan$\beta$=0.9 are excluded in the
H$^+$ mass range of 200-600 GeV obtained also in other
scenarios of MSSM, it would be interesting to
do the calculations of $\sigma\times Br$ for tan$\beta$=2. For the
studying of properties of charged Higgs bosons,
H$^{\pm}$, we have used the set of parameters of MSSM
model to calculate the cross-sections of tH$^+$
production with the help of the software program
PROSPINO \cite{20.}
with data implemented from the
latest computer program SOFTSUSY4.0.
The corresponding results for
$\sigma(pp\rightarrow tH^+)BR(H^+\rightarrow tb)$, obtained
for the parameter set of tan$\beta$=2 and for the
energy of 14 TeV in the mass range of m$H^+$=500-1200 GeV
are presented in Fig.11
\begin{center}
{\includegraphics[width=0.55\textwidth]{11}}\\
\emph{{Fig.11.}} {\emph{
$\sigma(pp\rightarrow tH^+)BR(H^+\rightarrow tb)$
for 14 TeV at the LHC in the mass range of m$_{H^+}$=500-1200 GeV.}}\\
\end{center}
Another most visible decay channel of a
charged Higgs boson is $H^+\rightarrow\tau\nu$. Its searches in
association with a single top quark were
performed by ATLAS Collaboration at the
LHC with proton--proton collision at $\sqrt{s}$=13 TeV
corresponding to an integrated luminosity
of 3.2 fb$^{-1}$. The analysis of experimental
data leads to 95$\%$ CL upper limits on
the $\sigma(pp\rightarrow [b]tH^{\pm})BR(H^{\pm}\rightarrow\tau\nu)$,
between 1.9 pb and 15 fb, for m$_{H^+}$=200-2000 GeV,
that is presented in Fig. 12.
\begin{center}
{\includegraphics[width=0.61\textwidth]{12}}\\
\emph{{Fig.12.}} {\emph{
Observed and expected 95$\%$ CL exclusion limits for heavy
charged Higgs boson production as a function
of m$_{H^+}$, from \cite{21.}.}}\\
\end{center}
From these experimental data tan$\beta$ = 42--60
for m$_{H^+}$=200 GeV and tan$\beta$=60 for the H$^+$ mass
range from 200 to 340 GeV were excluded. So we
have considered two cases of tan$\beta$=2 and 30
for comparison of the value of $\sigma(pp\rightarrow [b]tH^{\pm})
BR(H^{\pm}\rightarrow\tau\nu)$
for these two cases, presented in Fig. 13.
\begin{center}
{\includegraphics[width=0.84\textwidth]{13}}\\
\emph{{Fig.13.}} {\emph{
$\sigma(pp\rightarrow [b]tH^{\pm})
BR(H^{\pm}\rightarrow\tau\nu)$ for (a) tan$\beta$=30
in the mass range m$_H^+$= 1200-2650 GeV (b) and
tan$\beta$=2 in the mass range m$_H^+$=2200-4600 GeV
with the planned 14 TeV at the LHC.}}\\
\end{center}
From Fig. 13 the predominance in the value of
$\sigma(pp\rightarrow [b]tH^{\pm})
BR(H^{\pm}\rightarrow\tau\nu)$ for the variant (a)
is obvious but we can see the larger values of
$\sigma(pp\rightarrow [b]tH^{\pm})
BR(H^{\pm}\rightarrow\tau\nu)$ for tan$\beta$=30 in
the range of the mass intersection of charged
Higgs boson, m$_H^+$=2200-2650 GeV for (a) and (b) variants.
In addition, it is known that for $m_{H^+}>m_t$ the dominant
decay of H$^+$ is $H^+\rightarrow tb$, but for large values of tan$\beta$
is observed a substantial contribution from $H^+\rightarrow\tau\nu$
\cite{21.}. For comparison
we calculated $\sigma(pp\rightarrow tH^+)BR(H^+\rightarrow tb)$ for
tan$\beta$=30 for 14 TeV at the LHC, presented in Fig.14
\begin{center}
{\includegraphics[width=0.61\textwidth]{14}}\\
\emph{{Fig.14.}} {\emph{
$\sigma(pp\rightarrow tH^+)BR(H^+\rightarrow tb)$ for 14 TeV at the LHC
in the mass range of m$_H^+$=1200-2650 GeV.}}\\
\end{center}
From the Fig. 14 and 13 (a) it can be concluded
about the largest values of $\sigma(pp\rightarrow tH^+)BR(H^+\rightarrow tb)$
in contrast
with $\sigma(pp\rightarrow [b]tH^{\pm})BR(H^{\pm}\rightarrow\tau\nu)$
for the same tan$\beta$=30, but the increase of the value
$\sigma(pp\rightarrow [b]tH^{\pm})BR(H^{\pm}\rightarrow\tau\nu)$ for the larger
tan$\beta$ was stressed above.
\section{Conclusion}
Using the restricted parameter set of the hMSSM model,
presented in \cite{5.} and \cite{12.} for
the extended sector of Higgs bosons
as well as the latest experimental data on the observed and
expected CL exclusion limits for Higgs boson production,
performed by ATLAS Collaboration \cite{13.},
\cite{14.}, \cite{18.}, \cite{21.} with the help of software programs
SOFTSUSY4.0, SusHi and PROSPINO we have calculated $\sigma\times Br$
for CP-even Higgs boson, H, CP-odd Higgs boson,
A and charged Higgs bosons, H$^{\pm}$. From our calculations
we can conclude about the large values of the $\sigma\times Br$ at
small tan$\beta$=2 for chosen decay channels of Higgs bosons for the
energy at the LHC of 14 TeV. But for
the charged Higgs boson are obtained another results,
that are connected with larger values of tan$\beta$.
|
1,108,101,565,487 | arxiv | \section*{Introduction}
\noindent In optically pumped magnetometers (OPMs), measurements of external magnetic fields rely on detection of energy-level shifts/spin precession arising from Zeeman interaction \cite{budker2013optical}. However, if nonmagnetic interactions similarly affects the energy levels, OPMs can also be used to detect those interactions. This enables the application of optical magnetometry to searches for anomalous spin-dependent interactions.
Currently, OPMs are used to search for physics beyond the Standard Model in a variety of experiments (see, Ref.~\cite{Safronova2018Search} and references therein). A particular example of such OPMs application is the search for microscopic-range spin-dependent interactions, indicating a possibility of existence of axion-like particles (ALPs), which are one of prime candidates for the dark matter \cite{Safronova2018Search}. OPMs are also used to search for transient nonmagnetic spin couplings, which could arise due to interaction with macroscopic objects made of ALPs, in particular, Q-balls \cite{Kimball2018Searching}, topological defects (e.g., domain walls) of ALP field \cite{Pospelov2013Detecting}, or ALP-field pulses generated in cataclysmic astrophysical events (e.g., black-hole mergers) \cite{Dailey2021Quantum}. These transient couplings are targeted by the Global Network of Optical Magnetometers for Exotic physics searches (GNOME) \cite{Pustelny2013Global,Afach2018Characterization, afach2021search}. Heretofore, the GNOME consists of various OPMs, originally developed for ultra-sensitive magnetometry in globally distributed locations. This leads to several challenges when searching the data for global transient signals. First, due to a different nuclear-spin content of atoms used in specific sensors, they are characterised with different sensitivity to exotic spin couplings\cite{Kimball_2015} (coupling to protons and neutrons could be, in general, different). Second, the implemented OPMs are characterised with different bandwidths, sensitivities, and local noise floors, which complicates data analysis\cite{MASIAROIG2020_analysis_method}. Third, the magnetometers were designed to maximise the sensitivity to magnetic fields. Uncontrolled and uncompensated magnetic-field perturbations are detrimental to the sensitivity to other couplings. These issues triggered work to upgrade conventional OPMs in the GNOME with a sensor less sensitive to magnetic fields but highly sensitive to nonmagnetic spin couplings.
A specific example of a sensor, being predominantly sensitive to nonmagnetic spin couplings, hence well suited for searches for physics beyond the Standard Model, is an alkali-metal-noble-gas co-magnetometer originally developed by Romalis and coworkers \cite{Romalis2002Dynamic, Kornack_Nuclear_spin_gyro,Ghosh_Romalis_SEOP_Ne} and later studied extensively by other researchers \cite{Rot_sens_Li,fang2016low, chen2016spin, Co-mag_magn_field_resp_Fan, Shi:20}. Such a system operates based on coupled evolution of the magnetizations of noble gas (NG) and alkali-metal (AM) vapour. Both can achieve a high percentage of polarization as AM can be optically pumped, whereas polarisation of the NG can be generated via spin-exchange collisions with the optically polarized AM \cite{theory_of_he_SEOP}. In the co-magnetometer, the vapour cell is heated above 150$^\circ$C to achieve a sufficiently high AM density, such that relaxation due to spin-exchange collisions, which is one of the main mechanisms of AM polarisation relaxation and hence one of a limiting factor to spin-coupling sensitivity, is suppressed. This is the so-called Spin-Exchange Relaxation Free (SERF) regime \cite{allred2002high}. Furthermore, for the alkali-metal-noble-gas co-magnetometer the effect of low-frequency magnetic drifts can be suppressed by application of a carefully chosen bias magnetic field. If the bias field is approximately equal to the sum of AM and NG magnetization fields, the system retains high sensitivity to both electron and nuclear nonmagnetic spin couplings, but becomes insensitive to low-frequency magnetic-field changes.
Such self-compensating co-magnetometers have already been used for tests of the Lorentz symmetry \cite{Kornack_phdthesis, new_CPT_limit,New_test_of_local_lorentz_invariance}, setting limit on the neutron coupling to light pseudoscalar particles \cite{Limits_on_new_long_range_nuclear_spin_dep_forces_Romaslis}, and spin-mass interaction of fermions \cite{New_limit_spin_mass_lee}. In all of those applications, however, the signals of interests had low frequencies (typically below a few Hz), for which the response to magnetic fields is almost entirely suppressed and the system is only sensitive to nonmagnetic spin couplings. Since the GNOME targets transient signals, a question of the co-magnetometer's applicability to such searches is important and well motivated. Additional interest in co-magnetometer systems, in the context of searches for transient effects, arises from the possibility for quantitative distinguishing between magnetic and nonmagnetic transients. This was originally considered for precise rotation sensing with NG-AM co-magnetometers\cite{Kornack_Nuclear_spin_gyro} but can be extended for other scenarios.
In this work, we analyse the alkali-metal-noble-gas co-magnetometer in the context of its response to time-dependent electron and nuclear spin perturbations, and we compare the results with the response of the AM SERF magnetometer. Our theoretical analysis is based on numerical solution of differential equations describing the coupled evolution of the AM and NG. We use Ordinary Differential Equations (ODE) to simulate the response of atoms to spin perturbations in the absence of noise. The discussions concern the responses of both, SERF magnetometer and co-magnetometer, to magnetic fields and pseudo-magnetic spin perturbations, where in the latter case we independently consider the effects of electron, proton, and neutron spin perturbations. These allow us to determine the frequency and phase responses of both devices. Finally, we compare the response of the co-magnetometer and SERF magnetometer to transient effects of both magnetic and nonmagnetic nature. We show that differences between the co-magnetometer frequency responses for different spin perturbations allows identification of nonmagnetic transient effects even with a single sensor. These additional signatures of the nonmagnetic transient effects can be utilised to decrease the false-positive event rate in searches with the GNOME network.
\section*{Methods}
\subsection*{Numerical models}
\label{sect:NM}
In this section, we describe the theoretical models used to simulate the response of the SERF and co-magnetometers to magnetic and nonmagnetic spin perturbations.
\subsubsection*{SERF magnetometer model}
In a conventional SERF magnetometer, a spin-zero NG can be used as a buffer gas. The buffer gas limits diffusion of the atoms towards the cell walls (the AM atoms are depolarised in wall collisions), which increases the polarisation lifetime but, it does not produce any magnetisation. Thereby, to simulate a response of such a magnetometer to a spin perturbation, we implement an approach based on the solutions of the Bloch equation\cite{Romalis_Savukov_2005} with inclusion of nonmagnetic spin couplings (for more details see Supplementary Information (SI)). The equation describing the AM polarisation $\mathbf{P}$ of atoms subjected to an external magnetic field $\mathbf{B}$ and circularly polarised light can be written as
\begin{equation}
\label{eq:BE}
\frac{d \mathbf{P}}{d t} = \frac{1}{q}\bigg[\gamma_e \mathbf{(B+b_e)\times P}+(q-1)\gamma_e \mathbf{b_N^{AM}\times P}+\mathbf{(s-P)}R_p - {R}\mathbf{P}\bigg],
\end{equation}
where $\gamma_e$ is the electron gyromagnetic ratio, $\mathbf{s}$ is the optical pumping vector, $R_p$ is the pumping rate, $R$ is the polarisation-relaxation rate, $q$ is the slowing-down factor, which is a function of the nuclear spin of the AM and its polarisation $\mathrm{P}$. The vectors $\mathbf{b_e}$ and $\mathbf{b_N^{AM}}$ are the nonmagnetic electron and nuclear spin perturbations, respectively, given the magnetic units (pseudo-magnetic field).
\subsubsection*{Co-magnetometer model}
\label{subsec:Comag_mum_model}
In the co-magnetometer, a polarised AM and NG (in this case NG with nonzero nuclear spin is used, so its nuclei can be polarised) occupy the same volume inside of a spherical glass cell. Then, the response of the co-magnetometer to magnetic and nonmagnetic perturbations is determined by a set of coupled Bloch equations (the so-called Bloch-Hasegawa equations) \cite{Romalis2002Dynamic} with inclusion of nonmagnetic spin perturbations (for more details see the SI)
\begin{equation}
\begin{cases}
\frac{d \mathbf{P^e}}{d t} &= \frac{1}{q}\bigg[\gamma_e \mathbf{(B+b_e+}\lambda M^n\mathbf{ P^n)\times P^e}
+(q-1)\gamma_e \mathbf{b_N^{AM}\times P^e} +{\mathbf{(s-P^e)}}R_p- {R^e}\mathbf{P^e}\bigg], \\
\frac{d \mathbf{P^n}}{d t}& =\gamma_n \mathbf{(B+b_N^{NG}+}\lambda M^e\mathbf{ P^e)\times P^n+( P^n_0} - \mathbf{P^n})R^n,
\end{cases}
\label{eq:BHE}
\end{equation}
where $\mathbf{P^e}$ and $\mathbf{P^n}$ stand for the electron polarisation of the AM and nuclear polarisation of the NG, respectively, $\lambda$ is the coupling-strength factor for interaction between the two polarisations \cite{Schaefer_freq_shifts}, $M^e$ and $M^n$ are the maximal possible magnetisation of AM and NG, and $\gamma_n$ is the nuclear gyromagnetic ratio of NG, $R^e$ and $R^n$ are the electron and nuclear polarisation-relaxation rates, respectively. The nonmagnetic nuclear perturbation of the NG spins is denoted $\mathbf{b_N^{NG}}$.
In order to fully capitalise on the co-magnetometric capabilities (self-compensation of slow magnetic fields), here we consider the operation in the self-compensating regime, which is achieved when a static magnetic field $\mathbf{B_c}$
\begin{equation}
\mathbf{B_c =} -(\lambda M^e P_0^e+\lambda M^n P_0^n)\mathbf{z},
\label{eq:comp_point}
\end{equation}
is applied to the system\cite{Romalis2002Dynamic} (here we assumed that the initial AM and NG polarisations are oriented along the $\mathbf{z}$ axis).
\subsection*{Nuclear spin content and sensitivity to neutron and proton spin perturbations}
\label{subsection:Nuclear_spin_content}
Due to the composite nature of atomic nuclei, the nuclear response may arise due to coupling to protons, neutrons, or a combination of both. Therefore, the effective pseudo-magnetic fields $\mathbf{b_N^{AM}}$ and $\mathbf{b_N^{NG}}$ can be divided into parts: $\mathbf{b_p}$ affecting the protons and $\mathbf{b_n}$ acting on the neutrons \cite{Kimball_2015}
\begin{equation}
\begin{split}
\mathbf{b_N^{i}}&=\mathbf{b_n^{i}+b_p^{i}},
\end{split}
\end{equation}
where $i$ may stand for either AM or NG. These pseudo-magnetic fields are determined by the nonmagnetic field $\bm{\Xi}$ and coupling constants $\chi_n$ and $\chi_p$ characterising the coupling to neutrons and protons, respectively
\begin{subequations}
\begin{align}
\mathbf{b^{AM}_j} &=-\frac{\sigma^{AM}_j }{\mu_B g_s}\chi_j\bm{\Xi},\\
\mathbf{b^{NG}_j} &= \frac{ \sigma^{NG}_j}{\mu_Ng_{K}}\chi_j\bm{\Xi},
\end{align}
\label{eq:effective_fields_body}
\end{subequations}
where $j$ indicates either proton or neutron, $\sigma_j$ corresponds to the proton or neutron fraction of the nuclear spin polarisation of AM and NG (denoted with upper indices), $\mu_B$ is the Bohr magneton, $\mu_N$ is the nuclear magneton, $g_S$ is the AM Land\'{e} factor, and $g_K$ is the NG nuclear spin $g$-factor. Equations~\eqref{eq:effective_fields_body} show that the effective pseudo-magnetic fields for the NG are generally different from those for the AM. Thus, for the simulations or interpretation of results, it is convenient to introduce scaling factors $\eta_j$ which allows comparison between the response of the SERF magnetometer and the co-magnetometer to nonmagnetic spin couplings of the same strength
\begin{equation}
\mathbf{b^{AM}_j}= \eta_j \mathbf{b^{NG}_j},
\end{equation}
where the scaling factors are
\begin{equation}
\eta_j = - \frac{\sigma^{AM}_j}{\sigma^{NG}_j}\frac{\mu_N g_K}{\mu_Bg_S}.
\label{eq:sacaling_for_the_same_strength}
\end{equation}
In the simulations presented in this paper, we consider the responses of the magnetometers to the perturbation of the same coupling strength. This approach takes into account the scaling factors defined in Eq.~\eqref{eq:sacaling_for_the_same_strength}. One can find a more detailed discussion of the nonmagnetic spin couplings in the SERF and the co-magnetometer in the SI.
\subsection*{Simulation parameters}
\label{subsec:simulation_parameters}
In the case of both, the SERF and co-magnetometer, the spin polarisation is monitored through measurements of the AM-polarisation projection on a given direction (here it is the $\mathbf{x}$ axis). As in both systems the atomic species are initially polarised along the $\mathbf{z}$ axis, both magnetometers are primarily sensitive to perturbations along $\mathbf{y}$ (the sensitive direction is determined by the torque generated by the external fields, rotating the spins around the sensitive direction). Specifically, it can be shown from Eqs. (\ref{eq:BE}) and (\ref{eq:BHE}) that the magnetic or pseudo-magnetic field applied along $\mathbf{y}$ rotates the initial polarisation in the $\mathbf{xz}$ plane, which results in a change of the polarisation projection on the $\mathbf{x}$ axis. At the same time, a field applied along the $\mathbf{x}$ axis generates rotation in the $\mathbf{yz}$ plane, therefore the projection of the polarisation on the $\mathbf{x}$ axis remains unchanged.
In the simulations, we assume that the SERF magnetometer operates using $^{39}$K atoms and the co-magnetometer operates using a $^{39}$K-$^3$He mixture. Parameters of potassium vapour are chosen to be exactly the same for both systems, so we can properly compare the sensitivity. The concentration of the alkali metal is equal to $ 10^{14}$ cm$^{-3}$, which corresponds to saturated atomic K vapour at {190$^\circ$C}. The concentration of $^3$He is $10^{20}$ cm$^{-3}$, which corresponds to 3.5 amg. The assumed relaxation rates 600\,s$^{-1}$ for potassium and $5\cdot 10^{-5}$\,s$^{-1}$ helium, corresponding to a lifetime of about 53 min, well reproduce the experimental conditions. The steady-state polarisation of the AM is 0.5, which ensures the highest amplitude of the co-magnetometer response\cite{Kornack_phdthesis}, and also corresponds to typical experimental conditions. Polarisation of the NG is chosen to be 0.05, which corresponds to typical experimental conditions. These parameters lead to the compensation-field value of {$-131$~nT}. The other simulation parameters (a complete list) is given in the SI.
In case of $^{39}$K and $^3$He the relation between the effective magnetic fields for the AM and the NG generated by the same nonmagnetic perturbation defined in Eq. (\ref{eq:sacaling_for_the_same_strength}) leads to the following scaling factors
\begin{subequations}
\begin{align}
\mathbf{b_p^{^{39}K}} &= \eta_p\mathbf{b_p^{^{3}He}}\approx 10^{-3}\ \mathbf{b_p^{^{3}He}},\\
\mathbf{b_n^{^{39}K}} &= \eta_n \mathbf{b_n^{^{3}He}} \approx 10^{-5}\ \mathbf{b_n^{^{3}He}},
\end{align}
\label{eq:sacaling_for_the_same_strength_num_value}
\end{subequations}
where the nuclear spin content information was taken from Ref.~\cite{Kimball_2015}.
\section*{Results and discussion}
\subsection*{Frequency response to different spin perturbations}
\label{sect:bandwidths}
In this section, we compare responses of the SERF and co-magnetometer devices to various spin perturbations. We analyse the response of the devices by investigating their signals when perturbed with an either magnetic or nonmagnetic periodic $\mathbf{y}$-oriented field $\mathbf{A}$
\begin{equation}
\mathbf{A} = A_0 \sin(2\pi \nu t)\mathbf{y},
\end{equation}
where $A_0$ is the amplitude and $\nu$ is the frequency of the field. For the simulations, we assume that the amplitude of the perturbation is low enough so that the co-magnetometer continuously operates in the self-compensating regime.
To analyse the response, the simulated data are fitted with the function
\begin{equation}
S=S_0\sin(2\pi f t +\phi),
\end{equation}
where $S_0$, $\phi$ and $f$ are, respectively, the amplitude, phase, and frequency of the fitted signal. To avoid distortions in the fit, we ignore transient phenomena at the beginning of the simulations and just fit the dynamical steady-state data.
The fitted amplitude and phase of the signals arising due to magnetic, electron nonmagnetic, neutron nonmagnetic, and proton nonmagnetic perturbations are shown in Fig.~\ref{fig:same_coupling_bandwidth}.
\begin{figure}[h!]
\centering
\begin{minipage}{0.49\linewidth}
\textbf{(a)}
\begin{center}
\includegraphics[width = 0.9\linewidth]{images/subplots_magnetic.eps}
\end{center}
\end{minipage}
\hfil
\begin{minipage}{0.49\linewidth}
\textbf{(b)}
\begin{center}
\includegraphics[width = 0.9\linewidth]{images/subplots_electron.eps}
\end{center}
\end{minipage}
\hfil
\begin{minipage}{0.49\linewidth}
\textbf{(c)}
\begin{center}
\includegraphics[width = 0.9\linewidth]{images/subplots_neutron.eps}
\end{center}
\end{minipage}
\hfil
\begin{minipage}{0.49\linewidth}
\textbf{(d)}
\begin{center}
\includegraphics[width = 0.9\linewidth]{images/subplots_proton.eps}
\end{center}
\end{minipage}
\vspace{0.3 cm}
\caption{ Simulated amplitude and phase of the SERF (dashed lines) and co-magnetometer (solid lines) responses (normalised polarisation along $\mathbf{x}$) for magnetic (a), electron nonmagnetic (b) neutron nonmagnetic (c), and proton nonmagnetic (d) perturbations.}
\label{fig:same_coupling_bandwidth}
\end{figure}
A key feature of the co-magnetometer in searches for pseudo-magnetic spin couplings is its suppressed sensitivity to low-frequency magnetic-field perturbations. This is clearly visible in the data presented in Fig.~\ref{fig:same_coupling_bandwidth}(a). At the lowest frequencies, the response amplitude of the co-magnetometer to the magnetic field is roughly four orders of magnitude lower than the response of the SERF magnetometer. This difference decreases at higher frequencies reducing to zero at about 4~Hz. Since compensation is provided by the NG, which adiabatically follows the field changes, and, at the compensation point, the atoms are only experiencing a field proportional to their own magnetisation, $B^i\approx B_c+\lambda M^{i'}P^{i'}_0=\lambda M^iP^i_0$, the frequency at which the SERF and co-magnetometer magnetic responses are equal is determined by the NG Larmor frequency. Shifting the compensation point towards higher frequencies requires increasing the NG magnetisation, which can be achieved by either increasing the NG concentration or the polarisation. Both may be challenging experimentally. For higher frequencies the response of both systems has the same amplitude, since outside of the self-compensating regime the co-magnetometer response is predominantly determined by the AM. Thereby, the AM polarisation in the co-magnetometer starts to behave in the same way as the free-AM polarisation in a usual SERF magnetometer.
There is a remarkable difference in the phase response of the two devices [Fig.~\ref{fig:same_coupling_bandwidth}(a)]. For magnetic field frequencies below the NG Larmor frequency (4~Hz), the response of the co-magnetometer is phase shifted by about $\pi$, and it decreases sharply above that frequency, eventually becoming similar for both devices. For lower frequencies, the difference is due to the NG that compensates the magnetic field and no such compensation is present in the SERF magnetometer. For higher field frequencies the magnetic response of both devices is determined by the AM, so the observed dependencies are similar.
When analysing the response of the co-magnetometer to nonmagnetic nuclear perturbation, it should be first noted that, unlike in the case of magnetic perturbation, the nonmagnetic nuclear perturbations are not compensated. Therefore, in that case, there is no reduction of the response amplitude at lower frequencies. To the contrary, the amplitude response to pseudo-magnetic nuclear perturbations of the co-magnetometer is significantly stronger than the response of the SERF magnetometer [Fig.~\ref{fig:same_coupling_bandwidth}(c)\&(d)]. In particular, for frequencies below 4~Hz, the co-magnetometer response is roughly five orders of magnitude stronger for the neutron nonmagnetic perturbation [Fig.~\ref{fig:same_coupling_bandwidth}(c)], about three orders of magnitude stronger for the proton perturbation [Fig.~\ref{fig:same_coupling_bandwidth}(d)] and even though it deteriorates for higher frequencies, it still remains significantly larger than for the SERF system. Because of the high concentration of the NG, the response of the co-magnetometer to the nonmagnetic nuclear perturbation is predominantly determined by the gas. In turn, the large concentration difference between the NG and AM concentration (about six order of magnitude) is responsible for much higher sensitivity of the former to the nonmagnetic nuclear couplings. Moreover, the high magnetisation of the NG atoms in in the co-magnetometer ensures efficient transfer of the NG-spins perturbation to AM polarisation through the strong interaction between two species. Therefore, the mediation of the nuclear coupling by the NG magnetisation for the co-magnetometer leads to a significant increase of the amplitude of the response to nonmagnetic nuclear perturbations.
An additional cause of the difference in the response to nuclear perturbations stems from the different nuclear spin contents of the $^{39}$K and the $^{3}$He nuclei \cite{Kimball_2015}. Contribution of proton polarisation in $^{39}$K is roughly four times larger than in the case of $^{3}$He. In contrast, the neutron polarisation has an about 24 times stronger effect on the nuclear polarisation of $^{3}$He, than on the atomic polarisation of $^{39}$K. The difference in neutron and proton fraction in $^3$He also leads to different response amplitudes of the co-magnetometer for proton and neutron perturbations.
The phase response of the co-magnetometer is similar for both the neutron and proton pseudo-magnetic perturbations. Specifically, below {4~Hz}, the phase shift between perturbation and response is close to $\pi$ and it drops to about zero for higher frequencies. While for frequencies above 100~Hz the two responses differ, it should be noted that this frequency range is well beyond the bandwidth of the co-magnetometer, where amplitude of the response drops by several orders of magnitude. At the same time, the phase response of the SERF magnetometer is the same for magnetic and nuclear nonmagnetic perturbations, being zero at lower frequencies and monotonically shifting toward $-\pi/2$ for frequencies beyond the bandwidth of the magnetometer.
The response of both devices to the electron nonmagnetic perturbation is similar at most of the frequencies with a distinct exception of the frequency corresponding to the NG Larmor frequency (4~Hz) [Fig.~\ref{fig:same_coupling_bandwidth}(b)].
Such a behaviour is not surprising since, in both cases, the electron coupling perturbs the AM electron spins. On the other hand, differences in the response of the systems at the NG Larmor frequency arise due to the coupling between the perturbed AM and the NG atoms.
\subsection*{Co-magnetometer response to transients }
It was shown in the previous section that the response of the co-magnetometer to nonmagnetic nuclear couplings is much stronger than that of the SERF magnetometer. Therefore, below we only focus on analysis of the response of the co-magnetometer to transient magnetic and nonmagnetic perturbations.
As a generic example, we take a temporal Lorentzian perturbation of the amplitude $\Lambda_0$ and half-width $\Delta t$, centred at the time $t_0$, which is directed along the $\mathbf{y}$ axis (this can be easily generalised for any pulse shape)
\begin{equation}
\bm{\Lambda_t} = \mathbf{y} \frac{\Lambda_0 \Delta t^2}{(t-t_0)^2 + \Delta t^2}.
\label{eq:lorentzian_pulse}
\end{equation}
Such a definition allows keeping the amplitude of the pulse constant while varying its width (note that here the energy of the pulse is not preserved). Since here we are only interested in temporal parameters of the response, the proton and nuclear couplings are not considered independently but they are treated in as a generic nuclear coupling.
In the previous section, we have shown that the co-magnetometer frequency responses for magnetic and nonmagnetic spin couplings are different.
In particular, the compensation of low-frequency magnetic fields, the mechanism present in the co-magnetometer, results in a suppression of low-frequency components of the pulse. This may affect the shape of the response of the co-magnetometer to magnetic pulses and is well visible in Fig.~\ref{fig:pulses}(a), where the response of the co-magnetometer to 0.05-ms pulses of different nature is shown.
\begin{figure}[h!]
\centering
\begin{minipage}{0.49\linewidth}
\textbf{(a)}
\begin{center}
\includegraphics[width = 0.95\linewidth]{images/single_pulses.eps}
\end{center}
\end{minipage}
\hfil
\begin{minipage}{0.49\linewidth}
\textbf{(b)}
\begin{center}
\includegraphics[width = 0.95\linewidth]{images/bandwidth_subplots.eps}
\end{center}
\end{minipage}
\caption{(a) Temporal responses of the co-magnetometer to pseudo-magnetic electron (dotted line), pseudo-magnetic nuclear (solid line) and magnetic (dashed line) spin perturbations. The bottom subplot shows shape of the perturbation common for all types of coupling. (b) Frequency responses of the co-magnetometer to pseudo-magnetic electron (dotted line), pseudo-magnetic nuclear (solid line) and magnetic (dashed line) spin perturbations assuming the same amplitude of the perturbations in effective pseudo-magnetic magnetic field units. The bottom subplot shows spectra of the 1-s long (solid line) and 0.05-s long (dashed line) Lorentzian pulses.}
\label{fig:pulses}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 0.5\linewidth]{images/Pulse_responses.eps}
\caption{Energy and absolute value of the integral over temporal response of the co-magnetometer to the Lorentzian pulses of different width and coupling origin (electron nonmagnetic (dotted lines), nuclear nonmagnetic (solid lines), and magnetic (dashed lines) spin perturbations.}
\label{fig:final_result}
\end{figure}
In particular, the results show that the response significantly deviates from the Lorentzian shape of the magnetic pulse; the pulse is slightly longer and its shape is significantly distorted. At the same time the distortion is much smaller both in the case of electron and nuclear perturbations, which is a manifestation of absence of such compensation for pseudo-magnetic spin interaction. This is shown in Fig.~\ref{fig:final_result}, where pulse power and magnitude of the pulse integral is presented versus the pulse length. Comparing the frequency response of the co-magnetometer [Fig.~\ref{fig:pulses}(a)] with the spectrum of the Lorentzian pulses [Fig.~\ref{fig:pulses}(b)], one may expect a weaker response of the device to the magnetic pulse which spectrum is within the self-compensation band. To the contrary, no such behaviour is expected for nonmagnetic pulses, since the frequency response for nonmagnetic couplings have high amplitudes at low frequencies. These expectations are confirmed with the simulations; the longer the pulse, the more prominent is the divergence in energy between the response to magnetic and nonmagnetic perturbations [Fig.~\ref{fig:final_result}]. For 1-s long pulses, the difference is more than five orders of magnitude, and it grows for longer pulses. As for the pulse widths around 0.01 s, the response energy is comparable for all types of perturbations, since for the short pulses a significant fraction of the pulse spectrum is at higher frequencies, where response is comparable and large for both magnetic and nonmagnetic perturbations.
The cut-off of the low-frequency spectral magnetic components in the co-magnetometer provides another remarkable feature of the system; the integral over the co-magnetometer signal is significantly suppressed for magnetic field perturbations when integrated over time intervals significantly longer than the pulse width (for the presented results 200-s-long integration window have been used). In contrast, the integral is finite for nonmagnetic spin perturbations. For the simulated co-magnetometer system, assuming that the value of the effective pseudo-magnetic field is the same as the magnetic field, the difference between integrals over detected magnetic and nonmagnetic transients is about seven orders of magnitude for pulse widths between 0.01 s to 1 s [Fig.~\ref{fig:final_result}].
\section*{Conclusions}
Numerical simulations of SERF and AM-NG co-magnetometers show a significantly stronger response of the latter to nuclear nonmagnetic spin perturbations.
While the proton-coupling enhancement stems from the high sensitivity of the co-magnetometer to the nuclear spin perturbations,
a larger contribution of the neutron polarisation to the NG polarisation provides an additional enhancement.
At the same time, the response of both devices to the electron nonmagnetic spin perturbations is similar.
Our results demonstrate benefits of the co-magnetometer in searches for transient pseudo-magnetic spin couplings. On one hand, there is a suppressed response to low-frequency magnetic fields, which reduces noise of the device, on the other, due to ``high-pass'' magnetic-filter nature of the co-magnetometer, the device allows to differentiate between the magnetic and nonmagnetic transient responses, enabling a new way of identification of the observed signal nature. Specifically, the integral over time series signal for magnetic pulses has very small value, while it remains finite for nonmagnetic pulses. The features of the co-magnetometer presented at this work demonstrate the capabilities of the co-magnetometer in searches for transient nonmagnetic spin couplings, i.e., the signals that are being searched by the GNOME.
|
1,108,101,565,488 | arxiv | \section{Introduction}
Entangled Quantum Key Distribution \cite{Ekert} is a protocol to allow two remote parties to generate a shared secret. The key, or random bit string, is shared between the two parties, Alice and Bob, without a third party, Eve, gaining any information about the key. EQKD consists of three parts and three stages. There is a source of entangled two-level systems, generally pairs of entangled photons produced by down-conversion. Experimentally, this has been preformed for both pulsed and continuous-wave (CW) laser systems \cite{exp, exp-pulsed}. Additionally, there are two detectors each for Alice and Bob. The first phase of the protocol consists of data collection. The source emits a pair and sends one half to Alice and one half to Bob. Alice and Bob each make measurements in one of two orthogonal bases. In the second phase they reconcile the measurements, keeping only the data where they measured in the same basis and they both received a signal. They now share a noisy key between them. The final stage is error correction and privacy amplification. They compare information over an authenticated public classical channel and correct all errors calculating a bit-error rate ($BER$) value. It has been shown the Eve's information is bounded by this $BER$. They then calculate the amount of privacy amplification necessary and perform the privacy amplification to make Eve's information negligible.
The formula \cite{nothing} for the amount of secret key generated is
\begin{eqnarray}
K = S + D - (f_{\rm EC} + f_{\rm PA}) \cdot S \cdot H_2({\rm BER}) \label{rate}
\end{eqnarray}
\noindent where $S$ is the number of sifted bits, $D$ is the number of dark counts, $f_{EC/PA}$ is an efficiency factor greater than $1$ for error correction or privacy amplification resepctivly, and $H_2$ is the binary Shannon entropy. In the rate formula shown above, the $BER$ is the only variable that is difficult to simulate. \cite{lo} has shown simulations for the $BER$ and key rate in a system with a pulsed-source laser, which obeys thermal statistics. Here, we expand that technique from pulsed systems to allow for the analysis of continuous-wave (CW) laser systems as well as all intermediate ratios of thermal and Poissonian statistics. With the expanded technique in place, we compare CW and pulsed systems for a variety of realistic experimental parameters. The paper is organized as follows: section II describes how to simulate the $BER$ for a CW-laser. Section III shows the results and compares the situations where the source is in Alice's enclave or in the middle. Section IV shows how the procedure in section II can be adjusted for any value from a completely thermal state to a completely Poissionian state. Finally, in section V CW-laser systems are compared to pulsed-laser systems.
\section{Calculating the $BER$ for a CW-laser}
To calculate the $BER$ from a CW-laser system, we can use the same method as in \cite{lo}, except with a different initial state due to the fact that we are now considering a CW source. The initial state for a CW-laser over the time period given by the timing window of Alice's and Bob's detectors \cite{gisin} is
\begin{equation}
\ket{\Psi} = \frac{1}{C^N} \sum^{\infty}_{M = 0} T^M \ket{\overline{M}(N)}. \label{Gisin-State}
\end{equation}
where $\ket{\overline{M}(N)}$ is the unnormalized state of equal superpositions of all states with $M$ excitation pairs and $N$ modes, $N$ is the number of modes available, approximately $\frac{\delta t}{t_{coh}}$, $\delta t$ is the detector timing window, and $t_{coh}$ is the coherence time of the light field. The constants $C,T$ are defined as $C = \text{cosh}(\xi)$, $T = \text{tanh}(\xi)$ with $\xi$ being proportional to the amplitude of the laser pump field.
Expressing the unnormalized state $\ket{\overline{M}(N)}$ in terms of the normalized state $\ket{M(N)}$, requires counting all the ways that $M$ indistinguishable pairs may be placed into $N$ in principle distinguishable modes. The normalization is then the square root of the number of nonnegative solutions to $x_1 + x_2 + \ldots + x_N = M$,
\begin{eqnarray}
\ket{\overline{M}(N)} &=& \sqrt{{M+N-1}\choose{M}} \ket{M(N)} \nonumber \\
&=& \sum_{\substack{ x_i \geq 0 \\ \sum^{N}_{i=1} x_i = M } } \ket{x_1, x_2, \ldots, x_N}.
\end{eqnarray}
Since the timing window contains many modes, we count m clicks from all modes the same \cite{squash}. We can neglect which mode is which and define $\ket{\overline{M}_{\pi}(\vec{x})}$ as the unnormalized state corresponding to the symmetric polynomial M$_{\pi}$. Where $\pi$ is some partition of $M$ into $N$ pieces (some may be zero) representing a particular solution to the equation above. Expressing $\pi$ as a multiplicity vector\footnotemark:
\footnotetext{The multiplicity vector is one way of describing a partition. The $i^{\text{th}}$ entry in the vector tells how many times the number $i$ appears in the partition. For example, if $\pi \vdash 5 = \{1,1,3\}$ then the multiplicity vector would be $(2,0,1,0,0)$. Another way to represent partitions used in this paper is the Young Diagram. This is a left-justified array of boxes such that no row has a greater number of boxes than the rows above it. In the above example $\pi \rightarrow$
\tiny \yng(3,1,1)
}
\begin{eqnarray}
\braket{\overline{M}_{\pi}(\vec{x})}{\overline{M}_{\pi}(\vec{x})} &=& \frac{N!}{m_1! m_2! \ldots m_M! (n - l(\pi))!} \nonumber \\
&=& M_{\pi}(\vec{\bf{1}})
\end{eqnarray}
\noindent where $l(\pi)$ is the length of the partition. Putting it all together
\begin{eqnarray}
\ket{M(N)} &=& \sqrt{\frac{1}{ {{M+N-1}\choose{M}} }} \sum_{\pi \vdash M} \sqrt{\frac{N!}{m_1! m_2! . . m_M! (N - l(\pi))!}} \nonumber \\
&\times&\frac{1}{\prod_{i=1}^{l(\pi)}\pi_i!} M_{\pi}(\hat{a}_{1}^{\dag}, \hat{a}_{2}^{\dag}, \ldots, \hat{a}_{N}^{\dag} ) \ket{0} \nonumber \\
&=& \frac{1}{\sqrt{{M+N-1}\choose{M}}} \sum_{\pi \vdash M} \sqrt{M_{\pi}(\vec{\bf{1}})} \ket{M_{\pi}(\vec{x})}
\end{eqnarray}
The $BER$ of the state $\ket{\Psi}$ is a weighted sum of the $BER$s due to $\ket{M(N)}$ which in turn is a weighted sum of the $BER$s due to $\ket{M_{\pi}(\vec{x})}$.
For a state $\ket{M_{\pi}(\vec{x})}$ with $\pi\vdash M$, $\pi = \pi_1 \pi_2 \ldots \pi_{l(\pi)}$, we calculate the $BER$. $\displaystyle \ket{M_{\pi}(\vec{x})} = \frac{1}{\sqrt{C}} \sum_{m=0}^{M} \sqrt{A^{\pi}_{m}} \ket{m, M-m}$ where $\ket{m, M-m}$ is the state with $m$ pairs with horizontal polarization and $M-m$ pairs with vertical polarization. $A^{\pi}_{m}$ is the number of distinct ways you can have $m$ horizontally polarized pairs given a state with a partition $\pi$ of $M$ and $\sqrt{C}$ is the normalization.
If $\pi \rightarrow$ \tiny \yng(4) \normalsize ... (i.e. a thermal state) then $A^{\pi}_{m}$ is the number of nonnegative integral solutions to the equation
\begin{eqnarray}
\pi_1 = m && x_1 \in \{0,1, .., M\} \nonumber \\
&& x_{i > 1} \in \{0\} \label{thermal}
\end{eqnarray}
\noindent $A^{\pi}_{m} = 1$ and $C = M+1$.
If $\pi \rightarrow $ \tiny \yng(1,1,1,1) \normalsize ... (i.e. a Poissionian state) then $A^{\pi}_{m}$ is the number of nonnegative integral solutions to the equation
\begin{eqnarray}
\pi_1 + \pi_2 + \ldots + \pi_{l(\pi)} = m && x_i \in \{0, 1\} \nonumber \\
&& l(\pi) = M \label{poissionian}
\end{eqnarray}
\noindent $A^{\pi}_{m} = {{M}\choose{m}}$ and $\displaystyle C = \sum_{m=0}^{M} {{M}\choose{m}} = 2^M$.
For a general partition $A^{\pi}_{m}$ is the number of nonnegative integral solutions to the equation
\begin{eqnarray}
\pi_1 + \pi_2 + \ldots + \pi_{l(\pi)} = m && x_i \in \{0, 1, \ldots , \pi_i\} \nonumber \\
&& l(\pi) = M
\end{eqnarray}
\noindent To find $A^{\pi}_{m}$ let $S$ be the set of nonnegative integral solutions to the unbounded problem. Let $S_i$ be the set of nonnegative integral solutions to the problem where all variables are unbound except $x_i > \pi_i$. Further let $S_A$ be the set of nonnegative integral solutions to the problem where all variables are unbounded except $x_i > \pi_i$, $\forall i \in A$. Now, we can express $A^{\pi}_{m}$ in terms of $|S|$, $|S_i|$, $|S_A|$.
\begin{eqnarray}
A^{\pi}_{m} &=& |S| - \left|\bigcup_{i=1}^{l(\pi)} S_i\right| \nonumber \\
&&\text{and by the principle of inclusion and exclusion \cite{combin}} \nonumber \\
A^{\pi}_{m} &=& |S| + \sum_{r=1}^{l(\pi)} (-1)^r \sum_{\substack{ T_r: |T_r| = r \\ T_r \subseteq \{ 0,1, \ldots, l(\pi)\} } } |S_{T_r}| \nonumber \\
C &=& \sum_{m=0}^{M} A^{\pi}_{m}.
\end{eqnarray}
\noindent Calculating the cardinalities yields
\begin{eqnarray}
|S| &=& {{l(\pi) - 1 + M}\choose{l(\pi) - 1}} \nonumber \\
|S_i| &=& {{l(\pi) - 1 + M - \pi_i - 1}\choose{l(\pi) - 1}} , m \geq \pi_i +1 \nonumber \\
&& 0 \text{ , otherwise} \nonumber \\
|S_A| &=& {{l(\pi) - 1 + M - \sum_{i \in A} (\pi_i + 1)}\choose{l(\pi) - 1}} , m \geq \sum_{i \in A} (\pi_i + 1) \nonumber \\
&& 0 \text{ , otherwise} \nonumber \\
\end{eqnarray}
Now, we have a way of calculating the $BER$ of the state $\ket{M_{\pi}(\vec{x})}$ as a weighted sum of the $BER$ of the states $\ket{m, M-m}$. From \cite{lo} the $BER$ for a state $\ket{m, M-m}$ is
\begin{eqnarray}
e_{mM} &=& e_0 - (\frac{(e_0 - e_d)}{Y_n}\left[ (1 - \eta_A)^{m} - (1-\eta_A)^{M-m} \right] \nonumber \\
&\times& \left[ (1 - \eta_B)^{m} - (1-\eta_B)^{M-m} \right]),
\end{eqnarray}
\noindent where $e_0$ is the error rate of the dark counts ($\frac{1}{2}$), $e_d$ is the detector error rate, and $\eta_{A/B}$ is the detector efficiency of Alice or Bob including the losses in the optical path from the source to the detector.
\section{CW-laser Simulations}
With the $BER$ of a state in the form of (Eqn. \ref{Gisin-State}), we can easily simulate what the secret key generation rate will be for any set of system parameters. The results are shown in Figs. \ref{graph_AMdistance} and \ref{graph_AMS}. For the simulations we used a system visibility of $97\%$ and operated for $100$ s and set the optical loss within both Alice's and Bob's enclaves as $7$ dB including detector efficiency. The simulation parameters for the detectors were that they had a dead time of $1\ \mu$s and a dark and background count rate of $1500$ Hz. The timing window was set at $1$ ns. For the source, we used a variable $S$ for the pair generation rate per second. This was modified by a geometric factor of $G = (3\%)^2$, to get the number of pairs sent to Alice and Bob per second. The coherence time for the down-converted light was $100$ femtoseconds.
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{paperDistance.pdf}
\caption{A semi-log plot of the secret key generation rate (Eqn. \ref{rate}) vs. the distance for a variety of pair generation rates (S) and for the cases where the source is kept either in Alice's enclave or in the middle. For low strength there is not much difference between the two situations, but at high power the case where the source is in the middle is superior.}
\label{graph_AMdistance}
\end{figure}
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{paperS.pdf}
\caption{A log-log plot of the secret key generation rate (Eqn. \ref{rate}) vs the pair generation rate for a variety of transmission losses for the cases where the source is in Alice's enclave and when it is in the middle. The maximum pair generation rate cutoff is similar for all cases where the source is in Alice's enclave, but increases the losses when the source is placed in the middle.}
\label{graph_AMS}
\end{figure}
Fig. \ref{graph_AMdistance} shows the secret key generation rate vs distance between Alice and Bob for a variety of power levels for a system with the source in the middle and for a system with the source inside Alice's enclave. With low power, having the source in the middle doesn't improve the maximum rate significantly. Yet, when $S = 10^{10}$ pairs / s, having the source in the middle improves the distance by $50$ km. Another curious part of the plot is the line for $S = 10^{11}$ pairs / s and the source in the middle. The key rate is zero for low distances. Counter-intuitively, as the loss in the channel increases the key rate jumps. This is caused by the fact that at lower losses, the probability that a single timing window will have many photons at both Alice's and Bob's side is too high and thus the $BER$ is too high. It is the ability of CW systems to change the effective mean photon number by changing the timing window, combined with locating the source in the middle, that will give the CW system increased flexibility and further distances than a comparable pulsed system. As the losses increase, this problem disappears. When the source is in Alice's enclave, the high power will continue to reach her detector regardless of the distance, so the same phenomenon does not occur. In fact, for $S = 10^{11}$ pairs / s there is no key generated for the system with the source inside Alice's enclave.
The second plot (Fig. \ref{graph_AMS}) shows the secret key generation vs. the pair generation rate. Here, one can see the power cutoff for a particular distance more strikingly. Notice that for the system with the source inside Alice's enclave all the different distances have a similar cutoff, but for the system with the source in the middle, the cutoffs increase with distance.
\section{Comparing CW and Pulsed laser systems}
The approximation that $N = \frac{\Delta t}{t_{\text{coh}}}$ is an integer and thus that each timing window consists of an integral number of temporal modes is only justified for $\frac{\Delta t}{t_{\text{coh}}} >> 1$. It may be useful to use the results of section 2 when this is no longer the case. One can see that for a pulsed laser with $\frac{t_{\text{coh}}}{t_{\text{pulse}}} >> 1$, setting $N = 1$ in section $2$ should reproduce the pulsed laser system. Putting both of the systems in the same framework allows for easy comparisons between them.
So when $N=1$, the resultant state may only have one form of Young diagram, the horizontal (\ref{thermal}). As $N \rightarrow \infty$, the resultant state has a Young diagram that approaches vertical (\ref{poissionian}). For the mostly likely error, the 4-photon state, the Young diagrams are just \tiny \yng(2) \normalsize and \tiny \yng(1,1) \normalsize . A Calculation of the $BER$ yields $16\frac{2}{3}\%$ and $25\%$ for the thermal and Poissionian states, respectively. However, that doesn't mean that Poissionian states are worse for QKD. The other factor is with what probability these multi-pair states occur. To compare the mostly Poissionian CW-laser case and the thermal pulsed-laser case, we used the same system parameters as in section 3. With the addition of the fact that for the pulsed system $N=1$ and $t_{\text{pulsed}} = \Delta t = 1$ ns, and $S_{pulsed} = \frac{\mu_{pulsed}}{G \cdot t_{pulsed}}$, so that we can perform a comparison.
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{paperPulsedDistance.pdf}
\caption{A semi-log plot of the secret key generation rate (Eqn. \ref{rate}) vs the distance for a variety of pair generation rates, for both the CW and the pulsed systems, for the case where the source is in the middle. The results for the two systems lie on top over each other for all $S < 1 \times 10^{11}$ pairs / s.}
\label{graph_CWPdistance}
\end{figure}
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{paperPulsedS.pdf}
\caption{A log-log plot of the secret key generation rate (Eqn. \ref{rate}) vs the pair generation rate, for a variety of distances and for both the CW and the pulsed systems when the source is in the middle. For low pair generation rates the two are the same, but when the rates are high, the cutoff is slighty higher for the CW system. This separation increases with the transmission losses.}
\label{graph_CWPS}
\end{figure}
The third plot is again a plot of the secret key rate vs the distance for many pair generation rate $S$. Both systems had a source in the middle. The pulsed and CW systems are mostly identical except in the case of high power. The low loss cutoff is present for both systems, but is worse for the pulsed system. Also with high power the CW system has a positive rate for distances quite a bit further than pulsed systems. The fourth plot shows both that the improvement in cutoff as distance increases for both systems and that the separation between the cutoff for the pulsed and the CW systems increases as distance increases.
\section{Conclusions}
We have presented a method for calculating the expected $BER$ for a CW-laser EQKD system. The method consists of three reductions from the state of the light to easier to handle superpositions of states. First, the light state reduction proceeds from the laser state to a superposition of states with definite excitation number, then to a superposition of partition states. And finally, from a partition state to a superposition of polarization states. The computationally intensive step is calculating all the weights by means of a loop over all possible partitions of a positive integer. We find that for lower pair generation rate (power) the CW and pulsed systems are very close in terms of secret key generation rates and maximum losses. When the pair generation rates are high (much higher than currently available) there is an advantage to using CW laser systems, if all other experimental parameters for the two systems are the same. In addition, including both Poissonian and thermal statistics allows for more practical simulations of EQKD systems.
|
1,108,101,565,489 | arxiv | \section{Introduction}
\par
Speech enhancement aims to isolate a desired speech signal from the additive background noise, and increase the quality or intelligibility of the processed speech \cite{benesty2005}.
In the past decade, due to important theoretical advances, faster and cheaper computational resources, and the availability of large recorded data set for training, neural networks have been applied successfully to a variety of non-linear mapping problems, including speech enhancement.
For instance, \cite{xu2015} proposes a supervised speech enhancement system based on Deep Neural Network (DNN) that can outperform the conventional methods.
\par The Generative Adversarial Network (GAN) aims to generate more realistic output patterns that exhibit characteristics closer to the real data \cite{goodfellow2014}.
Adversarial training can also be employed in the field of speech enhancement.
Proposed by \cite{pascual2017}, Speech Enhancement GAN (SEGAN) works in time-domain and uses a one dimensional Convolutional Neural Network (CNN).
A similar architecture is investigated by \cite{donahue2017} using Short-Term Fourier Transform (STFT) features. Studies by \cite{soni2018} and \cite{pandey2018} use Gammatone spectrum and STFT features, respectively, and propose modified training targets.
Neural network systems require substantial training data to give the best performance.
Thus, having a reliable feature set which reduces memory requirements and training time is an important asset, especially for embedded systems and real-time applications. Speech enhancement with GAN can work in both time \cite{pascual2017,abdul2019} and frequency domains \cite{donahue2017,soni2018,pandey2018}. However, these works indicate that frequency-domain features have a clear advantage over the former, especially in terms of measures like Perceptual Evaluation of Speech Quality (PESQ) \cite{abdul2019}.
\par Frequency-domain features such as STFT, Gammatone spectrum and Mel-Frequency Cepstral Coefficients (MFCC) have been used frequently. In addition, a combination of STFT with MFCC is employed in \cite{ribas2019} for training wide residual networks for speech enhancement.
Compared to STFT, filter-based features like MFCC exhibit reduced dimensionality and are more suitable for learning algorithms,
as they can reduce memory and computational requirements
while maintaining comparable level of performance
\cite{pandey2018,razani2017,chen2014}.
MFCC belong to a larger family of so-called Audio Fingerprinting (AFP) features, which include the Spectral Subband Centroids (SSC) and Spectral Energy Peaks (SEP), and are used to compress data and extract essential patterns in audio frames \cite{duong2015}.
\par
The MFCC are computed by applying the Discrete Cosine Transform (DCT) to a set of weighted subband energies obtained from a Mel-spaced filterbank.
The filter-based energy computation of this process ignores important information about the audio signal in each subband, such as the locations of energy peaks corresponding to speech formants.
The SSC introduced by Paliwal \cite{paliwal1998}, provides crucial information about the centroid frequency in each subband, which has proven to be of great value in several applications. The SSC have been successfully employed in speech recognition, speaker identification and music classification, with non-learning or dictionary-based systems \cite{poh2004,nicolson2018,Kinnunen2007}. Besides a combination of MFCC and SSC was proposed for speaker authentication with non-learning methods in \cite{thian2004}.
\par In this paper, a state-of-art speech enhancement system based on GAN is implemented to predict the Ideal Ratio Mask (IRM) of the noisy speech, using a compact set of features obtained from the combination of MFCC, Normalized SSC (NSSC) and their time differences (i.e. delta versions).
The performance of the resulting systems is evaluated by means of standard objective measures, and compared to that of other possible combinations of features, including the STFT coefficients.
Our results show that the proposed combination of AFP features based on MFCC and NSSC can achieve best (or near best) performance under a wide range of SNR, while significantly reducing memory requirements and training time.
\section{Generative Adversarial Network}
\label{sec:gan}
\par
GANs are generative models designed to map noisy sample vectors $\mathbf{z}$ from a prior distribution into outputs that resemble those generated from the real (i.e. actual) data distribution.
To achieve this, a generator (G) learns to effectively imitate the real data distribution under adversarial conditions.
The adversary in this case is the discriminator (D) which is a binary classifier whose inputs are either samples from the real distribution, or \emph{fake} samples made up by G.
The training process is a game between G and D: G is trying to fool D to accept its outputs as \emph{real}, and D gets better in detecting fake inputs from G and distinguishing them from real data. As a result, G adjusts its parameters to move towards the real data manifold described by the training data \cite{goodfellow2014}.
The described adversarial training can be formulated as the following minmax problem,
\begin{equation}
\min_G\max_D V = ~\mathbb{E}[\log D(\mathbf{x})] + \mathbb{E}[\log (1-D(G(\mathbf{z})))]
\label{eq:GAN_obj}
\end{equation}
\noindent where $V \equiv V(D,G)$ is the value function of the system, referred to as sigmoid cross entropy loss function,
$\mathbf{x}$ is the feature vector from the real data distribution,
$\mathbf{z}$ is the latent vector generated from a noisy distribution,
$D(\mathbf{x})$ and $G(\mathbf{x})$ are the outputs of D and G, and $\mathbb{E}$ denotes expected value.
\par In speech enhancement applications, it has been observed that Conditional GAN (CGAN) \cite{mirza2014} results in better performance than conventional GAN \cite{pascual2017,soni2018,pandey2018}. CGAN uses an additional data vector $\mathbf{x_c}$ in both G and D for regression purposes.
Moreover, the GAN method from \eqref{eq:GAN_obj}
uses sigmoid cross entropy loss function which causes vanishing gradients problem for some fake samples far from the real data, which leads to saturation of the loss function. In the sequel, CGAN is combined with the Least-Squares GAN (LSGAN) \cite{mao2016} which solves this problem by stabilizing GAN training and increasing G's output quality.
This is achieved by substituting the cross-entropy loss with a binary-coded least-squares function, and training G and D individually. This modified GAN objective function is expressed by,
\begin{equation}
\nonumber
\min_D V(D) = \mathbb{E}[(D(\mathbf{x},\mathbf{x_c})-1)^2] + \\ \mathbb{E}[(D(G(\mathbf{z},\mathbf{x_c}),\mathbf{x_c}))^2]
\label{eq:lsgan_d}
\end{equation}
\vspace*{-2ex}
\begin{equation}
\min_G V(G) = \mathbb{E}[(D(G(\mathbf{z},\mathbf{x_c}),\mathbf{x_c})-1)^2]
\label{eq:lsgan}
\end{equation}
\vspace*{-1ex}
\section{Proposed System}
\label{sec:sys}
\subsection{Speech Model in the Frequency Domain}
Let $y[m]$ denote the observed noisy speech signal, where $m \in \mathbb{Z}$ is the discrete-time index.
The noisy speech results from the contamination of a desired,
clean speech signal $s[m]$ with an additive noise signal $n[m]$, i.e.,
\begin{equation}
y[m] = s[m] + n[m], \quad m\in \mathbb{Z}
\label{eq:model_time}
\end{equation}
We represent the signals of interest in the time-frequency domain, as obtained from application of the STFT to \eqref{eq:model_time}. Specifically, the STFT coefficients of the noisy speech signal $y[m]$ are defined as,
\begin{equation}
Y(k,f) = \sum_{m=0}^{M-1} y[m+kL] h[m] e^{-j2 \pi f m /M }
\label{eq:stft}
\end{equation}
where $k \in \mathbb{Z}$ is the frame index,
$L$ is the frame advance,
$f \in \{ 0,1,2,...,M/2\}$ is the frequency bin index,
$M$ is the frame size and
$h[m]$ is a window function.
In practice, the calculation in \eqref{eq:stft} is implemented by means of an $M$-point Fast Fourier Transform (FFT) algorithm.
Applying the STFT formula from \eqref{eq:stft} on the time-domain model \eqref{eq:model_time} yields the time-frequency model representation
\begin{equation}
Y(k,f) = S(k,f) + N(k,f)
\label{eq:model_stft}
\end{equation}
where $S(k,f)$ and $N(k,f)$ are the STFT of the clean speech and noise signals, respectively.
\subsection{Audio Fingerprinting Features}
\par To train the GAN architecture, we propose a new feature set obtained by combination of MFCC and NSSC.
In this part, we explain the calculation and combination of these AFP features.
\subsubsection{Mel-Frequency Cepstral Coefficients (MFCC)}
\par MFCC are widely used in speech recognition and enhancement due to their powerful compacting capabilities while preserving essential information in speech \cite{chen2014,razani2017,kolbaek2017}.
To calculate the MFCC features, the time-domain signal $y[m]$ is passed through a first order FIR filter to boost the highband formants in a so-called pre-emphasis stage, as given by,
\begin{equation}
y'[m] = y[m] - \alpha y[m-1]
\label{eq:preemph}
\end{equation}
where $\alpha$ is the pre-emphasis coefficient, with $0.95 \leq \alpha \leq 1$.
\par Next, the STFT of the filtered signal $y'[m]$ is calculated as in \eqref{eq:stft}, yielding the STFT coefficients $Y'(k,f)$. For each data frame, these STFT coefficients are used to calculate a set of Spectral Subband Energies (SSE) defined in terms of a bank of overlapping narrow-band filters.
Specifically, the SSE of the $k$-th frame are calculated as,
\begin{equation}
\text{SSE}_y(k,b) = \sum_{f=l_b}^{h_b} w_b(f) |Y'(k,f)|^2
\label{eq:sse}
\end{equation}
where $b \in \{0,1,\ldots,B-1\}$, $B$ is the number of subbands in the filterbank,
and $w_b(f) \ge 0$ is the spectral shaping filter of the $b$-th subband, with
$l_b$ and $h_b$ denoting the lower and upper frequency limits of $w_b(f)$.
More specifically, the filters $w_b(f)$ together form a mel-spaced filterbank, i.e., they are characterized by triangular shapes with peak frequencies distributed according to the mel-scale of frequency.
\par Finally, the Discrete Cosine Transform (DCT) - Type III is applied to the logarithm of the SSE to obtain the desired MFCC features, which is expressed as,
\begin{equation}
\text{MFCC}_y(k,p) = \sqrt{\frac{2}{B}}\sum_{b=0}^{B-1} \log_{10}{\text{SSE}_y(k,b)}\cos{(\frac{p\pi}{B}(b-0.5))}
\label{eq:mfcc}
\end{equation}
\noindent where $p \in \{0,1,\ldots,P-1\}$ and $P$ is the number of coefficients.
We define the MFCC feature vector of the current data frame as: $\textbf{MFCC}_y = [\text{MFCC}_y(k,0),...,\text{MFCC}_y(k,P-1)]$.
\subsubsection{Spectral Subband Centroids (SSC)}
\par The SSC were introduced in \cite{paliwal1998} to measure the center of mass of a subband spectrum in terms of frequency, using a weighted average technique.
These features exhibit robustness against the equalization, data compression and additive noise which do not significantly alter the peak frequencies at moderate to high Signal-to-Noise Ratio (SNR) \cite{duong2015}.
In \cite{seo2005}, the SSC outperform MFCC when used as inputs in a audio recognition task based on dictionary matching.
To generate SSC values, the noisy speech signal $y[m]$ is pre-emphasized as in \eqref{eq:preemph} and the corresponding STFT coefficients $Y'(k,f)$ are computed. For each frame, a set of SSC is obtained by calculating the centroid frequencies of a bank of narrowband filters as in the MFCC.
Specifically, the SSC of the $k$-th frame are calculated as,
\begin{equation}
\text{SSC}_y(k,b) = \frac{\sum_{f=l_b}^{h_b} f \, w'_b(f)|Y'(k,f)|^2}{\sum_{f=l_b}^{h_b}w'_b(f)|Y'(k,f)|^2}
\label{eq:ssc}
\end{equation}
where $b \in \{0,1,\ldots, B-1\}$ and $w'_b(f)$ is the corresponding subband filter.
In this work, to simplify implementation, we use the same bank of triangular mel-scale filters for both MFCC and SSC calculations, i.e. $w'_b(f)=w_b(f)$
\par Finally, following \cite{seo2005}, the SSC values are normalized within the range $[-1,1]$,
which is more convenient for use in neural network layers and activation functions. The normalized SSC (NSSC) features are obtained as,
\begin{equation}
\text{NSSC}_y(k,b) = \frac{\text{SSC}_y(k,b) - (h_b - l_b)}{h_b - l_b}
\label{eq:nssc}
\end{equation}
For later reference, we define the NSSC feature vector of signal $y[m]$ at the current frame $k$ as $\textbf{NSSC}_y = [\text{NSSC}_y(k,0),...,\text{NSSC}_y(k,B-1)]$.
\subsubsection{Feature Combination}
\par
In this paper, we propose to use the concatenation of MFCC and NSSC vectors, along with some of their first and second differences (i.e., delta and double-delta) for training the GAN architecture. In the sequel, we refer to this extended feature set as AFP Combination (AFPC).
The MFCC and their deltas have long been used as an efficient alternative to the STFT, as they contain crucial information about the spectral subband energies and their temporal evolution \cite{huang2001}.
Nevertheless, due to the smoothing nature of \eqref{eq:sse}, the MFCC ignore the dynamics of the formant present in each subband.
In contrast, the NSSC and their deltas can provide critical information about the formant locations and their temporal variations. At the same time, the NSSC tend to be more noise-robust, compared to the MFCC, since the formant locations are not significantly disturbed by the additive noise distortion \cite{paliwal1998}.
Thence, the proposed AFPC features have the ability to capture information about the distribution of energy, both across and inside spectral subbands.
\par
To obtain the AFPC, the MFCC and NSSC are both extracted from the STFT of the noisy signal, $Y(k,f)$ as described previously. The proposed AFPC feature vector at the $k$-th time frame for signal $y[m]$ is then defined as,
\begin{equation}
\begin{split}
\textbf{AFPC}_y = [\textbf{MFCC}_y, \Delta\textbf{MFCC}_y,\Delta^2\textbf{MFCC}_y,\\ \textbf{NSSC}_y, \Delta\textbf{NSSC}_y, \Delta^2\textbf{NSSC}_y]
\end{split}
\label{eq:afpc_1}
\end{equation}
where $\Delta\textbf{MFCC}_y$ and $\Delta^2\textbf{MFCC}_y$ are the deltas and double-deltas of the MFCC. Similarly, $\Delta\textbf{NSSC}_y$ and $\Delta^2\textbf{NSSC}_y$ are the deltas and double deltas of the NSSC.
\subsection{Incorporation of AFPC within GAN}
\par
We assume that the magnitude spectrum of the noisy speech can be approximated by the sum of the clean speech and noise magnitude spectra, i.e, $|Y(k,f)| \approx |S(k,f)| + |N(k,f)|$.
The generator in the adversarial setting is trained to predict a \emph{real} output, which is taken as the Ideal Ratio Mask (IRM) genera-ted from the known clean speech and noise signals \cite{narayanan2013}, i.e.,
\begin{equation}
\text{IRM}(k,f) = \sqrt{\frac{|S(k,f)|^2}{|S(k,f)|^2+|N(k,f)|^2}}
\label{eq:irm}
\end{equation}
We define the IRM vector at the current frame $k$ as $\textbf{IRM} = [\text{IRM}(k,0),...,\text{IRM}(k,M/2)]$ . Then, the generator produces the estimated IRM whose patterns and distribution should be close to the real IRM, as expressed by,
\begin{equation}
\widehat{\textbf{IRM}} = G(\mathbf{z},\textbf{AFPC}_y^j)
\label{eq:esirm}
\end{equation}
\noindent where $\textbf{AFPC}^j_y$ represents the AFPC feature vector at the current frame, obtained by concatenating the AFPC feature vectors from a subset of $2j+1$ consecutive context frames centered at the current one (i.e., by including the $j$ adjacent frames to its left and right). The estimated output $\widehat{\textbf{IRM}}$ in \eqref{eq:esirm} is only calculated for the current frame.
\par By examining $\widehat{\textbf{IRM}}$ and the $\textbf{AFPC}_{y}$ of the current frame, D decides whether its input is the real IRM from \eqref{eq:irm}, or the \emph{fake} output from \eqref{eq:esirm}. The estimated IRM for every frame and frequency index is used as a Wiener type of filter on the STFT magnitude of the noisy speech. This method only enhances the amplitude of the signal and uses the phase from the noisy speech to reconstruct the time-domain enhanced signal using the overlap-add and Inverse STFT (ISTFT) as shown in,
\begin{equation}
|\hat S(k,f)| = \widehat{\text{IRM}}(k,f)|Y(k,f)|
\label{eq:recon}
\end{equation}
\begin{equation}
\hat s[m] = \text{ISTFT}\{|\hat S(k,f)|e^{jk\angle Y(k,f)}\}
\label{eq:t-recon}
\end{equation}
\par In \cite{pascual2017}, it is reported that having an extra term in training the generator using CGAN is very useful. Pandey \emph{et al.} \cite{pandey2018} show that using the $L_1$ loss gives a better performance compared to the $L_2$ loss in speech enhancement applications.
This approach allows adversarial component to produce more refined and realistic results.
The weight of the $L_1$ component in the objective function is controlled by a parameter $\lambda>0$. Therefore, the objective functions from \eqref{eq:lsgan} are modified as,
\begin{equation}
\begin{split}
\min_D V(D) = & ~ \mathbb{E}[(D(\textbf{IRM},\textbf{AFPC}_y)-1)^2] \\ & +\mathbb{E}[(D(G(\mathbf{z},\textbf{AFPC}^j_y),\textbf{AFPC}_y))^2]
\end{split}
\label{eq:se_d}
\end{equation}
\begin{equation}
\begin{split}
\min_G V(G) = &~ \mathbb{E}[(D(G(\mathbf{z},\textbf{AFPC}^j_y),\textbf{AFPC}_y)-1)^2] \\& + \lambda \| G(\mathbf{z},\textbf{AFPC}^j_y)-\textbf{IRM}\|_1
\end{split}
\label{eq:se_g}
\end{equation}
\par A schematic of this adversarial training procedure is illustrated in Fig. \ref{fig:GAN_arch}. The training consists of three consecutive steps:
First, D is trained with a concatenation of the \textbf{IRM} vector and the $\textbf{AFPC}_{y}$ feature vector, in such a way that it recognizes the \textbf{IRM} as real (or output 1).
Next, D learns to categorize the concatenation of the $\widehat{\textbf{IRM}}$ and $\textbf{AFPC}_y$ feature vector as fake data distribution (or output 0). Finally, D variables are frozen and the G is trained with the $\textbf{AFPC}_y^j$ features to fool the D.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.7] {gan_arch.eps}
\caption{The Proposed GAN training procedure used with the AFPC.
}
\label{fig:GAN_arch}
\end{figure}
\par A block diagram of the system architecture is depicted in Fig. \ref{fig:arch}. The operation consists of two stages: training and enhancement. During the training stage, the system uses the AFPC features to train the D and G as shown in Fig. \ref{fig:GAN_arch} and learn the IRM. In the enhancement stage, the G from the GAN setting is inputted with the AFPC features to output the estimated $\widehat{\text{IRM}}$ and the speech spectrum is reconstructed using a Wiener type of filtering shown in \eqref{eq:t-recon}.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.81] {arch.eps}
\caption{Block diagram of the proposed AFPC training feature set and its incorporation into GAN.}
\label{fig:arch}
\end{figure*}
\section{Experimental Setup}
\label{sec:exp}
\subsection{Dataset}
We use the LibriSpeech \cite{panayotov2015} dataset which is an open corpus based on audio books and containing 1000 hours of relatively noise-free speech in English.
For training, 1755 utterances are randomly selected from 250 speakers (half male, half female) for a total of 6 hours of speech. For testing, 255 utterances are selected from 40 speakers (half male, half female), for a total of 30 minutes of speech. The clean files are contaminated with additive noise at -5dB, 0dB and 5dB SNRs for both training and testing sets, while two extra SNRs of 10dB and 15dB are added for testing. Five different noise types from NOISEX-92 \cite{varga1993} are used for both training and testing: babble, pink, buccaneer2, factory1 and hfchannel.
\par All the audio files are sampled at 16 KHz. The STFT coefficients are extracted with an $M=512$ STFT, using a 32ms Hanning window, overlap of 50\% ($L=256$) and three context frames (i.e. $j=1$). The MFCC and NSSC are computed from the STFT parameters using $B=64$ subbands with mel-frequency triangular filters $w_b(f)$ distributed between $0$Hz and $8$KHz.
The number of MFCC is set to $P=22$ while for NSSC, only the first 22 coefficients are kept in the feature vector. The pre-emphasis factor $\alpha = 0.97$ is used in \eqref{eq:preemph}. The delta and double-delta variations are included in the feature sets for each context frame \cite{paliwal1998}. The estimated IRM \eqref{eq:esirm} is calculated only for the middle STFT frame. For each feature set, one model is trained for all noise types, SNRs and speakers.
\subsection{Training and Evaluation}
\par The generator's architecture has three hidden layers, each including 512 nodes. The ReLU activation function is used after each hidden layer with a dropout rate of 0.2. The discriminator has the same structure as the generator but uses instead the leaky ReLU activation function. Both employ the sigmoid activation at the output layer because they predict the IRM. The latent vector $\mathbf{z}$ has 15 elements generated randomly from a normal Gaussian distribution. The GAN architecture is trained in 50 epochs with a learning rate of $10^{-4}$ for the first half and $10^{-5}$ for the second half of the epochs. The batch size is set to 128 and ADAM optimizer is used for training. We set $\lambda=100$ in \eqref{eq:se_g}, which provides good convergence.
We compare different combinations of the discussed features, i.e. STFT coefficients, MFCC and NSSC, and they are designated with "+", which means concatenation of the indicated feature vectors. Out of the seven distinct possible combinations, STFT+MFCC+NSSC combination is not included in the study, since it does not substantially improve the performance nor the computational efficiency. In each experiment, one GAN architecture is trained for each feature set using all SNRs and noise types and uses the same architecture, training and hyper-parameters.
The feature sets are compared objectively in terms of PESQ, which provides a measure of signal quality between -0.5 and 4.5, Signal-to-Distortion Ratio (SDR) which measures the speech quality in dB based on the introduced speech distortion, and Short-Time Objective Intelligibility (STOI), which provides a measure of intelligibility between 0 and 1.
Besides these performance measures, we also compare the different feature combinations in terms of system efficiency, i.e. feature vector size, training time per epoch, and number of network parameters.
\section{Results and Discussion}
\label{sec:res}
\par In this section, we present and discuss the experimental results. To select the number of context frames (i.e., $2j+1$), the PESQ performance of three selected feature sets is studied as demonstrated in Fig. \ref{fig:pesq}. When the number of context frames increases, the performance tend to improve for each feature set. However, since most of the gains for MFCC+NSSC and STFT+MFCC are obtained with 3 context frames, we use the value of $j=1$ for all subsequent experiments.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.95] {pesq.eps}
\caption{Average PESQ performance for three feature sets: STFT, MFCC+NSSC and STFT+MFCC in different context frames from 1 to 9.}
\label{fig:pesq}
\end{figure}
\par For each feature set, results are obtained for five different noise types at five SNR levels from -5dB to 15dB. Average PESQ, SDR and STOI measures over all noise types are reported in Tables \ref{tab:ls_pesq}-\ref{tab:ls_stoi}, where the best results (within the 2\% of the observed maximum) are highlighted for each SNR.
When used separately, MFCC and NSSC improve the overall speech quality compared to the noisy speech but do not generally outperform STFT.
Comparing STFT with STFT+NSSC and STFT+MFCC indicates that both AFP features add important information to the STFT features.
STFT+MFCC outperforms STFT+NSSC in terms of both PESQ and STOI, while achieving a similar SDR performance.
\par According to Tables \ref{tab:ls_pesq}-\ref{tab:ls_stoi}, the proposed AFPC, i.e., MFCC+NSSC, substantially increases the performance of the GAN-based speech enhancement system in all three measures compared to MFCC or STFT. Furthermore, MFCC+NSSC achieves the best PESQ performance (within the error margin) and demonstrates a performance close to STFT+MFCC in terms of SDR and STOI. In particular, MFCC+NSSC outperforms the other feature sets in all three measures at high unmatched SNR of 15dB. This is due to the fact that at such high SNR, the additive noise does not significantly corrupt the extraction of formant frequencies with NSSC.
\begin{table}[ht]
\centering
\caption{Average PESQ Results for all noise types at various SNRs}
\label{tab:ls_pesq}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Feature Set}} & \multicolumn{5}{c|}{\textbf{PESQ}} \\ \cline{2-6}
&-5dB&0dB&5dB&10dB&15dB\\ \hline
Noisy &1.13& 1.40& 1.72& 2.07& 2.43\\ \hline
STFT &1.71&2.12&2.52&2.82&2.99 \\ \hline
NSSC &1.56&2.07&2.48&2.80&3.07 \\ \hline
MFCC &1.69&2.11&2.50&2.84&3.12 \\ \hline
STFT+NSSC &1.77&2.20&2.60&2.90&3.04 \\ \hline
STFT+MFCC &\textbf{1.83}&\textbf{2.27}&\textbf{2.64}&\textbf{2.94}&3.14 \\ \hline
MFCC+NSSC &\textbf{1.82}&\textbf{2.25}&\textbf{2.63}&\textbf{2.96}&\textbf{3.21} \\ \hline
\end{tabular}
\end{table}
\begin{table}[ht]
\centering
\caption{Average SDR Results for all noise types at various SNRs}
\label{tab:ls_sdr}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Feature Set}} & \multicolumn{5}{c|}{\textbf{SDR(dB)}} \\ \cline{2-6}
&-5dB&0dB&5dB&10dB&15dB\\ \hline
Noisy &-5.21& -0.34& 4.62& 9.61& 14.6 \\ \hline
STFT &3.80&7.71&11.5&15.1&17.8 \\ \hline
NSSC &3.05&7.10&10.8&14.0&16.5 \\ \hline
MFCC &3.17&6.96&10.7&14.3&17.2 \\ \hline
STFT+NSSC &\textbf{4.16}&\textbf{7.95}&\textbf{11.7}&\textbf{15.2}&17.9 \\ \hline
STFT+MFCC &\textbf{4.18}&\textbf{7.96}&\textbf{11.7}&\textbf{15.3}&18.3 \\ \hline
MFCC+NSSC &\textbf{4.11}&\textbf{7.80}&\textbf{11.6}&\textbf{15.2}&\textbf{18.5} \\ \hline
\end{tabular}
\end{table}
\begin{table}[!ht]
\centering
\caption{Average STOI Results for all noise types at various SNRs}
\label{tab:ls_stoi}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Feature Set}} & \multicolumn{5}{c|}{\textbf{STOI}} \\ \cline{2-6}
&-5dB&0dB&5dB&10dB&15dB\\ \hline
Noisy &0.56&0.67&0.78&0.87&0.93 \\ \hline
STFT &0.69&0.79&\textbf{0.87}&\textbf{0.92}&\textbf{0.94}\\ \hline
NSSC &0.64&0.76&0.85&0.90&0.93\\ \hline
MFCC &0.68&0.79&0.86&0.91&\textbf{0.94}\\ \hline
STFT+NSSC &\textbf{0.70}&\textbf{0.80}&\textbf{0.88}&\textbf{0.92}&\textbf{0.94} \\ \hline
STFT+MFCC &\textbf{0.71}&\textbf{0.81}&\textbf{0.88}&\textbf{0.92}&\textbf{0.95} \\ \hline
MFCC+NSSC &\textbf{0.70}&\textbf{0.80}&\textbf{0.88}&\textbf{0.92}&\textbf{0.95} \\ \hline
\end{tabular}
\end{table}
\par While the bottom 3 feature sets in Tables \ref{tab:ls_pesq}-\ref{tab:ls_stoi} achieve the best performance in terms of average PESQ, STOI and SDR, the cost of this improvement for a GAN-based system using STFT+NSCC or STFT+MFCC is much more than for the proposed MFCC+NSSC (i.e., AFPC).
As shown in Table \ref{tab:complex}, the latter significantly outperforms the former in terms of feature size, training time and number of network parameters.
Specifically, MFCC+NSCC leads to reductions of
59.1\% in memory storage for the training data,
43.3\% in training time for the GAN system,
and 25.0\% in the number of network parameters. Compared to the STFT baseline, MFCC+NSCC requires 49.6\% less memory storage for features
and 30.1\% less training time, while achieving
significant performance improvements.
The savings in training time and network size with the proposed AFPC become larger when we add more context frames (i.e., $j>1$).
The testing time is not reported in Table IV since it is almost the same for all systems. In testing, most of the processing time is allocated to the STFT computation which is similar for all feature combinations.
\begin{table}[ht]
\centering
\caption{Feature Vector Size and Training Time per Epoch}
\label{tab:complex}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{\textbf{Feature Set}} &
\makecell{\textbf{Average} \\ \textbf{PESQ}} &
\makecell{\textbf{Feature} \\ \textbf{Size}} & \makecell{\textbf{Training Time} \\ \textbf{per epoch}} & \makecell{\textbf{Network} \\ \textbf{Param.}} \\ \hline
STFT & 2.43 & 257 & 17.6 mins & 1.06M\\ \hline
STFT+NSSC &2.50 & 323& 21.7 mins & 1.16M\\ \hline
STFT+MFCC& \textbf{2.56} & 323& 21.7 mins & 1.16M\\ \hline
\textbf{MFCC+NSSC} & \textbf{2.57} & \textbf{132}& \textbf{12.3 mins} & \textbf{870K}\\ \hline
\end{tabular}
\end{table}
\vspace{5mm}
\par Fig. \ref{fig:spec} shows the spectrograms of: (a) clean speech; (b) noisy speech after contamination with babble noise at 0dB SNR; (c) enhanced speech using GAN with STFT, and; (d) enhanced speech using proposed AFPC.
It can be seen that the proposed AFPC features preserve the speech formants while removing more noise during non-speech segments.
\begin{figure}[ht]
\scriptsize
\begin{tabular}{cc}
(a) Clean speech & (b) Noisy speech \\
\includegraphics[width=40mm]{clean.eps}&
\includegraphics[width=40mm]{noisy.eps}\\
(c) Processed with STFT & (d) Processed with AFPC\\
\includegraphics[width=40mm]{stft3.eps}&
\includegraphics[width=40mm]{afpc3.eps}
\end{tabular}
\caption{(a) Clean speech (b) Noisy speech (0dB babble noise) (c) Processed speech using STFT features (d) Processed speech using the AFPC features.} \label{fig:spec}
\end{figure}
\section{Conclusion}
\label{sec:con}
In this work, we proposed using a compact set of features obtained from the combination of two AFP techniques, i.e., MFCC and NSSC, to implement a speech enhancement system based on GAN and trained to predict the IRM of the noisy speech.
The NSSC capture the speech formants and the distribution of energy in each subband, and therefore complement the MFCC in a crucial way.
In experiments with diverse speakers and noise types, GAN-based speech enhancement with the proposed AFPC (MFCC+NSCC) achieved the best average performance in terms of PESQ, STOI and SDR objective measures.
Furthermore, compared to the STFT+MFCC combination with nearly similar performance,
AFPC led to reductions of about
60\% in memory storage,
45\% in training time,
and 25\% in network size.
Hence, the proposed AFPC set is a promising feature-extraction method in learning-based speech enhancement systems.
|
1,108,101,565,490 | arxiv | \section{Introduction}
\label{sec:intro}
We are concerned with the well-posedness of linear elliptic systems of
the form
\begin{align}
\label{eq:ellsys_strong}
-{\rm div}\, \mathbb{C} : \partial u =~& f, \\
\label{eq:farfield_bc_formal}
u(x) \sim~& 0, \quad \text{as } |x| \to \infty,
\end{align}
where $\mathbb{C} \in C(\mathbb{R}^d; \mathbb{R}^{m^2d^2})$ is bounded
and satisfies the Legendre--Hadamard condition,
\begin{displaymath}
\mathbb{C}_{i\alpha}^{j\beta}(x) v_i v_j k_\alpha k_\beta \geq c_0 |v|^2
|k|^2
\qquad \forall x, k \in \mathbb{R}^d, \quad v \in \mathbb{R}^m.
\end{displaymath}
The functions $f, u : \mathbb{R}^d \to \mathbb{R}^m$ (we will define the precise
function spaces to which $f$ and $u$ belong to later on), and $\mathbb{C} :
{\sf G} = (\mathbb{C}_{i\alpha}^{j\beta} {\sf G}_{j\beta})_{i\alpha}$ denotes the
contraction operator.
The concrete problem of interest, for which we require this theory,
arises from the linearization of the equations of anisotropic finite
elasticity in infinite crystals, however, our results are more
generally applicable to translation-invariant problem posed on
$\mathbb{R}^d$. Some of the main challenges to be overcome in
translation-invariant problems on infinite domains are the absence of
Poincar\'e-type inequalities, and the interpretation of boundary
conditions.
A common approach to PDEs on infinite domain, as well as for exterior
problems, is the formulation in weighted function spaces (see, e.g.,
\cite{Kufner, WaJaBe:1985}). Our aim in this note is to outline a
more straightforward existence, uniqueness, and regularity theory in
Sobolev spaces of Beppo Levi type (also called homogeneous Sobolev
spaces). Such spaces have previously been analyzed in detail in
\cite{DL} and used for the solution of elliptic PDEs (see, e.g.,
\cite{SamVar:2003, Sohr, GalSim:1990, KozSoh:1991}).
In the present work we describe a version of the homogeneous Sobolev
space approach. Variants (and sometimes generalisations) of most of
our results can be found in the cited literature; however, the
equivalence class viewpoint considered here is not normally taken and
the growth characterisation given in Theorem \ref{th:growth} appears
to be new. This research note is intended as an elementary
introduction to and reference for some key ideas.
We wish to define the homogeneous Sobolev space as a closure of smooth
functions with compact support. The following cautionary example was
discussed by Deny \& Lions \cite{DL}: let $u_n : \mathbb{R} \to \mathbb{R}$ be defined
by
\begin{displaymath}
u_n(x) := n \max(0, 1 - |x| / n^3),
\end{displaymath}
where $u_n$ has compact support and $u_n'(x) = \pm \frac1{n^2}$ in
$\pm (0, n^3)$, and hence $\|\partial u_n \|_{L^2} \to 0$ as $n \to
\infty$. However, $u_n$ clearly does not converge in the topology of
$D'$.
To avoid this difficulty, we will define spaces of equivalence
classes, or, factor spaces. Indeed, if we shift $u_n$ to obtain $v_n
:= u_n - n$, then it is straightforward to see that $v_n \to 0$ in the
sense of distributions, which is consistent with the convergence $\|\partial
u_n \|_{L^2} = \|\partial v_n\|_{L^2} \to 0$ as $n\to \infty$.
\subsection{Notation}
$B_R$ denotes the open ball, centre $0$, radius $R$ in $\mathbb{R}^d$, $d \in \{1,2,\dots\}$;
$p \in [1,\infty]$, $p'=p/(p-1)$, and $p^*$ denotes the Sobolev
conjugate of $p \in [1,d)$, $d>1$, defined by
$1/p^*= 1/p - 1/d$. For Lebesgue and Sobolev spaces of functions defined on the whole of $\mathbb{R}^d$ we shall suppress the symbol $\mathbb{R}^d$
in our notations for these function spaces, and will simply write $L^p$ and $W^{1,p}$, respectively, instead of
$L^p(\mathbb{R}^n)$ and $W^{1,p}(\mathbb{R}^n)$. We define the integral average $(u)_A$ of a locally integrable function $u \in L^1_{\rm loc}$
over a measurable set $A \subset \mathbb{R}^n$, $|A|:=\mbox{meas}(A)<\infty$, by $(u)_A:=|A|^{-1} \int_A u(x) {\rm d} x$. Throughout this
note $\int$ will signify $\int_{\mathbb{R}^d}$.
Assuming that $\Omega_k$, $k=1, 2, \dots$, in an increasing sequence of bounded open
sets in $\mathbb{R}^n$, $L^p_{\rm loc}(\Omega)$ is equipped with the family of seminorms
\[ \|u\|_{L^p(\Omega_k)} := \left(\int_{\Omega_k} |u(x)|^p {\rm d} x\right)^{\frac{1}{p}}.\]
The linear space $L^p_{\rm loc}(\Omega)$ is then a Fr\'echet space (i.e., a metrizable
and complete topological vector space).
\section{Sobolev spaces of equivalence classes}
For any measurable function $u : \mathbb{R}^d \to \mathbb{R}$, let $[u] := \{ u + c
\,|\, c \in R \}$ denote the equivalence class of all translations of $u$. Let
$D$ denote the space of test functions ($C^\infty$ functions with
compact support in $\mathbb{R}^d$), and let $\dot{\DD} := \{ [u] \,|\, u \in D \}$ be the
associated linear space of equivalence classes $[u]$ of translations of $u \in D$.
We denote the linear space of equivalence classes $[u]$ of functions $u \in W^{1,p}_{\rm loc}$
with $p$-integrable gradient by
\begin{displaymath}
\We{1,p} := \big\{ [u] \,\b|\, u \in W^{1,p}_{\rm loc}, ~ \partial u \in L^p \big\},
\end{displaymath}
equipped with the norm
\begin{displaymath}
\| [u] \|_{\We{1,p}} := | u |_{W^{1,p}} = \| \partial u \|_{L^p},\quad u \in [u].
\end{displaymath}
\begin{proposition}
$\We{1,p}$ is a Banach space.
\end{proposition}
\begin{proof}
It is clear that $\|\bullet\|_{\We{1,p}}$ is a semi-norm on $\We{1,p}$. To show
that it is a norm, suppose that $\|[u]\|_{\We{1,p}} = 0$. Then $\partial u
= 0$ and hence $u$ is a constant, that is, $[u] = [0]$.
To prove that $\We{1,p}$ is complete, suppose that $([u_j])_{j \in
\mathbb{N}}$ is a Cauchy sequence. Let $u_j \in [u_j]$ be defined through
the condition that $(u_j)_{B_1} = 0$. Then, it is straightforward to
show that there exist $u \in L^p_{\rm loc}$ and $g \in L^p$ such that
$u_j \to u$ in $L^p_{\rm loc}$ and $\partial u_j \to g$ in $L^p$. By the uniqueness of
the distributional limit, it then follows that $g = \partial u$. Hence we have shown that there exists $u \in
W^{1,p}_{\rm loc}$ such that $\partial u_j \to \partial u$ in $L^p$, that is, $[u_j]
\to [u]$ in $\We{1,p}$ as $j \to \infty$.
\end{proof}
The next result establishes that test functions are dense in
$\We{1,p}$. This result is a special case of \cite[Thm. 1]{Sohr}.
\begin{theorem}
Let $p \in (1, \infty)$ or $p = 1$ and $d > 1$; then, $\dot{\DD}$ is
dense in $\We{1,p}$.
\end{theorem}
\begin{proof}
Suppose first that $d > 1$. Fix $u \in [u] \in \We{1,p}$. Since
$D$ is dense in $W^{1,p}$ it is sufficient to show the existence
of a sequence $(u_n) \subset W^{1,p}$ such that $[u_n] \to [u]$ in
$\We{1,p}$.
Let $\eta \in C^1([0, \infty))$ be a cut-off function satisfying
\begin{displaymath}
\eta(r) = \cases{1, & r \leq 1, \\ 0, & r \geq 2.}
\end{displaymath}
For each $n \in \mathbb{N}$, let $A_n := B_{2n} \setminus B_n$ and define
\begin{displaymath}
u_n(x) := \eta(|x|/n)\, \big(u(x) - (u)_{A_n}\big).
\end{displaymath}
Hence,
\begin{displaymath}
\partial u_n(x) = n^{-1}\, \eta'(|x|/n)\, \smfrac{x}{|x|}\, \big(u - (u)_{A_n}\big) +
\eta(|x|/n)\, \partial u.
\end{displaymath}
Since $u \in W^{1,p}_{\rm loc}$ and $u_n$ has compact support, it is
clear that $u_n \in W^{1,p}$. Further, since $\eta'$ is uniformly bounded, we can estimate
\begin{align*}
\| \partial u - \partial u_n \|_{L^p}
%
\leq~& \big\| n^{-1} \eta' (u - (u)_{A_n}) \big\|_{L^p}
+ \big\| (1 - \eta) \partial u \|_{L^p} \\
%
\leq~& C n^{-1} \| u - (u)_{A_n} \|_{L^p(A_n)}
+ \| \partial u \|_{L^p(\mathbb{R}^d \setminus B_n)}.
\end{align*}
Poincar\'{e}'s inequality on $A_1$ and a standard scaling argument
then imply that
\begin{displaymath}
C n^{-1} \| u - (u)_{A_n} \|_{L^p(A_n)} \leq (C n^{-1}) (C_P n) \|
\partial u \|_{L^p(A_n)} \leq C \| \partial u \|_{L^p(\mathbb{R}^d \setminus B_n)},
\end{displaymath}
that is, $\| \partial u - \partial u_n \|_{L^p} \leq C \| \partial u \|_{L^p(\mathbb{R}^d
\setminus B_n)}$. Since $\|\partial u \|_{L^p}$ is finite it follows
that this upper bound tends to zero as $n \to \infty$.
Hence, we have constructed a sequence $(u_n) \subset W^{1,p}$ such
that $\partial u_n \to \partial u$ in $L^p$, or, equivalently $[u_n] \to [u]$ in
$\We{1,p}$.
If $d = 1$, then $A_n$ is not simply connected and hence the
Poincar\'e inequality does not hold. Instead, we prove that for any
$u \in W^{1,p}_{\rm loc}$ with $u' = \chi_{(a, b)}$ (the
characteristic function of an interval) we can construct a sequence
$[u_n] \in \dot{\DD}$ approching $[u]$. Density of the span of
characteristic functions in $L^p$ then implies the stated result for
$d = 1$. Let $u_n$ be defined by
\begin{displaymath}
u_n'(x) = \cases{ 1, & x \in (a, b), \\
-1/n , & x \in (b, b+ n(b-a)), \\
0, & \text{otherwise},}
\end{displaymath}
then it is a straightforward computation to show that $u_n' \to u' =
\chi_{a,b}$ in $L^p$ for any $p > 1$, but not in $L^1$.
\end{proof}
\begin{remark}
If $d = 1$ then $\dot{\DD}$ is not dense in $\We{1,1}$. If this were the
case, then all functions $u \in \We{1,1}$ would satisfy $\int_\mathbb{R} u'
\,{\rm d}x = 0$. However, it is clear that the equivalence class of the
function $u(x) = \max(0, \min(x, 1))$ belongs to $\We{1,1}$, but
does not satisfy this condition.
\end{remark}
Our next result classifies the growth or decay of classes $[u] \in
\We{1,p}$ at infinity. Case (i) is essentially contained in
\cite[Prop. 2.4(i)]{KozSoh:1991}; cases (ii) and (iii) are new to the
best of our knowledge.
\begin{theorem}
\label{th:growth}
There exist linear maps $J_\infty : \We{1,p} \to C^\infty$ and $J_0
: \We{1,p} \to W^{1,p}$ such that
%
\[[u] = [J_\infty[u] + J_0[u]], \qquad \text{for } [u] \in \We{1,p},\]
and
\begin{align*}
\| \partial J_\infty [u] \|_{L^p} \leq \|\partial u \|_{L^p},
\quad \| \partial J_\infty[u] \|_{L^\infty} \leq \| \partial u \|_{L^p} \quad
\text{and}
\quad \| J_0[u] \|_{W^{1,p}} \leq C \| \partial u \|_{L^p},
\end{align*}
where $C = C(d) > 0$.
Moreover, $J_\infty$ may be chosen to satisfy the following growth
conditions at infinity:
\begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})}
\item If $p < d$, then $\We{1,p}$ is continuously embedded in
$L^{p*}$, in the sense that, for each $[u] \in \We{1,p}$ there exists a unique $u_0 \in
[u]$ such that $u_0 \in L^{p*}$ and $\| u_0 \|_{L^{p*}} \leq C \| \partial u_0 \|_{L^p}$, where
$C$ is a positive constant independent of $u_0$.
In particular, $J_\infty[u](x) \to 0$ as $|x| \to \infty$, $x \in \mathbb{R}^d$.
\medskip
\item If $p > d$, then $|J_\infty[u](x)| \leq C \| [u] \|_{\We{1,p}}
|x|^{1/p'}$, $x \in \mathbb{R}^d$.
\medskip
\item If $p = d$, then $|J_\infty[u](x)| \leq C \|[u] \|_{\We{1,p}}
\log (2+ |x|)$, $x \in \mathbb{R}^d$.
\end{enumerate}
\end{theorem}
\begin{proof}
We shall assume throughout that $1\leq p < \infty$; in case (ii) the
choice of $p = \infty$ can be dealt with separately using an
analogous argument to the one for $d<p<\infty$.
Let $\eta \in D$, $0 \leq \eta \leq 1$, $\int \eta(x) \,{\rm d}x = 1$, fix
$u \in [u] \in \We{1,p}$ and define
\begin{displaymath}
v := \eta \ast u \in C^\infty, \quad \text{and} \quad
w := u - v.
\end{displaymath}
By Young's inequality for convolutions, $\| \partial v \|_{L^p} \leq \| \partial
u \|_{L^p}$, and, because of the assumption that $\eta \leq 1$, it
is also straightforward to show that $\| \partial v \|_{L^\infty} \leq \|
\partial u \|_{L^p}$:
\begin{align*}
\big| \partial v(x) \big| = \bigg| \int \eta(x - z) \partial u(z) \,{\rm d}z \bigg| \leq
\bigg( \int \eta(x-z) |\partial u(z)|^p \,{\rm d}z \bigg)^{1/p}
\leq \| \partial u \|_{L^p}.
\end{align*}
Next, we show that $w \in W^{1,p}$. It follows directly from the
definition of $w$ that $\partial w = \partial u - \partial v \in L^p$. Hence, $\| \partial w
\|_{L^p} \leq \| \partial u \|_{L^p} + \| \partial v \|_{L^p} \leq 2 \| \partial u
\|_{L^p}$. To show that $w \in L^p$, let $R > 0$ be such that ${\rm
supp}\,\eta \subset B_R$. For any $\xi \in \mathbb{R}^d$ we have
\begin{align*}
\int_{B_R(\xi)} |w(x)|^p \,{\rm d}x =~& \int_{B_R(\xi)} \bigg| \int \eta(x-z) (u(z) - u(x)) \,{\rm d}z
\bigg|^p \,{\rm d}x \\
\leq~& \int_{B_R(\xi)} \int \eta(x-z) \big| u(z) - u(x) \big|^p \,{\rm d}z
\,{\rm d}x \\
\leq~& \int_{B_{2R}(\xi)} \int_{B_{2R}(\xi)} \big| u(z) - u(x) \big|^p
\,{\rm d}z \,{\rm d}x \leq C(R) \| \partial u \|_{L^p(B_{2R}(\xi))}^p,
\end{align*}
where the last inequality is an immediate consequence of
Poincar\'{e}'s inequality on the ball $B_{2R}(\xi)$. We can cover
$\mathbb{R}^d$ with countably many balls $B_R(\xi)$, $\xi \in R \mathbb{Z}^d$, such
that the balls $B_{2R}(\xi)$ have finite overlap, that is, any $x
\in \mathbb{R}^d$ belongs to at most $m$ balls where $m$ is independent of
$x$. Summing over all balls gives the result that $\| w \|_{L^p}
\leq C \| \partial u \|_{L^p}$, where $C$ may depend on the support of
$\eta$ and hence on the dimension $d$, but is independent of the value of $p$.
\smallskip
We now distinguish between three cases, depending on the values of $p$
and $d$.
{\it (i) $p < d$: } Since $\dot{\DD}$ is dense in $\We{1,p}$ and by the
Gagliardo--Nirenberg--Sobolev Inequality, it follows that $\We{1,p}$
is embedded in $L^{p*}$ in the sense that for each $[u] \in \We{1,p}$
there exists a unique $u_0 \in [u]$ such that $u_0 \in L^{p*}$ and $\|
u_0 \|_{L^{p*}} \leq C_{\rm GNS} \| \partial u_0 \|_{L^p}$, where $C_{\rm
GNS}$ is the constant in the Gagliardo--Nirenberg--Sobolev
Inequality. We define
\[J_\infty[u] := v_0 := \eta \ast u_0.\]
It is an immediate consequence of this definition that $v_0 \in
L^{p*}$. Since $\| \partial v_0 \|_{L^\infty}$ is finite, it follows also
that the sequence $( \| v_0 \|_{L^{\infty}(B_1(\xi))})_{\xi \in \mathbb{Z}^d}$
belongs to $\ell^{p*}(\mathbb{Z}^d)$, and this implies that $\| v_0
\|_{L^{\infty}(B_1(\xi))} \to 0$ uniformly as $|\xi| \to \infty$. We
obtain statement (i) as a special case.
{\it (ii) $p > d$: } In this case we define $J_\infty[u](x) := (\eta
\ast u)(x) - (\eta\ast u)(0)$ for any element $u \in [u]$, which does
not of course change the foregoing results. If we define $v := \eta
\ast u$, then we obtain
\begin{align*}
|J_\infty[u](r)| =~& |v(r) - v(0)| \leq \bigg|\int_0^{|r|} \partial v\big(t
\smfrac{r}{|r|}\big) \smfrac{r}{|r|} \,{\rm d}t \bigg| \\
\leq~& |r|^{1/p'} \bigg(\int_{0}^{|r|} \big| \partial v\big(t \smfrac{r}{|r|}
\big)\big|^p \,{\rm d}t\bigg)^{1/p} \\
\leq~& |r|^{1/p'} \bigg( \int_0^{|r|} \bigg| \int \eta\big(t
\smfrac{r}{|r|} - z\big) \partial u(z) \,{\rm d}z\bigg|^p \,{\rm d}t \bigg)^{1/p} \\
\leq~& |r|^{1/p'} \bigg( \int \bigg[ \int_0^{|r|} \eta\big(t
\smfrac{r}{|r|} - z\big) \,{\rm d}t \bigg] |\partial u(z)|^p \,{\rm d}z \bigg)^{1/p}.
\end{align*}
Since the diameter of the support of $\eta$ is independent of $|r|$ it
follows that
\begin{displaymath}
\int_0^{|r|} \eta\big(t \smfrac{r}{|r|} - z\big) \,{\rm d}t
\leq C
\end{displaymath}
for some universal constant $C$, which implies {\it (ii)}.
{\it (iii) $p = d$: } In this critical case, we use the fact that
$\We{1,p}$ may be embedded in the space BMO of functions of bounded
mean oscillation (see \cite{JN}), though for simplicity we will not
refer to BMO directly.
If $Q$ is a cube in $\mathbb{R}^d$ with arbitrary orientation and $u \in [u]
\in \We{1,p}$, then
\begin{displaymath}
\Xint-_Q \big| u - (u)_Q \big| \,{\rm d}x \leq |Q|^{1/p'-1} \| u - (u)_Q
\|_{L^p(Q)} \leq C \| \partial u \|_{L^p(Q)} \leq C \|[u]\|_{\We{1,p}},
\end{displaymath}
where the second inequality follows on noting that $|Q|^{1/p'-1} =
|Q|^{-1/d} \leq ({\rm diam} Q)^{-1}$.
The key observation is that if $u \in W^{1,p}_{\rm loc}$ with $\partial u \in
L^p$, then for each $x \in \mathbb{R}^d$ there exist unit cubes $Q_x$ centred
at $x$ and $Q_0$ centred at $0$, such that
\begin{equation}
\label{eq:bmo_argument}
\big| (u)_{Q_x} - (u)_{Q_0} \big| \leq C \| \partial u \|_{L^p} \log(2+ |x|).
\end{equation}
The proof of inequality \eqref{eq:bmo_argument} will be given below,
after the end of the proof of this theorem. With this in hand, we can
define $J_\infty[u](x) := v_0 := (\eta \ast u)(x) - (\eta \ast u)(0)$
to obtain
\begin{align*}
\big| v_0(x) \big| \leq~& \big| v_0(x) - (v_0)_{Q_x} \big| + \big| (v_0)_{Q_0}
\big|+ \big| (v_0)_{Q_x}
- (v_0)_{Q_0} \big| \\
\leq~& C\, \| \partial v_0 \|_{L^\infty} + C\, \| \partial v_0 \|_{L^\infty}
+ C \,\| \partial v_0 \|_{L^p} \log (2+|x|)
\\
\leq~& C\, \| \partial u \|_{L^p} \log(2+ |x|),
\end{align*}
for a generic (dimension-dependent) constants $C$. This concludes the
proof of case (iii).
\end{proof}
\begin{figure}
\includegraphics[width=8cm]{bmo_argument}
\caption{\label{fig:bmo_argument} Visualization of an argument used
in the proof of \eqref{eq:bmo_argument}.}
\end{figure}
\begin{proof}[Proof of \eqref{eq:bmo_argument}]
We assume without loss of generality that $|x| \geq 1$. Let $Q_0,
Q_x$ be unit cubes centred, respectively, at $0$ and $x$ such that
one set of edges of each of the cubes $Q_0$ and $Q_x$ is aligned
with the direction $\vec{0x}$. There
exists $N \leq C (2+\log|x|)$ and cubes $Q_2, \dots, Q_{N-1}$ with
the same alignment as $Q_0, Q_x$ and with disjoint interior such
that, for any two neighbouring cubes, their sidelengths differ by at
most a factor $2$ and one face of the smaller cube is contained
within one face of the large cube. See Figure \ref{fig:bmo_argument}
for a visualization of this argument.
For any two neighbouring cubes $Q_j, Q_{j+1}$ we have
\begin{displaymath}
\big| (u)_{Q_j} - (u)_{Q_{j+1}} \big| \leq C \| \partial u \|_{L^p},
\end{displaymath}
which is a special case of \cite[Lemma 2]{JN}, but can
also be verified directly by enclosing $Q_j, Q_{j+1}$ in a larger
cube of approximately the same size. Hence, defining $Q_x = Q_N$, we
obtain
\begin{displaymath}
\big| (u)_{Q_x} - (u)_{Q_0} \big| \leq \sum_{j = 0}^{N-1} \big|
(u)_{Q_{j+1}} - (u)_{Q_j} \big| \leq N \| \partial u \|_{L^p}. \qedhere
\end{displaymath}
\end{proof}
\begin{remark}
The map $J' := J_\infty + J_0$ defines an embedding of $\We{1,p}$
into $D'$. Since $J_0$ is continuous to $W^{1,p}$ and $J_\infty$
is continuous to $W^{1,\infty}_{\rm loc}$ it follows that the embedding
$J'$ of $\We{1,p}$ into $\dot{\DD}$ is in fact continuous. However, it
is not particularly useful for our purposes since we are explicitly
interested in operations that are translation invariant, that is,
independent of the representative $u \in [u]$, whenever $[u] \in
\We{1,p}$.
\end{remark}
\section{Well-posedness and regularity}
From now on we restrict our presentation to the case $p = 2$ and hence
define $\He{1} := \We{1,2}$. Since we will take particular care that
all operators, linear functionals, and bilinear forms we consider are
translation invariant, we will drop the brackets in $[u] \in \He{1}$
and instead write simply $u \in \He{1}$ instead, by which we mean an
arbitrary representative from the class $[u]$. (For convenience one
may take $u = J_\infty[u] + J_0[u]$.)
Since we consider elliptic systems, we will from now on identify all
function spaces with spaces of vector-valued functions, that is, $L^p
= (L^p)^m$, $\He{1} = (\He{1})^m$, and so forth, for some fixed $m \in
\mathbb{N}$.
Before we embark on the analysis of the elliptic system
\eqref{eq:ellsys_strong} we briefly discuss admissible right-hand
sides $f$ for \eqref{eq:ellsys_strong} as well as the far-field
boundary condition \eqref{eq:farfield_bc_formal}.
\subsection{The dual of $\He{1}$}
\label{sec:dual}
We denote the topological dual of $\He{1}$ by $\He{-1}$. Since
$\He{1}$ is a Hilbert space with inner product $(\partial \cdot, \partial
\cdot)_{L^2}$ it follows that, for each $\ell \in \He{-1}$, there
exists $F \in L^2$ such that $\ell = -{\rm div} F$ in the
distributional sense. (For a generalisation of this result to
$\We{-1,p}$, $p \in (1,\infty)$ see \cite[Lemma 2.2]{KozSoh:1991}.)
If we wish to define $\ell$ via an $L^2$-pairing, then the following
two examples give concrete conditions:
\begin{enumerate}
\item Let $f \in L^1_{\rm loc}$; then we can define $\ell : \dot{\DD} \to \mathbb{R}$ by
\begin{equation}
\label{eq:linear_fcnl_ex1}
\ell([u]) := \int_{\mathbb{R}^d} f \cdot u^* \,{\rm d}x, \quad \text{where } u^*
\in [u], u^* \in D.
\end{equation}
If, moreover, $f = {\rm div} g$, where $g \in L^2$, then
\eqref{eq:linear_fcnl_ex1} can be extended to a bounded linear
functional on $\He{1}$.
\item More concretely, if $f \in L^1$ with $\int_{\mathbb{R}^d} f \,{\rm d}x = 0$ and
${\rm div} g = f$, $g \in L^2$, then we may define
\begin{equation}
\label{eq:linear_fcnl_ex2}
\ell([u]) := \int_{\mathbb{R}^d} f \cdot u \,{\rm d}x, \quad \text{for any } u
\in [u] \in \dot{\DD}.
\end{equation}
Again, \eqref{eq:linear_fcnl_ex2} can be extended to a bounded
linear functional on $\He{1}$.
\item Even more concretely, if $f \in L^1 \cap L^\infty$ with
$\int_{\mathbb{R}^d} f \,{\rm d}x = 0$, and $x \otimes f \in L^1$, then this is
sufficient to ensure that $\ell$ defined through
\eqref{eq:linear_fcnl_ex2} can be extended to a bounded linear
functional on $\He{1}$ (see Lemma \ref{th:linear_fcnl_ex3}
below). We note, however, that right-hand sides with such strong
decay assumptions may be more naturally treated within the framework
of weighted Sobolev spaces \cite{Kufner}.
\end{enumerate}
\begin{lemma}
\label{th:linear_fcnl_ex3}
Suppose that $f \in L^1 \cap L^2$ with $\int_{\mathbb{R}^d} f \,{\rm d}x = 0$,
$f \otimes x \in L^1$, and let $\ell : \dot{\DD} \to \mathbb{R}$ be defined
through \eqref{eq:linear_fcnl_ex2}; then
\begin{displaymath}
\ell(u) \leq C \| \partial u \|_{L^2} \qquad \forall u \in \dot{\DD}.
\end{displaymath}
\end{lemma}
\begin{proof}
Consider the Fourier transform of $f$, which is defined in a
pointwise sense since $f \in L^1$:
\begin{displaymath}
\hat{f}(k) = \int f(x) \exp(-i k \cdot x) \,{\rm d} k.
\end{displaymath}
Taking the formal derivative with respect to $k$ we obtain
\begin{displaymath}
\partial \hat{f}(k) = - i \int f(x) \otimes x \exp(-i k \cdot x) \,{\rm d} k.
\end{displaymath}
If $f \otimes x \in L^1$ then Lebesgue's differentiation theorem can
be used to make this rigorous. Hence we deduce that $\hat{f} \in
W^{1,\infty}$. Therefore, since $\hat{f}(0) = 0$, it follows that
$\hat{f}(k)/|k|$ is bounded as $k \to 0$. Because $f \in L^2$, it
follows that $\hat{f} \in L^2 \cap L^\infty$, from which is follows
easily that $\hat{f}(k)/|k| \in L^2$.
\end{proof}
\subsection{The far-field boundary condition}
In this section we interpret the far-field boundary condition
\eqref{eq:farfield_bc_formal} by showing that the space $\He{1}$ is a
natural ansatz space to make this condition rigorous. A simple
motivation for selecting $\He{1}$ as space of functions in which
a solution to \eqref{eq:ellsys_strong}, \eqref{eq:farfield_bc_formal}
is sought, is that this space can be understood as the closure of $\dot{\DD}$ in an
``energy-norm''. However, we can give a finer interpretation
of \eqref{eq:farfield_bc_formal} by employing Theorem~\ref{th:growth}.
Let $m = d$. Suppose that an elastic body occupies the reference
domain $\mathbb{R}^d$. Deformations of $\mathbb{R}^d$ are sufficiently smooth
invertible maps $y : \mathbb{R}^d \to \mathbb{R}^d$. Suppose we apply a far-field
boundary condition
\begin{equation}
\label{eq:farfield_bc_defm}
y(x) \sim {\sf A} x \quad \text{ as } |x| \to \infty
\end{equation}
for some non-singular matrix ${\sf A} \in \mathbb{R}^{d \times d}$, which is
usually understood to mean
\begin{displaymath}
y(x) = {\sf A} x + o(|x|), \quad \text{or,
equivalently} \quad \frac{|y(x) - {\sf A} x|}{|x|} \to 0 \qquad
\text{as } |x| \to \infty.
\end{displaymath}
Suppose now that we decompose $y(x) = {\sf A} x + u(x)$; then, the
far-field boundary condition \eqref{eq:farfield_bc_defm} for the
deformation, written in terms of the displacement $u$, becomes
\begin{equation}
\label{eq:farfield_bc_u_v2}
u(x) = o(|x|), \quad \text{or, equivalently,} \quad
\frac{|u(x)|}{|x|} \to 0
\qquad \text{as } |x| \to \infty.
\end{equation}
While the pointwise condition \eqref{eq:farfield_bc_u_v2} cannot
be satisfied for classes of Sobolev functions, Theorem
\ref{th:growth} indicates that \eqref{eq:farfield_bc_u_v2}
is satisfied ``on average'' for the representative $J_\infty[u] +
J_0[u]$, for all $[u] \in \He{1}$. Hence it is reasonable to take
$\He{1}$ as the function space in which a solution to
\eqref{eq:ellsys_strong} is sought subject to the far-field displacement
boundary condition \eqref{eq:farfield_bc_formal}.
\subsection{Weak form and well-posedness}
Let $\mathbb{C} = (\mathbb{C}_{i\alpha}^{j\beta})_{i,j = 1, \dots, m}^{\alpha, \beta
= 1, \dots, d} \in (L^\infty)^{m^2d^2}$; we then define the
symmetric bilinear form $a : \He{1} \times \He{1} \to \mathbb{R}$,
\begin{displaymath}
a(u, v) := \int \mathbb{C}_{i\alpha}^{j\beta} \partial_\alpha u_i \partial_\beta
u_j \,{\rm d}x;
\end{displaymath}
where, here and throughout, we employ the summation
convention. Clearly, $a$ is bounded,
\begin{displaymath}
a(u, v) \leq c_1 \| \partial u \|_{L^2} \| \partial v \|_{L^2} \qquad
\forall u, v \in \He{1},
\end{displaymath}
where $c_1 = \|\mathbb{C}\|_{L^\infty}$, hence we can pose
\eqref{eq:ellsys_strong}, \eqref{eq:farfield_bc_formal} in weak form:
\begin{equation}
\label{eq:weak_form}
a(u, v) = \ell(v) \qquad \forall v \in D(\mathbb{R}^d; \mathbb{R}^m),
\end{equation}
where $\ell$ is of the form of Example 1 discussed in
\S~\ref{sec:dual}. An application of the Lax--Milgram theorem gives
the following result.
\begin{theorem}
\label{th:wellposedness}
Suppose that $a$ is also coercive:
\begin{equation}
\label{eq:coercivity}
a(u, u) \geq c_0 \| \partial u \|_{L^2}^2 \qquad \forall u \in
\He{1}(\mathbb{R}^d; \mathbb{R}^m),
\end{equation}
for some constant $c_0 > 0$; then, \eqref{eq:weak_form} possesses a
unique solution.
\end{theorem}
\medskip We present three elementary examples of coercivity
\eqref{eq:coercivity}:
\begin{enumerate}
\item If $m = 1$ and ${\sf C} := (\mathbb{C}_\alpha^\beta)_{\alpha, \beta = 1,
\dots, d}$ is uniformly positive definite, i.e., $k^{\rm T} {\sf C}(x) k
\geq c_0 |k|^2$ for a.e. $x \in \mathbb{R}^d$ and for all $k \in \mathbb{R}^d$,
then \eqref{eq:coercivity} holds.
\item If $m \in \mathbb{N}$, $\mathbb{C}$ is a constant tensor and satisfies the
Legendre--Hadamard condition
\begin{equation}
\label{eq:LH}
\mathbb{C}_{i\alpha}^{j\beta} v_i v_j k_\alpha k_\beta \geq c_0 |v|^2
|k|^2 \qquad \forall v \in \mathbb{R}^m, \quad k \in \mathbb{R}^d,
\end{equation}
then \eqref{eq:coercivity} holds.
This result is classical if $u \in D$. Since $a$ is translation
invariant it follows that it also holds for all $u \in \dot{\DD}$. Since
$a$ is bounded and $\dot{\DD}$ is dense in $\He{1}$ coercivity holds also
in the full space $\He{1}$.
\item Let $\bar{\mathbb{C}} \in \mathbb{R}^{d^2m^2}$ be a constant tensor satisfying
\eqref{eq:LH} with $c_0 = \bar{c}_0$; then, \eqref{eq:coercivity}
holds with $c_0 = {\bar{c}}_0 - c_1 \| \bar{\mathbb{C}} - \mathbb{C} \|_{L^\infty}$.
\end{enumerate}
\begin{remark}
One may give more general conditions for coercivity of $a$ (or inf-sup conditions)
based on G\aa rding's inequality and conditions on the $L^2$-spectrum of $a$.
\end{remark}
\subsection{Regularity}
Higher regularity of the right-hand side leads to higher
regularity of the solution to \eqref{eq:weak_form}. For $s \in \{2, 3,
\dots\}$ we define
\begin{displaymath}
\He{s} := \big\{ u \in \He{1} \,\b|\, \partial u \in H^{s-1} \big\}.
\end{displaymath}
In the following theorem we present conditions for $\He{2}$ and
$\He{3}$ regularity. Regularity in $\He{s}$ for $s \geq 4$ can be
established similarly.
\begin{theorem}
\label{th:regularity}
Let all conditions of Theorem \ref{th:wellposedness} be satisfied,
and let $u$ denote the unique solution to \eqref{eq:weak_form}.
\begin{enumerate}\renewcommand{\labelenumi}{(\roman{enumi})}
\item Suppose, in addition, that $\mathbb{C} \in C^{1}$ and $f \in L^2$;
then, $u \in \He{2}$ and
\begin{displaymath}
\| \partial^2 u \|_{L^2} \leq C \big( \| f \|_{L^2} + c_2 \| \partial \mathbb{C}
\|_{L^\infty} \big).
\end{displaymath}
\item Suppose, in addition, that $\mathbb{C} \in C^2$, $\partial\mathbb{C} \in L^2$, and
$f \in H^1$; then, $u \in \He{3}$ and
\begin{displaymath}
\| \partial^3 u \|_{L^2} \leq C \big( \| \partial f \|_{L^2} +
\| \partial \mathbb{C} \|_{L^2} \| \partial^2 u \|_{L^2}
+ c_2 \| \partial^2 \mathbb{C} \|_{L^\infty} \big).
\end{displaymath}
\end{enumerate}
\end{theorem}
\begin{proof}
Let $u_* := J_\infty[u] + J_0[u]$ denote a concrete representative
of the solution $[u] \in \He{1}$ of the weak form
\eqref{eq:weak_form}. Then clearly $u_*$ satisfies
\eqref{eq:weak_form}. The finite difference technique \cite[Section
6.3.1]{evans} ensures that $u_* \in H^{s+2}_{\rm loc}$, and in particular
$\partial u_* \in H^{s+1}_{\rm loc}$. The latter property is independent of the
representative; hence we may say that $\partial u \in H^{s+1}_{\rm loc}$.
To obtain the global bound {\it (i)}, we test \eqref{eq:weak_form}
with $v' = \partial_\gamma v$ for some $v \in D$, $\gamma \in \{1,\dots,
d\}$. Then,
\begin{align*}
\int f \cdot \partial_\gamma v \,{\rm d}x =~& \int \mathbb{C}_{i\alpha}^{j\beta}
\partial_\alpha u_i \partial_\beta \partial_\gamma v_j \,{\rm d}x \\
=~& - \int \Big( \partial_\gamma \mathbb{C}_{i\alpha}^{j\beta} \partial_\alpha u_i +
\mathbb{C}_{i\alpha}^{j\beta} \partial_\alpha \partial_\gamma u_i\Big) \partial_\beta v_j \,{\rm d}x,
\end{align*}
which implies that
\begin{displaymath}
c_0 \| \partial(\partial_\gamma u) \|_{L^2} \leq \| f \|_{L^2} + \| \partial \mathbb{C}
\|_{L^\infty} \| \partial u \|_{L^2} \leq \| f \|_{L^2} + c_2 \| \partial \mathbb{C}
\|_{L^\infty}.
\end{displaymath}
To prove {\it (ii)} we test with $v' = \partial_\gamma \partial_\delta v$ and
perform a similar calculation.
\end{proof}
\bibliographystyle{siam}
|
1,108,101,565,491 | arxiv | \section{Introduction}
In this note we develop a new approach to certain results of Cheeger, Colding and Tian involving the curvature of Riemannian manifolds close, in the Gromov-Hausdorff sense, to a singular limit. In one aspect our results go a little beyond those in the literature but the main interest, for us, is that the arguments are substantially different and may be more amenable to certain generalisations.
Let $n,k$ be positive integers with $n\geq k>2$. Let $\Gamma\subset SO(k)$ be a non-trivial finite group which acts freely on the unit sphere $S^{k-1}$. Write $M_{n,\Gamma}$ for the singular space $\left( \bR^{k}/\Gamma \right)\times R^{n-k}$. Let $B_{n,\Gamma}$ be the unit ball centred at the equivalence class of $(0,0)$. In other words, $B_{n,\Gamma}$ is the quotient of the unit ball in $\bR^{n}$ by the action of $\Gamma$, embedded in $SO(n)$ in the obvious way.
We consider a complete Riemannian manifold $(X^{n},g)$, with bounded Ricci curvature: $\vert \Ric\vert\leq (n-1)$. (In practice $\vert \Ric \vert$ will usually be very small, by re-scaling.) We assume that we have a non-collapsing condition:
\begin{equation} \Vol (B(x,r))\geq C^{-1} r^{n} \end{equation}
for all balls in $X$, when $r$ is less than the diameter of $X$. Then we have
\begin{thm}
There are $\delta=\delta(n,C)>0$ and $\zeta=\zeta(n,C)>0$ such that if the Gromov-Hausdorff distance from a unit ball $B(x_{0},1)$ in $X$ to $B_{n,\Gamma}$ is less than $\delta$ then
$$ \int_{B(x_{0},1)} \vert \Riem \vert^{k/2} \geq \zeta. $$
\end{thm}
Next let $0<\beta<1$ and let $\bR^{2}_{\beta}$ be the standard cone with cone angle $2\pi \beta$. Thus in polar co-ordinates the metric is
$dr^{2}+ \beta^{2} r^{2} d\theta^{2}$. For $n\geq 2$ let $N_{n,\beta}$ be the product $\bR^{2}_{\beta}\times \bR^{n-2}$ and let $B_{n,\beta}$ be the unit ball centred at the origin, in the obvious sense.
\begin{thm}
There are $\delta=\delta(n,C,\beta)>0$ and $\zeta=\zeta(n,C,\beta)>0$ such that if the Gromov-Hausdorff distance from a unit ball $B(x_{0},1)$ in $X$ to $B_{n,\beta}$ is less than $\delta$ then
$$ \int_{B(x_{0},1)} \vert \Riem \vert \geq \zeta. $$
\end{thm}
In the case of K\"ahler metrics, Theorem 1 was proved by Cheeger, Colding and Tian in \cite{kn:CCT}. They also establish the result in the real case for many groups $\Gamma$. See also results of Cheeger and Tian in \cite{kn:CT}. In the co-dimension 2 case (Theorem 2), Cheeger proved in \cite{kn:C} a significantly stronger result than that stated here, involving only a lower bound on the Ricci curvature. In fact a conjecture of Anderson \cite{kn:A2} would, if true, mean that, in Theorem 2, if $\delta$ is sufficiently small there is no such ball in any $X$. It seems possible to us that the approach here could be extended to prove the analogues of Theorem 1 and 2 for metrics with Ricci curvature bounded below but this certainly involves significant new problems.
Our proof of Theorem 1 will use the co-dimension 2 result, Theorem 2. We also sketch a proof of Theorem 2, which is similar but involves some extra complications. So we will discuss Theorem 1 first, in Sections 2, 3 and 4 below, and then discuss Theorem 2 in Section 5. Of course we can also quote what we need from \cite{kn:C}.
The central notion in our proof is that of a point $x$ in $X$ around which the \lq\lq energy is small at all scales''. For any $r$ we set
$$ E(x,r)= r^{2-n} \int_{B(x,r)} \vert \Riem\vert. $$
To explain the point of the definition: given a ball $B(x,r)$ write $B(x,r)^{\sharp}$ for the same ball with the metric (i.e lengths) scaled by a factor $r^{-1}$. So $B(x,r)^{\sharp}$ is a unit ball in the new metric. Then $E(x,r)$ is the $L^{1}$ norm of the curvature of $B(x,r)^{\sharp}$. Set $\overline{E}(x)=\max_{r\leq 2} E(x,r)$. (The restriction to $r\leq 2$ is just because we are concerned with the local picture.) For $\epsilon>0$ let $\cA_{\epsilon}\subset X$ be the set of points $x$ where $\overline{E}(x)\leq \epsilon$. Thus
$\cA_{\epsilon}$ is the set of points where \lq\lq the energy is less than $\epsilon$ at all scales''(or, more precisely, at all small scales). For the proof of Theorem 1 we could work with a similar definition using $L^{k/2}$ norms. A disadvantage of that is that {\it a posteriori} the set corresponding to $\cA_{\epsilon}$ would be empty for small enough $\epsilon$ and the statements we prove about it would be vacuous. That is one reason why we prefer to use $L^{1}$. (A similar issue arises in the case of Theorem 2, see the discussion at the end of Section 5 below).
With this background we can attempt to explain the central idea of our proof in very vague informal terms, but perhaps sufficient that an expert could reconstruct the arguments. Suppose $\delta$ is very small and we start at distance roughly $1$ from the \lq\lq approximately singular set''. Then the metric appears very close to a flat cone with a codimension $k$ singularity. As we travel towards the the approximately singular set, viewing at smaller and smaller scales, the apparent singularity must eventually resolve into a smooth metric and at this scale we must see some curvature, measured in $L^{1}$ norm on a ball of the appropriate scale. Conversely at a point in $\cA_{\epsilon}$ where the energy is small at all scales we still see the same apparent singularity--or a flat space--at every scale. That is, staying in $\cA_{\epsilon}$ we can never reach the apparent singularity. This implies that the complement of $A_{\epsilon}$ must be \lq\lq large'' in a certain topological sense and that yields the desired lower bound by a standard covering argument.
We should emphasise that while our proofs, in so much as they are new, are elementary and self-contained they depend strongly on deep foundational results in this area due to Anderson, Cheeger, Colding and Tian.
\section{Set up and strategy}
We put some notation and conventions in place for our later arguments. First to simplify exposition slightly we will first give the proof of Theorem 1 in the case of an Einstein manifold $X$, so $\Ric=\lambda g$ with $\vert \lambda\vert \leq 1$. Then in Section 4.2 we explain the arguments needed to extend to bounded Ricci curvature.
We suppose throughout Sections 3 and 4 that we have some fixed $\Gamma\subset SO(k)$. (For fixed $n$ and $C$ there are only finitely many possible choices of $\Gamma$, up to conjugacy.) Note first that the complete manifold $X$---far away from the point $x_{0}$---- plays no real role since all our arguments will be local. Similarly there is no loss in supposing that a substantially larger ball $B(x_{0}, K)$ (with $K=10$, say) is also Gromov-Hausdorff close to the $K$-ball in the model. This is because we could always restrict attention to a smaller ball and re-scale. With this understood, we do not need to distinguish between the distance function in $B(x_{0},1)$ and that induced from the metric in $X$.
We write a point in $M_{n,\Gamma}$ as a pair $(\xi,\eta)$ with $\xi\in \bR^{k}/\Gamma$ and $\eta\in \bR^{n-k}$. We write $\vert \xi\vert$ with the obvious meaning. For $s>0$, let $H_{s}$ be the set
$$ H_{s}= \{ (\xi,\eta): \vert \xi \vert^{2}+\vert \eta\vert^{2}\leq 1-s \ , \ \vert \xi \vert\geq s\}. $$
Let $\Omega\subset H_{s}$ be the region where $\vert \eta\vert \leq (1/10)+ \vert \xi\vert/5$. Thus $\Omega$ meets the outer boundary $\vert \xi\vert^{2}+\vert\eta\vert^{2}=1-s$ in an \lq\lq equatorial region'' $E$. The crucial property is that $\xi$ is never zero on $E$.
Now recall that to say that the Gromov-Hausdorff distance between $B(x_{0},1)$ and $B_{n,\Gamma}$ is less than $\delta$ is to say that there is a metric on the disjoint union which extends the given metrics on the two subsets and such that the $\delta$-neighbourhood of either subset is the whole of the union. In our situation we can use a much more concrete notion. Define an
{\it $s$-chart} at $x_{0}$ to be a map $\chi_{s}: H_{s}\rightarrow B(x_{0},1)$ with the following properties
\begin{itemize}
\item $\chi_{s}$ is a diffeomorphism to its image.
\item The pull-back of the metric $g$ differs in $C^{2}$ norm from the given flat metric on $H_{s}$ by at most $s$.
\item $ B(x_{0},1)$ is contained in the $2s$ neighbourhood of the image of $\chi_{s}$.
\end{itemize}
Then under our hypotheses it is a fact that for any $s$ there is an $s$-chart if $\delta$ is small enough. This depends on a great deal of deep theory, notably Anderson's results on the volume ratio \cite{kn:A} and Colding's result on volume convergence \cite{kn:Co}. We refer to \cite{kn:C2}, \cite{kn:C3} for accounts of the whole theory. Of course an $s$-chart is not unique but it is essentially unique up to the isometries of $B_{n,\Gamma}$ and a small arbitrary error. Recall that for the time being we are considering Einstein metrics. This means that in the second item we could replace $C^{2}$ by $C^{r}$ for any $r$ using elliptic regularity of the Einstein equations in local harmonic co-ordinates.
Given an $s$-chart, let $\Omega'\subset B(x_{0}, 1)$ be the union of
$\chi_{s}(\Omega)$ and the ball $B(x_{0},1/10)$. This has an \lq\lq outer boundary''given by $\chi_{s}(E)$. A moments thought will show the reader that, in proving our main theorem, there is no loss of generality in assuming, for any given $\epsilon$, that $\chi_{s}(E)$ lies in $\cA_{\epsilon}$. (Here again we use the freedom to restrict to a smaller ball in the original problem.)
With all this preparation we can state the central result in our proof.
\begin{prop}
There is an $s_{0}$ and for all $s\leq s_{0}$ an $\epsilon(s)$ such that if an $s$-chart exists and $\epsilon<\epsilon(s)$ then there is a continuous retraction $R: \Omega'\cap \cA_{\epsilon}\rightarrow \chi_{s}(E)$ which restricts
to the identity map on $\chi_{s}(E)$.
\end{prop}
This is, in precise language, what we mean by saying that \lq\lq staying in $\cA_{\epsilon}$ we can never reach the singularity''.
In the remainder of this section we will explain the proof that Proposition 1 implies Theorem 1. This involves the construction of \lq\lq slices''. What we need is quite standard and elementary. (There is some similarity here with the approach of Cheeger, Colding and Tian but they need a much more sophisticated slicing, controlling higher derivatives.) Recall that we are supposing that the metric is Gromov-Hausdorff close, say distance $\delta$, to the flat model on the much larger ball $B(x_{0}, K)$.
For $i=1,\dots, n-k$ let $p_{i}$ be the point in $\bR^{n-k}$ with ith. coordinate $K$ and all others $0$. Choose a point $q_{i}\in B(x_{0},K)$ which is a distance less than $\delta$ from $(0,p_{i})$. Let $f_{i}$ be the function on $B(x_{0},1)$
$$ f_{i}(x)= d(x_{0}, q_{i})- d(x,q_{i})$$ and let $\uf:B_{x_{0}}(1)\rightarrow \bR^{n-k}$ be the map with components $f_{i}$. Clearly for the corresponding construction in the flat model this map approaches the projection from $\left(\bR^{k}/\Gamma\right)\times \bR^{n-k}$ to $\bR^{n-k}$ as $K$ tends to infinity. It follows that if $K$ is sufficiently large (how large being something one can determine by elementary geometry in the flat model) then we can find a $\alpha>0$ such that if we have an s-chart with sufficiently small $s$ then for any $\eta$ with $\vert \eta\vert \leq \alpha$ the fibre $\uf^{-1}(\eta)$ is contained in $\Omega'$.
The function $\uf$ is Lipschitz, with Lipschitz constant $1$. It is then standard that we can find a smooth map $\tuf$, arbitrarily close to $f$ in $C^{0}$ with derivative bounded in operator norm by $2$ say (or any number bigger than $1$). It is also clear that we can suppose that near the boundary region $\chi_{s}(E)$ the composite $\tuf \circ \chi_{s}$ is exactly equal to the projection map from $\left( \bR^{k}/\Gamma\right)\times
\bR^{n-k}$ to $\bR^{n-k}$ and that fibres $\tuf^{-1}(\eta)$ are contained in $\Omega'$, for $\vert \eta\vert\leq \alpha$.
Now let $\pi:\chi_{s}(E)\rightarrow \bR^{k}/\Gamma$ be the composite of $\chi_{s}^{-1}$ with the map $(\xi,\eta)\mapsto \xi/\vert \xi\vert$. For almost all $\eta$ with $\vert \eta\vert \leq \alpha$ the fibre $\tuf^{-1}(\eta)$ is a compact $k$ manifold with boundary and $\pi$ yields a diffeomorphism from the boundary to $S^{k-1}/\Gamma$. This means that $\tuf^{-1}(\eta)$ cannot be contained in $\cA_{\epsilon}$, for then the map $R\circ \pi$ would lead to a retraction of $\tuf^{-1}(\eta)$ onto its boundary. To sum up, we have established the following.
\begin{prop}
For suitable choice of $\delta,\epsilon$ there is an $\alpha>0$ and a smooth map $\tuf$ from $B_{x_{0}}(1)$ to $\bR^{n-k}$ with derivative bounded by $2$ and which maps the complement of $\cA_{\epsilon}$ onto the $\alpha$-ball in $\bR^{n-k}$.
\end{prop}
Now the proof is completed by a standard Vitali covering argument(as in \cite{kn:S}, Section 1.6 for example).
Call a ball $B(x,r)$ with $E(x,r)>\epsilon$ a \lq\lq high energy ball''.
We successively choose high energy balls $B_{1}, B_{2},\dots $ such that at stage $i$ the ball $B_{i}=B(x_{i}, r_{i})$ is disjoint from $B_{1}\dots, B_{i-1}$ and has maximal radius among all such possibilities. (If there are no such possibilities we stop.) Directly from the definition, any point of the complement of $\cA_{\epsilon}$ is the centre of some high energy ball. Then the selection scheme implies that the complement of $\cA_{\epsilon}$ is contained in the union of the closures of the twice-sized balls $B(x_{i}, 2r_{i})$.
Thus $\tuf$ maps $\bigcup_{i}\overline{ B(x_{i}, 2r_{i})}$ onto the $\alpha$- ball in $\bR^{n-k}$. Write $\omega_{n-k}$ for the volume of the unit ball in $\bR^{n-k}$. Then the derivative bound on $\tuf$ implies that the volume of $\tuf B(x_{i}, 2r_{i})$ is at most $\omega_{n-k} (2 r_{i})^{n-k}$. Since the volume of the $\alpha$ ball is $\omega_{n-k} \alpha^{n-k}$, we get
\begin{equation} 2^{n-k} \sum r_{i}^{n-k} \geq \alpha^{n-k} . \end{equation}
On the other hand since the balls $B_{i}$ are disjoint we have
$$ \sum \int_{B_{i}}\vert \Riem\vert^{k/2} \leq \int_{B_{x_{0}}(1)} \vert \Riem \vert^{k/2}. $$
By the definition of a high energy ball
$$ \int_{B_{i}} \vert \Riem \vert \geq \epsilon r_{i}^{n-2}. $$
The bound on the Ricci curvature implies an upper bound on the volume of each of these balls, say $\Vol(B_{i})\leq c r_{i}^{n}$. Then H\"older's inequality gives
$$ \int_{B_{i}} \vert \Riem \vert^{k/2} \geq c^{1-k/2}\epsilon^{k/2} r_{i}^{n-k}. $$
Using (2), we conclude that
$$ \int_{B(x_{0},1)} \vert \Riem \vert^{k/2}\geq c^{1-k/2}\epsilon \alpha^{n-k} 2^{k-n}. $$
\section{The main proof}
\subsection{Strategy}
The purpose of this Section is to prove Proposition 1. Our approach is to construct a vector field $v$ on $B(x_{0}, 1)$ such that the integral curve of $v$ starting at any point $x$ in $\cA_{\epsilon}\cap \Omega'$ hits $\chi_{s}(E)$ after some finite time and then define $R(x)$ to be the hitting point. In terms of our chosen $s$-chart $\chi_{s}$ on $B(x_{0},1)$ it is obvious how one might do this if one only considers points $x$ which also lie in the image $\chi_{s}(H_{s})$.
That is, in the flat model, define $\partial_{r}$ to be the unit radial vector field , given by the gradient of the function $\vert \xi\vert$. Then we could simply use the push forward of $\partial_{ r}$ under $\chi_{s}$. The whole problem is to show that we can extend this definition---or something like it---outside $\chi_{s}(H_{s})$.
This divides into two parts. One is to show that for a point in $\cA_{\epsilon}$ we see essentially the same picture at any scale. The ingredients for that are developed in this subsection 3.1, except for the proof of Theorem 3 below, which we postpone to Section 4. The other part is to devise some method of defining the vector field $v$ which, roughly speaking, looks like $\partial_{r}$ at all scales. The ingredients for that are developed in 3.2. In 3.3 we bring these ideas together to prove Proposition 1
We will say that a space $Z$ is an {\it $L^{1}$-flat limit ball} if \begin{itemize}
\item Z is the Gromov-Hausdorff limit of unit balls $B(p_{i}, 1)\subset X_{i}$ where $X_{i}$ are Einstein $n$-manifolds satisfying the conditions we considered in Section 1 (with a fixed $C$ in (1));
\item For some $r>1$
$$ \int_{B(p_{i}, r)} \vert \Riem \vert \rightarrow 0 , $$
as $i\rightarrow \infty$.
\end{itemize}
Thus we know that an open dense subset $Z_{\reg}$ of $Z$ has
a flat Riemannian metric. The definition on a $s$-chart extends in the obvious way to this situation---we just require that $\chi_{s}$ maps in to the smooth subset. We have the following rigidity result, proved in Section 4.
\begin{thm}
Suppose $B$ is an $L^{1}$-flat limit ball which admits an $s$-chart. If $s$ is sufficiently small then $B$ is isometric to a ball in $M_{\Gamma}$.
\end{thm}
Let $\Sigma$ be the set of points of the form $(\xi,0)$ in $H_{s}\subset M_{\Gamma,n}$. (Thus, roughly speaking, $\Sigma$ is a transversal to the singular set.) Recall that we are using the notation $B^{\sharp}$ for rescaled balls and we will write the rescaled metric as $d^{\sharp}$.
\begin{prop}
Suppose $B(p,1)$ is a ball in the manifold $X$ which admits an $s$-chart and let $q$ be a point with $d(q,p)\leq 1/10$. For $s$ sufficiently small there is an $\epsilon(s)>0$ such that if the $E(p,1)<\epsilon(s)$then there is a point $p'\in B(p,1)$ with $d(p',q)<1/2$ and such that the rescaled ball $B(p',1/2)^{\sharp}$ admits an $s$ chart $\chi_{s}$. This $s$-chart can be chosen such that if $d^{\sharp}(p',q)\geq 1/10$ then $q$ lies in the image $\chi_{s}(\Sigma)$.
\end{prop}
It is easy to see here that $d^{\sharp}(p',q)$ cannot be more than $1/5+O(s)$.
This proposition expresses, in precise language, what we mean by saying that
\lq\lq at a point in $\cA_{\epsilon}$ \dots we still see the same apparent singularity, or a flat space, at every scale''.
To prove this proposition we argue by contradiction. We can suppose that $s$ is less than the value given by Theorem 3. If the statement fails we get a sequence of such balls with $L^{1}$ norm of the curvature tending to zero. We get a flat limit ball
which admits an $s$-chart and hence, by the theorem, is itself isometric to a ball in $M_{\Gamma,n}$. Thus the half-sized ball in the limit is isometric to a ball in $M_{\Gamma,n}$ and this gives a contradiction. The condition that $q$ lies in the image of $\chi(\Sigma)$, if $d^{\sharp}(q,p')>1/10$ is achieved by making a small translation in the $\bR^{n-k}$ factor in $M_{\Gamma,n}$.
\subsection{The lasso function}
Given any ball $B\subset B(x_{0},1)$ such that the normalised ball $B^{\sharp}$ admits an $s$ chart, for some small $s$, it is clear, as we said in the previous subsection, how we could define an appropriate vector field on the image of the s-chart in $B$. The problem is that when $B$ is a tiny ball not contained in the image of the $s$-chart in $B(x_{0},1)$ (which is the case of interest) there is no {\it a priori} notion of compatability between the $s$ chart in $B$ and the s-chart in $B(x_{0},1)$. One way around this might be to build up a global vector field by a patching construction but instead we prefer to concoct a global definition and show that it behaves in the right way at every scale. This definition requires a digression.
\begin{defn}
Let $V$ be a complete Riemannian manifold and fix $\lambda>0$. For $p\in V$ a lasso based at $p$ is a pair $(\gamma_{0}, \gamma_{1})$ where $\gamma_{i}:[0,l_{i}]\rightarrow V$ are geodesics segments parametrised by arc length such that
\begin{itemize}
\item $l_{1}>0$;
\item $\gamma_{0}(0)=p$;
\item $\gamma_{0}(l_{0})=\gamma_{1}(0)=\gamma_{1}(l_{1})$.
\end{itemize}
We call $\gamma_{0}$ the {\it knot} of the lasso and define the $\lambda$-length of the lasso to be $l_{0}+ \lambda l_{1}$. If $l_{0}>0$ we will say the lasso is a {\it genuine lasso}.
\end{defn}
If $V$ is compact (say) it is clear that the set of lassos based at $p$ is not empty and we thus have a function $\rho_{\lambda}(p)$ defined by the infimum of the $\lambda$-length. It follows immediately from the definition that $\rho_{\lambda}$ is Lipschitz, with Lipschitz constant $1$. It is also clear that the infimum is realised by at least one {\it minimising lasso}. For a genuine lasso, the minimising condition implies (by the usual first variation formula about geodesics) that
\begin{equation} \gamma_{0}'(l_{0}) = \lambda\left( \gamma_{1}'(0)- \gamma_{1}'(l_{1})\right). \end{equation}
Thus the angle between the two tangent vectors of $\gamma_{1}$ at the knot is determined by $\lambda$. If $\lambda<1/2$ then this condition (3) can never be realised, so a minimising lasso is never genuine. In this case we have
$\rho_{\lambda}(p)= 2 \lambda I(p)$, where $I(p)$ is the injectivity radius at the point $p$. In fact in our application we could use this injectivity radius function but the variant $\rho_{\lambda}$ (for a suitable large value of $\lambda$) will make the argument simpler and more transparent. For any minimising lasso $\ugamma=(\gamma_{0},\gamma_{1})$ based at $p\in V$ we define the {\it tension vector} $\tau_{\ugamma}\in TV_{p}$ to be $-\gamma_{0}'(0)$ if the lasso is genuine and $\lambda( \gamma_{1}'(l_{1})- \gamma_{1}'(0))$ if not. (In our situation we will only really be concerned with the case of genuine lassos.)
The function $\rho_{\lambda}$ will not necessarily be smooth. We make a further digression to review a useful general principle by which we can get around this difficulty. Suppose $P,Q$ are manifolds and $F$ is a smooth function defined on an open set $U\subset P\times Q$. For simplicity we assume that $Q$ is compact. Let $\pi_{Q}:U\rightarrow Q$ be the restriction of the projection map and for each $q\in Q$ let $U_{q}=\pi_{Q}^{-1}(q)$. Let
$f(q)$ be the infimum of $F$ on $U_{q}$. Suppose that this is finite and that there is a compact subset $K_{q}\subset U_{q}$ and $\delta_{q}>0$ such that $ F\geq f(q)+\delta_{q}$ on $U_{q}\setminus K_{q}$. Thus the infimum
$f(q)$ is attained on a compact set $J_{q}\subset K_{q}$. It is clear that the function $f$ on $Q$ is continuous, but in general it may not be smooth. For each point $(p,q)$ in $J_{q}$ let $(D_{Q}F)(p,q)$ denote the partial derivative of $F$ at $(p,q)$ in the $Q$ variable. This can be regarded as an element of $T^{*}Q_{q}$.
So we have a map, say $\iota_{q}:J_{q}\rightarrow T^{*}Q_{q}$. Take the convex hull of the image in the vector space $T^{*}Q_{q}$. Then as $q$ varies over $Q$ the union of these yields a subset ${\cal D}\subset T^{*} Q$. In the next proposition we will regard the derivative of a function on $Q$ as a subset of $T^{*} Q$.
\begin{prop}
In this situation $f$ can be approximated arbitrarily closely in $C^{0}$ by a smooth function whose derivative lies in an arbitrarily small neighbourhood of ${\cal D}$.
\end{prop}
To see this, we can first approximate $F$ by a function $\tilde{F}$ whose restriction to slices $U_{q}$ has nondegenerate critical points, for $q$ outside a set $\Delta\subset Q$ of codimension at least $1$. We can further suppose that the minimum is unique, for $q$ outside $\Delta$. The minimiser lies arbitrarily close to $J_{q}$. Define $\tilde{f}$ on $Q$ using $\tilde{F}$ in place of $F$. Outside $\Delta$ it is clear that $\tilde{f}$ is smooth and by elementary calculus its derivative is given by the partial derivative of $\tilde{F}$ at the minimiser. Then smooth $\tilde{f}$ in the standard way using a suitable family of integral operators
$$ \tilde{f}_{\epsilon}= \int_{Q} \beta_{\epsilon}(q,q') \tilde{f}(q') dq'. $$
It is straightforward to verify that these approximations have the desired properties.
\
Now return to the lasso function $\rho_{\lambda}$ on $V$. For each $p\in V$ we take the convex hull of the tension vectors
$\tau_{\ugamma}$ as $\ugamma$ runs over all the mimimising lassos based at $p$ and let ${\cal D}\subset TV $ be the union of these, much as above. Then we have
\begin{prop}
The function $\rho_{\lambda}$ on $V$ can be approximated arbitrarily closely in $C^{0}$ by a smooth function whose gradient lies in an arbitrarily small neighborhood of ${\cal D}$.
\end{prop}
To see this we write the length of $\gamma_{i}$ as the square root of the \lq\lq energy''. There is a minor complication because the square root is not smooth when $\gamma_{0}$ has length zero. (In fact this case will not enter in our application below, so we will be rather sketchy.) We choose a family of functions $W_{\eta}(x)$ (for $\eta>0$), with $W_{\eta}(x)$ a smooth function of $x^{2}$ in a small neighbourhood of $x=0$ and $W_{\eta}(x)=x$ for $x\geq \eta$. Now perturb the length function
to $$ W_{\eta}(l_{0})+ \lambda l_{1}$$ and consider lassos which minimise this perturbed functions. The appropriate matching condition, generalising (3), is
$$ W_{\eta}'(l_{0})
\gamma'_{0}(l_{0}) = \lambda(\gamma'_{1}(0)- \gamma'_{1}(l_{1}))$$
where the left hand side is interpreted as zero if $\gamma_{0}$ is a constant map.
With the same convention, we define the tension vector to be $\tau_{\ugamma,\eta}= W_{\eta}'(l_{0})\gamma'_{0}(0)$. We want to prove the obvious analogue of Proposition 5 for the perturbed problem. Once this is done the proof of Proposition 5 follows easily by letting $\eta$ tend to zero.
Just as in the usual Morse Theory of geodesics based on the energy functional (\cite{kn:M}), we can construct a finite-dimensional space of \lq\lq piecewise geodesic lassos'' which can be identified with an open set in the product $V^{N}$ of a large number of copies of $V$. Here we parametrise curves by the fixed interval $[0,1]$. The minimisers for the perturbed problem are the minimisers of a function $\tau(\sqrt E_{1}) + W_{\eta}(\sqrt{E_{0}})$ where $E_{0}, E_{1}$ are the energies of the two piecewise-geodesic curves. We fit into the framework of Proposition 4, with $P=V$ and $Q=V^{N-1}$, and the result follows.
\subsection{The main argument}
In this subsection we bring together the strands developed above. We begin by considering the flat model space $M_{\Gamma,n}$. Consider a geodesic loop in the smooth part of $M_{\Gamma,n}$. This lifts to a line segment in $\bR^{n}$ joining two points in the same $\Gamma$ orbit. It is clear that the angle between the two tangent vectors at the base point cannot be too small. So if we choose the parameter $\lambda$ large enough (how large depending solely on $\Gamma$) then the matching condition (3) can never be satisfied. Next let $\nu$ be a point in $\bR^{k}/\Gamma$ with $\vert \nu\vert=1$.
It is clear that there is some $c>0$ such that, for all such $\nu$ there are distinct geodesics segments in the smooth part of $\bR^{k}/\Gamma$ emanating from $\nu$, of lengths less than $c$, with the same endpoints and whose distance from the origin is greater than $c^{-1}$.
Suppose that $B$ is a ball in $X$ centred at $p$, such that the normalised ball $B^{\sharp}$ admits an $s$-chart $\chi_{s}$.Recall that we write $d^{\sharp}$ for the normalised metric. Simplifying notation, we will write $\rho$ for $\rho_{\lambda}$ on $X$ and $\rho^{\sharp}$ for the normalised version in $B^{\sharp}$. We will now ignore the fact that $\rho$ is not smooth, since for our purposes below we can always pass to a smooth approximation.
Let $x=\chi_{s}(\xi,\eta)$ be a point in the image of $\chi_{s}$. Define $$ \varpi(x)= \chi_{s}( cs \vert \xi\vert^{-1} \xi, \eta). $$
Then the preceding discussion implies that the injectivity radius at $\varpi(x)$ (for the normalised metric) is not more than $c^{2}s$. Since the distance from $d^{\sharp}(x,\varpi(x))$ is at most $\vert \xi \vert +O(s)$ we see that there are lassos based at $x$ of $\lambda$-length (in the normalised metric) less than $\vert \xi \vert + C s$, for some fixed $C$. Thus $\rho^{\sharp}_{\lambda}(x)\leq \vert \xi \vert + Cs $. Now fix a constant $\kappa_{1}$ slightly less than $1/2$ and suppose that $d^{\sharp}(p,x)\leq \kappa_{1}$. Then it follows that, if $s$ is sufficiently small, any lasso based at $x$ of $\lambda$-length less than $\vert \xi\vert + Cs$ must lie in $B^{\sharp}$. On the other hand, by our choice of $\lambda$ the lasso cannot lie entirely in the image of $\chi_{s}$. It is then clear that $\rho^{\sharp}(x)= \vert \xi\vert+ O(s)$. If we consider points with $\vert \xi \vert$ bounded below by some fixed number then it is clear that (for small $s$) any minimising lasso based at $x$ is genuine and has a knot a (normalised) distance $O(s^{1/2})$ from $\varpi(x)$. Then Proposition 5 shows that the gradient of $\rho^{\sharp}$ at $x$ differs from $\partial_{r}$ (interpreted via the $s$-chart) by $O(s^{1/2})$. In particular what we shall need is that for any $\kappa_{0}>0, \kappa_{3}>1$ we can suppose (by making $s$ small) that at points $x$ with $d^{\sharp}(x,p)\leq \kappa_{1}, \rho(x)\geq \kappa_{0}$ we have
\begin{equation} \vert {\rm grad} \rho^{\sharp} \vert \geq \kappa_{3}^{-1}\end{equation}
We fix $\kappa_{0}$ slightly smaller than $1/10$.
Now fix $\kappa_{3}$ slightly bigger than $1$ and choose $\kappa_{2}>0 $ so that \begin{equation}\frac{1}{5}+ 2\kappa_{3}\kappa_{2}<\kappa_{1}\end{equation}
Thus $\kappa_{2}$ is approximately $3/20$ and in particular $\kappa_{2}>\kappa_{0}$.
Let $q$ be a point in $ B^{\sharp}$ with $d^{\sharp}(p,q)< \kappa_{1}-2\kappa_{2}\kappa_{3}$
We define a set
\begin{equation} S_{q}=\{ x\in B^{\sharp}: d^{\sharp}(x,q)< \kappa_{3} \rho^{\sharp}(x) \}.\end{equation}
Then we have
\begin{lem}
Let $x_{0}\in S_{q}$ and $\kappa_{0}\leq \rho^{\sharp}(x_{0})\leq \kappa_{2}$. Then there is a time interval $[0,T]$ so that gradient flow
$x_{t}$ of ${\rm grad} \rho^{\sharp}$ starting with $x_{0}$ is defined and lies in $S_{q}\subset B^{\sharp}$ for $t\leq T$ and such that $\rho^{\sharp}(x_{T})= 2 \kappa_{2}$.
\end{lem}
To see this, observe that the flow is defined and lies in $S_{q}$ for at least a short time. While the flow is in $S_{q}$ and while $\rho^{\sharp}(x_{t})\leq 2\kappa_{2}$ we have $d^{\sharp}(x,p)\leq d^{\sharp}(x,q)+ d^{\sharp}(q,p)\leq 1/5 + 2\kappa_{3}\kappa_{2}<\kappa_{1}$ by our choices. Since $\rho^{\sharp}(x_{t})\geq \rho^{\sharp}(x_{0})\geq \kappa_{0}$ the gradient bound (4) holds. This implies that, over this time interval, the function $d^{\sharp}(x_{t},q)-\kappa_{3}\rho^{\sharp}(x_{t})$ is decreasing, so we can never reach the boundary of $S_{q}$ and the result follows.
Now we can give the main argument. Write $B_{0}$ for our original unit ball $B(p_{0},1)$ which admits an $s$-chart for suitably small $s$. Suppose $q_{0}$ is a point in $B_{0}$ with $d(p,q_{0})\leq 1/10$ and that $q_{0}$ is in $\cA_{\epsilon}$ where $\epsilon=\epsilon(s)$ as determined by Proposition 3. Applying that Proposition we get another centre $p_{1}$ such that the rescaled ball $B(p_{1}, 1/2)^{\sharp}$ admits an $s$-chart with the same value of $s$. If $d^{\sharp}(p_{1}, q_{0})>1/10$ we stop, otherwise we can repeat the discussion to get a new centre $p_{2}$ and so on. Let $B^{\sharp}_{0}, B^{\sharp}_{1}, \dots$ be the resulting sequence of rescaled balls (so $B^{\sharp}_{0}=B_{0}$). If we reach stage $j$ then $\rho(q_{0})$ must be less than $2^{-j}$ so, since $\rho(q_{0})$ is strictly positive, we must stop at some stage. Thus we have rescaled balls $B^{\sharp}_{0}, \dots B^{\sharp}_{J}$, all admitting $s$ charts. In the rescaled metric $d^{\sharp}$ on $B^{\sharp}_{J}$ the distance $d^{\sharp}(p_{J}, q_{0})$ is bigger than $1/10$ but less than $1/5+O(s)$ so we can suppose it is less than the quantity $\kappa_{1}- 2\kappa_{3}\kappa_{2}$ considered above. Thus we can apply Lemma 1 with initial condition $q=x_{0}= q_{0}$ The condition that $q_{0}$ lies in the image of $\Sigma$ implies that $\rho(q_{0})$ is in roughly the same range.
In particular we can suppose that $\rho^{\sharp}(q_{0})$ is greater than $\kappa_{0}$ and less than $2\kappa_{2}$. Thus we can apply Lemma 1 with initial condition $q=x_{0}= q_{0}$, since obviously $q$ lies in $S_{q}$. Thus we can flow until $\rho^{\sharp}$ increases to $2\kappa_{2}$. Now pass to the ball $B^{\sharp}_{J-1}$, scaling by a factor of $2$. Our new initial condition has $\rho^{\sharp}=\kappa_{2}$ and we can again flow until $\rho^{\sharp}$ has increased to $2\kappa_{2}$. We continue this process through the sequence of balls $B^{\sharp}_{j}$ until in the original ball $B_{0}$ we flow to a point $x$ with $\rho(x) =2\kappa_{2}$ and $d(x,q)\leq \kappa_{3} \rho(x)$. This is not exactly what we asserted in Proposition 1, but it is clear that (by our assumption that $B_{0}$ lies in a larger ball which is close to the model) we can continue the process a little further until we hit $\chi_{s}(E)$.
\section{Completion of proof}
\subsection{Flat limit spaces: proof of Theorem 3}
Theorem 3 is not surprising and is very likely well-known to experts, but we did not find a precise statement in the literature. Notice that if we cast our net wider to consider general length spaces which are the completions of flat Riemannian manifolds then there is a huge amount of flexibility. For example, any $n$-dimensional simplicial complex with the property that all points lie in the closure of $n$-simplices can be given such a structure via an embedding in Euclidean space. In particular the analogue of Theorem 3 would fail in this larger class.
Let $B$ be a flat limit ball as considered in Theorem 3. The first step is to see that we can suppose that the s-chart $\chi_{s}:H_{s}\rightarrow B$ is actually an {\it isometry} (possibly after slightly shrinking the ball). Recall that if $V$ is any flat, connected, $n$-manifold (not necessarily complete) there is a {\it developing map}: an open immersion of the universal cover of $V$ in $\bR^{n}$. This yields a developing homomorphism from $\pi_{1}(V)$ to the isometry group ${\rm Euc}_{n}$ of $\bR^{n}$. We apply this with $V=\chi_{s}(H_{s})$. Then $\pi_{1}(V)=\Gamma$ is finite and the homomorphisms from $\Gamma$ to ${\rm Euc}_{n}$ are rigid, up to conjugacy. This easily implies the statement.
The next observation is that, assuming Theorem 2 is known, there can be no codimension $2$ singular points---that is, no points with a tangent cone of the form $N_{\beta,n}$.
More generally no iterated tangent cone in $B$ can have this form.
Write $H$ for the the set of points $(\xi,\eta)\in \bR^{k}\times \bR^{n-k}$ such that $\vert \xi\vert \geq 1$.
Let $\nu$ be a point in $S^{k-1}$, and let $r>0$. Define the subset
$U_{\nu,r}\subset H$ to be the points which can be joined to $(\nu,0)$ by a path {\it in $H$} of length at most $r$. This is a slightly complicated set but it is clear from the convexity of the complement of $H$ that the volume of $U_{\nu,r}$ strictly exceeds $\frac{\omega_{n}}{2} r^{n}$, where $\omega_{n}$ is the volume of the unit ball in $\bR^{n}$. (Note that $U_{\nu,r}$ is not quite the same as the intersection of $H$ with the ball $B(\nu,r)$, since $H$ is not convex.) Fix $r$ small enough that $U_{\nu,r}$ and $g(U_{\nu,r})$ are disjoint for all $\nu$ and for all $g\in \Gamma$, different from the identity. Now fix $c>1$ such that
\begin{equation} c^{n} \Vol(U_{\nu, r})> \frac{\omega_{n}}{2} (cr+c-1)^{n} \end{equation}
Thus $c$ depends only on $\Gamma$. We can assume that
\begin{equation} \frac{1}{4s}> \left(1+ \frac{1}{c-1}\right) \end{equation}
For $0<\sigma<2s$ let $A_{\sigma}$ be the \lq\lq annulus $\{ \sigma <\vert \xi \vert< 2s\}$ in $\bR^{k}/\Gamma$. If $\eta\in \bR^{n-k}$ with $\vert \eta\vert$ not too large we have an obvious isometric embedding
$\iota_{\eta}:A_{s}\rightarrow B$ defined by the $s$-chart. Define $d(y)$ to be the infimum of the set of $\sigma$ such that $\iota_{\eta}$ extends to an isometric immersion of $A_{\sigma}$ in the smooth part of $B$.
To simplify notation we suppose (as we obviously can by rescaling slightly) that $d(\eta)$ is actually defined for all $\eta$ with $\vert \eta \vert\leq 1 $.
\begin{lem}
Suppose that for all $\eta$ in some ball $D= \{ \eta: \vert \eta- \eta_{0}\vert <t\}$ we have $d(\eta)< \sigma $. Then the maps $\iota_{\eta}$ for $\eta$ in $D$ define an isometric embedding of $D\times A_{\sigma}$ in $B$.
\end{lem}
This is certainly true when $\sigma=s$: the map in question is just the restriction of the $s$-chart. We fix $\eta$ and let $\sigma_{\eta}$ be the infimum of the set of $\sigma$ such that the images $\iota_{\eta}(A_{\sigma})$ are disjoint, as $\eta$ ranges over $D$. If $\sigma_{\eta}>\sigma$ then there is a smooth point $q$ of $B$ which lies in the closure of $\iota_{\eta}(A_{\sigma_{\eta}})$ and of $\iota_{\eta'}(A_{\sigma})$ for some $\eta'\neq \eta$. (We allow the possibility that $\eta'$ is in the closure of $D$.) Elementary considerations involving the local geometry around $q$ show that this is impossible.(The essential point is the concavity of the boundaries of the sets in question.)
With all this preparation we reach the central step in the proof.
\begin{prop}For $\eta_{0}$ with $\vert \eta_{0}\vert<1/2$ we have $d(\eta_{0})=0$\end{prop}
Suppose we have any point $\eta$ with $\vert \eta\vert<1$ and $d(\eta)=d>0$. The limit of the isometric embeddings $\iota_{\eta}(A_{\sigma})$ as $\sigma$ tends to $d$ defines a map from the closure $\overline{A_{d}}$ to $B$ which we still denote by $\iota_{\eta}$. Clearly there must be a singular point $q$ of $B$ in $\iota_{\eta}(\overline{A_{d}})$.
Suppose, arguing for a contradiction, that for all points $\eta'$ with $\vert \eta-\eta'\vert \leq d $ we have $d(\eta')\leq c d(\eta)$. For a suitable choice of $\nu$ there is an copy of $U_{\nu,r}$ in $B$ scaled by a factor $cd$ and contained in the ball of radius $d(cr+c-1)$ centred at $q$. Thus
(7) implies that
\begin{equation} \Vol(B(q,cdr))> \frac{\omega_{n}}{2} (cdr)^{n}. \end{equation}
But now we have
\begin{lem} Suppose $q$ is a singular point in an n-dimensional $L^{1}$-flat limit space $Z$. Then for any $d>0$ we have $\Vol(B(q,d)\leq \frac{\omega_{n}}{2} d^{n}$.
\end{lem}
Suppose first that there is a tangent cone at $q$ of the form $\bR^{n-p}\times C(X)$ with $X$ smooth. Thus $X$ has the form $S^{p-1}/G$ where $G$ acts freely on the sphere, and the limit as $d$ tends to $0$ of $\frac{ \Vol(B(q,d))}{\omega_{n} d^{n}}$ is $1/\vert G\vert$. Then our result in this case follows from generalised Bishop-Gromov monotonicity.
Suppose next that there is a tangent cone at $q$ as above but with $X$ singular. Suppose however that there is a tangent cone of $C(X)$ of the form $\bR^{n-p} \times \bR^{p}/G$, where $G$ acts freely. Then we first apply volume monotonicity to see that $\Vol(X)\leq \omega_{n}/\vert G\vert$ and then argue as above. There is always some {\it iterated} tangent cone of the form $\bR^{n-p} \times \bR^{p}/G$, where $G$ acts freely, and we extend the argument in the obvious way.
At this stage, combining Lemmas 2 and 3, we see that in fact if $d(\eta)>0$ there must be some point $\eta'$ with $\vert \eta'-\eta\vert \leq d(\eta)$ and $d(\eta')\geq c d(\eta)$. Start with a point $\eta_{0}$ with $\vert \eta_{0}\vert \leq 1/2$ and suppose $d(\eta_{0})=d>0$.
There is another point $\eta_{1}$ with $\vert \eta_{1}-\eta_{0}\vert\leq d$ and $d(\eta_{1})\geq c d$. Repeating the argument we get $\eta_{i}$ with
$\vert \eta_{i+1}-\eta_{i}\vert \leq d c^{i}$ and $d(\eta_{i})\geq c^{i} d$. We have to stop at some stage $\eta_{N}$, when we approach the boundary, so $\vert \eta_{N}\vert \geq 1$. Then we have
\begin{equation} d + c d + c^{2} d+\dots + c^{N-1} d \geq \vert \eta_{0}-\eta_{N}\vert \geq 1/2. \end{equation}
On the other $d(\eta_{N-1})\leq s$ so $c^{N-1} d \leq s$. Combined with (10) this is a contradiction to our hypothesis (8).
Given Proposition 6, we argue just as in Lemma 2 that the half-sized ball in $Z$ is isometric to a ball in $M_{\Gamma}$.
Although we do not use this, Theorem 3 leads to a precise description of the local structure of $L^{1}$ flat limits.
\begin{prop}
An $L^{1}$ flat limit space is a Euclidean orbifold. Any point has a neighbourhood isometric to the quotient of the ball $B^{n}\subset \bR^{n}$ by a finite subgroup of $O(n)$.
\end{prop}
The holonomy of the flat metric defines a homomorphism $\pi_{1}(B_{\reg})\rightarrow
SO(n)$ and this in turn defines a covering of $B_{\reg}$. The basic point is that the metric completion of this covering is a smooth Riemannian manifold. It is easy to see that this implies Proposition 7. Now Theorem 3 has the following consequence. If there is an iterated tangent cone of $B$ of the form $M_{\Gamma,n}$ then there is a corresponding small ball in $B$ which is isometric to a ball centred at the origin in $M_{\Gamma,n}$. Now the assertion about the metric completion of the covering follows from an inductive argument on the codimension of the singular set which we leave to the interested reader to fill in.
The work of Joyce \cite{kn:J} provides many examples of such Euclidean orbifolds, often with intricate singular structures, which arise as the limits of non-collapsed Einstein manifolds.
There is a somewhat reciprocal relation between Theorem 3 and Proposition 7. In one direction, as we have sketched above, the first can be used to establish the second. Alternatively, if one knows Proposition 7 there are slightly simpler proofs of Theorem 3.
\subsection{Extension to bounded Ricci curvature}
Recall that Theorem 1 deals with metrics of bounded Ricci curvature while in the previous sections we have restricted attention to the Einstein case.
Here we discuss the modifications required to remove this restriction. These are of a fairly technical, but routine, nature.
We recall that, in general, the Cromov-Hausdorff limit $W$ of a sequence of non-collapsed metrics of bounded Ricci curvature contains a dense open subset $W_{\reg}$ which is a manifold of class $C^{2,\alpha}$ (for any $\alpha\in (0,1)$) and the limiting structure on $W_{\reg}$ is a Riemannian metric of class $C^{1,\alpha}$. This is shown by working in harmonic co-ordinates on suitable small balls before passing to the limit. Thus the first modification is to change the second item in the definition of an $s$-chart to say that the pull-back of $g$ differs in $C^{1,\alpha}$ norm from the flat metric by at most $s$. In fact all we need is $C^{1}$. Reviewing the proof we see that the only place where the regularity of the $s$-chart might be an issue occurs in Section 3.3, where we want to say that if the knot of a minimising lasso based at $x$ has a distance $O(s^{1/2})$ from $\varpi(x)$ then the tension vector differs from $\partial_{r}$ by $O(s^{1/2})$. But this is straightforward for metrics whose difference from the flat metric is small in $C^{1}$ since the Christoffel symbols are then small in $C^{0}$. Thus solutions of the geodesic equation
$$ \ddot{x}_{i}= \Gamma^{i}_{jk} \dot{x}^{j}\dot{x}^{k}, $$
written in the $s$-chart, are close to Euclidean lines which easily yields what we need.
The second issue arises in our discussion of $L^{1}$-flat limit balls.
We should now change the definition, replacing Einstein by bounded Ricci curvature. The Gromov-Hausdorff limit $Z$ still has an open dense subset $Z_{\reg}$ with a $C^{1,\alpha}$ metric, but it is not immediately clear that this is, in suitable co-ordinates, a amooth flat Riemannian metric. To see this we go back to working in harmonic c-ordinates over small balls, before taking the limit. We recall that for $p>n/2$ there is a good theory of $L^{p}_{2}$ Riemannian metrics, with curvature in $L^{p}$. Elliptic estimates in harmonic co-ordinates mean that an $L^{\infty}$ (hence $L^{p}$) bound on the Ricci curvature give an $L^{p}_{2}$ bound on the metric on interior balls, and in particular an $L^{p}$ bound on the curvature. In our situation, the $L^{1}$ norm of the curvature tends to zero in the sequence, so for any $p'<p$ the $L^{p'}$ norm of the curvature tends to zero, by H\"older's inequality.
Fix $p'>n/2$, then we get an $L^{p'}_{2}$ limit, in harmonic co-ordinates, with curvature zero. Elliptic regularity implies that this is smooth.
Thus $Z_{\reg}$ has a smooth Euclidean structure and all our arguments carry over.
\section{Codimension 2}
In this section we discuss extensions of the argument we have given above to establish the codimension 2 situation; Theorem 2. We emphasise again that Theorem 2 is a known result and because of that we will be content with a sketch. To simplify the discussion slightly we consider the Einstein case
The general set-up we considered in Section 2 and 3---the definition of an s-chart etc.---goes over in a obvious way. So now we write $H_{s}$ for the appropriate subset of $N_{\beta}= \bR^{2}_{\beta}\times \bR^{n-2}$. The new feature is that the analogue of Theorem 3 is false: the singularity is not rigid. Let $z_{1}, \dots z_{r}$ be points in $\bR^{2}$ such that we can find disjoint \lq\lq wedges'' $W_{i}\subset \bR^{2}$ with vertices at the $z_{i}$ and with angles
$\gamma_{i}$. We get a flat singular space $Z$ by cutting out these wedges from the plane and gluing along the resulting edges. Then $Z$ has $r$ singular points with cone angles $2\pi(1-\gamma_{i})$ and outside a compact set is isometric to the standard cone with angle $\beta=1-\gamma$ where $\gamma=\sum \gamma_{i}$. Consider a family of such wedges with fixed angles $\gamma_{i}$ and with vertices $z_{i}$ converging to the origin in $\bR^{2}$ and take the product with $\bR^{n-2}$. We get flat spaces which contain isometric copies of $H_{s}$ for arbitrarily small $s$, but which are not isometric to
$N_{\beta,n}$. One can also construct more complicated examples where we have a unit ball in a singular flat space which contains an arbitrarily small deformation of $H_{s}$. This is possible because the fundamental group of $H_{s}$ is ${\bf Z}$ and the developing homomorphism
from $ {\bf Z}$ to ${\rm Euc}_{n}$ can be deformed. However it seems likely that the general picture will be similar to that considered above, with the singular set breaking up into a number of \lq\lq almost parallel'' copies with cone angles $\beta_{i}=1-\gamma_{i}$ where $\sum \gamma_{i}$ is as close as we please to $\gamma=1-\beta$. Thus to explain the argument simply we will assume a rather weak version of this idea.
\
{\bf Assumption}
Fix $c>1/2$. Suppose there is an $s$-chart $\chi_{s}:H_{s}\rightarrow B$ in an $L^{1}$- flat limit ball. If $s$ is sufficiently small then either the half-sized ball $\frac{1}{2} B$ is isometric to a ball in $N_{\beta,n}$ or there is a point
$q\in \frac{1}{2} B$ with a tangent cone $N_{\beta',n}$ where $(1-\beta')\leq c (1-\beta)$.
\
Now we argue as follows. Write $B^{n}$ for the unit ball in $\bR^{n}$. Recall that, according to Anderson and Colding, there is a $\delta_{0}$ such that if our unit ball $B(x,1)\subset X$ has Gromov-Hausdorff distance less than $\delta_{0}$ from $B^{n}$ then we get a fixed bound on the curvature tensor in the half-sized ball. If the $L^{1}$ norm of the curvature tensor is small the same is true for all $L^{p}$ norms and then, by standard elliptic theory, for the $L^{\infty}$ norm. It follows that, given any $\delta_{1}$ we can find an $\epsilon(\delta_{1})$ so that if the $L^{1}$ norm of $\Riem$ is less than $\epsilon(\delta_{1})$ the Gromov-Hausdorff distance from the half sized ball $B(x,1/2)$ to $\frac{1}{2} B^{n}$ is less than $\delta_{1}$. Now suppose that $\beta$ is sufficiently close to $1$ that the distance from the unit ball in $B_{\beta,n}\subset M_{\beta,n}$ to $B^{n}$ is less than $\delta_{0}/2$. Set
$$\delta'= d_{GH}( \frac{1}{2} B_{\beta, n}, \frac{1}{2} B^{n}), $$
so $\delta'>0$, since $\beta<1$. Now if $\delta\leq \min(\delta_{0}/2, \delta'/2)$ it follows that we can {\it never} have the situation where $d_{GH}(B(x,1), B_{n,\beta})<\delta$ and the $L^{1}$ norm of the curvature is less than $\epsilon(\delta'/2)$.
This implies that we cannot have a point in a flat limit space with tangent cone of the form $N_{n,\beta}$ for such values of $\beta$. Say this covers a range $1-\gamma_{0}\leq \beta<1$.
Now consider a value of $\beta$ with $1-\gamma_{0}/c<\beta<1$. Invoking the \lq\lq assumption'', we see that a flat limit which admits an $s$-chart, for small enough $s$, is actually isometric to a ball in $N_{\beta,n}$ and our argument goes through.
So we know that in fact we cannot have a tangent cone in a flat limit space of the form $N_{n,\beta}$ for this larger range of $\beta$. Then repeat the argument until we cover the whole range $0<\beta<1$.
By making more complicated arguments it seems that one can prove Theorem 2 using this approach, without using the \lq\lq assumption'' . This involves another level of limits, taking $\delta\rightarrow 0$ and arguing with the resulting complete limit space.
For example, using volume monotonicity and the \lq\lq volume cone implies metric cone'' theorem one sees that if $\beta\geq 1/2$ then there are no higher codimension singularities in B, so certainly there must be {\it some} points with codimension 2 singularities. Similarly one sees that in proving the high codimension result, Theorem 1, it suffices to know Theorem 2 for the range $\beta\geq 1/2$. But the arguments become convoluted and, since the result is known, we leave the interested reader to fill in details.
There is an unsatisfactory aspect in the statement of the \lq\lq assumption''. {\it A posteriori}---
given Theorem 2---none of these spaces can arise as $L^{1}$-flat limits. Further, if Anderson's conjecture in \cite{kn:A2} is true then none can arise as limits of metrics with bounded Ricci curvature. On the other hand they certainly do arise as limits of metrics with Ricci curvature bounded below Thus we feel that, properly formulated, there should be an interesting classification problem of a deformations of the $N_{\beta,n}$ within a suitable class of flat singular spaces.
|
1,108,101,565,492 | arxiv | \section{Introduction}
\label{S:intro}
Graph-theoretic trees are abundant in mathematics and its applications, from computer science to theoretical
biology. A natural question is how to define limits and limit objects as the size of the trees tends to
infinity. On the one hand, there are \emph{local} approaches yielding countably infinite graphs, or generalized
so-called graphings with a Benjamini-Schramm-type approach (going back to \cite{BenjaminiSchramm2001}, see
\cite[Part~4]{Lovasz2012}). On the other hand, if one takes a more \emph{global} point of view, as we are doing
here, the predominant approach is to consider graph-theoretic trees as metric spaces equipped with the
(rescaled) graph distance. Then the limit objects are certain ``tree-like'' metric spaces, most prominently
so-called $\R$\nobreakdash-tree s introduced in \cite{Tits1977}.
They are also of independent interest, e.g.\ for studying isometry groups of
hyperbolic space (\cite{MorganShalen1984}), or as generalized universal covering spaces in the study of the
fundamental groups of one-dimensional spaces (\cite{FischerZastrow2013}). Characterizing the topological
structures induced by $\R$\nobreakdash-tree s has received considerable attention
(\cite{MayerOversteegen90,MayerNikielOversteegen92,Fabel15}). Here, instead of the topological structures, we are
more interested in the ``tree structures'' induced by $\R$\nobreakdash-tree s. We formalize the tree structure with a branch point
map and call the resulting axiomatically defined objects \emph{algebraic trees}.
While, unlike for metric spaces, we do not know any useful notion of convergence for topological spaces or
topological measure spaces, it is essential for us that we can define a very useful convergence of algebraic
measure trees.
Our main motivation lies in suitable state spaces for tree-valued stochastic processes.
The construction and investigation of scaling limits of tree-valued Markov chains within a metric space setup
started with the continuum analogs of the Aldous-Broder-algorithm for sampling a uniform spanning tree from the
complete graph (\cite{EvansPitmanWinter2006}), and of the tree-valued subtree-prune and regraft Markov chain used
in the reconstruction of phylogenetic trees (\cite{EvansWinter2006}).
It continued with the construction of evolving genealogies of infinite size populations in population genetics
(\cite{GrevenPfaffelhuberWinter2013,DepperschmidtGrevenPfaffelhuber2012,KliemLoehr2015,Piotrowiak2010,GrevenSunWinter2016}) and
in population dynamics (\cite{Gloede2012,KliemWinter19}). Moreover, continuum analogues of pruning procedures
were constructed
(\cite{AbrahamDelmasVoisin2010,AbrahamDelmas2012,LoehrVoisinWinter2015,HeWinkel2014,HeWinkel2017}).
All these constructions have in common that they encode trees as metric (measure) spaces or bi-measure
$\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-trees, and equip the respective space of trees with the Gromov-Hausdorff (\cite{Gromov2000}),
Gromov-weak (\cite{Fukaya1987,GrevenPfaffelhuberWinter2009,Loehr2013}), Gromov-Hausdorff-weak
(\cite{Villani2009,AthreyaLohrWinter16}), or leaf-sampling weak-vague topology
(\cite{LoehrVoisinWinter2015}).
In the present paper, we shift the focus from the metric to the tree structure for several reasons. First,
checking compactness or tightness criteria for (random) metric (measure) spaces is not always easy, and some natural
sequences of trees do not converge as metric (measure) spaces with a uniform rescaling of edge-lengths. At least
for the subspace of binary algebraic measure trees we introduce, the situation is much more favorable, because
it turns out to be compact. Second, the metric is often less canonical than the tree structure in situations
where it is not clear that every edge should have the same length, e.g.\ in a phylogenetic tree, where edges
might correspond to very different evolutionary time spans. Third, one might want to preserve certain
functionals of the tree structure in the limit. For instance, the limit of binary trees is not always binary in
the metric space setup, while this will be the case for our algebraic measure trees. Also, the centroid
function used in \cite{Aldous2000} is not continuous on spaces of metric measure trees, but it is continuous on
our space.
The starting point of our construction is the notion of an $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree (see \cite{Tits1977,DreMouTer96,Chiswell2001,Evans2008}).
There are many equivalent definitions, but the following one is the most convenient for us:
\begin{definition}[$\R$\nobreakdash-tree s]
A metric space $(T,r)$ is an \define{$\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree} iff it satisfies the following:
\begin{enumerate}[\axiom(RT1)]
\item $(T,r)$ satisfies the so-called \emph{$4$-point condition}, i.e., for all $x_1,x_2,x_3,x_4\in T$,
\begin{equation}
\label{e:4point}
r(x_1,x_2)+r(x_3,x_4)
\le
\max\big\{r(x_1,x_3)+r(x_2,x_4),\,r(x_1,x_4)+r(x_2,x_3)\big\}.
\end{equation}
\item $(T,r)$ is a connected metric space.
\end{enumerate}
\label{d:realGH}
\end{definition}
Notice that any metric space $(T,r)$ satisfying (RT1) and (RT2) admits a \emph{branch point map} $c\colon T^3\to
T$, i.e., for all $x_1,x_2,x_3\in T$ there exists a unique point $c(x_1,x_2,x_3)\in T$ such that
\begin{equation}
\label{e:bp}
\bigl\{c(x_1,x_2,x_3)\bigr\} = [x_1,x_2]\cap[x_1,x_3]\cap[x_2,x_3],
\end{equation}
where for $x,y\in T$ the \emph{interval} $[x,y]$ is defined as
\begin{equation}
\label{e:arc}
[x,y]:=\bset{z\in T}{r(x,z)+r(z,y)=r(x,y)}.
\end{equation}
Given the branch point map $c$, we can recover the intervals via the identity
\begin{equation}\label{e:arc2}
[x,y]=\bset{z\in T}{c(x,y,z)=z}.
\end{equation}
\xymatfig{f:4pt}{\bullet\ar@{-}[dr]^(0){x_1} & & & & \bullet\\
& \bullet\ar@{-}[rr]^<{c_1}^>{c_2} & & \bullet\ar@{-}[ur]^(1){x_3}\ar@{-}[dr]_(1){x_4} & \\
\bullet\ar@{-}[ur]_(0){x_2}& & & & \bullet}
{The only possible tree shape spanned by four points separates them into two pairs. Here,
$r(x_1,x_2)+r(x_3,x_4)<\max\{r(x_1,x_3)+r(x_2,x_4),\,r(x_1,x_4)+r(x_2,x_3)\}$, while any other permutation
yields equality. Furthermore, $c_1=c(x_1,x_2,x_3)=c(x_1,x_2,x_4)$ and $c_2=c(x_1,x_3,x_4)=c(x_2,x_3,x_4)$.
}
While condition (RT1) is crucial for trees as it reflects the fact that there is only one possible shape for the
subtree spanned by four points (as shown in Figure~\ref{f:4pt}), the assumption of connectedness can be relaxed.
In \cite{AthreyaLohrWinter17}, the notion of a
\emph{metric tree} was introduced to allow for a unified set-up in discrete and continuous situations.
A metric tree $(T,r)$ is defined as a metric space
which can be embedded isometrically into an $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree such that it contains all branch points
$c(x_1,x_2,x_3)$, $x_1,x_2,x_3\in T$, as defined by \eqref{e:bp}.
To exclude non-tree graphs satisfying the $4$\protect\nobreakdash-\hspace{0pt} point condition (see Figure~\ref{f:nontree}), we have to require the property of containing the branch points explicitly.
{
\xymatrixrowsep{1.6pc}
\xymatfig{f:nontree}{& \bullet\ar@{-}[dl]\ar@{-}[dr] & \\ \bullet\ar@{-}[rr] & & \bullet}
{The graph shown here is not a tree, but the vertices satisfy the $4$\protect\nobreakdash-\hspace{0pt} point condition
with respect to the graph-distance.
Condition~(MT\ref{MT:cex}) fails.}
}
\begin{definition}[metric trees]
A metric space $(T,r)$ is a \define{metric tree} if the following holds:
\begin{enumerate}[\axiom(MT1)]
\item\label{MT:4pt} $(T,r)$ satisfies the $4$-point condition (RT1).
\item\label{MT:cex} $(T,r)$ admits all branch points, i.e., for all $x_1,x_2,x_3\in T$ there exists a
(necessarily unique) $c(x_1,x_2,x_3)\in T$ such that
\begin{equation} \label{e:brapoi}
r\big(x_i,\,c(x_1,x_2,x_3)\big)+r\big(c(x_1,x_2,x_3),\,x_j\big)=r(x_i,x_j)\quad\forall\,i,j\in\{1,2,3\},\,
i\ne j.
\end{equation}
\end{enumerate}
\label{d:metrictree}
\end{definition}
Our main goal is to forget the metric while keeping the tree structure
encoded by the branch point map. To axiomatize the latter, notice that for metric trees the branch point map satisfies the following obvious properties:
\begin{enumerate}[\axiom(BPM1)]
\item\label{BPM:1} The map $c\colon T^3\to T$ is symmetric.
\item\label{BPM:2} The map $c\colon T^3\to T$ satisfies the \emph{$2$-point condition} that
for all $x,y\in T$
\begin{equation}
c(x,y,y)=y.
\end{equation}
\item\label{BPM:3} The map $c\colon T^3\to T$ satisfies the \emph{$3$-point condition} that
for all $x,y,z\in T$
\begin{equation}
\label{e:2pc}
c\big(x,y,\,c(x,y,z)\big)=c(x,y,z).
\end{equation}
\item\label{BPM:4} The map $c\colon T^3\to T$ satisfies the \emph{$4$-point condition} that
for all $x_1,x_2,x_3,x_4\in T$,
\begin{equation}\label{e:BPM4}
c(x_1,x_2,x_3)\in\big\{c(x_1,x_2,x_4),\,c(x_1,x_3,x_4),\,c(x_2,x_3,x_4)\big\}.
\end{equation}
\end{enumerate}
\begin{definition}[algebraic tree]
An \define{algebraic tree} $(T,c)$ consists of a set $T\not=\emptyset$ and a branch point map $c\colon T^3\to T$ satisfying (BPM1)--(BPM4).
\label{d:algebraic}
\end{definition}
We define a natural topology on an algebraic tree $(T,c)$ as follows.
For each $x\in T$, we define an equivalence relation $\sim_x$ on $T\setminus\{x\}$ such that for all
$y,z\in T\setminus\{x\}$,
$y\sim_x z$
iff $c(x,y,z)\not =x$.
For $y\in T\setminus\{x\}$, we denote by
\begin{equation}
\label{e:equiv}
\mathcal{S}_x(y):=\set{z\in T}{z\sim_x y}
\end{equation}
the equivalence class w.r.t.\ $\sim_x$ which contains $y$.
$\mathcal{S}_x(y)$ should be thought of as a subtree rooted at (but not containing) $x$.
We consider the topology generated by sets of the form \eqref{e:equiv} with $x\ne y$
and denote by $\mathcal{B}(T,c)$ the corresponding Borel $\sigma$\protect\nobreakdash-\hspace{0pt} algebra.
Our first main result (Theorem~\ref{t:algtreechar}) relates metric trees with algebraic trees. On the one
hand, if $(T,r)$ is a metric tree, then it is clear that $T$ together with the map $c$ from (MT\ref{MT:cex}) yields an algebraic tree.
On the other hand, we show that every order separable algebraic tree (Definition~\ref{d:ordersep})
is induced by a metric tree in this way.
More concretely, if $\nu$ is a measure on $\mathcal{B}(T,c)$ which is finite
and non-zero on non-degenerate intervals, i.e., on sets of the form
\begin{equation}
\label{e:001}
[x,y]:=\big\{z\in T:\,c(x,y,z)=z\big\}
\end{equation}
for $x,y\in T$, $x\ne y$, then a metric representation of $(T,c)$ is given by
\begin{equation}\label{e:rnu}
r_\nu(x,y)
:=
\nu\big([x,y]\big)-\tfrac{1}{2}\nu\big(\{x\}\big)-\tfrac{1}{2}\nu\big(\{y\}\big).
\end{equation}
Next, we equip an algebraic tree $(T,c)$ with a sampling probability measure $\mu$ on $\mathcal{B}(T,c)$, and call the
resulting triple $(T,c,\mu)$ \emph{algebraic measure tree}.
Two algebraic measure trees $(T,c,\mu)$ and $(T',c',\mu')$ are equivalent (compare with
Definition~\ref{d:amtequiv}) if there are $A\subseteq T$, $A'\subseteq T'$ and a bijection $\phi\colon A \to A'$
such that the following holds.
\begin{itemize}
\item $\mu(A)=\mu'(A')=1$, $c(A^3) \subseteq A$ and $c'((A')^3) \subseteq A'$.
\item $\phi$ is measure preserving, and $c'(\phi(x),\phi(y),\phi(z))=\phi(c(x,y,z))$ for all $x,y,z\in T$.
\end{itemize}
Denote by $\mathbb{T}$ the space of all equivalence classes of order separable algebraic measure trees. We equip
$\mathbb{T}$ with a topology based on the Gromov-weak topology
(introduced in \cite{GrevenPfaffelhuberWinter2009} and shown in \cite{Loehr2013} to be equivalent to Gromov's
$\underline\Box_1$-topology from \cite{Gromov2000}).
For that purpose, we introduce a particular
metric representation of an algebraic measure tree. As metric representations are far from being unique, we will
consider the intrinsic metric $r_\nu$ which comes from the branch point distribution, i.e.,
the image measure $\nu:=c_\ast\mu^{\otimes 3}$ of $\mu^{\otimes 3}$ under the branch point map $c$.
We declare that
\begin{equation}
\label{e:convergence}
\begin{aligned}
(T_n,c_n,\mu_n)\tno(T,c,\mu)\hspace{.2cm}\mbox{ iff }\hspace{.2cm}(T,r_{(c_n)_\ast\mu_n^{\otimes 3}},\mu_n)\to(T,r_{c_\ast\mu^{\otimes 3}},\mu)\mbox{ Gromov-weakly},
\end{aligned}
\end{equation}
or equivalently, $\Phi((T_n,c_n,\mu_n)) \tno \Phi((T,c,\mu))$
for all test functions of the form
\begin{equation}
\label{e:PhiGw}
\Phi(T,c,\mu)=\Phi^{n,\phi}(T,c,\mu):=\int_{T^n}\phi\big((r_{c_\ast\mu^{\otimes 3}}(x_i,x_j))_{1\le i,j\le n}\big)\,\mu^{\otimes n}(\mathrm{d}\underline{x}),
\end{equation}
where $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and $\phi\in\mathcal{C}_b(\mathbb{R}} \newcommand{\Z}{\mathbb{Z}^{n\times n})$.
We refer to this convergence as {branch point distribution distance Gromov-weak convergence}, or shortly,
\define{bpdd-Gromov-weak convergence}. It is important to keep in mind that---even though bpdd-Gromov-weak
convergence is defined via Gromov-weak convergence of particular metric representations---Gromov-weak
convergence of a sequence $(T_n,r_n,\mu_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of metric measure trees does not imply bpdd-Gromov-weak
convergence of the corresponding sequence of algebraic measure trees. For instance, if the diameters
$\sup_{x,y\in\mathbb{T}_n} r_n(x,y)$ converge to zero, the sequence of metric measure trees converges to the trivial
(one-point) tree, while the corresponding sequence of algebraic measure trees might or might not converge, to the
same or a different limit. The same reasoning also applies to the stronger Gromov-Hausdorff-weak topology.
A particular subclass of interest is the space of binary algebraic measure trees.
Similar to encoding compact $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-trees by a continuous excursion on the unit interval,
binary algebraic trees can be encoded by \emph{sub-triangulations of the circle} (see Figure~\ref{f:triangtree}), where a sub-triangulation of the circle $\S$ is a closed, non-empty subset $C$ of $\mathbb{D}$ satisfying the following two conditions:
\begin{enumerate}[\axiom(Tr{i}1)]
\item The complement of the convex hull of $C$ consists of open interiors of triangles.
\item $C$ is the union of non-crossing (non-intersecting except at endpoints), possibly
degenerate closed straight line segments with endpoints in $\S$.
\end{enumerate}
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}{A triangulation of the $12$-gon and the tree coded by it.}
Such an encoding was introduced by David Aldous in \cite{Aldous94,Aldous94b},
and there has since then been an increasing amount of research in the random tree community using this approach
(e.g.\ \cite{CurienLeGall11,BroutinSulzbach15,CurienKortchemski15}). Also more general ${}$-angulations and
dissections have been considered which allow for encoding not necessarily binary trees
(\cite{Curien,CurienHaasKortchemski15}). Note, however, that the relation between triangulations and trees has
never been made explicit, except for the finite case, where the tree is the dual graph.
Aldous originally defines a triangulation of the circle as a
closed subset of the disc the complement of which is a disjoint union of open triangles with vertices on the
circle (\cite[Definition~1]{Aldous94b}). We modify his definition in two respects.
First, we add Condition (Tri2) which enforces existence of branch points and under which triangulations of the
circle are precisely the Hausdorff-metric limits of triangulations of $n$\nobreakdash-gon s as $n\to\infty$.
Second, we extend the definitions to sub-triangulation of the circle (triangulations of a subset of
the circle) which allow for encoding algebraic measure trees with point masses on leaves. In fact,
triangulations of the whole circle encode binary trees with non-atomic measures,
which is relevant in the case of Aldous's CRT.
We formally construct the coding map that associates to a sub-triangulation of the circle the corresponding
binary algebraic measure tree with point-masses restricted to the leaves.
Furthermore, we show that---similar to the case of coding compact $\R$\nobreakdash-trees\ by continuous excursions---the
coding map is \emph{surjective} and \emph{continuous} when the set of sub-triangulations
is equipped with the Hausdorff metric topology and the set of binary algebraic measure trees with our
bpdd-Gromov-weak topology (Theorem~\ref{t:tree}).
We also analyze the subspace of binary algebraic measure trees with point-masses restricted to the leaves in more
detail. Our third main result (Theorem~\ref{t:topeq}) states that this space in the bpdd-Gromov-weak topology is
topologically as nice as it gets, namely a compact, metrizable space.
We also give two more notions of convergence which turn out to be equivalent to bpdd-Gromov-weak convergence on
this subspace.
One is of combinatorial nature and based on the weak convergence of test functions of the form
\begin{equation}
\label{e:Phicomb}
\Phi(T,c,\mu)=\Phi^{n,\mathfrak{t}}(T,c,\mu):=\mu\(\bset{(u_1,...,u_n)\in T^n}{\shape[(T,c)](u_1,...,u_n)=\mathfrak{t}}\),
\end{equation}
where $\mathfrak{t}$ is an $n$-cladogram (a binary graph-theoretic tree with $n$ leaves) and $\mathfrak{s}_{(T,c)}$
denotes the shape spanned by a finite sample in $(T,c)$ (Definitions~\ref{def:cladogram} and~\ref{def:treeshape}).
The other one is more in the spirit of stochastic analysis and based on weak convergence of the \emph{tensor of
subtree-masses} read off the algebraic measure subtree spanned by a finite sample (see
Definition~\ref{d:massdist}).
This equivalence allows to switch between different perspectives and turns out to be very useful for the following reasons:
\begin{itemize}
\item Using convergence of sample bpd-distance matrices allows to exploit well-known results about Gromov-weak convergence.
\item Showing convergence of graph theoretic tree-valued Markov chains as the number of vertices tends to
infinity is, due to the combinatorial nature of the Markov chains, often easiest by showing
convergence of the sample shape distributions. This has recently been successfully applied in the
construction of the conjectured continuum limit of the Aldous chain (\cite{Aldous2000}) in
\cite{LoehrMytnikWinter}, and of the continuum limit of the $\alpha=1$-Ford chain (\cite{Ford2005}) in
\cite{Nussbaumer}.
\item The convergence of sample subtree-mass tensor distributions allows to analyze the limit process with
stochastic analysis methods and gives more insight into the global structure of the evolving random
trees.
\end{itemize}
\medbreak
\noindent{\bf Related work. }As an alternative with better compactness properties to Gromov-Hausdorff
convergence of discrete trees, Curien suggested in \cite{Curien} to look at convergence of coding
triangulations (in Hausdorff metric topology). He also proposed to read off a measured, ordered,
\emph{topological tree} from the limit triangulation, and sketched the construction as quotient w.r.t.\ some
equivalence relation in the special case of the Brownian triangulation. Note, however, that the topological
information cannot be completely encoded by the triangulation, because the latter only encodes the algebraic
measure tree by Theorem~\ref{t:tree}, and the algebraic structure does not determine the topological structure
uniquely (see Example~\ref{ex:homhomeom}). Therefore, Curien did not obtain a general map from the space of
triangulations to a space of trees.
In order to turn the set of valuations on the ring $\mathbb{C}\gentree{x,y}$ into the so-called
\emph{valuative tree}, Favre and Jonsson use in \cite{FavreJonsson04} partial orders to define the tree structure.
Using partial orders is essentailly equivalent to using branch point maps, and under some additional
assumptions (separability, order completeness and edge-freeness), their \emph{nonmetric trees} are equivalent
to our algebraic trees. We want to stress, however, that for our theory the branch point map plays a much more
crucial role than the partial order.
The relation between partial orders and algebraic trees is further discussed in Section~\ref{S:trees}.
The random exchangeable \emph{didendritic systems} introduced recently by Evans, Gr\"ubel, and Wakolbinger in
\cite{EvansGruebelWakolbinger17} can be considered as rooted, ordered versions of binary algebraic measure trees
with diffuse measure on the set of leaves. A didendritic system is an equivalence relation on $\mathbb{N}} \newcommand{\M}{\mathbb{M}\times \mathbb{N}} \newcommand{\M}{\mathbb{M}$
together with two partial orders on the set of equivalence classes.
An exchangeable didendritic system is similar to our sequence of sample-shape distributions. The authors also
introduce a particular metric representation as an $\R$\nobreakdash-tree. Even though it is implicit in their work
that they think of the set of exchangeable didendritic systems as equipped with a kind of sample shape
convergence, they do not define it explicitly and do not analyze the resulting topological space.
Close relatives of algebraic measure trees have recently been studied independently by Forman in
\cite{Forman2018}. He uses ideas from \cite{FormanHaulkPitman2018} to represent rooted trees by so-called
\emph{hierarchies} (certain sets of subsets) on $\mathbb{N}} \newcommand{\M}{\mathbb{M}$, which are similar to the didendritic systems in
\cite{EvansGruebelWakolbinger17}, but unordered. Thus, exchangeable random hierarchies can be thought of as
rooted versions of algebraic measure trees. Forman shows that the resulting equivalence classes of rooted
measure $\R$\nobreakdash-tree s coincide with the so-called \emph{mass-structural} equivalence classes, which he defines by
bijections preserving intervals as well as masses of points, intervals and certain sub-trees.
He also singles out a particular representative, which he calls \emph{interval partition tree}, with the
essentially same metric as in \cite{EvansGruebelWakolbinger17} (not restricted to the binary case). This metric
follows a similar idea to but is different from our $r_\nu$.
Note that \cite{Forman2018} does not talk about convergence of trees or introduce a notion of ``continuum tree''
without a measure.
\medbreak
\noindent{\bf Outline. }The rest of the paper is organized as follows.
In Section~\ref{S:trees}, we introduce our concept of \emph{algebraic trees} by formalising the
branch point map as a tertiary operation on the tree. We also introduce an intrinsic Hausdorff topology and
characterize compactness (Proposition~\ref{p:compact}) and second countability (Proposition~\ref{p:separable}).
We show that under a separability constraint, algebraic trees can be seen as metric trees (subtrees of $\R$\nobreakdash-tree
s), where the metric structure has been ``forgotten'' (Theorem~\ref{t:algtreechar}), and give an example that
the separability condition cannot be dropped without replacement.
In Section~\ref{S:amt}, we introduce the space of (equivalence
classes of) order separable \emph{algebraic measure trees}, and equip it with the Gromov-weak topology with respect to
the metric associated with the branch point distribution. We show that the resulting space is separable and
metrizable (Corollary~\ref{c:bpdd-metrizable}).
Furthermore, we prove a Carath\'eodory-type extension theorem, which is helpful for constructing algebraic
measure trees (Propositions~\ref{p:caratheodory} and \ref{p:construction}).
In Section~\ref{S:triangulation}, we give a definition of \emph{triangulations of the circle}, and show that
they are precisely the limits of triangulations of $n$-gons (Proposition~\ref{p:fintriapp}).
We also formalize the notion of the algebraic measure tree associated with a given triangulation of the circle.
This correspondence has been allured to in the literature, but it has never been made
precise (except for finite trees), and it has never been shown a tree in what sense is coded by a
triangulation of the circle.
We show that the resulting \emph{coding map} (mapping triangulations to trees) is well-defined
and surjective onto the space of binary algebraic measure trees with non-atomic measure.
Furthermore, the coding map is \emph{continuous} if the space of triangulations is equipped with the Hausdorff metric
topology, and the space of trees with the bpdd-Gromov-weak topology (Theorem~\ref{t:tree}).
In Section~\ref{S:topo}, we consider the subspace of \emph{binary} algebraic measure trees, and introduce
two other, natural notions of convergence.
We use the construction of the coding map from Section~\ref{S:triangulation}
to show that on this subspace all three notions of convergence are actually equivalent and define the same
topology (Theorem~\ref{t:topeq}).
This topology turns the subspace of binary algebraic measure trees into a \emph{compact, metrizable} space,
which in particular implies that it is a closed subset of the space of algebraic measure trees.
In this section, we also finish the proof of Theorem~\ref{t:tree} by showing continuity of the coding map.
In Section~\ref{s:examples}, we consider the example of the continuum limits of sampling consistent families of
random trees and illustrate it with the example of so-called $\beta$-splitting trees introduced in
\cite{Aldous1996}. This family includes the uniform binary tree (converging to the Brownian CRT) and the Yule
tree (aka Kingman tree or random binary search tree).
\section{Algebraic trees}
\label{S:trees}
In this section we introduce algebraic trees. In Subsection~\ref{sub:trees}
we formalize the ``tree structure'' common to both graph-theoretic trees and metric trees by a function that
maps every triplet of points in the tree to the corresponding branch point. We show that the set of defining properties is rich enough to obtain known concepts such as leaves, branch points, degree, edges,
intervals, subtrees spanned by a set, discrete and continuum trees, etc. In Subsection~\ref{s:morphisms} we introduce the notion of structure preserving morphisms.
In Subsection~\ref{sub:astopological}
we equip algebraic trees with a canonical Hausdorff topology. We also characterize compactness and a concept we
call order separability, which is closely related to second countability of the topology.
Finally, in Subsection~\ref{sub:asRtree}, we show that any order
separable algebraic tree is induced by a metric tree (which is not true without order separability), and
establish the condition under which this metric tree can be chosen to be a compact $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree.
\subsection{The branch point map}
\label{sub:trees}
In this subsection we introduce algebraic trees. Recall from Definition~\ref{d:metrictree} the definition of a metric tree, and
the properties (BPM1)--(BPM4) of the map which sends a triplet of $3$ points in a metric tree to its branch point.
\begin{definition}[algebraic trees] \label{d:algebraic2}
An \define{algebraic tree} $(T,c)$ consists of a set $T\not=\emptyset$ and
a branch point map $c\colon T^3\to T$ satisfying (BPM1)--(BPM4).
\end{definition}
The following useful property reflects the fact that any four points in an algebraic tree can be associated with
a shape as illustrated in Figure~\ref{f:4pt} above.
\begin{lemma}[a consequence of (BPM4)]
Let\/ $(T,c)$ be an algebraic tree. Then for all\/ $x_1,x_2,x_3,x_4\in T$
the following hold:
\begin{enumerate}
\item\label{i:4ptsplit} If\/ $c(x_1,x_2,x_3)=c(x_1,x_2,x_4)$, then\/ $c(x_1,x_3,x_4)=c(x_2,x_3,x_4)$.
\item If\/ $c(x_1,x_2,x_3)=c(x_1,x_2,x_4)$, then\/ $c(x_1,x_2,x_3)=c(x_1,x_2,c(x_1,x_3,x_4))$.
\end{enumerate}
\label{l:4pt}
\end{lemma}
\begin{proof}
Let $x_1,x_2,x_3,x_4\in T$ with $c_1:=c(x_1,x_2,x_3)=c(x_1,x_2,x_4)$, and $c_2:=c(x_1,x_3,x_4)$.
\smallskip
\emph{(i) }
Condition (BPM4) implies that
\begin{equation}
c_2\in \big\{c_1=c(x_1,x_3,x_2),\, c(x_2,x_3,x_4),\, c_1=c(x_1,x_2,x_4)\big\}.
\end{equation}
Thus $c_1=c_2$, or $c_2=c(x_2,x_3,x_4)$. The second case is the claim.
In the first case, we apply Condition (BPM4) once more to find that
\begin{equation}\label{e:006}
c(x_2,x_3,x_4)
\in
\big\{c_1=c(x_1,x_2,x_3),\, c_2=c(x_1,x_3,x_4),\, c_1=c(x_1,x_2,x_4)\big\}
=\{c_1,c_2\}=\{c_2\},
\end{equation}
so that the claim also holds in this case.
\smallbreak
\emph{(ii) } Condition~(BPM3) implies that
\begin{equation}
c(x_1,x_3,c_2) = c\big(x_1,\, x_3,\, c(x_1,x_3,x_4)\big)=c(x_1,x_3,x_4)=c_2,
\end{equation}
and similarly also $c(x_2,x_3,c_2)=c(x_2,x_3,x_4)=c_2$.
Now part \emph{\ref{i:4ptsplit}} with $x_4$ replaced by $c_2$ yields $c(x_1,x_2,x_3)=c(x_1,x_2,c_2)$ as claimed.
\end{proof}
We have seen that the four axiomatizing properties of the branch point map are
necessary. In many respects they are also sufficient to capture the tree structure.
For example, in analogy to \eqref{e:arc} we can
define for each $x,y\in T$ the \emph{interval} $[x,y]$ by
\begin{equation}\label{e:path}
[x,y]:=\big\{w\in T:\,c(x,y,w)=w\big\}.
\end{equation}
We also use the notation $\openint xy := [x,y] \setminus\{x,y\}$, and similarly $\lopenint xy$, $\ropenint xy$.
The following properties of intervals are known to hold in $\R$\nobreakdash-trees\ (compare, e.g., to
\cite[Chapter~2]{Chiswell2001} or \cite[Chapter~3]{Evans2008}):
\begin{lemma}[properties of intervals]
Let\/ $(T,c)$ be an algebraic tree. Then the following hold:
\begin{enumerate}
\item If\/ $x,v,w,z\in T$ are such that\/ $w\in[x,z]$ and\/ $v\in[x,w]$, then\/ $v\in[x,z]$.
\item If\/ $x,y,z\in T$, then
\begin{equation} \label{e:010}
[x,y] \cap [y,z] = \bigl[c(x,y,z),\, y\bigr].
\end{equation}
In particular,
\begin{equation}
\label{e:011}
\bigl[x,\,c(x,y,z)\bigr]\cap \bigl[c(x,y,z),\,z\bigr] = \bigl\{c(x,y,z)\bigr\}.
\end{equation}
\item If\/ $x,y,z\in T$, then
\begin{equation}
\label{e:012}
[x,y] \cup [y,z] = [x,z] \uplus \blopenint{c(x,y,z)}{y}.
\end{equation}
In particular,
\begin{equation}
\label{e:013}
[x,y] \cup [y,z] = [x,z] \quad\mbox{iff}\quad y\in[x,z].
\end{equation}
\item For all $x,y,z\in T$,
\begin{equation} \label{e:014}
[x,y]\cap [y,z]\cap [z,x] = \bigl\{c(x,y,z)\bigr\}.
\end{equation}
\end{enumerate}
\label{l:intprop}
\end{lemma}
\begin{proof} \emph{(i) } Let $x,v,w,z\in T$ with $w=c(x,w,z)$ and $v=c(x,v,w)$. Then by Condition~(BPM4),
\begin{equation}
\label{e:009}
c(x,v,z) \in \bigl\{c(x,v,w),\,w=c(x,w,z),\,c(v,w,z)\bigr\}.
\end{equation}
We discuss the three cases separately.
If $c(x,v,z)=c(x,v,w)$, then $c(v,w,z)=c(x,w,z)=w$ by Lemma~\ref{l:4pt}(i).
It then follows that $c(x,v,z)=c(x,v,c(x,w,z))=c(x,v,w)=v$ by Lemma~\ref{l:4pt}(ii), which gives the claim in this case.
If $c(x,v,z)=w$ then $v=c(v,w,x)=c(v,w,z)$ by Lemma~\ref{l:4pt}(i). It then follows that $c(x,v,z)=c(x,z,c(z,w,v))=c(x,v,v)=v$ by Lemma~\ref{l:4pt}(ii), which gives the claim in this case.
If $c(x,v,z)=c(v,w,z)$ then $v=c(x,w,v)=c(x,w,z)=w$ by Lemma~\ref{l:4pt}(i).
Thus $v=w\in[x,z]$, and the claim holds also in this case.
\smallbreak
\emph{(ii) }Let $x,y,z\in T$, and $v\in[x,y]\cap[y,z]$. That is,
$v=c(x,v,y)=c(y,v,z)$. It follows from Lemma~\ref{l:4pt}(i) that
$c(x,z,v)=c(x,z,y)$, and then from Lemma~\ref{l:4pt}(ii) together with Condition~(BPM2) that
\begin{equation}
\label{e:008}
v=c(x,v,y)=c\big(v,y,c(y,x,z)\big).
\end{equation}
Equivalently, $v\in[c(x,y,z),y]$. This proves the inclusion $[x,y]\cap[y,z]\subseteq[c(x,y,z),y]$. The other inclusion follows from (i).
Notice that \eqref{e:011} follows from \eqref{e:010} with the special choice $y=c(x,y,z)$.
\smallbreak
\emph{(iii) } Notice first that it follows immediately from (i) that the union on the right hand side is disjoint.
We claim that
\begin{equation}
\label{e:sclaim}
[x,z] \subseteq [x,y] \cup [y,z].
\end{equation}
Indeed, let $v\in [x,z]$, i.e.\ $c(x,z,v)=v$. Then by (BPM4) applied to $\{v,x,y,z\}$,
\begin{equation}
\label{e:004}
v=c(x,z,v) \in \big\{c(x,y,v),\, c(x,y,z),\, c(y,z,v)\big\},
\end{equation}
which implies that $v\in[x,y]$ (if $v=c(x,y,v)$) or $v\in[x,z]\cap[x,y]$ (if $v=c(x,y,z)$) or $v\in[y,z]$ (if $v=c(y,z,v)$).
Second, we claim that for all $x,y,z\in T$,
\begin{equation}
[x,z] \cup \bigl[c(x,y,z),\,y\bigr] \subseteq [x,y] \cup [y,z].
\end{equation}
To see this, recall from (ii) that $[c(x,y,z),y]= [x,y]\cap[z,y]\subseteq [x,y]\cap[z,y]$.
As $[x,c(x,y,z)]\subseteq[x,y]$ by (i), we have $[x,y]\subseteq [x,c(x,y,z)]\cup
[c(x,y,z),y]\subseteq[x,y]\uplus\lopenint{c(x,y,z)}{y}$. The corresponding inclusion for $[y,z]$
is shown in the same way, and we have proven Equation \eqref{e:012}.
\smallbreak
\emph{(iv) } This follows immediately from (ii).
\end{proof}
We say that $\{x,y\}\subseteq T$ with $x\not=y$ is an \emph{edge} of $(T,c)$ if and only if there is ``nothing
in between'', i.e.\ $[x,y]=\{x,y\}$, and denote by
\begin{equation}\label{e:ed}
\edge(T,c) := \bset{\{x,y\}\subseteq T}{x\ne y,\;[x,y] = \{x,y\}}
\end{equation}
the \emph{set of edges}. The following example explains that there is no need to distinguish between finite
algebraic trees and graph-theoretical trees, and the definitions of edges are consistent.
\begin{example}[finite algebraic trees correspond to graph-theoretic trees]
Finite algebraic trees are in one to one correspondence with finite (undirected) graph-theoretic trees.
Let $(T,E)$ be a graph-theoretic tree with vertex set $T$ and edge set $E$. Then $(T,E)$ corresponds to
the algebraic tree $(T,c_E)$ with $c_E(u,v,w)$ defined as the unique vertex that is on the
(graph-theoretic) path between any two of $u,v,w$. Conversely, if $(T,c)$ is an algebraic tree with $T$
finite, then $(T,c)$ corresponds to the graph-theoretic tree $(T,E_c)$ with $E_c:=\edge(T,c)$.
Obviously, $c_{E_c}=c$.
\label{ex:finite}
\end{example}
For a graph-theoretic tree $(T,E)$, we can allow the vertex set $T$ to be countably infinite, and still obtain a
corresponding algebraic tree as in the previous example.
Note, however, that countable algebraic trees do not necessarily correspond to graph-theoretic trees. Indeed, it
is possible that $T$ is countably infinite and $\edge(T,c)=\emptyset$. This can be seen by taking $T=\mathbb{Q}$ in the
following example, which shows that every totally ordered space naturally corresponds to an algebraic tree.
\begin{example}[totally ordered spaces as algebraic trees]\label{ex:totord}
For a totally ordered space $(T,\le)$, define $c_\le(x,y,z) := y$ whenever $x\le y \le z$, ($x,y,z\in
T$). Then it is trivial to check that $(T,c_\le)$ is an algebraic tree and the interval $[x,y]$
coincides with the order interval $\set{z\in T}{x\le z \le y}$.
\end{example}
Conversely, given an algebraic tree $(T,c)$ and a distinguished point $\rho$ (often referred to as \emph{root}), we can
define a \emph{partial order} $\le_\rho$ by letting for $x,y\in T$,
\begin{equation}
\label{e:partialrho}
x\le_\rho y\quad\mbox{ iff }x\in[\rho,y].
\end{equation}
Partial orders provide an equivalent way of defining algebraic trees.
\begin{proposition}[algebraic trees and semi-lattices]\label{p:semilat}
\begin{enumerate}
\item Let\/ $(T,c)$ be an algebraic tree, and\/ $\rho\in T$. Then\/ $(T,\le_\rho)$ is a partially ordered
set, and a meet semi-lattice with infimum
\begin{equation}\label{e:inf}
x\land y = c(\rho,x,y) \qquad \forall x,y \in T.
\end{equation}
Furthermore, $\le_\rho$ is a total order on\/ $[\rho,x]$ for all\/ $x\in T$.
\item Let\/ $(T,\le)$ be a partially ordered set, such that all initial segments\/ $\set{y\in T}{y\le x}$, $x\in
T$, are totally ordered (in particular a meet semi-lattice). Then for\/ $x,y,z\in T$
\begin{equation}\label{e:cfrominf}
c(x,y,z) := \max\{x\land y,\, y\land z,\, z\land x\}
\end{equation}
exists, and\/ $(T,c)$ is an algebraic tree.
\end{enumerate}
\end{proposition}
\begin{proof}\proofcase{(i)}
Let $x,y\in T$ with $x\le_\rho y$ and $y\le_\rho x$. That is, $x=c(\rho,x,y)$ and $y=c(\rho,y,x)$ which implies that $x=y$, and proves that $\le_\rho$ is \emph{antisymmetric}.
As $x=c(\rho,x,x)$, $x\le_\rho x$ which proves that $\le_\rho$ is \emph{reflexive}. Finally, to show \emph{transitivity}, let $x,y,z\in T$ with $x\le_\rho y$ and $y\le_\rho z$. That is $x\in[\rho,y]$ and $y\in[\rho,z]$, which implies that $x\in[\rho,z]$ by Lemma~\ref{l:intprop}(i). Equivalently, $x\le_\rho z$
which proves the transience, and thus that $\le_\rho$ is a partial order.
For the \emph{infimum}, notice that $v\le_\rho x$ and $v\le_\rho y$ if and only if $v\in[\rho,x]\cap[\rho,y]$, or equivalently by Lemma~\ref{l:intprop}(ii), $v\in[\rho,c(\rho,x,y)]$.
As for all $v\in[\rho,c(\rho,x,y)]$ we have $v\le c(\rho,x,y)$, the claim \eqref{e:inf} follows.
Fix $x\in T$. For \emph{totality} on $[\rho,x]$, let $v,w\in[\rho,x]$, i.e., $v=c(\rho,v,x)$ and $w=c(\rho,w,x)$. Applying Condition (BPM4) to $\{\rho,v,w,x\}$ we find that one of the following three cases must occur: $c(\rho,v,w)=c(\rho,v,x)$ (which implies that $v=c(\rho,v,w)$, or equivalently, $v\le_\rho w$), $c(\rho,w,v)=c(\rho,w,x)$ (which implies that $w=c(\rho,w,v)$, or equivalently, $w\le_\rho v$), or $c(\rho,x,v)=c(\rho,x,w)$ (which implies that $w=v$).
\proofcase{(ii)} The maximum in \eqref{e:cfrominf} is over a totally ordered set (because initial segments are
totally ordered), thus exists. Furthermore, if $x\land y\le x\land z \le y\land z$, say, we also obtain $x\land y =
x\land y \land z \ge x\land z$. This means that (at least) two of $x\land y$, $y\land z$, $z\land x$ are
identical, and the maximum $c(x,y,z)$ is the third one. That $v$ satisfies (BPM1)--(BPM3) is obvious. To see the
4-point condition (BPM4), let $x_1, \ldots, x_4\in T$ and assume w.l.o.g.\ that $x_2\land x_3=c(x_1, x_2, x_3)=:v$, and hence $x_1\land x_2 = x_1\land x_3 \le v$.
We distinguish cases: If $x_2\land x_4 <v$, then $c(x_2, x_3, x_4)=\max\{x_2\land x_4, v, x_3\land x_4\} = v$,
and \eqref{e:BPM4} is satisfied.
Otherwise, $x_2\land x_4, x_3\land x_4 \ge v$ and at most one of them can be strictly larger.
If $x_2 \land x_4 > v \ge x_1 \land x_2$, then $x_1\land x_2 = x_1 \land x_4 = x_1\land x_3$,
and $c(x_1, x_3, x_4)=x_3\land x_4 = v$. The case $x_3 \land x_4 >v$ is analogous.
In the last case $x_2\land x_4 = x_3 \land x_4 = v$, and $c(x_2, x_3, x_4)=v$.
\end{proof}
\begin{remark}[Favre and Jonsson's nonmetric trees]
In \cite{FavreJonsson04}, \emph{rooted nonmetric trees} are introduced as partially ordered sets with
global minimum, totally ordered initial segments, and the additional property that all full, totally ordered
subsets are order isomorphic to a real interval. Proposition~\ref{p:semilat} shows that they naturally
induce algebraic trees.
\end{remark}
\begin{cor} Let\/ $(T,c)$ be an algebraic tree, and\/ $\rho,x,y\in T$. If\/ $v\in[x,y]$, then\/ $v\ge_\rho c(x,y,\rho)$.
\label{l:vgec}
\end{cor}
\begin{proof} Let $\rho,x,y\in T$ and $v\in[x,y]$. That is, $v=c(x,v,y)$.
We need to show that $c(\rho,v,c(\rho,x,y))=c(\rho,x,y)$.
By Condition~(BPM4) applied to $\{x,y,\rho,v\}$ we have one of the following three cases:
$c(x,y,\rho)=c(x,y,v)$ (in which case $c(\rho,x,y)=v$) or $c(\rho,y,x)=c(\rho,y,v)$ (in which case $c(x,v,\rho)=c(x,v,y)=v$ by Lemma~\ref{l:4pt}(i) and thus $v\in[\rho,x]$; the claim then follows since this implies that $v\in[\rho,x]\cap[x,y]=[c(\rho,x,y),y]$ by Lemma~\ref{l:intprop}(ii)), or $c(\rho,x,y)=c(\rho,x,v)$ (in which we conclude similar to the second case that $v\in[c(\rho,x,y),x]$).
\end{proof}
The partial orders $\le_\rho$ allow us to define a notion of completeness of algebraic trees.
\begin{definition}[directed order completeness]
Let $(T,c)$ be an algebraic tree. We call $(T,r)$ \define{(directed) order complete} if for all $\rho\in
T$ the supremum of every totally ordered, non-empty subset exists in the partially ordered set
$(T,\le_\rho)$.
\label{d:ordercomplete}
\end{definition}
Obviously, in an order complete algebraic tree, infima of totally ordered sets exists, because they are
either $\rho$ if the set is empty or a non-empty supremum w.r.t.\ a different root.
This notion of completeness allows us to define the analogs of complete $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-trees.
\begin{definition}[algebraic continuum tree] \label{d:aCT}
We call an algebraic tree $(T,c)$ \define{algebraic continuum tree} if the following two conditions hold:
\begin{enumerate}[\axiom({A}CT1)]
\item $(T,c)$ is order complete.
\item $\edge(T,c)=\emptyset$.
\end{enumerate}
\end{definition}
\subsection{Morphisms of algebraic trees} \label{s:morphisms}
Like any decent algebraic structure (or in fact mathematical structure), algebraic trees come with a notion of
structure-preserving morphisms.
\begin{definition}[morphisms]
Let $(T,c)$ and $(\widehat{T},\hat{c})$ be algebraic trees. A map $f\colon T\to \widehat{T}$
is called a \define{tree homomorphism} (from $T$ into $\widehat{T}$) if for all $x,y,z\in T$,
\begin{equation}
\label{e:treehom}
f\bigl(c(x,y,z)\bigr) = \hat{c}\big(f(x),f(y),f(z)\big).
\end{equation}
We refer to a bijective tree homomorphism as \define{tree isomorphism}.
\label{d:treehom}
\end{definition}
As we have seen, the tree structure can be expressed also in terms of
intervals or partial orders rather than the branch point map. This also works for the morphisms.
\begin{lemma}[equivalent definitions of morphisms]
Let\/ $(T,c)$ and\/ $(\widehat{T},\hat{c})$ be algebraic trees, and\/ $f\colon T\to\widehat{T}$. Then
the following are equivalent:
\begin{enumerate}[1.]
\item\label{i:homom} $f$ is a tree homomorphism.
\item\label{i:order} For all\/ $\rho\in T$, $f$ is an order preserving map from\/ $(T,\le_\rho)$ to\/ $(\widehat{T},\le_{f(\rho)})$.
\item\label{i:interval} For all\/ $x,y\in T$, $f([x,y])\subseteq[f(x),f(y)]$.
\end{enumerate}
\label{l:treehom}
\end{lemma}
\begin{proof}
\istep{\ref{i:homom}}{\ref{i:order}} Let $x,y,\rho\in T$ with
$x\le_\rho y$. Then $x=c(\rho,x,y)$ and thus
$f(x)=\hat{c}(f(\rho),f(x),f(y))$. Therefore $f(x)\le_{f(\rho)} f(y)$.
\istep{\ref{i:order}}{\ref{i:interval}} Let $x,y,z\in T$ with $z\in [x,y]$. Then $z\le_x y$ and thus $f(z) \le_{f(x)} f(y)$, i.e.\ $f(z) \in [f(x),f(y)]$.
\istep{\ref{i:interval}}{\ref{i:homom}} Let $x,y,z\in T$. Then $\{c(x,y,z)\}=[x,y]\cap[x,z]\cap[y,z]$. Hence
\begin{equation}
\big\{f(c(x,y,z))\big\}\subseteq \big[f(x),f(y)\big]\cap \big[f(y),f(z)\big]\cap \big[f(x),f(z)\big]
= \bigl\{ \hat{c}\(f(x),f(y),f(z)\)\bigr\}.
\end{equation}
Therefore, $f(c(x,y,z))=\hat{c}(f(x),f(y),f(z))$.
\end{proof}
Lemma~\ref{l:treehom} shows that our notion of morphisms of algebraic trees is weaker than the morphisms of
nonmetric trees used in \cite{FavreJonsson04}, but the notion of isomorphism is the same.
The image of an algebraic tree under a tree homomorphism is a subtree in the following sense.
\begin{definition}[subtree]
Let $(T,c)$ be an algebraic tree, and $\emptyset\ne A\subseteq T$.
$A$ is called a \define{subtree (of\/ $(T,c)$)} if
\begin{equation}
\label{e:subtree}
c(A^3)\subseteq A.
\end{equation}
We refer to $c(A^3)$ as the \define{algebraic subtree generated by\/ $A$}.
\label{d:subtree}
\end{definition}
Obviously, a subtree $A$ of $(T,c)$, implicitly equipped with the restriction of $c$ to $A^3$, is an algebraic
tree in its own right. Furthermore, the following lemma is easy to check.
\begin{lemma}[tree homomorphisms]
Let\/ $(T,c)$ and\/ $(\widehat{T},\hat{c})$ be two algebraic trees, and\/ $f\colon T \to \widehat{T}$ a homomorphism.
Then the image\/ $f(T)$ is a subtree of\/ $\widehat{T}$.
If\/ $f$ is injective, $f^{-1}$ is a tree homomorphism from\/ $f(A)$ into\/ $T$.
In particular, if\/ $(\tilde{T},\tilde{c})$ is another algebraic tree, and\/ $g$ is a homomorphism form\/
$(\widehat{T},\hat{c})$ to\/ $(\tilde{T},\tilde{c})$, then\/ $g\circ f$ is a homomorphism from\/ $(T,c)$ to\/
$(\tilde{T},c_{\tilde{T}})$.
\label{l:aa}
\end{lemma}
\subsection{Algebraic trees as topological spaces}
\label{sub:astopological}
In contrast to metric trees, there is a priori no topology defined on a given algebraic tree.
In this section, we therefore equip algebraic trees with a canonical topology.
For each $x\in T$, we introduce a (component) relation $\sim_x$ by letting $y\sim_x z$
if and only if $x\not\in[y,z]$, where $y,z\in T$. Let for each $y\in T\setminus\{x\}$
\begin{equation} \label{e:005}
\mathcal{S}_x(y) = \mathcal{S}^{(T,c)}_x(y) := \bset{z\in T\setminus \{x\}}{z\sim_x y}
\end{equation}
be the equivalence class of $T\setminus\{x\}$ containing $y$, and
note that $\mathcal{S}_x(y)$ is a subtree for all $x,y\in T$, and $\mathcal{S}_x(y)=\mathcal{S}_x(z)$ whenever $z\in
\mathcal{S}_x(y)$.
We refer to $\mathcal{S}_x(y)$ as the \emph{component} of $T\setminus x$ containing $y$.
Now and in the following, we equip $(T,c)$ with the topology
\begin{equation}
\label{e:tau}
\tau := \tau\(\bset{\mathcal{S}_x(y)}{x,y\in T,\; x\ne y}\)
\end{equation}
generated by the set of components, i.e.\ with the coarsest topology such that all components are open sets. We
call $\tau$ the \define{component topology} of $(T,c)$.
\begin{example}[on totally ordered trees, $\tau$ is the order topology]
If $(T,\le)$ is a totally ordered space, and $(T,c_\le)$ the corresponding algebraic tree as in
Example~\ref{ex:totord}, then $\tau$ coincides with the \emph{order topology}
(i.e.\ the one generated by sets of the form $\set{y\in T}{y>x}$ and $\set{y\in T}{y<x}$ for $x\in T$).
\end{example}
\begin{example}[intervals are closed sets] \label{exp:002}
Let $(T,c)$ be an algebraic tree, and $x,y\in T$. Then
\begin{equation}
\label{e:030}
T\setminus [x,y] = \bigcup\bset{\mathcal{S}_u(v)}{u\in[x,y],\,v\in T,\, \mathcal{S}_u(v)\cap[x,y]=\emptyset}\in\tau.
\end{equation}
This means that $[x,y]$ is closed in the component topology $\tau$.
\end{example}
\begin{lemma}\label{l:ccont}
Let\/ $(T,c)$ be an algebraic tree. Then\/ $c$ is continuous w.r.t.\ the component topology\/ $\tau$.
\end{lemma}
\begin{proof}
By definition of $\tau$, it is sufficient to show that for any $x,y\in T$, $x\ne y$, the set
$c^{-1}(\mathcal{S}_x(y))$ is open in $T^3$. By definition of $\mathcal{S}_x(y)$ and the property $c(u,v,w)\in
[u,v]\cap [v,w]\cap [w,u]$ shown in Lemma~\ref{l:intprop}, $c(u,v,w) \in \mathcal{S}_x(y)$ if and only if (at
least) two of $u,v,w$ are in $\mathcal{S}_x(y)$. Because $\mathcal{S}_x(y)$ is open, the same is true for
$\set{(u,v,w)\in T^3}{u,v\in \mathcal{S}_x(y)}$ in the product topology. Hence $c^{-1}(\mathcal{S}_x(y))$ is a union
of open set and thus open.
\end{proof}
Next, we show that $\tau$ is a Hausdorff topology and characterize compactness of algebraic trees in this
topology.
\begin{lemma}[$\tau$ is Hausdorff]\label{l:Hausdorff}
Let\/ $(T,c)$ be an algebraic tree. Then the component topology\/ $\tau$ defined in \eqref{e:tau} is a
Hausdorff topology on\/ $T$.
\end{lemma}
\begin{proof}
To show that $(T,\tau)$ is Hausdorff, let $x,y\in T$ be distinct. If $\mathcal{S}_y(x) \cap \mathcal{S}_x(y) =
\emptyset$, then $\mathcal{S}_y(x)$ and $\mathcal{S}_x(y)$ are clearly disjoint neighbourhoods of $x$ and $y$,
respectively. Assume that this is not the case, and choose $z\in \mathcal{S}_x(y)\cap \mathcal{S}_y(x)$. Then
$\rho:=c(x,y,z)\not\in\{x,y\}$. Furthermore, $c(x,\rho,y)=c(x,y,z)=\rho$, and hence $x\not\sim_\rho y$.
Thus $\mathcal{S}_\rho(x)$ and $\mathcal{S}_\rho(y)$ are disjoint neighbourhoods of $x$ and $y$, respectively.
Hence $\tau$ is Hausdorff.
\end{proof}
\begin{proposition}[characterizing compactness]\label{p:compact}
Let\/ $(T,c)$ be an algebraic tree with component topology $\tau$.
Then\/ $(T,\tau)$ is compact if and only if\/ $(T,c)$ is directed order complete.
\end{proposition}
\begin{proof}
\pstep{``only if''} Assume first that $(T,c)$ is not order complete. Then we can choose $\rho\in T$ and
$\emptyset\ne A\subseteq T$ such that $A$ is totally ordered w.r.t.\ $\le_\rho$ but does not have a
supremum in $(T,\le_\rho)$.
For $x,y\in T$, let $U_x:= \set{z\in T}{z\not\ge_\rho x}$ and $V_y:=\set{z\in T}{z>_\rho y}$. Then $U_x$
and $V_y$ are open sets.
We claim that $\mathcal{U}:=\set{U_x}{x\in A} \cup \set{V_y}{y\ge A}$ is an open cover of $T$. Indeed, if
$z\ge_\rho A$, then, because $A$ has no supremum, there is $y\in T$ with $A\le_\rho y \le_\rho z$, hence
$z\in V_y \in \mathcal{U}$. Otherwise, if $z\not\ge_\rho A$, there is $x\in A$ with $z\in U_x \in \mathcal{U}$. Thus $\mathcal{U}$
is a cover of $T$.
$\mathcal{U}$ has no finite sub-cover, because if $\mathcal{U}' = \{U_{x_1},...,U_{x_n},V_{y_1},...,V_{y_m}\}$
were such a finite sub-cover, then $\{U_{x_1},...,U_{x_n}\}$ would cover $A$. This, however, would imply that
$\max\{x_1,...,x_n\}$ would be a supremum of $A$, contradicting our assumption. Hence $(T,\tau)$ is not
compact.
\pstep{``if''} Assume that $(T,c)$ is order complete. Consider a cover $\mathcal{U}$ of $T$ with components,
i.e.\ $\mathcal{U}\subseteq \set{\mathcal{S}_y(x)}{x,y\in T,\, x\ne y}$. By the Alexander subbase theorem, for
compactness of $\tau$, it is sufficient to show that $\mathcal{U}$ has a finite sub-cover.
To this end, fix an element $\rho\in T$ and consider the set $\mathcal{U}_\rho:=\set{U\in \mathcal{U}}{\rho \in
U}\not=\emptyset$. By Hausdorff's maximal chain theorem (or Zorn's lemma), there is a maximal chain $I$
in the partially ordered set $(\mathcal{U}_\rho, \subseteq)$.
For every $U\in I$, we have $\rho\in U$, and thus there is $x_U\in T$ such that $U=\mathcal{S}_{x_U}({\rho})$.
We claim that $U\subseteq V$ implies $x_U \le_\rho x_V$. Indeed, $x_V\not\in V$
and hence $x_V\not\in U$ which is equivalent to $x_V \ge_\rho x_U$. Therefore, $z:= \sup \set{x_U}{U\in I}$
exists in $(T,\le_\rho)$ by directed order completeness of $T$.
Because $\mathcal{U}$ is a cover, there is
$V\in \mathcal{U}$ with $z\in V$, hence $V=\mathcal{S}_y(z)$ for some $y\in T$.
Because $V\not\in I$ and $I$ is a maximal chain, $y\not\ge_\rho z$.
Hence there is $U\in I$ with
$y\not\ge_\rho x_U=:x$. We claim that $T=\mathcal{S}_y(z)\cup \mathcal{S}_x(\rho)$.
Indeed, let $w\in T\setminus \mathcal{S}_x(\rho)$. Then $w\ge_\rho x$. Using $z\ge_\rho x$ and $c(w,z,y)\in [w,z]$,
we obtain $c(w,z,y) \ge_\rho x$, and hence $c(w,z,y) \ne y$. Thus $w\in \mathcal{S}_y(z)$ as claimed, and
$\{\mathcal{S}_y(z),\,\mathcal{S}_x(\rho)\}$ is the desired sub-cover.
\end{proof}
It turns out that the following separability condition, which we call order separability,
is crucial for us.
\begin{proposition}[order separability]\label{p:separable}
Let\/ $(T,c)$ be an algebraic tree with component topology $\tau$.
Then the following are equivalent:
\begin{enumerate}[1.]
\item\label{i:dense} There exists a countable set\/ $D$ such that for all\/ $x,y\in T$ with\/ $x\not=y$,
\begin{equation} \label{e:sep}
D\cap \ropenint{x}{y} \ne \emptyset.
\end{equation}
\item\label{i:A2} The topological space\/ $(T,\tau)$ is second countable (i.e.\ $\tau$ has a countable base),
and\/ $\edge(T,c)$ is countable.
\item\label{i:sepa} The topological space\/ $(T,\tau)$ is separable, and\/ $\edge(T,c)$ is countable.
\end{enumerate}
\end{proposition}
\begin{proof}
\istep{\ref{i:sepa}}{\ref{i:dense}} Assume that $\edge(T,c)$ is countable, and
that $(T,\tau)$ is separable. Then there exists a countable, dense subset $\tilde{D}\subseteq T$. We claim that
\begin{equation}\label{e:orderdense}
D:=c\big(\tilde{D}^3\big)\cup\big\{z\in T:\,\exists\,x\in T\mbox{ such that }\{x,z\}\in\edge(T,c)\big\}
\end{equation}
satisfies \eqref{e:sep}. Indeed, $D$ is countable by assumption. Moreover,
let $x,y\in T$. Then two cases are possible:
either $\mathcal{S}_x(y)\cap\mathcal{S}_y(x)=\emptyset$. In this case, $\{x,y\}\in\edge(T,c)$, which implies that
$\ropenint{x}{y} \cap D\not=\emptyset$.
Or $\mathcal{S}_x(y)\cap\mathcal{S}_y(x)\not=\emptyset$.
In this case, as $\mathcal{S}_x(y)\cap\mathcal{S}_y(x)$ is open by definition of $\tau$, there is $z\in
\tilde{D}\cap \mathcal{S}_x(y)\cap\mathcal{S}_y(x)$. Let $v:=c(x,y,z)$. Then $v\in\openint{x}{y}$, and either
$v=z\in D$, or the three components $\mathcal{S}_v(x)$, $\mathcal{S}_v(y)$, $\mathcal{S}_v(z)$ are distinct. In the second case, we
can choose $x'\in \tilde{D} \cap \mathcal{S}_v(x)$ and $y'\in \tilde{D}\cap \mathcal{S}_v(y)$ to see that $v=c(x',y',z) \in
D$. In any case, $v\in \ropenint{x}{y} \cap D$.
\istep{\ref{i:dense}}{\ref{i:A2}}
Let $D$ be a countable set satisfying \eqref{e:sep}. Then for all $\{x,y\}\in\edge(T,c)$, $D\cap
\ropenint{x}{y}=\{x\}$. This implies that $\edge(T,c)$ is countable.
We consider the countable set $\mathcal{U}=\bset{\mathcal{S}_v(u)}{u,v\in D} \subseteq \tau$ and claim that it is a subbase for
$\tau$ (i.e.\ generates $\tau$). To this end, let $x,y\in T$. We show that $U:=\mathcal{S}_x(y)$ is a union of sets
from $\mathcal{U}$, i.e.\ for every $z\in U$ we construct $V\in\mathcal{U}$ with $z\in V \subseteq U$. By assumption on $D$, there
is $v\in D \cap \ropenint{x}{z}$ and $u\in D \cap \lopenint{v}{z}$. Let $V:=\mathcal{S}_v(u) \in \mathcal{U}$. Because
$c(u,v,z)=u\ne v$, we have $z\in V$. Let $w\in T\setminus U$. Because $u\in U$, we have $U=\mathcal{S}_x(u)$ and
therefore $x\in[u,w]$. Similarly, $x\in [v,w]$. In particular, by Lemma~\ref{l:4pt},
$c(u,v,w)=c(u,v,c(u,x,w))=c(u,v,x)=v$, and thus $w\not\in V$.
Because $w\in T\setminus U$ is arbitrary, we obtain $V\subseteq U$.
\istep{\ref{i:A2}}{\ref{i:sepa}} Trivial, because every second countable topological space is separable.
\end{proof}
\begin{definition}[order separability]\label{d:ordersep}
We call an algebraic tree $(T,c)$ \define{order separable} if the equivalent conditions of
Proposition~\ref{p:separable} are satisfied. We call a set $D\subseteq T$ \define{order dense} if it
satisfies \eqref{e:sep}.
\end{definition}
\begin{example}[uncountable star tree] This example shows that in \eqref{e:sep}
we can not replace $\ropenint{x}{y}$ by $[x,y]$. Let $T:=\{0\}\cup [1,2]$ with $c(x,y,z):=0$ whenever $x,y,z\in
T$ are distinct. Then if $D\subseteq T$ is such that $D\cap\ropenint{x}{0}\not =\emptyset$ for all $x\in[1,2]$
then $[1,2]\subseteq D$, and thus $D$ is uncountable and $(T,c)$ not order separable.
On the other hand, $D:=\{0\}$ has the property that $D\cap[x,y]\not=\emptyset$ for all $x,y\in T$ with $x\not
=y$.
\label{exp:star}
\end{example}
An order complete, order separable algebraic tree is, in its component topology $\tau$, a compact, second
countable Hausdorff space by Propositions~\ref{p:compact} and \ref{p:separable}. In particular, it is metrizable.
In fact, order separability already implies metrizability, as we will see in Subsection~\ref{sub:asRtree}.
The following example shows that (topological) separability of $(T,\tau)$ alone, without requiring the number of
edges to be countable, is neither sufficient for order separability nor for metrizability of $(T,\tau)$.
\begin{example}[a continuum ladder]\label{ex:topnotalgsep}
Let $T=[0,1]\times\{0,1\}$ with the lexicographic order $\le$ on $T$, and define the canonical branch
point map $c_\le$ as in Example~\ref{ex:totord}. Then $\edge(T,c_\le) =
\bset{\{(x,0),\,(x,1)\}}{x\in [0,1]}$ is uncountable, and hence $(T,c_\le)$ is not order separable.
Because $(\mathbb{Q}\cap[0,1]) \times \{0,1\}$ is a countable dense set, $(T,\tau)$ is (topologically) separable.
The topological subspace $[0,1] \times \{1\}$ is the \emph{Sorgenfrey line},
which is known to be non-metrizable (see \cite[Counterexample~51]{SteenSeebach78}). Thus also $(T,\tau)$
cannot be metrizable.
\end{example}
\begin{definition}[Borel $\sigma$-algebra $\mathcal{B}(T,c)$]
Let $(T,c)$ be an algebraic tree. We denote the Borel $\sigma$-algebra of the component topology $\tau$
by $\mathcal{B}(T,c)$ and call it \define{Borel\/ $\sigma$-algebra of\/ $(T,c)$}.
\end{definition}
In general, $\mathcal{B}(T,c)$ is not generated by the set of components. Order separability, however, is sufficient to
ensure this property because it implies second countability of the component topology.
\begin{cor}[Borel $\sigma$-algebra generated by components]\label{c:borel}
Let\/ $(T,c)$ be an order separable algebraic tree, and\/ $D\subseteq T$ an order dense set.
Then its Borel\/ $\sigma$-algebra is generated by the set of components indexed by\/ $D$, i.e.
\begin{equation}
\mathcal{B}(T,c) = \sigma\(\bset{\mathcal{S}_x(y)}{x,y\in D,\; x\ne y}\).
\end{equation}
\end{cor}
\begin{proof}
Define $\mathcal{U}:=\bset{\mathcal{S}_x(y)}{x,y\in D,\; x\ne y}$.
By Proposition~\ref{p:separable}, $(T,\tau)$ is second countable. Hence $\mathcal{B}(T,c)$ is generated by any
subbase of $\tau$. If $D$ is order dense, $\mathcal{U}$ is such a subbase as shown in the proof of
Proposition~\ref{p:separable}.
\end{proof}
\subsection{Metric tree representations of algebraic trees}
\label{sub:asRtree}
In this subsection, we discuss the connection of metric trees with algebraic trees.
Let $(T,r)$ be a metric tree (recall from Definition~\ref{d:metrictree}). Then by (MT\ref{MT:cex}),
there exists to any three points $x_1,x_2,x_3\in T$ a unique branch point $c_{(T,r)}(x_1,x_2,x_3)$
satisfying \eqref{e:brapoi}. We refer to $(T,c_{(T,r)})$ as the algebraic tree \define{induced by} $(T,r)$, and
to $(T,r)$ as a \define{metric representation} of $(T,c_{(T,r)})$.
\begin{lemma}[the algebraic tree induced by a metric tree]
Let\/ $(T,r)$ be a metric tree, and\/ $c_{(T,r)}$ the map which sends any three distinct points to their branch point.
Then the following hold:
\begin{enumerate}
\item $(T,c_{(T,r)})$ is an algebraic tree.
\item $(T,c_{(T,r)})$ is order separable if and only if\/ $(T,r)$ is separable.
\item $(T,c_{(T,r)})$ is directed order complete if and only if\/ $(T,r)$ is bounded and complete. In
particular, it is an algebraic continuum tree if and only if\/ $(T,r)$ is a bounded, complete\/ $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree.
\end{enumerate}
\label{l:cinduce}
\end{lemma}
\begin{proof} \emph{(i) } It can be easily checked that $(T,c_{(T,r)})$ is an algebraic tree.
\smallbreak
\emph{(ii) } Let $(T,r)$ be separable. Then $\edge(T,c_{(T,r)})$ is countable. The topology induced by $r$ is obviously
stronger than the topology $\tau$ introduced in \eqref{e:tau}, hence $\tau$ is separable and therefore the algebraic tree
$(T,c_{(T,r)})$ is order separable. Conversely, if $(T,c_{(T,r)})$ is order separable, then any countable set $D$ satisfying
\eqref{e:sep} is also dense in $(T,r)$.
\smallbreak
\emph{(iii) }
Clearly, $(T,c_{(T,r)})$ admits suprema along any linearly ordered set with respect to some root if and only if $(T,r)$
is bounded and complete. The ``in particular'' follows because a complete metric tree $(T,r)$
is an $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree if and only if $\edge(T,r):=\edge(T,c_{(T,r)})=\emptyset$
(\cite[Remark~1.2]{AthreyaLohrWinter17}).
\end{proof}
Our first main result states that under the assumption of order separability any algebraic tree can be embedded
by an injective homomorphism into a compact $\R$\nobreakdash-tree\ and hence is isomorphic to (the algebraic tree induced by) a
totally bounded metric tree.
\begin{theorem}[characterisation of order separable algebraic trees]
Let\/ $T$ be a set, $c\colon T^3 \to T$.
\begin{enumerate}
\item\label{i:converse} $(T,c)$ is an order separable algebraic continuum tree if and only if there
exists a metric\/ $r$ on\/ $T$ such that\/ $(T,r)$ is a compact\/ $\R$\nobreakdash-tree\ with
\begin{equation}
\label{e:cTrTc}
c=c_{(T,r)}.
\end{equation}
\item $(T,c)$ is an order separable algebraic tree if and only if there is an order separable algebraic continuum
tree\/ $(\overline{T},\bar{c})$ such that\/ $(T,c)$ is a subtree of\/ $(\overline{T},\bar{c})$. In particular, every order
separable algebraic tree is induced by a totally bounded metric tree.
\end{enumerate}\label{t:algtreechar}
\end{theorem}
The separability hypothesis in Theorem~\ref{t:algtreechar} is crucial and cannot be dropped.
In Example~\ref{ex:topnotalgsep}, we have already seen an algebraic tree where the component topology $\tau$ is
not metrizable. Moreover, in this example, $\tau$ coincides with the order topology which is also the case for the
metric topology of any metric tree without branch points. Thus the algebraic tree cannot be induced by a metric tree.
The following example shows that also algebraic continuum trees need not be induced by metric trees.
\begin{example}[algebraic continuum tree that is not induced by a metric tree]
Let $T=[0,1]\times[0,1]$ with lexicographic order, and $(T,c)$ the corresponding algebraic tree as in
Example~\ref{ex:totord}. It is easy to check that $(T,c)$ is an algebraic continuum tree.
It cannot be induced by a metric tree because in its order topology $\tau$, it is connected but not
path-wise connected. These two properties are equivalent for metric trees
(see \cite[Theorem~2.20]{Evans2008}).
\label{exp:001}
\end{example}
In order to prove Theorem~\ref{t:algtreechar}, given an algebraic tree $(T,c)$, we need to provide a metric
$r$ such that \eqref{e:cTrTc} holds.
For that purpose, we consider for any measure $\nu$ on $(T,\mathcal{B}(T,c))$ such that $\nu$ is finite on
every interval, the following pseudometric,
\begin{equation}\label{e:rnu2}
r_\nu(x,y) := \nu([x,y]) - \tfrac12\nu(\{x\}) - \tfrac12\nu(\{y\}), \quad x,y\in T.
\end{equation}
\begin{lemma}[$r_\nu$ is a pseudometric]
Let\/ $(T,c)$ be an algebraic tree, and\/ $\nu$ a measure on\/ $(T,c)$ with\/
$\nu([x,y])<\infty$ for all\/ $x,y\in T$. Then\/ $r_\nu$ is a pseudometric on\/ $T$.
\label{l:pseudo}
\end{lemma}
\begin{proof} By Lemma~\ref{l:intprop} for all $x,y,z\in T$,
\begin{equation}
\label{e:017}
\begin{aligned}
\nu\big([x,y]\big) + \nu\big([y,z]\big)
&=
\nu\big([x,y] \cup [y,z]\big) + \nu\big([x,y]\cap [y,z]\big)
\\
&=
\nu\big([x,z]\big) + \nu\big(\blopenint{c(x,y,z)}{y}\big)+\nu\big(\bigl[c(x,y,z),\,y\bigr]\big).
\end{aligned}
\end{equation}
Hence
\begin{equation}
\label{e:018}
\begin{aligned}
\MoveEqLeft r_\nu(x,y)+\tfrac{1}{2}\nu\{x\}+\tfrac{1}{2}\nu\{y\}+r_\nu(y,z)
+\tfrac{1}{2}\nu\{y\}+\tfrac{1}{2}\nu\{z\}
\\
&=
r_\nu(x,z)+\tfrac{1}{2}\nu\{x\}+\tfrac{1}{2}\nu\{z\}
+2r_\nu\big(c(x,y,z),\,y\big)+\nu\{c(x,y,z)\}+\nu\{y\}-\nu\{c(x,y,z)\},
\end{aligned}
\end{equation}
or equivalently,
\begin{equation}
\label{e:019}
\begin{aligned}
r_\nu(x,y)+r_\nu(y,z)
&=
r_\nu(x,z)+2r_\nu\big(c(x,y,z),\,y\big).
\end{aligned}
\end{equation}
This implies that $r_\nu$ satisfies the triangle inequality.
\end{proof}
We denote the quotient metric space by $(T_\nu,r_\nu)$, i.e.\ $T_\nu$ is the set of equivalence classes of
points in $T$ with $r_\nu$-distance zero, and the quotient metric on $T_\nu$ is again denoted by $r_\nu$.
Furthermore, let $\pi_\nu \colon T \to T_\nu$ be the canonical projection.
\begin{lemma}[$(T_\nu,r_\nu)$ is a metric tree]
Let\/ $(T,c)$ be an algebraic tree, and\/ $\nu$ a measure on\/ $(T,c)$ with\/
$\nu([x,y])<\infty$ for all\/ $x,y\in T$.
Then the quotient space\/ $(T_\nu, r_\nu)$ is a metric tree, and the canonical projection\/ $\pi_\nu$ is a tree
homomorphism.
\label{l:o0hyper}
\end{lemma}
\begin{proof}
Let $x_1,\ldots,x_4\in T$. By Condition~(BPM4), we can assume w.l.o.g.\ that $c(x_1,x_2,x_3)=c(x_1,x_2,x_4)$.
Then by Lemma~\ref{l:4pt}(ii),
$c(x_1,x_2,x_3)\in [x_1,x_2]\cap[x_1,x_3]\cap[x_2,x_3]\cap[x_1,x_4]\cap [x_2,x_4]$, and \eqref{e:019} yields that
for $\{i,j\} \in \{\{1,2\},\,\{1,3\},\,\{1,4\},\,\{2,3\},\,\{2,4\}\}$,
\begin{equation}\label{e:cinint}
r_\nu(x_i,x_j) = r_\nu\(x_i,c(x_1,x_2,x_3)\) + r_\nu\(c(x_1,x_2,x_3),x_j\).
\end{equation}
Therefore,
\begin{equation}
\begin{aligned}
\label{e:015}
&r_\nu(x_1,x_3)+r_\nu(x_2,x_4)
\\
&=
r_\nu\big(x_1,\,c(x_1,x_2,x_3)\big)+r_\nu\big(c(x_1,x_2,x_3),\,x_3\big) +
r_\nu\big(x_2,\,c(x_1,x_2,x_3)\big)+r_\nu\big(c(x_1,x_2,x_3),\,x_4\big)
\\
&=r_\nu(x_1,x_2)+r_\nu\big(c(x_1,x_2,x_3),\,x_3\big)+r_\nu\big(c(x_1,x_2,x_3),\,x_4)
\\
&\ge
r_\nu(x_1,x_2)+r_\nu(x_3,x_4),
\end{aligned}
\end{equation}
and analogously,
\begin{equation}
\begin{aligned}
\label{e:016}
r_\nu\big(x_1,x_4\big) + r_\nu\big(x_2,x_3)
&=
r_\nu\big(x_3,\,c(x_1,x_2,x_3)\big)+r_\nu\big(c(x_1,x_2,x_3),\,x_4\big)+r_\nu(x_1,x_2)
\\
&\ge
r_\nu(x_1,x_2)+r_\nu(x_3,x_4).
\end{aligned}
\end{equation}
This means that the four point Condition~(MT\ref{MT:4pt}) is satisfied. Moreover, \eqref{e:cinint} implies
Condition~(MT\ref{MT:cex}) with branch point $\pi_\nu(c(x_1,x_2,x_3))$. In particular, $\pi_\nu$ is a tree
homomorphism.
\end{proof}
\begin{remark} Lemma~\ref{l:o0hyper} also explains why we had defined
$r_\nu$ as in \eqref{e:rnu2} and not just as $r_\nu':=\nu([x,y])$ for $x\ne y$.
Namely, in the latter case we would still have (MT\ref{MT:4pt}), but (MT\ref{MT:cex}) might fail.
Take, for example, $T:=\{1,2,3\}$, $c(1,2,3)=2$, and $\nu=\delta_2$. In this case, $r_\nu'$ is the
discrete metric on $T$, thus $2$ does not lie on the interval $[1,3]$ anymore.
\label{r:020}
\end{remark}
Let $(T,c)$ be an algebraic tree.
For all $v\in T$, define the \emph{degree} of $v$ in $(T,c)$ by
\begin{equation}
\label{e:degree}
\deg(v):=\deg_{(T,c)}(v):=\#\bset{\mathcal{S}_v(y)}{y\in T}.
\end{equation}
We say that $v\in T$ is
a \emph{leaf} if $\deg_{(T,c)}(v)=1$, and
a \emph{branch point} if $\deg_{(T,c)}(v)\ge 3$.
Notice that
\begin{equation}
\label{e:leaf}
\lf(T,c):=\big\{u\in T:\,c(u,v,w)\ne u \;\forall v,w\in T\setminus\{u\}\big\}
\end{equation}
equals the set of leaves of $T$, and
\begin{equation}
\label{e:br}
\br(T,c):=\big\{u\in T:\,c(x,v,w)= u \;\mbox{ for some } x,v,w\in T\setminus\{u\}\big\}
\end{equation}
the set of branch points.
Moreover, note that any $\nu$-mass on $\lf(T,c)$ that is not atomic does not contribute to $r_\nu$.
\begin{proposition}[metric representations of algebraic trees]
\label{p:properI}
Let\/ $(T,c)$ be an algebraic tree, $\nu$ a measure on\/ $(T,\mathcal{B}(T,c))$ with\/ $\nu([x,y])<\infty$ for
all\/ $x,y\in T$, and\/ $r_\nu$ defined by \eqref{e:rnu}. Then the following hold:
\begin{enumerate}
\item If\/ $(T,c)$ is order separable and\/ $\nu$ has at most countably many atoms, then\/ $(T_\nu,r_\nu)$ is separable.
\item If\/ $\# T>1$, $(T,c)$ is order complete, and\/ $[x,y]$ is order separable for every $x,y\in T$, then
$(T_\nu,r_\nu)$ is connected if and only if\/ $\nu$ is non-atomic.
In this case, $(T_\nu,r_\nu)$ is a complete\/ $\R$\nobreakdash-tree.
\end{enumerate}
\end{proposition}
\begin{proof} Throughout the proof denote by $\pi_\nu\colon T\to T_\nu$ the canonical projection.
\smallbreak
\emph{(i) }
It is easy to see that if a set $A\subseteq T$ satisfies \eqref{e:sep} and contains all atoms of $\nu$, then
$\pi_\nu(A)$ is dense in $(T_\nu,r_\nu)$. Therefore, by Proposition~\ref{p:separable} order separability of
$(T,c)$ implies separability of $(T_\nu,r_\nu)$.
\smallbreak
\emph{(ii) }
For all $x,y\in T$ with $x\ne y$, $r_\nu(x,y)\ge \frac12 \nu\{x\}$. Hence $(T_\nu,r_\nu)$ cannot be connected if $\nu$ has atoms.
Conversely, assume that $\nu$ is non-atomic. For $x,z\in T$, consider $([x,z], \le_x)$, which is a
totally ordered space according to Proposition~\ref{p:semilat}, and define $y:=\sup\set{v\in
[x,z]}{2\nu([x,v]) \le \nu([x,z])}$. The supremum exists due to order completeness of $(T,c)$.
Because of the order separability of $[x,z]$ and the non-atomicity of $\nu$, we obtain
$2\nu([x,y])=\nu([x,z])=2\nu([y,z])$ and therefore $2r_\nu(x,y)=r_\nu(x,z)=2r_\nu(y,z)$.
From this equality, connectedness follows once we have shown completeness, and every connected metric
tree is an $\R$\nobreakdash-tree.
Recall from Lemma~\ref{l:o0hyper} that $(T_\nu,r_\nu)$ is a metric tree. The same holds for its metric
completion $\overline{T}_\nu$. Assume for a contradiction that there is a sequence $\folge{x}$ in $T_\nu$
converging to some $x\in \overline{T}_\nu\setminus T_\nu$.
Then $x$ cannot be a branch point and one of the at most two components of $\overline{T}_\nu \setminus \{x\}$
contains infinitely many $x_n$. Thus we may assume w.l.o.g.\ that $x\in \lf(\overline{T}_\nu)$.
Define $y_n := c_{\overline{T}_\nu}(x_1, x_n, x)$. Then $y_n\to x$ and, for large enough $m$,
we have $y_n=c_{\overline{T}_\nu}(x_1, x_n, x_m)$.
Hence $y_n\in T_\nu$ for all $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and we may choose representatives
$x_n'\in \pi_\nu^{-1}(y_n)$ such that $x_n'=c(\rho, x_n', x_m')$ for $\rho:=x_1'$ and all sufficiently
large $m$.
By Proposition~\ref{p:semilat}, $\set{x_n'}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ is totally ordered w.r.t.\ $\le_\rho$, and hence
$x':=\sup\set{x_n'}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} \in T$ exists by order completeness. Obviously, $\pi_\nu(x')=x$ and $x\in
T_\nu$.
\end{proof}
In order to prove Theorem~\ref{t:algtreechar}\ref{i:converse} using Proposition~\ref{p:properI}, we need a
non-atomic probability measure $\nu$ (to ensure connectedness of $(T_\nu, r_\nu)$) charging all intervals (so
that $\pi_\nu$ is injective). Such a measure always exists in the case of \emph{order separable} algebraic
continuum trees.
\begin{lemma}\label{l:133}
Let\/ $(T,c)$ be an order separable algebraic continuum tree with\/ $\# T>1$. Then there exists a non-atomic
probability measure\/ $\nu$ on\/ $(T,\mathcal{B}(T,c))$ with\/ $\nu(\lf(T,c))=0$ and
\begin{equation}\label{e:fullsupp}
\nu\big([x,y]\big)>0\quad\forall\, x,y\in T,\; x\ne y.
\end{equation}
\end{lemma}
\begin{proof} Fix $\rho\in T$. Then, for every $x\in T\setminus\{\rho\}$, the interval $([\rho,x], \le_\rho)$ is
a separable \emph{linear continuum} in the sense of order theory, i.e.\ a totally ordered space (proven in
Proposition~\ref{p:semilat}) without \emph{jumps} (what we call here edges) or \emph{gaps} (which follows from
directed order completeness).
Due to Cantor's order characterisation of $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$ (e.g.\ \cite[Theorem~560]{Dasgupta14}),
this means that $[\rho,x]$ is order isomorphic to the unit interval. Obviously, every order isomorphism
is measurable and bijective, and the image of Lebesgue measure on the unit interval is a non-atomic
probability measure $\nu_x$ on $[\rho,x]$. Then $\sum_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} 2^{-n} \nu_{x_n}$, where
$\set{x_n}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ satisfies \eqref{e:sep}, is a non-atomic
probability satisfying \eqref{e:fullsupp} and $\nu(\lf(T,c))=0$.
\end{proof}
Any separable $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-tree $(T,r)$ comes with an intrinsic measure, called length measure, that generalizes the
Lebesgue-measure on $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$. More generally, if $(T,r)$ is a complete, separable metric tree and $\rho\in T$ a
fixed root, the \emph{length measure} $\lambda=\lambda^{(T,r,\rho)}$ is uniquely defined by the two properties
$\lambda([\rho,x])=r(\rho, x)$ for all $x\in T$, and $\lambda(\lf_0(T,r))=0$, where $\lf_0$ is the set
of non-isolated leaves (see \cite[Section~2.1]{AthreyaLohrWinter17}).
Note that the total mass $\lambda(T)$ (the ``total length'' of the metric tree) does not depend on the choice of
$\rho$.
\begin{proposition}[total length of $(T_\nu,r_\nu)$]
Let\/ $(T,c)$ be an order separable, order complete algebraic tree, $\nu$ a measure on\/
$(T,\mathcal{B}(T,c))$ with\/ $\nu([x,y])<\infty$ for all\/ $x,y\in T$ and such that\/
$\nu\restricted{\lf(T,c)}$ is purely atomic, and\/ $r_\nu$ be defined by \eqref{e:rnu}. Then the following hold:
\begin{enumerate}
\item The total length of the metric tree\ $(T_\nu,r_\nu)$ is given by
\begin{equation}
\label{e:Tnulen}
\lambda(T_\nu) = \tfrac12 \int_T \deg_{(T,c)} \,\mathrm{d}\nu.
\end{equation}
\item $\int_T\deg_{(T,c)} \,\mathrm{d}\nu = \int_{T_\nu} \deg_{(T_\nu,r_\nu)}\circ \pi_\nu \,\mathrm{d} \nu.$
\end{enumerate}
\label{p:Tnuprop}
\end{proposition}
\begin{proof} \emph{(i) }
Let $D:=\set{v_n}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ be a subset of $(T,c)$ which contains the atoms of $\nu$ and satisfies
\eqref{e:sep}, and $\pi_\nu\colon T\to T_\nu$ be the canonical projection. We use $\rho:=\pi_\nu(v_1)$
as the root of $(T_\nu,r_\nu)$. Then
\begin{equation} \label{e:130}
T\setminus \lf(T,c) \subseteq \gentree{D}=\bigcup_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}\gentree{v_1,...,v_n},
\end{equation}
where $\gentree{A}:=\bigcup_{x,y\in A}[x,y]$.
Hence $\nu(T\setminus \gentree{D})=0$, and
\begin{equation} \label{e:Tnapprox}
\lambda^{(T_\nu,r_\nu,\rho)}(T_\nu)
= \lim_{n\to\infty}\lambda^{(T_\nu,r_\nu,\rho)}\big(\pi_\nu(\gentree{v_1,...,v_n})\big).
\end{equation}
Abbreviate $T_n:=\gentree{v_1,...,v_n}$ and $\ell_n:=\lambda^{(T_\nu,r_\nu,\rho)}\big(\pi_\nu(\gentree{v_1,...,v_n})\big)$.
If $v_{n+1}\in T_n$, then $T_{n+1}=T_n$ and
$\lambda^{(T_\nu,r_\nu,\rho)}\big(\pi_\nu(T_{n+1})\big) =\lambda^{(T_\nu,r_\nu,\rho)}\big(\pi_\nu(T_n)\big)$.
Otherwise, there exists a unique $u_n\in T$ with $T_{n+1}=T_n\uplus\lopenint{u_n}{v_{n+1}}$, and thus
\begin{equation}
\ell_{n+1} = \ell_n + r_\nu(u_n,v_{n+1})
= \ell_n + \nu\(\lopenint{u_n}{v_{n+1}}\) - \tfrac12 \nu\{v_{n+1}\} + \tfrac12 \nu\{u_n\}.
\end{equation}
For $v\in T_n$, let $\deg_n(v)$ be the degree of $v$ in the tree $(T_n, c\restricted{T_n})$.
In the case $v_{n+1}\not\in T_n$, we have $\deg_{n+1}(v)=\deg_n(v)$ for $v\in T_n\setminus\{u_n\}$,
and $\deg_{n+1}(u_n)=\deg_n(u_n)+1$.
By induction over $n$, we obtain
\begin{equation}\label{e:Tnlen}
\ell_n = \tfrac12 \int_{T_n} \deg_n \,\mathrm{d}\nu
\end{equation}
Note that $\deg_n(v)$ is monotonically increasing in $n$, and $\deg(v)=\lim_{n\to\infty} \deg_n(v)$
holds for all $v\in \gentree{D}$. Thus using the monotone convergence theorem,
combining \eqref{e:Tnapprox} and \eqref{e:Tnlen} yields \eqref{e:Tnulen}.
\smallbreak
\emph{(ii) }If $\deg_{(T,c)}(v) \ne \deg_{(T_\nu,r_\nu)}(\pi_\nu(v))$,
then either $\pi(\mathcal{S}_v(y))=\{\pi(v)\}$ for some $y\in T$
(and thus $\deg_{(T,c)}(v) > \deg_{(T_\nu,r_\nu)}(\pi_\nu(v))$), or
$\pi(v)=\pi(v')$ for some $v'\in \mathrm{Br}(T,c)$ (and thus
$\deg_{(T,c)}(v) <\deg_{(T_\nu,r_\nu)}(\pi_\nu(v))$). In both cases we have that
$\nu\{v\}=\nu\{\pi_\nu(v)\}=0$,
and thus the claim follows.
\end{proof}
\begin{cor}[compactness for bounded degree trees]
Let\/ $(T,c)$ be an order separable algebraic tree, and\/ $\nu$ a finite measure on\/ $(T,\mathcal{B}(T,c))$
with\/ $\nu\{v\in T:\;\deg(v)>d\}=0$ for some\/ $d\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$. Then the completion of\/ $(T_\nu,r_\nu)$ is compact.
\label{c:Tcompact}
\end{cor}
\begin{proof}
W.l.o.g.\ assume that $\nu\restricted{\lf(T,c)}$ is non-atomic (if $\nu\restricted{\lf(T,c)}$ has a
non-atomic part, we can remove it without changing $r_\nu$).
Then by Proposition~\ref{p:Tnuprop}(i), $(T_\nu,r_\nu)$ has finite total length.
As complete metric trees with finite total length are necessarily compact, the statement follows.
\end{proof}
We are now in a position to prove Theorem~\ref{t:algtreechar}.
\begin{proof}[Proof of Theorem~\ref{t:algtreechar}.]
\emph{(i) ``$\Longleftarrow$'' }
Since every compact metric space is bounded, complete and separable, this step follows from Lemma~\ref{l:cinduce}.
\emph{``$\Longrightarrow$'' } Let $(T,c)$ be an order separable algebraic continuum tree. To avoid
trivialities, assume that $T$ contains more than two points. By Lemma~\ref{l:133}
we can choose a non-atomic probability measure $\nu$ on $(T,\mathcal{B}(T,c))$ satisfying \eqref{e:fullsupp}.
Define $r_\nu$ by \eqref{e:rnu}. Then the equivalence classes in $T_\nu$ are
singletons by \eqref{e:fullsupp}, and we may identify $T_\nu$ with $T$.
By Proposition~\ref{p:properI}, $(T,r_\nu)$ is a complete $\R$\nobreakdash-tree\ and the identity is a tree homomorphism by Lemma~\ref{l:o0hyper}.
Thus $c$ is induced by $r_\nu$. Moreover, $\nu(\br(T,c))=0$ because $\br(T,c)$ is countable and $\nu$
is non-atomic. We can therefore conclude with Corollary~\ref{c:Tcompact} that
$(T,r_\nu)$ is also compact. \smallskip
\smallbreak
\emph{(ii) ``$\Longleftarrow$'' } This is obvious because every order separable algebraic continuum tree is
induced by a separable $\R$\nobreakdash-tree\ according to part(i), and subspaces of separable metric spaces are
separable.
\emph{``$\Longrightarrow$'' }
Let $(T,c)$ be an order separable algebraic tree and $D\subseteq T$ a countable set satisfying
\eqref{e:sep}.
Let $\nu$ be any probability measure on $D$ with $\nu\{x\}>0$ for all $x\in D$, and $r_\nu$ defined by
\eqref{e:rnu}. The equivalence classes in $T_\nu$ are singletons, and we may again identify $T_\nu$ with
$T$. By Proposition~\ref{p:Tnuprop}, $(T, r_\nu)$ is a metric tree with \eqref{e:cTrTc}. As $(T,c)$ is
order separable, $(T,r_\nu)$ is separable by Proposition~\ref{p:properI}(i). Moreover, the diameter of
$(T,r_\nu)$ is bounded by $1$.
Hence, by \cite[Theorem~3.38]{Evans2008} (known since \cite{Dress84}), there is a bounded, separable
$\R$\nobreakdash-tree\ $(\overline{T},\bar{r})$ such that $T\subseteq \overline{T}$ and $r_\nu$ is the restriction of $\bar{r}$ to $T$.
By Lemma~\ref{l:cinduce}, this $\R$\nobreakdash-tree\ induces an algebraic continuum tree $(\overline{T},\bar{c})$, and $T$ is a
subtree of $\overline{T}$.
\emph{``in particular''.} According to part \emph{(i)}, there is a metric $\tilde{r}$ on $\overline{T}$ such that
$(\overline{T},\tilde{r})$ is a compact $\R$\nobreakdash-tree\ inducing $(\overline{T},\bar{c})$. Let $r$ be the restriction of $\tilde{r}$
to $T$. Then $(T,r)$ is a totally bounded metric tree inducing $(T,c)$.
\end{proof}
\subsection{Tree homomorphisms versus homeomorphisms}
\label{s:hom(e)o}
Since order separable algebraic continuum trees are $\R$\nobreakdash-trees\ where we have ``forgotten'' the metric, the question
arises how homeomorphisms of $\R$\nobreakdash-trees\ relate to tree homomorphisms of the corresponding algebraic trees. A
first observation is that homeomorphisms are necessarily tree homomorphisms. This statement relies on
connectedness of the $\R$\nobreakdash-trees\ and we cannot replace ``$\R$\nobreakdash-tree'' by ``metric tree'': every bijection between finite
metric trees is obviously a homeomorphism because the topologies are discrete, but not necessarily a tree
homomorphism.
\begin{lemma}[homeomorphisms are tree isomorphisms]\label{l:homeomhom}
Let\/ $(T,r),\, (\widehat{T},\hat{r})$ be\/ $\R$\nobreakdash-trees, and\/ $f\colon T \to \widehat{T}$ a homeomorphism. Then\/ $f$ is a tree homomorphism.
\end{lemma}
\begin{proof}
The branch point map can be expressed in terms of intervals by \eqref{e:bp}. In an $\R$\nobreakdash-tree\ $(T,r)$, the interval
$[x,y]$, $x,y\in T$, is the unique simple path from $x$ to $y$, which is a purely topological notion, and
hence preserved by homeomorphisms.
\end{proof}
\begin{example}[tree isomorphisms need not be homeomorphisms] In Lemma~\ref{l:homeomhom},
the converse is not true: bijective tree homomorphisms need not be
homeomorphisms, even if the trees are order separable.
To see this, let $r,\hat{r}$ the metrics on $\mathbb{N}} \newcommand{\M}{\mathbb{M}$ defined by $r(n,m)=\frac1n+\frac1m$,
$\hat{r}(n,m)=2$ for distinct $n,m\in \mathbb{N}} \newcommand{\M}{\mathbb{M}$. Let $T$ and $\widehat{T}$ be the $\R$\nobreakdash-trees\ generated by $(\mathbb{N}} \newcommand{\M}{\mathbb{M},r)$ and
$(\mathbb{N}} \newcommand{\M}{\mathbb{M},\hat{r})$, respectively. Then both $\widehat{T}$ and $T$ are the countable star with set $\mathbb{N}} \newcommand{\M}{\mathbb{M}$ of leaves. In $T$,
the distance from the branch point to leaf $n$ is $\frac1n$, while it is $1$ in $\widehat{T}$. Hence $T$ is
compact while $\widehat{T}$ is not.
The identity on $\mathbb{N}} \newcommand{\M}{\mathbb{M}$ can be extended to a bijective tree homomorphism $f\colon T \to \widehat{T}$ which cannot be
continuous.
\label{ex:homhomeom}
\end{example}
Example~\ref{ex:homhomeom} shows that it is possible for non-homeomorphic (topologically non-equivalent) $\R$\nobreakdash-tree s
to induce isomorphic (equivalent) algebraic continuum trees. This can only happen if at least one of the
trees is non-compact.
\begin{proposition}[tree isomorphisms of compact $\R$\nobreakdash-trees\ are homeomorphisms]
Let\/ $T,\widehat{T}$ be $\R$\nobreakdash-trees, and\/ $f \colon T \to \widehat{T}$.
\begin{enumerate}
\item\label{i:homcont} If\/ $\widehat{T}$ is compact, $f(T)$ is connected, and\/ $f$ a tree homomorphism, then\/ $f$ is
continuous.
\item\label{i:isomhomeom} If both\/ $T$ and\/ $\widehat{T}$ are compact and\/ $f$ is bijective, then\/ $f$ is a
homeomorphism if and only if it is a tree homomorphism.
\end{enumerate}
\label{p:isomhomeom}
\end{proposition}
\begin{proof}
\ref{i:isomhomeom} is obvious from \ref{i:homcont} and Lemma~\ref{l:homeomhom}.
Assume $f$ is a tree homomorphism, $f(T)$ is connected, and $\widehat{T}$ is compact. Choose a root $\rho \in T$.
Let $v_n \to v$ be a convergent sequence in $T$, and $w\in \widehat{T}$ an accumulation point of $f(v_n)$.
Then there is a subsequence $(n_k)_{k\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ with $f(v_{n_k}) \to w$. We have
\begin{equation}\label{e:supinf}
v = \sup_{k\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} \inf_{i>k} v_{n_i} \qquad\text{and}\qquad
w = \sup_{k\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} \inf_{i>k} f(v_{n_i}),
\end{equation}
where $\sup$ and $\inf$ are w.r.t.\ the partial orders $\le_\rho$ and $\le_{f(\rho)}$ in the first and
second equality, respectively. In the following, we show $w=f(v)$.
Because $f$ is order preserving for these partial orders due to Lemma~\ref{l:treehom}, we obtain $w
\le_{f(\rho)} f(v)$. Assume for a contradiction $w\ne f(v)$. Because $f(T)$ is connected, there is
$y\in \widehat{T}$ with $w<_{f(\rho)} y<_{f(\rho)} f(v)$ and $x \in T$ with $y=f(x)$. For $u:=c(\rho,x,v)$, we
have $f(u) = \hat{c}(f(\rho),y,f(v)) = y$, $u \le_\rho v$, and $u\ne v$.
Therefore, $u \le_\rho v_{n_i}$ for all sufficiently large $i$, and thus $y=f(u) \le_{f(\rho)}
f(v_{n_i})$ for those $i$. Now \eqref{e:supinf} implies $y\le_{f(\rho)} w$ in contradiction to the
choice of $y$, finishing the proof of $w=f(v)$.
Compactness of $\widehat{T}$ and uniqueness of accumulation points implies $f(v_n) \to f(v)$, and $f$ is
continuous.
\end{proof}
In view of Theorem~\ref{t:algtreechar}, Proposition~\ref{p:isomhomeom} implies that order separable algebraic
continuum trees are in one-to-one correspondence with homeomorphism classes of compact $\R$\nobreakdash-trees. Furthermore,
the unique metric topology induced by the compact $\R$\nobreakdash-tree\ coincides with the component topology $\tau$
introduced in Subsection~\ref{sub:astopological}.
But be aware that there may be other, non-homeomorphic, non-compact $\R$\nobreakdash-trees\ inducing the same order separable
algebraic continuum tree, as shown in Example~\ref{ex:homhomeom}.
\begin{cor}[uniqueness of inducing $\R$\nobreakdash-tree]\label{c:unique}
Every order separable algebraic continuum tree is induced by a compact\/ $\R$\nobreakdash-tree\ that is unique up to
homeomorphism, and the unique induced topology coincides with the component topology\/ $\tau$ defined in
\eqref{e:tau}.
\end{cor}
\begin{proof}
That an order separable algebraic continuum tree is induced by a compact $\R$\nobreakdash-tree\ is
Theorem~\ref{t:algtreechar}\ref{i:converse}.
Any two such compact $\R$\nobreakdash-trees\ are isomorphic as algebraic trees, hence homeomorphic by
Proposition~\ref{p:isomhomeom}. The component topology is a Hausdorff topology and clearly weaker than the
topology induced by the $\R$\nobreakdash-tree, because components are open sets of $\R$\nobreakdash-tree s.
Hence, by compactness of the $\R$\nobreakdash-tree, the two topologies coincide.
\end{proof}
\section{The space of algebraic measure trees}
\label{S:amt}
In this section, we define algebraic measure trees, and equip the space of (equivalence classes of) algebraic
measure trees with a topology.
In what follows, the order separability of the underlying algebraic tree is crucial. Therefore, we include it
already in the following definition of algebraic measure trees.
\begin{definition}[algebraic measure trees]
An \define{algebraic measure tree} $(T,c,\mu)$ is an order separable algebraic tree $(T,c)$ together with a
probability measure $\mu$ on $\mathcal{B}(T,c)$.
\label{d:amt}
\end{definition}
\begin{definition}[equivalence of algebraic measure trees]
\begin{enumerate}
\item We call two algebraic measure trees $(T_i,c_i,\mu_i)$, $i=1,2$, \define{equivalent} if there exist
subtrees $A_i$ of $T_i$ with $\mu_i(A_i)=1$, and a measure preserving tree isomorphism $f$ from
$A_1$ onto $A_2$. In this case, we call $f$ \define{isomorphism} of the algebraic measure trees.
\item A metric measure tree $(T,r,\mu)$ is called a \define{metric representation} of the algebraic measure
tree $(T',c',\mu')$ if its induced algebraic measure tree $(T,c_{(T,r)},\mu)$ is equivalent to
$(T',c',\mu')$.
\end{enumerate}
\label{d:amtequiv}
\end{definition}
In the following, we denote for an algebraic measure tree $\mathpzc{x}:=(T,c,\mu)$ by $\supp(\mathpzc{x})$
the algebraic subtree generated by the support of $\mu$, i.e.\
\begin{equation}\label{e:suppsmallx}
\supp(\mathpzc{x}):=c\(\supp(\mu)^3\),
\end{equation}
and by
\begin{equation}
\label{e:brsmallx}
\br(\mathpzc{x}):=\br(T,c)\cap \supp(\mathpzc{x})
\end{equation}
the set of \emph{branch points} of $\mathpzc{x}$.
It is easy to check that an isomorphism $f$ from $\mathpzc{x}=(T,c,\mu)$ to $\mathpzc{x}'=(T',c',\mu')$ induces a
bijection between $\br(\mathpzc{x})$ and $\br(\mathpzc{x}')$ (although it need neither be defined nor injective on all
of $\supp(\mathpzc{x})$). Also note that $\mathpzc{x}$ is equivalent to $\supp(\mathpzc{x})$ equipped with the appropriate
restrictions of $c$ and $\mu$.
\begin{remark}[a note on our definition of equivalence]
Every algebraic measure tree is equivalent to an algebraic continuum measure tree, and has a metric
representation with a compact $\R$\nobreakdash-tree\ by Theorem~\ref{t:algtreechar}. For the definition of equivalence
of algebraic measure trees it is important that we do not require the whole trees to be isomorphic (see
Example~\ref{e:toptree} below). On the other hand, it is also important that the isomorphism is injective
on a subtree (as opposed to only a subset) of full measure, because otherwise it would not be an
equivalence relation and every tree with $n$ leaves and uniform distribution on them would be equivalent
to the $n$-star.
\label{r:algcont}
\end{remark}
\begin{example}[the linear non-atomic measure tree]\label{ex:lintree}
There is only one equivalence class of linearly ordered algebraic measure trees with non-atomic measure.
Indeed, let $(T,c,\mu)$ be an algebraic measure tree with $\br(T,c)=\emptyset=\at(\mu)$. Then, by
Theorem~\ref{t:algtreechar}, there is a tree isomorphism from $T$ into $[0,1]$ and we may assume
$T\subseteq [0,1]$ to begin with. Let $F_\mu\colon [0,1]\to[0,1]$ be the
distribution function of $\mu$. Then $F_\mu$ is continuous and maps $\mu$ to Lebesgue-measure
$\lambda_{[0,1]}$.
Let $A:=\set{x\in\supp(\mu)}{\text{there is no }y_n\in[0,1]\setminus \supp(\mu): y_n<x,\; y_n\to x}$ be
the support of $\mu$ with left boundary points removed. Then $F_\mu$ restricted to $A$ is bijective and
hence a measure preserving tree isomorphism onto $[0,1]$ (with Lebesgue measure and canonical branch
point map).
Thus $(T,c,\mu)$ is equivalent to $[0,1]$.
\end{example}
Let
\begin{equation}
\label{e:bbT}
\mathbb{T}:=\{\text{equivalence classes of algebraic measure trees}\}.
\end{equation}
Next, we equip $\mathbb{T}$ with a topology. We shall base this notion of convergence on the fact that algebraic
measure trees allow for metric representations (see Theorem~\ref{t:algtreechar}), and require convergence in
Gromov-weak topology of particular representations.
To this end, let
\begin{equation}
\label{e:bbH}
\H := \{\text{equivalence classes of (separable) metric measure trees}\},
\end{equation}
where we consider two metric measure trees $(T,r,\mu)$ and $(T',r',\mu')$ as \emph{equivalent} if there exists a
measure preserving isometry between the metric completions of $\supp(\mu)$ and $\supp(\mu')$.
In order to get a useful topology on $\mathbb{T}$, we cannot take arbitrary (optimal) metric representations. Instead,
given an algebraic measure tree $(T,c,\mu)$, we use the metric $r_\nu$ defined in \eqref{e:rnu2} for the
\emph{branch point distribution} $\nu$, namely the distribution of the random branch point obtained by sampling
three points with the sampling measure $\mu$.
\begin{definition}[branch point distribution]
The \define{branch point distribution} of an algebraic measure tree $(T,c,\mu)$ is the push-forward of
$\mu^{\otimes 3}$ under the branch point map,\label{d:bpd}
\begin{equation}
\label{e:bpd}
\nu:=c_\ast\mu^{\otimes 3}.
\end{equation}
\end{definition}
Note that the branch point distribution is not necessarily supported by $\br(T,c)$. For instance, every atom of
$\mu$ is also an atom of $\nu$.
If $(T,c,\mu)$ and $(T',c',\mu')$ are equivalent algebraic measure trees with branch point distributions $\nu$
and $\nu'$, respectively, then the isomorphism is also an isometry w.r.t.\ $r_\nu$ and
$r_{\nu'}$.
Therefore, the following selection map, which associates a particular metric representation to every algebraic
measure tree, is well-defined.
\begin{definition}[selection map $\iota$]
Define the map $\iota\colon \mathbb{T} \to \H$ by
\begin{equation}\label{e:iota}
\iota(T,c,\mu) := (T_\nu,r_\nu,\mu_\nu),
\end{equation}
where $\nu=c_*\mu^{\otimes 3}$ is the branch point distribution of $(T,c,\mu)$, $(T_\nu,r_\nu)$ is the quotient
metric space, and $\mu_\nu$ is the image of $\mu$ under the canonical projection $\pi_\nu$.
\label{d:iota}
\end{definition}
The topology we use on $\mathbb{T}$ is the Gromov-weak topology w.r.t.\ the branch point distribution distance. That is,
it is the topology induced by the selection map $\iota$, i.e., the weakest (coarsest) topology on $\mathbb{T}$ such that
$\iota$ is continuous.
\begin{definition}[bpdd-Gromov-weak topology]
Let $\H$ be equipped with the Gromov-weak topology.
We call the topology induced on $\mathbb{T}$ by the selection map $\iota$
\define{branch point distribution distance Gromov-weak topology} (\define{bpdd-Gromov-weak
topology}).
\label{d:bpddGw}
\end{definition}
The following reconstruction theorem is crucial for the usefulness of bpdd-Gromov-weak convergence. It shows that
the selection map $\iota$ is an embedding and indeed selects metric representations.
\begin{proposition}[$\iota$ is injective]
The selection map\/ $\iota\colon \mathbb{T} \to \H$ is injective, and\/
$\iota(\mathpzc{x})$ is a metric representation of\/ $\mathpzc{x} \in \mathbb{T}$.
\label{p:injective}
\end{proposition}
\begin{proof}
If we show that $\iota(\mathpzc{x})$ is a metric representation of $\mathpzc{x}=(T,c,\mu)\in \mathbb{T}$, it is
obvious that $\iota$ is injective, because equivalence of metric measure spaces implies equivalence of
the corresponding algebraic measure trees by Lemma~\ref{l:homeomhom}.
Choosing an appropriate representative, we can assume that $\nu\{v\}>0$ for all $v\in\br(T,c)$.
The canonical projection $\pi_\nu\colon T \to T_\nu$ is a tree homomorphism by Lemma~\ref{l:o0hyper}.
To show equivalence of $(T,c,\mu)$ and $(T_\nu, c_{(T_\nu,r_\nu)},\mu_\nu)$, we have to show that
$\pi_\nu$ is injective on a subtree $A\subseteq T$ with $\mu(A)=1$.
Let $N:=\set{v\in T}{\pi_\nu(v) \ne \{v\}}$. Then $\mu(\pi_\nu(v)) = 0$ for all $v\in N$, and
$w\in\pi_\nu(v)$ implies $[v,w]\subseteq \pi_\nu(v)$ because $\pi_\nu$ is a tree homomorphism.
Because there are at most countably many non-degenerate, disjoint closed intervals in $T$ due to
order separability, this implies that $\pi_\nu(N)$ is countable, and thus $\mu(N)=0$.
Define $A=T\setminus N$. Then $\mu(A)=1$, and $\pi_\nu$ is injective on $T\setminus N$.
To see that $A$ is a subtree, pick $x,y,z\in A$. If $v:=c(x,y,z)\in\{x,y,z\}$, then $v\in A$.
Otherwise, $v\in \br(T,c)$, and hence $\nu\{v\}>0$. This implies $\pi_\nu(v)=\{v\}$, i.e.\ $v\in A$.
\end{proof}
\begin{cor}[metrizability]\label{c:bpdd-metrizable}
$\mathbb{T}$ equipped with bpdd-Gromov-weak topology is a separable, metrizable space.
\end{cor}
\begin{proof}
The Gromov-weak topology on $\H$ is separable, and metrizable, e.g.\ by the Gromov-Prohorov metric
$d_\mathrm{GP}$ (see \cite{GrevenPfaffelhuberWinter2009}).
Because $\iota$ is injective by Proposition~\ref{p:injective},
$d_\mathrm{BGP}(\mathpzc{x},\mathpzc{y}) := d_\mathrm{GP}(\iota(\mathpzc{x}),\iota(\mathpzc{y}))$, $\mathpzc{x},\mathpzc{y}\in \mathbb{T}$, is a metric
on $\mathbb{T}$ inducing bpdd-Gromov-weak topology.
\end{proof}
\begin{remark}[distance polynomials]\label{rem:bpddGw}
By definition, a sequence $(\mathpzc{x}_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ in $\mathbb{T}$ converges to $\mathpzc{x}\in \mathbb{T}$ bpdd-Gromov-weakly
if and only if $\iota(\mathpzc{x}_n) \to \iota(\mathpzc{x})$ Gromov-weakly.
It has been shown that the Gromov-weak convergence is equivalent to the convergence
of the distribution of the distance matrix (\cite[Theorem~5]{GrevenPfaffelhuberWinter2009}). Therefore, the
bpdd-Gromov-weak convergence is equivalent to
\begin{equation}
\Phi(\mathpzc{x}_n) \tno \Phi(\mathpzc{x})
\end{equation}
for all so-called \emph{polynomials} $\Phi\colon \mathbb{T}\to\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$, which are test functions of the form \eqref{e:PhiGw}.
Note that the set $\Pi_\iota$ of all polynomials is an
algebra, and therefore also convergence determining for $\mathbb{T}$-valued random variables (see
\cite{Loehr2013,BlountKouritzin2010}).
\end{remark}
As pointed out in Remark~\ref{r:algcont}, the equivalence class of every algebraic measure tree contains an
algebraic \emph{continuum} measure tree.
The following example shows that $\iota$ would not be injective if we had defined it on the set of algebraic
continuum measure trees with the stricter notion of equivalence where the whole algebraic continuum trees
have to be measure preserving isomorphic.
\begin{example}\label{e:toptree}
For $x\ge 0$, let $T_x$ be the $\R$\nobreakdash-tree\ generated by the interval $I_x=[-x,1]$ together with additional
leaves $\{v_n\}$, $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, where $c(0,1,v_n)=\frac1n$ and $r(\frac1n,v_n)=\frac1n$, i.e.\ at each point
$\frac1n \in I_x$ there is a branch of length $\frac1n$ attached.
Then $T_x$ is a compact $\R$\nobreakdash-tree\ for every $x\ge 0$, hence induces an algebraic continuum tree by
Theorem~\ref{t:algtreechar}. Let $\mu_x\{-x\}=\frac12$, and $\mu_x\{v_n\}=2^{-n-1}$ for $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$. Then
$\mathpzc{x}_x := (T_x,\mu_x) \in \T_{2}$. Now $\iota(\mathpzc{x}_x)=\iota(\mathpzc{x}_y)$ for every $x,y\ge 0$, but
$T_x$ and $T_0$ are not homeomorphic, hence not isomorphic by Proposition~\ref{p:isomhomeom}.
Note that $A_x:=\{x\} \cup\set{v_n}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} \cup \set{\frac1n}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ is a subtree of $T_x$ with
$\mu_x(A_x)=1$, and $A_x$ is isomorphic (although not homeomorphic) to $A_0$.
\end{example}
In order to construct algebraic measure trees, it is of course not necessary to specify the mass of every Borel
subset. On the contrary, we can use the following Carath\'eodory-type extension result.
To this end, recall for $x,y\in T$ with $x\not =y$ from \eqref{e:005} the component
$\mathcal{S}_x(y)=\mathcal{S}_x^{(T,c)}(y)$ of $T\setminus\{x\}$ which contains $y$. In this section, it is convenient to define
\begin{equation}\label{e:Subxx}
\mathcal{S}_x(x) := \{x\}.
\end{equation}
Then $T$ is the disjoint union of the $\deg(x)+1$ sets in
\begin{equation}
\mathcal{C}_x := \bset{\mathcal{S}_x(y)}{y\in T}.
\end{equation}
Note that $\mathcal{C}_x=\set{\mathcal{S}_x(y)}{y\in V}$ for order dense $V\subseteq T$ with $x\in V$. In particular,
$\mathcal{C}_x$ is countable if $(T,c)$ is order separable.
For $y\in T$, $V\subseteq T$, we call a function $f\colon V \to \mathbb{R}} \newcommand{\Z}{\mathbb{Z}$ \define{order left-continuous} on $V$ w.r.t.\
$\le_y$ if the following holds.
For all $x,x_n\in V$ with $x_1\le_y x_2 \le_y\cdots$ and $x=\sup_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}} x_n$ w.r.t.\ $\le_y$ (in short $x_n
\uparrow x$), we have $\lim_{n\to\infty} f(x_n) = f(x)$.
Recall the notion of algebraic continuum tree from Definition~\ref{d:aCT}.
\begin{proposition}[extension to a measure]\label{p:caratheodory}
Let\/ $(T,c)$ be an order separable algebraic continuum tree, and\/ $V\subseteq T$ order
dense. Then a set-function\/ $\mu_0 \colon \mathcal{C}_V:= \bigcup_{x \in V} \mathcal{C}_x \to [0,1]$ has a unique extension to a
probability measure on\/ $\mathcal{B}(T,c)$ if it satisfies
\begin{enumerate}[1.]
\item For all\/ $x\in V$, $\sum_{A \in \mathcal{C}_x} \mu_0(A) = 1$
\item For all\/ $x,y\in V$ with\/ $x\ne y$,
\begin{equation}\label{e:posint}
\mu_0\(\mathcal{S}_x(y)\) + \mu_0\(\mathcal{S}_y(x)\) \ge 1
\end{equation}
\item For every\/ $y\in V$, the function\/ $\psi_y\colon x\mapsto \mu_0(\mathcal{S}_x(y))$
is order left-continuous on\/ $V$ w.r.t.\ $\le_y$.
\end{enumerate}
\end{proposition}
\begin{proof}
Note that $\psi_y(x) = \psi_z(x)$ for $z\in \mathcal{S}_x(y)$. We therefore may write $\psi_A(x) := \psi_y(x)$
for any $A\subseteq \mathcal{S}_x(y)$.
Define the $\cap$-stable set system
\begin{equation}\label{e:A}
\mathcal{A} := \Bset{\bigcap_{k=1}^n A_k}{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M},\, A_k \in \mathcal{C}_V}.
\end{equation}
By Corollary~\ref{c:borel}, $\mathcal{A}$ generates the Borel $\sigma$-algebra $\mathcal{B}(T,c)$.
Let $\emptyset \ne A \in \mathcal{A}$ and $y \in A$. Because $(T,c)$ has no edges and is order complete, we have
$A = \bigcap_{x\in \partial A} \mathcal{S}_x(y)$, where $\partial$ denotes the boundary w.r.t.\ the component
topology $\tau$, which is a finite set in the case of $A$.
Using \eqref{e:posint}, we obtain for $v\in V$, $x_0,\ldots,x_n \in V\setminus\{v\}$ such that
$\mathcal{S}_v(x_0),\ldots,\mathcal{S}_v(x_n)$ are distinct, that
\begin{equation}
\psi_{x_0}(v) \le 1 - \sum_{k=1}^n \psi_{x_k}(v) \le 1 - \sum_{k=1}^n \(1 - \psi_v(x_k)\).
\end{equation}
This implies for $\emptyset \ne A\in \mathcal{A}$, by induction over $\#\partial A$, that
\begin{equation}\label{e:muA}
\mu(A) := 1 - \sum_{x\in \partial A} \(1-\psi_A(x)\) \ge 0,
\end{equation}
hence $\mu$ is a non-negative extension of $\mu_0$ to $\mathcal{A}$.
We claim that $\mu$ is super-additive, additive and inner regular for compact sets. From this it follows
by standard arguments that it has a unique extension to a measure on the generated $\sigma$-algebra
$\sigma(\mathcal{A})=\mathcal{B}(T,c)$.
\pstep{Additivity}
Let $n\in \mathbb{N}} \newcommand{\M}{\mathbb{M}\setminus\{1\}$, and $A_1,\ldots, A_n \in \mathcal{A}\setminus \{\emptyset\}$ disjoint with
$A:=\biguplus_{k=1}^n A_k \in \mathcal{A}$. Define $D:= \bigcup_{k=1}^n \partial A_k$.
Then $\partial A \subseteq D$ and there is $x\in D \setminus \partial A \subseteq A$.
Let $I_x:=\set{k\in \{1,\ldots,n\}}{x\in \partial A_k}$ and choose $y_k\in A_k$.
Then, because the $A_k$ are disjoint, the $\mathcal{S}_x(y_k)$, $k\in I$, are distinct, and because the $A_k$
cover $A$, we have $\set{\mathcal{S}_x(y_k)}{k\in I_x} = \mathcal{C}_x$.
In particular, $\sum_{k\in I_x} \psi_{y_k}(x) = 1$, and $B_x:=\bigcup_{k\in I_x} A_k \in \mathcal{A}$ with
$\partial B_x = \biguplus_{k\in I_x} \partial A_k\setminus \{x\}$.
We obtain
\begin{equation}\label{e:lokad}
\begin{aligned}
\sum_{k\in I_x} \mu(A_k)
&= \sum_{k\in I_x} \Bigl( 1 - \(1-\psi_{y_k}(x)\) - \sum_{z\in \partial A_k \setminus\{x\}} \(1-\psi_{y_k}(z)\) \Bigr) \\
&= \sum_{k\in I_x} \psi_{y_k}(x) - \sum_{z\in \partial B_x} \(1-\psi_x(z)\)
= \mu(B_x).
\end{aligned}
\end{equation}
By induction over $n$, this implies additivity of $\mu$.
\pstep{Super-additivity}
Let $A_1,\ldots, A_n \in \mathcal{A} \setminus \{\emptyset\}$ be disjoint and $\biguplus_{k=1}^n A_k \subseteq A
\in \mathcal{A}$. The case $n=1$
is trivial, and we proceed by induction over $n$.
Choose $y\in A_1$ and let $D:=\partial A_1 \setminus \partial A$.
For $x\in D$, $C\in \mathcal{C}_x':=\mathcal{C}_x \setminus \mathcal{S}_x(y)$
and $k\in \{2,\ldots,n\}$, either $A_k \subseteq C$, or $A_k \cap C = \emptyset$.
Therefore, we have the decomposition $\{2,\ldots,n\}=\biguplus_{x\in D}\biguplus_{C\in \mathcal{C}_x'} I_C$ with
$I_C:=\set{k}{A_k \subseteq C}$. Because $C\cap A \in \mathcal{A}$, and $A_k \subseteq C\cap A$ for $k\in I_C$,
we can use the induction hypothesis to obtain
\begin{equation}
\sum_{k\in I_C} \mu(A_k) \le \mu(C\cap A) = \psi_C(x) - \sum_{z\in \partial A \cap C} \(1-\psi_A(x)\).
\end{equation}
Therefore,
\begin{equation}\begin{aligned}
\mu(A_1) &= 1 - \sum_{x\in \partial A_1 \cap \partial A} \(1- \psi_y(x)\)
- \sum_{x\in D} \(1-\psi_y(x)\) \\
&= \mu(A) + \sum_{x\in \partial A \setminus \partial A_1} \(1-\psi_y(x)\)
- \sum_{x\in D} \sum_{C\in \mathcal{C}_x'} \psi_C(x) \\
&\le \mu(A) - \sum_{x\in D} \sum_{C\in \mathcal{C}_x'} \sum_{k\in I_C} \mu(A_k) \\
&= \mu(A) - \sum_{k=2}^n \mu(A_k).
\end{aligned}\end{equation}
\pstep{Compact regularity}
According to Proposition~\ref{p:compact}, all closed subsets of $T$ are compact. Let $y\in A\in \mathcal{A}$.
Because $(T,c)$ is an order separable algebraic continuum tree, and $V$ is order dense, we find for $z\in
\partial A$ a sequence $(x_n(z))_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ in $A \cap V$ with $x_n(z) \uparrow z$ w.r.t.\ $\le_y$ as
$n\to\infty$. Define $A_n:= \bigcap_{z\in \partial A} \mathcal{S}_{x_n(z)}(y) \in \mathcal{A}$ and $K_n := A_n \cup
\partial A_n$. Then $K_n$ is compact, $A_n \subseteq K_n \subseteq A$, and because $\partial A$ is
finite, we have $\partial A_n = \set{x_n(z)}{z\in \partial A}$ for sufficiently large $n$. Thus, by
order left-continuity of $\psi_y$,
\begin{equation}
\lim_{n\to\infty} \mu(A_n) = 1 - \lim_{n\to\infty} \sum_{z\in \partial A} \(1-\psi_y(x_n(z))\)
= 1 - \sum_{z\in \partial A} \(1-\psi_y(z)\) = \mu(A),
\end{equation}
and $\mu$ is inner compact regular as claimed.
\end{proof}
We conclude this section with an extension result, which will be very useful for reading off algebraic measure
trees from (sub\nobreakdash-)triangulations of the circle in Section~\ref{S:triangulation} below.
In Proposition~\ref{p:caratheodory}, we assumed the whole tree to be known, and considered the question of
constructing a probability measure on it. Now, we assume that not the whole tree is given a priori, but only the
(countably many) branch points. The question is, whether there is an extension of the tree which is rich
enough to carry a measure with the specified masses of components.
\begin{proposition}[construction of algebraic measure trees]\label{p:construction}
Let\/ $(V,c_V)$ be a countable algebraic tree, and for each\/ $x\in V$, let\/ $A \mapsto \psi_A(x)$ be a
probability measure on\/ $\mathcal{C}_x$. Define\/ $\psi_y(x) := \psi_{\mathcal{S}_x(y)}(x)$.
Assume that for $x,y\in V$ with $x\ne y$,
\begin{equation}\label{e:psisum}
\psi_x(y)+\psi_y(x) \ge 1.
\end{equation}
Then there is a unique (up to equivalence) algebraic measure tree\/ $\mathpzc{x}=(T,c,\mu)$ such that
\begin{enumerate}
\item\label{i:tree} $V\subseteq T$, $\br(T,c) = \br(V,c_V)$,
\item\label{i:mupsi} $\mu\(\mathcal{S}^{(T,c)}_{x}(y)\) = \psi_y(x)$ for all\/ $x,y\in V$,
\item\label{i:at} $\at(\mu) \subseteq V$,
where\/ $\at(\mu)$ denotes the set of atoms of\/ $\mu$.
\end{enumerate}
\end{proposition}
Note that in general we cannot obtain $\lf(T,c) \subseteq \lf(V,c_V)$. To the contrary, $\lf(T,c)$ can be
uncountable (for every representative of $\mathpzc{x}$).
\begin{proof} \pstep{Existence}
First note that for $y\in V$, $\psi_y$ is monotonic w.r.t. $\le_y$. Indeed,
$z\le_y x$ implies $\psi_y(z) \le 1 - \psi_x(z) \le \psi_z(x) = \psi_y(x)$.
We need to enlarge the tree to make $\psi_y$ order left-continuous.
Because $V$ is countable, we may consider one $y$ and one point $x$ at a time.
If $x,y \in V$ are such that there exists $x_n \in V$ with $x_n \uparrow x$, then by
monotonicity $\phi_y(x):=\lim_{n\to\infty} \psi_y(x_n) \le \psi_y(x)$ exists and is independent of the
choice of $x_n$. If $\phi_y(x) \ne \psi_y(x)$, we extend the tree by adding one extra point $z\not\in
V$, i.e.\ we consider $\tilde{V} := V \uplus \{z\}$ with the unique extension $\tilde{c}$ of $c_V$
such that $(\tilde{V},\tilde{c})$ is an algebraic tree with $x_n \le_y z \le_y x$ for all $n$.
Furthermore, we extend $\psi$ to $\tilde{\psi}$ on $\tilde{V}$ by defining $\tilde\psi_y(z):=\phi_y(x)$,
$\tilde\psi_z(z)=0$ and $\tilde\psi_x(z)=1-\phi_y(x)$.
It is easy to check that $(\tilde{V},\tilde{c})$ together with
$\tilde\psi$ satisfies the prerequisites of the Proposition, $\br(\tilde{V},\tilde{c})=\br(V,c)$, and
$\set{x\in \tilde{V}}{\tilde\psi_x(x) > 0} = \set{x\in V}{\psi_x(x)>0} \subseteq V$.
Now assume w.l.o.g.\ that $\psi_y$ is already order left-continuous for all $y\in V$. Because $V$ is
countable, it is in particular order separable and according to Theorem~\ref{t:algtreechar}, there is an
order separable algebraic continuum tree $(T,c)$ such that $(V,c_V)$ is a subtree. We can choose
$(T,c)$ such that $\br(T,c) = \br(V,c_V)$. Consider the closure $\overline{V}$ of $V$ w.r.t.\ the component
topology $\tau$. For $x\in \overline{V}\setminus V$, we define
\begin{equation}
\psi_y(x) := \sup\set{\psi_y(z)}{z\in V\cap \mathcal{S}_x(y)}.
\end{equation}
Then \eqref{e:psisum} holds for $x,y\in\overline{V}$, $x\ne y$, and $\psi_y$ is order left-continuous.
For every $\{x,y\} \in \edge(\overline{V},\bar{c})$, where $\bar{c}$ is the restriction of $c$ to $\overline{V}^3$, we fix an
order isomorphism $\varphi_{x,y}\colon [x,y] \to [0,1]$, which exists by Cantor's order characterization of $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$
because $[x,y]$ is a linearly ordered, separable algebraic continuum tree.
For every $z\in T \setminus \overline{V}$, there exists $\{x,y\} \in \edge(\overline{V},\bar{c})$, with $z\in[x,y]$. We define
\begin{equation}
\psi_y(z) := (1-\varphi_{x,y}(z)) \psi_y(x) + \varphi_{x,y}(z) \( 1 - \psi_x(y)\),
\end{equation}
$\psi_x(z):=1-\psi_y(z)$ and $\psi_z(z):=0$.
Now we can use Proposition~\ref{p:caratheodory} to see that
\begin{equation}
\mu_0\(\mathcal{S}_x(y)\) := \psi_y(x)
\end{equation}
has a unique extension to a probability measure $\mu$ on $\mathcal{B}(T,c)$.
The last step in the construction is to remove point-masses outside $V$ by expanding them to intervals.
To this end, let $P:=\at(\mu) \setminus V$, and $\overline{T}:=(T \setminus P) \uplus (P \times [0,1])$. Because
$P\subseteq T\setminus V$ contains no branch points, we can extend the restriction
of $c$ to $T\setminus P$ to a branch point map $\tilde{c}$ on $\overline{T}$ in a canonical way such that
$[(x,0),(x,1)]=\{x\}\times[0,1]$ for $x\in P$. Define the Markov kernel $\kappa$ from $T$ to $\overline{T}$ by
\begin{equation}
\kappa(x) := \begin{cases} \delta_x, & x\in T\setminus P,\\
\delta_x \otimes \lambda_{[0,1]}, & x \in P, \end{cases}
\end{equation}
where $\delta_x$ is the Dirac measure in $x$ and $\lambda_{[0,1]}$ is Lebesgue measure.
Let $\bar{\mu}:=\kappa_*(\mu)$ be the push-forward of $\mu$ under $\kappa$.
Then $(\overline{T},\tilde{c},\bar{\mu})$ is a separable algebraic
measure tree, and by construction $\br(\overline{T},\tilde{c}) = \br(V,c_V)$ as well as $\at(\bar{\mu}) = \at(\mu)\cap V
\subseteq V$.
Furthermore, for $x,y\in V$, we have $\bar{\mu}(\mathcal{S}_x^{(\overline{T},\tilde{c})}(y)) = \mu(\mathcal{S}_x^{(T,c)}(y)) = \psi_y(x)$
as claimed.
\pstep{Uniqueness} Follows similarly, where we note that it does not matter how we distribute the mass on an
edge of $(\overline{V},\bar{c})$ in a non-atomic way, because all algebraic measure trees without branch points and
non-atomic measure are equivalent by Example~\ref{ex:lintree}.
\end{proof}
\section{Triangulations of the circle}
\label{S:triangulation}
In this section, we encode binary algebraic measure trees by triangulations of subsets of the circle.
This is comparable with the encoding of compact (ordered, rooted) metric (probability) measure trees by
excursions over the unit interval, where the height profile encodes the branch point map as well as the metric
distances. Moreover, also the measure can be encoded by the excursion by identifying the lengths of
sub-excursions with the mass of the corresponding subtrees. Similarly, it turns out that we can encode binary
algebraic measure trees by what we call sub-triangulations of the circle. As in the case of coding metric
measure trees with excursions, the resulting \emph{coding map} associating to a sub-triangulation the algebraic
measure tree is continuous.
In Subsection~\ref{s:spacetriang}, we introduce the space of sub-triangulations of the circle. In
Subsection~\ref{s:coding}, we construct the coding map.
\subsection{The space of sub-triangulations of the circle}
\label{s:spacetriang}
Let $\mathbb{D}$ be a (fixed) closed disc of circumference $1$, and $\S:=\partial\mathbb{D}$ the circle.
As usual, for a subset $A\subseteq\mathbb{D}$, we denote by $\bar{A}$, $\mathring{A}$, $\partial A$ and $\conv(A)$ the
closure, the interior, the boundary and the convex hull of $A$, respectively.
Furthermore, let
\begin{equation}
\label{e:tri}
\Delta(A):=\bigl\{\text{connected components of $\conv(A)\setminus A$}\bigr\},
\end{equation}
and
\begin{equation}
\label{e:nabla}
\nabla(A):=\bigl\{\text{connected components of $\mathbb{D}\setminus\conv(A)$}\bigr\}.
\end{equation}
Then we have the disjoint decomposition $\mathbb{D} = A \uplus \bigcup\Delta(A) \uplus \bigcup\nabla(A)$.
\begin{definition}[(sub-)triangulations of the circle]
A \define{sub-triangulation of the circle} is a closed, non-empty subset $C$ of $\mathbb{D}$
satisfying the following two conditions:
\begin{enumerate}[\axiom(Tr{i}1)]
\item $\Delta(C)$ consists of open interiors of triangles.
\item $C$ is the union of non-crossing (non-intersecting except at endpoints), possibly
degenerate closed straight line segments with endpoints in $\S$.
\end{enumerate}
We denote the set of sub-triangulations of the circle by $\mathcal{T}$, i.e.\
\begin{equation}
\label{e:tatT}
\mathcal{T} := \bigl\{\text{sub-triangulations of the circle}\},
\end{equation}
and call $C\in\mathcal{T}$ \emph{triangulation of the circle} if and only if $\S\subseteq C$.
\label{d:triangfinite}
\end{definition}
In particular, (Tri1) implies that $\partial\conv(C) \subseteq C$, and we may call $C$ triangulation of
$\partial\conv(C)$. Given (Tri1), (Tri2) implies that $\nabla(A)$ consists of circular segments with the
bounding straight line excluded and the rest of the bounding arc included.
We want to point out that our definition of triangulation of the circle differs from the one given by Aldous in
\cite[Definition~1]{Aldous94b}. Namely, Aldous required only Condition~(Tri1).
For the characterization of triangulations of the circle as limits of triangulations of $n$-gons given in
Proposition~\ref{p:fintriapp} below, however, Condition~(Tri2) is necessary.
See Figure~\ref{f:nontriang} for
an example of a triangulation in the sense of Aldous that is excluded by Condition~(Tri2), a sub-triangulation
of the circle that is no triangulation of the circle, and a triangulation of the circle.
\begin{figure}[t]
\begin{center}
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\end{center}
\caption{\emph{left:} An Aldous-triangulation of the circle that is not a triangulation of the circle (Condition~(Tri2) does not hold as the
black triangle in the middle is not the union of non-crossing straight lines with endpoints on the circle).
\emph{middle:} A sub-triangulation of the circle (compare with Example~\ref{ex:compbin}).
\emph{right:} A triangulation of the circle. It is a realisation of the Brownian triangulation (compare with
Example~\ref{ex:CRT}).
}
\label{f:nontriang}
\end{figure}
For a metric space $(X,d)$, let
\begin{equation}
\mathcal{F}(X):= \set{F\subseteq X}{F\ne \emptyset,\, F\text{ closed}},
\end{equation}
and equip $\mathcal{F}(X)$ with the \emph{Hausdorff metric topology}. That is, we say that a
sequence $(F_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ converges to $F$ in $\mathcal{F}(X)$ if and only if for all $\varepsilon>0$ and all large
enough $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$,
\begin{equation}\label{e:Hausdorff}
F^\eps_n\supseteq F \qquad\mbox{and}\qquad F^\eps\supseteq F_n,
\end{equation}
where for all $A\in\mathcal{F}(X)$, as usual, $A^\eps:=\set{x\in X}{d(x,A)<\eps}$. It is well-known that if $(X,d)$ is
compact, then $\mathcal{F}(X)$ is a compact metrizable space as well.
As sub-triangulations of the circle are elements of $\mathcal{F}(\mathbb{D})$, we naturally equip $\mathcal{T}$ with the Hausdorff
metric topology. A first observation is that $\mathcal{T}$ is actually a closed, and therefore compact subspace of
$\mathcal{F}(\mathbb{D})$.
\begin{lemma}[compactness of $\mathcal{T}$] \label{l:triangcomp}
Both the space of triangulations of the circle, and the space\/ $\mathcal{T}$ of sub-triangulations of the
circle, are compact metrizable spaces in the Hausdorff metric topology.
\end{lemma}
\begin{proof}
Because $\mathbb{D}$ is compact, $\mathcal{F}(\mathbb{D})$ is compact as well, and it is sufficient to show that $\mathcal{T}$
and the set of triangulations of the circle are closed subsets of $\mathcal{F}(\mathbb{D})$.
Let $C_n\in\mathcal{T}$ with $C_n \tno C \in \mathcal{F}(\mathbb{D})$ in the Hausdorff metric topology.
(Tri1) is easily seen to be a closed property, thus $C$ satisfies (Tri1).
Let $L_n$ be a set of non-crossing line segments with endpoints in $\S$ such that $C_n=\bigcup L_n$.
The closure of $L_n$ in $\mathcal{F}(\mathbb{D})$ has the same property (it possibly differs from $L_n$ by a set of
degenerated one-point segments contained in non-degenerate segments of $L_n$), so we may assume $L_n$ is
closed to begin with, so that $L_n \in \mathcal{F}(\mathcal{F}(\mathbb{D}))$. Because $\mathcal{F}(\mathcal{F}(\mathbb{D}))$ is compact, we may
assume, taking a subsequence if necessary, that $L_n \to L$ for some $L\in\mathcal{F}(\mathcal{F}(\mathbb{D}))$.
Obviously, $L_n$ consists of non-crossing line segments with endpoints in $\S$.
Because the union operator $\bigcup \colon \mathcal{F}(\mathcal{F}(\mathbb{D})) \to \mathcal{F}(\mathbb{D})$ is continuous, we have
$\bigcup L = C$. In particular, (Tri2) holds for $C$, and $C\in \mathcal{T}$.
Obviously, also the property that $\S\subseteq C$ is preserved by Hausdorff metric limits, thus the
set of triangulations of the circle is closed as well.
\end{proof}
We now show two characterizations of sub-triangulations of the circle. Namely, condition (Tri2) can be replaced
by existence of ``triangles in the middle'' which is the major technical ingredient for the construction of the
branch point map in the next subsection.
Furthermore, they are precisely the limits of finite sub-triangulations, where we consider a sub-triangulation
$C$ as \emph{finite} if $C\cap \S$ is a finite set, or equivalently, $C$ consists of finitely many line
segments.
\begin{proposition}[characterization of (sub-)triangulations]\label{p:fintriapp}
Let\/ $\emptyset\ne C\subseteq\mathbb{D}$ be closed. Then the following are equivalent.
\begin{enumerate}[1.]
\item\label{it:sub} $C$ is a sub-triangulation of the circle.
\item\label{it:br} Condition~(Tri1) holds, all extreme points of\/ $\conv(C)$ are contained in\/ $\S$, and
\begin{enumerate}[\bf(Tr{i}1)']\setcounter{enumii}{1}
\item For\/ $x,y,z\in\Delta(C)\cup \nabla(C)$ pairwise distinct, there exists a unique\/
$c_{xyz}\in\Delta(C)$ such that\/ $x,y,z$ are subsets of pairwise different connected
components of\/ $\mathbb{D}\setminus\partial c_{xyz}$.
\end{enumerate}
\item\label{it:approx} There exists a sequence\/ $(C_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of finite sub-triangulations of the
circle with\/ $C_n\tno C$ in the Hausdorff metric topology.
\end{enumerate}
Furthermore, $C$ is a triangulation of the circle if and only if\/ $C_n$ in \ref{it:approx}.\ can be chosen as a
triangulation of a regular\/ $n$-gon inscribed in\/ $\S$.
\end{proposition}
\begin{remark}[condition (Tri2)']
That $x,y,z$ are subsets of different connected components of $\mathbb{D}\setminus \partial c_{xyz}$ means
that either\/ $c_{xyz}\in\{x,y,z\}$ and the two elements of\/ $\{x,y,z\}\setminus\{c_{xyz}\}$ are
subsets of different connected components of\/ $\mathbb{D}\setminus\overline{c_{xyz}}$,
or\/ $c_{xyz}\not\in\{x,y,z\}$ and\/ $x,y,z$ are subsets of pairwise different connected components of\/
$\mathbb{D}\setminus\overline{c_{xyz}}$.
\end{remark}
\begin{proof}[Proof of Proposition~\ref{p:fintriapp}]
\istep{\ref{it:sub}}{\ref{it:br}}
Because $C$ is the union of line segments with endpoints on $\S$, it is obvious that the extreme points
of $\conv(C)$ are contained in $\S$.
We have to show (Tri2)', so let $x,y,z\in\Delta(C)\cup\nabla(C)$ be pairwise distinct and note that
uniqueness is obvious. If one of the elements of $\{x,y,z\}$, say $x$, is such that the other two are
subsets of two different connected components of $\mathbb{D}\setminus\bar{x}$, then necessarily $x\in
\Delta(C)$, and $c_{xyz}:=x$ has the desired properties. So assume this is not the case.
Fix a set $L$ of non-crossing, closed lines with endpoints in $\S$ such that $C=\bigcup L$. Define
\begin{equation}
L_x:=\set{\ell \in L}{\ell\text{ separates $x$ from $y \cup z$ in $\mathbb{D}$}},
\end{equation}
note that $L_x \ne \emptyset$ because $y$ and $z$ are in the same connected component of
$\mathbb{D}\setminus\bar{x}$ by assumption, and order $L_x$ by distance from $x$.
Similarly, define $L_y$ as set of lines separating $y$ from $x\cup z$ ordered by distance from $y$, and
$L_z$ as set of lines separating $z$ from $x\cup y$, ordered by distance from $z$.
Define $\ell_x := \sup L_x$, $\ell_y:=\sup L_y$, and $\ell_z := \sup L_z$, which exist because $C$ is
closed. In particular, they are non-crossing, and because $\conv(C)\setminus C$ may only consist of
triangles, they have to be the sides of some $c_{xyz}\in\Delta(C)$ which has the desired properties.
\istep{\ref{it:br}}{\ref{it:approx}}
Because the extreme points of $\conv(C)$ are on the circle, for every $x\in \nabla(C)$,
the boundary $\partial_\mathbb{D} x$ in $\mathbb{D}$ is a single straight line with endpoints in $\S$.
Let $(V_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ be an increasing sequence of finite subsets of $\Delta(C)\cup\nabla(C)$ such that
$c_{xyz} \in V_n$ for pairwise distinct $x,y,z \in V_n$, and $V_n \uparrow \Delta(C)\cup \nabla(C)$.
Let $A_n:=\mathbb{D}\setminus \bigcup V_n$. Then $A_n \to C$ in the Hausdorff metric topology.
Because $c_{xyz} \in V_n$ for distinct $x,y,z \in V_n$, the boundary of each of the finitely many
connected components of $A_n \setminus \S$ consists of one or two line segments and one or two
connected sub-arcs of $\S$. Therefore, there is a finite sub-triangulation $C_n\subseteq A_n$ of the
circle with Hausdorff distance from $A_n$ less than $e^{-n}$. Thus $C_n \to C$.
\istep{\ref{it:approx}}{\ref{it:sub}}
Obvious, because $\mathcal{T}$ is a closed subset of $\mathcal{F}(\mathbb{D})$ by Lemma~\ref{l:triangcomp}.
\pstep{``Furthermore''}
If $C_n$ is a triangulation of the $n$-gon, it contains the $n$-gon, and hence any Hausdorff metric
limit as $n\to \infty$ contains the circle, and hence is a triangulation of the circle.
That triangulations of the circle can be approximated by triangulations of regular
$n$-gons is a slight modification of the arguments above. Details are left to the reader.
\end{proof}
The most prominent random tree is Aldous's Brownian CRT, which is the limit of uniform random trees. Similarly,
one can define the Brownian triangulation of the circle.
\begin{example}[Brownian triangulation]
The uniform random triangulation of the $n$-gon converges in law with respect to the Hausdorff metric
topology to the so-called \emph{Brownian triangulation} $C_{\mathrm{CRT}}$,
see \cite{Aldous94,Aldous94b,CurienKortchemski14}. A realisation is shown in the right of
Figure~\ref{f:nontriang}. It has a.s.\ Hausdorff dimension $\frac32$ (see \cite{Aldous94}).
\label{ex:CRT}
\end{example}
\subsection{Coding binary measure trees with (sub-)triangulations of the circle}
\label{s:coding}
Given an algebraic tree $(T,c)$, recall the set of leaves $\lf(T,c)$, and the degree $\deg_{(T,c)}(v)$ of $v\in T$
from \eqref{e:leaf} and \eqref{e:degree}, respectively.
In this section, we are interested in the following subspace of the space of all binary algebraic measure trees.
\begin{definition}[our space $\T_{2}$]\label{d:Tbin}
Let $\T_{2} \subseteq \mathbb{T}$ be the set of (equivalence classes of) algebraic measure trees
$(T,c,\mu)$ with $(T,c)$ binary (i.e.\ $\deg_{(T,c)}(v)\le 3$ for all $v\in T$) and
$\at(\mu)\subseteq \lf(T,c)$.
\end{definition}
The space $\T_{2}$ is of particular interest to us, as it is invariant under the dynamics of the Aldous diffusion
on cladograms, the construction of which was one of the motivations for studying algebraic measure trees, and
because, as we will see, it is precisely the space of algebraic measure trees that can be coded by
sub-triangulations of the circle.
To illustrate the construction of the tree coded by a sub-triangulation, we first consider a triangulation $C$
of the regular $n$-gon into necessarily $n-2$ triangles (see Figure~\ref{f:triangtree}).
Here, the coded tree is the dual graph. That is, every triangle corresponds to a branch point of the tree,
and two branch points are connected by an edge if and only if the triangles share a common edge.
We then add a leaf for every edge of the $n$-gon and obtain a graph-theoretic binary tree with $n$ leaves
and $n-2$ internal vertices. Recall from Example~\ref{ex:finite} that the finite graph-theoretic tree
corresponds to a unique algebraic tree. We finally assign to each leaf mass $n^{-1}$ (which corresponds to the
length of the arcs of the circle connecting two endpoints of edges of the $n$-gon if we inscribe it in a circle
of unit length), and obtain an algebraic measure tree.
The main result of this section is that there is a natural, surjective coding map from $\mathcal{T}$ onto $\mathbb{T}_2$,
which is also continuous.
To state that formally, we need further notation.
Given a sub-triangulation $C\subseteq\mathbb{D}$, recall $\Delta(C)$ and $\nabla(C)$ from \eqref{e:nabla} and
\eqref{e:tri}, respectively.
For $x\in\Delta(C)\cup \nabla(C)$, and $y\subseteq\mathbb{D}$ connected and disjoint from $\partial_\mathbb{D} x$, where
$\partial_\mathbb{D}$ denotes the boundary in the space $\mathbb{D}$, let
\begin{equation}
\label{s:comp}
\comp{x}{y} := \text{the connected component of $\mathbb{D}\setminus\partial_\mathbb{D} x$ which contains $y$}.
\end{equation}
For $x\in \Delta(C)$, let $p_i(x)$, $i=1,2,3$, be the mid-points of the three arcs of $\S\setminus \partial x$,
and define
\begin{equation}
\label{e:Box}
\Box(C) := \bset{\{p_i(x)\}}{x\in \Delta(C),\, i\in \{1,2,3\},\, \comp{x}{\{p_i(x)\}} \subseteq C},
\end{equation}
as well as $\comp{p}{p}:=p$ for $p\in \Box(C)$ (see Figure~\ref{f:Box}).
Recall the definition of components $\mathcal{S}_v(w)$ in an algebraic tree from \eqref{e:005}.
\cfigure{f:Box}{
\ifpdf
\includegraphics{pictBox}
\else
\psset{unit=0.09\textwidth}
\pictBox
\fi
}{Triangulation $C$ with $\#\Delta(C)=1$, $\nabla(C)=\emptyset$, and $\#\Box(C)=3$. The coded tree consists of
three line segments with non-atomic measure of $\frac13$ each, glued together at one branch point.}
\begin{lemma}[induced branch point map]\label{l:branchtriang}
For\/ $C\in \mathcal{T}$, let\/ $V_C:=\Delta(C) \cup \nabla(C) \cup \Box(C)$.
If\/ $V_C\ne \emptyset$, then there is a unique branch point map\/ $c_V\colon V_C^3\to V_C$ such that\/
$(V_C, c_V)$ is an algebraic tree with\/ $\mathcal{S}_x^{(V_c,c_V)}(y) = \set{v\in V_C}{\comp{x}{y} =
\comp{x}{v}}$ for\/ $x,y\in V_C$.
Furthermore, $\deg(x) = 3$ for all\/ $x\in \Delta(C)$, and\/ $\deg(x)=1$ for\/ $x\in \nabla(C)\cup \Box(C)$.
\end{lemma}
\begin{proof}
Recall from Proposition~\ref{p:fintriapp} that for a sub-triangulation $C$ of the circle and pairwise distinct
$x,y,z\in \Delta(C)\cup \nabla(C)$, there is a triangle $c_{xyz}\in \Delta(C)$ ``in the middle''.
It is straight-forward to see that this defines a branch point map and can naturally be extended to $V_C^3$.
\end{proof}
The following theorem states that all sub-triangulations $C$ of the circle can be associated with an element in $\mathbb{T}_2$ for which
$\Delta(C)$ corresponds to the set of branch points, $\nabla(C)$ corresponds to the set
\begin{equation}
\label{e:033}
\lf_\mathrm{atom}(\mathpzc{x}):=\bset{x\in\lf(T,c)}{\mu(\{x\})>0}
\end{equation}
of leaves which carry an atom,
and $\comp{v}{w}$ corresponds to the component $\mathcal{S}_v(w)$.
\begin{theorem}[algebraic measure tree associated to a sub-triangulation] \label{t:tree}
\begin{enumerate}[(i)]
\item For every sub-triangulation\/ $C\subseteq\mathbb{D}$ of the circle, there is a unique (up to equivalence)
algebraic measure tree\/ $\mathpzc{x}_C=(T_C,c_C,\mu_C)\in \T_{2}$ with the following properties:
\begin{enumerate}[\bf(CM1)]
\item $V_C \subseteq T_C$, $\br(\mathpzc{x}_C)=\Delta(C)$, and\/ $c_C$ is an extension
of\/ $c_V$, where\/ $(V_C,c_V)$ is defined in Lemma~\ref{l:branchtriang}.
\item $\mu_C\big(\mathcal{S}^{(T_C,c_C)}_x(y)\big) = \lambda_\S\big(\S\cap\comp{x}{y}\big)$
for all\/ $x,y\in V_C$, where\/ $\lambda_\S$ denotes the Lebesgue measure on\/ $\S$.
\item $\at(\mu_C) = \nabla(C)$.
\end{enumerate}
\item The \emph{coding map} $\tau\colon \mathcal{T}\to\T_{2}$, $C \mapsto \mathpzc{x}_C$ is surjective and continuous,
where\/ $\mathcal{T}$ is equipped with the Hausdorff metric topology and\/ $\T_{2}$ with the bpdd-Gromov-weak
topology.
\end{enumerate}
\end{theorem}
\begin{proof}
\proofcase{(i)} Let $C$ be a sub-triangulation of the circle.
If $C=\mathbb{D}$, then $\Delta(C)=\nabla(C)=\emptyset$, which requires by (CM1) that $\br(\mathpzc{x}_C)=\emptyset$, and
by (CM3) that $\at(\mu)=\emptyset$.
There is a unique algebraic measure tree without branch points and atoms, namely the line segment with no atoms
(see Example~\ref{ex:extremal}). We may therefore assume w.l.o.g.\ that $C\ne \mathbb{D}$, and consequently that
$T_C\ne\emptyset$.
We claim that $(V_C,c_V)$ together with $\psi_y(x):= \lambda_\S\(\S\cap\comp{x}{y}\)$ satisfies the assumptions
of Proposition~\ref{p:construction}. Indeed, $V_C$ is obviously countable and an algebraic tree by
Lemma~\ref{l:branchtriang}, $\psi_y(x)$ depends on $y$ only through its equivalence class w.r.t.\ $\sim_x$, and
the lengths of all the arcs add up to the total length of $\lambda_\S(\S)=1$. Furthermore, $\psi_x(y) +
\psi_y(x) \ge \lambda_\S(\S)=1$, and Proposition~\ref{p:construction} yields existence and uniqueness of the
desired algebraic measure tree.
\proofcase{(ii)}
Let $\mathpzc{x}=(T,c,\mu) \in \T_{2}$. We construct a sub-triangulation $C$ such that $\tau(C)=\mathpzc{x}$.
Fix $\rho\in\lf(T,c)$, and recall that $\rho$ induces a partial order relation $\le_\rho$.
We can extend this partial order to a total (planar) order $\le$ by picking for every $v\in \br(T,c)$ an order
of the two components of $T\setminus\{v\}$ that do not contain $\rho$. That is, we define
$S_0(v):=\mathcal{S}_v(\rho)$, denote the two remaining components of $T\setminus\{v\}$ by $S_1(v)$, $S_2(v)$, and
define
\begin{equation}\label{e:totord}
v\le w \quad:\Leftrightarrow\quad v\le_\rho w \text{ or } v\in S_1\(c(x,y,\rho)\),\, w\in S_2\(c(x,y,\rho)\).
\end{equation}
Identify $\S$ with $[0,1]$, where the endpoints are glued. For $a\in [0,1]$ and $b,c>0$ with
$a+b+c\le 1$, let $\Delta(a,b,c)\subseteq \mathbb{D}$ be the open triangle with vertices $a,a+b,a+b+c\in \S$,
$\ell(a,b)\subseteq \mathbb{D}$ the straight line from $a$ to $a+b$, and $L(a,b)$ the connected component of
$\mathbb{D}\setminus \ell(a,b)$ containing $a + \frac b2\in \S$.
The first vertex of the triangle or circular segment corresponding to $v\in \br(T,c)\cup \lf_\mathrm{atom}(\mathpzc{x})$ is given
by the total mass before (w.r.t.\ $\le$ defined in \eqref{e:totord}), i.e.\ by
\begin{equation}\label{e:alpha}
\alpha(v) := \mu\(\set{u\in T}{u<v}\).
\end{equation}
Define
\begin{equation}\label{e:025}
\begin{aligned}
\mathbb{D}\setminus C
&:=
\biguplus_{v\in\br(T,c)}\Delta\big(\alpha(v),\mu(S_1(v)),\mu(S_2(v))\big)
\uplus
\biguplus_{v\in\lf_\mathrm{atom}(\mathpzc{x})}L\big(\alpha(v),\mu\{v\}\big)
\end{aligned}
\end{equation}
By definition of $C$, $\conv(C)\setminus C$ consists of open triangles, i.e.\ condition (Tri1) is satisfied.
Furthermore, the extreme points of $\conv(C)$ are contained in $\S$, and for $x,y,z \in \Delta(C) \cup \nabla(C)$
distinct, there are corresponding $u,v,w\in T$, and a triangle $c_{xyz} \in \Delta(C)$ corresponding to
$c(u,v,w)$, which satisfies the requirements of (Tri2)'. Thus, by Proposition~\ref{p:fintriapp}, $C$ is a
sub-triangulation of the circle.
It is straight-forward to check that $\tau(C)=\mathpzc{x}$.
We defer the proof of continuity of $\tau$ to the next section, where we prove it in Lemma~\ref{l:Fcont}.
\end{proof}
The following is obvious now.
\begin{lemma}[non-atomicity] A sub-triangulation\/ $C$ of the circle is a triangulation of the circle if and only
if, for\/ $(T_C,c_C,\mu_C):=\tau(C)$, the measure\/ $\mu_C$ is non-atomic.
\label{l:CRT}
\end{lemma}
\begin{corollary}[finite tree approximation]\label{c:fintreeapp}
Let\/ $\mathpzc{x}=(T,c,\mu) \in \T_{2}$.
Then there is a sequence\/ $(\mathpzc{x}_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of finite algebraic measure trees in\/ $\T_{2}$ with\/
$\mathpzc{x}_n \to \mathpzc{x}$ bpdd-Gromov-weakly. Furthermore, if\/ $\mu$ is non-atomic, then\/ $\mathpzc{x}_n$
can be chosen as a tree with\/ $n$ leaves and uniform distribution on the leaves.
\end{corollary}
\begin{proof}
By Theorem~\ref{t:tree}, there is a sub-triangulation $C\in \mathcal{T}$ with $\tau(C)=\mathpzc{x}$, and by
Proposition~\ref{p:fintriapp}, there are finite sub-triangulations $C_n$ with $C_n\to C$. Obviously,
$\mathpzc{x}_n:=\tau(C_n)$ is a finite algebraic measure tree and by continuity of $\tau$ we have $\mathpzc{x}_n
\to \mathpzc{x}$. If $\mu$ is non-atomic, then, by Lemma~\ref{l:CRT}, $C$ is a triangulation of the circle,
and hence, by Proposition~\ref{p:fintriapp}, $C_n$ can be chosen as triangulation of the $n$-gon, which
means that $\mathpzc{x}_n$ has $n$ leaves and uniform distribution on them.
\end{proof}
We conclude this section with a few illustrative examples.
\begin{figure}[t]
\begin{center}
\ifpdf
\includegraphics{nobr1}\hfil
\includegraphics{nobr2}\hfil
\includegraphics{nobr3}\hfil
\includegraphics{nobr4}\hfil
\includegraphics{nobr5}
\else
\picttrinobr
\fi
\end{center}
\caption{Sub-triangulations of the circle which correspond to the five cases of algebraic measure trees without branch points as explained in Example~\ref{ex:extremal}.}
\label{f:nobranch}
\end{figure}
\begin{example}[coding algebraic measure trees without branch points]\label{ex:extremal}
Let $\mathpzc{x}$ be an algebraic measure tree without branch points. If $\mathpzc{x}=\mathpzc{x}_C$ for some
sub-triangulation $C$, then $\Delta(C)= \br(\mathpzc{x}_C) = \emptyset$ and the following five cases can occur (see Figure~\ref{f:nobranch}):
a) $\mathpzc{x}_C$ consists of one single point of mass $1$. Then $C=\{x\}$ for some $x\in\S$.
b) $\mathpzc{x}_C$ consists of an interval with two leaves, where each carries positive mass adding up to $1$,
in which case $C$ is a single line segment dividing the circle into two arcs with length corresponding to the
masses of the two leaves.
c) $\mathpzc{x}_C$ consists of an interval with two leaves, where each has positive mass adding up to $a<1$.
In this case, $C$ is the area of the disc bounded by two distinct line segments and two arcs (possibly one of
them degenerated) of $\S$, and the lengths of the remaining two arcs are given by the masses of the leaves.
d) $\mathpzc{x}_C$ consists of an interval with two leaves, where one has positive mass $a<1$ and the other one has
zero mass. Then $C$ is a circular segment with arc length $1-a$.
e) $\mathpzc{x}_C$ consists of an interval with no atoms on the leaves, which implies $C=\mathbb{D}$.
\end{example}
\begin{example}[a complete binary tree]\label{ex:compbin}
Let $C$ be the sub-triangulation of the circle drawn in the middle of Figure~\ref{f:nontriang}. Then
$\#\nabla(C)=\#\lf_\mathrm{atom}(\tau(C))=1$. We refer to this only leaf with positive mass as the root $\rho$,
and obtain $\mu(\{\rho\})=\frac13$, corresponding to the length of the dotted arc.
Moreover, $\tau(C)$ consists of a complete rooted binary tree in the sense of graph theory (with the
convention that the root has degree one), together with an uncountable set of leaves given by the ends
at infinity and carrying the remaining $\frac23$ of the mass.
\end{example}
\begin{example}[coding the Brownian CRT]
Recall the Brownian triangulation $C_{\mathrm{CRT}}$ from Example~\ref{ex:CRT}, which is defined as the limit in
distribution of uniform random triangulations $C_n$ of the $n$-gon. A realization is shown in the right
of Figure~\ref{f:nontriang}.
It is easy to see that $\tau(C_n)$ is the uniform binary tree with $n$ leaves and uniform distribution on
the leaves. Thus, by Theorem~\ref{t:tree}, the uniform binary tree converges bpdd-Gromov-weakly to
$\tau(C_{\mathrm{CRT}})$.
At this point it is not entirely clear that $\tau(C_{\mathrm{CRT}})$ is the algebraic measure tree
induced by the metric measure Brownian CRT. We will see in Section~\ref{s:examples} that this
is indeed the case.
\end{example}
\section{The subspace of binary algebraic measure trees}
\label{S:topo}
In this section we introduce in Subsections~\ref{s:convshape} and~\ref{s:convmass} with the \emph{sample shape
convergence} and the \emph{sample subtree-mass convergence} two more notions of convergence of algebraic measure
trees which seem more natural when thinking of algebraic trees as combinatorial objects. We then show in
Subsection~\ref{s.equivalence} that on $\T_{2}$, both of these notions are equivalent to the bpdd-Gromov-weak
convergence. The main tools are a uniform Glivenko Cantelli argument, and that the coding map sending a
sub-triangulation of the circle to an element in $\T_{2}$ is continuous.
\iffalse{
\begin{remark}[convergence of algebraic trees versus convergence of $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$-trees]\label{r:convcomp}
Let $\hat{\smallx}_n \in \H$, and $\mathpzc{x}_n \in \mathbb{T}$ the corresponding algebraic measure tree.
Since the equivalence classes of $\mathbb{T}$ are strictly coarser than those of $\H$, it is obvious that
any kind of convergence of $\mathpzc{x}_n$ does not imply convergence of $\hat{\smallx}_n$ in any Hausdorff topology
on $\H$.
But also conversely, convergence of $\hat{\smallx}_n$ in Gromov-weak or Gromov-Hausdorff-weak topology cannot imply
convergence of $\mathpzc{x}_n$ in any Hausdorff topology as shown by the following simple example.
Let $\hat{\smallx}_n = (T,\frac1n r, \mu)$ for any $(T,r,\mu)\in\H$ with $\mu$ not supported by a single point.
Then $\hat{\smallx}_n$ converges to the one-point space. All $\mathpzc{x}_n$, however, are identical (equivalent as
algebraic measure trees), but different from the one-point space.
\end{remark}
}\fi
\subsection{Convergence in distribution of sampled tree shapes}
\label{s:convshape}
The basic idea behind Gromov-weak convergence for metric measure spaces is to sample finite metric sub-spaces
with the sampling measure $\mu$ and then require these to converge in distribution. In this section, we propose
a corresponding construction for binary algebraic measure trees, where we sample finite tree shapes with $\mu$.
First, we have to make precise what we mean by ``tree shape'', which we understand to be a cladogram with the
peculiarity that leaves may carry more than one label. The multi-label case is necessary to allow for
sampling the same point several times due to a possible atom at that point.
\xymatfig{f:shape}
{&u_1\ar@{-}[d] &&&&&&&&
\\
\ar@{-}[r]&\bullet\ar@{-}[dr] & & &\bullet\ar@{-}[ul]\ar@{-}[ur] & &&&&& u_1\ar@{-}[dr] & & u_3\ar@{-}[d] &
\\
&&{v}\ar@{-}[r]&u_3\ar@{-}[r]&\bullet\ar@{-}[r]\ar@{-}[u]&u_4&&&&& &{v_1}\ar@{-}[r]&{v_2}\ar@{-}[r]&u_4
\\
&u_2\ar@{-}[ur] & & & & &&&&& u_2\ar@{-}[ur] & & &}
{A tree $T$ and the shape $\shape(u_1,u_2,u_3,u_4)$. Here, we are considering the homomorphism $f\colon C\to
c^3(\{u_1,...,u_4\}^3)$ given by $f(u_i):=u_i$, $i=1,...,4$, and then necessarily $f(v_1)=v$, $f(v_2)=u_3$.
$f$ is clearly no isomorphism, and the cladogram is not isomorphic to the subtree
$c(\{u_1,u_2,u_3,u_4\}^3)$ because $c(u_1,u_4,u_3)=u_3$.}
\begin{definition}[\nclad] For $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, an \define{\nclad} is a binary, finite algebraic tree $C=(C,c)$ together with a surjective
labelling map $\ell\colon \{1,...,m\} \to \lf(C)$.
Two \nclads\ $(C_1,\ell_1)$ and $(C_2,\ell_2)$ are equivalent if they are label preserving
isomorphic i.e., there exists a tree isomorphism $f\colon C_1\to C_2$ with $f(\ell_1(i))=\ell_2(i)$ for all $i=1,...,m$.
\label{def:cladogram}
\end{definition}
Define
\begin{equation}
\Clad := \{ \text{isomorphism classes of \nclads} \}.
\end{equation}
In the following we will use cladograms to encode the shape of a subtree spanned by a finite sample of leaves.
\begin{definition}[tree shape]
For a binary algebraic tree $(T,c)$, $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, and $u_1,...,u_m\in T\setminus \br(T,c)$,
the \define{tree shape} $\shape(u_1,...,u_m)$
of the $m$-labelled cladogram spanned by $(u_1,...,u_m)$ in $(T,c)$
is the unique (up to isomorphism) \nclad\ $\shape(u_1,...,u_m)=(C,c_C,\ell)$ with
$\lf(C)=\{u_1,...,u_m\}$ and $\ell(i)=u_i$ for all $i=1,...,m$, and such that the identity on $\lf(C)$ extends to a tree
homomorphism from $C$ onto $c\(\{u_1,...,u_m\}^3\)$.
\label{def:treeshape}
\end{definition}
\begin{remark}[spanned subtree and cladogram are not necessarily isomorphic]
The tree homomorphism from $\shape(u_1,...,u_m)$ onto $c(\{u_1,...,u_m\}^3)$ does not need to be
injective. This is the case if (and only if) $u_i\in \openint{u_j}{u_k}$ for some $i,j,k\in\{1,...,m\}$.
See Figure~\ref{f:shape}.
\label{rem:002}
\end{remark}
\begin{example}[shape of a totally ordered algebraic tree]
Let $(T,c)$ be a totally ordered algebraic tree, and $u_1,...,u_m\in T$.
Then $\shape(u_1,...,u_m)$ is a so-called \emph{comb tree} which has a totally ordered spine of binary
branch points with attached leaves (see Figure~\ref{f:ordered}).
\label{exp:003}
\end{example}
In the following, we build a topology on the convergence of tree shapes of $m$ randomly sampled points. We
therefore need the measurability of the shape map.
\begin{lemma}[measurability of the shape map]
For every binary algebraic tree\/ $(T,c)$ and\/ $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$,
the tree shape map\/ $\shape\colon (T\setminus\br(T,c))^m \to \Clad$ is a measurable function.
\label{l:001}
\end{lemma}
\begin{proof}
Restricted to the open subset $\bset{v\in (T\setminus\br(T,c))^m}{v_1,...,v_m \text{ distinct}}$,
$\shape$ is locally constant, hence continuous. The same is true on the set $\bset{v\in
(T\setminus\br(T,c))^m}{v_1={v_2},\; v_2,..., v_m \text{ distinct}}$, which is an intersection of a closed
and an open set, hence measurable. We can continue this way to see that $\shape$ is measurable on
$(T\setminus \br(T,c))^m$.
\end{proof}
\xymatfig{f:ordered}
{\ar@{-}[r] & u_1 \ar@{-}[r] & u_2 \ar@{-}[r] & u_3 \ar@{-}[r] & u_4 \ar@{-}[r] & &&
& u_1 \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & u_4 & &&
& u_1,u_5 \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & u_4
\\
& & & & & &&
& & u_2\ar@{-}[u] & u_3\ar@{-}[u] & & &&
& & u_2\ar@{-}[u] & u_3\ar@{-}[u] &
}
{The left shows a totally ordered binary algebraic tree and four distinct points $u_1,...,u_4$.
The middle shows the shape $\shape(u_1,...,u_4)$ of the cladogram which forms a comb tree. The right illustrates
what happens if a fifth point is equal to $u_1$. Now one of the leaves of $\shape(u_1,...,u_5)$ has two labels.}
\begin{definition}[tree shape distribution]
For $\mathpzc{x}=(T,c,\mu) \in \T_{2}$ and $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, the $m$\protect\nobreakdash-\hspace{0pt} \define{tree shape distribution} of $\mathpzc{x}$ is
defined by
\begin{equation}
\label{e:131}
\shapedist(\mathpzc{x}):=\mu^{\otimes m} \circ \shape^{-1}\in\mathcal{M}} \newcommand{\CP}{\mathcal{P}_1(\Clad).
\end{equation}
\end{definition}
\begin{example}[shape of the linear non-atomic measure tree] \label{exp:005}
Let $\mathpzc{x}=(T,c,\mu)$ be the linear non-atomic algebraic measure tree
(Example~\ref{ex:lintree}). Then any sample $(u_1,...,u_m)$ with $\mu$ consists of pairwise different
points, and $\shapedist(\mathpzc{x})$ is the mixture of Dirac measures on labelled comb trees where the
mixture is over all (up to isometry) permutations of the labels.
\end{example}
We refer to the weakest topology on $\T_{2}$ such that for every $m\in \mathbb{N}} \newcommand{\M}{\mathbb{M}$ the $m$-tree shape distribution is
continuous as sample shape topology.
\begin{definition}[sample shape topology]\label{d:shapeconv}
The topology induced on $\T_{2}$ by the set $\set{\shapedist}{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of tree shape
distributions is called \define{sample shape topology}.
\end{definition}
We say that a sequence $(\mathpzc{x}_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ is \define{sample shape convergent} to $\mathpzc{x}$ in $\T_{2}$ if it
converges w.r.t.\ the sample shape topology, i.e.\ if $\shapedist(\mathpzc{x}_n)$ converges to $\shapedist(\mathpzc{x})$
as $n\to\infty$ for every $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$.
In analogy to the set $\Pi_\iota$ of polynomials introduced in Remark~\ref{rem:bpddGw}, we also introduce a set of
test functions which evaluate the tree shape distributions. We refer to
$\Phi=\Phi^{m,\varphi}\colon \T_{2}\to\mathbb{R}} \newcommand{\Z}{\mathbb{Z}$,
\begin{equation}
\label{e:spol}
\Phi(\mathpzc{x}) = \inta{\Clad}{\varphi\,}{\shapedist(\mathpzc{x})}
=\intamu{T^m}{\varphi\circ\shape},
\end{equation}
where $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and $\varphi\colon \Clad\to \mathbb{R}} \newcommand{\Z}{\mathbb{Z}$, as \define{shape polynomial}. We also define
\begin{equation} \label{e:141}
\Pi_{\shape[]} := \{\,\text{shape polynomials on $\T_{2}$}\,\}.
\end{equation}
Obviously, the sample shape topology is induced by the set $\Pi_{\shape[]}$ of shape polynomials.
\begin{proposition}[sample shape implies bpdd-Gromov-weak convergence]\label{p:shapestrongerGw}
On\/ $\T_{2}$, the sample shape topology is stronger than the bpdd-Gromov-weak topology (i.e.\ any
open set in the bpdd-Gromov-weak topology is open in the sample shape topology).
\end{proposition}
\begin{proof}
The bpdd-Gromov-weak topology is induced by the set $\Pi_\iota$ of polynomials (see Remark~\ref{rem:bpddGw}).
Because the set of $\phi\in\mathcal{C}_b(\mathbb{R}} \newcommand{\Z}{\mathbb{Z}^{m\times m})$ which are Lipschitz continuous is convergence determining for
probability measures on $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}^{m\times m}$, the subset of those $\Psi\in\Pi_\iota$ with
\begin{equation}\label{e:132}
\Psi(T,c,\mu) =
\intamuu{T^m}{\phi\((\nu[u_i,u_j] - \tfrac12\nu\{u_i\} - \tfrac12\nu\{u_j\})_{i,j=1,\ldots,m}\)}{\underline{u}}
\end{equation}
for some $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and Lipschitz continuous $\phi\in\mathcal{C}_b(\mathbb{R}} \newcommand{\Z}{\mathbb{Z}^{m\times m})$ also induces the
bpdd-Gromov-weak topology. Therefore, it is enough to show that such a $\Psi$ is continuous on $\T_{2}$
w.r.t.\ the sample shape topology. We do so by showing that the restriction to $\T_{2}$ of $\Psi$ is in
the uniform closure of $\Pi_{\shape[]}$. Let $L$ be the Lipschitz constant of $\phi$ w.r.t.\ the
$\ell_\infty$-norm on $\mathbb{R}} \newcommand{\Z}{\mathbb{Z}^{m\times m}$.
For $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ with $3n\ge m$, we define
\begin{equation} \label{e:133}
\Phi_n(T,c,\mu) := \intamuu[3n]{T^{3n}}
{\phi\((\nu_{n,\underline{u}}[u_i,u_j] - \tfrac12\nu_{n,\underline{u}}\{u_i\} - \tfrac12\nu_{n,\underline{u}}\{u_j\})_{i,j=1,...,m}\)}{\underline{u}},
\end{equation}
with the empirical branch point distribution
\begin{equation}
\label{e:134}
\nu_{n,\underline{u}} := \tfrac1n \sum_{k=0}^{n-1} \delta_{c(u_{3k+1},u_{3k+2},u_{3k+3})}.
\end{equation}
Note that the restriction of $\Phi_n$ to $\T_{2}$ belongs to $\Pi_{\shape[]}$ because whether or not $c(u_{k+1},u_{k+2},u_{k+3})$ lies on $[u_i,u_j]$,
$k\in\{0,...,n-1\},\,i,j\in\{1,...,m\}$ only depends on the shape $\shape[3n](\underline{u})$.
Finally, we observe
\begin{equation}
\| \Psi - \Phi_n\|_\infty \le \sup_{(T,c,\mu)\in \T_{2}} \intamuu[3n]{T^{3n}}
{ L\cdot 3\sup_{I\in\I} |\nu(I) - \nu_{n,\underline{u}}(I)| }{\underline{u}}
\le 3L\cdot \eps_n \tno 0,
\end{equation}
with $\I:=\{[x,y];\,x,y\in T\}$ and $(\eps_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}\tno 0$, where we have used a uniform
Glivenko-Cantelli estimate which upper bounds the distance of the empirical branch point distribution to
the branch point distribution. Such an estimate should be known, but as we could not come up with a
reference, we show it in Lemma~\ref{l:VCestim} in the appendix.
We note that $\dimVC(\I)=2$ (compare Example~\ref{exp:004}).
\end{proof}
\begin{corollary}[metrizability] \label{c:shapemetrizable}
The sample shape topology is metrizable.
\end{corollary}
\begin{proof}
Because the sample shape topology is induced by a countable family of functions $(\shapedist)_{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$
with values in metrizable spaces, it is pseudo-metrizable. By Proposition~\ref{p:shapestrongerGw}, it is
stronger than the bpdd-Gromov-weak topology, hence a Hausdorff topology. Therefore, it is metrizable.
\end{proof}
\subsection{Convergence in distribution of sampled subtree masses}
\label{s:convmass}
In this subsection, we introduce yet another notion of convergence of algebraic measure trees which, in contrast to
sampling tree shapes, is based on sampling branch points and evaluating the masses of the subtrees that are
joined at these branch points.
This approach might be more similar to the case of metric measure spaces and distance matrix distributions,
because we sample a tensor of real numbers (masses of subtrees) as opposed to a combinatorial object (tree shape).
Thus, the typical tools of analysis are more readily applicable for the corresponding class of test functions.
Let $(T,c,\mu)\in\T_{2}$, and recall from (\ref{e:005}) for $u,v,w\in T$ the subtree component $\mathcal{S}_{c(u,v,w)}(x)$ of
$T\setminus\{c(u,v,w)\}$ which contains $x\not=c(u,v,w)$. Here, we always take the component containing $x=u$,
and consider its mass
\begin{equation}\label{e:eta}
\eta(u,v,w) := \mathds{1}_{u\ne c(u,v,w)} \cdot \mu\(\mathcal{S}_{c(u,v,w)}(u)\).
\end{equation}
\begin{lemma}[measurability of the subtree masses] \label{l:002}
For every binary algebraic measure tree\/ $\mathpzc{x}=(T,c,\mu)\in \T_{2}$ and\/ $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$,
the function\/ $\eta\colon T^3\to [0,1]$ is measurable.
\end{lemma}
\begin{proof}
First, we claim that the map $\psi\colon T^2 \to [0,1]$,
\begin{equation}
\psi(u,v) := \mathds{1}_{u\ne v} \cdot \mu\(\mathcal{S}_v(u)\)
\end{equation}
is lower semi-continuous. Indeed, let $(u_n,v_n)$ be a sequence converging to $(u,v)$. We may assume
w.l.o.g.\ that $v\ne u$, $u_n\in\mathcal{S}_v(u)$, and either $v_n \not\in \mathcal{S}_v(u)$ for all $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, or
$v_n \in \mathcal{S}_v(u)$ for all $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$.
In the first case, $\mathcal{S}_v(u) \subseteq \mathcal{S}_{v_n}(u_n)$, and hence $\psi(u,v) \le \psi(u_n,v_n)$.
In the second case, for every $x\in \mathcal{S}_v(u)$ and $n \ge n_x$ sufficiently large, we have $u\in
\mathcal{S}_{v_n}(u_n)$ and $v_n\not\in [x,u]$. This means $x\in \mathcal{S}_{v_n}(u)=\mathcal{S}_{v_n}(u_n)$ and hence
\begin{equation}\label{e:}
\psi(u,v) - \liminf_{n\to\infty} \psi(u_n,v_n) \le \lim_{n\to\infty}\mu\(\mathcal{S}_v(u)\setminus \mathcal{S}_{v_n}(u_n)\) = 0.
\end{equation}
Therefore, $\psi$ is lower semi-continuous.
Because the branch point map $c$ is continuous due to Lemma~\ref{l:ccont}, the same applies to
$\eta(u,v,w) = \psi((u,\, c(u,v,w))$, and $\eta$ is measurable.
\end{proof}
Given a vector $\underline{u}=(u_1,\ldots,u_m)\in T^m$, $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, we consider the masses of all the subtrees we obtain as branch
points of entries of $\underline{u}$. To this end, let
\begin{equation}\label{e:ueta}
\underline{\eta}(u,v,w) := \(\eta(u,v,w),\, \eta(v,u,w),\, \eta(w,u,v)\)
\end{equation}
and define the function\/ $\m\colon T^m\to \tensor$, given by
\begin{equation} \label{e:m}
\m(\underline{u}) := \( \underline{\eta}(u_i,u_j,u_k) \)_{1\le i < j < k \le m}
\end{equation}
\begin{definition}[subtree-mass tensor distribution]\label{d:massdist}
For $\mathpzc{x}=(T,c,\mu) \in \T_{2}$ and $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, the $m$\protect\nobreakdash-\hspace{0pt}\define{subtree-mass tensor distribution} of $\mathpzc{x}$ is
defined by
\begin{equation}\label{e:massdist}
\massdist(\mathpzc{x}):=\mu^{\otimes m}\circ \m^{-1}\in \mathcal{M}} \newcommand{\CP}{\mathcal{P}_1\(\tensor\),
\end{equation}
\end{definition}
\begin{example}[symmetric binary tree] Let for each $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, $\mathpzc{x}_n=(T_n,c_n,\mu_n)$ the symmetric binary
tree with $N=2^n$ leaves and the uniform distribution on the set of leaves. Then the $3$\protect\nobreakdash-\hspace{0pt} \define{subtree-mass tensor distribution} of $\mathpzc{x}_n$ is equal to
\begin{equation}
\label{e:139}
\begin{aligned}
\massdist[3](\mathpzc{x}_n) &= \mu_n^{\otimes 3}\circ \m^{-1}
\\
&=\sum_{k=1}^{n-1} \,\frac{1-2^{-k}}{2^{k+1}}\,
\Bigl( \delta_{(\tfrac{1}{2^{k+1}},\tfrac{1}{2^{k+1}},1-\tfrac{1}{2^k})}
+ \delta_{(\tfrac{1}{2^{k+1}},1-\tfrac{1}{2^k},\tfrac{1}{2^{k+1}})}
+ \delta_{(1-\tfrac{1}{2^k},\tfrac{1}{2^{k+1}},\tfrac{1}{2^{k+1}})} \Bigr)
\\
&\phantom{{}={}} + \tfrac1N (1-\tfrac1N) \( \delta_{(\frac{1}{N},\frac{1}{N},1)}
+ \delta_{(\frac{1}{N},1,\frac{1}{N})} + \delta_{(1,\frac{1}{N},\frac{1}{N})} \)
+\tfrac1{N^2}\delta_{(\frac{1}{N},\frac{1}{N},\frac{1}{N})}
\\
&\tno \sum_{k=1}^\infty \,\frac{1-2^{-k}}{2^{k+1}}\,
\Bigl( \delta_{(\tfrac{1}{2^{k+1}},\tfrac{1}{2^{k+1}},1-\tfrac{1}{2^k})}
+ \delta_{(\tfrac{1}{2^{k+1}},1-\tfrac{1}{2^k},\tfrac{1}{2^{k+1}})}
+ \delta_{(1-\tfrac{1}{2^k},\tfrac{1}{2^{k+1}},\tfrac{1}{2^{k+1}})} \Bigr)
\end{aligned}
\end{equation}
\label{exp:007}
\end{example}
\begin{remark}[$3$-subtree-mass tensor distribution is not enough]
It is not enough to consider only the $3$\protect\nobreakdash-\hspace{0pt} \define{subtree-mass tensor distribution}. Indeed, $\massdist[3]$ cannot distinguish all
non-isomorphic binary algebraic measure trees, i.e.\ it does not separate the points of $\T_{2}$. To see
this, take the tree from Figure~\ref{f:3nonsep} with uniform distribution on its $12$ leaves, and the
same tree with the subtrees marked by $\times$ and $\circ$, respectively, exchanged.
These two trees are clearly non-isomorphic, and because the
two marked subtrees have the same number of leaves, every vertex in one tree corresponds to a vertex in
the other with the same value for $\m$.
\label{rem:004}
\end{remark}
\xymatfig{f:3nonsep}{
&\leaf\ar@{-}[dr]& &\leaf\ar@{-}[dl]& & & & &\leaf\ar@{-}[d] &\ & &\leaf\ar@{-}[dl]\\
\leaf\ar@{-}[dr]& &\Bnode\ar@{-}[d] &\leaf\ar@{-}[dr]& &\leaf\ar@{-}[dl]&\leaf\ar@{-}[dr]& &\Rnode\ar@{-}[dl]& &\Rnode\ar@{-}[dl]& \\
&\Bnode\ar@{-}[r] &\Bnode\ar@{-}[dr]& &\node\ar@{-}[dl]&\leaf\ar@{-}[dr]& &\node\ar@{-}[dl]& &\Rnode\ar@{-}[ul] & &\leaf\ar@{-}[ul]\\
\leaf\ar@{-}[ur]& & &\node\ar@{-}[rrr]& & &\bullet& & & &\leaf\ar@{-}[ul] &
}{$\mu$ is the uniform distribution on the leaves. Swap the $\circ$-part with the $\times$-part to obtain a
non-isomorphic tree giving the same value for $\massdist[3]$.}
We consider the weakest topology on $\T_{2}$ such that for every $m\in \mathbb{N}} \newcommand{\M}{\mathbb{M}$ the $m$-subtree-mass tensor distribution is
continuous. Here, as usual, we equip $\mathcal{M}} \newcommand{\CP}{\mathcal{P}_1(\tensor)$ with the weak topology.
\begin{definition}[sample subtree-mass topology]\label{d:massconv}
The topology induced on $\T_{2}$ by the set $\set{\massdist}{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of subtree-mass tensor
distributions is called \define{sample subtree-mass topology}.
\end{definition}
We say that a sequence $(\mathpzc{x}_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ is \define{sample subtree-mass convergent} to $\mathpzc{x}$ in $\T_{2}$ if it
converges w.r.t.\ the sample subtree-mass topology, i.e.\ if $\massdist(\mathpzc{x}_n)$ converges to $\massdist(\mathpzc{x})$
as $n\to\infty$ for every $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$.
To see that the sample subtree-mass topology is a Hausdorff topology on $\T_{2}$, we need the following reconstruction theorem.
\begin{proposition}[reconstruction theorem]
The set of subtree-mass tensor distributions\/ $\set{\massdist}{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ separates points of\/ $\T_{2}$,
i.e., if\/ $\mathpzc{x}_1,\mathpzc{x}_2\in\T_{2}$ are such that\/ $\massdist(\mathpzc{x}_1)=\massdist(\mathpzc{x}_2)$ for all\/
$m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$, then\/ $\mathpzc{x}_1=\mathpzc{x}_2$.
\label{p:reconstruction}
\end{proposition}
\begin{proof}
We always assume that the representative $(T,c,\mu)$ of an algebraic measure tree is chosen such that
$\mu(\mathcal{S}_{v}(u))>0$ whenever $u,v\in T$, $u\ne v$.
Because the set $\set{\shapedist}{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of tree shape distributions separates points by
Corollary~\ref{c:shapemetrizable}, it is enough to show that $\shapedist$ is determined by the $m$-subtree-mass tensor
distribution $\massdist$ for every $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$. We do so by showing that there exists a (non-continuous) function
$h\colon \tensor\to\Clad$ such that for every $\mathpzc{x}=(T,c,\mu)\in\T_{2}$ we have $\shape=h\circ \m$ on
$\(T\setminus \br(T,c)\)^m$.
This is enough, because $\mu(\br(T,c))=0$ by countability of $\br(T,c)$ and the assumption that $\at(\mu)\subseteq \lf(T,c)$.
Fix $\underline{u}=(u_1,...,u_m)\in (T\setminus\br(T,c))^m$ and set $C=(C,c_C,\ell):=\shape(\underline{u})$. For $i\ne j$,
we have $u_i=u_j$ if and only if $\eta(u_i,u_j,u_k)=\eta(u_j,u_i,u_k)=0$ for any and hence all $k\in
\{1,...,m\}\setminus \{i,j\}$. Thus, we can determine multiple labels of $C$ by $\m(\underline{u})$ and may assume
in the following that $u_1,...,u_m$ are distinct.
Then, the \nclad\ $C$ is uniquely determined by
the set of pairs $(\underline{x}_1,\underline{x}_2)$ of triples $\underline{x}_i=(x_{i,1},x_{i,2},x_{i,3}) \in\{u_1,...,u_m\}^3$, $x_{i,j} \ne
x_{i,k}$ for $j\ne k$, $i=1,2$, such that
\begin{equation} \label{e:ceq}
c_C\big(x_{1,1},\,x_{1,2},\,x_{1,3}\big) = c_C\big(x_{2,1},\, x_{2,2},\, x_{2,3}\big).
\end{equation}
We claim that \eqref{e:ceq} holds if and only if we can reorder the three entries of $\underline{x}_2$ such that we
can replace every entry of $\underline{x}_1$ by the corresponding entry of $\underline{x}_2$ and obtain the same masses of
subtrees. More precisely,
\begin{equation}\label{e:samem}
\underline{\eta}(x_{1,1},x_{1,2},x_{1,3}) = \underline{\eta}(x_{i,1},x_{j,2},x_{k,3}) \quad\forall i,j,k\in\{1,2\}.
\end{equation}
Indeed, if $c_C(\underline{x}_1)=c_C(\underline{x}_2)$, then $c(\underline{x}_1)=c(\underline{x}_2)$ by definition of $\shape$.
Because none of the $u_i$ is a branch point, every component of $T\setminus \{c(\underline{x}_1)\}$ contains
precisely one of the $x_{1,i}$, as well as one of the $x_{2,i}$. We can reorder the entries of
$x_2$ such that $x_{1,i}$ is in the same component as $x_{2,i}$, $i=1,\ldots,3$. Then it is easy to
check that \eqref{e:samem} holds.
Conversely, assume that $c_C(\underline{x}_1) \ne c_C(\underline{x}_2)$. Because the
restriction of the tree homomorphism $C \to c(\{u_1,\ldots,u_m\}^3)$ to the branch points of $C$ is
injective, this implies $v_1 := c(\underline{x}_1) \ne c(\underline{x}_2)=:v_2$. There must be an $i$ with
$x_{1,i} \in \mathcal{S}_{v_1}(v_2)$, say $i=3$. Also, $x_{2,j}\in \mathcal{S}_{v_1}(v_2)$ for at least two different
$j$, so at least one which is different from $i$, say $j=2$ (see Figure~\ref{f:twocase}). Then
$v_3:=c(x_{1,1}, x_{2,2}, x_{1, 3}) \in \mathcal{S}_{v_1}(v_2)$, and in particular, $x_{1,1},x_{1,2} \in
\mathcal{S}_{v_3}(x_{1,1})$. Thus $\eta(\underline{x}_1) < \eta(x_{1,1}, x_{2,2}, x_{1,3})$, and \eqref{e:samem} does not hold.
\xymatfig{f:twocase}{
x_{1,1}\ar@{-}[dr] & & x_{1,3}\ar@{-}[d] & & x_{2,2}\ar@{-}[dl] \\
& v_1\ar@{-}[r] & v_3\ar@{-}[r] & v_2\ar@{-}[dr] \\
x_{1,2}\ar@{-}[ur] & & & &
}{The situation in the proof of Proposition~\ref{p:reconstruction}.}
\end{proof}
\begin{corollary}[metrizability]\label{c:massmetrizable}
The sample subtree-mass topology is metrizable.
\end{corollary}
\begin{proof}
Because the sample subtree-mass topology is induced by a countable family of functions $(\massdist)_{m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$
with values in metrizable spaces, it is pseudo-metrizable. By Proposition~\ref{p:reconstruction}, it is
a Hausdorff topology, hence it is metrizable.
\end{proof}
In analogy to the sets $\Pi_\iota$ and $\Pi_{\shape[]}$ of polynomials and shape polynomials, respectively, the sample
subtree-mass topology also comes with a canonical set of test functions.
We call $\Psi\colon \T_{2}\to \mathbb{R}} \newcommand{\Z}{\mathbb{Z}$ \define{subtree-mass polynomial} if there is $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and
$\psi\in \mathcal{C}_b(\tensor)$ with
\begin{equation}\label{e:mpol}
\Psi(\mathpzc{x})
= \inta{\tensor}{\psi\,}{\massdist(\mathpzc{x})}
= \intamu{T^m}{\psi\circ \m}
\end{equation}
We also define
\begin{equation} \label{e:140}
\Pi_{\m[]} := \{\,\text{subtree-mass polynomials on $\T_{2}$}\,\}.
\end{equation}
Obviously, the sample subtree-mass topology is induced by the set $\Pi_{\m[]}$ of subtree-mass polynomials.
\begin{proposition}[sample shape convergence implies sample subtree-mass convergence]
The sample shape topology is stronger than the sample subtree-mass topology.
\label{p:shapestrongermass}
\end{proposition}
\begin{proof}
The proof is similar to that of Proposition~\ref{p:shapestrongerGw}.
We will show that each subtree-mass polynomial in $\Psi \in \Pi_{\m[]}$,
\begin{equation}\label{e:142}
\Psi(T,c,\mu) = \intamuu{T^m}{\psi\(\(\underline{\eta}(u_i,u_j,u_k)\)_{1\le i<j<k\le m}\)}{\underline{u}},
\end{equation}
with $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ and $\psi\in\mathcal{C}(\tensor)$ Lipschitz continuous w.r.t.\ the
$\ell_\infty$-Norm on $\tensor$ is in the uniform closure of $\Pi_{\shape[]}$.
Let $L$ be the Lipschitz constant of $\Psi$.
For $n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ with $n\ge m$, we define
\begin{equation}
\label{e:143}
\Phi_n(T,c,\mu) := \intamuu[n]{T^n}{\psi\(\(\underline{\eta}^{\mu_{n,\underline{u}}}(u_i,u_j,u_k)\)_{1\le i<j<k\le m}\)}{\underline{u}},
\end{equation}
where $\underline{\eta}^{\mu_{n,\underline{u}}}$ is defined in the same way as $\underline{\eta}$, but with $\mu$ replaced by
the empirical sample distribution
\begin{equation} \label{e:144}
\mu_{n,\underline{u}}:= \tfrac1n\sum_{\ell =1}^{n}\delta_{u_\ell}.
\end{equation}
Note that $\Phi_n\in\Pi_{\shape[]}$ because whether or not $u_\ell\in \mathcal{S}_{c(u_i,u_j,u_k)}(u_i)$ for some
$\ell\in\{1,...,n\},\; i,j,k\in\{1,...,m\}$ depends only on the shape $\shape(\underline{u})$.
Finally, applying the uniform Glivenko-Cantelli estimate Lemma~\ref{l:VCestim}, we have
\begin{equation}
\label{e:145}
\begin{aligned}
\| \Psi -\Phi_n \|_\infty \le \sup_{(T,c,\mu)\in \T_{2}}\,
\intamuu[n]{T^n}{L\cdot\sup_{S\in\Sset} \bigl|\mu(S)-\mu_{n,\underline{u}}(S)\bigr|\,}{\underline{u}} \le L \eps_n \tno 0,
\end{aligned}
\end{equation}
where $\Sset:=\bset{\mathcal{S}_v(u)}{u,v\in T}$ and $(\eps_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}\tno 0$.
We note that $\dimVC(\Sset)\le 3$ (compare Example~\ref{exp:006}).
\end{proof}
\subsection{Equivalence and compactness of topologies}
\label{s.equivalence}
In this section, we show that sample shape convergence (Definition~\ref{d:shapeconv}), sample subtree-mass
convergence (Definition~\ref{d:massconv}) and branch point distribution distance Gromov-weak convergence
(Definition~\ref{d:bpddGw}) on $\T_{2}$ are equivalent.
While spaces of metric measure spaces are usually far from being locally compact, $\T_{2}$ is in this topology
even a compact metrizable space.
\begin{theorem}[equivalence of topologies and compactness]
The sample shape topology, the sample subtree-mass topology, and the bpdd-Gromov-weak topology
coincide on\/ $\T_{2}$.
Furthermore, $\T_{2}$ is compact and metrizable in this topology.
\label{t:topeq}
\end{theorem}
Because compact subsets of a Hausdorff space are closed, a direct corollary is that unlike the situation in the
space of metric measure trees (with Gromov-weak or Gromov-Hausdorff-weak topology), the set of binary trees is
closed w.r.t.\ the bpdd-Gromov-weak topology. In particular, Gromov(-Hausdorff)-weak convergence does not imply
bpdd-Gromov-weak convergence of the induced trees.
\begin{cor}
The subspace\/ $\T_{2}$ of binary algebraic measure trees with atoms restricted to leaves is closed
in\/ $\mathbb{T}$ (with bpdd-Gromov-weak topology).
\end{cor}
As a preparation of the proof for the theorem, we show that binary algebraic measure trees depend continuously on
their encoding as sub-triangulations of the circle. Together with Proposition~\ref{p:shapestrongerGw}, this also
finishes the proof of Theorem~\ref{t:tree}. Recall the space $\mathcal{T}$ of sub-triangulations of the circle
equipped with the Hausdorff metric topology from \eqref{e:tatT}, and the coding map
$\tau\colon \mathcal{T} \to \T_{2}$ from Theorem~\ref{t:tree}.
\begin{lemma}[continuity of the coding map]
Let\/ $\T_{2}$ be equipped with the sample shape topology, and\/ $\mathcal{T}$ with the
Hausdorff metric topology. Then the coding map\/ $\tau\colon \mathcal{T}\to\T_{2}$ is continuous.
\label{l:Fcont}
\end{lemma}
\begin{proof} Fix $C\in\mathcal{T}$ and $m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$. By definition of the sample shape topology, it
is enough to show that $\shapedist\circ\tau\colon \mathcal{T}\to\mathcal{M}} \newcommand{\CP}{\mathcal{P}_1(\Clad)$ is continuous at $C$.
Let $U_1,...,U_m$ be i.i.d.\ points on the circle $\S$ chosen with the Lebesgue measure.
Recall from \eqref{e:nabla} the set $\nabla(C)$ of connected components of $\mathbb{D}\setminus\conv(C)$,
from \eqref{s:comp} the connected component $\comp{x}{y}$ of $\mathbb{D}\setminus\partial_\mathbb{D} x$ which contains $y$,
where $x\in\Delta(C)\cup \nabla(C)$, and $y\subseteq\mathbb{D}$ connected and disjoint from $\partial_\mathbb{D} x$.
Furthermore, recall the set $\Box(C)$ from \eqref{e:Box}, and the subtree components $\mathcal{S}_x(y)$ from \eqref{e:equiv}.
For $\eps>0$, there exists $N=N_{C,m,\eps}\in \mathbb{N}} \newcommand{\M}{\mathbb{M}$ and $v_1,...,v_N\in
\Delta(C) \cup \nabla(C)$ distinct such that with probability at least $1-\eps$ the following holds:
\begin{itemize}
\item if $\{U_1,...,U_m\}\cap v\not=\emptyset$ for $v\in\nabla(C)$, then $v\in\{v_1,...,v_N\}$, and
\item if $\{U_1,...,U_m\}\cap \comp{v}{w}\not=\emptyset$ for some $v\in\Delta(C)$ and all $w\in\Delta(C)\cup\nabla(C)\cup\Box(C)$ with $w\not=v$, then $v\in\{v_1,...,v_N\}$.
\end{itemize}
\iffalse{
for all $v\in \nabla(C)\setminus\{v_1,...,v_N\}$, we have $\{U_1,...,U_m\}\cap v = \emptyset$,
and
for all $v\in \Delta(C) \setminus \{v_1,...,v_N\}$ there is $w\in (\Delta(C)\cup \nabla(C)\cup
\Box(C))\setminus\{v\}$ with $\{U_1,...,U_m\}\cap \comp{v}{w} = \emptyset$.
}\fi
Put $\eps':=\eps\cdot (12mN)^{-1}$. Then
\begin{equation}
\label{e:444}
\mathbb{P}\big(\big\{d(U_i,\partial v_j)\ge\eps',\,\forall\,i=1,...,m;j=1,...,N\big\}\big)\ge 1-\eps.
\end{equation}
There is a $\delta=\delta(\eps)>0$ sufficiently small such that for any $C'\in \mathcal{T}$ with Hausdorff metric
$\dH(C,C')<\delta$ there are distinct $v_1',...,v_N'\in \Delta(C')\cup\nabla(C')$ such that
$\dH(v_i,v_i')\le\eps'$ for $i=1,...,N$.
Let $\mathpzc{x}=(T,c,\mu):=\tau(C)$, and $V_1,...,V_m$ be i.i.d.\ $\mu$\protect\nobreakdash-\hspace{0pt} distributed, coupled to
$U_1,...,U_m$ such that $V_k\in\mathcal{S}_v(w)$ if and only if $U_k \in \comp{v}{w}$, which is possible due to the
properties of $\tau$ established in Theorem~\ref{t:tree}.
Define $\mathpzc{x}'$ and $V_1',...,V_m'$ similarly with $C'$ instead of $C$.
Then
\begin{equation}
\label{e:120}
\mathbb{P}\big(\big\{\shape(V_1,...,V_m)=\shape[T'](V_1',...,V_m')\big\}\big)\ge 1-2\eps,
\end{equation}
which implies that
$d_\mathrm{Pr}\(\shapedist(\tau(C)),\, \shapedist(\tau(C'))\) \le 2\eps$ (with $d_\mathrm{Pr}$ denoting the Prokhorov distance).
This shows that $\shapedist\circ\tau$ is continuous at $C$ and, since $m$ and $C$ are arbitrary, that $\tau$
is continuous.
\end{proof}
Now we are in a position to combine our results to a proof of the main theorem of Section~\ref{S:topo}.
\begin{proof}[Proof of Theorem~\ref{t:topeq}]
The space $\mathcal{T}$ of sub-triangulations of the circle with Hausdorff metric topology is compact according to
Lemma~\ref{l:triangcomp}.
The coding map $\tau \colon \mathcal{T} \to \T_{2}$ is surjective by Theorem~\ref{t:tree},
and continuous when $\T_{2}$ is equipped with the sample shape topology by Lemma~\ref{l:Fcont}.
Therefore, the sample shape topology is a compact topology on $\T_{2}$. Moreover, the sample shape topology is Hausdorff by
Corollary~\ref{c:shapemetrizable}. As the sample subtree-mass topology is a weaker
Hausdorff topology by Proposition~\ref{p:shapestrongermass} and Corollary~\ref{c:massmetrizable}, it coincides with the sample shape topology.
The same is true for the bpdd-Gromov-weak topology by Proposition~\ref{p:shapestrongerGw}.
\end{proof}
Recall from Remark~\ref{rem:bpddGw} that the set of distance polynomials is convergence determining for measures on $\T_{2}$.
It directly follows from the construction that the same is true for the sets of shape polynomials and subtree-mass polynomials.
This property is very useful for proving
convergence in law of random variables.
\begin{cor}[convergence determining classes of functions] \label{c:convdet}
The sets\/ $\Pi_{\shape[]}\subseteq \mathcal{C}_b(\T_{2})$ (defined in \eqref{e:spol}) and\/ $\Pi_{\m[]}$ (defined in \eqref{e:mpol}) are
convergence determining for measures on\/ $\T_{2}$ with bpdd-Gromov-weak topology.
\end{cor}
\begin{proof}
$\T_{2}$ is a compact metrizable space, and both $\Pi_{\shape[]}$ and $\Pi_{\m[]}$ induce the bpdd-Gromov-weak topology
on $\T_{2}$ by Theorem~\ref{t:topeq}. Furthermore, each of $\Pi_{\shape[]}$ and $\Pi_{\m[]}$ is closed under multiplication.
Thus the claim follows by the Stone-Weierstrass theorem.
\end{proof}
\section{Example: sampling consistent families}
\label{s:examples}
Consider a family $(T_n, c_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of random, finite binary (algebraic) trees, where $(T_n,c_n)$ has $n$ leaves.
Let $K_n$ be the Markov kernel that takes such a tree and removes a leaf uniformly chosen at random,
together with the branch point it is attached to, thus obtaining a binary tree with $n-1$ leaves.
We say that the family is \define{sampling consistent} if $K_n(T_n,\,\cdot\,)=\mathcal{L}(T_{n-1})$, where $\mathcal{L}$
denotes the law of a random variable.
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\end{center}
\caption{Realisations of $\beta$-splitting trees for (from left to right)
$\beta=-1$, $\beta=0$ (Yule tree), $\beta=10$.}
\label{f:betasplit}
\end{figure}
\begin{example}[$\beta$-splitting trees]
For every $\beta\in [-2,\infty]$, let $T^\beta_n$ be the $\beta$\protect\nobreakdash-\hspace{0pt} splitting tree on $n$ leaves
from \cite{Aldous1996} (with forgotten labels). For $-2<\beta<\infty$, the $\beta$\protect\nobreakdash-\hspace{0pt} splitting tree
$T^\beta_n$ can be constructed recursively as follows.
$T^\beta_2$ consists of two leaves connected by a distinguished root edge. If $n>2$, choose $i\in
\{1,\ldots,n-1\}$ with probability
\begin{equation}
q_n^\beta(i) = \frac1{a_n(\beta)} \binom ni \int_0^1 x^{i+\beta}(1-x)^{n-i+\beta} \,\mathrm{d} x,
\end{equation}
where $a_n(\beta)$ is a normalisation constant. Then construct two independent $\beta$\protect\nobreakdash-\hspace{0pt} splitting trees $T^\beta_i$ and
$T^\beta_{n-i}$, introduce a new branch point in the middle of each of the two root edges, and connect
these new branch points with the new root edge to obtain $T^\beta_n$.
It is easy to see (and observed in \cite{Aldous1996}) that $(T^\beta_n)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ is sampling consistent.
Note the special cases $\beta=-2$ which is the {\em comb tree}, $\beta=-\frac32$ which is the \emph{uniform cladogram},
$\beta=0$ which is the \emph{Yule tree} and $\beta=\infty$ which is the {\em symmetric binary tree}.
See Figure~\ref{f:betasplit} for triangulations of a realization of
$\beta$\protect\nobreakdash-\hspace{0pt} splitting trees for different values of $\beta$ and large $n$. The Aldous Brownian CRT, which is the
limit for $\beta=-\frac32$, is shown in Figure~\ref{f:nontriang}.
\label{exp:009}
\end{example}
\begin{lemma}[convergence of sampling consistent families]\label{l:sampconsist}
Let\/ $((T_n,c_n))_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ be a sampling consistent family of random binary trees, and\/ $\mu_n$ the uniform
distribution on\/ $\lf(T_n,c_n)$. Then we have the convergence in law
\begin{equation}
(T_n,c_n,\mu_n) \,\xrightarrow[n\to\infty]{\mathcal{L}}\, (T,c,\mu) \quad\text{on\/ $\T_{2}$ with bpdd-Gromov-weak topology}
\end{equation}
for some random algebraic measure tree\/ $(T,c,\mu) \in \T_{2}$ with non-atomic measure\/ $\mu$.
\end{lemma}
\begin{proof}
Recall the $m$\protect\nobreakdash-\hspace{0pt} tree shape distribution $\shapedist$ from Definition~\ref{d:shapeconv}.
Let $n,m\in\mathbb{N}} \newcommand{\M}{\mathbb{M}$ with $m<n$ and define
\begin{equation}
\eps_{n,m} := \mu_n^{\otimes m}\bset{x\in T^m}{x_1,\ldots,x_m \text{ not distinct}}
\le \tfrac{m^2}{n}.
\end{equation}
Because $(T_n)$ is sampling consistent, we obtain for the annealed shape distribution
\begin{equation}\label{e:shapeconv}
\mathbb{E}\(\shapedist(T_n,c_n,\mu_n)\) = (1-\eps_{n,m}) \mathcal{L}(T_m^*) + \eps_{n,m} \mu_{n,m},
\end{equation}
where $T_m^*$ is obtained from $T_m$ by randomly labelling the leaves, and $\mu_{n,m} \in \mathcal{M}_1(\Clad)$
is some law of \nclad s supported by cladograms where at least one leaf has more than one label.
This shows that, for every fixed $m$, the expected $m$\protect\nobreakdash-\hspace{0pt} tree shape distribution converges as
$n\to\infty$. Because the $m$\protect\nobreakdash-\hspace{0pt} tree shape distribution is convergence determining for the bpdd-Gromov-weak
topology by Corollary~\ref{c:convdet}, all limit points of $\mathcal{L}(T_n,c_n,\mu_n)$ in $\mathcal{M}_1(\T_{2})$ coincide.
According to Theorem~\ref{t:topeq}, $\T_{2}$, and hence $\mathcal{M}_1(\T_{2})$, is compact and thus a unique
limit exists. That the limiting measure is non-atomic is obvious, because the probability that a sampled
shape is single-labelled tends to one by \eqref{e:shapeconv}.
\end{proof}
In the parameter range $\beta \in \ropenint{-2}{-1}$, the height (in graph distance) of the $\beta$-splitting
tree with $n$ leaves is asymptotically of power-law order $\Theta(n^{-\beta -1})$.
In this case, after rescaling edge-lengths with the factor $n^{\beta+1}$, Gromov-Hausdorff convergence in law to
a fragmentation tree is shown in \cite[Corollary~16]{HaasMiermontPitmanWinkel08}.
In the case $\beta>-1$, the height of the tree is only of logarithmic order $\Theta(\log(n))$, and it is easy to
see that no non-trivial Gromov-Hausdorff scaling limit (with uniform edge rescalings) exists. Seen as algebraic
measure trees, however, it easily follows from sampling consistency that the bpdd-Gromov-weak limit exist in the
full parameter range $\beta\in [-2,\infty]$.
\begin{example}[$\beta$-splitting trees continued]
By Lemma~\ref{l:sampconsist}, for every $\beta \in [-2,\infty]$, the sequence $(T_n^\beta, c_n^\beta,
\mu_n^\beta)_{n\in\mathbb{N}} \newcommand{\M}{\mathbb{M}}$ of increasing $\beta$\protect\nobreakdash-\hspace{0pt} splitting trees converges in distribution to some
limiting random algebraic measure tree $(T^\beta, c^\beta, \mu^\beta)$. In the case of the uniform cladogram
($\beta=-\frac32$), the limit is the Brownian algebraic continuum random tree which can be obtained as
tree $\tau(C_{\mathrm{CRT}})$ coded by the Brownian triangulation (see Example~\ref{ex:CRT}), or as the algebraic
measure tree induced by the metric measure Brownian CRT which is known to have uniform shape
distribution (\cite{Aldous1993}).
In the case of the comb tree ($\beta=-2$), the limit is the unit interval with Lebesgue measure (a
coding triangulation is shown in the very right of Figure~\ref{f:nobranch}).
\label{exp:008}
\end{example}
|
1,108,101,565,493 | arxiv | \section{Introduction}
The propagation of optical waves in random media, including atmospheric turbulence, is governed by the stochastic Hemholtz equation. Applying weak perturbation theory, it is possible to achieve closed form expressions for most field statistics as outlined by Tatarskii\cite{tatarski1967wave}. Similarly, asymptotic theory\cite{hill1981theory},\cite{hill1982theory}, provides an excellent match to experiment when fluctuations are very large. In between these two regions Extended Rytov Theory\cite{andrews1999theory} (ERT) provides a heuristic analytical model that has found a good match to some experimental data. Attempts at providing solutions in this region that do not rely on approximations involve complex multi-dimensional integrals and are difficult to generalize\cite{Barakat:99},\cite{flatte1987path},\cite{dashen1979path},\cite{dashen1984distribution}.
While not analytical, Wave Optics Simulations (WOS) are known to be an accurate approximation to the stochastic Hemholtz equation\cite{flatte2000irradiance}. This approximation is achieved by discretizing the propagation volume and collapsing the media in each segment into a thin phase screen. The optical field is then propagated between adjacent segments via a Fresnel propagation operator. The relative fluctuations at each screen are weak such that the approximation is equivalent to the small perturbation approximation used by Tatarskii. The main strength of WOS is that they can provide insight into wave propagation statistics and the performance of systems that attempt to compensate for the effects of the media and are relatively straightforward to execute.
Though the technique was previously described by others, relative to optical wave propagation Martin and Flatt\'e \cite{martin1988intensity} were the first to explore the use WOS for optical waves. Their studies \cite{martin1988intensity} of plane and spherical waves\cite{martin1990simulation} in random media described by a power-law examines the intensity scintillation as a function of fluctuation strength in the media. Since that time WOS have become the default tool used in modeling the performance of adaptive optics, beam projection, and Free Space Optical (FSO) communication systems \cite{schmidt2010numerical},\cite{voelz2011computational},\cite{belmonte2000feasibility},\cite{johansson1994simulation},\cite{frehlich2000simulation}. Simulations involving phase screens are also commonly used to create synthetic image data of scenes blurred by turbulence \cite{Hardie2017},\cite{bos2012technique},\cite{carrano2003anisoplanatic}. Until relatively recently it has been common to model atmospheric turbulence in WOS as purely Kolmogorov with an infinite outer scale and zero inner scale.
In 1994 Dauldier \cite{dalaudier1994direct} published evidence of turbulence in the upper atmosphere from balloon-borne measurements described as non-Kolmogorov in nature. Soon after Beland \cite{beland1995some} examined its implications to existing turbulence theory. Strilbing\cite{stribling1995optical} then extended those results to weak fluctuation theory. In this context, non-Kolmogorov turbulence refers to deviation of the power-law away from the 2/3 power-law structure function description of turbulence predicted by Kolmogorov in the inertial subrange between the energy input region (outer-scale) and dissipation region (inner scale). The equivalent two-dimensional energy spectral density of turbulent index of refraction fluctuations shows an -11/3 slope as a function of increasing spatial frequency. This latter quantity is more often found in modern analytical models that begin with the refractive index fluctuation energy spectrum starting with Toselli\cite{toselli2007scintillation},\cite{Cui:12},\cite{Yi:12},\cite{deng2012scintillation}. A common thread of all the works cited below and the work that followed is the prominence of analytical methods as opposed to numerical methods like WOS in exploring phenomenology.
While Martin and Flatt\`{e} did use WOS to examine the effect of random media on intensity scintillation it lacks a connection to modern atmospheric propagation work. It is also useful to understand the limitation of WOS and the effect of a change of power-law on those limits. In an earlier work \cite{Grulke} we made an early attempt at making these connections and understanding the relationship between the power-law exponent of a random media and scintillation as function of turbulence strength. Our conclusions, though, were that an explicit inner and outer scale were necessary.
In this work, we describe the results of a comprehensive WOS campaign, started in \cite{bos2015simulation} for plane waves propagating in uniform non-Kolmogorov turbulence volumes. Results are presented here for intensity scintillation as a function both Rytov number and power-law exponent. Results are limited to a single propagation geometry featuring a finite inner and outer scale. Recent results have shown that failure to account for these features can result in inaccuracies when evaluating some quantities\cite{beck2022wave}\cite{beck2021saturation}. We find that peak Normalized Intensity Variance (NIV) increases with power-law. Similarly, the Rytov number where peak NIV occurs also increases. This confirms earlier results \cite{bos2015simulation} indicating a shift of the NIV curve up and to the right as power-law increases. Across the campaign WOS are run to their practical limit for this propagation geometry assuming an upper bound of 8192 x 8192 phase screen samples. Thus, this work also describes the upper bound of WOS as a function of non-Kolmogorov power-law using Martin and Flatt\'e’s sampling constraints. Interesting here is that smaller-power laws are limited to Rytov numbers of about 7 compared to 12 for Kolmogorov turbulence and larger power-laws exponents.
The remainder of this paper is outlined as follows, following this introduction we provide an overview of WOS model and the sampling constraints used to define our simulation campaign. In section \ref{sec:Methods} we describe the campaign in detail and describe our scoring quantities. Results are provided in Section \ref{sec:Results} followed by conclusions and directions for future work in Section \ref{sec:Conclusions}.
\section{BACKGROUND}
\label{sec:back}
A goal of this work is to understand the practical limits of WOS via phase screens generated by filtering white Gaussian noise by the Power Spectral Density (PSD) spectrum of the turbulence fluctuations and using the split-step propagation method. Excellent descriptions of the split-step propagation technique can be found elsewhere\cite{voelz2011computational},\cite{schmidt2010numerical} and are not included here.
In approaching this work, we use the sampling constraints outlined by Martin and Flatt\'e\cite{martin1988intensity} for plane waves to ensure that the constraints account a non-Kolmogorov power-law medium. This limit is based upon the spatial bandwidth of the power-spectrum as sampled by the phase screen. As pointed out indirectly by Martin and Flatt\'e and elsewhere the power-law affects the prominence of high versus low frequency fluctuations in the media. For this reason, power-law random media with smaller power law are likely to have more high frequency fluctuations and therefore require a higher sampling rate compared to the Kolmogorov default.
Before describing those constraints let us first define the three-dimensional PSD of index of refraction fluctuations for a generalized turbulence volume described by a power-law with an inner and outer scale.
\begin{equation}
\Phi_{n}(\kappa, \alpha, z) = A(\alpha) \beta(z) \exp( -\kappa^2 / \kappa_{m}^2) (\kappa^2 + \kappa_{0}^2)^{(-\alpha/2)}
\label{eq:nokSpectrum}
\end{equation}
In Eq.\ref{eq:nokSpectrum} $A(\alpha )=(1/4{{\pi }^{2}})\cos \left( \frac{\pi \alpha }{2} \right)\Gamma [\alpha -1]$. In this description, the power-law exponent, $\alpha$, is restricted to values $3 < \alpha < 4$. The term $\beta(z)$ is a stand-in for the index of refraction structure constant, $C_{n}^2$ and has units of $m^{3-\alpha}$. The term $\kappa_m = c_1(\alpha)/l_0$, represents the inner scale of turbulence, $l_0$, and is defined as
${{c}_{1}}(\alpha )=2{{\left( \frac{8}{\alpha -2}\Gamma \left[ \frac{2}{\alpha -2} \right] \right)}^{\frac{\alpha -2}{2}}}$
In Kolmogorov turbulence, $\alpha = 11/3$, $A(\alpha) = 0.033$, $c_1(\alpha) = 6.88$. The term $\kappa_0$ sets the outer scale of turbulence, $L_0$, and is either set directly as $\kappa_0 = 1/L_0$ or as $\kappa = 2\pi/L_0$ and Eq.\ref{eq:nokSpectrum} reduces to the modified von Karman spectral model.
It is common for WOS to use the Fried parameter, $r_0$, to set turbulence strength for phase screens. In a previous work\cite{bos2016simulation}, one of us compared two methods of setting the phase screen turbulence strength for studies of non-Kolmogorov turbulence. One method used the non-Kolmogorov equivalent Fried parameter so that each screen has the same effective spatial coherence properties. The other technique relied on first normalizing the spectral energy and then scaling the energy in the screen by the equivalent fluctuation energy that would be found in a Kolmogorov turbulence volume. A finding of this work \cite{bos2016simulation} was that the latter method provides a better match to ERT when evaluating scintillation index in terms of the plane wave Rytov number ${{\sigma }_{R}}={{(1.23C_{n}^{2}{{k}^{7/6}}{{L}^{11/6}})}^{1/2}}$ where $k$ is the optical wavenumber and $L$ is the propagation distance.
In this work, phase screens are generated using the technique described originally in \cite{bos2015anisotropic} and extended to WavePy\cite{beck2016wavepy}. The current version of WavePy uses the sub-harmonic method described by Johansson and Gavel\cite{johansson1994simulation}. In a previous work\cite{bos2015anisotropic} one of us noted a practical limit of around $\alpha = 3.8$ for accurately modeling non-Kolmogorov phase screens with an infinite outer scale. Anecdotally, we find that this new method improves phase screen accuracy in terms of a structure function match and allows us to explore out to $\alpha = 3.9$. Though, as we will discuss later as $\alpha \to 4$ the turbulent disturbance becomes equivalent to a pure tilt \cite{fried1965statistics}. It follows then that improved subharmonic modeling would allow for larger power-laws to be modeled more accurately.
In determining sampling rates for our WOS campaign we use the spatial bandwidth definition described by Martin and Flatt\'e to determine the required spatial bandwidth, ${R}_{\chi }$ , in each phase screen as
\begin{equation}
{{R}_{\chi }}={{2}^{\left( 1-\frac{4}{\alpha }+\frac{1}{2+\alpha } \right)}}{{5}^{\left( \frac{1}{2+\alpha } \right)}}{{\sigma }^{2/\alpha }}{{\left( \frac{\left( 2+\alpha \right)\sec \left[ \frac{\pi \alpha }{4} \right]}{\alpha \Gamma \left[ 1+\frac{\alpha }{2} \right]} \right)}^{2/\alpha }} %
\label{eq:spatialBW}
\end{equation}
here
\begin{equation}
{{\sigma }_{R}}^{2}=C(\alpha )\beta {{k}^{1-\alpha /2}}{{L}^{2+\alpha /2}}
\label{eq:sigmaNOK}
\end{equation}
and
\begin{equation}
C(\alpha )=-\frac{2\Gamma \left[ -\frac{\alpha }{2} \right]\Gamma \left[ \alpha +3 \right]}{\left( \alpha +2 \right)\left( \alpha /2+1 \right)}\cos \left( \frac{\pi \alpha }{2} \right)\sin \left( \frac{\pi \alpha }{2} \right)
\end{equation}
In Eq.\ref{eq:spatialBW} power spectrum index has bounds $3<\alpha <4$. Note that we have used the more common 2D isotropic power spectrum representation as opposed to the 1D representation used by Martin and Flatt\'e where $1<\alpha <2$ and have adjusted the expressions presented here accordingly. From Eq.\ref{eq:spatialBW} we can find the required spatial sampling as $\Delta x=1/2{{R}_{\chi }}$. For the single simulation scenario considered in this work we fix the side length, $D = 1$ m, so that the required number of samples for each phase screen is $N=1/\Delta x=2{{R}_{\chi }}$.
Another requirement described by Martin and Flatt\'e \cite{martin1988intensity} and others\cite{schmidt2010numerical} limits the log-amplitude variance to not exceed 0.1 or 10\% between propagation steps. This is effectively a forward-scattering requirement or the Rytov approximation and can be set in terms of $\alpha$ as
\begin{equation}
\Delta L=\sqrt[1+\alpha /2]{C(\alpha )\beta {{k}^{2-\alpha /2}}{{\sigma }_{R}}^{2}}
\label{eq:nScreen}
\end{equation}
With the consequence that the minimum number of screens for a fixed propagation distance, $L$, is
${{n}_{scr}}= \textnormal{ceil}\left( \frac{L}{\Delta L} \right)+1$
Having laid out these requirements we can now define a set of propagation parameters for our fixed scenario laid out in the next section.
\section{Methods}
\label{sec:Methods}
The main result of this work is an extensive WOS campaign that evaluates the scintillation index or NIV as a function of the plane wave Rytov number in non-Kolmogorov turbulence. Here we intend to hold the path length fixed and vary only the turbulence strength of the volume in terms of the plane-wave Rytov number, $\sigma_R$, with the goal of performing simulations out to the limit allowed by the sampling constraints described in Section \ref{sec:back}. For the purposes of this work the maximum number of samples per screen will be $2^{13} = 8192$ square. While $2^{14} = 16384$ is feasible, for the scenarios here that sampling is required only for conditions well-within the saturation region and where wall-clock executions times are considerable.
The WOS campaign was run in two batches, the first was completed out to the maximum Rytov number allowed for one of three power laws, $\alpha$. Power laws values of $3.1$ and $3.9$ cover the lower and upper limits of practical simulation, while the Kolmogorov power value of $11/3$ $(3.66)$ was included as a baseline. A second, more limited, campaign was conducted for twenty-five values of $\alpha$ between $3$ and $4$. Values started at $\alpha = 3.02$ and increased increments of $0.02$ up to a value of $\alpha = 3.2$. From there $\alpha$ was incremented in steps of $0.1$ up to $3.8$ over the regime including the Kolmogorov value ($\alpha = 3.66$) included as a special case. From $\alpha = 3.8$ to $3.98$ steps were reduced again to $0.02$. The objective here being to identify values of power-law exponent where the simulation is no longer valid, or fidelity is reduced.
As mentioned in the previous section we fixed the side length $D =1$ m and varied the sampling rate $N$ as required. For the other WOS parameters, the wavelength is set to $\lambda = 1$ $\mu$m, and the path length to $L = 5$ km. For Kolmogorov turbulence these parameters allow for a full exploration of the various propagation regimes described by theory without resorting to values of $C_{n}^2$ unlikely to be measured in the field; in the range of $10^{-17}$ to $10^{-12}$ $m^{-2/3}$.
Using the Eqns.\ref{eq:sigmaNOK} and \ref{eq:nScreen} in Section \ref{sec:back} we are able to evaluate the maximum Rytov number that can be evaluated for a given sampling rate, $N$. Simultaneously, for a given value of $\alpha$, we can find the minimum sampling rate needed to evaluate a specific Rytov number to ensure accurate results. As outlined in \cite{Grulke} it is possible to model atmospheres with power-laws very near to the lower-bound of $\alpha=3$ via WOS. However, as the value of $\alpha$ approaches $3$ the number of samples required to simulate even weak turbulence is very large. This both unnecessarily limits the range of volume turbulence strengths we can explore and increases wall-clock simulation time. Using a value of $\alpha = 3.1$ as our lower-limit case provides some relief in these regards while allowing a detailed examination of the behavior at smaller power law exponents.
In our previous, related works \cite{beck2022wave},\cite{beck2021saturation}, we found that it is likely that WOS simulations without finite inner, $l_0$, and outer, $L_0$, scales are likely not valid. For this work, the outer scale size was set to $L_0=1$ m and the inner scale to $l_0 = 0.005$m. The former matches the fixed screen size, $D$, while the latter is slightly larger than the largest spatial sampling rate of $D/N$ for $N=256$ of $\Delta x = 0.0039$ m.
\subsection{Limits on WOS in non-Kolmogorov turbulence}
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width = 0.4\columnwidth]{limits_p.png}& \includegraphics[width = 0.4\columnwidth]{lr_limits.png}\\
(a) & (b)
\end{tabular}
\end{center}
\caption[Limits on WOS in NOK turbulence]{\label{fig:Limits} Limits on the maximum plane wave Rytov number, $\sigma_R$ that can be simulated for a specific power-law exponent, $\alpha$ (a) or screen size, N (b). The three values of $\alpha$ indicated are those chosen for detailed evaluation.}
\end{figure}
In Fig.\ref{fig:Limits} we evaluate Eq.\ref{eq:sigmaNOK} and \ref{eq:nScreen} in order to visualize the limits on WOS in non-Kolmogorov turbulence as described in this paper. In subfigure (a) the maximum Rytov number allowable for each screen size of $N=2^n$ where $n = 1,2,..14$ is presented. Of note here is that the maximum Rytov number that can be simulated with a screen size of $N=2^{14}$ is close to $\sigma_R = 24$. Also, that for all values of sampling rate the power-law allowing the highest turbulence strength is close to $\alpha=3.8$ it is also at this point where additional sampling most increases the range of Rytov numbers covered. While as $\alpha \to 3$ the same benefit is not conveyed, and the maximum turbulence strength is $\sigma_R = 6$ for $N=2^{13}$ and $\sigma_R = 7$ for $N=2^{14}$. Though not captured in our campaign we also note that as $\alpha \to 4$ the max Rytov number drops to zero. Again, this is because media with a power law of exactly $4$ are pure tilts and cannot be described via a structure function. This also explain the difficulty observed in \cite{bos2012technique} of generating accurate phase screen statistics at larger values of $\alpha$ and further underscores the practical limit of $\alpha = 3.9$ for WOS modeling.
\subsection{Simulation Banding}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
N / $\alpha$ & 256 & 512 & 1024 & 2048 & 4096 & 8192 \\
\hline
3.9 & 0-1.6 (10) &1.8-2.8 (15) & 2.9-4 (20) & 5 (20) & 6,7 (25) & 8,9 (35), 10-12 (60) \\
\hline
3.66 & 0-1.8 (10) & 2 - 3 (15), 3-4 (25) & 4,5 (25) & 6 (30) & 7-9 (40) & 10-12 (60) \\
\hline
3.1 & 0-2.6 (15) & 2.6 - 3 (20) & 3.2-3.8 (25) & 4,5 (35) & 6 (60) & 7 (60) \\
\hline
\end{tabular}
\caption{\label{tab:banding} Range of turbulence strengths $\sigma_R$ for each set of screen sizes, $N$, and ($n_{scr}$) per propagation. For each $\sigma_R$, 100 turbulence volumes were modeled for most cases. In the focusing regions 200 runs were performed in some instances. }
\end{center}
\end{table}
The figure on the right in Fig. \ref{fig:Limits} shows maximum $\sigma_R$ for the three values of $\alpha$ considered in the first batch of simulations as a function of sampling rate. In Table \ref{tab:banding} we lay out the banding for the first batch of our WOS campaign. For each value of $\alpha$ and $N$ the range of Rytov numbers is indicated. The number of WOS steps is indicated in parentheses. In the region approximately bound by $1.2 < \sigma_R < 2.$ for $\alpha = 3.66, 3.9$ a total of $200$ Monte-Carlo runs were executed for each value of $\sigma_R$ specified. Runs in the weak turbulence regime and saturation region required only $100$ runs to acheive statistical convergence.
\subsection{Calculation of NIV}
As defined by Andrews and Phillips\cite{andrews2005laser} the NIV or scintillation index, $\sigma_{I}^2$, is defined as the normalized variation in intensity such that
\begin{equation}
\sigma_{I}^2 = \frac{<I^2>}{<I>^2}-1
\label{eq:scintIdx}
\end{equation}
where the angle brackets indicate an ensemble average. In our simulations here, we wish to evaluate the scintillation index in the receiver plane for a plane wave source. The outcome of each simulation Monte Carlo trial is an intensity distribution in the receiver plane. The question, then, is how and where to evaluate scintillation in each case.
For a plane wave, defined as uniform in amplitude and infinite in extent, propagating in a vacuum the expected intensity profile in any receiver plane is similarly uniform in amplitude, and therefore, intensity. Consequently, we can evaluate the scintillation index at any point in the receiver plane over the ensemble of turbulence volumes or runs. Doing so increases the amount of averaging allowing evaluation of the fluctuation statistics with fewer Monte Carlo trials. Practically, though, we must account for the fact that our propagator \cite{beck2016wavepy} incorporates a super-Gaussian absorbing boundary to prevent wrap-around artifacts. For this reason, and to avoid the influence of wraparound or boundary effects we evaluate scintillation index for each point inside a circle with a radius of $D/4 = 0.25$m from on axis center of the receive plane.
\section{Results}
\label{sec:Results}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[]{new_full_scint.png}
\end{tabular}
\end{center}
\caption[Plane Wave Results Full Range or turbulence strength]{\label{fig:PWResults} Plane wave simulation of scintillation index, $\sigma_{I}^2$, as a function of Rytov number, $\sigma_R$, for $\alpha= 3.1, 3.66, 3.9$. For each power-law simulations were conducted out to the maximum Rytov number specified for a sampling rate of $N$=8192. In this scenario the pathlength was $5000$ m and the wavelength was $1$ $\mu$m. Error bars indicate the variance about the mean at each data point.For all values of $\alpha$ the outer-scale size was $L_0 = 1$ m and inner-scale was $l_0 = 5$ mm.}
\end{figure}
Fig.\ref{fig:PWResults} provides the results of the first of our two WOS campaigns. For each specified value of $\alpha$ scintillation as measured in our WOS is plotted out to the maximum Rytov number allowed according to the Eq.\ref{eq:spatialBW} for a screen with $N=8192$ samples. As described in Table \ref{tab:banding}, $\alpha = 3.1$ is limited to a maximum of $7$ while $\alpha = 3.66, 3.9$ go to $\sigma_R = 12$. The results here are similar to those presented elsewhere and previously \cite{bos2015simulation},\cite{Grulke},\cite{toselli2007scintillation} but in this instance include an explicit inner and outer scale in the turbulence spectrum sized based on the simulation geometry.
Examining Fig. \ref{fig:PWResults} we see the that the curves shift up and to the right relative to power-law. Earlier onset of saturation for the smaller power law case is also observed though without the sharp peak in the transition in the focusing region observed in \cite{bos2015simulation}. We attribute this difference to the explicit, finite, inner-scale in the turbulence spectrum in Eq.\ref{eq:nokSpectrum}. This spectral feature filters, or limits, the degree of small-scale fluctuations that drive scintillation and loss of spatial coherence resulting in saturation. Note that even if an inner scale is not an explicit feature of the turbulence power spectrum one is included, implicitly, by the simulation sampling rate. In WOS campaigns, like ours, where the sampling rate increases with turbulence strength, the inner scale would decrease with each change in sampling rate. Analytic theory indicates \cite{andrews2005laser}that this change should hasten onset of saturation and reduce peak scintillation. Here, our chosen value of $l_0 = 5$ mm is larger than the largest sampling rate, $\Delta x$, used in the WOS to avoid this complication.
In some previous works\cite{bos2015simulation},\cite{toselli2007scintillation} it was noted that scintillation may increase without bound as $\alpha \to 4$, and also that this behavior is missing in Fig.\ref{fig:PWResults}. Here, also, the inclusion of an explicit finite outer scale likely plays a role by limiting the relative energy in large-scale fluctuations\cite{Yi:12}. However, the observation holds that, all things being equal, peak scintillation increases with power-law and occurs at higher equivalent turbulence strength. So that, relative to Kolmogorov turbulence, an increase power law increases peak scintillation and results in relatively higher scintillation in the saturation region. Conversely, if the power-law of the medium is decreased peak focusing scintillation and scintillation in the saturation region are smaller.
On the other hand, for plane waves propagating in weak-to-moderate strength turbulence volumes ($\sigma_R < 1$) the situation is reversed. In this regime the relationship between turbulence strength and intensity scintillation is driven by the power-law of the medium. Consequently, we observe that mediums with smaller power laws experience higher fluctuations in intensity. Likewise, mediums with smaller power-laws experience less focusing and enter saturation sooner as a function of turbulence strength.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width = 0.4\columnwidth]{plot_all.png}& \includegraphics[width = 0.4\columnwidth]{scint_surf.png}\\
(a) & (b)
\end{tabular}
\end{center}
\caption[Plane Wave Results full range of alpha]{\label{fig:PWAlpha} (a) result of WOS for $25$ values of $\alpha$ in the range $3 < \alpha$ <4 between $0 < \sigma_R < 4$. (b) surface plot of (a).}
\end{figure}
The results of the second batch of our WOS campaign are presented in Fig.\ref{fig:PWAlpha} and confirm these results further. All parameters are the same those in Fig.\ref{fig:PWResults} but the value of $\sigma_R$ is limited to a maximum of $4$ while $\alpha$ was varied as described in Section \ref{sec:back}. Subfigure (a) can be compared directly to Fig.\ref{fig:PWResults} and confirms that the behavior observed there is approximately continuous and increases monotonically as a function of $\alpha$.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width = 0.4\columnwidth]{max_rytov_scint_neq.png}& \includegraphics[width = 0.4\columnwidth]{max_scint_neq.png}\\
(a) & (b)
\end{tabular}
\end{center}
\caption[Scintillation as a function of $\alpha$]{\label{fig:trends}. Value of $\sigma_R$ where maximum scintillation occurs (a) and peak value of $\sigma_I^2$ (b) as a function of power-law index $\alpha$}
\end{figure}
In Fig.\ref{fig:trends} the peak value of scintillation index (right) and the plane wave Rytov number where the peak scintillations were observed (left) are plotted as power-law, $\alpha$, is varied. In both figures, the least squares linear fit to the data is also plotted. The relationship between $\alpha$ and maximum scintillation is observed to have a slope of $0.9$ while the slope of the Rytov number where peak scintillation occurs and $\alpha$ is $1.8$. In this latter case, values necessarily are restricted to the specific values of Rytov number simulated during the campaign resulting in binning or quantization errors. In both cases, as we noted earlier, WOS accuracy is not certain as we exceed the bounds of $3.1 < \alpha < 3.9$ also contributing some uncertainty to these results. Regardless, the evidence of a positive linear relationship between power-law and both peak intensity scintillation and the volume turbulence strength at which that peak scintillation occurs is clear. For this propagation scenario it is also clear that the turbulence value of peak scintillation moves to the right at about double the rate of the peak scintillation increases.
To our knowledge, this linear relationship between power-law of the medium and peak scintillation in terms of Rytov number has not been reported elsewhere. Toselli\cite{toselli2007scintillation} used ERT to generate analytical models of plane wave scintillation as a function of power-law and Rytov number. However, this model does not include finite inner and outer scales. Though, that work is consistent with these findings and our previous work \cite{bos2015simulation}\cite{bos2016simulation}\cite{Grulke}. In subsequent works by Toselli \cite{toselli2009free} and others \cite{deng2012scintillation},\cite{Yi:12}, \cite{Cang:11}, \cite{Cui:12},\cite{YI2013199} on non-Kolmogorov turbulence findings are often reported for a fixed value of $\beta$ in the weak to moderate regime while varying propagation distance instead of normalized volume turbulence strength as is done here. As pointed out by Charnotskii \cite{charnotskii2011twelve} because the units of $\beta$, or $\tilde{C}_{n}^2$ in some other works, vary with $\alpha$ it is not possible to use a common length scale for comparing results in any meaningful way. As far as we can ascertain, all previous works generally report a relationship similar to \cite{toselli2009free} where peak scintillation occurs at a $\alpha < 11/3$ in the region of $3.2 < \alpha < 3.3$ and is lower on either side of the peak going to zero as $\alpha \to 3$ and to a small value as $\alpha \to 4$.
\begin{figure}
\begin{center}
\includegraphics[]{fix_rytov.eps}
\end{center}
\caption[Scintillation for fixed $\sigma_R$]{\label{fig:fixed}. Scintillation index for three fixed values of integrated turbulence strength in terms of $\sigma_R$ as a function of $\alpha$. Each trace is normalized to the maximum value of scintillation index over all values of $\alpha$}
\end{figure}
To place our work in this context, in Fig.\ref{fig:fixed}, we have plotted the normalized scintillation index, $\sigma_{I}^2/Max(\sigma_{I,\alpha})^2$, as a function of power-law index, $\alpha$. As pointed out earlier, we see that in the weak turbulence regime, for a fixed volume turbulence strength, scintillation decreases with power-law. In contrast, scintillation increases if the volume is characterized as strong or deep ($\sigma_R >> 1$). However, in this figure when $\sigma_R = 1.5$ scintillation peaks near the center of the range at $\alpha = 3.5$ and decreases if the power-law is increased or decreased.
Thus, relative to those other works the relationship between $\alpha$ and $\sigma_{I}^2$ varies with integrated turbulence strength in the volume. Also, because of the ambiguity introduced via a fixed $C_{n}^2$ this behavior may change depending on the power-law so that any of the behaviors observed in Fig. \ref{fig:fixed} may apply. For example, if $\lambda = 1.55$ $\mu m$, $L = 1000$ m, and $\beta = 7 \times 10^{-14}$ m$^{3-\alpha}$, as in \cite{toselli2009free}, the Rytov number varies from $3.8$ to $1.18$ to $0.73$ for $\alpha = 3.1, 3.66,$ and $3.9$. Consider then that for this one example all three of the behaviors in Fig. \ref{fig:fixed} may be observed. This observation further emphasizes that some measure of equivalent volume turbulence strength must be when trying to understand the impact of turbulence power-law exponent on beam propagation and imaging.
\section{Conclusions and Future Work}
\label{sec:Conclusions}
In this work we explored the limits of WOS plane-waves propagating in non-Kolmogorov turbulence and with power-law exponents in the range $3 < \alpha < 4$. At the upper bound the medium becomes a pure tilt and therefore WOS are limited by the size of the screen or the fidelity of the low-spatial frequency compensation mechanism; sub-harmonics for example. Conversely, as the power-law approaches the lower-bound the medium become spatially uncorrelated, and the spatial bandwidth required to properly simulate the medium becomes very large. Accordingly, the number of samples required by the simulation is also large. Both WOS requirements, can be ameliorated by the inclusion of a finite inner and outer scale. Indeed, as we have shown elsewhere\cite{beck2021saturation} if these values are not defined explicitly in the power-spectrum model they are implicit in the simulation model. Finally, if we take the practical limit of WOS to limited to $N = 16384$ samples, the maximum plane wave Rytov number that can be simulated in any power-law medium is $\sigma = 24$ when $\alpha = 3.8$.
The results of our analysis on the limits of WOS were used to inform two simulation campaigns exploring the interplay between Rytov number, $\alpha$, and scintillation of intensity or NIV. The first campaign aimed to evaluate scintillation as a function of Rytov number for values of power law $\alpha = 3.1, 3.66, 3.9$. For each value of $\alpha$ simulations were undertaken out the maximum $\sigma_R$ allowed for a screen size of $N = 8192$. Consistent with other results the maximum scintillation observed and the Rytov number where the peak occurs increase with power-law. At small power-law exponents scintillation quickly saturates as the volume turbulence strength increases. In this work we did not observe a strong focusing peak seen in our previous works as $\alpha \to 3$. We attribute this to the use of a finite inner scale larger than the sampling rate. Similarly, in previous works scintillation appeared to increase without bound at power-laws near $\alpha = 4$. However, if we include an explicit outer scale on the order of the screen size peak scintillation increases but eventually rolls off into saturation. These findings are consistent with theory that attributes small-scale fluctuations to scintillation strength and large-scale fluctuations to the peak of intensity fluctuations in the focusing region.
Our second simulation campaign aimed to empirically evaluate the effect of medium power-law index on scintillation. We found here that, in contrast to some previous works, intensity scintillation increases monotonically with power law. Likewise, the $\sigma_R$ where peak scintillation occurs moves to the right at twice the rate peak scintillation increases. We assert that the differences between these results and previous works is due to the proper scaling of turbulence strength with power-law used here. The aforementioned inner and outer-scale may also be contributing factors.
Over the course of this work, the contradictory nature of our findings and previous works led us to revisit our results to ensure our WOS parameters were correct. As a result of these explorations\cite{Grulke}, we were led to conclude that, at least for the simulation scenario explored here, these parameters are a mostly a second order effect. That is to say, the overall trends remain as long even if the sampling rate or number of screens changes. This finding has been recently confirmed by Wijerathna \cite{wijerathna2021numerical}. In one of his last works Flatte\cite{flatte2000irradiance} indicated that WOS are an exact solution to the stochastic Hemholtz equation over all bounds when properly configured. Much has been made of the qualification but based on this work and others we feel WOS will usually provide accurate results even if they are not precise \cite{Grulke}.
There is more work to be done in this area. For example, it may be worthwhile to further explore Fresnel zone effects by varying wavelength and path length. This work looked only at plane-wave propagation. Therefore, it may be interesting to see if further accommodations\cite{schmidt2010numerical} are needed to account for beam expansion and beam-wander for as diverging sources. This work also explicitly does not include a Hill ``bump'' or other features in the dissipation range. This exclusion is purposeful as it is not at all clear the nature of this feature when the medium is non-Kolmogorov. Finally, a straight-forward extension would be to explore the effect of larger values of $l_0$ and smaller values of $L_0$ for different power-laws as $\alpha$ is varied.
\acknowledgments
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0201. Any opinions, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the United States Air Force. This work was began when the main author was a NRC Postdoctoral associate with the United States Air Force Research Laboratory under the supervision of Dr. Venkata (Rao) Gudimetla and he is credited for have first suggested the WOS campaigns described herein.
|
1,108,101,565,494 | arxiv | \section{Introduction}
The problem of factoring a univariate polynomial $P(x)$ over a finite
field $F$ often arises in computational
algebra~\cite{Cantor92,Gathen02,Gathen92}.
An important case is
when $F$ has small characteristic and $P(x)$ has high degree but is
{\em sparse}, that is $P(x)$ has only a small number of nonzero terms.
To simplify the exposition we
restrict attention to the case where
$F = \mathrm{GF}(2)$ and $P(x)$ is a {\em trinomial}
\[P(x) = x^r + x^s + 1, \;\; r > s > 0, \]
although the ideas apply more generally and should be useful for factoring
sparse polynomials over fields of small characteristic.
Our aim is to give an algorithm with good {\em amortized complexity}, that
is, one that works well {\em on average}. Since we are restricting
attention to trinomials, we average over all trinomials of fixed degree~$r$.
Our motivation is to speed up previous algorithms for searching for
irreducible trinomials of high degree~\cite{rpb199,rpb214,Kumada00}.
For given degree $r$, we want to find all irreducible trinomials
$x^r + x^s + 1$.
In our examples the degree $r$ is a {\em Mersenne exponent}, i.e., $2^r-1$ is
a Mersenne prime. In this case an irreducible trinomial of degree~$r$ is
necessarily primitive. In general, without the restriction to Mersenne
exponents, we would need the prime factorisation of $2^r-1$ in order to test
primitivity (see e.g.,
\cite{Gathen99}).
We are only interested in Mersenne exponents $r = \pm 1 \bmod 8$, because
in other cases Swan's theorem~\cite{Pellet78,Stickelberger97,Swan62}
rules out irreducible trinomials of degree~$r$ (except for $s = 2$ or $r-2$,
but these cases are usually easy to handle:
for example if $r = 13466917$ or $20996011$
we have $r = 1 \bmod 3$,
so $x^r + x^2 + 1$ is divisible by $x^2 + x + 1$).
Mersenne exponents can be found on the GIMPS website~\cite{GIMPS}. At the
time of writing, the five largest known Mersenne exponents $r$ satisfying
the condition $r = \pm 1 \bmod 8$ are $r = 6972593$, $24036583$, $25964951$,
$30402457$ and $32582657$. In the smallest case $r = 6972593$, a primitive
trinomial was found by Brent, Larvala and Zimmermann~\cite{rpb214} using an
efficient implementation of the naive algorithm. However, it was not
feasible to consider the larger Mersenne exponents $r$
using the same algorithm,
since the time complexity of this algorithm is roughly of order $r^3$,
and the next case $r = 24036583$ would take about 41 times longer than
$r = 6972593$.
With the new ``fast'' algorithm described in this paper we have been able to
find two primitive trinomials of degree $r = 24036583$ in less time than
the naive algorithm took for $r = 6972593$. The speedup over the
naive algorithm for $r = 24036583$ is about a factor of $560$.
If $x^r + x^s + 1$ is reducible then we want to provide an
easily-checked {\em certificate} of reducibility. The certificate can simply
be an encoding of an irreducible factor $f$ of $x^r + x^s + 1$.
We choose the factor $f$ of smallest
degree $d > 0$. In case there are several factors of equal smallest degree
$d$, we give the one that is least in lexicographic order,
e.g., $x^3 + x + 1$ is preferred to $x^3 + x^2 + 1$.
\subsection{Distinct degree factorization}
Our basic algorithm performs
{\em distinct degree factorization}~\cite{Flajolet01,Gathen99,Gathen02}.
That is, if $P(x)$ has several factors of the same degree $d$, the algorithm
will produce the product of these factors. The Cantor-Zassenhaus algorithm
is used to split this product into distinct factors. This is cheap
because the product usually consists of just one irreducible factor or is a
product of irreducible factors of small (equal) degree.
In the complexity analysis we only consider the time required to find {\em
one} nontrivial factor (it will be a factor of smallest degree) or output
``irreducible'', since that is what is required in the search for
irreducible trinomials.
\subsection{Factorization over $\mathrm{GF}(2)$}
It is well-known that $x^{2^d}+x$
is the product of all irreducible polynomials of degree dividing~$d$.
For example,
\[x^{2^3} + x = x(x+1)(x^3 + x + 1)(x^3 + x^2 + 1)\enddot\]
Thus, a simple algorithm to find a factor of smallest degree of $P(x)$ is
to compute $\mathrm{GCD}(x^{2^d} + x, P(x))$ for $d = 1, 2, \ldots$
The first time that the GCD is nontrivial, it contains a factor of minimal
degree~$d$. If the GCD has degree $>d$, it must be a product of factors
of degree~$d$.
If no factor has been found for $d \le r/2$, where $r = \deg(P(x)$), then
$P(x)$ must be irreducible.
Some simplifications are possible when $P(x) = x^r + x^s + 1$ is a
trinomial over $\mathrm{GF}(2)$ with $r$ or $s$ odd (otherwise $P(x)$ is trivially
reducible):
\begin{enumerate}
\item We can skip the case $d=1$ because a trinomial can not have
a factor of degree~$1$.
\item Since $x^rP(1/x) = x^r + x^{r-s} + 1$,
we only need consider $s \le r/2$.
\item We can assume that $P(x)$ is square-free.
\item By applying Swan's theorem, we can often show that the trinomial under
consideration has an odd number of irreducible factors; in this case we only
need check $d \le r/3$ before claiming that $P(x)$ is irreducible.
\end{enumerate}
\section{Complexity of the algorithm}
Note that $x^{2^d}$ should not be computed explicitly;
it is much better to compute
$x^{2^d} \bmod P(x)$ by repeated squaring. The complexity of squaring
modulo a trinomial of degree $r$ is only $S(r) = O(r)$ bit-operations.
\subsection{Complexity of polynomial multiplication and squaring}
As well as performing GCD computations we need to perform multiplications in
$\mathrm{GF}(2)[x]/P(x)$, and an important special case is squaring a polynomial
modulo $P(x)$, so we first consider the bit-complexity of these operations.
Multiplication of polynomials of
degree $r$ over $\mathrm{GF}(2)$ can be performed in time
$M(r) = O(r \log r \log\log r)$. We have implemented an algorithm of
Sch\"onhage~\cite{Schonhage77} that achieves this bound. The algorithm uses
a radix-$3$ FFT and is different from the better-known
Sch\"onhage-Strassen algorithm~\cite{SS71}. We remark that the $\log\log r$
term in the time-bound for the Sch\"onhage-Strassen algorithm has been
reduced by F\"urer~\cite{Furer07}, but it is not clear if a similar idea
can be used to improve Sch\"onhage's algorithm~\cite{Schonhage77}.
In any event the
$\log\log r$ term comes from the number of levels of recursion and is
a small constant for the values of~$r$ that we are considering.
In practice, Sch\"onhage's algorithm is not the fastest unless $r$ is quite
large. We have also implemented classical, Karatsuba and Toom-Cook
algorithms that have $M(r) = O(r^\alpha)$, $1 < \alpha \le 2$, since these
algorithms are easier to implement and are faster for small~$r$. Our
implementations of the Toom-Cook algorithms TC3 and TC4 are based on recent
ideas of Bodrato~\cite{Bodrato07}.
For brevity we assume that $r$ is
large and Sch\"onhage's algorithm is used.
On a 64-bit machine the crossover versus TC4 occurs near degree $r = 108000$.
In the complexity estimates we assume that $M(r)$ is a sufficiently smooth
and well-behaved function.
By {\em Squaring} we mean squaring a polynomial of degree $<r$ and reduction
mod $P(x)$. Squaring in $\mathrm{GF}(2)[x]/P(x)$ can be performed in time
$S(r) = \Theta(r) \ll M(r)$ (assuming, as usual, that $P(x)$ is a trinomial).
Our algorithm takes advantage of the fact that squaring is much faster than
multiplication.
Where possible we use the memory-efficient squaring algorithm of Brent,
Larvala and Zimmermann~\cite{rpb199}, which in our implementation
is about $2.2$ times faster than the naive squaring algorithm.
\subsection{Complexity of GCD}
For GCDs we use a sub-quadratic algorithm that runs in time
$G(r) = \Theta(M(r)\log r)$.
More precisely,
\[G(2r) = 2G(r) + \Theta(M(r))\endcomma\]
so for $\alpha > 1$,
\[M(r) = \Theta(r^\alpha) \Rightarrow G(r) = \Theta(M(r))\endcomma\]
and
\[M(r) = \Theta(r \log r \log\log r) \Rightarrow
G(r) = \Theta(M(r) \log r)\enddot\]
In practice, for $r \approx 2.4\times 10^7$ and
our implementation on a 2.2~Ghz Opteron,
$S(r) \approx 0.005$ second,
$M(r) \approx 2$ seconds,
$G(r) \approx 80$ seconds,
so $M(r)/S(r) \approx 400$,
and $G(r)/M(r) \approx 40$.
\subsection{Avoiding GCD computations}
In the context of integer factorization, Pollard~\cite{Pollard75}
suggested a blocking
strategy to avoid most GCD computations and thus reduce the amortized cost;
von zur Gathen and Shoup~\cite{Gathen92} applied the same idea to polynomial
factorization.
The idea of blocking is to choose a parameter $\ell > 0$ and,
instead of computing
\[\mathrm{GCD}(x^{2^d} + x, P(x)) \;\;\mbox{for}\;\; d \in [d', d'+\ell)\endcomma\]
compute
\[\mathrm{GCD}(p_\ell(x^{2^{d'}}, x), P(x))\endcomma\]
where the {\em interval polynomial}
$p_\ell(X,x)$ is defined by
\[p_{\ell}(X,x) = \prod_{j=0}^{\ell-1} \left( X^{2^j} + x \right)\enddot\]
In this way we replace $\ell$ GCDs by one GCD and $\ell-1$ multiplications
mod $P(x)$.
The drawback of blocking is that we may have to backtrack if $P(x)$ has
more than one factor with degree in the interval $[d', d'+\ell)$,
since the algorithm produces the product of these factors.
Thus $\ell$ should not be too
large. The optimal strategy depends on the expected size distribution of
factors and the ratio of times for GCDs and multiplications.
\subsection{Multi-level blocking}
Our (apparently new) idea is to use
a finer level of blocking to replace most multiplications by squarings,
which speeds up the computation
in $\mathrm{GF}(2)[x]/P(x)$ of the above interval polynomials.
The idea is to split the interval $[d', d'+\ell)$ into $k \geq 2$ smaller
intervals of length $m$ over which
\begin{equation} \label{pofm}
p_{m}(X,x) = \prod_{j=0}^{m-1} \left( X^{2^j} + x \right)
= \sum_{j=0}^m x^{m-j} s_{j,m}(X)\endcomma
\end{equation}
where
\begin{equation}
s_{j,m}(X) = \sum_{0 \le k < 2^m,\; w(k) = j} X^k\endcomma \label{eq:s}
\end{equation}
and $w(k)$ denotes the {\em Hamming weight} of $k$,
that is the number of nonzero bits in the binary representation of~$k$.
For example, for $m=3$, we have:
\[ p_m(X,x) = x^3 + x^2 (X^4 + X^2 + X) + x (X^6 + X^5 + X^3) + X^7, \]
where $s_{0,3}(X) = 1$, $s_{1,3}(X) = X^4 + X^2 + X$,
$s_{2,3}(X) = X^6 + X^5 + X^3$, and $s_{3,3}(X) = X^7$.
Note that
\[s_{j,m}(X^2) = s_{j,m}(X)^2 \;\;\mbox{in}\;\; \mathrm{GF}(2)[x]/P(x)\enddot\]
Thus,
$p_{m}(x^{2^d},x)$
can be computed with cost $m^2S(r)$ if we already know
$s_{j,m}(x^{2^{d-m}})$ for $0 < j \le m$.
(The constant polynomial $s_{0,m}(X)=1$ is computed only once.)
Continuing the example with
$m=3$, and assuming that we know $s_{1,3}(x^{2^{d-3}})$,
$s_{2,3}(x^{2^{d-3}})$, and $s_{3,3}(x^{2^{d-3}})$,
squaring each of these $m=3$ times gives
$s_{1,3}(x^{2^d})$, $s_{2,3}(x^{2^d})$, and $s_{3,3}(x^{2^d})$,
from which we can easily get $p_3(x^{2^d},x)$ using the sum in
Eq.~(\ref{pofm}).
In this way we replace $m-1$ multiplications and $m$ squarings~---
if we used the product in Eq.~(\ref{pofm})~---
by $m^2$ squarings.
Each $s_{j,m}$, $0 < j \le m$, requires $m$ squarings to be shifted from
argument $x^{2^{d-m}}$ to argument $x^{2^d}$.
The summation in Eq.~(\ref{pofm}) costs only $O(mr)$, which is negligible.
Choosing $m \approx \sqrt{M(r)/S(r)}$
(about $20$ if
$M(r)/S(r) \approx 400$), the speedup over single-level blocking is about
$m/2 \approx 10$ (not counting the cost of GCDs).
Von zur Gathen and Gerhard~\cite[p.~1685]{Gathen02} suggested using the
same idea with $m=2$ (thus reducing the number of multiplications by a
factor of two), but did not consider choosing an optimal $m > 2$.
At first sight initialization of the polynomials $s_{j,m}(X)$ for $X = x$
might appear to be expensive, since the
definition~(\ref{eq:s}) involves $O(2^m)$ terms.
However, the polynomials $s_{j,m}(X)$
satisfy a ``Pascal triangle'' recurrence relation
\[ s_{j,m}(X) = s_{j,m-1}(X^2) + X s_{j-1,m-1}(X^2)\]
with boundary conditions
\[s_{j,m}(X) = \left\{ \begin{array}{ll}
0 &\mbox{if $j > m \ge 0$,}\\
1 &\mbox{if $m \ge j = 0$.}
\end{array} \right.
\]
Using this recurrence,
it is easy to compute $s_{j,m}(x) \bmod P(x)$ for $0 \leq j \leq m$
in time $O(m^2r)$. Thus, the initialization is cheap.
To summarise, we use two levels of blocking:
\begin{enumerate}
\item
The outer level replaces most GCDs by multiplications.
\item
The inner level replaces most multiplications by squarings.
\item
The parameter $m \approx \sqrt{M(r)/S(r)}$ is used for
the inner level of blocking.
\item
A different parameter $\ell = km$ is used
for the outer level of blocking.
\end{enumerate}
For example, suppose
$S = 1/400$, $M = 1$, $G = 40$ (where we have normalised so $M = 1$).
We could choose $\ell = 80$ and $m = 20$.
With no blocking, the cost for an interval of length $80$ is
$80G + 80S = 3200.2$;
with 1-level blocking the cost is $G + 79M + 80S = 119.2$;
with 2-level blocking the cost is $G + 3M + 1600S = 47.0$.
\subsection{Sieving out small factors} \label{sieving}
We define a
{\em small} factor to be one with degree $d < \frac{1}{2}\log_2 r$,
so $2^d < \sqrt{r}$. The constant $\frac{1}{2}$ in the definition is
arbitrary and could be replaced by any fixed constant in $(0,1)$.
A {\em large} factor is a factor that is not small.
It would be inefficient to find small factors in the same way as large
factors.
Instead, let $D = 2^d-1$, $r' = r \bmod D$, $s' = s \bmod D$.
Then
\[P(x) = x^r + x^s + 1 = x^{r'} + x^{s'} + 1 \bmod (x^{D} - 1)\endcomma\]
so we only need compute
\[\mathrm{GCD}(x^{r'} + x^{s'} + 1, x^{D} - 1)\enddot\]
Because $r', s' < D < \sqrt{r}$, the cost of finding
small factors is negligible (both theoretically and in practice),
so can be neglected.
\subsection{Outer level blocking strategy} \label{subsec:outer}
The blocksize in the outer level of blocking is $\ell = km$. We take
a linearly increasing sequence of block sizes
\[k = k_0 j \;\;\mbox{for}\;\; j = 1, 2, 3, \dots\endcomma\]
where the first interval starts at about
$\log r$ (since small factors will have
been found by sieving).
The choice $k = k_0 j$ leads to a quadratic polynomial for the interval
bounds;
other
possibilities
are discussed by von zur Gathen
and Gerhard~\cite{Gathen02}.
In principle, using the data that we have obtained on the distribution of
degrees of smallest factors of trinomials (see \S\ref{sec:dist}),
and assuming that this
distribution is not very sensitive to the degree $r$, we could obtain a
strategy that is close to optimal. However, the choice
$k_0 j$ with suitable $k_0$ is easy to implement and not too far from
optimal. The number of GCD and sqr/mul operations is usually
within a factor of $1.5$ of the minimum possible
in our experiments.
\section{Distribution of degrees of factors} \label{sec:dist}
In order to predict the expected behaviour of our algorithm, we need
to know
the expected distribution of degrees of smallest irreducible factors.
{From} Swan's theorem~\cite{Swan62}, we know that there are
significant differences between the distribution of factors of trinomials
and of all polynomials of the same degree.
Our complexity estimates are based on the heuristic assumption that this
difference is not too large, in a sense made precise by
Hypothesis~\ref{hyp1}.
\begin{hypothesis} \label{hyp1}
Over all trinomials $x^r+x^s+1$ of degree $r$ over $\mathrm{GF}(2)$,
the probability $\pi_d$ that a trinomial has no nontrivial factor of degree
$\le d$, $1 < d \le r$, is at most $c/d$, where $c$ is a constant.
\end{hypothesis}
Hypothesis~\ref{hyp1} implies that there are at most $c$ irreducible
trinomials of degree~$r$. This is probably false, as there may well be
a sequence of exceptional $r$ for which the number of irreducible trinomials
is unbounded. Thus, we may need to replace the constant $c$ in
Hypothesis~\ref{hyp1} by a slowly-growing function $c(r)$.
Nevertheless, in order to give realistic complexity estimates that are
in agreement with experiments, we assume below that Hypothesis~\ref{hyp1}
is correct. Under this assumption
we use an amortized model to obtain
the total complexity over all trinomials of degree $r$.
{From} Hypothesis~\ref{hyp1}, the probability that a trinomial does not
have a small factor (as defined in \textsection\ref{sieving}) is $O(1/\log r)$.
Table~\ref{table:3M24M} gives the observed values of $d\pi_d$ for
$r=3021377$, $r=6972593$, and $r=24036583$. The maximum values for
each $r$ are given in bold.
The table shows that the values of $d\pi_d$
are remarkably stable for small~$d$, and bounded by~$4$ for large~$d$
(this is because there are four irreducible trinomials of degree $3021377$
and also four of degree $24036583$, when we count
both trinomials $x^r + x^s + 1$ and
their reciprocals $x^r + x^{r-s} + 1$).
\begin{table}[ht]
\centering
\caption{$d\pi_d$ for various degrees $r$.}
\label{table:3M24M}
\begin{tabular}{|c|c|c|c|}
\hline
$d$ & $r=3021377$ & $r=6972593$ & $r=24036583$ \\
\hline
2 & 1.333 & 1.333 & 1.333\\
3 & 1.429 & 1.429 & 1.429\\
4 & 1.524 & 1.524 & 1.524\\
5 & 1.536 & 1.536 & 1.536\\
6 & 1.598 & 1.598 & 1.598\\
7 & 1.600 & 1.600 & 1.600\\
8 & 1.667 & 1.667 & 1.667\\
9 & 1.642 & 1.642 & 1.642\\
10 & 1.652 & 1.652 & 1.652\\
100 & 1.763 & 1.771 & 1.770\\
1000 & 1.783 & 1.756 & 1.786\\
10000 & 1.946 & 1.873 & 1.786\\
100000 & 1.986 & 1.606 & 1.880\\
279383 & 1.480 & {\bf 2.084} & 1.813\\
1000000 & 1.324 & 1.147 & 1.831\\
10000000& -- & -- & 1.664\\
$r-1$ & {\bf 4.000} & 2.000 & {\bf 4.000}\\
\hline
\end{tabular}
\end{table}
\subsection{Consequences of the hypothesis}
Define $p_k = \pi_{d-1} - \pi_d$ to be the
probability that the smallest nontrivial factor $f$ of a randomly
chosen trinomial has degree $d = \deg(f)$. In order to estimate the
running time of our algorithm, we use the following Lemma, which gives
the expectation $E_\beta$ of $d^\beta$.
\begin{lemma} \label{expectA}
If $\beta > 0$ is constant and Hypothesis~\ref{hyp1} holds,
then
\[E_{\beta} := \sum_{d=1}^r d^\beta p_d =
\left\{ \begin{array}{ll}
O(1) &\mbox{if $\beta < 1$,}\\
O(\log r) &\mbox{if $\beta = 1$,}\\
O(r^{\beta-1}) &\mbox{if $\beta > 1$.}
\end{array} \right.
\]
\end{lemma}
\begin{proof}
We use summation by parts.
Note that a trinomial has no factor of degree~$1$, so $p_1 = 0$
and $\pi_0 = \pi_1 = 1$. Thus
\begin{eqnarray*}
E_\beta &=& \sum_{d=1}^r d^\beta p_d
\;\;=\;\; \sum_{d=1}^r d^\beta (\pi_{d-1} - \pi_d) \\
&=& \sum_{d=1}^{r-1} \left((d+1)^\beta - d^\beta\right)\pi_d
+ \pi_0 - r^\beta\pi_r \\
&\le& 1 + c\sum_{d=1}^{r-1} \frac{(d+1)^\beta - d^\beta}{d}
\;\;\mbox{(by Hypothesis~\ref{hyp1})} \\
&\le& 1 + O\left(\sum_{d=1}^{r-1} d^{\beta-2}\right)
\end{eqnarray*}
and the result follows.
\end{proof}
The following Lemma gives a stronger result in the case $\beta < 1$.
\begin{lemma} \label{expectB}
If $0 < \beta < 1$, $0 < D \le r$, and Hypothesis~\ref{hyp1} holds,
then
\[\sum_{d=D}^r d^\beta p_d = O\left(D^{\beta-1}\right)\enddot
\]
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma~\ref{expectA}. We end with the
upper bound
\[
\sum_{d=D}^{r-1}\frac{(d+1)^\beta - d^\beta}{d} + D^{\beta}\pi_{D-1}
\enddot\]
{From} Hypothesis~\ref{hyp1},
$\pi_{D-1} = O(1/D)$, and the sum over $d$ is $O(D^{\beta-1})$,
so the result follows.
\end{proof}
\section{Expected cost of sqr/mul and GCD}
Recall that the inner level of blocking replaces $m$ multiplications by
$m^2$ squarings and one multiplication, where the choice
$m \approx \sqrt{M(r)/S(r)}$ makes the total cost of squarings about equal to
the cost of multiplications.
For a smallest factor of degree $d$, the number of squarings is
$m(d + O(\sqrt{d}))$,
where the $O(\sqrt{d})$ term follows from our choice of outer-level
blocksizes (see~\S\ref{subsec:outer}).
Averaging over all trinomials of degree~$r$,
the expected number of squarings is
\[O\left(m\;\sum_{d \le r/2} (d + O(\sqrt{d}))p_d\right)\endcomma\]
and from Lemma~\ref{expectA} this is
$O(m\log r)$.
Thus, the expected cost of sqr/mul operations per trinomial is
\begin{eqnarray}
O\left(S(r)\log r \sqrt{M(r)/S(r)}\right)
&=& O\left(\log r \sqrt{M(r)S(r)}\right) \nonumber \\
&=& O\left(r (\log r)^{3/2}(\log\log r)^{1/2}\right)\enddot \label{eq:S1}
\end{eqnarray}
If we used only a single level of blocking, then
the cost of multiplications would dominate that of
squarings, with an expected cost per trinomial of
$O\left(\log r M(r) \right) = O\left(r (\log r)^2 \log\log r \right)$.
(\ref{eq:S1}) is correct as $r \to \infty$. However, in practice,
at least for $r < 6.4 \times 10^7$,
our implementation of Sch\"onhage's FFT-based polynomial multiplication
algorithm~\cite{Schonhage77} calls a different multiplication
routine (usually TC4) to perform smaller multiplications,
rather than recursively calling itself. TC4 has
exponent $\alpha' = \ln(7)/\ln(4) \approx 1.4$, so the effective
exponent for FFT multiplication is
$\alpha = (1 + \alpha')/2 \approx 1.2 > 1$. In this case, the
expected cost of sqr/mul operations per trinomial is
\begin{equation}
O\left(\log r \sqrt{M(r)S(r)}\right) =
O(r^{(1+\alpha)/2}\log r) = O(r^{1.1\cdots}\log r) \label{eq:S2}
\end{equation}
\subsection{Expected cost of GCDs}
Suppose that $P(x)$ has a smallest factor of degree~$d$. The number of GCDs
required to find the factor, using our (quadratic polynomial)
blocking strategy,
is at least $1$, and $O(\sqrt{d})$ if $d$ is large. By Hypothesis~\ref{hyp1},
the expected number of GCDs for a trinomial with {\em no small factor} is
\[1 + O\left(\sum_{\log_2 r < 2d \le r} d^{1/2}\;p_d \right)\endcomma\]
and by Lemma~\ref{expectB} this is
\[1 + O\left(\frac{1}{\sqrt{\log r}}\right)\enddot\]
Thus the expected cost of GCDs per trinomial is
\begin{equation}
O(G(r)/\log r) = O(M(r)) = O(r\log r \log\log r)\enddot \label{eq:G1}
\end{equation}
(\ref{eq:G1}) is asymptotically less than the expected cost
(\ref{eq:S1}) of sqr/mul operations.
However,
if $M(r) = O(r^\alpha)$ with $\alpha > 1$,
then the expected cost of GCDs is $O(r^\alpha/\log r)$, which is
asymptotically greater than the expected cost~(\ref{eq:S2})
of sqr/mul operations.
Note the expected cost of GCDs does not depend on whether we use
one or two levels of blocking.
For $r \approx 2.4 \times 10^7$,
GCDs take about 65\% of the time versus 35\% for sqr/mul.
\subsection{Comparison with previous algorithms}
For simplicity we use the $\makebox{$\widetilde O$}$ notation which ignores $\log$
factors. For example, $M(r) = \makebox{$\widetilde O$}(r)$.
The ``naive'' algorithm, as implemented by Brent, Larvala and
Zimmermann~\cite{rpb199} and earlier authors, takes an expected time
$\makebox{$\widetilde O$}(r^2)$ per trinomial, or $\makebox{$\widetilde O$}(r^3)$ to cover all trinomials of
degree~$r$.
The single-level blocking strategy and the new algorithm both take
expected time $\makebox{$\widetilde O$}(r)$ per trinomial,
or $\makebox{$\widetilde O$}(r^2)$ to cover all trinomials of degree~$r$.
In practice, the new algorithm is faster over the naive algorithm
by a factor of about $160$ for
$r = 6972593$, and by a factor of about $560$ for $r = 24036583$.
For $r = 24036583$, where
sqr/mul operations take 35\% of the total time in the new algorithm,
and the corresponding speedup is about 10, this gives a global
speedup of more than 4 over the single-blocking strategy.
\subsection{Some details of our implementation}
We first implemented the\linebreak
2-level blocking strategy in NTL \cite{NTL}.
To get full efficiency, we rewrote all critical routines
and tuned them efficiently on the target processors.
Our squaring routine implements the algorithm described in \cite{rpb199},
which is more than twice as fast as the corresponding optimized
NTL routine for trinomials.
Our multiplication routine implements Toom-Cook $3$-way, $4$-way, and
Sch\"onhage's algorithm \cite{Schonhage77}.
We also improved the basecase multiplication code; more details
concerning
efficient multiplication in $\mathrm{GF}(2)[x]$ will be published in \cite{BrGaThZi07}.
Finally, we implemented a subquadratic GCD routine, since NTL only
provides a classical GCD for binary polynomials.
\subsection{Primitive trinomials}
The largest published primitive trinomial is
\[x^{6972593} + x^{3037958} + 1\endcomma\]
found by Brent, Larvala and Zimmermann~\cite{rpb199}
in 2002 using a naive (but efficiently implemented) algorithm.
In March--April 2007, we tested our new program by verifying
the published results on primitive
trinomials for Mersenne exponents $r \le 6972593$, and in the process produced
certificates of reducibility
(lists of smallest factors for each reducible trinomial). These are
available from the first author's website~\cite{rpbweb}.
In April--August 2007, we ran our new algorithm to search for
primitive trinomials of degree $r = 24036583$. This is the next Mersenne
exponent, apart from two that are trivial to exclude by Swan's theorem.
It would take about 41
times as long as for $r = 6972593$ by the naive algorithm,
but our new program is 560 times
faster than the naive algorithm.
Each trinomial takes on average about 16 seconds
on a 2.2~Ghz Opteron.
The complete computation was performed in four months, using
about 24 Opteron and Core~2 processors located
at ANU
and INRIA.
We found two new primitive trinomials of (equal) record degree:
\begin{equation}
x^{24036583} + x^{8412642} + 1 \label{Eugenie}
\end{equation}
and
\begin{equation}
x^{24036583} + x^{8785528} + 1\enddot \label{Judy-anne}
\end{equation}
\subsection{Verification}
Allan Steel~\cite{Steel} kindly verified irreducibility of
(\ref{Eugenie})--(\ref{Judy-anne}) using Magma~\cite{Magma}.
Each verification took
about 67 hours on an 2.4~GHz
Core~2 processor.
Independent verifications using our {\tt irred V3.15}
program~\cite{rpb199,rpb214}
took about 35 hours on a 2.2~Ghz Opteron.
The difference in speed is mainly due to the fast squaring algorithm
implemented in {\tt irred}.
Primitivity of (\ref{Eugenie})--(\ref{Judy-anne}) follows from irreducibility
provided that the degree $24036583$ is a Mersenne exponent. We have not
verified this, but rely on computations performed by the GIMPS
project~\cite{GIMPS}.
Reducibility of the remaining trinomials of degree $24036583$ can be
verified using the certificate (or {\em extended log},
a list of smallest irreducible factors) available from our
website~\cite{rpbweb}.
The verification takes less than $10$ hours using Magma on a 2.66~Ghz Core~2
processor.
\section{Conclusion}
The new double-blocking strategy, combined with fast
multiplication and GCD algorithms, has allowed us to find new primitive
trinomials of record degree.
The same ideas should work over finite fields $\mathrm{GF}(p)$ for small prime $p > 2$,
and for factoring sparse polynomials $P(x)$ that are
not necessarily trinomials:
all we need is that the time for $p$-th powers (mod $P(x)$)
is much less than the time for multiplication (mod $P(x)$).
\subsection*{Acknowledgements}
We thank Allan Steel for verifying irreducibility of the
trinomials~(\ref{Eugenie})--(\ref{Judy-anne}),
and Marco Bodrato, Pierrick Gaudry and Emmanuel Thom\'e for
their assistance in implementing fast algorithms for multiplication
of polynomials over $\mathrm{GF}[2]$.
ANU and INRIA provided computing facilities. The first author's
research was supported by MASCOS and the Australian Research Council.
\bibliographystyle{amsalpha}
|
1,108,101,565,495 | arxiv | \section{Introduction}
\IEEEPARstart{S}{peech-driven}
human-machine interaction has progressed up to real-world scenarios today, with applications to smart speakers, intelligent conference room and virtual assistant, etc. Speech separation has become a major concern as the front-end for robust speech interaction.
Leveraging the advances in deep learning, close-talk monaural speech separation has achieved a great progress in recent years \cite{hershey2016deep, yu2017permutation, chen2017deep,wang2018alternative,luo2018tasnet, luo2019convtasnet}. Most existing approaches formulate speech separation problem in time-frequency (T-F) domain, owing to T-F sparsity and auditory masking effect \cite{colle1976acoustic, pfander2010sparsity}. Specifically, the network is trained to learn a mapping from mixture features to a T-F mask, where each element indicates the dominance of a source at each T-F bin of the mixture spectrogram \cite{wang2018supervised}.
Due to the complexity of phase reconstruction, most methods decouple the magnitude and phase part. The mask is estimated only based on the magnitude spectra and employed to reconstruct the speech waveform along with the mixture phase. However, many researches show that there is still nonnegligible error even reconstructing speech with ideal magnitude and mixture phase \cite{le2019phasebook, takahashi2018phasenet, choi2019phase}. This introduces a performance upper bound for methods based on T-F magnitude masking. Although the significance of phase retrieval has been addressed by many recent works and achieved improved performance \cite{le2019phasebook, takahashi2018phasenet, choi2019phase, wang2018end, wichern2018phase}, the estimated phase remains suboptimal on speech separation task. Also, complex phase reconstruction algorithms may increase the system latency.
To avoid the phase problem and speed up the separation process, an increasing number of researchers focus on time-domain end-to-end speech separation \cite{luo2018tasnet, luo2019convtasnet, shi2019furcax}. One of the representative work is fully convolutional time-domain audio separation network (Conv-TasNet) \cite{luo2019convtasnet}. Conv-TasNet formulates the separation task in a real-valued high-dimensional space, which is assumed to contain both magnitude and phase information. Specifically, Conv-Tasnet employs a linear encoder to map the mixture chunk to a high-dimensional representation that optimized for speech separation. Separation is achieved by estimating the weight (mask) of each speaker in the high-dimensional space. See Section \ref{sec:single-channel_tss} for details.
Despite the significant improvement achieved by these end-to-end close-talk speech separation methods, there remains two main problems towards real-world applications: \emph{i}) The close-talk multi-speaker mixture model originates from the linear instantaneous mixture model, where the observed mixtures are linear combinations of the sources. While at far-field, the general mixing model is reformulated as convolution process of room impulse responses and sources \cite{cardoso1998blind, vincent2014blind}. The observed speech mixture is corrupted by reverberation and potential noise, which diminish the speech quality and increase the separation difficulty \cite{naylor2010speech}. Therefore, the performance of these close-talk separation methods is hard to fulfill the practical requirement. \emph{ii}) Most methods need to know the number of sources in the mixture and the output dimension is fixed to a specific source count. When the actual number of sources mismatches the output dimension, termed as \emph{output dimension mismatch} problem \cite{huang2019research}, these methods are not facilitated to adjust their outputs. Furthermore, it's never easy to know the separated speech corresponds to which speaker in the mixture. Hence, in applications such as conference transcription, additional speaker verification or classification module is required to recognize the identity of separated speech \cite{rao2019target}, which we define as \emph{blind source allocation}.
First, to tackle with the first problem, far-field speech separation, it's a promising research direction to combine multi-channel speech separation (MSS) approaches with end-to-end approaches. MSS approaches include beamforming techniques based on microphone array signal processing \cite{gannot2017consolidated, adel2012beamforming,gannot2001signal,markovich2009multichannel}, blind source separation (BSS) \cite{sawada2006blind} and deep learning based methods \cite{chen2018multi, wang2019combining, wang2018integrating,wang2018multi,lianwu2019multi,yoshioka2018multi,drude2017tight}. With the direction of arrival (DOA) information, beamforming techniques conducts spatial filtering by means of the appropriate configuration of microphone array. The source from the desired direction is enhanced while interferences from other directions are suppressed. Since beamforming techniques employ spatial information to perform speech separation, when the sources are located closely to each other, these algorithms become less effective. BSS methods assume the sources are statistically independent to each other, and independent component analysis (ICA) is often used to model the separation process. However, when the reverberation is strong or the number of microphones is less than the number of sources, these methods degrade significantly. The deep learning-based methods employ the powerful feature learning capability of deep neural networks (DNN) to learn the T-F mask for the target source, using spatial binaural cues, e.g., interaural phase/time/level difference (IPD, ITD, ILD). The rationale lies in that, when sources are (W-) disjoint orthogonal, the binaural cues will form clusters within each frequency band for spatially separated directional sources with different time delays. This is also the theoretical basis of spatial clustering technique \cite{mandel2017multichannel, sawada2010underdetermined}. These spatial cues have been proven effective in deep learning-based frequency domain separation methods, especially when combined with spectral feature (e.g., logarithm power spectra, LPS) at input level \cite{wang2018multi,wang2018integrating,chen2018multi,lianwu2019multi,wang2019combining}.
Next, a common solution to output dimension mismatch and blind source allocation problem is to separate a target speaker once, instead of separating them all. To inform the network of which speaker needs to be extracted, target related prior knowledge needs to be incorporated. For instance, the voice characteristics of the target speaker. \cite{wang2018voicefilter,xu2018modeling,zmolikova2017speaker,vzmolikova2017learning,wang2018deep} employ speaker embeddings extracted from a reference signal from the target speaker to speaker-condition the separation system’s output. Apart from the target voice information, \cite{xiao2019single} proposed to utilize interference speakers’ embeddings as well, to increase the discrimination between speakers in the mixture. However, the main limitation for these speaker-aware methods is that they require the identity or the reference recordings from the target speaker. Other than the voice characteristics, direction information can be used to associate with a specific speaker in multi-channel speech separation. The direction information can either be estimated from acoustic/visual signals or predefined based on real usage scenario. Chen et al. \cite{chen2018multi} proposed a location-based angle feature which computes the cosine distance between the steering vector and inter-channel phase difference (IPD) for each source in mixture. But the phase ambiguities create difficulties for precisely discriminating one speaker from another in certain frequency bands. In \cite{wang2018spatial,wang2019combining}, Wang et al. developed two directional features to improve speech separation. One is the compensated IPD that shares the similar concept with angle feature and the other derives from beamforming outputs. Although impressive improvement is achieved with the directional features, \cite{wang2019combining} is a multi-stage system with high computational complexity. Also, it may neglect spatial ambiguity issue when speakers are close to each other, the directional features show less discrimination and it's hard to tell which speaker is corresponding to.
Taking two points discussed above into consideration, this study addresses the task of separating the target speech from multi-channel mixture in reverberant environments, assisted with directional information. We propose to fully exploit the spatio-temporal structure of multi-channel speech and integrate complementary separation cues into an end-to-end network for better performance while maintaining the real-time property.
First, we jointly model the temporal, spectral and spatial discriminability to create a more complete representation. The motivations are: 1) the separability of sources can be obtained from different views and they are complementary to each other. For instance, spectral sparsity in frequency domain, spatial diversity in spatial domain; 2) The discriminative capability of features from different domains depends on the conditions. For example, if sources come from close directions, then the spatial discrimination is not effective or even noisy. Under this condition, the model should rely on other features towards better separation.
Second, to reduce the latency, we adopt a purely end-to-end and single-pass separation network. It is a waveform-in, waveform-out separation system in a single neural network architecture, which inherited from Conv-TasNet. The model takes multi-channel mixture waveform and the directions of speaker(s) as input, and directly outputs estimated target speech waveform. We name this model as Temporal-Spatial Neural Filter, since it performs speech separation in the time-domain and separates the target speaker based on the informed directional information.
The rest of paper is organized as follows. Section \ref{sec:physical_model} first introduces the physical model of far-field multi-speaker mixture. Then, section \ref{sec:single-channel_tss} reviews the monaural Conv-TasNet. Next, we elaborate on our proposed system in section \ref{sec:proposed_system}, which is illustrated in figure \ref{fig:tsnf_framework}. We present the experimental setup and evaluation results in section \ref{sec:exp} and \ref{sec:result}, respectively, and conclude this paper in section \ref{sec:conclusion}.
\section{Physical Model}
\label{sec:physical_model}
In this study, we assume that the speakers do not move during speaking. The number of speakers and microphones are denoted as $C$ and $J$, respectively. The physical model for a reverberant mixture in time-domain is formulated as \cite{cardoso1998blind,vincent2014blind}:
\begin{equation}
\mathbf{y}[n] = \underset{c=1}{\overset{C}{\sum}}{\mathbf{s}_c[n]}
\label{eq:time_mixture}
\end{equation}
where $\mathbf{y}[n]=\left[ y^1[n],...,y^J[n] \right]^T$ is the $J$-dimensional vector of the mixture signal captured by the $J$-element microphone array, and $\mathbf{s}_c[n]=\left[ s_c^1[n],...,s_c^J[n] \right]^T=(\mathbf{h}_c\circledast x_c)[n]$ denotes the contribution of each source $c$ to microphone, where $\circledast$ is the convolution operator, $x_c$ is the dry clean speech of source $c$, $\mathbf{h}_c$ is the vector of room impulse responses associated with sound propagation from source $c$ to each microphone.
Formulate Eq. \ref{eq:time_mixture} in the time-frequency domain by means of the complex-valued STFT:
\begin{equation}
\mathbf{Y}(t,f) = \underset{c=1}{\overset{C}{\sum}}{\mathbf{S}_{c}(t,f)}
\label{eq:freq_mixture}
\end{equation}
where $\mathbf{Y}(t,f)$ and $\mathbf{S}_{c}(t,f)$ respectively denotes the complex spectrogram of mixture and reverberant image of source $c$ at time index $t$ and frequency band $f$.
In this study, given the multi-channel mixture signal $\mathbf{y}$, we aim to estimate individual signal $\hat{s}^{ref}_tgt$ for the target speaker, where $ref$ is the index of the reference microphone and $tgt$ is the target speaker's index in the mixture. It should be noted that our proposed system concentrate on speech separation task and do not take dereverberation into account. As a result, the model learns to estimate the reverberant speech $\hat{s}_c$ rather than dry clean speech $\hat{x}_c$.
\section{Time-domain Monaural Speech Separation}
\label{sec:single-channel_tss}
In this section, we will review Conv-TasNet \cite{luo2019convtasnet, luo2018surpass}, a deep learning framework for time-domain close-talk speech separation. Unlike the T-F masking based methods, Conv-TasNet replaces the STFT with a convolutional encoder-decoder architecture (figure \ref{fig:tasnet}).
Firstly, a short mixture segment $y$ is mapped to a high-dimensional representation $\mathbf{W}$ in the feature space by a linear encoder, which is a convolution 1d (conv1d) layer:
\begin{equation}
\mathbf{W} = ReLU(y\circledast \mathbf{B})
\label{eq:encoder}
\end{equation}
where $\mathbf{B} \in \mathbb{R}^{G\times L}$ is the encoder basis matrix, which contains $G$ convolution channels, each with window length $L$, $\circledast$ is the convolution operator, and $ReLU$ is rectified linear unit activation function. The number of convolution channels $G$ represents the number of basis functions. The kernel size $L$ and stride are the window length and hop size, respectively.
Then, the separation module adopts a temporal fully-convolutional network (TCN) \cite{lea2016temporal, feichtenhofer2016convolutional}, which computes a mask $\mathbf{M}_c$ for each source $c$, similar to the T-F masking. As a result, the speaker representation can obtain by a multiplicative product $\mathbf{D}_c=\mathbf{M}_c \odot \mathbf{W}$. The TCN consists of stacked dilated conv1d blocks, where each layer in the block features with exponentially increasing dilation factors. Meanwhile, in these dilated conv1d blocks, the traditional convolution is substituted with depthwise separable convolution to further reduce the parameters.
\begin{figure}[th]
\centering
\includegraphics[width=\linewidth]{tasnet_diagram.pdf}
\caption{The diagram of Conv-TasNet.}
\label{fig:tasnet}
\end{figure}
Finally, the decoder invert the speaker's representation $\mathbf{D}_c$ back to the time-domain signal, using 1d linear deconvolution:
\begin{equation}
\hat{x}_c = \mathbf{D}_c\mathbf{V}
\label{eq:decoder}
\end{equation}
where $\mathbf{V} \in \mathbb{R}^{G\times L}$ is the decoder basis matrix. To optimize the network, instead of using a time-domain mean squared error (MSE) loss, the speech separation metric scale-invariant signal-to-distortion (SI-SDR) is used to directly optimize the separation performance, which is defined as:
\begin{equation}
\left\{
\begin{array}{lr}
x_{\text{target}}:=\frac
{\left<\hat{x}, x\right>x}
{\left\|x\right\|_{2}^{2}} \\
e_{\text{noise}}:=\hat{x}-x_{\text{target}} \\
\text{SI-SDR}:=10\log_{10}\frac
{\left\|x_{\text{target}}\right\|_{2}^{2}}
{\left\|e_{\text{noise}}\right\|_{2}^{2}}
\end{array}
\right.
\label{eq:si_sdr}
\end{equation}
where $x$ and $\hat{x}$ are the dry clean and estimated source waveform, respectively. The zero-mean normalization is applied to $x$ and $\hat{x}$ to guarantee the scale invariance.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{tsnf_framework.pdf}
\caption{The framework of our proposed temporal-spatial neural filter. First, the multi-channel mixture waveform is passed to a conv1d layer that realizes the STFT function and produces multi-channel complex spectrogram. Meanwhile, the encoder transforms the reference channel's mixture $\mathbf{y}^{ref}$ to the mixture representation in a high-dimensional feature space. Second, along with the input directions $\mathbb{\phi}$, the feature computation module computes the spectral, spatial and directional features. Third, a joint acoustic representation is then obtained by concatenating the mixture representation and computed features in feature fusion module. Fourth, the separation module learns to estimate a mask for the target source, therefore the multiplicative product of the mask and mixture representation is the estimated target speech representation. Finally, the decoder reconstructs the target waveform from the masked mixture representation.
}
\label{fig:tsnf_framework}
\end{figure*}
\section{Proposed system}
\label{sec:proposed_system}
The proposed model is an integrated waveform-in waveform-out separation system in a single neural network architecture. The model consists of six processing modules: STFT conv1d layer, encoder, feature computation, feature fusion, separation and decoder. We have elaborated the encoder, separation module and decoder in section \ref{sec:single-channel_tss}, all of which inherited from Conv-TasNet. In this section, we will give the details of rest modules. First, to enable end-to-end training, the STFT operation is implemented with convolution layer and used to extract frequency domain features, described in Sec \ref{subsec:STFT_layer}; Second, the formulation and online computation of spatial and directional features are respectively presented in \ref{subsec:SI_features} and \ref{subsec:SR_features}. Finally, the fusion details are elucidated in \ref{subsec:fusion}.
\subsection{STFT convolution layer}
\label{subsec:STFT_layer}
Given a window function $w$ with length $N$, the spectrum $Y$ can be calculated by standard STFT:
\begin{equation}
\begin{split}
y[n] \xrightarrow[]{\tt STFT} Y(t,f)
&=\overset{N-1}{\underset{n=0}{\sum}}y[n]w[n-t]\exp{\left(-i\frac{2 \pi n}{N}f\right)}
\end{split}
\label{eq:STFT}
\end{equation}
To speed up the separation procedure, we intend to implement all the feature computation with network layers and operations. Following \cite{wichern2018phase, gu2019end}, we reformulate the standard STFT as a convolution kernel in Eq.~\ref{eq:STFT}. To distinguished with the standard STFT, we use $\tau$ and $m$ as the frame and frequency band index, which is actually the feature map height and width index after convolution, respectively:
\begin{equation}
y[n] \xrightarrow[]{\tt STFT} Y_{\tau,m}
=\overbrace{e^{-i\frac{2\pi \tau}{N}m}}^{\tt{phase~factor}} (y[\tau] \circledast \overbrace{w[\tau]e^{-i\frac{2\pi \tau}{N}m}}^{\tt{STFT~kernel}} )
\label{eq:STFT_conv}
\end{equation}
Note the phase factor in Eq.~\ref{eq:STFT_conv} is constant vector, which means it will neither affect the magnitude ($|\cdot|=1$) nor IPD. The complex STFT kernel can be splited in real and imaginary parts:
\begin{equation}
\begin{split}
\mathbf{K}^{\tt real}_{\tau,m} &=w[\tau] \cos(2\pi \tau m/N) \\
\mathbf{K}^{\tt imag}_{\tau,m} &=-w[\tau] \sin(2\pi \tau m/N)
\end{split}
\label{eq:STFTkernel}
\end{equation}
The shape of the kernel is determined by the $w[n]$. The size of the kernel is actually the window length $N$ of $w[n]$. The stride of convolution equivalents to the hop size in the STFT operation. We can also customize the number of kernels and use preferred kernel sizes other than $N$. The stride in convolution is also now configurable other than just using the hop size of STFT. In this work, to match the encoder configuration, the length and stride of STFT kernel is set the same as encoder kernel length $L$ and stride.
The reference channel's LPS is served as the spectral feature, calculated by:
\begin{equation}
\textit{LPS}_{\tau,m}=10\log \left (
\left ( \mathbf{y}^{ref} \circledast \mathbf{K}^{\tt real}_{\tau,m} \right )^2 +
\left (\mathbf{y}^{ref} \circledast \mathbf{K}^{\tt imag}_{\tau,m} \right )^2
\right )
\label{eq:lps}
\end{equation}
\subsection{Enhancing Speech Separation with spatial features}
\label{subsec:SI_features}
As discussed in introduction, well-established spatial cues like IPDs have shown great beneficial for T-F masking based MSS methods \cite{lianwu2019multi,wang2018multi,chen2018efficient,wang2018spatial}. The standard IPD is computed by the phase difference between channels of complex spectrogram as:
\begin{equation}
\textit{IPD}^{(u)}(t,f)=\angle\mathbf{Y}^{u_1}(t,f)-\angle\mathbf{Y}^{u_2}(t,f)
\label{eq:ipd_ori}
\end{equation}where $u_1$ and $u_2$ represents two microphones' indexes of the $u$-th microphone pair.
In our study, given the STFT kernel $\mathbf{K}$, the $u$-th pair of IPD can be computed by:
\begin{equation}
\label{eq:kernel_IPD}
\textit{IPD}^{(u)}_{\tau,m}=\arctan\left( \frac{\mathbf{y}^{u_1} \circledast \mathbf{K}^{\tt real}_{\tau,m} }{\mathbf{y}^{u_1} \circledast \mathbf{K}^{\tt imag}_{\tau,m} }\right)
-\arctan\left(
\frac{\mathbf{y}^{u_2} \circledast \mathbf{K}^{\tt real}_{\tau,m} }{\mathbf{y}^{u_2} \circledast \mathbf{K}^{\tt imag}_{\tau,m} }
\right)
\end{equation}
\begin{comment}
inter-channel cue
Besides the well-defined interaural features, we also propose to automatically learn inter-channel cues from the data. Interaural features are defined as representation deviation between channels, such as IPD is the deviation in phase. Similarly, the difference of automatically learned representation between channels can also be viewed as a kind of inter-channel cue. We achieve this by a customized convolution 2d layer and the details are presented as follows.
Given the $G$-channel mixture signal $\mathbf{y}$ which can be reviewed as a 2d matrix, convolving it with a Conv2d kernel $\mathbf{Q}$ with $F$ convolution channels and kernel size of $(H, L)$. The kernel height $H$ is the number of crossed signal channels and the kernel width $L$ indicates the window length operated on each channel's time-domain signal, the same length as that in waveform encoder. The $i$-th row of $H$ channels' encoded representation is computed by
\begin{equation}
\label{eq:conv2d}
\mathbf{Rep}_{i} = \sum_{h=0}^{H-1}{\mathbf{y}^{di+eh} \odot \mathbf{q}_h}
\end{equation}
where $h$ denotes the height index and $di+eh$ is the signal channel index, $d$ and $e$ respectively represents the padding and dilation on the height axis. Different value combinations of $d, e, h$ can obtain cues extracted from different groups of signal channels, each group corresponds to a row of $\mathbf{Rep}$. For example, setting $d=2, e=1, h=2$ for a 6-channel signal, we can get the three groups of channels: $i=0$: (0, 1), $i=1$: (2, 3) and $i=2$: (4, 5) and setting $d=1, e=2, h=3$, we can get two groups of channels, $i=0$: (0, 2, 4) and $i=1$: (1, 3, 5). Specially, $\mathbf{q}_h= \mathbf{q} \cdot \lambda_h \in \mathbb{R}^{L\times F}$ is a customized kernel, where $\mathbf{q} \in \mathbb{R}^{L\times F}$ is a basic learnable weight kernel shared for each signal channel. This basic kernel is meant to ensure that each channel of the time-domain signal shares the same encoding and the encoded representations are in the same high-dimensional embedding space. Meanwhile, for every height index $h$, there is a trainable difference factor $\lambda_h \in \mathbb{R}^L$. It plays a similar role as analysis window $w[n]$ in STFT
Compared to IPD that formulated in the frequency domain, this learned representation difference may work better with mixture representation produced by waveform encoder, since they are jointly trained and optimized under a purely data-driven fashion. Figure \ref{fig:conv2d+ipd} illustrates a sample of the automatically learned inter-channel cues and IPDs using same pair selection.
\end{comment}
\subsection{Target speech separation with directional features}
\label{subsec:SR_features}
\begin{figure*}[th]
\centering
\includegraphics[width=14cm]{all_features_v3.pdf}
\caption{A example in WSJ0 2-mix of the mixture logarithm power spectrum (LPS), directional features DPR and AF for the target speaker's direction and the interference speaker's direction, the LPS of ground truth target speech and the LPS of the recovered target speech. }
\label{fig:all_features}
\end{figure*}
Spatial feature such as IPD successfully extracts the spatial information of all the sources in the mixture signal. Moreover, with some or all of the speaker directions, features of specific speaker-dependent direction could be extracted to improve the performance of separation further. In this paper, it is assumed that the oracle location of each speaker is known by the separation system. This is an reasonable assumption in some real applications, for example, the speaker location could be detected by face detection techniques with very high accuracy.
In this work, we use an improved version of angle feature (AF) \cite{chen2018multi} and our proposed directional power ratio (DPR) as directional features.
The location-guided angle feature (AF) is first introduced in \cite{chen2018multi}, specially designed for the seven-element circular microphone array. We reformulate AF so that it can be applied to general microphone array topology. AF measures the cosine distance between the steering vector, which is formed according to the direction of the target speaker, and IPDs:
\begin{equation}
\textit{AF}_{\phi,\tau,m}=Real \left (
\overset{U}{\underset{u=1}{\sum}}
\frac{
\mathbf{a}_{\phi,m}^{u}
\exp(\textit{IPD}^{u}_{\tau,m})
}
{
\left |\mathbf{a}_{\phi,m}^{u}
\exp(\textit{IPD}^{u}_{\tau,m}) \right |
}
\right )
\label{eq:AF}
\end{equation}
where $\mathbf{a}_{\theta,m}^{u1}$ is the steering vector coefficient for target speaker from $\phi$ at the $m$-th frequency band for first microphone of $u$-th pair. Also, the pre-masking step in \cite{chen2018multi} is also applied to increase the discrimination of AF. The design principle of AF lies in that if the T-F bin is dominated by the source from desired direction, then the similarity between the steering vector and IPD will be close to 1, otherwise close to 0.
This feature provides the desired speaker's directional information to the network so that the network is expected to attend to the target speech. However, since the steering vector is computed with the exact $\phi$, the tiny estimation error may lower the quality of AF and cause the degradation of separation performance.
In our previous study, DPR was defined based on the output power of multi-look Cardioid Filters that carefully designed for a specific microphone array. However, the design can be complex and not easy to implement inside the network. In this work, for simplicity and reproducibility, we replace the Cardioid filters with a set of delay-and-sum beamformers (DAS-BF) steered at different directions. For a given microphone array and a pre-defined direction grid $\{\theta_1,\theta_2,...\theta_P\}$, a set of DAS-BFs are denoted as $\mathbf{w}_{p,m}\in\mathbb{C}^{J}$, which aims to enhance sound sources from direction $\theta_p$ for $m$-th frequency band.
Under far-field and free-field conditions, the DAS-BF steered at $\theta_p$ is computed by
\begin{equation}
\label{eq:ds_bf}
\mathbf{w}_{p,m} = \exp(-2\pi im \triangle t_{\theta_p,j}) / J
\end{equation}
where
$\triangle t_{\theta_p,j}$
denotes the time difference of arrival (TDOA) from microphone $ref$ to $j$ when the source is from direction $\theta_p$.
Assuming these beamformers can provide well enough spatial separation and the multiple speakers are not closely located in the space, we can use the processing output power of $\mathbf{w}_{p,m}$ as a reasonable estimation of the signal power from direction $\theta_p$. Therefore, the DPR can be considered as an indicator of how well is a T-F bin $(\tau,m)$ dominated by the signal from direction $\theta_p$, defined as follows:
\begin{equation}
\textit{DPR}_{\theta_p,\tau, m}=\frac{
\left \|\mathbf{w}_{p,m}^{H}\mathbf{Y}_{\tau, m} \right \|_{F}^{2}
}
{
\sum_{p'=1}^{P}
{
\left \|\mathbf{w}_{p',m}^{H}\mathbf{Y}_{\tau, m} \right \|_{F}^{2}
}
}
\label{eq:dpr}
\end{equation}
where $\mathbf{Y}_{\tau, m}$ is the complex spectral vector of multi-channel mixture signal in T-F bin $(\tau,m)$, computed as $\mathbf{Y}_{\tau, m} = \mathbf{y} \circledast \mathbf{K}^{\tt real}_{\tau,m} + i * ( \mathbf{y} \circledast \mathbf{K}^{\tt imag}_{\tau,m} )$. In our implementation, the beamformer coefficients of $\mathbf{w}_p$ are set as weights of a linear layer. As a result, the DPR feature can be calculated by a set of linear layers and safe divide operation.
Although the resolution of DPR, i.e., $\theta_{p+1}-\theta_{p}$, is lower than that of AF, one significant advantage is that it is more robust to the direction estimation error since it covers a particular range of directions. Therefore, combining AF and DPR as directional features can be a good choice to promise both the robustness and precision for separation.
Figure \ref{fig:all_features} illustrates the directional features when applied to a sample in spatialized dataset WSJ0 2-mix. It can be interpreted from the figure that both reformulated AF and the proposed DPR that focus at target direction can provide clues for the target source's contribution. Although DPR's saliency is weaker on lower frequency bands, the model can refer to AF as supplementary clues. Also, discrimination between target and interference speech can be achieved by combining the directional features for target and interference speaker.
\begin{comment}
\subsection{Attention mechanism}
\label{subsec:attention}
Spatial ambiguity issue is first addressed in \cite{chen2019multi}, when the speakers locate close to each other or the angle difference between speakers is small. The angle difference between two speakers is defined as:
\begin{equation}
\label{eq:ad}
ad(\phi_1, \phi_2)=\min(|\phi_1-\phi_2|, |360\degree-\phi_1+\phi_2|)
\end{equation}
where $\phi_1$ and $\phi_2$ respectively denotes the direction (in degree) of two speakers. The spatial ambiguity issue results in dramatically performance degradation of multi-channel T-F masking network under small angle difference range. This issue is mainly caused by the increasing dependency that network has on spatial features, since spatial features are more discriminative than spectral features under large angle difference ranges. Therefore, the network compromises on performances of small angle samples and put too much weight on spatial features in order to achieve overall improvement. Our recent study \cite{NSFpaper} proposed to apply an attention mechanism to guide the network to selectively focus on spectral, spatial or directional features under different angle difference ranges. The key idea is to learn a relative higher weight for the features that are more discriminative, and let the network focus on these features and neglect others.
The attention is a function of the angle difference $ad$ that multiplies to all spatial and directional features before feeding them to the upper network layers:
\begin{equation}
att(ad)=2*\max
\left (\sigma(ad)-0.5,0 \right)
\label{eq:global_attention}
\end{equation}
where $\sigma(ad)=1/(1+exp(-w(ad-b)))$ is the sigmoid score denotes how much emphasis should be put on spatial and directional features, $w$ and $b$ are trainable parameters. However, the attention weight is shared among all the spatial and directional feature, where they may be not equally significant always. Moreover, the attention mechanism in \cite{NSFpaper} is defined on angle difference between two speakers and do not generalize to more speaker mixed condition.
In this section, we first generalize angle difference definition to any mixing number, then propose two more fine-grained mechanisms to extend the global attention mechanism in \cite{NSFpaper}.
\subsubsection{Angle difference}
When there is more than one interference speaker, we define the angle difference as the degree difference between the target speaker and the closest speaker, i.e., $ad({\phi_1, ..., \phi_C})=\min_{c,c\ne tgt}(ad(\phi_{tgt}, \phi_{c}))$.
\subsubsection{Pair-wise attention}
\label{subsubsec:pairwise}
\begin{figure}[t]
\centering
\includegraphics[width=2.5in]{mic_att.pdf}
\caption{An illustration of two-speaker location setup when a 6-element uniform circular array is used. The direction of speaker 1 and 2 are respectively 270\degree and 90\degree and their angle difference is 180\degree. The time delay between speaker 1 and 2 to microphone pair 1-4 is zero, i.e., $r_{11}=r_{21}, r_{14}=r_{24}$. While for pair 3-6, the time }
\label{fig:mic_att}
\end{figure}
The extracted interaural features of different pairs may have different discriminative power, since the speaker location setup relative to each microphone pair is different. Figure \ref{fig:mic_att} shows an example when a 6-element uniform circular array is used for signal capturing and two speakers are simultaneously talking. Although the angle difference between these two speakers is 180\degree, which will leads to a very high attention weight in Eq. \ref{eq:global_attention}, the time delay between speaker 1 and speaker 2 to microphone 1 and 4 is zero. This suggests that little effective interaural information can be utilized between these two channels of signal. For pair 3-6, the time delay between from speaker 1 and speaker 2 to microphone 3 and 6 is different. Therefore, interaural features extracted from this pair could imply the cues for separating spatially distributed sources. (Here can also attach a fig, plot the IPD of pair 1-4 and pair 3-6.)
We define the relative angle difference $ad^u$ for each pair $u$ as follows:
\begin{equation}
\label{eq:relative_ad}
ad^{u}(\phi_1, \phi_2)=|ad(\phi_1,\alpha^{u1,u2})-ad(\phi_2,\alpha^{u1,u2})|
\end{equation}
where $\alpha^{u1,u2}$ is the angle difference between directed line $u1\rightarrow u2$ and standard 0\degree line. For example, in Figure \ref{fig:mic_att}, $\alpha^{1,4}=180\degree$ and $\alpha^{3,6}=60\degree$.
\subsubsection{Feature-wise attention}
\label{subsubsec:featurewise}
In \cite{NSFpaper}, spatial and all kinds of directional features are assigned with equally attention weight, which may not always the case. We propose to associate weights separately for each feature based on its contribution measured in two ways: automatic learned . The contribution is decided by
\end{comment}
\subsection{Feature fusion}
\label{subsec:fusion}
Before fed into the separation module, the mixture representation $\mathbf{W}$ and computed spectral, spatial, directional features are fused to form a joint acoustic representation. Specifically, with the same convolution kernel size and stride, the numbers of time steps of $\mathbf{W}$, $\textit{LPS}$, $\textit{IPD}$, $\textit{AF}$ and $\textit{DPR}$ are the same. Then, these features are concatenated along the feature axis and passed to the proceeding layers.
It should be noted that, in this work, we have tried to introduce an attention module to automatically learn the time-varying contribution of each feature, like our previous trial \cite{gu2019neural}. However, the gain is minor and extra network parameters are introduced. The reason may lie in that the first layer in the separation module is a conv1$\times$1 (fully-connected) layer that has associated the weight with each feature. Also, the learned contribution of directional features is always high. Therefore, we do not report the corresponding results in this paper and leave it to discussion.
\section{Experiments procedures}
\label{sec:exp}
\subsection{Dataset}
We simulated a spatialized reverberant dataset derived from Wall Street Journal 0 (WSJ0) 2-mix and 3-mix corpus, which are open and well-studied datasets used in monaural and multi-channel speech separation \cite{hershey2016deep,yu2017permutation,luo2019convtasnet,lianwu2019multi,wang2019combining}. In 2-mix corpus, there are 20,000, 5,000 and 3,000 multi-channel, reverberant, two-speaker mixed speech in training, development and test set respectively. The amount of three-speaker speech mixture in training, development and test set of 3-mix corpus is respectively the same as that in 2-mix corpus. The performance evaluation is all done on test set, the speakers in which are all unseen during training. The mixing signal-to-noise ratio (SNR), pairs, dataset partition are exactly coincident with anechoic monaural WSJ0 2-mix and 3-mix. For the selection of microphone array, we consider a 6-microphone circular array of 7cm diameter with speakers and the microphone array randomly located in the room, as illustrated in figure \ref{fig:mic}. The two speakers and microphone array are on the same plane and all of them are at least 0.3m away from the wall. The image method [23] is employed to simulate RIRs randomly from 3000 different room configurations with the size (length-width-height) ranging from 3m-3m-2.5m to 8m-10m-6m. The reverberation time T60 is sampled in a range of 0.05s to 0.5s.
For 2-mix spatialized dataset, samples with angle difference between two simultaneous speakers of 0-15$\degree$, 15-45$\degree$, 45-90$\degree$ and 90-180$\degree$ respectively account for 16\%, 29\%, 26\% and 29\%. The angle difference between two speakers is defined as: $ad(\phi_1, \phi_2)=\min(|\phi_1-\phi_2|, |360\degree-\phi_1+\phi_2|)$, where $\phi_1$ and $\phi_2$ respectively denotes the direction (in degree) of two speakers. For 3-mix spatialized dataset, the angle difference is defined as the smallest angle difference between the target speaker and other interference speakers. The proportion for each angle difference range is 29\%, 36\%, 23\% and 12\%. When there is more than one interference speaker, the angle difference is defined as the degree difference between the target speaker and the closest speaker, i.e., $ad({\phi_1, ..., \phi_C})=\min_{c,c\ne tgt}(ad(\phi_{tgt}, \phi_{c}))$.
\begin{figure}[h]
\centering
\includegraphics[width=3cm]{6mic.pdf}
\caption{The microphone configuration and space grid for the experiments.}
\label{fig:mic}
\end{figure}
\subsection{Feature extraction and Network hyper-parameters}
The selected pairs for IPDs are (1, 4), (2, 5), (3, 6), (1, 2), (3, 4) and (5, 6) in all experiments. These pairs are selected considered that the distance between each pair is either the furthest or nearest.
The reference channel is set to the first channel of speech mixture waveform as default. To match the encoder, all frequency domain features, including LPS, IPDs and directional features, are extracted with 2.5ms window length and 1.25ms hop size with 64 FFT points. That means that the length and stride of STFT kernel is the same as encoder filter length $L$ and stride, i.e., 40 and 20 points. The number of filters is set to 33 since we round the window length to the closest exponential of 2, i.e., 64.
For DPR computation, we use 36 fixed spatial filters and the $p$-th filter is steered at azimuth $10p\degree$. This resolution is selected empirically considering the balance between precision and robustness.
All the data is sampling at 16kHz. All hyper-parameters are the same with the best setup of Conv-TasNet in \cite{luo2018surpass}, except $L$ is set to 40 and encoder stride is 20. Batch normalization (BN) is adopted because it has the most stable performance. Note that we do not adopt the update version of Conv-TasNet \cite{luo2019convtasnet}, although it exhibits better performance and smaller model size. Also, adopting global layer normalization (gLN) instead of BN could further improve the performance a lot. Since our target is to approach real-time separation, we prefer the early version of Conv-TasNet without densely connected structure and global normalization.
\subsection{Training paradigms}
For the output of separation system, there are two cases: i) the system outputs the estimated speech of all speakers in the mixture. When there is no directional information, permutation invariant training \cite{yu2017permutation} is adopted to tackle with label permutation problem. This means at the inference, the output-speaker assignment is unknown and additional speaker recognition procedure is required. While all speakers' directions are given, the output speech order is exactly the order of input speakers' directions. Therefore, the target speech can be obtained by selecting the corresponding index. However, under small angle angle range, i.e, $<15\degree$, the output order become vague due to spatial ambiguity issue. For 2-targets case, we evaluate the performance with the permutation that achieves the best result.
ii) Only the target speech is estimated. For this 1-target case, we investigate two types of input: 1) only target speaker's directional feature is provided as target-related information, denoted as \emph{tgt}. This system is more applicable in practical since it can perform target speech separation with only the target speaker's direction; 2) Following our recent study \cite{gu2019neural} and \cite{xiao2019single}, interference speaker's directional feature is also attached to the target speaker's to enhance the discrimination between different directions, denoted as \emph{tgt+intf}. For target speech separation task, the performance is obtained by running the separation network by the times of speaker number, each time selecting one speaker in mixture signal as target.
SI-SDR (Eq. \ref{eq:si_sdr}) is utilized as training objective. The training uses chunks with 4.0 seconds duration. The batch size is set to 32.
\subsection{Evaluation metrics}
Following the recent advances in speech separation metrics \cite{le2019sdr}, average SI-SDR is adopted as the main evaluation metric. Also, following the common practice, average SDR computed using BSS\_EVAL toolbox \cite{vincent2006performance} and perceptual estimation of speech quality (PESQ) are used to measure the speech quality and intelligibility. Since we do not perform speech dereverberation, we consider the reverberant image of source $s^{ref}_c$ for metric computation.
Instead of the overall performance, we also report the performances under different ranges of angle difference between speakers. A good model should perform excellent overall and balanced under all angle difference ranges. We intend to investigate the relative performance difference under different ranges and give a more comprehensive assessment for the model.
\section{Results and analysis}
\label{sec:result}
\subsection{Window length selection for frequency domain features}
As discussed in \ref{subsec:STFT_layer}, the features that extracted in frequency domain using different analysis window length and hop size may mismatch the encoder output.
Considering the short-term stationary characteristics of speech, the typical window length for T-F masking based methods is 15-32ms (240-512 FFT points for speech sampled at 16kHz). Also, a larger window length can provide higher frequency resolution for more refined spectrum analysis. However, using this window length, the time resolution is much lower than that of mixture representation output by the encoder, i.e., 2.5ms (40 points). Therefore, we investigate the window length for extracting the frequency domain features that works the best with encoded representation. To ensure the synchronization between frequency domain features and the mixture representation, upsampling operation is applied to the frequency domain features when the stride (hop size) is larger than 40.
\begin{table}[b]
\caption{SDRi (dB) and SI-SDRi (dB) performances of 2-target speech separation with different FFT size, window length and stride (hop size) on far-field WSJ0 2-mix.
}
\label{tab:fft_size}
\centering
\begin{tabular}{c|ccc|cc}
\hline
\textbf{Features} & \textbf{FFT} & \textbf{Win.} & \textbf{stride} & \textbf{SI-SDRi (dB)} & \textbf{SDRi (dB)} \\
\hline
Conv-TasNet & - & 40 & 20 & 9.1 & 9.4 \\
\hline
+cosIPD & 512 & 512 & 256 & 10.1 & 10.5 \\
+cosIPD & 256 & 256 & 128 & 10.6 & 11.0\\
+cosIPD & 128 & 128 & 64 & 10.8 & 11.2\\
+cosIPD & 64 & 40 & 20 & \textbf{11.2} & \textbf{11.6}\\
+cosIPD & 40 & 40 & 20 & 9.1 & 9.5\\
\hline
\end{tabular}
\end{table}
Table \ref{tab:fft_size} reports the results. As we can observe, the configuration of 64-point FFT along with window length of 40 achieves the best performance, since its window length and stride is coincident with those of encoder. In the following experiments, we all adopt the configuration of 64-point FFT, 2.5ms window length and 1.25ms hop size for computing frequency domain features.
\subsection{Ablation study on feature combination}
\begin{table*}[t]
\caption{SDRi (dB) and SI-SDRi (dB) performances of target separation systems on far-field WSJ0 2-mix.}
\label{tab:ablation}
\centering
\begin{tabular}{c|l|cccc|c|c}
\hline
\multirow{2}{*}{\textbf{\# of target}} &
\multirow{2}{*}{\textbf{Features \& Setup}} &
\multicolumn{5}{c}{\textbf{SI-SDRi (dB)}} &
\multirow{2}{*}{\textbf{SDRi (dB)}} \\
& &$<$15\degree &15\degree-45\degree &45\degree-90\degree &$>$90\degree & Ave. \\
\hline
2 & Single-channel Conv-TasNet & 8.5 & 9.0 & 9.1 & 9.3 &9.1 & 9.4 \\
\hline
2 & +LPS & 9.2 & 9.7 & 9.6 & 10.0 & 9.7 & 10.0 \\
2 & +cosIPD & 8.5 & 11.8 & 12.0 & 11.6 &11.2 & 11.6 \\
2 & +cosIPD,sinIPD & 7.7 & 11.6 & 12.3 & 12.6 & 11.5 & 11.9\\
2 & +LPS, cosIPD, sinIPD &8.1 &11.9 &12.7 &13.1 & 11.9 & 12.2\\
\hline
2 & +cosIPD, AF &10.7 &12.9 &13.4 &13.7 &12.9 & 13.3\\
2 & +cosIPD, DPR & 9.2 & 12.8 & 13.3 & 13.6 & 12.6 & 13.0\\
2 & +cosIPD, AF, DPR & 10.5 & 13.0 & 13.5 & 13.8 & 13.0 & 13.4 \\
2 & +cosIPD, sinIPD, AF, DPR & 10.6 & 13.1 & 13.6 & 13.9 &\textbf{13.1} & \textbf{13.4}\\
\hline
1 & +cosIPD, AF (tgt) & 9.6 & 12.9 & 13.3 & 13.6 & 12.7 & 13.1\\
1& +cosIPD, AF (tgt+intf) & 9.8 & 12.9 & 13.4 & 13.7 & 12.8 & 13.2\\
1& +cosIPD, DPR (tgt+intf) & 5.3 & 12.5 & 13.0 & 13.3 & 11.8 & 12.2\\
1&+cosIPD, AF, DPR (tgt) & 9.4 & 12.7 & 13.1 & 13.5 & 12.5 & 12.9\\
1&+cosIPD, AF, DPR (tgt+intf) & 9.8 & 13.0 & 13.5 & 13.8 &\textbf{12.9} & \textbf{13.3} \\
1& +cosIPD, sinIPD, AF (tgt+intf) & 9.5 & 12.5 & 13.0 & 13.3 & 12.4 & 12.8\\
\hline
\end{tabular}
\end{table*}
In order to evaluate the contribution of different features and their combinations, we conduct an ablation study on the feature selection on the task of speech separation from a mixture of two speakers.
Table \ref{tab:ablation} shows the results of our ablation study. The table includes evaluation using SI-SDRi and SDRi with three input setups: 1) temporal encoded representation only (Single-channel Conv-TasNet); 2) adding spectral (LPS) and spatial features (IPD); 3) adding spectral, spatial and directional features (DPR, AF).
The single-channel 2-speaker separation network performs with SI-SDRi of 9.1dB. Adding the LPS of the mixture's reference channel elevates the performances under all angle difference ranges for about 0.7dB. Further adding spatial features (cosIPD and sinIPD) boosts the overall performance to 11.9dB. Particularly, the inclusion of spatial features greatly enhances the performance under large angle difference range for about 2.8dB.
With the aid of directional features, better performance can be achieved compared to systems only with spatial and spectral features, i.e., 13.1dB versus 11.9dB.
As for single target separation, compared to feeding both target's and interference's AF, the performance of model that only provides target's AF drops only 0.1dB. Adding DPR feature further boost overall performance, especially under large angle difference range. However, the inclusion of sinIPD may weaken the contribution of AF and reduces the performance.
\begin{comment}
\subsection{Attention mechanism}
\begin{table*}[t]
\caption{SDRi (dB) and SI-SDRi (dB) performances of target separation systems on far-field WSJ0 2-mix.}
\label{tab:attention}
\centering
\begin{tabular}{c|l|c|cccc|c|c}
\hline
\textbf{\# of output} &
\multirow{2}{*}{\textbf{Features \& Setup}} &
\multirow{2}{*}{\textbf{attention}} &
\multicolumn{5}{c}{\textbf{SI-SDRi (dB)}} &
\multirow{2}{*}{\textbf{SDRi (dB)}} \\
\textbf{/\# of target} & & &$<$15\degree &15\degree-45\degree &45\degree-90\degree &$>$90\degree & Ave. \\
\hline
2 / 2 & 1ch wav, cosIPD, sinIPD & - & 7.7 & 11.6 & 12.3 & 12.6 & 11.5 & 11.9\\
2 / 2 & 1ch wav, cosIPD, sinIPD & global & 7.5 & 11.7 & 12.6 & 12.9 & 11.6 & 12.0\\
2 / 2 & 1ch wav, cosIPD, sinIPD & pair-wise & 7.9 & 12.1 & 13.0 & 13.3 & 12.0 & 12.4\\
\hline
2 / 1 & 1ch wav, cosIPD, AF, DPR (tgt+intf) & - & 10.0 & 13.0 & 13.4 & 13.8 & 12.9 & 13.2 \\
2 / 1 &1ch wav, cosIPD, AF, DPR (tgt+intf) &global & ? & & & & & \\
2 / 1 &1ch wav, cosIPD, AF, DPR (tgt+intf) &feature-wise & 10.1 & 13.0 & 13.5 &13.8 &13.0 & \\
2 / 1 &1ch wav, cosIPD, AF, DPR (tgt+intf) &pair-wise & ? & & & & & \\
\hline
\end{tabular}
\end{table*}
\end{comment}
\subsection{Varying mixing number}
To illustrate the flexibility of our proposed model, in table \ref{tab:any_mix}, we summarize the single target separation performance of the models respectively trained on WSJ0-2mix, WSJ0-3mix and both WSJ0-2mix and WSJ0-3mix. The computed features are cosIPD, AF and DPR (tgt+intf). All the models are tested on both two- and three-speaker mixtures without knowing the source number in the mixture.
For 3-speaker mixture, the interference speakers’ directional features are merged as one, either selecting the closer interference speaker to target speaker, or summing/averaging the directional features of both interference speakers. In this experiment, the closest interference speaker's direction is used to compute the interference speaker's directional feature.
For the model trained on only 2-speaker mixtures to perform 3-speaker separation, we always chose the closest interference speaker to the target speaker to compute the directional feature.
Compared with models that trained with the dataset mixed by a fixed number of speakers, the model trained with both 2- and 3-mix dataset shows superior performance both under 2-speaker and 3-speaker mixing condition. This result indicates that a single model can handle a varying and unknown number of speakers' mixture. This is of great practical importance, since a priori knowledge about the number of speakers is not needed at test time.
\begin{table}[t]
\caption{SDRi (dB) and SI-SDRi (dB) performances of single target speech separation on far-field WSJ0 2-mix and 3-mix. The extracted features are cosIPD, AF, DPR (\emph{tgt+intf}) for all models.
}
\label{tab:any_mix}
\centering
\begin{tabular}{c|cc|cc|cc}
\hline
\multirow{2}{*}{\textbf{Tr. Dataset}} &
\multicolumn{2}{c}{\textbf{SI-SDRi (dB)}} &
\multicolumn{2}{c}{\textbf{SDRi (dB)}} &
\multicolumn{2}{c}{\textbf{PESQ}} \\
& 2spk &3spk &2spk &3spk &2spk &3spk\\
\hline
Mixture & 0 & 0 & 0 & 0 & 1.98 & 1.62\\
\hline
2-mix & 12.9 & 5.9 & 13.3 & 6.3 & - & -\\
3-mix & 11.7 & 9.5 & 12.0 & 9.8 & - & -\\
2+3-mix & \textbf{13.5} & \textbf{11.1} & \textbf{13.9} & \textbf{11.5} &3.28 &2.74\\
\hline
IBM & 12.2 & 12.7 & 12.6 & 13.1 & 3.11 & 2.70\\
IRM & 12.0 & 12.6 & 12.4 & 13.0 & 3.79 & 3.66\\
IPSM & 15.0 & 15.8 & 15.4 & 16.3 & 3.95 & 3.84\\
\hline
\end{tabular}
\end{table}
\begin{comment}
\begin{table}[t]
\centering
\caption{PESQ scores for the ideal T-F masks and proposed temporal-spatial neural filter on the entire WSJ0-2mix and WSJ0-3mix test sets. }
\label{tab:pesq}
\begin{tabular}{c|ccccc}
\hline
\multirow{2}{*}{\textbf{Dataset}} &
\multicolumn{5}{c}{\textbf{PESQ}}\\
& Mixture & IRM & IRM & IPSM & Proposed \\
\hline
WSJ0-2mix & 1.98 &3.11 &3.79 &3.95 &3.28\\
WSJ0-3mix & 1.62 & & & &2.74\\
\hline
\end{tabular}
\end{table}
\end{comment}
\subsection{Comparisons with other approaches}
We take two types of rival systems into consideration:
1) monaural end-to-end system with state-of-the-art performance, i.e., single-channel Conv-TasNet \cite{luo2018surpass};
2) deep learning-based MSS methods, including Freq-BLSTM based speech separation methods, multi-channel deep clustering (DC) \cite{wang2018multi} and neural spatial filter \cite{gu2019neural}.
For all the Freq-BLSTM based methods, the input is the concatenation of cosIPD and LPS, and the output is the T-F mask for each source. When the training objective is phase-sensitive spectrum approximation (PSA), the estimated mask is referred as phase-sensitive mask (PSM), which is a commonly used training target in speech separation task \cite{kolbaek2017multitalker, wang2014training}. We also investigate the end-to-end training objective, SI-SDR, which is the same as that of our model. MISI-5 proposed by \cite{wang2018end} is a phase reconstruction algorithm that implemented with iterative network layers. As for multi-channel DC, it integrates conventional spatial clustering with DC \cite{hershey2016deep} by including IPD patterns in the input of deep clustering network. First, the 2-channel DC model is trained on interaural patterns extracted from different microphone pairs. Then, the embeddings of each pair that produced by the DC network is stacked along the embedding dimension. Finally, perform K-means on these embeddings and obtain the estimated T-F binary mask. Neural Spatial Filter takes LPS, cosIPD, DPR, Directional signal-to-noise ratio (DSNR) and AF as the LSTM network input, where both DPR and DSNR are calculated by Cardioid Filters. The network is trained with spectrum approximation objective to estimate a ideal ratio mask (IRM).
Table \ref{tab:all} also report the performances of oracle masks for reference, including ideal binary mask (IBM), IRM and ideal PSM (IPSM) \cite{wang2014training}. These masks are computed with 256-point FFT, 16ms hanning window. The processing speed is evaluated as the average processing time for the systems to separate each frame in the mixtures, which referred as time per frame (TPF) \cite{luo2018surpass}. If a system can be implemented in real time, the required TPF should be smaller than the frame length. For the CPU configuration, we tested all the systems with one processor on an Intel Xeon E5-2680 CPU. For the GPU configuration, the systems are tested on one Tesla M40 GPU.
Specially, the frame length for both single-channel Conv-TasNet and our proposed temporal-spatial neural filter is 2.5ms. While for other deep learning based multi-channel separation methods, the frame length is 32ms.
Note that we do not include a state-of-the-art multi-channel speech separation system \cite{wang2019combining} for comparison, which also proposes to enhance the separation with directional features. The main reason is that it is a multi-stage system that adopts the strategy of \emph{Separate-Localize-Enhance}. The products of separation stage are also utilized in enhancement stage. Also, authors claimed that this system focused on offline processing so that the computational cost may be high.
As can be observed in Table \ref{tab:all}, our proposed Temporal-Spatial Neural Filter exhibits the best performance among all the comparison methods under all the angle difference ranges. Also, our model's size is smaller and the processing speed is faster, which means it is a more promising approach for real-time applications. With the same end-to-end training objective and similar network structure, our method outperforms single-channel Conv-TasNet by 4.4dB of SI-SDR. This demonstrates the benefits of the joint representation of temporal, spectral, spatial and directional features. With the similar input features, the performance of end-to-end Temporal-Spatial Neural Filter obtains 4.4dB improvement over the LSTM-based Neural Spatial Filter. Furthermore, the processing speed is greatly reduced, which indicates the effectiveness of on-the-fly feature extraction and end-to-end network.
\begin{table*}[h]
\caption{Performances of target separation systems on simulated far-field WSJ0 2-mix test set.}
\label{tab:all}
\centering
\begin{tabular}{c|c|l|c|cc|c|c}
\hline
\multirow{2}{*}{\textbf{\# params}} &
\multirow{2}{*}{\textbf{\# mic.}}&
\multirow{2}{*}{\textbf{Approach \& Setup}} &
\multirow{1}{*}{\textbf{CPU/GPU}} &
\multicolumn{3}{c}{\textbf{SI-SDRi (dB)}} &
\multirow{2}{*}{\textbf{SDRi (dB)}}\\
& & &\textbf{TPF \#ms} &$<$15\degree &$>$15\degree & Ave. & \\
\hline
- & - & Mixture & - & 0 & 0 & 0 & 0 \\
\hline
8.8M & 1 & Single-channel Conv-TasNet \cite{luo2019convtasnet} & 0.4 / 0.02 & 8.5 & 9.1 & 9.1 & 9.5 \\
21.7M & 6 & Multi-channel Freq-BLSTM (PSA) & 22 / 8.0 & 6.5 & 9.8 & 9.3 & 9.7 \\
30.0M & 6 & Multi-channel Freq-BLSTM (SI-SDR) & 22 / 8.0 & 6.5 & 10.1 & 9.5 & 9.9 \\
30.2M & 6 & Multi-channel Freq-BLSTM (SI-SDR+MISI-5) & 24 / 8.2 & 7.3 & 10.5 & 10.0 & 10.5 \\
33.8M & 6 & Multi-channel DC* \cite{wang2018multi} &39 / 14 &9.1 & 10.1 & 9.9 & 10.3 \\
\hline
21.5M & 6 & Neural Spatial Filter \cite{gu2019neural} &23 / 7.7 &4.8 & 10.1 & 9.1 &9.5 \\
8.8M & 6 & Temporal-Spatial Neural Filter (proposed) &0.5 / 0.03 &\textbf{10.8} &\textbf{14.0} &\textbf{13.5} &\textbf{13.9} \\
\hline
- & - & IBM & - & 12.1 & 12.1 & 12.2 & 12.6 \\
- & - & IRM &- & 12.0 &12.0 &12.0 & 12.4 \\
- & - & IPSM &- & 15.0 &15.0 &15.0 & 15.4 \\
\hline
\end{tabular}
\end{table*}
\subsection{Sensitivity of proposed system to direction estimation error}
In above experiments, we assume the oracle direction of each speaker is known without any estimation error. However, in real-world scenarios, the DOA estimation algorithms or face location techniques may suffer from interferences from acoustic and visual signals and bring about the estimation error. Therefore, in order to investigate how the error affects the proposed model's performance, we introduce random error to the input target speaker's direction. Specifically, we deviate the ground truth target direction for ±1 to ±10$\degree$ for all the testing samples of WSJ0 2-mix and 3-mix, where the positive or negative deviation is random. We compare two systems that integrated with different directional features, one with AF only, the other one with AF and DPR. All the directional features are computed using deviated directions.
The results for two systems are illustrated in Fig. \ref{fig:dee_2mix} and \ref{fig:dee_3mix}, where the full lines indicate the cosIPD+AF+DPR system while the dotted lines represent the cosIPD+AF system. We can see from the figure that as the direction estimation error increases, the performance of both systems drops drastically when angle difference between 0-15$\degree$. It is reasonable because that, in order to ensure the spatial discrimination within a small angle range, the space grid should be finely divided to distinguish directional sources well. So, even a slight estimation error may lead to a wide deviation to the proper estimation. Also, the estimation error aggravates the spatial ambiguity issue and brings difficulty in determining which speaker should be separated. Encouragingly, figure \ref{fig:dee_2mix} and figure \ref{fig:dee_3mix} tell that when the angle difference is larger than 15$\degree$, the direction estimation error within ±10$\degree$ almost has no influence on the performance of cosIPD+AF+DPR system, i.e., smaller than 0.1dB of SI-SDRi for 2-mix and 0.4dB for 3-mix. While for cosIPD+AF system, the performance drops about 0.4dB for 2-mix and 0.7dB for 3-mix. We achieve this with the aids of DPR feature since it covers a relatively large direction grid (10$\degree$ in our configuration). Also, the overall performance decrease of cosIPD+AF+DPR system is less than 1.2dB on WSJ0-2mix and 1.7dB on WSJ0-3mix. These results confirms the robustness of our proposed spatial filter.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{dee_2mix+af.pdf}
\caption{The SI-SDRi performance versus different direction estimation error on spatialized reverberant WSJ0 2-mix.}
\label{fig:dee_2mix}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{dee_3mix+af.pdf}
\caption{The SI-SDRi performance versus different direction estimation error on spatialized reverberant WSJ0 3-mix.}
\label{fig:dee_3mix}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we introduce a temporal-spatial neural filter, which directly estimates the time-domain target speech from multi-speaker mixture in reverberant environments, assisted with directional information. The main contributions lie in:
1) We propose to jointly model the temporal, spectral, spatial and directional discriminability to create a more complete representation for better separation accuracy;
2) We design two effective directional features to indicate the dominance of directional source at each T-F bin. At meanwhile, experiments demonstrate the robustness of these directional features when existing a direction estimation error.
3) The proposed speaker-independent model can handle mixtures with up to 3 simultaneous speakers and associate the output to the speaker by specifying the speaker's corresponding direction.
Our evaluations showed that the proposed temporal-spatial neural filter outperforms other deep learning based multi-channel speech separation approaches. Also, it has a smaller model size and fast processing speed. Furthermore, we demonstrate that the proposed model is robust to the direction estimation error when speakers are not closely located, i.e., the angle difference between speakers is larger than 15$\degree$.
We point out that this work still has some limitations that will be addressed in future work.
First, one significant flaw of proposed filter is the spatial ambiguity issue. Under small angle different case, the performance is still far from satisfactory. Other target-related information can be introduced to complement the representation, such as speaker embedding \cite{wang2018voicefilter} and visual signals \cite{ephrat2018looking}.
Second, our approach needs to know the direction(s) of speaker(s) in advance, which is a strong assumption. A direction estimation module can be further integrated to build a more complete system.
Third, improved attention mechanism can be introduced to automatically extract the time-varying contribution of each feature. As shown in Fig. \ref{fig:dee_2mix}, when the target direction is precisely estimated, the performance of cosIPD+AF system is about 0.3dB better than that of cosIPD+AF+DPR under small angle difference. This implies that AF alone can provide more separation precision under small angle case. We can further make the model selectively attend to AF and neglect DPR under special scenarios.
Fourth, the proposed method only focuses on speech separation and do not deal with the dereverberation or denoising task. In the future, we will consider training these task jointly.
Finally, we assume that the speaker does not move during speaking and the directional feature is computed with a fixed direction. However, in some practical applications, the directions of speakers may change or jitter during an utterance. For this propose, the directional features should be carefully redesigned and reformulated.
\section*{Acknowledgment}
Thanks to Shixiong Zhang, Lianwu Chen, Yong Xu, Meng Yu and Dong Yu from Tencent AI Lab for significant guidance and suggestions. Thanks to Jian Wu from Northwestern Polytechnical University, Xi’an, China, and Fahimeh Bahmaninezhad from University of Texas at Dallas, Richardson, USA for very helpful discussions.
\ifCLASSOPTIONcaptionsoff
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\fi
\bibliographystyle{IEEEtran}
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1,108,101,565,496 | arxiv | \section{Introduction}
There is considerable evidence from astrophysical observations that there is more mass in the universe than can be accounted for with only standard model particles \cite{Clowe:2006eq,planck,structure}.
The most popular theory that explains this discrepancy introduces dark matter particles called WIMPs \cite{Jungman1996195}.
In past years, the sensitivity of direct dark matter search experiments has increased by orders of magnitude, lead by the development of large dual-phase xenon time-projection chambers~(TPCs) \cite{RevModPhys.82.2053, 1748-0221-8-04-R04001, Baudis201450}.
In the context of dark matter research, a small-scale liquid xenon TPC, called XAMS (Xenon Amsterdam), has been designed, built and commissioned at Nikhef in Amsterdam.
The setup described in this work is similar to small-scale dual-phase xenon setups, such as described in \cite{1344272, PhysRevLett.97.081302, Kwong2010328}.
Dual-phase TPCs detect a particle interaction using two distinct signals.
The first comes from excitations and recombined electron-ion pairs.
Bound excited states of two atoms form, and subsequent decays of these excitons causes scintillation light that is detected by photomultiplier tubes (PMTs).
This signal is called S1.
The second signal is caused by ionization electrons that do not recombine with ions.
These are drifted up by an electric field and extracted by a second, stronger field into the gas phase, where secondary scintillation (S2) is caused and measured by the same PMTs.
The drift time between these signals is proportional to the interaction depth (z).
In addition to this, the ratio of S2/S1 provides a powerful discrimination between electronic and nuclear recoils.
In large-scale TPCs, such as XENON100 \cite{Aprile:2011dd} and LUX \cite{Akerib:2012ys}, the light distribution of the S2 in the PMTs gives the coordinates in the plane of the PMTs, so that a three-dimensional resolution is obtained.
This article has the following structure.
In section~\ref{SEC2}, the XAMS setup and the TPC are introduced.
Section~\ref{SEC3} discusses the data processing and gives results based on the main S1- and S2-signals.
In section~\ref{SEC4}, S2-signals from single electrons are analyzed and a PMT calibration technique based on these signals is presented.
In section~\ref{SEC5}, we give a summary of the analyses in these sections.
\begin{figure*}[th]
\begin{center}
\includegraphics[width= 0.95 \linewidth]{FIG1.jpg}
\caption{
Cross-section of the XAMS TPC and the source in the collimator (drawn to scale).
All elements of the TPC are contained in a cylindrical PTFE structure made from stackable disks, as indicated by the gray color.
The electric field is defined by five meshes and seven copper rings, serving to homogenize the drift field.
The top and bottom screening meshes are held at the cathode potential of the PMTs.
The active volume is defined by the cylindrical volume between the gate and cathode mesh, measuring \SI{100}{mm} (height) $\times$ \SI{44}{mm} (diameter).
The $^{22}$Na gamma source (described in section~\ref{SEC2_3}) is mounted in a collimator that is made of two lead blocks with cylindrical holes.
These are positioned on the outside of the outer vessel (vessels not shown) and allow for a beam size of \SI{11}{mm} at the closest edge of the active volume.
A two-inch NaI(Tl)-detector (not depicted) is used for triggering, and is positioned \SI{100}{mm} to the left edge of the collimator.
The z-position of the collimator is adjustable.
}
\label{FIG1}
\end{center}
\end{figure*}
\section{The XAMS setup}\label{SEC2}
\subsection{The XAMS TPC}
The XAMS TPC features a cylindrically-shaped active volume of \SI{154}{cm^{3}}, which holds \SI{434}{\gram} of liquid xenon at a temperature of \SI{-90}{\celsius}, as shown in Fig.~\ref{FIG1} \cite{erik_master,maria_master,rolf}.
There are two PMTs, one at the top and one at the bottom, that view the active volume and record the S1- and the S2-signals.
Five meshes define the electric field: the drift field of \SI{0.52}{kV/cm} is between the cathode and the grounded gate mesh, whereas the extraction field is between the gate and anode mesh, where a voltage of \SI{2.5}{kV} is applied over \SI{5}{mm}.
The meshes were made by chemical etching of a \SI{150}{\micro m} thick stainless steel sheet.
They have a square pattern with a pitch of \SI{2.45}{mm} and a wire thickness of \SI{150}{\micro m}, giving a head-on optical transparency of \num{88}\%.
The drift field is shaped by a series of copper rings connected to a resistor chain between the cathode and gate mesh.
Two additional meshes shield the PMTs from the TPC's electric fields.
The distance between the cathode and the gate mesh, which defines the maximum drift length, is \SI{100}{mm}.
The PMTs are circular two-inch UV-sensitive low-temperature Hamamatsu PMTs of type R6041-406.
The low transit-time spread of \SI{0.75}{ns} in combination with the fast 500~MSa/s digitizer type CAEN V1730D makes XAMS well-suited for fast-timing applications, such as pulse-shape discrimination studies.
\subsection{Cryogenics and gas system}
For the successful operation of a dual-phase xenon TPC, a cryogenic cooling system is required in combination with a purification and storage system.
The piping and instrumentation diagram of the XAMS setup is included in~\ref{APP_A}.
The cryogenic part of the system consists of double-walled stainless steel vessels.
The insulation volume between the vessels is continuously pumped out during normal operation, and pressures of \SI{3e-7}{mbar} are reached.
In addition, aluminum-coated Mylar foil is inserted in the insulation volume to shield from radiative heat transfer.
The cooling is provided by an Iwatani PDC08 pulse tube refrigerator (PTR), which gives an effective cooling power of \SI[separate-uncertainty = true]{22 \pm 2}{\watt} at \SI{-90}{\celsius}.
We apply the cooling to a copper cold finger, where the xenon condenses and droplets fall down into a funnel leading into the TPC.
A resistive heating band wrapped around the cold finger enables us to regulate the temperature.
The current to the heating band is controlled by a PID controller based on the temperature read by a Pt100 temperature sensor at the cold finger.
A cooling power failure may result in a rising pressure in the TPC.
A burst valve with a pressure limit of $\sim$~\SI{4.0}{bar} is connected to the inner volume to ensure no higher pressure can build.
We provide emergency cooling with a pressurized liquid nitrogen dewar, with the flow controlled by a solenoid valve that is switched by a pressure sensor.
In addition, text and email warning messages are automatically sent in case of abnormal behavior of the system.
The pressure sensor, the solenoid valve and the computer that sends the messages are powered by an uninterruptible power supply.
The required xenon purity level is achieved by continuous circulation through a high-temperature SAES MonoTorr PS3-MT3-R-2 getter with a maximum flow rate of 5~standard liters per minute.
We use a heat exchanger at the cryogenic part of the system to achieve this flow rate with only modest cooling power.
In section~\ref{SEC3_2}, we show that we achieved an impurity level of \SI[separate-uncertainty = true]{1.2 \pm 0.1}{ppb} (oxygen-equivalent).
We use an EMP MX-808ST-S diaphragm pump to establish the flow in the recirculation circuit.
The flow is controlled with a needle valve and measured with a thermal mass flow meter.
For the measurements described in this work, no buffer volumes were installed at the inlet or outlet of the pump, causing oscillatory behavior in the flow.
The presumed effect on the measurements is described in detail in section~\ref{SEC3_3_1}.
We recognize this as a design flaw, which we have since adjusted by installing gas bottles as buffer volumes in the system.
The liquid level in the TPC is monitored by a stainless steel cylindrical capacitive level meter, which is read out by a custom-programmed Arduino board.
The flow control of the needle valve is used to set the liquid level, as we noticed that the liquid level decreased as we increased the flow rate.
We assume that this effect is due to a changing thermal equilibrium in the heat exchanger, where a nonnegligible amount of liquid xenon is kept.
The total xenon content in the XAMS setup is roughly \SI{6}{kg}, most of which surrounds the PTFE structure of the TPC.
The time required to fill the TPC, limited by the maximum cooling power of the PTR, is roughly \SI{10}{hours}.
We perform recuperation by immersing gas bottles into liquid nitrogen dewars and allowing gas to deposit on the walls of the cylinder.
The time for a full recuperation is roughly \SI{8}{hours}.
\subsection{Trigger and DAQ} \label{SEC2_3}
We use a $^{22}$Na gamma source with an activity of \SI[separate-uncertainty = true]{368 \pm 11}{kBq} to perform our studies.
The source is mounted in a lead collimator (see Fig.~\ref{FIG1}) on the outside of the insulation vacuum vessel, with an opening angle of \SI{2.9}{\degree} such that the beam has a width of \SI{11}{mm} at the closest edge of the active volume.
The direction of the beam is horizontal, giving lateral irradiation of the TPC.
We change the z-position of the collimator by varying the height of the platform on which the collimator is mounted.
To reach the active volume, the gamma rays have to cross the walls of the inner and outer vessels, a thin layer of liquid xenon and the PTFE holding structure of the TPC, so that the total material traversed is \SI{6}{mm} of stainless steel, \SI{2}{mm} of liquid xenon and \SI{46}{mm} of PTFE, respectively.
$^{22}$Na decays by positron emission (branching ratio 90.4\%) or electron capture (branching ratio 9.6\%).
The decay is almost always followed by the emission of a \SI{1274}{keV} gamma ray from its $^{22}$Ne daughter.
In the case of positron emission, two additional back-to-back gamma rays of \SI{511}{keV} are produced from positron annihilation.
By using thallium-doped sodium iodide~(NaI(Tl)) as a coincidence detector that measures one of the \SI{511}{keV} gamma rays, the other \SI{511}{keV} gamma ray going directly toward the active volume is tagged.
This increases the fraction of events where all the energy is absorbed, since the number of events where gamma rays enter the active volume after Compton scattering on the material surrounding the detector is reduced.
The trigger is based on a threefold coincidence of the two PMTs in the TPC and the external NaI(Tl) detector.
If the trigger condition is satisfied, all three channels are digitized by a CAEN V1730D digitizer board.
This board has 8 channels that are digitized with a time resolution of \SI{2}{\nano s} and a voltage resolution of \num{14} bits, distributed over a dynamic range of \SI{2}{V}.
We choose an event window of \SI{163}{\micro s}: more than twice as long as the maximum drift time of \SI{60}{\micro s}.
We place the trigger position in the middle, such that an (accidental) trigger on an S2-signal will always contain the S1 in the same window.
A cut in post-processing ensures that there was a true coincidence with the S1 and the external NaI(Tl) (and not, for example, a coincidence with the S2-signal and an uncorrelated interaction in the NaI(Tl) crystal).
The simple coincidence means that all three channels must exceed the threshold \emph{at the same time}; no coincidence window was used.
The time offset between the two PMTs in the TPC is negligible, however, the start of the peak of the NaI(Tl) detector output was shown to occur \SI[separate-uncertainty = true]{22 \pm 6}{ns} later than that of the PMT signals.
The trigger condition was therefore satisfied only if both PMT signals were still above threshold at this time after the peak amplitude.
In the case of high energy recoils, the pulses are sufficiently large and this causes no problems.
However, for low energy recoils we observe a low trigger efficiency, which we identify in the comparison to Monte Carlo simulation in section~\ref{SEC3_3_2} at energies below~\SI{150}{keV}.
\section{Data reduction and results} \label{SEC3}
\subsection{Peak finding, clustering, identification} \label{SEC3_1}
The data of each event consist of the waveforms of the two PMTs with a duration of \SI{163}{\micro s} (Fig.~\ref{FIG2}).
The data processor, which is the same software developed for XENON1T \cite{pax_url}, analyzes the waveforms in each individual PMT channel by looking for significant excursions above the baseline.
These are called \emph{hits}.
In XENON100 and XENON1T, \emph{zero-length encoding} is used: the data consists of small chunks of data around a significant excursion from the baseline, so that the baseline is suppressed and the data volume is reduced \cite{zle}.
In order to be compatible with this structure, we apply a software zero-length encoding with a very low threshold.
The hitfinder threshold is dynamically determined as \num{4.5} times the standard deviation of the noise in the first 40 digitizer samples (\SI{80}{ns}) of the zero-length encoded chunk containing the hits.
The hits from both channels are then clustered into \emph{peaks} based on the gap between the edges of the hits: if this exceeds \SI{450}{\nano s}, the hits are clustered into separate peaks.
The area and the width of the peak are computed based on the summed gain-corrected waveform properties.
The width metric uses the range containing \num{50}\% of the peak area with \num{25}\% on either side.
The peak position is defined as the amplitude-weighted mean time of the samples in the peak.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{FIG2.pdf}
\caption{Typical gamma-ray-induced sum-signal of the two PMT channels, showing the S1 (green box) due to prompt scintillation and the S2 (red box), delayed by the drift time.
The inset shows a detailed view of the S1-signal and an exponential fit to the falling slope of the S1, with a decay time of \SI[separate-uncertainty = true]{22.8 \pm 0.1}{\nano s}.
Additional peaks are found (in the blue boxes), mostly happening after the S2.
Details of this kind of signal can be found in Fig.~\ref{FIG7}.
The data processing software finds the hits in each channel, clusters them into peaks, determines peak properties and classifies each peak based on the width and area.
}
\label{FIG2}
\end{center}
\end{figure}
As seen in Fig.~\ref{FIG2}, the main signals are the S1- (highly peaked signal at~\SI{84}{\micro s}) and the S2-signal (the broad signal at~\SI{108}{\micro s}).
After the S2, some peaks with low area and high width are found (shown in blue boxes).
These signals are due to secondary emission of electrons caused by photo-ionization of S2 UV photons and drifted up to produce an additional, much smaller S2.
These signals will be discussed in section~\ref{SEC4}.
All peaks are classified as either `S1', `S2' or `other' based on their width and area.
\subsection{S1 and S2 corrections} \label{SEC3_2}
After data processing, the following selection criteria are applied to the events.
First of all, only events with a single S1 and S2 are kept.
This cut rejects pileup events, double scatter events (which cause two S2-signals), or events where no S2 is generated (for instance, where the interaction occurs below the cathode mesh).
Events where the S1 is not in coincidence (difference of peak center position less than \SI{200}{ns}) with a signal in the NaI(Tl)-crystal are also cut.
In addition, the energy deposition in the NaI(Tl)-crystal is required to be less than \SI{600}{keV}, so that the triggers on \SI{1274}{keV} gamma ray are cut.
We impose no lower bound other than the trigger threshold on the NaI(Tl) energy, so that we keep events where the \SI{511}{keV} Compton scattered in the NaI(Tl)-crystal.
A summary of all the cuts and the number of events surviving each successive cut is given in table~\ref{TABLE1}.
The fraction of events surviving all cuts for the analysis presented here is 47.0\%.
Most events cut are due to multiple S1s or S2s.
\begin{table}[tp]
\caption{Data selection cuts and the number and fraction of events surviving each cut.
The cuts are applied successively.}
\label{TABLE1}\centering
\begin{tabular}{lcc}
\toprule%
{\bf Cut } & {\bf Events} & {\bf Fraction} \\ \toprule
No cuts & \num{215831} & 100.0\% \\
At least one S1 & \num{205417} & 95.2\% \\
Only one S1 & \num{166353} & 77.1\% \\
At least one S2 & \num{158586} & 73.5\%\\
Only one S2 & \num{115005} & 53.3\%\\
Coincidence S1 and NaI(Tl) & \num{105062} & 48.7\%\\
NaI(Tl) $<$ \SI{600}{keV} & \num{101381} & 47.0\%\\
\midrule
{\bf Total} & {\bf 101~$\!$381} & {\bf 47.0\%} \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width= 1.0 \linewidth]{FIG3A.pdf}
\caption{ }\label{FIG3A}
\end{subfigure}%
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width=1.0 \linewidth]{FIG3B.pdf}
\caption{ }\label{FIG3B}
\end{subfigure}
\caption{Density plot of the area of the sum-signal of the S1 {\bf (a)} and the S2 {\bf (b)} signal for different z-positions in the TPC, corresponding to different drift times.
The data shown in these figures were taken with the collimated source pointing at five different positions in the TPC.
The thick white lines are fits to the photo-peak.
For the S1, an overall increase is found due to LDE effects, a second degree polynomial fit gives the correction.
For the S2, an exponential fit provides a correction for loss of electrons during the drift time.
}
\label{FIG3}
\end{figure*}
For both the sum-signals of the S1- and the S2-signals, the area of the peak is proportional to the recoil energy.
However, the response to a mono-energetic energy deposition is not uniform throughout the TPC, requiring spatial corrections.
Since the XAMS TPC has only two PMTs, the position in the x,y-plane cannot be determined, but the z-coordinate is calculated based on the drift time that is defined by the difference of the weighted mean times of the S2 and the S1.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width= \linewidth]{FIG4A.pdf}
\caption{ }\label{FIG4A}
\end{subfigure}%
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width= \linewidth]{FIG4B.pdf}
\caption{ }\label{FIG4B}
\end{subfigure}
\caption{{\bf (a)}: Density plot of the area of the S1 and the area of the S2 in the same event for a \SI{511}{keV} gamma-ray source.
The ellipse shows the anti-correlation between the S1- and the S2-signal at the expected photo-peak.
A superior energy resolution is found by fitting the photo-peak and projecting along the short axis of the ellipse.
The shoulder at low energy is due to Compton-scatter events; the second peak at higher S2 area than the photopeak is discussed in the text.
{\bf (b)}: The spectra using the S1, S2 and the combined signal.
The energy resolution at \SI{511}{keV} improves from (\num[separate-uncertainty = true]{14.5\pm 0.2})\% and (\num[separate-uncertainty = true]{10.8\pm 0.4})\%, respectively, for the S1- and the S2-signal alone to (\num[separate-uncertainty = true]{5.8\pm 0.2})\% for the combined spectrum.
}\label{FIG4}
\end{figure*}
A z-dependent scale factor is applied to the S1-signal to eliminate differences in light detection efficiency (LDE).
The amount of light detected by the PMTs for different interaction positions depends on optical properties of the TPC, such as the reflection properties of the walls of the TPC, optical transparency of the meshes and reflection on the liquid-to-gas interface.
The secondary scintillation light of the S2-signal is always produced in the small region between the liquid-to-gas interface, so no z-correction for LDE has to be applied.
However, the number of electrons that create the S2-signal decreases with increasing drift time due to attachment of electrons to impurities in the xenon.
Assuming~$n_0$ electrons are produced at the interaction position, the number of electrons~$n_e$ left after a drift time~$t_d$ can be calculated with
\begin{equation}
n_e = n_0 \exp{\left(-t_d/ \tau_e\right)} {\rm ,}
\label{EQN1}
\end{equation}
where $\tau_e$ is the \emph{electron lifetime}, which is an indirect measure of the purity of the xenon.
Five datasets were taken with a different z-position of the collimator.
Fig.~\ref{FIG3} shows the area of the sum-signal of the S1- and the S2-signal for all datasets, each containing a prominent peak at high energy and a broad shoulder for lower energies.
The former is attributed to the full absorption peak (mostly due to photoelectric absorption, or multiple scatter events where the S2s are too close together to be separated), whereas the latter is due to Compton-scatter events.
For the S1, uncertainties on optical parameters limit the use of a detailed LDE model.
We therefore use a data-driven approach, modeling the correction function as a second degree polynomial.
We determine this function in two steps.
We first fit a Gaussian function to the photopeak for several slices in drift time, and then fit the photopeak position as a function of drift time with a second-degree polynomial.
This polynomial function, shown by the white line in Fig.~\ref{FIG3A}, provides the correction factor for the LDE.
The average value of the fit function, which gives the volume-averaged light yield for \SI{511}{keV} gamma rays, is \num[separate-uncertainty = true]{1.29 \pm 0.07 e3}~p.e., or (\num[separate-uncertainty = true]{2.5 \pm 0.1})~p.e./keV in this configuration.
This is equivalent to (\num[separate-uncertainty = true]{5.6 \pm 0.3})~p.e./keV at zero field and \SI{122}{keV} using data from NEST \cite{Szydagis:2011tk}, which is comparable to TPCs like XENON100 (4.3~p.e./keV) and LUX (8.8~p.e./keV) \cite{Aprile:2011dd,PhysRevLett.112.091303}.
An overall increase of LDE with drift time is found, since most of the scintillation light is detected by the bottom PMT.
For the S2, the correction function is expected to be an exponential (see equation~\ref{EQN1}).
The electron lifetime as determined from the fit is \SI[separate-uncertainty = true]{429 \pm 26}{\micro s}, similar to the average lifetime of \SI{514}{\micro s} during the year-long science run of XENON100 \cite{xe100_225}, and was achieved in only \SI{7}{days} of continuous purification with the high-temperature getter.
We observed that the electron lifetime rapidly increases in the first \SI{6}{days}, but levels off after this \cite{maria_master}.
For XAMS, this electron lifetime means that even at the maximum drift time, only 13\% of the S2-signal is lost.
Using the values in \cite{noble_gas_detectors}, this electron lifetime corresponds to an impurity level of \SI[separate-uncertainty = true]{1.2 \pm 0.1}{ppb} (oxygen-equivalent).
We kept the recirculation flow rate constant over the full duration of all measurements described here (one day).
\subsection{Energy calibration}
After the corrections for the S1- and the S2-signal have been applied, the absorbed energy can be determined.
Both signals provide a measurement of the deposited energy, since the area of the S1-signal is proportional to the number of photons produced in the interaction and the area of the S2-signal is proportional to the number of electrons produced.
The total energy deposited in these events is always identical: the ionization and scintillation signals are therefore anti-correlated.
In Fig.~\ref{FIG4A}, this anti-correlation is clearly visible as the ellipse with a downward slope.
The best energy resolution is achieved by using a projection along the short axis of the ellipse, which is known as the combined energy scale (CES) \cite{Shutt2007451, PhysRevB.76.014115}.
We use the same projection for all energies, which is a good approximation for energies greater than roughly \SI{100}{keV} \cite{Aprile:2011dd}.
In Fig.~\ref{FIG4B}, the spectra obtained from the S1, the S2 and the CES are shown.
The energy resolutions, as defined by $\sigma_E/E$ for a Gaussian fit, are (\num[separate-uncertainty = true]{14.5\pm 0.2})\%, (\num[separate-uncertainty = true]{10.8\pm 0.4})\% and (\num[separate-uncertainty = true]{5.8\pm 0.2})\%, respectively.
\subsubsection{High-S2 population} \label{SEC3_3_1}
In addition to the photopeak, a second peak at the same S1-area but larger S2-area was found, see Fig.~\ref{FIG4A}.
We also find this effect for the Compton-scatter events, and throughout all datasets.
The appearance of the high-S2 events is highly correlated in time, with a frequency of \SI[separate-uncertainty = true]{0.110 \pm 0.006}{Hz}, i.e., roughly a \SI{9}{s} period (see Fig.~\ref{FIG6A}).
The cause of a varying S2 size can be related to only few parameters.
Since the S1-signal is unaffected, the PMT gain, cathode voltage, DAQ problems or processing errors can be excluded.
Possible detector parameters changing the S2 size, but not the S1 size, are the xenon purity, the anode voltage, and the liquid level.
The anode voltage was not monitored by the slow control system, but the display showed a stability of better than \SI{1}{V}.
Unfortunately, we cannot correlate the detector parameters monitored by the slow control to the time behavior found in the high-S2 population appearance.
This is because the variables were read out only every two minutes; a decision that was taken because the readout of temperature sensors in the TPC caused noise in the PMT signals by electronic pick-up.
A plausible explanation found is a time-varying liquid level in the TPC.
The S2 size is highly dependent on this, so that only a small change in liquid level can still give significant effects.
Alternatively, there could be ripples on the liquid surface, appearing every \SI{9}{s}.
One of the mechanisms that could cause either a changed liquid level or ripples on the surface is related to the recirculation flow.
During the measurements, we observed that the gas flow rate in the recirculation system was constantly varying.
To investigate this effect further, we did a test where nitrogen gas was pumped through the system.
We observed a highly periodic behavior of the flow rate, with a period depending on recirculation speed.
Fig.~\ref{FIG5} shows the flow rate for a mass flow similar to the flow used during the measurements with liquid xenon.
The typical frequencies found in these tests are higher than the \SI[separate-uncertainty = true]{0.110 \pm 0.006}{Hz} found in the data, but it should be noted that the systems with liquid xenon and with nitrogen gas are not equivalent, and that the frequency found in the nitrogen gas tests depends on the pressure and the recirculation flow.
The reason for this oscillatory behavior is related to the absence of a buffer volume at the recirculation pump.
Buffer volumes were installed, and subsequent tests showed a significant increase in the stability of the flow rate.
Future measurements with liquid xenon will show if the effect is related to the instability of the flow rate.
\begin{figure}
\begin{center}
\includegraphics[width= \linewidth]{FIG5.pdf}
\caption{Power spectrum of the flow rate as measured in a test where nitrogen gas was pumped through the detector volume.
Shown in the inset is the flow as a function of time for the first \SI{30}{seconds} of this measurement.
A clear peak at a frequency of \SI[separate-uncertainty = true]{0.228 \pm 0.004}{Hz} is visible, along with several harmonics of this frequency.}
\label{FIG5}
\end{center}
\end{figure}
As illustrated in Fig.~\ref{FIG6A}, we can use a time cut to remove a large fraction of the events with a large S2-signal.
Whenever more than six events with an S2 size larger than \num{150000}~p.e.\ are found within one second, the events from one second before to one second after this bin are cut.
This removes \num{41.1}\% of all events passing previous cuts.
\subsubsection{Comparison to simulation} \label{SEC3_3_2}
The resulting spectrum was compared to a GEANT4 \cite{geant4} Monte Carlo simulation, where the energy deposition was registered when there is a simultaneous energy deposition in the NaI(Tl) crystal and the liquid xenon active volume.
The result was then smeared with an energy resolution function according to
\begin{equation}
\frac{\sigma_E}{E} = \frac{a}{\sqrt{E}} {\rm ,}
\end{equation}
where $a$ is fixed by the requirement that $\sigma_E/E = \num{5.8}\%$ at \SI{511}{keV}.
The comparison is shown in Fig.~\ref{FIG6B}, where the green points are from data and the blue line is from simulation.
The data points are scaled to the total rate observed before any cuts of \SI{26.6}{Hz}, which agrees well with the rate of \SI{25.9}{Hz} from simulation.
The contribution at S2 sizes larger than \SI{600}{keV} is still visible.
At energies below $\sim$\SI{150}{keV}, the simulation predicts a higher rate than observed in measurement.
This difference is due to a timing offset between the NaI(Tl) and the S1-signal, which causes a trigger on the falling edge of the S1 instead of on its peak amplitude and deteriorates the trigger efficiency for low energy recoils, as described in section~\ref{SEC2_3}.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width= \linewidth]{FIG6A.pdf}
\caption{ }\label{FIG6A}
\end{subfigure}%
\begin{subfigure}[b]{.5\linewidth}
\centering \includegraphics[width= \linewidth]{FIG6B.pdf}
\caption{ }\label{FIG6B}
\end{subfigure}
\caption{{\bf (a)}: The rate of events with an S2 larger than \num{150000}~p.e.\ (blue) and the rate of other events (green) for the first 60 seconds of the dataset.
A clear time-correlation is visible.
The shaded regions show the times that are cut.
{\bf (b)}: The CES spectrum from data after applying the time cut (green points) compared to a smeared spectrum from a Monte Carlo simulation (blue line).
The data points are scaled to reflect the rate before any cuts.
The mismatch between simulation and data at low energy is due to a decreased trigger efficiency, as described in section~\ref{SEC2_3}.
At high energy, this is due to the partly cut high-S2 population described in section~\ref{SEC3_3_1}.
}\label{FIG6}
\end{figure*}
\section{Single-electron S2-signals} \label{SEC4}
A distinct signal that is found in dual-phase xenon TPCs is that of S2-signals produced by single electrons \cite{Edwards200854,Santos2011,0954-3899-41-3-035201}.
The scintillation light of xenon, at \SI{178}{\nano m}, can liberate electrons in the TPC.
In general, the electrons come from impurities in the xenon that have a low ionization potential, such as O$^{-}$~ions, or from exposed metallic surfaces.
If these electrons are somewhere in the active volume, they will in turn drift upward and create very small S2-signals.
Since the main S2-signal is the dominant source of UV photons in the TPC, it causes the large majority of single-electron peaks.
Fig.~\ref{FIG7} shows an example of a single-electron signal found in the data.
As described in section~\ref{SEC3_1}, the waveform is cut into small sections analogous to zero-length encoded data, so that the hitfinder threshold is dynamically updated based on the local noise level.
The blue and green parts of the waveform show the hits that are found, when a threshold of \num{4.5} times the standard deviation of the baseline noise is crossed (indicated by the dashed lines).
The width of the signal is around \SI{1}{\micro s}, comparable to normal S2-signals.
\begin{figure}[h]
\begin{center}
\includegraphics[width= \linewidth]{FIG7.pdf}
\caption{
Example of a single-electron S2-signal, shown for both PMTs separately.
The data is cut into small pieces based on the crossing of a very low threshold.
This is indicated by the dark gray part of the waveforms.
The noise level is determined on the first 40~samples of these pieces, yielding a hitfinder threshold of \num{4.5} times the standard deviation of the noise (indicated by the dashed lines).
When the waveform crossed this threshold, a hit is found, indicated by the blue and green waveforms.
}
\label{FIG7}
\end{center}
\end{figure}
Fig.~\ref{FIG8} shows the time distribution of the peak position relative to the position of the S2 for all candidate single-electron S2-signals, namely all peaks that are not classified as S1 or S2 and have a coincidence of both PMTs.
A large fraction of the peaks occurs between \SIrange{0}{60}{\micro s} (as defined by the maximum drift time).
We observe a clear increase at \SI{60}{\micro s}, which is due to the S2 light impinging on the cathode mesh, where electrons are liberated relatively easily due to the low ionization potential of the iron in the stainless steel.
Before the S2 ($\Delta t <$~\num{0}), as well as after the full drift length ($\Delta t >$~\SI{60}{\micro s}), there is a nonzero contribution, which is partly due to noise hits clustered into peaks, but partly shows the same properties as the single-electron signals in the drift region.
These peaks can be caused by a delayed extraction phenomenon, as discussed in \cite{Santos2011} and \cite{0954-3899-41-3-035201}.
Single-electron S2s can be effectively used as `calibration sources': the detector response to just one electron can be probed in this way.
This enables the direct determination of various parameters, such as the secondary scintillation gain.
In addition, these signals can be used for a PMT gain calibration, since they consist of single-photoelectron hits.
\subsection{Gain calibration}
The PMT gain is defined as the average number of electrons at the anode responding to one electron emerging from the photocathode.
Often PMT calibrations are done using external pulsed light sources.
Although such calibration provides a direct and usually accurate gain calibration, it requires an interruption during dark matter data taking.
In addition, a dedicated LED calibration system and calibration measurements are necessary.
Finally, the LED calibration is usually performed at a higher wavelength than the xenon scintillation light of \SI{178}{nm}, because it is technically challenging to guide UV light through an optical fiber.
This makes it impossible to probe effects like double-photoelectron emission, which occurs only at short wavelengths \cite{1748-0221-10-09-P09010}.
In this section, we discuss a method to use single-electron peaks for PMT gain calibration.
These consist of well-separated single-photon hits and are abundant in all data, so they can be used to measure the PMT gain continuously.
The LUX collaboration already uses single-electron signals as part of their gain calibration, which operates on different principles~\cite{new_lux}.
\subsubsection{Hit data selection}
Single-electron S2s typically have the same width as ordinary S2-signals (about \SI{1}{\micro s} wide), but consist of a small number of photoelectrons.
For example, for XENON100, these signals consists of roughly 20 photoelectrons \cite{0954-3899-41-3-035201}.
\begin{figure}[h]
\begin{center}
\includegraphics[width= \linewidth]{FIG8.pdf}
\caption{Distribution of peak positions relative to the position of the S2.
A large fraction of the peaks occurs between \SIrange{0}{61}{\micro s}, which is expected for single-electron signals that are caused by the S2 light.
The large peak at \SI{60}{\micro s} is due to the cathode mesh at this drift length: electrons are easily liberated from the stainless steel.}
\label{FIG8}
\end{center}
\end{figure}
Since the hits in single-electron S2s are spread out over a relatively long duration, the PMT hits of the detected photons can be found individually (see Fig.~\ref{FIG7}).
This means that the single-electron S2s provide a source of single photoelectron hits, which can be used for an \emph{in-situ} gain calibration.
We apply cuts on the event, peak and hit level to select proper single-photon hits in proper single-electron signals.
Events are selected by the same criteria as in section~\ref{SEC3}.
For the peaks, defined as clusters of hits, we introduce the following cuts.
Since single-electron S2s are primarily caused by S2 photons, only peaks that occur within \SIrange{5}{60}{\micro s} after the S2 are selected.
The lower bound ensures no fragments from accidentally split S2s are selected, and the upper bound cuts peaks beyond the maximum drift time.
Both PMTs are required to contribute to the peak, to suppress peaks consisting of noise and dark current hits.
In order to reduce contamination from common-mode noise clustering, we apply a cut on the average area of the hits in a peak: since noise hits on average have a smaller area than particle-induced PMT hits, a cluster of noise hits will have a low average area.
Finally, at the hit level, we do not consider hits with an extremely small width, indicative of noise hits rather than real PMT hits.
The width parameter used here is the sum absolute deviation~(SAD), given by
\begin{equation}
\rm{SAD} = \sum_i \frac{A_i}{A_{tot}} \left| t_i-t_c \right| {\rm ,}
\end{equation}
where $i$ runs over all samples in the hit, $t_c$ is the amplitude-weighted mean time of the hit, and $A$ denotes the area.
This parameter takes continuous values greater than or equal to zero, which has the advantage of discriminating different widths even if this is at the same order as the sampling time.
Typical values for a single-photoelectron hit are about \SI{3.5}{ns} for the XAMS PMTs.
We cut hits with an SAD less than \SI{0.5}{ns}, which mostly consist of hits that are just one sample wide (such that $\rm{SAD} = 0$ exactly) or where the hit is two samples wide but the area is dominated by just one sample.
With the above selection of hits we proceed to calibrate the gain of each PMT.
For gain calibration, the parameter of interest is the \emph{area} of the hits (given by $\int V dt$), which is proportional to the number of electrons~$n_e$ at the PMT output according to
\begin{equation}
n_e = Q/q_e = \frac{1}{q_e} \int I dt = \frac{1}{q_e R} \int V dt {\rm ,}
\end{equation}
where $Q$ is the charge at anode, $q_e$ is the charge of the electron and where $R$ denotes the termination resistance of the digitizer (\SI{50}{\ohm}).
Because of this relation, the area of a hit can be expressed as number of electrons equivalent area.
If the hit results from one photoelectron, the average number of electrons at the output is simply $\mean{n_e} = G$, and the gain~$G$ can be computed.
\subsubsection{Acceptance correction} \label{SEC4_1_2}
The hitfinding algorithm preferentially detects high-area hits, since these are more likely to exceed the hitfinding amplitude threshold.
Fig.~\ref{FIG9} shows the correlation between the amplitude, measured in units of $\sigma_{\rm noise}$, and the area of the hit.
The distribution is sharply cut at \num{4.5}$\sigma_{\rm noise}$, the hitfinder threshold.
This was chosen to limit the contribution of noise hits, which are visible in the bottom left corner of Fig.~\ref{FIG9}.
\begin{figure}
\begin{center}
\includegraphics[width = \linewidth ]{FIG9.pdf}
\caption{The distribution of selected single-electron hits in amplitude and area.
There is a clear correlation between the hit area and amplitude.
The distribution is cut at \num{4.5}$\sigma_{\rm noise}$ as defined by the hitfinding threshold.
A correction for hits below this threshold is calculated and used in the analysis.
At small area and low amplitude, a second band coming from noise hits can be seen (indicated by the black circle).
}
\label{FIG9}
\end{center}
\end{figure}
To correct for this loss of hits below the threshold, we must estimate the acceptance $\epsilon$ of the hitfinder, i.e.\ the fraction of photon hits found by the hitfinder, as a function of the hit area.
This function can be estimated by studying the amplitude distribution for a sample of hits with similar area (equal up to a difference of \num{0.1e6}~electrons).
This distribution is cut at the hitfinder threshold, but since it is well-described by a Gaussian distribution above the threshold, we will assume that it follows a Gaussian function also below this threshold.
By evaluating the fraction of the area under the fitted distribution below the threshold, the acceptance can be calculated, as illustrated in Fig.~\ref{FIG10}.
\begin{figure}[h]
\begin{center}
\includegraphics[width= \linewidth]{FIG10.pdf}
\caption{Example of a fit used to determine the acceptance for an area slice.
The fraction of the hits that is not found is inferred from the area under the fit in the region of the function that extends below the threshold. In this example, this fraction is \num{19.5}\%, so the acceptance is \num{80.5}\%.
The bottom panel shows the deviation from this fit in units of the error on the points.
The dashed and solid lines indicate a deviation of $1\sigma$ and $2 \sigma$, respectively.
Up to a threshold of \num{9}$\sigma_{\rm noise}$, the distribution is well described by the fit.
}
\label{FIG10}
\end{center}
\end{figure}
The distribution of hits and Gaussian fits are shown for all area slices in Fig.~\ref{FIG11}.
For low-area hits, only a tail of the Gaussian distribution exceeds the hitfinding threshold of \num{4.5}$\sigma_{\rm noise}$, making it difficult to fit the distribution.
We instead infer the parameters $\mu$ and $\sigma$ by extrapolation.
Since the shape of PMT hits is to a good approximation independent of the area, the mean~$\mu$ is extended linearly to zero \cite{erik_master}.
We assume that the standard deviation~$\sigma$ is constant at low area, since this should be dominated by baseline noise on the highest bin and is therefore independent of the hit area.
\begin{figure}[h]
\begin{center}
\includegraphics[width= \linewidth]{FIG11.pdf}
\caption{
Stacked hit amplitude histograms for each area slice (blue points), together with Gaussian fits (blue lines).
The data as well as the fits are scaled such that the maximum amplitude of all distributions is the same.
Red dashed lines indicate the mean and standard deviations of the fitted Gaussians.
For low area slices, the amplitude distribution is estimated by extrapolating the mean and standard deviations found in higher-area slices as described in the text.}
\label{FIG11}
\end{center}
\end{figure}
With the amplitude distribution specified by $\mu(A)$ and $\sigma(A)$, the acceptance for every area and for different hitfinding thresholds can be computed.
For a hitfinding threshold of $n_{tr} \sigma$, the acceptance~$\epsilon$ as a function of peak area~$A$ is
\begin{equation}
\epsilon(A) = \int_{n_{tr} \sigma}^{\infty} g\left( x ; \mu(A), \sigma(A) \right) dx
\label{EQN5}
\end{equation}
where $g$ is a normalized Gaussian distribution and $x$ denotes the amplitude.
To correct the area spectrum, it is divided by~$\epsilon(A)$.
Although a hitfinder threshold that is as low as possible is desired for determining the acceptance, it is not necessarily ideal for determining the gain.
This is because low-amplitude noise hits are too dominant in the area spectrum, so that the gain will be underestimated.
We therefore use a higher amplitude threshold to remove any possible bias due to noise hits, which we compensate by a corresponding change in the acceptance function.
In Fig.~\ref{FIG12}, the uncorrected and corrected area spectrum for a hitfinding threshold of 6.5$\sigma_{\rm noise}$ is shown, together with the acceptance function for this threshold.
A clear peak is visible, which is fit in the area around the peak to determine the gain.
The fit is limited to a part around the maximum; at low area, the noise contribution becomes dominant, while at high area the contribution from two-photoelectron hits cannot be excluded.
Similar features are found in other PMT calibrations, such as in~\cite{1502.01000}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{FIG12.pdf}
\caption{
The area distribution for the hits with (dark blue points, left scale) and without (light blue points, left scale) correction from the acceptance function (red dashed line, right scale) as determined from the model described in section~\ref{SEC4_1_2}.
A Gaussian fit (blue solid line) is used to determine the gain.
At low area, noise hits give a large contribution to the corrected spectrum, since they are highly amplified by the acceptance correction.
Points below \num{0.3e6} electrons area, where the acceptance drops below \num{0.1}\%, are omitted from the plot.
}
\label{FIG12}
\end{center}
\end{figure}
In Fig.~\ref{FIG13}, the determined gain is plotted as a function of the hitfinding threshold.
For low thresholds, the noise contribution becomes too pronounced to properly fit the spectrum.
For higher thresholds, the gain that is determined converges to a final value, which we infer to be the true gain of the PMT.
We allow a range around this value from uncertainty on the convergence of the final points, which we estimate to be 5\% for PMT~1 and 10\% for PMT~2.
The PMT gains found in this analysis were $(1.30 \pm 0.07) \times 10^6$ for PMT~1 and $(0.71 \pm 0.07) \times 10^6$ for PMT~2; both close to the typical gain of \num{1.0e6} quoted by Hamamatsu for this type of PMT \cite{hamamatsu}.
The error bars in Fig.~\ref{FIG13} originate from systematic errors on the acceptance function, which we calculate by perturbing the fit parameters $\mu(A)$ and $\sigma(A)$ in equation~\ref{EQN5}.
\begin{figure}[h]
\begin{center}
\includegraphics[width= \linewidth]{FIG13.pdf}
\caption{The gain for different hitfinding thresholds (in units of standard deviation of the noise), for the two PMTs in XAMS.
As the hitfinding thresholds rises far enough above the low-amplitude noise, the gain that is determined levels off to one value, which is the gain of the PMT.
The error bars are calculated by allowing varying values for $\mu(A)$ and $\sigma(A)$ for the acceptance function.
At high thresholds, the result of the fit is very sensitive to variations in the acceptance function, causing the large error bars.
The systematical errors are estimated to be 5\% for PMT~1 and 10\% for PMT~2, as shown with the bands, based on the convergence of the final points.
}
\label{FIG13}
\end{center}
\end{figure}
\subsection{Discussion}
The method used in this work relies on modeling the PMT hits.
In particular, the amplitude of hits of a given area is assumed to be normally distributed.
For large-area hits, this assumption can be verified since most hits are above the hitfinding threshold, and the distribution follows a Gaussian distribution to a high degree.
For smaller areas, a significant part of the distribution is inaccessible and needs to be inferred from the visible part of the distribution (as in Fig.~\ref{FIG10}).
Moreover, the parameters~$\mu$ and~$\sigma$ of the distribution are extrapolated from larger areas where fits can be made (Fig.~\ref{FIG11}).
The validity of the extrapolation can break down at small area, although this will not affect the gain determination if the approximation is valid sufficiently far below the average single-photoelectron hit area.
In the case of low PMT gains, the separation of noise and true PMT hits becomes a serious issue.
This is the case for PMT~2 in our analysis, where at high thresholds the errors increase and we eventually fail to fit the distribution because of low statistics.
Moreover, we cannot confirm if convergence is reached before this effect starts to dominate.
We therefore estimate the systematic errors to be higher for PMT~2 than for PMT~1.
It should be noted that these limitations becomes less important if the PMT gain is higher, so that the PMT hits are more separated from the noise.
A PMT calibration requires a source of single photoelectron hits.
For single-electron S2s, a few photoelectron signals are seen in the PMT channels over a time window of typically \SI{1}{\micro s}.
There is a finite probability of having two or more PMT signals clustered together into one hit, so that there is a contribution of two-photoelectron hits in the data.
The importance of this effect could be different for other TPCs, as it depends on several parameters such as the transient time spread of the PMTs, the sampling time of the ADCs, the width of the S2 and the anode voltage.
These effects will thus need to be studied further if this method is to be used for other TPCs.
Compared to the normal PMT calibration with LED pulsed light, there are some definite advantages to using single-electron S2s.
One of these is they are usually readily available and easily identified in ordinary (energy) calibration or dark matter data; no extra dedicated calibration runs are required.
This means that the drift of PMT gains can be monitored on timescales far shorter than with ordinary PMT calibration runs.
A second advantage is that the response to the scintillation light is directly probed.
The scintillation light of xenon has a wavelength of \SI{178}{nm}, but since this is technically challenging to provide for a calibration, higher wavelengths are used.
For example, XENON100 uses an LED at \SI{470}{nm} \cite{Akerib:2012ys}.
The method described here makes it possible to study, for example, the possibility of two-photoelectron emission due to one scintillation photon at the photocathode.
\section{Summary} \label{SEC5}
In this work, the first data of the XAMS TPC were presented.
An energy resolution of (\num[separate-uncertainty = true]{5.8 \pm 0.2})\% was achieved at \SI{511}{keV}.
The electron lifetime was found to be \SI[separate-uncertainty = true]{429 \pm 26}{\micro s}, which is sufficient for this TPC, after only 7 days of purification.
An average light yield of (\num[separate-uncertainty = true]{5.6 \pm 0.3})~photoelectrons/\si{keV} (recalculated to zero field and \SI{122}{keV}) was found, which is comparable to TPCs like XENON100 and LUX.
A new PMT calibration method based on single-electron S2-signals was explored.
Since single-electron S2-signals are very abundant in dual-phase xenon TPCs, this method of PMT calibration can give an important independent cross-check of the normal PMT calibration, with the advantage of superior time resolution and no need for dedicated PMT calibration data.
\section{Acknowledgments}
This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO).
We gratefully acknowledge the technical support from to the mechanical, electrical and computing departments at Nikhef.
\section{References}
|
1,108,101,565,497 | arxiv | \section{Introduction}
\label{sec:intro}
Our understanding of exoplanet atmospheres has improved dramatically over the last two decades. A wide range of spectroscopic observations have been used extensively to constrain atmospheric chemical compositions, temperature structures, clouds, hazes and other atmospheric properties \citep{Crossfield2015, Kreidberg2018, Madhu2019}. The method of transmission spectroscopy \citep{Charbonneau2002} has proven to be particularly effective in characterising atmospheric chemical compositions as well as clouds and hazes. The Hubble Space Telescope (HST) has been a key driver in this direction, starting with the detection of atomic species in the optical range using the Space Telescope Imaging Spectrograph (STIS) \citep{Charbonneau2002, VidalMadjar2003} and later the detections of H$_2$O with infrared observations from the Wide Field Camera 3 (WFC3) \citep{Deming2013, Huitson2013, Mccullough2014,Sing2016}. Additionally, advances in atmospheric modelling and retrievals led to the ability to place important constraints on atmospheric abundances and cloud/haze parameters for a number of giant exoplanets using such HST spectra \citep[e.g.][]{Madhu2014, Kreidberg2014,Barstow2017,Pinhas2019,Welbanks2019b}.
With the advent of the James Webb Space Telescope (JWST), the field will experience a generational shift in observational capabilities, starting from its very first observations \citep{Beichman2014, Stevenson2016, Batalha2017b, Bean2018, Kalirai2018, Sarkar2020}. Promising a virtually complete coverage of the near-infrared with better sensitivity than HST, JWST has much to offer. Notably, however, JWST's coverage does not include the full optical part of the spectrum. This makes it difficult to rely on optical observations to constrain the possible impact of clouds/hazes on the transmission spectrum, as has been pursued for hot Jupiters with HST \citep{ Sing2016, Barstow2017, Pinhas2019, Welbanks2019a}.
One of the most exciting applications for JWST is in the characterisation of temperate mini-Neptunes, planets which have no solar system equivalent but are the most common type of exoplanet, along with super-Earths \citep{Fulton2017, Hardegree-Ullman2020}. Here we refer to mini-Neptunes as planets smaller than Neptune with volatile-rich interiors and H$_2$-rich atmospheres. Nominally, such planets could have radii in the range $\sim$1.6 - 4 R$_\oplus$, considering that most planets with radii larger than 1.6 R$_\oplus$ are unlikely to be rocky \citep{Rogers2015,Lozovsky2018}. One such example is K2-18~b \citep{Montet2015, Cloutier2017}, which has been shown with HST observations to contain H$_2$O in its hydrogen-dominated atmosphere \citep{Tsiaras2019, Benneke2019} and among the possible surface conditions is the existence of a habitable liquid ocean layer \citep{Madhusudhan2020}. For K2-18~b and many other such planets, more precise abundance estimates of key molecules, obtained from JWST observations, will lead to better constraints on surface conditions \citep{Hu2021a, Madhusudhan2021, Yu2021}. Additionally, the greater spectral coverage will help determine if CH$_4$ and NH$_3$, which are expected to be present from equilibrium calculations and so far have been undetected \citep{Benneke2019, Madhusudhan2020}, are indeed present and at what abundances. K2-18~b in particular is set to be observed as part of JWST Cycle 1 GO programs 2372 (PI: Renyu Hu) and 2722 (PI: Nikku Madhusudhan) for a total of 9 transits, the most of any mini-Neptune.
Given the multitude of instruments appropriate for transit spectroscopy aboard JWST, much effort has already been made to understand its observational capabilities. Several general studies \citep{Greene2016, Howe2017, Batalha2017a} considering a range of planets and employing various approaches to noise modelling and parameter estimation found that the $\sim$1-3$\mu$m range stands to be the most informative in constraining the atmosphere's metallicity. However, \citet{Guzman-Mesa2020}, using a random forest-based approach, find that for a relatively hot Neptune-like planet, the $\sim$3-5$\mu$m wavelength range is optimal instead, in the context of establishing atmospheric abundances and the C/O ratio.
In addition to the above, there have been numerous studies specifically examining how JWST observations can be used to characterise super-Earths \citep[e.g.][]{Deming2009, Benneke2012, Molliere2017, Morley2017, Tremblay2020, Gialluca2021}. While JWST's generational improvements in wavelength coverage and resolution will be boons to super-Earth observation campaigns \citep{Tremblay2020, Gialluca2021}, they remain a sizeable undertaking, requiring many transits to be observed for their atmosphere to become detectable \citep{Barstow2016}. On the other hand, Mini-Neptunes, even temperate ones lying in or close to their host star's habitable zone, are significantly more amenable to spectroscopic characterisation than super-Earths, aided in no small part by their more extended, hydrogen-dominated atmospheres.
Observing mini-Neptune atmospheres, however, is not without risks. Several previous attempts to constrain a mini-Neptune's atmospheric composition have been foiled by the possible presence of high-altitude clouds, most notably GJ~1214~b \citep{Bean2011,Desert2011,Kreidberg2014}. Depending on their altitude clouds can mask spectral signatures from molecules in the atmosphere, preventing their detection. Even if the molecular absorption features are not completely obscured by clouds, a lack of optical measurements could lead to potential degeneracies between any nominal cloud effects and the chemical abundances.
Given the limited observing time available with JWST, the threat of clouds may scuttle attempts to fully explore the properties of temperate mini-Neptunes with JWST. Such a problem also exists for larger and hotter planets, and recent studies have explored the possibility of constraining atmospheric properties of warm sub-Neptunes/Neptunes and hot Jupiters in the presence of clouds/hazes \citep[][]{Schlawin2018, Kawashima2019, Mai2019}. However, the problem is arguably more acute for temperate ($\lesssim$500~K) mini-Neptunes on which we focus here, whose lower temperatures decrease the extent of their atmospheres, resulting in weaker spectral signatures.
In this work we present an optimal retrieval-driven approach to take advantage of JWST's observing capabilities to constrain atmospheric abundances of temperate mini-Neptunes in the presence of high-altitude clouds. We focus on temperate mini-Neptunes, as they are prime targets for JWST characterisation, with our findings also being relevant to larger and hotter planets offering an even better Signal-to-Noise Ratio (SNR). Using K2-18~b and TOI-732~c/LTT~3780~c as nominal examples, we generate cloudy and cloud-free synthetic transmission spectra and carry out atmospheric retrievals to determine how well atmospheric parameters can be constrained for different instrument configurations. We show that, thanks to the precision, spectral resolution and large wavelength coverage available with JWST, atmospheric retrievals can overcome high-altitude clouds and obtain precise abundance constraints for the dominant oxygen-, carbon- and nitrogen-carrying molecules in temperate mini-Neptunes.
In Section \ref{sec:methods} we present our retrieval methodology, describing how we generate synthetic data as well as the retrieval approach used to analyse the data. Our results are presented in section \ref{sec:results}. We begin by considering a reference cloud-free atmosphere for the case of K2-18~b in section \ref{sec:cloud_free_reference}. We then investigate the more difficult cloudy case, again for K2-18~b in section \ref{sec:wavelength_coverage}, examining the performance of retrievals on observations combining one, two and three instruments with a single transit per instrument. Having established what abundance constraints are possible for K2-18~b, we then consider a more favourable target, TOI-732~c, as a case study in section \ref{sec:case_study}. We first establish how single-transit single-instrument observations of a cloudy atmosphere perform for this more favourable case, comparing our findings to those obtained for K2-18~b. We then benchmark the observing capabilities of JWST with a three-instrument configuration spanning the complete 1-5~$\mu$m range, seeking the highest altitude cloud deck with which molecular abundances can still be constrained for two atmospheric composition scenarios. Finally in section \ref{sec:conclusion}, we summarise and discuss our results about what can be expected from JWST observations of temperate, cloudy mini-Neptunes.
\section{Methods}
\label{sec:methods}
Our goal is to determine how JWST observations can be used to successfully characterise mini-Neptune atmospheres despite the presence of high-altitude clouds. We explore a range of instrument configurations and atmospheric properties, seeking to establish what atmospheric parameter constraints can be obtained in each case. In this section, we present our approach.
\subsection{Case Studies and Canonical Atmospheric Model}
\label{sec:canonical_model}
\begin{table}
\caption{ Planetary and stellar properties for the two case studies.}
\centering
\begin{tabular}{c|c|c}
Properties & K2-18~b & TOI-732~c \\
\hline
$R_{\mathrm{P}}$ / R$_{\earth}$ & 2.61 & 2.42 \\
$M_{\mathrm{P}}$ / M$_{\earth}$ & 8.63 & 6.29 \\
$g / \mathrm{m}\mathrm{s}^{-2}$ & 12.4 & 10.5 \\
$T_{\mathrm{eq}}$ / K & 282 & 363 \\
$(\frac{R_\mathrm{P}}{R_\mathrm{\**}})^2$ (\%) & 0.289 & 0.337 \\
\multicolumn{3}{c}{\emph{Host Stars}}\\
$T_{\mathrm{eff}}$ / K & 3503 & 3360 \\
$R_{\mathrm{\**}}$ / R$_{\mathrm{\sun}}$ & 0.445 & 0.382 \\
$M_{\mathrm{\**}}$ / M$_{\mathrm{\sun}}$ & 0.495 & 0.379 \\
$J$ mag & 9.8 & 9.0 \\
\hline
\end{tabular}
\newline
\footnotesize{Values for K2-18~b are from \citet{Cloutier2019} and \citet{Benneke2019}, while values for TOI-732~c are from \citet{Nowak2020}. $T_{\mathrm{eq}}$ is calculated assuming zero Bond albedo and full day-night redistribution.}
\label{tab:planet_properties}
\end{table}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/K2-18b_contributions.pdf}
\caption{Contributions to a model transmission spectrum of K2-18~b corresponding to the canonical model described in section \ref{sec:canonical_model}. Each coloured line is the transmission spectrum produced exclusively by absorption from H$_2$O (blue), CH$_4$ (red) or NH$_3$ (pink). The grey dashed line shows the featureless spectrum with only the cloud deck present at 3~mbar, which sets the spectral baseline in the cloudy canonical model. The resulting transmission spectrum, with contributions from all chemical species and the cloud deck is shown in black. In all cases, we also include the effects of H$_2$-H$_2$ and H$_2$-He collision-induced absorption.}
\label{fig:contribution_plot}
\end{figure*}
For our investigation, we consider two mini-Neptune planets as nominal examples, whose properties are summarised in table \ref{tab:planet_properties}. The first is K2-18~b \citep{Montet2015,Cloutier2017}, which we use for the majority of this work. Orbiting a J~=~9.8~mag M dwarf, K2-18~b has a mass of 8.63~$\pm$~1.35~M$_{\earth}$ \citep{Cloutier2019} and multiple radii reported in literature, with a value 2.51$^{+0.13}_{-0.18}$~R$_{\earth}$ obtained from K2 observations \citep{Hardegree-Ullman2020} and 2.610$\pm 0.087$~R$_{\earth}$ obtained using K2 and Spitzer observations \citep{Benneke2019}. Both are consistent to within 1-$\sigma$. For this work, we use a radius of 2.610$\pm 0.087$~R$_{\earth}$ \citep{Benneke2019}, for the sake of consistency with previous studies \citep[e.g.,][]{Madhusudhan2020}. We choose K2-18~b as a conservative case, with a low zero-albedo equilibrium temperature of 282~K and orbiting a star of intermediate brightness compared to more recently-discovered TOI targets such as TOI-175~d \citep{Kostov2019}, TOI-270~c and d \citep{Gunther2019, VanEylen2021}, TOI-732~c/LTT~3780~c \citep{Nowak2020, Cloutier2020} and TOI-776~b and c \citep{Luque2021}.
Our second case is the more recently-discovered TOI-732~c/LTT~3780~c \citep{Nowak2020, Cloutier2020}, with a mass of 6.29${^{+0.63}_{-0.61}}$~M$_{\earth}$ and radius of 2.42${^{+0.10}_{-0.10}}$~R$_{\earth}$, orbiting a relatively brighter J~=~9.0~mag host star \citep{Nowak2020}. It is also slightly hotter than K2-18~b, with an equilibrium temperature of 363~K. We choose TOI-732~c for our second case study as a more optimistic scenario, representative of the above mentioned population of mini-Neptunes that are more amenable to transmission spectroscopy, which JWST will likely be observing extensively over its lifetime.
For both planets, we consider a canonical model of a 1D plane-parallel atmosphere in hydrostatic equilibrium, dominated by H$_2$ and He in solar ratio. We additionally include H$_2$O, CH$_4$ and NH$_3$, the dominant O- C- and N-carrying molecules expected in temperate H$_2$-rich atmospheres in thermochemical equilibrium \citep{Burrows1999, Lodders2002, Madhu2011, Moses2013}. We use a nominal atmospheric composition arising from 10$\times$ solar elemental abundances \citep{Asplund2009} at chemical equilibrium. This corresponds to volume mixing ratios of $10^{-2}$, $5 \times 10^{-3}$ and $10^{-3}$ for H$_2$O, CH$_4$ and NH$_3$, respectively. We opt for 10$\times$~solar elemental abundances so that the H$_2$O mixing ratio matches the median H$_2$O estimate of $\sim10^{-2}$ obtained from HST observations of K2-18~b \citep{Benneke2019, Madhusudhan2020}. While CH$_4$ and NH$_3$ were not detected in K2-18~b, we use chemical equilibrium abundances rather than assuming any kind of depletion due to disequilibrium for this case study \citep[e.g.][]{Yu2021, Hu2021a, Hu2021b, Madhu2011}.
We assume an isothermal terminator atmospheric profile, in keeping with self-consistent P-T profile calculations in literature \citep{Benneke2019, Piette2020} finding that in the pressure range probed by transmission spectroscopy in K2-18~b the temperature structure is effectively isothermal. We set the terminator temperature profile to an isotherm at 300~K for K2-18~b, at the lower end of the temperate mini-Neptune range. Similarly to our choice of atmospheric composition, this choice is motivated by both the actual properties of K2-18~b (T$_{\mathrm{eq}}$~=~282~K), as well as seeking the most difficult case for retrievals, as a low temperature suppresses the atmospheric scale height. For TOI-732~c (T$_{\mathrm{eq}}$~=~363~K), we opt for an isotherm temperature of 350~K. While higher than what we use for K2-18~b, it is still relatively low, especially in comparison to many other similar mini-Neptunes, e.g. K2-3~b, TOI-270~c, etc. We nominally set the reference pressure ($P_{\rm ref}$) to be 0.1 bar for all the models in this work.
For this canonical model, the H$_2$O spectral amplitude of a K2-18~b transmission spectrum in the 1.1-1.7 $\mu$m range is comparable to that observed in the HST WFC3 band \citep{Benneke2019, Madhusudhan2020}, modulo a constant offset in transit depth due to our choice of $P_{\rm ref}$. We also consider additional contributions from CH$_4$ and NH$_3$ in the models which are below the detection threshold with current HST data of K2-18~b.
For TOI-732~c, we additionally consider a second atmospheric model, where H$_2$O is present at a volume mixing ratio of $10^{-2}$, the same as the canonical model, but CH$_4$ and NH$_3$ are depleted by 1 dex to volume mixing ratios of $5 \times 10^{-4}$ and $10^{-4}$, respectively. Such a composition therefore corresponds to either a solar metallicity atmosphere where H$_2$O has been enhanced, or a 10$\times$ solar metallicity atmosphere where CH$_4$ and NH$_3$ have been depleted. We do not include CO and CO$_2$ in our model atmospheres as their abundances in temperate H$_2$-rich atmospheres are expected to be very low under chemical equilibrium, compared to the prominent molecules we consider here. In principle, non-equilibrium mechanisms can enhance the amount of CO and CO$_2$ in the atmosphere to limits that may be detectable \citep{Hu2021a,Hu2021b,Yu2021}. We do not consider this aspect in the present study.
Some temperate mini-Neptunes could be effectively cloud-free in their observable atmospheres, while others may have high-altitude cloud decks. For the specific case of K2-18~b, \citet{Benneke2019} infer clouds in the lower atmosphere of K2-18~b using HST observations, with a model preference of 2.6~$\sigma$, and infer a cloud deck pressure range of 7.74–139~mbar with their retrievals and 10-1000~mbar using self-consistent atmospheric temperature models. \citet{Madhusudhan2020}, however, do not find strong evidence for clouds with the same data, reporting only a 1.1~$\sigma$ model preference.
We begin by considering the cloud-free case of K2-18~b as a reference. We then consider cases with clouds present, which we model as a grey opacity present at all pressures below the cloud-top pressure. We adopt a conservative cloud deck pressure of 3~mbar for our canonical model, motivated by both the findings of \citet{Benneke2019} mentioned above, as well as \citet{Blain2021} who found that optically-thick clouds in K2-18~b may form at pressures of 10~mbar or higher (i.e. deeper in the atmosphere), depending on metallicity. Our 3~mbar cloud deck therefore lies at a pressure $\sim$0.5~dex lower (i.e. at a higher altitude) than the 10~mbar theoretical estimates of \citet{Benneke2019} and \citet{Blain2021}. The spectral contributions of all three molecules as well as a 3~mbar cloud deck for K2-18~b are shown in figure \ref{fig:contribution_plot}. For TOI-732~c, we additionally consider even lower cloud deck pressures, seeking to benchmark the capabilities of JWST for the two atmospheric composition cases described above. We consider cloud deck pressures ranging from 100~bar, i.e., effectively cloud-free, to 0.03~mbar (3$\times 10^{-5}$~bar), i.e. high up in the atmosphere. For both planets, we use the worst-case scenario of 100$\%$ cloud coverage.
\subsection{Simulating JWST observations}
\label{sec:simulating_data}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/K2-18b_forwardmodels.png}
\caption{High resolution model transmission spectra of K2-18~b corresponding to the canonical atmospheric model described in section \ref{sec:canonical_model} with a cloud deck at 3~mbar (grey) and no clouds (blue). Also shown are simulated observations generated from the cloudy model using NIRSpec gratings G140H, G235H and G395H at native resolution (green points) and binned to R=50 (black errorbars). For visual clarity the simulated data are centered on the binned model points.}
\label{fig:K2-18b_data_generation}
\end{figure*}
We use the forward model-generating component of the AURA retrieval code \citep{Pinhas2018}, described in section \ref{sec:retrieval_methods} to generate a high-resolution model spectrum. This spectrum is input to the JWST simulator Pandexo \citep{Batalha2017b}, which we use to generate the wavelength bins and corresponding uncertainties for our simulated JWST observations. We do not explicitly set a noise floor, as the uncertainty per spectral element is never low enough in our case studies for it to be of concern. Throughout this work, we simulate the observation of one transit per instrument.
We consider simulated observations with the Near Infrared Spectrograph (NIRSpec) \citep{Ferruit2012, Birkmann2014} as well as the Near Infrared Imager and Slitless Spectrograph (NIRISS) \citep{Doyon2012}. For NIRSpec, we consider all three high-resolution gratings: G140H, G235H and G395H, paired with the F100LP, F170LP and F290LP filters, respectively. With this configuration, observations are obtained over a wavelength range of 1.0-1.8~$\mu$m for G140H, 1.7-3.1~$\mu$m for G235H and 2.9-5.1~$\mu$m for G395H. For all three gratings we use the SUB2048 subarray for maximal wavelength coverage and the NRSRAPID readout mode. The three gratings achieve a resolution of R~$\sim$2700. Additionally, we use NIRISS in Single Object Slitless Spectroscopy mode (SOSS), with the GR700XD grism, SUBSTRIP96 and NISRAPID readout mode , yielding observations with a wavelength coverage of 0.9-2.8~$\mu$m and a resolution of R~$\sim$700 with order 1.
We do not consider observations with the Near Infrared Camera as in its Wide Field Slitless Spectroscopy mode it offers lower wavelength coverage and resolution than those provided by the high-resolution NIRSpec gratings. We also do not consider NIRSpec PRISM in our investigations, despite its large wavelength coverage, due to its faint saturation threshold (J $\gtrsim$ 10.5). Lastly, we do not consider pairing the NIRSpec G140H grating with the F070LP filter, as this achieves a comparatively small wavelength coverage between 0.8-1.3~$\mu$m, which largely overlaps with that obtained with G140H/F100LP.
Throughout this work, we conduct retrievals on all simulated observations at their native resolution, rather than binning down to a particular resolution. We find that retrievals on native resolution observations achieve more accurate and robust abundance estimates, especially in the case of very high altitude cloud decks as discussed in section~\ref{sec:benchmark_canonical}. Figure \ref{fig:contribution_plot} shows the wavelength coverage of NIRISS and the three NIRSpec gratings, as well as the spectral contributions from the three molecules considered in this work, H$_2$O, CH$_4$ and NH$_3$. The masking effect of clouds at 3~mbar is shown as a grey dashed line.
To generate simulated JWST observations, we first convolve the model high-resolution spectrum with each instrument's spectral point-spread function \citep{Perrin2012, Sarkar2021}. We then bin the convolved spectrum down to the Pandexo-generated wavelength bins, taking into account each instrument's sensitivity function. We lastly add Gaussian noise to the spectrum, offsetting each datapoint according to its Pandexo-provided uncertainty. As this step may introduce artefacts in the data that are specific to a particular noise instance, we carry out several retrievals for each case presented, re-generating the data with a different noise instance each time. Figure \ref{fig:K2-18b_data_generation} shows such simulated observations for K2-18~b with our canonical cloudy model, simulated for the three high-resolution NIRSpec gratings at native resolution and at R~=~50. The data shown has no noise added for visual clarity. Also shown are the cloudy and cloud-free high-resolution transmission spectra we use to generate synthetic observations.
\subsection{Retrieval Methodology}
\label{sec:retrieval_methods}
For this work we use the latest version of the AURA retrieval code \citep{Pinhas2018}. AURA comprises of two parts. The first is a forward model generator, which was also used in creating the simulated JWST observations described in section \ref{sec:simulating_data}. The second is a robust Bayesian parameter estimator based on the PyMultiNest Nested Sampling package \citep{Buchner2014}.
Our forward model generator computes radiative transfer for a 1D plane-parallel atmosphere under hydrostatic equilibrium to create transmission spectra. The atmosphere's chemical composition is treated as uniform in altitude, with the mixing ratio of each constituent chemical species being a free parameter, avoiding any assumptions of equilibrium chemistry. As mentioned in section \ref{sec:canonical_model}, we model the temperature profile as an isotherm whose temperature is a single free parameter.
We include opacity contributions from H$_2$O \citep{Rothman2010}, CH$_4$ \citep{Yurchenko2014} and NH$_3$ \citep{Yurchenko2011}, as well those arising from H$_2$-H$_2$ and H$_2$-He collision-induced absorption (CIA) \citep{Richard2012}. These are included as they are the most dominant species expected in thermochemical equilibrium in temperate H$_2$-rich atmospheres \citep{Burrows1999, Madhu2011, Moses2013}. As discussed in section \ref{sec:canonical_model}, we do not include less abundant species such as CO or CO$_2$ in our retrievals. Clouds are treated as a grey opacity across the entire wavelength range, described by the pressure at the top of the cloud deck, as well as the fraction of the terminator atmosphere covered by clouds. In total, our atmospheric model has 7 free parameters: 3 of which are the log-mixing ratios of H$_2$O, CH$_4$ and NH$_3$, while the remaining four are the isotherm temperature $T_{\mathrm{iso}}$, the reference pressure, $P_{\mathrm{ref}}$, corresponding to the pressure at the given planet's radius, the cloud deck pressure, $P_{\mathrm{c}}$ and fractional cloud coverage ${\bar \phi}$.
For our retrievals, we use log-uniform volume mixing ratio priors between 10$^{-12}$-10$^{-0.3}$ for all three chemical species. For the isotherm temperature, we use a relatively uninformative uniform prior between 50-600~K. We note that while retrievals on actual data will most likely use a smaller prior range whose upper limit will be informed by the planet's equilibrium temperature, we do not do so here so as to fully capture any potential degeneracies involving atmospheric temperature. We note that for each forward model generated by our retrieval, the atmospheric scale height is recomputed for its specific mean molecular weight and atmospheric temperature. For both the reference pressure and cloud deck pressure, we use priors that span the full extent of our model atmosphere. We additionally use a uniform prior for the cloud coverage fraction, ranging from 0 to 1.
\section{Results}
\label{sec:results}
In this section we present our findings on how JWST observations can provide abundance constraints for temperate mini-Neptune atmospheres in the presence of high-altitude clouds. As mentioned in section \ref{sec:methods}, we consider single, double and triple instrument configurations consisting of NIRISS, covering the 0.9-2.8~$\mu$m wavelength range, as well as NIRSpec G140H, G235H and G395H configurations spanning wavelength ranges of 1.0-1.8, 1.7-3.1 and 2.9-5.1~$\mu$m, respectively. We simulate observing only one transit with each instrument considered.
We first examine the cloud-free case for K2-18~b in section \ref{sec:cloud_free_reference}, establishing a reference against which results from retrievals on cloudy atmospheres will be compared. We then examine the effects of instrument choice and wavelength coverage for retrievals on a cloudy atmosphere in section \ref{sec:wavelength_coverage}, again for the case of K2-18~b. We subsequently present retrieval results for TOI-732~c in section \ref{sec:case_study}, a more spectroscopically amenable target, starting with the minimal single-transit single-instrument setup in section \ref{sec:TOI-732_single_instruments}. We then consider observations combining all three NIRSpec gratings for atmospheric conditions ranging from cloud free to the highest-altitude cloud deck with which molecular abundances can still be constrained, thereby benchmarking what is achievable with JWST observations. We carry this investigation out for both the canonical model composition used for K2-18~b in section \ref{sec:benchmark_canonical}, as well as a second case, where the abundances of CH$_4$ and NH$_3$ are reduced by 1~dex relative to the canonical model in section \ref{sec:benchmark_depleted}.
\begin{table}
\centering
\caption{ Retrieved log-mixing ratio constraints for K2-18~b for all atmospheric and instrumental configurations considered.}
\begin{tabular}{l|c|c|c}
& \multicolumn{3}{c}{log-Mixing Ratios} \\[0.5mm]
Case & H$_2$O & CH$_4$ & NH$_3$ \\[0.5mm]
\hline
\hline
True Values& -2 & -2.3 & -3 \\[0.5mm]
\hline
\multicolumn{4}{c}{Cloud-Free} \\[0.5mm]
\hline
\multicolumn{4}{l}{\emph{Single-Instrument Configurations}} \\[0.5mm]
NIRSpec G140H & $(-1.80)$ & $-2.69^{+0.87}_{-0.65}$ & $-4.37^{+0.93}_{-0.73}$ \\[0.5mm]
NIRSpec G235H & $-1.73^{+0.53}_{-0.82}$ & $-2.34^{+0.37}_{-0.49}$ & $-2.78^{+0.39}_{-0.49}$ \\[0.5mm]
NIRSpec G395H & $(-2.18)$ & $-3.06^{+1.00}_{-1.04}$ & $-3.58^{+1.08}_{-1.04}$ \\[0.5mm]
NIRISS & $-2.06^{+0.54}_{-0.72}$ & $-2.19^{+0.47}_{-0.46}$ & $-3.30^{+0.48}_{-0.51}$ \\[0.5mm]
\\[0.5mm]
\multicolumn{4}{l}{\emph{Three-Instrument Configuration}} \\[0.5mm]
G140H + G235H + G395H & $-1.93^{+0.27}_{-0.36}$ & $-2.34^{+0.24}_{-0.26}$ & $-3.01^{+0.24}_{-0.27}$ \\[0.5mm]
\hline
\multicolumn{4}{c}{3~mbar Cloud Deck} \\[0.5mm]
\hline
\multicolumn{4}{l}{\emph{Single-Instrument Configurations}} \\[0.5mm]
NIRSpec G140H & $(-1.93)$ & $-3.85^{+1.65}_{-0.87}$ & $-4.70^{+1.71}_{-0.89}$ \\[0.5mm]
NIRSpec G235H & $(-1.53)$ & $-1.51^{+0.75}_{-1.04}$ & $-2.63^{+0.97}_{-1.46}$ \\[0.5mm]
NIRSpec G395H & $(-2.47)$ & $-4.81^{+2.39}_{-0.96}$ & $(-1.58)$ \\[0.5mm]
NIRISS & $(-2.95)$ & $-3.38^{+1.20}_{-0.84}$ & $(-2.02)$ \\[0.5mm]
\\[0.5mm]
\multicolumn{4}{l}{\emph{Two-Instrument Configurations}} \\[0.5mm]
NIRSpec G140H + G235H & $-2.72^{+0.79}_{-0.85}$ & $-2.45^{+0.49}_{-0.48}$ & $-3.30^{+0.55}_{-0.55}$ \\[0.5mm]
NIRSpec G140H + G395H & $(-1.51)$ & $-2.51^{+0.98}_{-1.08}$ & $-3.10^{+0.99}_{-1.16}$ \\[0.5mm]
NIRSpec G235H + G395H & $-2.48^{+0.87}_{-0.89}$ & $-2.67^{+0.59}_{-0.55}$ & $-3.18^{+0.60}_{-0.57}$ \\[0.5mm]
NIRISS + NIRSpec G395H & $-2.86^{+0.88}_{-1.60}$ & $-2.37^{+0.56}_{-0.58}$ & $-3.12^{-0.60}_{-0.62}$ \\[0.5mm]
\\[0.5mm]
\multicolumn{4}{l}{\emph{Three-Instrument Configuration}} \\[0.5mm]
G140H + G235H + G395H & $-2.22^{+0.55}_{-0.77}$ & $-2.51^{+0.45}_{-0.52}$ & $-3.16^{+0.48}_{-0.58}$ \\[0.5mm]
\hline
\end{tabular}
\newline
\footnotesize{Note: In cases where the lower 1-$\sigma$ interval spans more than 2 dex, we instead list the 2-$\sigma$ (95\%) upper estimate in brackets.}
\label{tab:K2-18b_results}
\end{table}
\subsection{A Reference Cloud-free Case of K2-18~b}
\label{sec:cloud_free_reference}
The best-case scenario for transmission spectroscopy is when the planet's cloud deck lies below the observable slant photosphere i.e. at altitudes that do not contribute to its transmission spectrum. Such atmospheres are therefore effectively "cloud-free" from a spectroscopic characterisation standpoint. As a result, molecular absorption features are not masked by a cloud deck's opacity contributions, illustrated in figures \ref{fig:contribution_plot} and \ref{fig:K2-18b_data_generation}, making such spectra most favourable for constraining atmospheric abundances and other properties.
We begin by considering such a cloud-free case for K2-18~b, establishing a reference for what is achievable without the truncating effect of clouds. In this section we first consider observations spanning the $\sim$1-5~$\mu$m range by combining NIRSpec G140H, G235H and G395H. We then consider single-instrument observations with the three NIRSpec gratings as well as NIRISS. As in the rest of this work, we simulate {the observation of a single transit} per instrument. The retrieved abundance constraints are summarised in table \ref{tab:K2-18b_results}.
\begin{figure*}
\centering
\includegraphics[width=0.995\textwidth]{Figures/K2-18b_cloudfree_cornerplot.png}
\caption{Posterior probability distribution for all 7 retrieved parameters obtained by a retrieval on combined NIRSpec G140H, G235H and G395H simulated observations of cloud-free K2-18~b. The data were generated for a cloud-free atmosphere with a composition corresponsing to 10$\times$ solar elemental abundances, as described in section \ref{sec:canonical_model}. The blue squares with error bars denote the median and 1-$\sigma$ intervals, while red solid and dashed lines denote the true values used to generate the data.}
\label{fig:threeinstrument_clear_corner}
\end{figure*}
\subsubsection{Three-Instrument Observations}
We first consider an instrument configuration combining observations from NIRSpec G140H, G235H and G395H. Figure \ref{fig:threeinstrument_clear_corner} shows the posterior probability distribution for all 7 retrieved parameters and the retrieved mixing ratio constraints are summarised in table \ref{tab:K2-18b_results}. The retrieval obtains log-mixing ratio estimates of $-1.93^{+0.27}_{-0.36}$ for H$_2$O, $-2.34^{+0.24}_{-0.26}$ for CH$_4$ and $-3.01^{+0.24}_{-0.27}$ for NH$_3$. It can be seen that the retrieval constrains the cloud deck pressure, $P_{\mathrm{c}}$, to high values. While there is some spread towards lower pressures, the correlation plot of $P_{\mathrm{c}}$ with the cloud fraction, $\bar \phi$, shows that they are associated with cloud fractions near zero and thus have minimal impact on the spectrum.
Our retrievals achieve abundance constraints mostly below $\sim$0.3 dex with only 3 transits observed in total. Moreover, such abundance constraints have been achieved without any supporting observations in the optical to constrain the spectrum's baseline. As can be seen in figure \ref{fig:contribution_plot}, the large wavelength coverage JWST offers encompasses numerous absorption peaks from all three species present in our models. This enables retrievals to use the relative heights of several peaks and the depths of troughs between them to implicitly ascertain the spectrum's baseline.
We conduct repeat retrievals, each time generating the synthetic data anew with a different noise instance, to ensure our results are not caused by noise-specific features. We find that the majority of retrievals find abundance estimates that are within 1-$\sigma$ of the true values while a minority obtain abundance estimates that are between 1 and 2 $\sigma$ away from the true values.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/K2-18b_cloudfree_singleinstrument.pdf}
\caption{Posterior distributions of retrievals carried out on simulated single-instrument observations of K2-18~b without high-altitude clouds, using NIRSpec G140H (orange), G235H (green) and G395H (blue) as well as NIRISS order 1 (purple), retrieving on data from one instrument at a time. All data were generated for a nominal atmosphere at 10$\times$ solar elemental abundances, as described in section \ref{sec:canonical_model}. Black vertical lines denote the true values used to generate the data, while horizontal errorbars denote the median value and 1-$\sigma$ intervals. If a retrieval produces a lower 1-$\sigma$ interval spanning more than 2~dex, the errorbar is replaced with an arrow denoting the 2-$\sigma$ (95\%) confidence upper limit.}
\label{fig:singleinstrument_clear_posteriors}
\end{figure*}
\subsubsection{Single-Instrument Observations}
Figure \ref{fig:singleinstrument_clear_posteriors} shows posterior distributions obtained from retrievals conducted on single-instrument observations, again without clouds present. The substantial decrease of wavelength coverage compared to the full $\sim$1-5~$\mu$m coverage used previously leads to less precise parameter constraints and even two cases where the mixing ratio of H$_2$O is largely unconstrained. Our findings indicate that observations solely from either NIRISS or NIRSpec G235H can precisely retrieve the correct abundances for H$_2$O, CH$_4$ and NH$_3$. In the case of NIRISS, we obtain an H$_2$O mixing ratio estimate with 0.5 and 0.7~dex upper and lower 1-$\sigma$ intervals, respectively. CH$_4$ and NH$_3$ are retrieved even more precisely, with a 1-$\sigma$ uncertainty of 0.5~dex for both molecules. Using NIRSpec G235H, we obtain constraints comparable to those obtained with NIRISS, finding upper and lower 1-$\sigma$ intervals of 0.5 and 0.8~dex, respectively, for H$_2$O and 0.4 and 0.5~dex for both CH$_4$ and NH$_3$. As can be seen in figure \ref{fig:singleinstrument_clear_posteriors}, the retrieval on NIRSpec G235H data achieved comparable abundance constraints, despite yielding a less precise estimate for the terminator temperature and reference pressure.
With single-transit NIRSpec G140H observations, we find that the abundances of CH$_4$ and NH$_3$ are still successfully constrained, but somewhat less precisely. Specifically, our retrieval produces upper and lower 1-$\sigma$ uncertainties of 0.9 and 0.7~dex for both CH$_4$ and NH$_3$. The NH$_3$ median estimate lies slightly more than 1-$\sigma$ away from the true value, but is still well within 2-$\sigma$. The H$_2$O posterior, however, shows a peak but it is not constrained, indicating that while H$_2$O is potentially present in the atmosphere, only an upper limit can be placed on its mixing ratio.
NIRSpec G395H observations also lead to constraints for the mixing ratios of CH$_4$ and NH$_3$ but not for H$_2$O. The retrieval produces mixing ratio estimates for CH$_4$ and NH$_3$ with uncertainties larger than those obtained with NIRSpec G140H, at $\sim$1.0 and $\sim$1.1~dex, respectively. As seen in figures \ref{fig:contribution_plot}, the NIRSpec G395H wavelength range contains significant spectral features from all three molecules in our model. However, the comparatively lower SNR that NIRSpec G395H achieves means that observing a single transit is insufficient to constrain all three molecules for our limiting case of K2-18~b.
As before, we conduct repeat retrievals on synthetic data with different noise instances to ensure the robustness of our findings. In the case of G140H, we find that subsequent retrievals sometimes under-estimate the abundances of CH$_4$ and NH$_3$, offering median values that are slightly more than 1-$\sigma$ below from the true value. Some G395H retrievals yield an H$_2$O posterior which is peaked at the true value, but nevertheless are poorly constrained and display a sizeable ``tail'' towards lower abundances. Retrievals on G235H and NIRISS show little variability, consistently yielding abundance estimates that are within 1-$\sigma$ of the true values most times and occasionally between 1- and 2-$\sigma$, as expected.
The generally less precise constraints of our single-transit NIRSpec G140H and G395H retrievals indicate that for observations of planets with comparable precision to K2-18~b and with atmospheric properties similar to our canonical model, multiple transits may have to be observed, or be combined with observations from NIRISS or NIRSpec G235H. On the other hand, both NIRSpec G235H and NIRISS consistently produce abundance constraints of comparable precision to those shown in table \ref{tab:K2-18b_results}, thanks to both covering a feature-rich part of the IR spectrum. This is in spite of NIRSpec G235H offering a smaller wavelength coverage than NIRISS, as it achieves a higher precision (accounting for resolution) as well as resolution, which is enough to compensate for its narrower bandpass.
From our investigation of cloud-free K2-18~b, we find that JWST observations, even those from a single instrument such as NIRISS or NIRSpec G235H observing a single transit, can lead to precise abundance constraints. Moreover, this is done without any supporting optical data. In the following section, we examine whether the same is true when high-altitude clouds obscure part of the planet's transmission spectrum.
\subsection{Case Study: K2-18~b with Clouds}
\label{sec:wavelength_coverage}
High-altitude clouds pose a significant challenge for retrievals. Opacity contributions from clouds mask spectral features, resulting in shallower, truncated troughs between absorption peaks. Consequently, retrievals may be unable to distinguish between features that are masked by clouds and those that are merely the result of a smaller scale height or low abundances. This can manifest as poorly constrained posterior distributions, or even as doubly-peaked posteriors, indicating two competing explanations for the data between which the retrieval is unable to distinguish.
We now investigate abundance constraints for K2-18~b in the presence of a high-altitude grey opacity cloud deck at 3~mbar, motivated by theoretical and observational grounds, as discussed in section \ref{sec:canonical_model}. We systematically investigate whether instrument combinations up to the three-instrument NIRSpec G140H+G235H+G395H combination, used in Section \ref{sec:cloud_free_reference}, can successfully constrain atmospheric properties using different combinations of wavelength coverage. Using the same nominal K2-18~b model as above, we examine all possible non-overlapping combinations of observations with NIRSpec G140H (F100LP filter), G235H and G395H as well as NIRISS. The precise configurations used to simulate each instrument are detailed in Section \ref{sec:simulating_data}. As before, we simulate observing a single transit with each instrument.
We first consider observations with individual instruments in section \ref{sec:single_transit_obs}, as was done for the cloud-free case in section \ref{sec:cloud_free_reference}, before moving on to two-instrument configurations in section \ref{sec:two_transit_obs}. Since we seek to ascertain the effect of wavelength coverage on retrieved parameters, we do not consider two-instrument cases where both largely cover the same wavelength range, i.e. NIRISS + NIRspec G140H or NIRISS + NIRspec G235H. For the same reason, we do not consider three-instrument configurations other than NIRSpec G140H + G235H + G395H presented in section \ref{sec:three_instrument_obs} All retrieved abundance constraints are summarised in table \ref{tab:K2-18b_results}, in addition to those obtained above for the cloud-free case.
\subsubsection{Single-Instrument Observations}
\label{sec:single_transit_obs}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/K2-18b_cloudy_singleinstrument.pdf}
\caption{Posterior distributions for retrievals carried out on simulated single-instrument observations for the conservative case of K2-18~b with clouds, using NIRSpec G140H (orange), G235H (green) and G395H (blue) as well as NIRISS (purple), retrieving on data from one instrument at a time. All data were generated for a nominal atmosphere at 10$\times$ solar elemental abundance with a cloud deck at 3~mbar. Black vertical lines denote the true values used to generate the data, while horizontal errorbars denote the median value and 1-$\sigma$ intervals. If a retrieval results in a lower 1-$\sigma$ uncertainty larger than 2~dex, the errorbar is replaced with an arrow denoting the 2-$\sigma$ (95\%) confidence upper limit.}
\label{fig:singleinstrument_clouds_posteriors}
\end{figure*}
We first consider single-instrument observations with a cloud deck at a pressure of 3~mbar. As seen in the posterior distributions shown in Figure \ref{fig:singleinstrument_clouds_posteriors}, clouds cause a significant deterioration to the parameter constraints obtained. Three of the four retrievals are unable to meaningfully constrain the cloud deck pressure, while the retrieval on NIRISS data somewhat constrains the cloud deck pressure to lower values. As a result, poor constraints are also obtained for the other parameters, as the retrievals are unable to ascertain if the spectral features are truncated by clouds or were instead produced by a small scale height or low abundances.
The retrieval on NIRSpec G140H observations produces estimates for the mixing ratios of NH$_3$ and CH$_4$ that are within 1-$\sigma$ of the true values, with both having upper and lower 1-$\sigma$ intervals of +1.7 and -0.9 dex. The retrieval only yields an upper bound for the mixing ratio of H$_2$O. The findings of the retrieval on NIRSpec G235H are similar to what was obtained by the retrieval on NIRSpec G140H data: H$_2$O is again largely unconstrained but its posterior distribution is more significantly peaked near the true value, while CH$_4$ and NH$_3$ mixing ratios are estimated accurately. CH$_4$ was retrieved with upper and lower 1-$\sigma$ intervals of +0.8 and -1.0~dex, respectively, while NH$_3$ was constrained with upper and lower 1-$\sigma$ intervals of +1.0 and -1.5~dex, respectively.
Retrieving on NIRISS data resulting in the log-mixing ratio of CH$_4$ being constrained with upper and lower 1-$\sigma$ intervals of 1.2 and 0.8~dex, respectively. The retrieval was only able to produce upper bounds for the mixing ratios of H$_2$O and NH$_3$. This is due to NIRISS achieving a lower spectroscopic precision (accounting for resolution) as well as a lower resolution than the high-resolution NIRSpec gratings, despite offering the largest wavelength coverage of the four instruments considered. Its larger wavelength coverage does not compensate for its lower precision, unlike in the cloud-free case. Due to its larger wavelength coverage, however, it is the only retrieval that led to a tentative indication that clouds may be present in the planet's atmosphere, somewhat constraining $P_\mathrm{c}$ to lower values that have an impact on the transmission spectrum. It is therefore possible to conclude that the the poorer constraints relative to the cloud-free case are most likely caused by high-altitude clouds.
The retrieval on NIRSpec G395H also successfully detects CH$_4$, producing a mixing ratio estimate with upper and lower 1-$\sigma$ intervals of 2.4 and 1.0~dex, respectively. It also indicates that NH$_3$ may potentially be also present, but the notable spread shown by its posterior distribution towards low abundances implies that models where NH$_3$ does not significantly contribute to the spectrum are also plausible. Additionally, H$_2$O is unconstrained. Notably, the retrieval produced doubly-peaked posterior distributions for the mixing ratios of CH$_4$ and NH$_3$ as well as $T_\mathrm{iso}$, indicating that there are two competing explanations for the data.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/K2-18b_cloudy_twoinstruments.pdf}
\caption{Posterior distributions for retrievals for the conservative case of K2-18~b with clouds, carried out on two-instrument combinations: NIRSpec G140H + G235H (orange), G140H + G395H (green), G235H + G395H (blue) as well as NIRISS + NIRSpec G395H. The input synthetic spectral data were generated for a nominal K2-18~b atmosphere at 10$\times$ solar elemental abundance with a cloud deck at 3~mbar. Black vertical lines denote the true values used to generate the data, while horizontal errorbars denote the median retrieved value and 1-$\sigma$ intervals. If a retrieval results in a lower 1-$\sigma$ uncertainty larger than 2~dex, the errorbar is replaced with an arrow denoting the 2-$\sigma$ (95\%) confidence upper limit.}
\label{fig:posterior_clouds_2transits}
\end{figure*}
As done previously, we run repeat retrievals to determine the variability of our results with different noise instances. We find that repeat retrievals on NIRSpec G140H data produce similar posterior distributions to those seen in figure \ref{fig:singleinstrument_clouds_posteriors}, where CH$_4$ and NH$_3$ mixing ratios are retrieved accurately while producing poorer H$_2$O constraints. Follow-up retrievals on NIRSpec G395H data were also unsuccessful at detecting H$_2$O, in some cases producing a posterior that is peaked at the true value but still has a substantial low abundance tail. The relative heights of the two peaks displayed by posteriors of CH$_4$, NH$_3$ and $T_\mathrm{iso}$ varied with different noise instances, with some retrievals obtaining only one of the two peaks.
Repeat retrievals on NIRSpec G235H and NIRISS also yield similar results to those shown in figure \ref{fig:singleinstrument_clouds_posteriors}. In a minority of cases, they succeed in fully constraining the abundances of H$_2$O and NH$_3$ in addition to CH$_4$, generally achieving precisions of $\sim$1~dex. This indicates that the success of retrievals on NIRISS and NIRSpec G235H observations in constraining the abundances of H$_2$O and NH$_3$ is dependent on specific spectral features not being degraded by random noise.
Overall, single-transit observations of K2-18~b consistently lead to CH$_4$ detections with all instrument configurations considered. Additionally, NH$_3$ is also detectable with some instruments, but its mixing ratio is not constrained as robustly. H$_2$O is the most challenging to retrieve for our particular model atmosphere, with the large majority of our retrievals only producing upper limits for its mixing ratio. We emphasise that these results are obtained with a minimal observing configuration, with only a single transit observed in each case. By observing more transits with each instrument or alternatively observing a planet that is more spectroscopically amenable than K2-18~b, better results can be obtained. Despite this, the mixing ratios of CH$_4$ and NH$_3$ can still be constrained when using NIRSpec G140H or G235H observations, although neither retrieval offers any firm indication that H$_2$O might also be present.
\subsubsection{Two-Instrument Observations}
\label{sec:two_transit_obs}
We now examine whether the addition of a second observation with a different, non-overlapping instrument is enough to allow retrievals to overcome the masking effects of clouds and robustly measure the abundances of H$_2$O, CH$_4$ and NH$_3$. In Figure \ref{fig:posterior_clouds_2transits}, we present the results of retrievals carried out on 4 instrument combinations: NIRSpec G140H + G235H, NIRSpec G140H + G395H, NIRspec G235H + G395H and NIRISS + NIRSpec G395H. We note that the NIRISS + NIRSpec G395H configuration achieves the same wavelength coverage as the three-instrument NIRSpec G140H + G235H + G395H combination considered in section \ref{sec:three_instrument_obs}, but at the cost of a lower precision and resolution over the $\sim$1-3~$\mu$m wavelength range covered by NIRISS.
As can be seen in figure \ref{fig:posterior_clouds_2transits}, all four retrievals perform better than those on single-transit observations, although ``tails'' towards low abundances can be seen in two of the H$_2$O posterior distributions. Retrieving on combined observations from NIRSpec G140H + G235H, which both individually led to constraints on the mixing ratios of CH$_4$ and NH$_3$ but not H$_2$O, again accurately constrained the mixing ratios of CH$_4$ and NH$_3$, this time with uncertainties of $\sim$0.5 and $\sim$0.6~dex, respectively. The most notable improvement is that the retrieval now also provides an estimate of the mixing ratio of H$_2$O with an uncertainty of $\sim$0.8~dex. Additionally, the retrieval is also more successful at constraining the cloud deck pressure, $P_\mathrm{c}$ compared to the individual retrievals on NIRSpec G140H or G235H observations, albeit with significant spread towards higher pressures that correspond to ``cloud free'' atmospheres.
The retrieval on NIRSpec G140H + G395H data produces tentative signs that H$_2$O may be present, with a peaked but unconstrained posterior distribution, which only offers an upper limit on the H$_2$O mixing ratio consistent with the true value. This retrieval does obtains accurate mixing ratio estimates for CH$_4$ and NH$_3$, but with the largest uncertainty of all two-instrument retrievals considered, at $\sim$~1.0 and $\sim$1.1~dex, respectively. Compared to the NIRSpec G140H + G235H retrieval, it obtains a more precise $P_\mathrm{c}$ estimate, but produces bimodal isotherm temperature and reference pressure posteriors, with one of the two modes corresponding to the true value. Notably, all other retrievals produce posteriors that have peaks corresponding to one of the two modes obtained by the NIRSpec G140H + G395H retrieval. This indicates that the retrievals are still susceptible to degeneracies between cloud truncation and scale height.
The retrieval on NIRSpec G235H + G395H data, obtains abundance constraints with a comparable precision to those obtained from NIRSpec G140H + G235H for all three molecules. The mixing ratio estimates obtained are $\mathrm{log}(X_{\mathrm{H}_{2} \mathrm{O}}) = -2.48^{+0.87}_{-0.89}$, $\mathrm{log}(X_{\mathrm{CH}_4}) = -2.67^{+0.59}_{-0.55}$ and $\mathrm{log}(X_{\mathrm{NH}_3}) = -3.18^{+0.60}_{-0.57}$. All are within 1-$\sigma$ of the respective true values.
Lastly, the retrieval on observations from NIRISS + NIRSpec G395H produces an estimate for the mixing ratio of H$_2$O, but with a slight tail in its posterior towards lower abundances. This leads to a lower 1-$\sigma$ interval to -1.6~dex, and a much smaller upper 1-$\sigma$ interval of +0.9~dex. Similarly to the other three retrievals, it also accurately retrieved the mixing ratios of CH$_4$ and NH$_3$, achieving precisions of $\sim$0.6~dex for both. Notably, it produces the most precise cloud deck constraint of all 2-instrument configurations considered, although the corresponding posterior distribution still shows a slight spread towards higher and lower pressures. This retrieval was also the most successful in constraining the isotherm temperature and reference pressure. This is due to the NIRISS + NIRSpec G395H configuration offering the largest wavelength coverage, encompassing the full $\sim$1-5$\mu$m range.
Across the 4 retrievals shown in figure \ref{fig:posterior_clouds_2transits}, the mixing ratios of CH$_4$ and NH$_3$ are consistently constrained, with all four retrievals producing estimates within 1-$\sigma$ of the true value. Two of the four retrievals produced H$_2$O mixing ratio posterior distributions with low abundance ``tails'', which indicate that the data do not entirely rule out models where the H$_2$O abundance is undetectably low. The NIRSpec G140H + G235H and G235H + G395H retrievals produces an H$_2$O posterior distribution without such a tail, corresponding to a precise abundance measurement.
\begin{figure*}
\centering
\includegraphics[width=0.995\textwidth]{Figures/K2-18b_cloudy_cornerplot.png}
\caption{Posterior probability distribution for all 7 retrieved parameters obtained by a retrieval on combined NIRSpec G140H, G235H and G395H simulated observations of our cloudy K2-18~b case. The data were generated for an atmosphere which has a composition corresponding to 10$\times$ solar elemental abundances and a cloud deck present at 3~mbar, as described in section \ref{sec:canonical_model}. Horizontal errorbars denote the median and 1-$\sigma$ intervals, while red solid and dashed lines denote the true values used to generate the data.}
\label{fig:threeinstrument_cloudy_corner}
\end{figure*}
As before we conduct repeat retrievals to assess the robustness of our results. Repeating the NIRSpec G140H + G235H retrieval produces similarly accurate CH$_4$ and NH$_3$ mixing ratio estimates, but sometimes also yield $\mathrm{log}(X_{\mathrm{H}_{2} \mathrm{O}})$ posteriors with low abundance tails. Repeat retrievals on observations with NIRSpec G140H + G395H obtain H$_2$O posteriors with low abundance tails of varying severity. Additionally, the relative heights of the bimodal isotherm temperature and reference pressure vary from run to run, with some exclusively finding one of the two peaks. Follow up retrievals on different noise instances of NIRISS and NIRSpec G395H data again produce H$_2$O posteriors with varying degrees of spread to lower mixing ratios, while reliably constraining the cloud deck pressure.
Repeat retrievals on re-generated observations with NIRSpec G235H + G395H also show some variability particularly for the retrieved H$_2$O abundance constraints. As can be seen in figure \ref{fig:contribution_plot}, there are two significant H$_2$O features that primarily drive the H$_2$O constraints from a NIRSpec G235H + G395H retrieval. Our findings therefore indicate that precision of the H$_2$O mixing ratio is conditional on these particular features not being too degraded by random noise. This risk can be mitigated by either observing a greater wavelength range, so as to include more molecular features, or repeat observations with the same instrument to improve the spectrophotometric precision achieved. This configuration however consistently retrieves the H$_2$O abundance more reliably and robustly than the NIRISS + NIRSpec G395H configuration. This indicates that in addition to wavelength coverage, the higher resolution and precision that NIRSpec G235H offers compared to NIRISS are also important in obtaining abundance estimates for cloudy atmospheres.
Given the significant challenge presented by our canonical cloudy model atmosphere of K2-18~b, with a cloud deck present at 3~mbar, we additionally consider cloud decks at lower altitudes. We find that with a cloud deck at 10~mbar all retrievals successfully constrain the mixing ratios of H$_2$O, CH$_4$ and NH$_3$ to within 1-$\sigma$ of the correct values. We therefore conclude that two-transit observations are viable for cloudy atmospheres where it is known a-priori that the cloud deck is at pressures greater than 3~mbar e.g. for K2-18~b \citep{Benneke2019, Madhusudhan2020}. We caution however that in the absence of such information, two-transit observations run the risk of failing to yield constraints on atmospheric properties, should a cloud deck at pressures less than 3~mbar (i.e., at higher altitudes) indeed be present.
Our findings therefore indicate that even with our highly challenging cloudy canonical model, two-instrument configurations, where each instrument is used to observe one transit, can successfully lead to constraints for the atmosphere's abundances and properties. Moreover, we find that this is possible with several two-instrument configurations, allowing observing programs to opt for whichever is optimal for their specific objectives. The NIRSpec G235H + G395H configurations however was the best performing, achieving the most precise abundance constraints and doing so more consistently than the other configurations. Our findings highlight the importance of wavelength coverage as well as spectroscopic precision in characterising cloudy atmospheres. In the following section, we examine to what extent retrieved atmospheric constraints are improved by increasing the number of instruments used to three, combining all three NIRSpec gratings.
\subsubsection{Three-Instrument Observations}
\label{sec:three_instrument_obs}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/K2-18b_cloudy_vs_noclouds.pdf}
\caption{ Posterior distributions for retrievals for K2-18~b with no clouds (blue) and with a cloud deck present at 3~mbar (orange). Both retrievals were carried out on combined single-transit observations with three instrument settings: NIRSpec G140H, G235H and G395H. For both cases, the input simulated spectral data were generated for the canonical K2-18~b atmosphere at 10$\times$ solar elemental abundance with a cloud deck at 3~mbar as described in section \ref{sec:canonical_model}. Black vertical lines denote the true values used to generate the data, while horizontal errorbars denote the median retrieved value and 1-$\sigma$ intervals.}
\label{fig:posteriors_cloudfree_vs_noclouds}
\end{figure*}
We now consider the three-instrument configuration comprising of NIRSpec G140H + G235H + G395H, which achieves a complete coverage of the $\sim$1-5~$\mu$m range, as shown in figure \ref{fig:contribution_plot}. We do not consider any other three-instrument combination, as NIRISS has a wavelength coverage that almost entirely overlaps with the NIRSpec G140H and G235H bands. We seek to understand the extent to which abundance constraints are improved in going from two- to three-instrument configurations. We therefore focus on two particular comparisons with results from section \ref{sec:two_transit_obs}. The first is with the NIRISS + NIRSpec G395H combination, which offers effectively the same $\sim$1-5~$\mu$m wavelength coverage as NIRSpec G140H + G235H + G395H, at the cost of NIRISS achieving a lower resolution and precision. The second is with NIRSpec G235H + G395H, which we found to be the best-performing two-instrument configuration in constraining atmospheric abundances, which does not cover the feature-right $\sim$1-2~$\mu$m range that NIRISS and NIRSpec G140H cover.
The full posterior distribution obtained from our retrieval on the combined NIRSpec G140H + G235H + G395H observations is shown in figure \ref{fig:threeinstrument_cloudy_corner}. The retrieval successfully obtains accurate mixing ratio estimates, retrieving values of -2.22$^{+0.55}_{-0.77}$ for H$_2$O, -2.51$^{+0.45}_{-0.52}$ for CH$_4$ and -3.16$^{+0.48}_{-0.58}$ for NH$_3$. Additionally, the retrieval also successfully constrains the cloud deck pressure, with upper and lower 1-$\sigma$ intervals of +0.5 and -0.4~dex, respectively. The isotherm temperature and log-reference pressure are estimated to be 262$^{+32}_{-46}$~K and -0.48$^{+0.65}_{-0.41}$, both marginally more than 1-$\sigma$ away from the true value, but well within 2-$\sigma$.
Retrieving on the three-instrument NIRSpec G140H + G235H + G395H configuration produces substantially more precise abundance estimates relative to any of the two-instrument configurations presented in section \ref{sec:two_transit_obs}, especially for H$_2$O. This is particularly notable when comparing to NIRSpec G235H + G395H, as the only difference is the inclusion of NIRSpec G140H, which on its own did not lead to any H$_2$O abundance constraints in section \ref{sec:single_transit_obs} or when combined with NIRSpec G395H in section \ref{sec:two_transit_obs}. The improvement in the H$_2$O mixing ratio precision that NIRSpec G140H provided indicates that it does contain information about the H$_2$O abundance, which can be accessed when combined with additional observations over a larger wavelength range. Additionally, the inclusion of NIRSpec G140H allows for more precise constraints for the mixing ratios of CH$_4$ and NH$_3$, which can indirectly lead to better H$_2$O constraints by alleviating degeneracies between the three molecules. The retrieval on the full three-instrument configuration also obtained more precise isotherm temperature and reference pressure estimates, than the NIRSpec G235H + G395H configuration. This supports earlier indications that observing a greater wavelength range and, hence, a greater number of absorption features, helps in establishing the spectrum's baseline in the absence of supporting optical data.
Relative to the retrieval on NIRISS + NIRSpec G395H, we find that the improvements are mainly in the abundance estimates, due to the greater resolution and precision achieved by replacing NIRISS with NIRSpec G140H + G235H. Meanwhile, the isotherm temperature, reference pressure and cloud deck pressure are retrieved with comparable precision, again supporting the deduction that a wide wavelength range is beneficial in establishing the spectral baseline.
Repeat retrievals with new noise instances show similar results, with comparable uncertainties to those shown in figure \ref{fig:threeinstrument_clear_corner}. In a minority of cases, the posterior distribution for the H$_2$O mixing ratio has a low abundance tail, indicating that H$_2$O is not well-retrieved. We find that H$_2$O being less precisely constrained tends to correlate with the cloud pressure and/or both the isotherm temperature and reference pressure being poorly retrieved as well. Moreover, we find that H$_2$O is well-retrieved more often than in the NIRSpec G235H +G395H case.
We therefore find that there are meaningful gains to be had by observing with the NIRSpec G140H + G235H + G395H configuration over any two-instrument configuration. We note, however, that two-instrument configurations also succeed in constraining the abundances of all three dominant molecules, but less reliably and with a greater risk of the abundance estimates not being robust.
We superpose the posterior distributions obtained for the cloudy and cloud-free cases in figure \ref{fig:posteriors_cloudfree_vs_noclouds}. It is evident that while the introduction of a cloud deck at 3~mbar has resulted in less precise constraints, all atmospheric parameters are retrieved well enough to allow for meaningful conclusions to be reached about the atmospheric properties of the planet. Moreover, observing additional transits, beyond the minimal single transit per instrument configuration explored in this work, as is the case for K2-18~b in Cycle 1 with programmes 2372 and 2722 (PI: Renyu Hu and Nikku Madhusudhan, respectively), will lead to even more precise constraints, approaching those obtained for the cloud-free case shown.
Overall, our results indicate that even when a high-altitude cloud deck at 3~mbar is present in a temperate mini-Neptune like K2-18~b, orbiting a moderately bright M dwarf, a minimal JWST observing setup consisting of 1 transit per instrument and judiciously combining at least two instruments, can result in retrievals overcoming clouds and establishing highly precise abundance constraints. We note that our choice of cloud deck pressure in our canonical model is conservative, with the cloud top lying at lower pressures (i.e. higher altitudes) than the constraints obtained from HST observations and theoretical studies \citep{Benneke2019, Blain2021}. It is therefore likely that actual JWST observations of K2-18~b or similar planets with clouds will yield more precise constraints than those obtained in this work. As noted above, our choice of atmospheric composition consistently results in H$_2$O often having the poorest abundance constraints. The precision with which each molecule's mixing ratio is retrieved can be expected to vary depending on the specific atmospheric composition, which we explore in section \ref{sec:benchmark_depleted}.
\subsection{Case Study: TOI-732~c With Clouds}
\label{sec:case_study}
\begin{figure*}
\includegraphics[width=\textwidth]{Figures/TOI-732c_labelled_spectrum.pdf}
\caption{ Model transmission spectra of TOI-732~c a promising temperate mini-Neptune orbiting a bright M dwarf. The blue spectrum corresponds to the cloud free case of our canonical model described in section \ref{sec:canonical_model}, with the additional inclusion of CO and CO$_2$, both at 100 parts-per-million mixing ratio, for illustrative purposes. The grey spectrum is the same as the one in blue but with the addition of a 1~mbar cloud deck, which is one of the cloud deck pressures considered in section \ref{sec:benchmark_canonical}. The prominent spectral features are labelled with the corresponding molecules. Also shown are the corresponding JWST observations for the cloud-free model with NIRSpec G140H, G235H and G395H, each observing 1 transit, binned down to a resolution of R~=~50 for clarity.}
\label{fig:TOI-732c_labelled_spectrum}
\end{figure*}
In this section, we investigate the observing capabilities of JWST for a target more amenable to spectroscopic observations than K2-18~b. We focus on TOI-732~c \citep{Nowak2020, Cloutier2020}, a more recently-discovered temperate mini-Neptune, orbiting a star twice as bright as K2-18 in the $J$ band, as shown in table \ref{tab:planet_properties}. TOI-732~c has a somewhat higher equilibrium temperature compared to K2-18~b, at 363~K (for zero Bond albedo and full redistribution), as well as lower gravity. As a result, it has a larger atmospheric scale height compared to K2-18~b, and hence larger spectral features, which in addition to its brighter host star, leads to higher SNR observations. This can be seen in figure \ref{fig:TOI-732c_labelled_spectrum}, which shows our canonical model and corresponding simulated data for TOI-732~c, with each spectral feature labelled with the molecules giving rise to it. Motivated by the planet's higher equilibrium temperature, we use a nominal terminator temperature of 350~K, which is a 50~K increase from what was used for K2-18~b.
We first examine how the improved SNR that TOI-732~c offers over K2-18~b affects the viability of single-transit single-instrument observations of a cloudy atmosphere in section \ref{sec:TOI-732_single_instruments}, using the same composition and 3~mbar cloud deck as used in section \ref{sec:wavelength_coverage}. In section \ref{sec:benchmark_canonical}, we use the instrument configuration that led to the best abundance constraints for K2-18~b, specifically NIRSpec G140H + G235H + G395H, to examine how the detectability of H$_2$O, CH$_4$ and NH$_3$ varies with cloud deck pressure. We now also include cases where the cloud deck is at lower pressures (i.e. higher altitudes) than 3~mbar, the value used so far in this work. We then carry out the same investigation in section \ref{sec:benchmark_depleted} for an atmospheric composition where CH$_4$ and NH$_3$ are depleted by 1~dex relative to the canonical model. In doing so, we explore how atmospheric composition affects the retrieved abundance constraints of the three dominant molecules considered, as well as how severely spectral features are masked by clouds. All retrieved mixing ratio constraints for TOI-732~c are summarised in table \ref{tab:TOI-732c_results}.
\begin{table}
\centering
\caption{Retrieved log-mixing ratio constraints for TOI-732~c for all atmospheric and instrumental configurations considered.}
\begin{tabular}{l|c|c|c}
& \multicolumn{3}{c}{log-Mixing Ratios} \\[0.5mm]
Case & H$_2$O & CH$_4$ & NH$_3$ \\[0.5mm]
\hline
\hline
\multicolumn{4}{c}{Canonical Abundances} \\[0.5mm]
\hline
True Values& -2 & -2.3 & -3 \\[0.5mm]
\hline
\multicolumn{4}{l}{\emph{Single-Instrument Configurations}} \\[0.5mm]
\emph{3~mbar Cloud Deck} \\[0.5mm]
NIRSpec G140H & $(-1.89)$ & $-2.39^{+0.97}_{-0.93}$ & $-3.41^{+0.98}_{-0.96}$ \\[0.5mm]
NIRSpec G235H & $-2.45^{+0.71}_{-0.60}$ & $2.62^{+0.49}_{-0.41}$ & $-3.33^{+0.56}_{-0.49}$ \\[0.5mm]
NIRSpec G395H & $(-0.88)$ & $-1.50^{+0.49}_{-1.23}$ & $-2.85^{+0.78}_{-1.32}$ \\[0.5mm]
NIRISS & $-1.93^{+0.65}_{-0.70}$ & $-2.48^{+0.50}_{-0.47}$ & $-3.46^{+0.51}_{-0.50}$ \\[0.5mm]
\\[0.5mm]
\multicolumn{4}{l}{\emph{Three-Instrument Configuration}} \\[0.5mm]
Cloud-Free & $-1.95^{+0.29}_{-0.40}$ & $2.26^{+0.20}_{-0.25}$ & $-2.89^{+0.20}_{-0.26}$\\[0.5mm]
10$^{-3}$~bar Cloud Deck & $-1.69^{+0.37}_{-0.87}$ & $-2.35^{+0.31}_{-0.60}$ & $-3.45^{+0.39}_{-0.66}$ \\[0.5mm]
10$^{-4}$~bar Cloud Deck & $-2.63^{+0.64}_{-0.79}$ & $-2.91^{+0.55}_{-0.65}$ & $-3.82^{+0.68}_{-0.87}$ \\[0.5mm]
10$^{-4.5}$~bar Cloud Deck & $(-2.62)$ & $-2.76^{+0.65}_{-0.86}$ & $(-3.37)$ \\[0.5mm]
\hline
\hline
\multicolumn{4}{c}{Depleted CH$_4$ and NH$_3$} \\[0.5mm]
\hline
True Values& -2 & -3.3 & -4 \\[0.5mm]
\hline
\multicolumn{4}{l}{\emph{Three-Instrument Configuration}} \\[0.5mm]
Cloud-Free & $-2.07^{+0.28}_{-0.28}$ & $-3.34^{+0.19}_{-0.19}$ & $-4.01^{+0.19}_{-0.20}$ \\[0.5mm]
10$^{-2}$~bar Cloud Deck & $-2.11^{+0.62}_{-0.59}$ & $-3.21^{+0.40}_{-0.39}$ & $-4.31^{+0.40}_{-0.41}$ \\[0.5mm]
10$^{-3}$~bar Cloud Deck & $-2.42^{+0.76}_{-0.80}$ & $-3.24^{+0.59}_{-0.58}$ & $-3.83^{+0.60}_{-0.60}$ \\[0.5mm]
10$^{-4}$~bar Cloud Deck & $-2.97^{+0.70}_{-1.24}$ & $-4.16^{+0.71}_{-1.10}$ & $(-3.92)$ \\[0.5mm]
\hline
\end{tabular}
\newline
\footnotesize{Note: In cases where the lower 1-$\sigma$ interval spans more than 2 dex, we instead list the 2-$\sigma$ (95\%) upper estimate in brackets.}
\label{tab:TOI-732c_results}
\end{table}
\subsubsection{Single-Instrument Observations}
\label{sec:TOI-732_single_instruments}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/TOI-732c_singleinstrument.pdf}
\caption{ Posterior distributions of retrievals carried out for the more optimistic case of TOI-732~c with our canonical cloudy model atmosphere, retrieving on simulated single-instrument observations with NIRSpec G140H (orange), G235H (green) and G395H (blue) as well as NIRISS (purple), retrieving on data from one instrument at a time. All data were generated for a nominal atmosphere at 10$\times$ solar elemental abundance with a cloud deck at 3~mbar. Black vertical lines denote the true values used to generate the data, while horizontal errorbars denote the median value and 1-$\sigma$ intervals. If a retrieval produces a lower 1-$\sigma$ interval spanning more than 2~dex, the errorbar is replaced with an arrow denoting the 2-$\sigma$ (95\%) confidence upper limit. }
\label{fig:TOI-732c_singleinstruments}
\end{figure*}
In this section, we explore how retrievals on the same single-transit single-instrument configurations perform for the more representative case of TOI-732~c. Figure \ref{fig:TOI-732c_singleinstruments} shows the retrieved posterior distributions for all four instrument configurations. Comparing these results to the equivalent for K2-18~b in figure \ref{fig:singleinstrument_clouds_posteriors}, we find that there is significant improvement in the abundance constraints the retrievals produce, particularly for H$_2$O and NH$_3$.
Both NIRSpec G235H and NIRISS observations lead to precise constraints for all three molecules. With NIRSpec G235H, the retrieved log-mixing ratio estimates for H$_2$O, CH$_4$ and NH$_3$ are, $-2.45^{+0.71}_{-0.60}$, $-2.62^{+0.49}_{-0.41}$ and $-3.33^{+0.56}_{-0.49}$, respectively. Using NIRISS data, the log-mixing ratio estimates obtained of H$_2$O, CH$_4$ and NH$_3$ are $-1.93^{+0.65}_{-0.70}$, $-2.48^{+0.50}_{-0.47}$ and $-3.46^{+0.51}_{-0.50}$, respectively, which are marginally less precise than those obtained using NIRSpec G235H. NIRSpec G140H and NIRSpec G395H also lead to CH$_4$ and NH$_3$ detections. With NIRSpec G140H data, our retrievals constrained CH$_4$ and NH$_3$ with an uncertainty of $\sim$1~dex for both molecules. The retrieval on NIRspec G395H data produced upper and lower 1-$\sigma$ intervals of +0.5 and -1.2~dex, respectively, for CH$_4$ and +0.8 and -1.3~dex, respectively, for NH$_3$. While neither retrieval was as successful in constraining the mixing-ratio of H$_2$O as the NIRISS and NIRSpec G235H retrievals, the NIRSpec G140H and G395H posterior distributions obtained are more strongly peaked close to the correct value than the equivalent posteriors obtained for K2-18~b.
The better performance of the retrievals shown in this section compared to those in section \ref{sec:single_transit_obs} is driven by a substantial improvement in SNR, thanks to TOI-732~c having a brighter host star than K2-18~b, as well as having larger spectral features. We highlight that a similar improvement can also be achieved for a K2-18~b-like planet by observing more than 1 transit with a given instrument. Repeat retrievals on new data instances for all three NIRSpec gratings produce similar results to those shown in figure \ref{fig:TOI-732c_singleinstruments}. Retrievals on NIRISS data show some variability, in some cases obtaining H$_2$O posteriors with spread towards lower abundances.
\subsubsection{Three Instruments}
\label{sec:benchmark_canonical}
We now consider retrievals on TOI-732~c observations with the three-instrument NIRSpec G140H + G235H + G395H configuration for different cloud deck pressures. The posterior distributions for four retrievals, carried out for different cloud deck pressures up to $10^{-4.5}$~bar (0.03~mbar), are presented in figure \ref{fig:TOI-732c_benchmark_equilibrium}. Starting with the base case of a cloud-free atmosphere, we find that our retrieval obtains atmospheric parameter constraints with better precision than those obtained for K2-18~b in section \ref{sec:cloud_free_reference}. Specifically, we retrieve log-mixing ratio estimates of -1.95$^{+0.29}_{-0.40}$ for H$_2$O, -2.26$^{+0.20}_{-0.25}$ for CH$_4$ and -2.89$^{+0.20}_{-0.26}$ for NH$_3$. For the case where the cloud deck is at 10$^{-3}$~bar (1~mbar), an even lower pressure than the 3~mbar considered in section \ref{sec:wavelength_coverage}, we once again obtain good constraints for the mixing ratios of all three molecules, with upper and lower 1-$\sigma$ intervals of +0.4 and -0.9 for H$_2$O, +0.3 and -0.6 for CH$_4$ and +0.4 and -0.7 for NH$_3$. These uncertainties are comparable to those obtained in section \ref{sec:three_instrument_obs} for K2-18~b with a less obstructive cloud deck at 3~mbar, thanks to the significantly improved SNR that TOI-732~c offers.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/TOI-732c_threeinstruments_equilibrium.pdf}
\caption{Posterior distributions for retrievals carried out on synthetic data for TOI-732~c, a more spectroscopically favourable planet than K2-18~b, with H$_2$O, CH$_4$ and NH$_3$ at 10$\times$ solar abundances and cloud decks at pressures of 10$^{-3}$~bar (green), 10$^{-4}$~bar (blue) and 10$^{-4.5}$~bar (purple), as well as without a cloud deck present (orange). All four retrievals were carried out on combined simulated observations from NIRSpec G140H, G235H and G395H. Black vertical lines denote the true values used to generate the data, while errorbars denote each parameter's median retrieved values and corresponding 1-$\sigma$ intervals obtained in each of the retrievals. In cases where the retrieval only finds an upper limit for a molecular abundance, the errorbar is replaced with a arrow denoting the 2-$\sigma$ confidence upper limit.}
\label{fig:TOI-732c_benchmark_equilibrium}
\end{figure*}
We find that reliable abundance constraints for all three molecules in TOI-732~c can be obtained for cloud deck pressures as low as 10$^{-4}$~bar (0.1~mbar). As shown in blue in figure \ref{fig:TOI-732c_benchmark_equilibrium}, we find that the mixing ratios of all three molecules are well retrieved, albeit at a lower precision than in the 1~mbar case. Our retrieval obtains upper and lower 1-$\sigma$ interval values of +0.6 and -0.8~dex for H$_2$O, +0.6 and -0.7~dex for CH$_4$ and +0.7 and -0.9~dex for NH$_3$. Additionally, NH$_3$ shows a slight low abundance tail.
Having a cloud deck at $10^{-4.5}$~bar (0.03~mbar), shown in purple in figure \ref{fig:TOI-732c_benchmark_equilibrium}, precludes good constraints on H$_2$O and NH$_3$. Nevertheless, the CH$_4$ mixing ratio is still retrieved with uncertainties that are only slightly larger than those obtained in the 0.1~mbar case. We note that the retrieved isotherm temperature is now showing two broad peaks, one at $\sim$200~K and one at $\sim$500~K. This indicates that the retrieval is being led astray by degeneracies, which is why it fails to constrain both H$_2$O and NH$_3$, which are both less spectrally dominant than CH$_4$. We additionally considered the case of a cloud-deck at 10$^{-5}$~bar and found that none of the three mixing ratios can be meaningfully constrained.
We highlight that these abundance constraints in all cases were obtained by retrieving on unbinned native-resolution (R$\sim$2700) NIRSpec data. By avoiding binning, retrievals can take advantage of strong absorption features from H$_2$O, CH$_4$ and NH$_3$ that remain observable at the native resolution despite the significant truncation from the cloud deck. This is particularly the case for $P_c = $ 0.1 mbar which provides good constraints on all the three molecules despite the presence of a high altitude cloud-deck. Our findings are the medium-resolution equivalent of prior reports in literature that high-resolution (R$\gtrsim$25,000) observations are capable of overcoming clouds in mini-Neptune atmospheres \citep{Gandhi2020, Hood2020}.
We also explore the impact of binning the observed spectra on the retrieved abundance constraints for cloudy atmospheres. We find that for cases with very high altitude clouds, excessive binning of spectra could lead to information loss resulting in less precise and less accurate abundance estimates. For example, considering the $P_\mathrm{c} = $ 0.1 mbar case of TOI-732~c, we find that binning to R=50 results in a largely unconstrained H$_2$O abundance and an NH$_3$ posterior with a significant low abundance tail. On the other hand, binning to R$\gtrsim$500 allows meaningful constraints on all three molecules approaching those achieved with native resolution spectra. The effect of binning is reduced for atmospheres that are cloud-free or with lower-altitude clouds.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/TOI-732c_threeinstruments_depletion.pdf}
\caption{ Posterior distributions for retrievals of TOI-732~c with depleted CH$_4$ and NH$_3$ abundances relative to the canonical model. The volume mixing ratios of H$_2$O, CH$_4$ and NH$_3$ are $10^{-2}$, 5$\times 10^{-4}$ and $10^{-4}$, respectively, the latter two being lower by 1~dex compared to the canonical model used in rest of this work. Each retrieval corresponds to a different cloud deck pressure: 10$^{-3}$~bar (green), 10$^{-4}$~bar (blue), 10$^{-4.5}$~bar (purple) and a cloud-free case (orange). All four retrievals were carried out on combined simulated observations with NIRSpec G140H, G235H and G395H. Black vertical lines denote the true values used to generate the data, while points with error bars denote each parameter's median retrieved values and corresponding 1-$\sigma$ intervals. In cases where the retrieval only finds an upper limit for a molecular abundance, the error bar is replaced with an arrow denoting the 2-$\sigma$ upper limit. }
\label{fig:TOI-732c_benchmark_depletion}
\end{figure*}
We carry out repeat retrievals on different noise instances to ensure that our results are not the result of noise-specific features. We confirm that repeat retrievals carried out with no clouds present are consistently successful at retrieving all atmospheric parameters, with the expected statistical variability, i.e. the majority of retrieved estimates lying within 1-$\sigma$ of true values while a minority between 1- and 2-$\sigma$. The same holds true for the 10$^{-3}$~bar case. Repeating retrievals with a cloud deck at 10$^{-4}$~bar, we find that in some cases H$_2$O and NH$_3$ display low abundance ``tails'' of varying severity, indicating that both are at the limit of observability. In a small minority of noise instances, H$_2$O is wholly unconstrained. Additionally, repeat retrievals with a $10^{-4.5}$~bar cloud deck sometimes find slight indications that H$_2$O or NH$_3$ may be present in the form of posterior distributions that are largely unconstrained but notably peaked near the true value. Encouragingly, the cloud deck pressure is consistently retrieved successfully, which means that it is possible to diagnose that the H$_2$O and NH$_3$ non-detections are due to high altitude clouds, and the solution to detecting them would be to observe additional transits to improve the spectroscopic precision of the data.
We therefore find that native-resolution JWST observations of mini-Neptunes more amenable to spectroscopic observation than K2-18~b, such as TOI-732~c, can lead to atmospheric abundance constraints even in the presence of clouds at pressures lower than the 3~mbar considered in section \ref{sec:wavelength_coverage}. The minimum cloud deck pressure that each molecule is detectable at, however, varies. For the atmospheric composition considered here, the mixing ratio of H$_2$O and NH$_3$ can no longer be constrained for cloud deck pressures of $10^{-4.5}$ (0.03~mbar) and below, while our retrievals can still estimate the abundance of CH$_4$ even with a cloud deck at the same cloud deck pressure. The minimum cloud deck pressure that each molecule is detectable is also expected to vary depending on the specific atmospheric composition. Exploring this is the focus of the next section.
\subsubsection{Three Instruments + Depleted CH$_4$ and NH$_3$}
\label{sec:benchmark_depleted}
We now carry out the same benchmark as before, this time deviating from the canonical abundances used so far in this work. Specifically, we consider a new atmospheric composition, with CH$_4$ and NH$_3$ at mixing ratios of 5$\times$10$^{-4}$ and 10$^{-4}$, respectively, i.e. depleted by 1~dex relative to the canonical model, and the same 0.01 H$_2$O mixing ratio as before. As discussed in section \ref{sec:canonical_model}, this scenario corresponds to an atmosphere where atmospheric processes have either enhanced the abundance of H$_2$O or reduced the abundances of CH$_4$ and NH$_3$.
As seen in figure \ref{fig:contribution_plot}, which corresponds to the canonical 10$\times$~solar composition, CH$_4$ and NH$_3$ have significant spectral contributions which often mask those from H$_2$O. As a result, the H$_2$O mixing ratio is consistently the least precisely constrained in sections \ref{sec:TOI-732_single_instruments} and \ref{fig:TOI-732c_benchmark_equilibrium}. Now that CH$_4$ and NH$_3$ are less abundant, it can therefore be expected that the H$_2$O mixing ratio will be more precisely retrieved. Conversely, CH$_4$ and especially NH$_3$, which has weaker spectral features than CH$_4$, are now expected to be the ones that are less easily constrained.
Figure \ref{fig:TOI-732c_benchmark_depletion} shows the obtained posterior distributions for the cloud free case as well as when clouds are present at pressures of 10$^{-2}$, 10$^{-3}$ and 10$^{-4}$~bar. In the cloud-free case, the retrieved log-mixing ratios are -2.07$^{+0.28}_{-0.28}$ for H$_2$O, -3.34$^{+0.19}_{-0.19}$ for CH$_4$ and -4.01$^{+0.19}_{-0.20}$ for NH$_3$. Notably, all three are retrieved more precisely than in the non-depleted case presented in section \ref{sec:benchmark_canonical}. This indicates that H$_2$O features being more prominent not only help in obtaining more precise H$_2$O abundances, but also help improve the precision of the CH$_4$ and NH$_3$ estimates. This is due to the inherent degeneracy between molecular abundances when absorption features partially mask each other, which is alleviated when CH$_4$ and NH$_3$ are at lower abundances.
Both the 10$^{-2}$ and 10$^{-3}$~bar cloud deck retrievals succeeded in constraining the mixing ratios of all three molecules. With a cloud deck present at 10$^{-2}$~bar, the H$_2$O mixing ratio is retrieved with a precision of 0.6~dex while CH$_4$ and NH$_3$ are both retrieved with 0.4~dex precision. This is a significant deterioration compared to the constraints obtained in the cloud-free case. Setting the cloud deck pressure to 10$^{-3}$~bar, the retrieval again obtain accurate mixing ratio estimates, with an uncertainty 0.8~dex for H$_2$O and 0.6~dex for CH$_4$ and NH$_3$. These uncertainties are greater than for the equivalent 10$^{-3}$~bar cloud deck retrieval for the canonical abundance case, indicative of the greater impact that a cloud deck can have on a transmission spectrum at lower atmospheric abundances.
Lastly, when the cloud deck lies at 10$^{-4}$~bar, the retrieval is no longer able to constrain the mixing ratio of NH$_3$. The retrieval is able to offer constraints for the H$_2$O and CH$_4$ mixing ratios, with both lying more than 1-$\sigma$ away from the true values but are still within 2-$\sigma$. Specifically, H$_2$O is retrieved with upper and lower 1-$\sigma$ intervals of +0.7 and -1.2~dex, respectively, while for CH$_4$ the upper and lower 1-$\sigma$ intervals are +0.7 and -1.1.
Similarly to before, we carry out repeat retrievals on new noise instances to establish the repeatability of our findings. Both the cloud-free and 10$^{-2}$~bar cloud deck cases produce consistent results, with all three molecules retrieved with precisions comparable to those described above, and the majority of estimates lying within 1-$\sigma$ of the true value and a minority lying between 1- and 2-$\sigma$. For the 10$^{-3}$~bar cloud deck case, retrievals occasionally produce a low abundance tail of varying severity in the NH$_3$ posterior distribution. This indicates that in order to constrain the abundance of NH$_3$, there must be no significant degradation of NH$_3$ spectral features by random noise. Lastly, repeat retrievals with a 10$^{-4}$~bar cloud deck are sometimes able to produce partially constrained NH$_3$ posteriors that are peaked close to the true values, offering a tentative indication that NH$_3$ may be present. In several cases,we obtain H$_2$O and CH$_4$ posteriors that also display low abundance ``tails'', indicating that both are at the limit of observability.
Our results therefore show that abundance constraints for all three dominant molecules are still achievable when CH$_4$ and NH$_3$ are depleted relative to H$_2$O, for cloud top pressures down to $\sim 10^{-3}$~bar. We find that in the non-cloudy case, depletion of CH$_4$ and NH$_3$ leads to even more precise abundance constraints not just for H$_2$O but also for CH$_4$ and NH$_3$. The precision to which these abundance measurements are made however is more sensitive to cloud deck pressure, showing more rapid deterioration as clouds are placed at higher and higher altitudes. We find that NH$_3$ remains detectable down to a cloud deck pressure of 10$^{-3}$~bar, while H$_2$O and CH$_4$ can be detected in atmospheres with 10$^{-4}$~bar cloud decks. Extrapolating from our findings, it can be expected that for mini-Neptune atmospheres that are more enriched than those used in this work, high-altitude clouds will have a comparatively lesser impact on the transmission spectrum.
\section{Summary and Discussion}
\label{sec:conclusion}
The atmospheric characterisation of temperate, low-mass exoplanets is a major frontier of the JWST era. In this work, we investigate the potential of transit spectroscopy with JWST for characterising the atmospheric compositions of temperate, cloudy mini-Neptunes. We first examine how precisely atmospheric abundances can be retrieved in the cloud-free case, and how these constraints are affected when high-altitude clouds are present. Additionally, we investigate what abundance constraints are achievable with different JWST instrument combinations. We also explore the constraints possible for different atmospheric compositions, highlighting the interplay between key molecular features. We pursue these investigations for two mini-Neptune prototypes, K2-18~b and TOI-732~c, simulating observations with JWST NIRISS and the three high-resolution NIRSpec gratings G140H, G235H and G395H. In all the cases, we assume a canonical atmospheric composition with 10$\times$~solar metallicity and single-transit observations with each instrument considered.
Our primary finding is that JWST transmission spectroscopy of temperate mini-Neptunes orbiting bright M dwarfs can provide precise constraints on their atmospheric compositions with modest observing time even in the presence of clouds at significantly high altitudes. In what follows, we summarise our main results, beginning with K2-18~b, a habitable-zone mini-Neptune orbiting an M2.5 dwarf, chosen as a relatively conservative case.
\begin{itemize}
\item For a cloud-free K2-18~b, single-transit, single-instrument observations with NIRISS or NIRSpec G235H can provide abundance constraints for H$_2$O, CH$_4$ and NH$_3$ with precisions better than $\sim$0.8~dex. A three-instrument combination using NIRspec G140H + G235H + G395H, in the 1-5 $\mu$m range provides constraints better than $\sim$0.3~dex for all three molecules.
\item For a cloudy K2-18~b multiple observations are required to obtain precise and robust abundance constraints. Single-transit three-instrument observations using NIRSpec G140H + G235H + G395H lead to precise abundance constraints for cloud top pressures as low as 3~mbar. H$_2$O, CH$_4$ and NH$_3$ are retrieved with precisions of $\sim$0.7, $\sim$0.5 and $\sim$0.6~dex, respectively.
\item Considering two-instrument combinations, NIRSpec G235H + G395H provides the best constraints, with precisions of $\sim$0.9~dex for H$_2$O and $\sim$0.6~dex for both CH$_4$ and NH$_3$, assuming the same cloud-top pressure as above.
\end{itemize}
We then consider a cloudy TOI-732~c, a more observationally favourable mini-Neptune due to its higher temperature, lower gravity, and smaller and brighter host star compared to K2-18~b.
\begin{itemize}
\item With a cloud deck at 3~mbar, single-instrument observations with NIRISS or NIRSpec G235H lead to good constraints on all three molecules, with precisions $\lesssim$0.7~dex for H$_2$O and $\lesssim$0.5~dex for CH$_4$ and NH$_3$.
\item Furthermore, using a three-instrument combination with NIRSpec G140H + G235H + G395H, H$_2$O and NH$_3$ can be constrained precisely with cloud decks at pressures as low as 0.1~mbar, while CH$_4$ constraints remain possible for cloud deck pressures down to 0.03~mbar.
\item Depleting CH$_4$ and NH$_3$ by 1~dex relative to the canonical 10$\times$~solar case, the mixing ratio of NH$_3$ can still be constrained with a cloud deck at 1~mbar, while H$_2$O and CH$_4$ constraints are still possible with a 0.1~mbar cloud deck.
\end{itemize}
We note that throughout this work, we have assumed a worst-case scenario of a grey cloud opacity. In reality, this may not be the case as clouds may have a non-gray opacity with lower impact on the spectrum at longer wavelengths. Our findings are therefore conservative in that regard, as any reduction in cloud opacity will lead to improved constraints on atmospheric parameters, provided such considerations are included in retrieval models. In principle, the shorter wavelengths below $\sim$1~$\mu$m accessible with NIRISS could provide additional constraints on the scattering slope due to clouds/hazes. Additionally, the NIRSpec G395H spectral range contains some opacity windows between $\sim$4-5 $\mu$m in the absence of other molecular absorption (e.g. CO or CO$_2$) beyond those considered here. Such opacity windows could also provide constraints on contributions of clouds/hazes to the spectral continuum in the infrared.
We have also assumed that the planet's terminator is fully covered by clouds, again as a worst-case scenario. In reality there may only be partial cloud coverage, resulting in the troughs between spectral features lying on a continuum from ``V''-shaped in the cloud-free case to ``U''-shaped in the fully cloudy case. With high-precision JWST spectra, atmospheric retrievals would enable accurate inferences of both the chemical abundances as well as the cloud parameters.
All our retrievals have been carried out on simulated observations at their native resolutions. We note however that it is normal practice to bin spectra down to a specific resolution. As we mention in section \ref{sec:benchmark_canonical}, excessive binning may result in loss of information, particularly if there are very high-altitude clouds present, leading to poorly constrained or unconstrained abundance estimates. However, we find that binned NIRSpec spectra with R$\gtrsim$500 provide constraints approaching those obtained with native resolution spectra for our limiting case with a 0.1 mbar cloud deck in TOI-732~c. While our focus throughout this work has been on the high-resolution NIRSpec gratings, NIRSpec also offers medium-resolution equivalents. Given our findings, the medium-resolution gratings are expected to yield constraints that are comparable to those from the high-resolution gratings particularly for atmospheres that are cloud-free or with low-altitude cloud decks.
K2-18~b in particular is set to be extensively observed during Cycle 1 as part of GO programs 2372 and 2722 (PI: Renyu Hu and Nikku Madhusudhan, respectively). Program 2372 is set to observe 2 transits with NIRSpec G235H and 4 with NIRSpec G395H, while program 2722 will combine observations from NIRISS, NIRSpec G395H and MIRI LRS, allocating one transit to each. Given both programs will achieve an extensive wavelength coverage, both can be expected to lead to abundance constraints for H$_2$O, CH$_4$ and NH$_3$ with precisions comparable to or better than those we find in this work. Moreover, CO and CO$_2$, which may be present in trace amounts, could also be detectable under certain conditions. This is thanks to the large number of transits set to be observed with NIRSpec G395H, which offers a wavelength range encompassing the prominent spectral features of both molecules.
It can be noted that throughout our results, the mixing ratios of CH$_4$ and NH$_3$ are more readily constrained than that of H$_2$O. This is evident from both the tighter abundance constraints obtained for CH$_4$ and NH$_3$ compared to H$_2$O with the canonical model in section \ref{sec:wavelength_coverage}, as well in section \ref{sec:benchmark_canonical}, where the retrieved constraints for CH$_4$ and NH$_3$ deteriorate less rapidly than those for H$_2$O with increasing cloud deck altitudes. Even when CH$_4$ and NH$_3$ were depleted by 1~dex in section \ref{sec:benchmark_depleted}, CH$_4$ remained similarly detectable compared to the significantly more abundant H$_2$O. This is in contrast to the failure so far in detecting either CH$_4$ or NH$_3$ in a temperate exoplanet atmosphere, the so-called ``Missing Methane'' Problem \citep{Stevenson2010, Madhu2011, Madhusudhan2020}. Given the ease with which both CH$_4$ and NH$_3$ can be detected, it is evident that JWST is crucial in resolving this problem.
In this work, we focus on the two mini-Neptunes K2-18~b and TOI-732~c, in both cases using atmospheric compositions where H$_2$O is always at super-solar abundances, while CH$_4$ and NH$_3$ are at either solar or super-solar abundances. In reality, compositions that significantly deviate from those considered here, such as a more extreme depletion of CH$_4$ and NH$_3$, are possible and would be consistent with current HST observations of K2-18~b \citep{Benneke2019, Madhusudhan2020}. Such depletions may either lead to even more precise constraints, by further alleviating degeneracies arising from overlapping spectral features as seen in section \ref{sec:benchmark_depleted}, or in more extreme cases, make CH$_4$ and NH$_3$ harder to detect as their spectral contributions may be masked by stronger H$_2$O features.
Our study benchmarks and showcases the ability of JWST to constrain atmospheric abundances and other properties of temperate mini-Neptunes with unprecedented precision, vastly outperforming what is currently possible with HST. This is thanks to a generational improvement in wavelength coverage, sensitivity and resolution, which together allow for atmospheres to be characterised even with high-altitude clouds and in the absence of supporting data in the optical.
\section*{Acknowledgements}
We thank the anonymous referee for their valuable review and feedback. We additionally thank Luis Welbanks and Subhajit Sarkar for helpful discussions. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (\url{www.csd3.cam.ac.uk}), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1), and DiRAC funding from the Science and Technology Facilities Council (\url{www.dirac.ac.uk}).
\section*{Data Availability}
No new data were generated or analysed in support of this research.
\bibliographystyle{mnras}
|
1,108,101,565,498 | arxiv | \section{Introduction}
The ultimate fate of the star (when it undergoes catastrophic phase
of collapse) is one of the most important questions in gravitation
theory today. When a star has exhausted all of its nuclear fuel, it
collapses under the influence of its own gravity and releases large
amount of energy. In fact, it is a highly dissipative process, i.e.,
energy is not conserved in it, rather due to various forces and with
the passage of time, it becomes lesser. Dissipative process plays
dominant role in the formation and evolution of stars.
The initial discussion over this problem was given by Oppenheimer
and Snyder \cite{1} who assumed a spherically symmetric distribution
of matter. They took the most simplest form of matter, i.e., dust
and the flow is considered to be adiabatic. It is somewhat
unrealistic to ignore the pressure as it cannot be overlooked in the
formation of singularity. Misner and Sharp \cite{2} adopted a better
approach by considering an ideal fluid which gave a more realistic
analysis of gravitational collapse. Both of them assumed vacuum in
the exterior region. Vaidya \cite{3} introduced a non-vacuum
exterior by giving the idea of outgoing radiation in collapse. It
was physically a quite reasonable assumption as radiation is a
confirmation that dissipative processes are occurring, causing loss
of thermal energy of the system which is an effective way of
decreasing internal pressure.
The Darmois junction conditions \cite{4} gave a way to obtain
exact models of an interior spacetime with heat flux to match with
exterior Vaidya spacetime. Sharif and Ahmad \cite{5} considered
the perfect fluid with positive cosmological constant to discuss
the junction conditions with spherical symmetry. The same authors
\cite{6} also worked on junction conditions for plane symmetric
spacetimes.
Goswami \cite{7} made an attempt in search of a more physical
model of collapse. He considered dust like matter with heat flux
to conclude that dissipation causes a bounce in collapse before
the formation of singularity. Nath at el. \cite{8} investigated
dissipation in the form of heat flow and formulated junction
conditions between charged Vaidya spacetime in exterior and
quasi-spherical Szekeres spacetime in interior regions. They also
discussed apparent horizons and singularity formation. Ghosh and
Deshkar \cite{9} studied gravitational collapse of radiating star
with plane symmetry and pointed out some useful results. A lot of
work is being done over gravitational collapse by considering
shear free motion of the fluid. Although, it leads to
simplification in obtaining exact solutions of the field
equations, yet it is an unrealistic approach. Shear viscosity is a
source of dissipating energy and plays an important role in
collapse. Chan \cite{10} investigated gravitational collapse, with
radial heat flow, radiation and shear viscosity. He showed how the
pressure became anisotropic due to shear viscosity.
Herrera and Santos \cite{11} discussed the dynamics of
gravitational collapse which undergoes dissipation in the form of
heat flow and radiation. Di Prisco et al. \cite{12} extended this
work by adding charge and dissipation in the form of shear
viscosity. Herrera \cite{13} provided comprehensive details of
inertia of heat and how it plays an effective role in dynamics of
dissipative collapse. Herrera and Martinez \cite{14} presented
relativistic model of heat conducting collapsing object and
debated over the effect of a parameter which occurs in dynamical
equation on collapse. Herrera and collaborators
\cite{15}-\cite{16} proposed a model of shear free conformally
flat collapse and focused on the role of relaxation process, local
anisotropy and relation between dissipation and density
inhomogeneity.
Recently, Herrera et al. \cite{17} threw light on behavior of
non-equilibrium massive object which lost energy due to heat flow,
radiation, shear and bulk viscosity. Matter under consideration was
distributed with spherical symmetry. It has become quite clear that
when mass and energy densities involved in the physical phenomenon
are sufficiently high as in gravitational collapse, gravitational
field plays an important and dominant role. The gravitational
dynamics then must be taken into account for a meaningful
description of such ultra high energy objects. This fact motivated
us to elaborate the above mentioned paper in the context of plane
symmetries. Matter under consideration is a complicated fluid which
suffers through dissipation. Misner and Sharp's prescription is used
to work out dynamical equations. Transport equations are obtained in
the context of M$\ddot{u}$ller Israel Stewart theory \cite{18},
\cite{19} which is a causal theory for dissipative fluids.
Thermodynamic viscous/heat coupling coefficients are taken to be
non-vanishing which is expected to be quite plausible in non-uniform
stellar models of universe. One of the dynamical equations is then
coupled to transport equations in order to figure out the influence
of dissipation over collapse.
The paper is written in the following manner. The next section is
about the matter distribution in the interior region and some
physical quantities relevant to matter under consideration. The
Einstein field equation are worked out in section \textbf{3} and
junction conditions are discussed in section \textbf{4}. Dynamical
equations are formulated in section \textbf{5} and are coupled to
transport equations in section \textbf{6}. The last section
discusses and concludes the main results of the paper.
\section{Interior matter distribution and some physical quantities}
A $4$-dimensional spacetime is split into two regions: interior
$V^{-}$ and exterior $V^{+}$ through a hypersurface $\Sigma$ which
is the boundary of both regions. We assume the matter distribution
in the interior region to be consistent with plane symmetry. The
interior region $V^{-}$ admits the following line element
\begin{equation}\label{1}
ds^{2}_{-}=-f(t,z)dt^{2}+g(t,z)(dx^{2}+dy^{2})+h(t,z)dz^{2},
\end{equation}
where ${\{\chi^{-\mu}\}}\equiv\{t,x,y,z\}~(\mu=0,1,2,3)$. The fluid
is presumed to dissipate energy in terms of heat flow, radiation,
shearing and bulk viscosity.
The energy-momentum tensor for such a fluid is defined as
\begin{equation}\label{2}
T_{ab}=(\mu+p+\Pi)V_{a}V_{b}+(p+\Pi)g_{ab}+q_{a}V_{b}+q_{b}V_{a}+\epsilon
l_{a}l_{b}+\pi_{ab},
\end{equation}
where $\mu,~p,~\Pi,~q_a,~l_a$ and $\pi_{ab}$ are the energy
density, pressure, bulk viscosity, heat flow, null four-vector in
$z$-direction and shear viscosity tensor respectively. Heat flow
$q_{a}$ is taken to be orthogonal to velocity $V^{a}$, i.e.,
$q_{a}V^{a}=0$. Moreover, we have
\begin{eqnarray}\label{3}
V^{a}V_{a}=-1,\quad l^{a}V_{a}=-1,\quad\pi_{ab}V^{b}=0, \quad
\pi_{[ab]}=0,\quad\pi^{a}_{a}=0,\quad l^{a}l_{a}=0.
\end{eqnarray}
In the standard irreversible thermodynamics by Eckart, we have the
following relation \cite{20}
\begin{equation}\label{4}
\pi_{ab}=-2\eta\sigma_{ab},\quad \Pi=-\zeta\Theta,
\end{equation}
where $\eta$ and $\zeta$ stand for coefficients of shear and bulk
viscosity, $\sigma_{ab}$ is the shear tensor and $\Theta$ is the
expansion. The algebraic nature of Eckart constitutive equations
causes several problems but we are concerned with the causal
approach of dissipative variables. Thus we would not assume
(\ref{4}) rather we shall resort to transport equations of
M$\ddot{u}$ller-Israel-Stewart theory.
The shear tensor $\sigma_{ab}$ is defined as
\begin{equation}\label{5}
\sigma_{ab}=V_{(a;b)}+a_{(a}V_{b)}-\frac{1}{3}\Theta h_{ab},
\end{equation}
where the acceleration $a_{a}$ and the expansion $\Theta$ are
given by
\begin{equation}\label{6}
a_{a}=V_{a;b}V^{b},\quad \Theta=V^{a}_{;a}
\end{equation}
and $h_{ab}=g_{ab}+V_{a}V_{b}$ is the projection tensor. The shear
tensor $\sigma_{ab}$ satisfies
\begin{equation}\label{7}
V_{a}\sigma^{ab}=0,\quad\sigma^{ab}=\sigma^{ba},\quad\sigma^{a}_{a}=0.
\end{equation}
In co-moving coordinates, one can take
\begin{equation}\label{8}
V^{a}=\frac{1}{\sqrt{f}}\delta^{a}_{0},\quad
q^{a}=\frac{q}{\sqrt{h}}\delta^{a}_{3},\quad
l^{a}=\frac{1}{\sqrt{f}}\delta^{a}_{0}+\frac{1}{\sqrt{h}}\delta^{a}_{3},
\end{equation}
here $q$ is a function of $t$ and $z$.
Using Eq.(\ref{8}), the non-vanishing components of the shear
tensor $\sigma_{ab}$ turn out to be
\begin{equation}\label{9}
\sigma_{11}=-\frac{g}{3}\sigma=\sigma_{22},\quad
\sigma_{33}=\frac{2h}{3}\sigma,
\end{equation}
where
\begin{equation}\label{10}
\sigma=\frac{1}{2\sqrt{f}}\left(\frac{\dot{h}}{h}-\frac{\dot{g}}{g}\right).
\end{equation}
Thus we have
\begin{equation}\label{11}
\sigma_{ab}\sigma^{ab}=\frac{2}{3}\sigma^{2}.
\end{equation}
Also, in view of Eqs.(\ref{3}) and (\ref{4}), it yields
\begin{equation}\label{12}
\pi_{0a}=0, \pi^{3}_{3}= -2\pi^{2}_{2}=-2\pi^{1}_{1}.
\end{equation}
In compact form, it can be written as
\begin{equation}\label{13}
\pi_{ab}=\Omega({\chi}_{a}\chi_{b}-\frac{1}{3}h_{ab}),
\end{equation}
where $\Omega=\frac{3}{2}\pi^{3}_{3}$ and $\chi^{a}$ is a
unit four-vector in $z$-direction satisfying
\begin{equation}\label{14}
\chi^{a}\chi_{a}=1,\quad \chi^{a}V_{a}=0,\quad
\chi^{a}=\frac{1}{\sqrt{h}}\delta^{a}_{3}.
\end{equation}
In view of Eqs.(\ref{6}) and (\ref{8}), it follows that
\begin{equation}\label{15}
a_{3}=\frac{f'}{2f},\quad
\Theta=\frac{1}{\sqrt{f}}\left(\frac{\dot{g}}{g}+\frac{\dot{h}}{2h}\right),
\end{equation}
where dot and prime represent derivative with respect to time $t$ and $z$
respectively.
The Taub's mass for plane symmetric spacetime is defined by
\cite{21}
\begin{equation}\label{16}
m(t,z)=\frac{(g)^{3/2}}{2}R^{12}_{12}=
\frac{1}{8\sqrt{g}}\left(\frac{\dot{g}^2}{f}-\frac{g'^2}{h}\right).
\end{equation}
\section{The Einstein field equations}
The Einstein field equations for the metric (\ref{1}) yield the
following set of equations
\begin{equation}\label{17}
\frac{\dot{g}}{2g}\left(\frac{\dot{g}}{2g}+\frac{\dot{h}}{h}\right)
+\frac{fg'}{2gh}\left(\frac{h'}{h}+\frac{g'}{2g}\right)-\frac{fg''}{gh}
=8\pi(\mu+\epsilon)f,
\end{equation}
\begin{eqnarray}\label{18}
&& \frac{\dot{g}}{2f}\left(\frac{\dot{f}}{2f}+\frac{\dot{g}}{2g}
-\frac{\dot{h}}{2h}\right)
+\frac{g'}{4h}\left(\frac{f'}{f}-\frac{h'}{h}
-\frac{g'}{g}\right)-\frac{f'g}{4fh}\left(\frac{h'}{h}+\frac{f'}{f}\right)\nonumber\\
&+&\frac{\dot{h}g}{4fh}\left(\frac{\dot{h}}{h}+\frac{\dot{f}}{f}\right)
-\frac{\ddot{g}}{2f}+\frac{g''}{2h}+\frac{g}{2fh}(f''-\ddot{h})
=8\pi(p+\Pi-\frac{1}{3}\Omega)g,\nonumber\\
\end{eqnarray}
\begin{equation}\label{19}
\frac{g'}{2g}\left(\frac{g'}{2g}+\frac{f'}{f}\right)+\frac{\dot{g}h}{2fg}\left
(\frac{\dot{g}}{2g}+\frac{\dot{f}}{f}\right)-\frac{\ddot{g}h}{fg}
=8\pi(p+\Pi+\epsilon+\frac{2}{3}\Omega)h,
\end{equation}
\begin{equation}\label{20}
\frac{\dot{g}}{2g}\left(\frac{g'}{g}+\frac{f'}{f}\right)+\frac{g'\dot{h}}{2gh}-
\frac{\dot{g}'}{g}=-8\pi(q+\epsilon)\sqrt{fh}.
\end{equation}
After some manipulation, we can also write Eq.(\ref{20}) in the
following form
\begin{equation}\label{21}
4\pi(q+\epsilon)\sqrt{h}=\frac{1}{3}(\Theta-\sigma)'-\sigma\frac{\sqrt{g}~'}{\sqrt{g}}.
\end{equation}
\section{Junction conditions}
We discuss junction conditions for the interior region $V^{-}$ given
by Eq.(\ref{1}) and the exterior region $V^+$ which is taken as
plane symmetric Vaidya spacetime ansatz given by the line element
\cite{22}
\begin{equation}\label{22}
ds^{2}_{+}=\frac{2m(\nu)}{Z}d\nu^{2}-2d{\nu}dZ+Z^{2}(dX^{2}+dY^{2}),
\end{equation}
where $\chi^{+\mu}\equiv \{\nu,X,Y,Z\}~(\mu=0,1,2,3)$, $\nu$ is the
retarded time and $m(\nu)$ represents total mass inside $\Sigma$.
The line element for the hypersurface $\Sigma$ is defined as
\begin{equation}\label{23}
(ds^{2})_{\Sigma}=-{d{\tau}}^2+A^{2}(\tau)(dx^{2}+dy^{2}),
\end{equation}
where $\xi^{i}\equiv(\tau,x,y)~(i=0,1,2)$ are the intrinsic
coordinates of $\Sigma$.
The Darmois junction conditions \cite{4} are
\begin{itemize}
\item The continuity of the line elements over the hypersurface $\Sigma$ gives
\begin{equation}\label{24}
(ds^{2})_{\Sigma}=(ds^{2}_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}.
\end{equation}
This is called continuity of the first
fundamental form.
\item The continuity of the extrinsic curvature $K_{ab}$ over the hypersurface $\Sigma$
yields
\begin{equation}\label{25}
[K_{ij}]=K^{+}_{ij}-K^{-}_{ij}=0,\quad(a,b=0,1,2).
\end{equation}
This is known as continuity of the second fundamental form.
\end{itemize}Here $K^{\pm}_{ij}$ is the extrinsic curvature defined as
\begin{equation}\label{26}
K^{\pm}_{ij}=-n^{\pm}_{\sigma}(\frac{{\partial}^2\chi^{\sigma}_{\pm}}
{{\partial}{\xi}^i{\partial}{\xi}^j}+{\Gamma}^{\sigma}_{{\mu}{\nu}}
\frac{{{\partial}\chi^{\mu}_{\pm}}{{\partial}\chi^{\nu}_{\pm}}}
{{\partial}{\xi}^i{\partial}{\xi}^j}),\quad({\sigma},
{\mu},{\nu}=0,1,2,3).
\end{equation}
where $n^{\pm}_{\sigma}$ are the components of outward unit normal
to hypersurface $\Sigma$ in the coordinates $\chi^{{\pm}\mu}$.
The equations of hypersurface $\Sigma$ in terms of coordinates $\chi^{{\mp}\mu}$
are given as
\begin{eqnarray}\label{27}
k_{-}(t,z)&=&z-z_{\Sigma}=0,\\
k_{+}(\nu,Z)&=&Z-Z_{\Sigma}(\nu)=0,\label{28}
\end{eqnarray}
where $z_{\Sigma}$ is taken to be an arbitrary constant. Using
Eqs.(\ref{27}) and (\ref{28}), the interior and exterior metrics
take the following form over hypersurface $\Sigma$
\begin{eqnarray}\label{29}
(ds^{2}_{-})_{\Sigma}&=&-f(t,z_{\Sigma})dt^{2}+g(t,z_{\Sigma})(dx^{2}+dy^{2}),\\
\label{30} (ds^{2}_{+})_{\Sigma}&=&2\left(\frac{m(\nu)}
{Z_{\Sigma}}-\frac{dZ_{\Sigma}}{d\nu}\right)d\nu^{2}+Z^{2}_{\Sigma}
(dX^{2}+dY^{2}).
\end{eqnarray}
In view of junction condition (\ref{24}), we get
\begin{eqnarray}\label{31}
Z^{2}_{\Sigma}&=&g(t,z_{\Sigma}),\\ \label{32}
\frac{dt}{d\tau}&=&\frac{1}{\sqrt{f}},\\ \label{33}
\frac{d\nu}{d\tau}&=&\left(2\frac{dZ_{\Sigma}}{d\nu}-\frac{2m(\nu)}{Z_{\Sigma}}\right)^{-1/2}.
\end{eqnarray}
Using Eqs.(\ref{27}) and (\ref{28}), the unit normals in $V^{-}$ and $V^{+}$
respectively, turn out to be
\begin{eqnarray}\label{34}
n^{-}_{\mu}&=&\sqrt{h}(0,0,0,1),\\ \label{35}
n^{+}_{\mu}&=&\left[2\left(\frac{dZ}{d\nu}-\frac{m(\nu)}{Z}\right)\right]
^{-1/2}\left(-\frac{dZ}{d\nu},0,0,1\right).
\end{eqnarray}
The non-zero components of the extrinsic curvature $K^{\pm}_{ij}$ are
\begin{eqnarray}\label{36}
K^{-}_{00}&=&-\left(\frac{f'}{2f\sqrt{h}}\right)_{\Sigma},\\ \label{37}
K^{+}_{00}&=&\left[\frac{d^{2}\nu}{d\tau^{2}}\left(\frac{d\nu}{d\tau}\right)^{-1}
-\frac{m}{Z^{2}}\frac{d\nu}{d\tau}\right]_{\Sigma},\\ \label{38}
K^{-}_{11}&=&K^{-}_{22}=\left(\frac{g'}{2\sqrt{h}}\right)_{\Sigma},\\ \label{39}
K^{+}_{11}&=&K^{+}_{22}=\left[Z\frac{dZ}{d\tau}-2m\frac{d\nu}{d\tau}\right]_{\Sigma}.
\end{eqnarray}
Now, by the junction condition (\ref{25}), i.e., continuity of
extrinsic curvatures, it follows that
\begin{equation}\label{40}
\left[\frac{d^{2}\nu}{d\tau^{2}}\left(\frac{d\nu}{d\tau}\right)^{-1}
-\frac{m}{Z^{2}}\frac{d\nu}{d\tau}\right]_{\Sigma}=
-\left(\frac{f'}{2f\sqrt{h}}\right)_{\Sigma},
\end{equation}
\begin{equation}\label{41}
2m\frac{d\nu}{d\tau}=\frac{\dot{g}}{2\sqrt{f}}-\frac{g'}{2\sqrt{h}}.
\end{equation}
Using Eqs.(\ref{33}) and (\ref{41}), we obtain
\begin{equation}\label{43a}
\left(\frac{d\nu}{d\tau}\right)^{-1}=\frac{1}{\sqrt{g}}
\left[\frac{\dot{g}}{2\sqrt{f}}+\frac{g'}{2\sqrt{h}}\right].
\end{equation}
Inserting Eq.(\ref{43a}) in (\ref{41}), it follows that
\begin{equation}\label{42}
m(\nu)=\frac{1}{8\sqrt{g}}\left(\frac{\dot{g}^2}{f}-\frac{g'^2}{h}\right)
\end{equation}
and hence
\begin{equation}\label{43}
m(t,z)\overset{\Sigma}{=}m(\nu).
\end{equation}
Differentiating Eq.(\ref{43a}) with respect to $\tau$, and making use of Eqs.(\ref{42})
and (\ref{43a}), we can write Eq.(\ref{40}) as
\begin{eqnarray}\label{43b}
\frac{1}{2\sqrt{fhg}}\left[\frac{-\dot{g}'}{\sqrt{g}}+\frac{g'\dot{h}}{2h\sqrt{g}}
+\frac{f'\dot{g}}{2f\sqrt{g}}+\frac{\sqrt{h}}{\sqrt{f}}\left\{\frac{{-\ddot{g}}}{\sqrt{g}}
+\frac{\dot{g}\dot{f}}{2f\sqrt{g}}+\sqrt{g}\left(\frac{\dot{g}}{2g}\right)^{2}
\nonumber\right.\right.\\\left.\left.+\frac{f}{4g^{3/2}}
\left(\frac{g'}{\sqrt{h}}\right)^{2}+\frac{f'g'}{2h\sqrt{g}}
+\frac{\sqrt{f}}{\sqrt{h}}\left(\frac{g'\dot{g}}{2g^{3/2}}\right)
\right\}\right]\overset{\Sigma}{=}0.
\end{eqnarray}
Comparing Eq.(\ref{43b}) with Eqs.(\ref{19}) and (\ref{20}), it yields
\begin{equation}\label{43c}
p+\Pi+\frac{2}{3}\Omega=q.
\end{equation}
\section{Dynamical equations}
The energy-momentum conservation, $T^{ab}_{;b}=0$, gives
\begin{eqnarray}\label{44}
T^{ab}_{;b}V_{a}&=&\frac{(\dot{\mu}+\dot{\epsilon})}
{\sqrt{f}}+\frac{(q'+\epsilon')}{\sqrt{h}}
+\frac{\dot{g}}{g\sqrt{f}}(\mu+p+\Pi+\epsilon-\frac{1}{3}\Omega)\nonumber\\
&+&\frac{\dot{h}}{2h\sqrt{f}}(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega)
+\frac{(fg)'}{fg}\frac{(q+\epsilon)}{\sqrt{h}}=0
\end{eqnarray}
and
\begin{eqnarray}\label{45}
T^{ab}_{;b}\chi_{a}&=&\frac{1}{\sqrt{f}}(\dot{q}+\dot{\epsilon})
+\frac{1}{\sqrt{f}}(q+\epsilon)\frac{(hg\dot{)}}{hg}+\frac{1}{\sqrt{h}}
(p'+\Pi'+\epsilon'+\frac{2}{3}\Omega')\nonumber\\
&&+\frac{f'}{2f\sqrt{h}}(p+\Pi+\mu+2\epsilon
+\frac{2}{3}\Omega)+\frac{g'}{g\sqrt{h}}(\epsilon+\Omega)=0.
\end{eqnarray}
Now we investigate the dynamical properties of the system using the
Misner and Sharp's \cite{2} perspective. For this purpose, we take
the proper time derivative as
\begin{equation}\label{46}
D_{T}=\frac{1}{\sqrt{f}}\frac{\partial}{\partial{t}},
\end{equation}
and the proper derivative in $z$-direction as
\begin{equation}\label{47}
D_{\tilde{Z}}=\frac{1}{\tilde{Z}'}\frac{\partial}{\partial{z}},
\end{equation}
where
\begin{equation}\label{48}
\tilde{Z}=\sqrt{g}.
\end{equation}
The velocity $U$ of the collapsing fluid can be defined as the
variation of $\tilde{Z}$ with respect to the proper time
\begin{equation}\label{49}
U=D_{T}(\tilde{Z})=\frac{1}{2\sqrt{g}}D_{T}g.
\end{equation}
In the case of collapse, the velocity of the collapsing fluid must
be negative. In view of Eq.(\ref{49}), Eq.(\ref{16}) can take the
following form
\begin{equation}\label{50}
E=\frac{\sqrt{g}~'}{\sqrt{h}}=[U^{2}-\frac{2}{\sqrt{g}}m(t,z)]^{1/2}.
\end{equation}
Making use of Eq.(\ref{47}) in Eq.(\ref{21}), it follows that
\begin{equation}\label{51}
4\pi(q+\epsilon)=E\left[\frac{1}{3}D_{\tilde{Z}}
(\Theta-\sigma)-\frac{\sigma}{\tilde{Z}}\right].
\end{equation}
In case of no dissipation, using Eqs.(\ref{10}), (\ref{15}) and (\ref{49}), the
above equation becomes
\begin{equation}\label{52}
D_{\tilde{Z}}\left(\frac{U}{\tilde{Z}}\right)=0.
\end{equation}
This implies that $U\sim \tilde{Z}$ depicting that now collapse will
be homologous. The rate of change of Taub's mass, using
Eqs.(\ref{16}), (\ref{19}), (\ref{20}) and (\ref{46}), turn out to
be
\begin{equation}\label{53}
D_{T}m=-4{\pi}\tilde{Z}^{2}[(p+\Pi+\epsilon+\frac{2}{3}\Omega)U+(q+\epsilon)E].
\end{equation}
Thus the rate of change of Taub's mass represents variation of total
energy inside the collapsing plane surface. Since this variation is
negative, it shows that total energy is being dissipated during
collapse. The first round brackets on the right hand side stand for
energy due to work being done by the effective isotropic pressure
$(p+\Pi+\frac{2}{3}\Omega)$ and the radiation pressure $\epsilon$.
The second brackets describe energy leaving the system due to heat
flux and radiation. Similarly, using Eqs.(\ref{16}), (\ref{17}),
(\ref{20}) and (\ref{47}), we get
\begin{equation}\label{54}
D_{\tilde{Z}}m=4{\pi}\tilde{Z}^{2}[\mu+\epsilon+(q+\epsilon)\frac{U}{E}].
\end{equation}
This equation describes about the variation of energy between
adjoining plane surfaces inside the fluid distribution. On the right
hand side, $(\mu+\epsilon)$ stands for energy density of the fluid
element plus the energy of null fluid showing dissipation due to
radiation. Moreover, $(q+\epsilon)\frac{U}{E}$ is negative (as
$U<0$), telling that energy is leaving due to outflow of heat and
radiation.
Making use of Eqs.(\ref{16}), (\ref{19}), (\ref{48}) and (\ref{50}),
the acceleration $D_{T}U$ of the collapsing matter inside the
hypersurface $\Sigma$ is given as
\begin{equation}\label{55}
D_{T}U=-4\pi(p+\Pi+\epsilon+\frac{2}{3}\Omega)\tilde{Z}
-\frac{m}{\tilde{Z}^2}+\frac{Ef'}{2f\sqrt{h}}.
\end{equation}
Substituting the value of $\frac{f'}{2f}$ from the above equation
into Eq.(\ref{45}), it follows that
\begin{eqnarray}\label{56}
(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega)D_{T}U&=&-(p+\Pi
+\mu+2\epsilon+\frac{2}{3}\Omega)\nonumber\\
&&\times[4\pi{\tilde{Z}}(p+\Pi+\epsilon+\frac{2}{3}\Omega)+\frac{m}{{\tilde{Z}}^2}]\nonumber\\
&&-E^2[D_{\tilde{Z}}(p+\Pi+\epsilon+\frac{2}{3}\Omega)
+\frac{2}{\tilde{Z}}(\epsilon+\Omega)]\nonumber\\
&&-E[D_{T}q+D_{T}\epsilon+4(q+\epsilon)\frac{U}
{\tilde{Z}}+2(q+\epsilon)\sigma].\nonumber\\
\end{eqnarray}
This equation has the form of Newton's second law, i.e.,
\begin{equation*}
Force=Mass\quad density\quad\times\quad Acceleration.
\end{equation*}
The term within the brackets on the left hand side stands for
"effective" inertial mass and the remaining term is acceleration.
The first term on the right hand side represents gravitational
force. Since by the equivalence principle, inertial mass is
equivalent to passive gravitational mass and passive gravitational
mass is equivalent to active gravitational mass. Thus the factor
within round brackets stands for active gravitational mass and the
factor within the square brackets shows how dissipation effects
active gravitational mass. The second square brackets firstly
include gradient of effective pressure which involves radiation
pressure and the collective effect of shear and bulk viscosity. The
second contribution is of local anisotropy of pressure which is the
result of radiation and shear viscosity. The last square brackets
entirely depend upon dissipation. The hydrostatic equilibrium can be
obtained from the above equation by substituting
$U=0,~q=0,~\epsilon=0,~\Pi=0$ and $\Omega=0$.
\begin{equation*}
D_{\tilde{Z}}p=-(\mu+p)\frac{h}{\tilde{Z'}^2}\left[\frac{m}{\tilde{Z}^2}
+4\pi \tilde{Z}p\right].
\end{equation*}
\section{Transport equations}
The general expression for entropy $4$-current is given as \cite{20}
\begin{equation}\label{57}
S^{\mu}=SnV^{\mu}+\frac{q^{\mu}}{T}-(\beta_{0}\Pi^{2}+\beta_{1}q_{\nu}q^{\nu}+
\beta_{2}\pi_{\nu\kappa}\pi^{\nu\kappa})\frac{V^{\mu}}{2T}+\frac{\alpha_{0}\Pi
q^{\mu}}{T}+\frac{\alpha_{1}\pi^{\mu\nu}q_{\nu}}{T},
\end{equation}
where $n$ is particle number density, $T$ is temperature,
$\beta_A(\rho,n)\geq{0}$ are thermodynamic coefficients for
scalar, vector and tensor dissipative contributions to the entropy
density and $\alpha_{A}(\rho,n)$ are thermodynamic viscous/heat
coupling coefficients. The divergence of extended current (follows
from Gibbs equation and Bianchi identities) is given by
\begin{eqnarray}\label{58}
TS^\alpha_{;\alpha}&=&-\Pi\left[V^{\alpha}_{;\alpha}
-\alpha_{0}q^\alpha_{;\alpha}+\beta_{0}\Pi_{;\alpha}
V^{\alpha}+\frac{T}{2}\left(\frac{\beta_{0}}{T}
V^{\alpha}\right)_{;\alpha}\Pi\right]\nonumber\\
&-&q^{\alpha}[h^{\mu}_{\alpha}(\ln{T})_{,\mu}(1
+\alpha_{0}\Pi)+V_{\alpha;\mu}V^{\mu}-\alpha_{0}
\Pi_{;\alpha}-\alpha_{1}\pi^{\mu}_{\alpha;\mu}\nonumber\\
&+&\alpha_{1}\pi^{\mu}_{\alpha}h^{\beta}_{\mu}(\ln{T})_{,\beta}
+\beta_{1}q_{\alpha;\mu}
V^{\mu}+\frac{T}{2}\left(\frac{\beta_{1}}{T}V^{\mu}\right)_{;\mu}
q_\alpha]\nonumber\\
&-&\pi^{\alpha\mu}\left[\sigma_{\alpha\mu}
-\alpha_{1}q_{\mu;\alpha}+\beta_{2}\pi_{\alpha\mu;\nu}V^{\nu}
+\frac{T}{2}\left(\frac{\beta_{2}}{T}V^{\nu}\right)_{;\nu}\pi_{\alpha\mu}\right].
\end{eqnarray}
The $2$nd law of thermodynamics requires that
$S^{\alpha}_{;\alpha}\geq{0}$. This leads to the following transport
equations for our dissipative variables
\begin{equation}\label{59}
{\tau}_{0}\Pi_{,\alpha}V^{\alpha}+\Pi=-\zeta\Theta+\alpha_{0}
{\zeta}q^{\alpha}_{;\alpha}-\frac{1}{2}{\zeta} T
\left(\frac{\tau_{0}}{{\zeta}T}V^\alpha\right)_{;\alpha}\Pi,
\end{equation}
\begin{eqnarray}\label{60}
\tau_{1}h^{\beta}_{\alpha}q_{\beta;\mu}V^{\mu}
+q_{\alpha}&=&-k[h^{\beta}_{\alpha}T_{,\beta}
(1+\alpha_{0}\Pi)+\alpha_{1}\pi^{\mu}_{\alpha}
h^{\beta}_{\mu}T_{,\beta}+T(a_{\alpha}\nonumber\\
&-&\alpha_{0}\Pi_{;\alpha}
-\alpha_{1}\pi^{\mu}_{\alpha;\mu})]-\frac{1}{2}
kT^{2}\left(\frac{\tau_{1}}{kT^2}V^{\beta}\right)_{;\beta}q_{\alpha}
\end{eqnarray}
and
\begin{equation}\label{61}
\tau_{2}h^{\mu}_{\alpha}h^{\nu}_{\beta}\pi_{\mu\nu;\rho}
V^{\rho}+\pi_{\alpha\beta}=-2\eta\sigma_{\alpha\beta}
+2\eta\alpha_{1}q_{<\beta;\alpha>}-{\eta}T\left(\frac{\tau_{2}}
{{2\eta}T}V^{\nu}\right)_{;\nu}\pi_{\alpha\beta},
\end{equation}
where
\begin{equation}\label{62}
q_{<\beta;\alpha>}=h^{\mu}_{\beta}h^{\nu}_{\alpha}\left(\frac{1}{2}
(q_{\mu;\nu}+q_{\nu;\mu})-\frac{1}{3}q_{\sigma;\kappa}
h^{\sigma\kappa}h_{\mu\nu}\right),
\end{equation}
with $k$ as the thermal conductivity. The relaxation times are given
by
\begin{equation}\label{63}
\tau_{0}=\zeta\beta_{0},\quad\tau_{1}=kT\beta_{1},\quad\tau_{2}=2\eta\beta_{2}.
\end{equation}
Notice that if the thermodynamic coupling coefficients are assumed
to be zero, Eqs.(\ref{59})-(\ref{61}) turn to be
Eqs.(2.21)-(2.23) as given in \cite{20}.
The independent components of Eqs.(\ref{59})-(\ref{61}) are
calculated as follows.
\begin{eqnarray}\label{64}
\tau_{0}\dot{\Pi}&=&-\left(\zeta+\frac{\tau_{0}\Pi}{2}\right)\Theta\sqrt{f}
+\alpha_{0}\zeta\frac{\sqrt{f}}{\sqrt{h}}\left[q'+q\left(\frac{f'}{2f}
+\frac{g'}{g}\right)\right]\nonumber\\
&-&\left[\frac{\zeta T}{2}\left(\frac{\tau_{0}}{\zeta
T}\right)^{.}+\sqrt{f}\right]\Pi,
\end{eqnarray}
\begin{eqnarray}\label{65}
\tau_{1}\dot{q}&=&-k\frac{\sqrt{f}}{\sqrt{h}}
\left[T'(1+\alpha_{0}\Pi+\frac{2}{3}\alpha_{1}\Omega)
+T\left\{\frac{f'}{2f}-\alpha_{0}\Pi''\right.\right.\nonumber\\
&-&\left.\left.\alpha_{1} \left(\frac{2}{3}\Omega'+\frac{f'}{3f}\Omega
+\frac{g'}{g}\Omega\right)\right\}\right]\nonumber\\
&-&q\left[\frac{kT^{2}}{2}\left(\frac{\tau_{1}}{kT^{2}}\right)^{.}
+\frac{\tau_{1}}{2}\Theta\sqrt{f}+\sqrt{f}\right],
\end{eqnarray}
\begin{eqnarray}\label{66}
\tau_{2}\dot{\Omega}&=&-2\sqrt{f}\eta\sigma+\eta\alpha_{1}\frac{\sqrt{f}}{\sqrt{h}}
(2q'-\frac{g'}{g}q)\nonumber\\
&-&\left[\eta T\left(\frac{\tau_{2}}{2\eta
T}\right)^{.}\Omega+\frac{\tau_{2}}{2}\Theta\sqrt{f}\Omega+\Omega\sqrt{f}\right].
\end{eqnarray}
Now we discuss the action of dissipation over dynamics of collapsing
object. We couple these transport equations to dynamical equation
(\ref{56}). Using Eq.(\ref{65}) in Eq.(\ref{56}), it follows that
\begin{eqnarray}\label{67}
&&(\mu+p+\Pi+2\epsilon+\frac{2}{3}\Omega)(1-\Lambda)D_{T}U=
(1-\Lambda)F_{grav}\nonumber+F_{hyd}\\&&+\frac{kE^2}{\tau_1}\left[
D_{\tilde{Z}}T(1+\alpha_{0}\Pi+\frac{2}{3}\alpha_{1}\Omega)
-T\left\{\alpha_{0}D_{\tilde{Z}}\Pi+\frac{2}{3}\alpha_{1}\left(
D_{\tilde{Z}}\Omega\nonumber+\frac{3}{\tilde{Z}}\Omega\right)
\right\}\right]\\&&+E\left[\frac{kT^2q}{2\tau_1}\nonumber
D_{T}\left(\frac{\tau_{1}}{kT^2}\right)-D_{T}
\epsilon\right]-E\left[\left(\frac{3q}{2}+2\epsilon\right)
\Theta-\frac{q}{\tau_1}-2(q+\epsilon)\frac{U}{\tilde{Z}}\right],\\
\end{eqnarray}
where $F_{grav}$ and $F_{hyd}$ are given by
\begin{eqnarray}\label{68}
F_{grav}&=&-(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega)\nonumber\\
&\times&\left[m
+4\pi(p+\Pi+\epsilon+\frac{2}{3}\Omega){\tilde{Z}}^3\right]\frac{1}{\tilde{Z}^{2}},
\end{eqnarray}
\begin{equation}\label{69}
F_{hyd}=-E^2\left[D_{\tilde{Z}}(p+\Pi+\epsilon+\frac{2}{3}\Omega)
+2(\epsilon+\Omega)\frac{1}{\tilde{Z}}\right]
\end{equation}
and
\begin{equation}\label{70}
\Lambda=\frac{kT}{\tau_{1}}\left(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega\right)^{-1}
\left(1-\frac{2}{3}\alpha_{1}\Omega\right).
\end{equation}
Inserting Eq.(\ref{64}) in Eq.(\ref{67}), we obtain
\begin{eqnarray}\label{71}
&&(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega)(1-\Lambda+\Delta)
D_{T}U=(1-\Lambda+\Delta)F_{grav}+F_{hyd}\nonumber\\
&&+\frac{kE^2}{\tau_1}\left[D_{\tilde{Z}}T\left(1+\alpha_{0}
\Pi+\frac{2}{3}\alpha_{1}\Omega\right)-T\left\{\alpha_{0}
D_{\tilde{Z}}\Pi+\frac{2}{3}\alpha_{1}\left(D_{\tilde{Z}}
\Omega+\frac{3}{\tilde{Z}}\Omega\right)\right\}\right]\nonumber\\
&&-E^2\left(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega\right)
\Delta\left(\frac{D_{\tilde{Z}}q}{q}+\frac{2}{\tilde{Z}}\right)\nonumber\\
&&+E\left[\frac{kT^{2}q}{2\tau_1}D_{T}
\left(\frac{\tau_1}{kT^2}\right)-D_{T}\epsilon\right]
+E\left[\frac{q}{\tau_1}+2(q+\epsilon)\frac{U}{\tilde{Z}}\right]\nonumber\\
&&+E\frac{\Delta}{\alpha_{0}\zeta{q}}\left(p+\Pi+\mu+2\epsilon
+\frac{2}{3}\Omega\right)\left[\left\{1+\frac{\zeta{T}}{2}
D_{T}\left(\frac{\tau_{0}}{\zeta{T}}\right)\right\}\Pi
+\tau_{0}D_{T}\Pi\right],\nonumber\\
\end{eqnarray}
where $\Delta$ is given by
\begin{equation}\label{72}
\alpha_{0}\zeta{q}\left(p+\Pi+\mu+2\epsilon+\frac{2}{3}\Omega\right)
^{-1}\left(\frac{3q+4\epsilon}{2\zeta+\tau_{0}\Pi}\right).
\end{equation}
Here we see that $(1-\Lambda+\Delta)$ is the major factor that
appears in the dynamical equation after coupling it with the
transport equations. We would like to mention here that
Eq.(\ref{71}) is the plane symmetric version of Eq.(55) in
\cite{17}.
\section{Summary and Conclusion}
Gravitational collapse in a star is an irreversible phenomenon.
Dynamics (such as transport processes) of such non-equilibrium
objects and connection between their dynamics and thermodynamics are
of extensive significance in order to have a better visualization of
this problem. Thus we have studied the dynamics of dissipative
collapse, i.e., what role does dissipation play with passing time as
star collapses under the influence of its own gravity. The most
realistic model of matter, i.e, complicated fluid is assumed in the
interior region and is taken to be consistent with plane symmetry.
To see how system evolves with time, dynamical equations for the
plane symmetric spacetime are obtained using Misner and Sharp
formalism. In the dynamical equation (\ref{56}), we see that the
gravitational force represented by the first term on the right hand
side is expected to be much effective as compared to non-dissipative
fluid and so gravitational collapse is expected to be faster in this
case. Moreover, since the pressure gradient is negative in the
second term on right hand side of this equation, which combined with
the minus sign preceding that term makes a positive contribution,
thereby reducing the rate of collapse. The last square brackets
entirely depend on dissipation and one cannot expect any such
contribution in a dynamical equation for non-dissipative collapse.
The third term in this bracket is positive due to negative sign of
velocity of collapsing fluid U. It shows that outflow of heat flux
$q>0$ and radiation $\epsilon>0$ reduces the total energy of the
system and hence reduces the rate of collapse.
Transport equations in the context of M$\ddot{u}$ller, Israel and
Stewart theory of dissipative fluids are obtained and coupled to
dynamical equation in order to see the influence of dissipation over
dynamics of a collapsing plane. After this union of dynamical and
transport equations, we get equation (\ref{71}) where the factor
$(1-\Lambda+\Delta)$ appears in the dynamical equation. We see the
effect of this factor for different possible values.
\begin{itemize}
\item If $0<(\Lambda-\Delta)<1$, inertial and gravitational
mass densities will be reduced.
\item If $(\Lambda-\Delta)$ tends to $1$, inertial mass density tends to zero.
\item If $(\Lambda-\Delta)>1$, gravitational force will become positive
and it will lead to the reversal of collapse. Another possibility
for reversal of collapse is to take $(\Lambda-\Delta)<1$ such that
$(1-\Lambda+\Delta)$ is sufficiently small. Consequently, it will
significantly decrease the gravitational force.
\end{itemize}
|
1,108,101,565,499 | arxiv |
\section{Conclusion}
\label{sec:conclusion}
In this paper we demonstrated how to create a highly competitive video deblurring model by revisiting details of an otherwise fairly standard CNN baseline architecture.
We show that despite a lot of effort being put into finding a good video deblurring architecture by the community, some benefits could possibly be even due to seemingly minor model and training details.
The resulting difference in terms of PSNR is surprisingly significant: In our study we improve the baseline network of \cite{Su:2017:DVD} by over 2dB compared to the original results in the paper, and 3.15dB over our initial implementation, which allows this simple network to outperform more recent and much more complex models.
This poses the question whether existing experimental comparisons in the deblurring literature actually uncover systematic accuracy differences from the architecture, or whether the differences may be down to detail engineering.
Future work thus needs to shed more light on this important point.
\section{Experiments}
\label{sec:experiments}
\paragraph{Evaluation on GOPRO by Su~\etal\cite{Su:2017:DVD}.}
As shown in the previous section, the proposed changes to Su's baseline strikingly boosted its deblurring accuracy by over 3dB compared to our basic baseline implementation.
We next consider how the improved baseline fares against the state-of-the-art.
We evaluate three variants:
Our best model without an optical flow backbone, trained under the same patch size ($128^2$) and sequence length (5) as competing methods (\cref{tab:dbn-ablation}\red{(h)}), denoted as {$\text{DBN}_{128,5}$}\@\xspace.
Our improved baseline, which includes optical flow pre-warping (\cref{tab:dbn-ablation}\red{(p)}), denoted as {$\text{FlowDBN}_{128,5}$}\@\xspace.
And our best performing model trained under large patches and two more input images (\cref{tab:dbn-ablation}\red{(v)}), denoted as {$\text{FlowDBN}_{192,7}$}\@\xspace.
\Cref{tab:gopro-su-short} shows the quantitative evaluation on the GOPRO testing dataset of \cite{Su:2017:DVD}.
Surprisingly, even our {$\text{DBN}_{128,5}$}\@\xspace model without optical flow already beats the highly competitive methods from Chen~\etal \cite{Chen:2018:R2D} by 0.11dB, which utilizes optical flow.
{
Our variants including optical flow, {$\text{FlowDBN}_{128,5}$}\@\xspace and {$\text{FlowDBN}_{192,7}$}\@\xspace are also highly competitive \wrt the recurrent approach of Nah~\etal \cite{Nah:2019:RNN} and the spatio-temporal transformer (STT) networks \cite{Kim:2018:STT}, \ie~{$\text{FlowDBN}_{128,5}$}\@\xspace yields a higher average PSNR than STT applied to the same DBN backbone.
}
Finally, we improve the authors' results of \cite{Su:2017:DVD} by more than 2dB.
While our best performing {$\text{FlowDBN}_{192,7}$}\@\xspace cannot quite reach the accuracy of methods based on the OVD backbone \cite{Kim:2017:OVD}, the OVD model exploits a dynamic temporal blending layer and uses recurrent predictions from previous iterations.
In contrast, our model is based on the conceptually simpler DBN, a plain feed-forward CNN.
{
We expect similar improvements when applying our insights in training details to the OVD backbone.
}
\input{tables/gopro-nah-final}
\input{figures/comparison-nah/figure}
\myparagraph{Evaluation on GOPRO by Nah~\etal \cite{Nah:2017:DMC}.}
To see whether the benefits we gain on our baseline generalize to other datasets, we also quantitatively evaluate on the GOPRO dataset of Nah~\etal \cite{Nah:2017:DMC}.
Note that the training set by \cite{Nah:2017:DMC} has roughly a third of the size of \cite{Su:2017:DVD}, hence our training schedule is three times as long, \ie 608 epochs and halving the learning rate at epochs $[308, 358, 408, 458, 508, 558]$.
The other details are as described in \cref{sec:baseline}.
We compare against DeblurGAN \cite{Kupyn:2018:DGB}, Nah~\etal's DMC baseline \cite{Nah:2017:DMC}, and the two highly competitive scale-recurrent models SRN+color/lstm by Tao~\etal \cite{Tao:2018:SRN}.
{
As these methods do not exploit multiple images, we additionally include {$\text{DBN}_{192,1}$}\@\xspace, a single-image variant of our baseline.}
The detailed results are shown in \cref{tab:gopro-nah-detailed}.
{
Interestingly, {$\text{DBN}_{192,1}$}\@\xspace already outperforms the highly competitive SRN+lstm model, a multiscale recurrent neural network, despite being trained on a smaller crop size (Tao~\etal \cite{Tao:2018:SRN} apply $256^2$ crops).
Both {$\text{FlowDBN}_{128,5}$}\@\xspace and {$\text{FlowDBN}_{192,7}$}\@\xspace perform even better, outperforming the best competing method by a very significant $\sim$0.8dB in PSNR.}
Qualitative results are shown in \cref{fig:comparison-nah}.
When inspecting the visual results, we find that both our FlowDBN models show perceptually better results, \eg they exhibit clearer text deblurring (\cf the plates in the \nth{1} row).
For moving people, faces can be problematic due to their small-scale details, as for instance shown in the results of the \nth{2} row, \ie DeblurGAN, DMC, and SRN+LSTM all show artifacts in the face of the person.
While the results for both FlowDBN models are far from perfect, they show significantly fewer artifacts.
We observe another subtle improvement in blob-like structures such as the orange repetitive structure in the advertisement (last row).
{
Here, our FlowDBN models reconstruct a sharper texture than all competing methods.
}
\section{Introduction}
\label{sec:introduction}
Blind image deblurring -- the recovery of a sharp image given a blurry one -- has been studied extensively \cite{Krishnan:2011:BDN,Levin:2006:BMD,Perrone:2014:TVB,Shan:2008:HQM,Sun:2013:EBK,Xu:2010:TPK,Xu:2013:ULS}.
However, more recently and perhaps with the increasing popularity of hand-held video cameras, attention has shifted towards deblurring videos \cite{Nah:2017:DMC,Su:2017:DVD}.
With the (re-)emergence of deep learning and the availability of large amounts of data, the best performing methods today are usually discriminatively trained CNNs
\cite{Chen:2018:R2D}, RNNs \cite{Tao:2018:SRN}, or a mixture thereof \cite{Kim:2017:OVD,Kim:2018:STT}.
While the ``zoo'' of video deblurring models differs quite significantly, explanations as to why one network works better than another often remain at an unsatisfactory level.
While the performance of state-of-the-art video deblurring methods is usually validated by training \textit{within}-paper models under the same conditions, the specifics of the training settings \textit{between} papers remain rather different.
In this work we show that some of these seemingly small details in the model setup and training procedure add up to astonishing quantitative and visual differences.
In fact, our quantitative evaluation raises the question whether the benefit for some state-of-the-art models comes from the proposed architectures or perhaps the setup details.
This mirrors observations in other areas of computer vision, where the significance of choosing the right training setup is crucial to achieve highly competitive models \cite{Sun:2019:MMT}.
Henceforth, we conduct a study on how the model setup and training details of a comparatively simple baseline CNN drastically influence the resulting image quality in video deblurring.
By finding the right settings, we unlock a significant amount of hidden power of this baseline, and achieve state-of-the art results on popular benchmarks.
Our systematic analysis considers the following variations:
\emph{(1)} We investigate the use of $\linear$ output layers instead of the typical $\sigmoids$ and consider different initialization methods.
Our new $\fanmax$ initialization combined with $\linear$ outputs already yields a substantial $2$dB benefit over our $\sigmoid$ baseline.
\emph{(2)} While recent work proposes to deblur in YCbCr color space \cite{Zhang:2019:ASL}, we show that there is no significant benefit over RGB.
Instead, a simple extension of the training schedule can lead to an additional $0.4$dB benefit.
\emph{(3)} We uncover that both photometric augmentations as well as random image scaling in training hurt deblurring results due to the mismatch of training \vs test data statistics.
The misuse of augmentations can diminish the generalization performance up to a severe 0.44dB.
\emph{(4)} We explore the benefits of using optical flow networks for pre-warping the inputs, which yields another 0.4dB gain.
Concatenating pre-warped images to the inputs improves over a simple replacement of the temporal neighbors by up to $0.27$dB.
This is in contrast to previous work, which either claimed no benefit from using pre-warping \cite{Su:2017:DVD}, or applied a complex spatio-temporal subnetwork with additional trainable weights \cite{Kim:2018:STT}.
\emph{(5)} We explore the influence of training patch size and sequence length.
Longer sequences yield only a minor benefit, but large patch sizes significantly improve over small ones by up to $0.9$dB.
Taken together, we improve our baseline by a striking $3.15$dB and the published results of \cite{Su:2017:DVD} by $2.11$dB, reaching and even surpassing the quality of complex state-of-the-art networks on standard datasets.
\section{Related Work}
\label{sec:relatedwork}
\paragraph{Classic uniform and non-uniform deblurring.}
Classic uniform deblurring methods that restore a sharp image under the assumption of a single blur kernel usually enforce
sparse image statistics, and are often combined with probabilistic, variational frameworks \cite{Cho:2009:FMD,Fergus:2006:RCS,Levin:2009:UEB,Levin:2011:EML,Miskin:2000:ELB}.
Less common approaches include the use of self-similarity \cite{Michaeli:2014:BDI}, discriminatively trained regression tree fields \cite{Schelten:2015:IRT}, a dark-channel prior \cite{Pan:2016:BID}, or scale normalization \cite{Jin:2018:NBD}.
Moving objects or a moving camera, on the other hand, significantly complicate deblurring, since the motion varies across the image domain.
Here, usually restricting assumptions are enforced on the generative blur, either in the form of a candidate blur model \cite{Couzinie:2013:LER,Kim:2013:DSD}, a linear blur model \cite{Gast:2016:POM,Kim:2014:SFD}, or a more generic blur basis \cite{Gupta:2010:SID,Hirsch:2010:FRN,Whyte:2010:NDS,Zheng:2013:FMD}.
\myparagraph{Classic video deblurring.}
Early work on video blurring \cite{Cho:2012:VDH,Matsushita:2006:FVS} proposes to transfer sharp pixels from neighboring frames to the central reference frame.
While Matsushita~\etal \cite{Matsushita:2006:FVS} apply a global homography, Cho~\etal \cite{Cho:2012:VDH} improve on this by local patch search.
Overall, the averaging nature of these approaches tends to overly smooth results \cite{Delbracio:2015:HVD}.
Delbracio~\etal \cite{Delbracio:2015:HVD} overcome this via a weighted average in the Fourier domain, but rely on a registration of neighboring frames, which may fail for large blurs.
Kim~\etal \cite{Kim:2015:GVD} propose an energy-based approach to jointly estimate optical flow along a latent sharp image using piece-wise linear blur kernels.
Later, Ren~\etal \cite{Ren:2017:VDS} incorporate semantic segmentation into the energy.
Both approaches rely on primal-dual optimization, which is computationally demanding.
\myparagraph{Deep image deblurring.}
Among the first deblurring methods in the light of the recent renaissance of deep learning has been the work by Sun~\etal~\cite{Sun:2015:LCN} who train a CNN to predict pixelwise candidate blur kernels.
Later, Gong~\etal \cite{Gong:2017:FMB} extend this from image patches to a fully convolutional approach.
Chakrabarti~\cite{Chakrabarti:2016:ANA} tackles uniform deblurring in the frequency domain by predicting Fourier coefficients of patch-wise deconvolution filters.
Note that, as in the classical case, all aforementioned methods are still followed by a standard non-blind deconvolution pipeline.
This restriction is lifted by Schuler~\etal \cite{Schuler:2016:LTD} who replace both the kernel and image estimator module of classic pipelines by neural network blocks, respectively.
Noroozi~\etal \cite{Noroozi:2017:MDW} propose a multi-scale CNN, which directly regresses a sharp image from a blurry one.
Tao~\etal \cite{Tao:2018:SRN} suggest a scale-recurrent neural network (RNN) to solve the deblurring problem at multiple resolutions in conjunction with a multi-scale loss.
\myparagraph{Deep image deblurring via GANs.}
Other approaches draw from the recent progress on generative adversarial networks (GANs).
Ramakrishnan~\etal \cite{Ramakrishnan:2017:DGF} propose a GAN for recovering a sharp image from a given blurry one; the generator aims to output a visually plausible, sharp image, which fools the discriminator into thinking it comes from the true sharp image distribution.
Nah~\etal \cite{Nah:2017:DMC} propose a multi-scale CNN accompanied by an adversarial loss in order to mimic traditional course-to-fine deblurring techniques.
Similarly, Kupyn~\etal \cite{Kupyn:2018:DGB} apply a conditional GAN, where the content (or perceptual) loss is notably defined in the domain of CNN feature maps rather than output color space.
We do not consider the use of adversarial networks here, as we argue that the accuracy of feed-forward CNNs is not yet saturated on the deblurring task.
Note that despite the simplicity of our baseline, we outperform the model of \cite{Kupyn:2018:DGB} by a large margin, \cf \cref{sec:experiments}.
\myparagraph{Deep video deblurring.}
Deep learning approaches to video deblurring have yielded tremendous progress in speed and image quality.
Kim~\etal \cite{Kim:2017:OVD} focus on the temporal nature of the problem by applying a temporal feature blending layer within an RNN.
{
Similarly, Nah~\etal \cite{Nah:2019:RNN} apply an RNN to propagate intra-frame information.
}
While RNNs are promising, we note that these are often difficult to train in practice \cite{Pascanu:2013:OTT}.
We do not rely on a recurrent architecture, but a plain CNN, achieving very competitive results.
Zhang~\etal \cite{Zhang:2019:ASL} use spatio-temporal 3D convolutions in the early stages of a deep residual network.
Chen~\etal \cite{Chen:2018:R2D} extend \cite{Kupyn:2018:DGB} with a physics-based reblurring pipeline, which constructs a reblurred image from the sharp predictions using optical flow, and subsequently enforces consistency between the reblurred image and the blurry input image.
{
Wang~\etal \cite{Wang:2019:VRE} apply deformable convolutions along an attention module to tackle general video restoration tasks.
}
The DBN model of Su~\etal \cite{Su:2017:DVD} serves as baseline model in our study.
DBN is a simple encoder-decoder CNN with symmetric skip connections; its input is simply the concatenation of the temporal window of the video input sequence.
Later, Kim~\etal \cite{Kim:2018:STT} extend the DBN model by a 3D spatio-temporal transformer, which transforms the inputs to the reference frame.
Note that this requires training an additional subnetwork that finds 3D correspondences of the inputs to the reference frame.
We find that we can outperform \cite{Kim:2018:STT} based on the same backbone network without the need of a spatial transformer network.
More generally, we uncover crucial details in the model and training procedure, which strikingly boost the accuracy by several dB in PSNR, yielding a method that is highly competitive.
\section{The Details of Deep Video Deblurring}
\label{sec:deep-video-deblurring}
As has been observed in papers in several areas of deep learning and beyond, careful choices of the architecture, (hyper-)parameters, training procedure, and more can significantly affect the final accuracy \cite{Chatfield:2014:RDD,Lucic:2018:AGC,Perrone:2014:TVB,Sun:2019:MMT}.
We show that the same holds true in deep video deblurring.
Specifically, we revisit the basic deep video deblurring network of Su~\etal \cite{Su:2017:DVD} and will uncover step-by-step, how choices made in mode, training, and preprocessing affect the deblurring accuracy.
All together, these details add up to a very significant $3.15$dB difference on the test dataset.
\subsection{Baseline network}
The basis architecture of our study is the DBN network of Su \etal~\cite{Su:2017:DVD} (\cf Table~1 therein), a fairly standard CNN with symmetric skip connections.
We closely follow the original training procedure in as far as it is specified in the paper \cite{Su:2017:DVD}.
Since we focus on details including the training procedure here, we first summarize the basic setup.
The baseline model and all subsequent refinements are trained on the 61 training sequences and tested on the 10 test sequences of the GOPRO dataset \cite{Su:2017:DVD}.
The sum of squared error (SSE) loss
is used for training and minimized with Adam \cite{Kingma:2015:AAM}, starting at a learning rate of $0.005$.
Following \cite{Su:2017:DVD}, the batch size is taken as 64 where we draw 8 random crops per example.
For all convolutional and transposed convolutional layers, 2D batch normalization \cite{Ioffe:2015:BNA} is applied and initialized with unit weights and zero biases.
While this simple architecture has led to competitive results when it was published in 2017, more recent methods \cite{Chen:2018:R2D,Kim:2018:STT} have strongly outperformed it.
In the following, we explore the potential to improve this baseline architecture and perform a step-by-step analysis.
\Cref{tab:dbn-ablation} gives an overview.
\input{figures/linear-activations/figure}
\subsection{Detail analysis}
\label{sec:baseline}
\paragraph{Output activation.}
The DBN network \cite{Su:2017:DVD} uses a $\sigmoid$ output layer to yield color values in the range $[0,1]$.
Given the limited range of pixel values in real digital images, this appears to be a prudent choice at first glance.
We question this, however, by recalling that $\sigmoid$ nonlinearities are a common root of optimization issues due to the well-known vanishing gradient problem.
We thus ask whether we need the $\sigmoid$ nonlinearity.
To that end, we replace it with a simple $\linear$ output.
As we can see in \cref{tab:dbn-ablation}\red{(a \vs d)}, this yields a very substantial 1dB accuracy benefit, highlighting again the importance of avoiding vanishing gradients.
In fact, the restriction to the unit range does not pose a significant problem even without output nonlinearity, since the SSE loss largely limits the linear outputs to the correct range anyway.
This is illustrated in \cref{fig:linear-activations}, which shows the linear activations on the test dataset after training with linear output activations under a $\sse$ loss; only very few values lie outside the valid color value range.
This can be easily addressed by clamping the outputs to $[0,1]$ at test time.
\myparagraph{Initialization.}
The choice of initialization is not discussed in \cite{Su:2017:DVD}.
However, as for any nonlinear optimization problem, initialization plays a crucial role.
Indeed, we find that good initialization is necessary to reproduce the results reported in \cite{Su:2017:DVD}.
Perhaps, the most popular initialization strategy for $\relu$-based neural networks today is the $\msra$ method of He~\etal \cite{Kaiming:2015:DDR}.
It ensures that under $\relu$ activations, the magnitudes of the input signal do not exponentially increase or decrease.
The $\msra$ initialization method typically comes in two variants, $\msra + \fanin$, and $\msra + \fanout$, depending on whether signal magnitudes should be preserved in the forward or backward pass.
In practice, $\fanin$ and $\fanout$ correspond to the number of gates connected to the inputs and outputs.
We additionally propose $\fanmax$, which we define as the maximum number of gates connected to either the inputs or outputs,
{
providing a trade-off between $\fanin$ and $\fanout$.
For hourglass architectures, it is typical to increase the number of feature maps in the encoding part; here, $\fanmax$ adapts to the increasing number of feature maps via $\fanout$ initialization.
The decoder is effectively initialized by $\fanin$ to accommodate the decreasing number of feature maps.
}
\Cref{tab:dbn-ablation}\red{(a -- f)} evaluates these initializations in conjunction with $\linear$ and $\sigmoid$ outputs layers.
Due to the attenuated gradient, all three $\sigmoid$ variants are worse than any $\linear$ output layer.
On the other hand, $\linear$ in conjunction with $\fanmax$ initialization works much better than the traditional $\fanin$ and $\fanout$ initializations, yielding a $\sim$0.7dB benefit.
The visual results in \cref{fig:ablation-outputinit} also reveal that the $\linear$ output contains fewer visual artifacts.
\vspace{0.25em}\noindent\textit{\underline{Verdict:}~}
For a color prediction task such as deblurring, $\sigmoids$ should be replaced by $\linear$ outputs.
{
We recommend considering a $\fanmax$ initialization as an alternative to $\fanin$ and $\fanout$.
}
\input{figures/ycbcr-oracle/figure}
\myparagraph{Color space.}
{
In classic deblurring color channels are typically deblurred separately.}
While this is clearly not necessary in deep neural architectures -- we can just output three color channels simultaneously -- the question remains whether the RGB color space is appropriate.
Zhang~\etal \cite{Zhang:2019:ASL} propose to convert the blurry input images to YCbCr space, where Y corresponds to grayscale intensities and CbCr denotes the color components, \cf \cref{fig:ycbycr-oracle}.
The sharp image is subsequently reconstructed from the deblurred Y channels and the blurry input CbCr channels.
{
This effectively enforces a natural upper bound on the problem, \ie computing the average PSNR value of the test dataset yields}%
\begin{subequations}
\begin{align}
\text{PSNR}(\text{RGB}_{\text{input}}, \text{RGB}_{\text{gt}}) &= 27.23\text{dB} \\
\text{PSNR}(\text{cat}(\text{Y}_\text{gt}, \text{CbCr}_\text{input}), \text{RGB}_{\text{gt}}) &= 56.26\text{dB}.
\end{align}
\end{subequations}
That is, an oracle with access to the ground truth $Y$ channel can achieve at most $56.26$dB PSNR.
Hence, the natural upper bound does not pose a real quantitative limitation, since $56.26$dB is much better than any current method can achieve.
In practice, however, we found that the benefit of solving the problem in YCbCr space is not significant.
\Cref{tab:dbn-ablation}\red{(f, g)} show a minimal $\sim$0.01dB benefit of using YCbCr over RGB.
YCbCr can still be useful as it allows for models with a smaller computational footprint, since fewer weights are required in the first and last layer.
Here, we want to raise another problem of YCbCr deblurring:
For very blurry regions, the reconstruction even from the ground truth Y channel may contain halo artifacts as depicted in \cref{fig:ycbcr-oracle-recon}.
\input{figures/scaled-gradient-stats/figure}
\myparagraph{Training schedule.}
As observed in other works, \eg \cite{Ilg:2017:FN2}, longer training schedules can be beneficial for dense prediction tasks.
Here, we apply two different training schedules, a short one with 116 epochs resembling the original schedule \cite{Su:2017:DVD} by halving the learning rate at epochs $[32, 44, 56, 68, 80, 92, 104]$, as well as a long schedule with 216 epochs, halving the learning rate at epochs $[108, 126, 144, 162, 180, 198]$.
To obtain the long training schedule, we initially inspected the results of running PyTorch's \texttt{ReduceLROnPlateau} scheduler (with patience=10, factor=0.5) for an indefinite time, where we subsequently scheduled the epochs in which learning rates drop in equidistant intervals (here 18).
The longer training schedule improves both the RGB and YCbCr networks roughly by 0.4dB, \cf \cref{tab:dbn-ablation}\red{(f -- i)}.
Since the benefit of YCbCr is rather small for both short and long schedule, we conduct the remaining experiments in RGB space.
\Cref{fig:ablation-colorspaceschedule} shows the visual differences between RGB and YCbCr deblurring.
While the perceptual differences between RGB and YCbCr are not significant, the long schedules improve the readability of the letters over the short ones.
\input{figures/ablation-colorspaceschedule/figure}
\input{figures/ablation-augmentations/figure}
\vspace{0.25em}\noindent\textit{\underline{Verdict:}~}
YCbCr does not present a significant benefit over RGB; it is, however, viable for very large models, if model size is an issue.
Very blurry training examples may be suboptimal, since even the oracle $Y$ channel yields halo artifacts.
Similar to other dense prediction tasks, long training schedules yield significant benefits.
\myparagraph{Photometric augmentation and random scales.}
Data augmentation plays a crucial role in many dense prediction tasks such as optical flow \cite{Dosovitskiy:2015:FN}.
However, it is often disregarded from the analysis of deblurring methods.
More precisely, while our baseline \cite{Su:2017:DVD} and recent work \cite{Kim:2018:STT, Zhang:2019:ASL} all train under random rotations ($0^\circ, 90^\circ, 180^\circ, 270^\circ$), random horizontal and vertical flips, and random crops (usually of size $128^2$), other types of augmentations such as photometric transformations and random scaling are not agreed upon.
Su \etal~\cite{Su:2017:DVD} train their model under random image scales of $[\nicefrac{1}{4}, \nicefrac{1}{3}, \nicefrac{1}{2}]$, yet Zhang~\etal \cite{Zhang:2019:ASL} do not rescale the training images.
Here, we explore the influence of both random photometric transformations and random scales.
\input{figures/ablation-flow/figure}
\input{figures/ablation-crops/figure}
We use four settings: No augmentations (other than random orientations and crops, \cref{tab:dbn-ablation}\red{(h)}), random photometric transformations (using PyTorch's popular random color jitter on hue, contrast, and saturation with p=0.5, \cref{tab:dbn-ablation}\red{(j)}), random scales (with a random scale factor in $[0.25, 1.0]$, \cref{tab:dbn-ablation}\red{(k)}), and with both augmentations (\cref{tab:dbn-ablation}\red{(l)}).
We find that these augmentations significantly hurt image quality; the quantitative difference between no and both augmentations (\cref{tab:dbn-ablation}\red{(h \vs l)}) amounts to a surprising 0.44dB.
Here, the photometric augmentations alone decrease the accuracy by 0.26dB (\cref{tab:dbn-ablation}\red{(h \vs j)}).
While we do not argue that any photometric augmentation will hurt accuracy, our results suggest that the common color jitter is counterproductive in deblurring; we attribute this to the fact that commonly applied photometric co-transforms obfuscate the ground truth signal for general non-uniform blur.
{
To illustrate this issue, let $\bm{P}$ be a photometric operator (applied to sharp images), and $\bm{K}$ be a non-uniform blur operator, respectively.
If $\bm{P}$ was linear, we could derive the appropriate photometric operator $\tilde{\bm{P}}$ for blurry images as%
\begin{align}
\tilde{\bm{P}} \bm{K} = \bm{K} \bm{P} \quad \Rightarrow \quad \tilde{\bm{P}} = \bm{K} \bm{P} \bm{K}^{-1}.
\end{align}
As there is no ground truth $\bm{K}$ available for the GOPRO datasets, the correct photometric transformation $\tilde{\bm{P}}$ to be applied to blurry images is not available.
}
{
The performance drop induced by random scales roots in a change of relative image statistics between blurry and sharp images.}
To that end, consider the gradient histogram statistics of 300 training image crops shown in \cref{fig:scaled-gradient-stats-unscaled} as well as the statistics for rescaled crops (scale factor 0.25) in \cref{fig:scaled-gradient-stats-scaled}.
The comparison reveals two points:
First, the original statistics are sparser than the rescaled ones.
Second, rescaling renders the gap of statistics between the blurry and sharp gradients less pronounced.
This difference manifests in a quantitative difference of 0.22dB, \cf \cref{tab:dbn-ablation}\red{(h \vs k)}.
Visually, the difference is most apparent for blob-like regions, \cf the leaves of the tree in \cref{fig:ablation-augmentations}.
Not applying any photometric or scale augmentation \subref{fig:ablation-augmentations-none} yields slightly clearer results than either random photometric transformations \subref{fig:ablation-augmentations-photometrics}, random scales \subref{fig:ablation-augmentations-scales}, or both \subref{fig:ablation-augmentations-photometrics-scales}.
\vspace{0.25em}\noindent\textit{\underline{Verdict:}~}
In contrast to other dense prediction problems, where photometric augmentations and random rescaling in training help to improve generalization, these augmentations can hurt the generalization performance of deblurring models.
One should thus be careful in choosing augmentation methods, as they may obfuscate the data statistics.
\myparagraph{Optical flow warping.}
Su~\etal \cite{Su:2017:DVD} experimented with pre-warping input images based on classic optical flow methods such as \cite{Perez:2013:TVL} to register them to the reference frame.
Surprisingly, they did not observe any empirical benefit, hence abandoned flow warping.
Yet, Chen~\etal \cite{Chen:2018:R2D} use a flow network after the deblurring network to predict an output sequence of sharp images, which is subsequently registered to the reference frame.
This consistency is worked into the loss function, which allows them to improve over the DBN baseline (\cf \cref{tab:gopro-su-short}).
Kim~\etal \cite{Kim:2018:STT} propose to put a spatio-temporal transformer network in front of the DBN baseline to transform 3D inputs (the stack of blurry input images) to the reference frame;
the synthesized images and the reference frame are then fed into the baseline network.
In contrast to \cite{Su:2017:DVD}, they observed the temporal correspondence to improve the deblurring accuracy.
While using a spatio-temporal transformer is elegant, we argue that the underlying correspondence estimation problem is itself very hard and requires a lot of engineering to achieve high accuracy \cite{Sun:2019:MMT}.
Hence, we consider pre-warping with the output from standard optical flow networks.
To avoid any efficiency concerns \cite{Kim:2018:STT}, we rely on pre-trained flow networks, which obviates backpropagating through them.
We experiment with two different backbones that we put in front of our baseline:
FlowNet1S (denoted as f1s) \cite{Dosovitskiy:2015:FN} and PWC-Net (denoted as pwc) \cite{Sun:2018:PWC}.
For both backbones, we warp the neighboring frames to the reference frame, and either input the reference frame along the replaced, warped neighbors (+ rep), or we concatenate the warped neighbors with the original input (+ cat).
{
Note that while concatenation allows the network to possibly overcome warping artifacts using the original inputs, this is not possible without the original input.
}
Our experiments in \cref{tab:dbn-ablation}\red{(m -- p)} show that, in contrast to the conclusions in \cite{Su:2017:DVD}, simple flow warping already helps (0.15dB improvement in \red{(m -- n)} over the no-flow baseline \red{(i)}).
A more substantial benefit of $\sim$0.4dB comes from concatenating the warped images along the original inputs (\cref{tab:dbn-ablation}\red{(o -- p)}).
Perhaps surprisingly, the FlowNet1S backbone performs only slightly worse than PWC-Net.
The visual results in \cref{fig:ablation-flow} reveal that flow-based methods clearly improve upon the no-flow variant, which exhibits artifacts at the horizontal structures of the house.
Also note how the PWC-Net backbone is clearer in deblurring the horizontal structures than the FlowNet1S variant, despite the small quantitative difference.
Visually, pwc+cat further improves over pwc+rep, \eg note the boundaries of the windows.
\input{tables/dbn-ablation}
\vspace{0.25em}\noindent\textit{\underline{Verdict:}~}
While previous work proposes a sophisticated treatment of temporal features, we find that pre-trained optical flow networks perform quite well.
Concatenating warped neighbors to the inputs works significantly better than just replacing inputs.
While a good flow network may not quantitatively improve over a simple one, deblurred images may show subtle improvements upon visual inspection.
\myparagraph{Patch size and sequence length.}
Much of previous work \cite{Chen:2018:R2D,Kim:2018:STT,Su:2017:DVD,Zhang:2019:ASL} is trained on random crops of size $128^2$, yet the significance of this choice is not further justified.
{%
In general, larger crops are beneficial as they reduce the influence of boundaries, given the typically big receptive fields.}
Here we explore additional patch sizes of $64^2, 96^2, 160^2$, and $192^2$, which we apply when training our pwc+cat model.
\Cref{tab:dbn-ablation}\red{(q -- t)} reveals that the choice of patch size -- when comparing to the baseline patch size of $128^2$ -- is quite important with a relative performance difference spanning from $-0.68$dB when using the smallest patch size $64^2$ to $+0.23$dB when using the largest ($192^2$).
While the performance difference between patch sizes is more significant for smaller absolute sizes, the performance gain from very large patches is still substantial.
This can also be seen in the visual results in \cref{fig:ablation-crops}.
Note the clearer poles.
Overall, the relative visual improvement becomes smaller with larger patch sizes, yet is still apparent.
\cite{Su:2017:DVD} proposed to use input sequences with $5$ images, which is kept in follow up work \cite{Zhang:2019:ASL,Kim:2018:STT}.
We include one more dimension in our case study, and test whether longer sequences can help.
In \cref{tab:dbn-ablation}\red{(u--v)}, we increased the number of input images to $7$ and retrained our pwc+cat model (with patch sizes $128^2$ and $192^2$).
The results reveal that $5$ input images largely suffice; two additional input images only yield a small benefit of $\sim 0.05$dB.
\vspace{0.25em}\noindent\textit{\underline{Verdict:}~}
Training patches should be chosen as big as the hardware limitations allow, since larger patch sizes provide clear benefits in accuracy.
Future GPUs may allow training at full resolution and improve results further.
Inputting more than $5$ images currently yields only minimal benefit.
|
1,108,101,565,500 | arxiv | \section{Introduction}
At a distance of 16.7~Mp
, M87 is the dominant galaxy of the Virgo Cluster. It is one of the
nearest radio galaxies and was the first extragalactic X-ray source to
be identified. Because of its proximity, many interesting
astrophysical phenomena can be studied in more detail in M87 than in
other comparable object
. Particularly remarkable is the prominent jet extending from the
nucleus, visible throughout the electromagnetic spectrum. The central
regions of M87, in particular the structure of the jet, have been
studied and compared intensively at radio, optical, and X-ray
wavelengths. \citep[e.g.][]{1991AJ....101.1632B, 1996A+A...307...61M,
2001A+A...365L.181B, 2001ApJ...551..206P, 2004ApJ...607..294S,
2005ApJ...627..140P, 2007ApJ...668L..27K, 2008A+A...482...97S}.
Compared to the available information at these wavelengths, our
knowledge of M87 at far-infrared (FIR) wavelengths is rather poor. A
controversial issue is the origin of the FIR emission in M87, i.e.,
the question of whether the FIR emission is caused entirely by
synchrotron emission or whether there is an additional contribution
from dust associated with either the global interstellar medium or a
nuclear dust component.
Several papers, based on IRAS, ISO and Spitzer observations, arrive at
different conclusions \citep{2007ApJ...663..808P, 2009ApJ...705..356B,
2004A+A...416...41X, 2007ApJ...655..781S, 2008ApJ...689..775T}.
\begin{figure}
\centering
\includegraphics[width=0.56\textwidth]{Baes_fg1a.pdf}%
\includegraphics[width=0.44\textwidth]{Baes_fg1b.pdf}
\caption{Left: the Herschel view of the central regions of M87. The
bottom right image is a VLA 20~cm image from the FIRST survey. The
20~cm radio contours have been overlaid on the Herschel
images. The field of view of all images is $160"\times90"$, beam
sizes are indicated in the bottom right corner. Top right: the
global SED of M87 from mid-infrared to radio wavelengths. Where no
error bars are seen, they are smaller than the symbol size. The
solid line in the plot is the best-fit power law of the ISOCAM,
IRAS, MIPS, SCUBA, GBT, WMAP, and VLA data; the dotted line has
only been fitted to the SCUBA, GBT, WMAP, and VLA data. Bottom
right: residual between data and the best-fit synchrotron model in
the infrared-submm wavelength range. The cyan line is a modified
black-body model with $T=23$~K and
$M_{\text{d}}=7\times10^4~M_\odot$.}
\label{RawData.pdf}
\end{figure}
We investigate the nature of the FIR emission of M87 using new FIR
data from the Herschel Space Observatory \citep{2010A&A...518L...1P},
obtained as part of the science demonstration phase (SDP) observations
of the Herschel Virgo Cluster Survey
\citep[HeViCS,][]{2010A&A...518L..48D}. HeViCS is an approved open
time key program, which has been awarded 286~h of observing time in
parallel mode with the PACS and SPIRE instruments. We will ultimately
map four $4\times4$ square degree regions of the cluster at 100, 160,
250, 350 and 500~$\mu$m, down to the 250~$\mu$m confusion limit of
about 1~MJy~sr$^{-1}$. The HeViCS SDP observations consisted of a
single cross-scan of one $4\times4$~deg$^2$ field at the centre of the
Virgo Cluster. While these SDP observations comprise only 6\% of the
total HeViCS observations, the analysis based on these observations
already gives a prelude to the primary HeViCS science goals including
the effects of the environment on the dust medium of galaxies, the FIR
luminosity function, the complete SEDs of galaxies and a detailed
analysis of the dust content of dwarf and early-type galaxies
\citep{2010A&A...518L..48D, 2010A&A...518L..49C, 2010A&A...518L..50C,
2010A&A...518L..51S, 2010A&A...518L..52G, 2010A&A...518L..54D,
2010A&A...518L..61B}. The observations also enabled us to study the
intensity and nature of the FIR emission of M87
\citep{2010A&A...518L..53B}.
\section{Analysis and conclusion}
\label{Analysis.sec}
The left part of Figure~{\ref{RawData.pdf}} shows the PACS and SPIRE
images of the central $160"\times90"$ region of M87, which is clearly
detected in all five bands. The top right panel in
Figure~{\ref{RawData.pdf}} shows the integrated SED in the
infrared-submm-radio region between 15~$\mu$m and 100~cm, with the new
Herschel data as well as ISOCAM, IRAS, MIPS, and SCUBA, GBT, WMAP, and
VLA data gathered from the literature \citep{2004A+A...416...41X,
1988AJ.....95...26G, 2007ApJ...655..781S, 2004A+A...424..531H,
2009ApJ...701.1872C, 2009ApJS..180..283W}; the actual flux densities
can be found in \citet{2010A&A...518L..53B}. The solid line is the
best-fit power law for all literature data and has a slope
$\alpha=-0.76$; the dotted line fits only the submm and radio data and
has a slope $\alpha=-0.74$. The bottom right panel in
Figure~{\ref{RawData.pdf}} shows the residual from the best-fit power
law in the infrared-submm wavelength region; clearly, the integrated
Herschel fluxes are in full agreement with synchrotron radiation. The
cyan line in this figure is a modified black-body fitted with $T=23$~K
and $M_{\text{d}}=7\times10^4~M_\odot$. This temperature is the mean
dust equilibrium temperature in the interstellar radiation field of
M87, determined using the SKIRT radiative transfer code
\citep{2003MNRAS.343.1081B, 2005AIPC..761...27B} and based on the
photometry from \citet{2009ApJS..182..216K}. The dust mass was
adjusted to fit the upper limits of the residuals. It is clear that
the SED of M87 is incompatible with dust masses higher than
$10^5~M_\odot$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Baes_fg2.pdf}
\caption{A comparison between the synchrotron model image and the
observed image at 250~$\mu$m. The left panel shows the synchrotron
image at the model resolution, the central panel shows the same
model convolved to the SPIRE 250~$\mu$m beam and pixel size. The
right panel shows the observed SPIRE 250~$\mu$m image. In all
panels, the green lines are the contours of the synchrotron model
at the model resolution.}
\label{H250.pdf}
\end{figure}
Although indicative, the analysis of the integrated SED does not
definitively identify the origin of the FIR emission in
M87. Approximating the global SED as a single power-law synchrotron
model is indeed an oversimplification of the complicated structure of
M87. Several studies have shown that M87 contains three distinct
regions of significant synchrotron emission, each with their own
spectral indices: the nucleus, the jet and associated lobes in the NW
region, and the SE lobes \citep[e.g.,][]{1991AJ....101.1632B,
1996A+A...307...61M, 2001ApJ...551..206P, 2007ApJ...655..781S}. We
have constructed a synchrotron model for the central regions of M87,
based on a newly reduced MIPS 24~$\mu$m map and archival MUSTANG
90~GHz and 15, 8.2, 4.9, 1.6, and 0.3~GHz maps
\citep{2009ApJ...701.1872C}. We fitted a second-order polynomial
synchrotron model to each pixel of the MIPS + radio data cube and used
this synchrotron model to predict the emission of M87 at 250~$\mu$m
(the SPIRE 250~$\mu$m image provides the optimal compromise between
S/N and spatial resolution). Figure~{\ref{H250.pdf}} shows the
comparison between the synchrotron model prediction at 250~$\mu$m and
the SPIRE observations. At the model resolution, the three distinct
components are visible, but when we convolve this synchrotron model
image with the SPIRE 250~$\mu$m beam, the three different components
merge into a single extended structure with one elongated peak
slightly west of the nucleus. Comparing the central and right panels
of Figure~{\ref{H250.pdf}}, we see that the synchrotron model is
capable of explaining the observed SPIRE 250~$\mu$m image
satisfactorily.
We conclude that for both the integrated SED and the SPIRE 250~$\mu$m
map, we have found that synchrotron emission is an adequate
explanation of the FIR emission. We do not detect a FIR excess that
cannot be explained by the synchrotron model. In particular, we have
no reason to invoke the presence of smooth dust emission associated
with the galaxy interstellar medium, as advocated by
\citet{2007ApJ...655..781S}. For a dust temperature of 23~K, which is
the expected equilibrium temperature in the interstellar radiation
field of M87, we find an upper limit to the dust mass of
$7\times10^4~M_\odot$.
Our conclusion is that, seen from the FIR point of view, M87 is a
passive object with a central radio source emitting synchrotron
emission, without a substantial diffuse dust component.
|
1,108,101,565,501 | arxiv | \section{Introduction}
\setcounter{equation}{0} Quantum systems with defects or boundaries
often show interesting physical behaviors. For example, impurities
in various materials have been one of the major subjects of research
in condensed matter physics. In such systems, we expect that the
whole system (both the bulk and the boundary) flows towards a fixed point of the renormalization
group in the low energy limit, where we can employ the powerful
method of conformal field theory. The boundary or
defect preserves a part of the conformal symmetry at the fixed
point.
One of the important quantities which characterizes properties of the
defects or boundaries is known as the boundary entropy (or, equivalently, the ground
state degeneracy $g$) \cite{Affleck:1991tk}. This measures the
degrees of freedom localized at a given defect and is a boundary
analogue of the central charge $c$. Recently, it has also been
pointed out that the boundary entropy can be regarded as the finite
part of the entanglement entropy \cite{Calabrese:2004eu}.
In general, the theory is strongly interacting at the RG fixed point and
sometimes it is very difficult to calculate physical quantities
like the boundary entropy. However, if the theory has a holographic
dual, we can compute many quantities rather simply by using the dual gravity
description. The most tractable examples will be the ones for which
we can apply the AdS/CFT correspondence.
The purpose of this paper is to holographically compute the boundary
entropy of 2d conformal field theories with defects using several
methods (for early discussions refer to \cite{Yamaguchi:2002pa}). The holographic calculation of entanglement entropy has been
recently formulated \cite{Ryu:2006bv,Ryu:2006ef}. This allows us to find the boundary
entropy from the entanglement entropy, in
addition to using a probe computation of the boundary entropy at high
temperature.
The organization of this paper is as follows. In section
\ref{BEandEE}, we present a brief summary of the definition and
properties of the boundary entropy. We will also work out a close
relation between the g-theorem and the strong subadditivity of
entanglement entropy. In section \ref{holographicBE}, we perform the
holographic computations of the boundary entropy using both
probe configurations and fully back-reacted geometries (in particular for the Janus solution).
In section \ref{conclusion} we summarize our conclusions. In
appendix \ref{appendix2point}, we present the calculations of two
point functions and the boundary entropy of a free scalar field in the
presence of the interface.
\section{Boundary Entropy and Entanglement Entropy}
\label{BEandEE} \setcounter{equation}{0} In this paper, we are
interested in two dimensional conformal field theories (2d CFTs) in
the presence of a conformal defect. If we define the time and space
coordinate by $(t,x)$, then we can consider a time-like defect which
is situated at $x=0$. The defect is called conformal if a linear
combination of two Virasoro symmetries in the bulk is preserved. We
will refer to a CFT with such a defect as a defect conformal field
theory (DCFT) (e.g. see the review part of \cite{Bachas:2001vj}).
Generically, there are extra propagating degrees of freedom
localized on the defect. However, it is also possible to construct a
system with no new degrees of freedom on the defect. Such a theory
is called an interface CFT (ICFT). A simple example of an ICFT is a
compactified scalar field $\phi(t,x)$ whose radius jumps at the
defect. An interesting quantity which characterizes a system with a
conformal boundary or defect is the boundary entropy, $S_{bdy}$.
$S_{bdy}$ is related to the ground state degeneracy $g$
\cite{Affleck:1991tk} as we explain below.
\subsection{Boundary Entropy and the $g$-function}
Consider a 2d CFT with periodic Euclidean time, $t\sim
t+2\pi$. We assume $x$ is also compactified on a large circle with
radius $L\gg1$. When we introduce a defect at $x=0$, the
partition function of this system on a torus behaves as
\begin{equation} \lim_{L\to \infty} Z_{torus}=
e^{-LE_0+S_{bdy}}, \end{equation} where $E_0$ is the ground state energy when
we regard $x$ as the Euclidean time direction. The quantity
$S_{bdy}$ is called the boundary entropy. This is motivated by the
observation that $S_{bdy}$ is the entropy when we artificially treat
$L$ as the temperature and $-\frac{\log Z_{torus}}{L}$ as the free
energy. Originally, the boundary entropy was defined in a 2d system
with a conformal boundary in \cite{Affleck:1991tk}. However, the
DCFT can be equivalently described by two copies of the system with
boundary via the doubling trick, as discussed, for example, in
\cite{Bachas:2001vj} or in appendix A of this paper.
The quantity $g$, defined by \begin{equation} g \equiv e^{S_{bdy}}, \end{equation} represents the
ground state degeneracy. We can extend the idea of the boundary
entropy to non-conformal systems and define the $g$-function.
According to the $g$-theorem \cite{Affleck:1991tk}, the $g$-function
is a monotonically decreasing function with respect to the length
scale $l$, \begin{equation} \frac{d}{dl}\log g(l)\leq
0, \label{gthfor} \end{equation} in analogy to the $c$-function and $c$-theorem.
\subsection{Boundary Entropy from Entanglement Entropy}
\label{befromee}
Recently, it was found that the boundary entropy is actually related
to a physical entropy, the entanglement entropy
\cite{Calabrese:2004eu}. To define the entanglement entropy, we
first divide the system into two parts $A$ and $B$. Accordingly, the
total Hilbert space is factorized as $H=H_A\otimes H_B$. Next we
introduce the reduced density matrix $\rho_A=\mbox{Tr}_B\rho$ for
the subsystem $A$ by tracing out $H_B$. Finally, the entanglement
entropy is defined as the von-Neumann entropy for $\rho_A$ \begin{equation}
S_A=-\mbox{Tr}\rho_A\log \rho_A. \end{equation}
Consider an infinitely long system and define the subsystem $A$ by
the finite interval with length $l$. The subsystem $B$ is defined to
be the complement of $A$. Then the entanglement entropy $S_A$ can be
computed to be \cite{Holzhey:1994we} \begin{equation} S_A=\frac{c}{3}\log \frac{l}{a},
\label{bulk} \end{equation} where $c$ is the central charge of the total system
and $a$ is the UV cut off (i.e. lattice spacing).
In the presence of a conformal boundary with boundary entropy
$\log g$, this is modified as follows \cite{Calabrese:2004eu}
\begin{equation}
S_A=\frac{c}{6}\log \frac{l}{a}+\log g . \label{entbg} \end{equation} Because the
boundary cuts off half of the space, we have the coefficient
$\frac{c}{6}$ instead of $\frac{c}{3}$.
When we consider a conformal defect which is situated at the middle
of the interval $A$, we can regard the system as two copies of a
BCFT by the doubling trick. This leads to
the following result
\begin{equation}
S_A=\frac{c}{3}\log \frac{l}{a}+\log g . \label{entg} \end{equation} To see the
relation (\ref{entg}) quickly, let us remember that in the 2d CFT
$S_A$ can be found from the formula \begin{equation} S_A=-\frac{\partial}{\partial
n}\mbox{Tr}\rho_A^n\bigl|_{n=1} =-\frac{\partial}{\partial
n}\left[\frac{Z_n}{(Z_1)^n}\right]\Bigl|_{n=1}, \label{formulaz}\end{equation}
where $Z_n$ is the partition function on the n-sheeted Riemann
surface with a cut along the interval $A$ \cite{Calabrese:2004eu}.
The important point is that both the original two dimensional space
and the $n$-sheeted one both have a single connected boundary. Thus
the ratio $\frac{Z_n}{(Z_1)^n}$ is proportional to the factor
$g^{1-n}$, which leads to the formula (\ref{entg}).
On the other hand, if the defect is not located at the midpoint
of the interval, the entanglement entropy cannot be determined only
from $c$ and $g$, but rather it depends on the details of
the theory. This is because we cannot relate this DCFT setup to the
BCFT setup by the folding trick, as the quantity we are interested in
does not have the reflection symmetry about the defect. In other
words, it is not possible to find a conformal map from the
$n$-sheeted Riemann surface defined by $v^n=\frac{w-l_1}{w+l_2}$ with
the defect at Re$~w=0$, to a single complex plane ${\bf C}$ with a
straight defect line except for $l_1=l_2$, which means that the defect
is at the midpoint of the interval.
\subsection{Strong Subadditivity and g-theorem}
It is intriguing to see if we can obtain useful properties of the
boundary entropy from the basic properties of entanglement entropy.
One of the most important inequalities satisfied by any entanglement
entropy is the strong subadditivity constraint (e.g. refer to the review
part of \cite{Casini:2004bw,Hirata:2006jx}). It is represented by
the inequality \begin{equation} S_{A}+S_B \geq S_{A\cap B}+S_{A\cup B}.
\label{SS} \end{equation} The holographic derivation of strong subadditivity
has been given in \cite{Headrick:2007km,Hirata:2006jx}.
It was shown in \cite{Casini:2004bw} that the entropic analogue of
the $c$-theorem follows from this relation. Therefore, it is natural
to ask if the $g$-theorem can also be derived from this condition.
Let us start with a simple setup (see fig.\ref{gth}.) in a defect
CFT. $A$ is defined by $[-l_a-l_c,\ l_a+l_c]$ and $B$ is defined by
the two intervals $[-l_a-l_b-l_c,-l_a]$ and $[l_a,\ l_a+l_b+l_c]$.
In this case, by substituting (\ref{entg}) into the strong
subadditivity constraint (\ref{SS}), we obtain \begin{eqnarray} && \frac{c}{3}\log
2(l_a+l_c) +\log g(2(l_a+l_c))+2\cdot \frac{c}{3}\log (l_b+l_c) \nonumber \\ &&\
\ \ \ \geq 2\cdot \frac{c}{3}\log l_c +
\frac{c}{3}\log(2(l_a+l_b+l_c))+\log g(2(l_a+l_b+l_c)).\label{ineq} \end{eqnarray}
Then in the limit $l_b\to 0$ we find \begin{equation} \frac{d}{dl}\log
g(l)\bigl|_{l=2(l_a+l_c)}\leq
\frac{c}{6}\left(\frac{2}{l_c}-\frac{1}{l_a+l_c}\right). \end{equation} By taking the
limit $l_a\to 0$, we obtain the bound \begin{equation} \frac{d\log g(l)}{dl}\leq
\frac{c}{3l}.\end{equation} Even though this is not enough to prove the
$g$-theorem (\ref{gthfor}), we can at least say that the $g$-theorem
is non-trivially
consistent with the strong subadditivity.\\
To see the relation to the g-theorem more clearly, we need to cancel
the log terms in (\ref{ineq}). This can be done by considering a
relativistic setup as in fig.\ref{gboost}. Notice that by requiring
Lorentz invariance, the Hilbert space $H_A$ for $A$ depends only on
the causal future (or past) of $A$ and remains the same under any
deformation which preserves it. If the Lorentz invariant length of
$A$ is denoted by $|A|$, we can easily show $|A|\cdot |B|=|A\cap
B|\cdot |A\cup B|$. Owing to this relation, the authors in
\cite{Casini:2004bw} were able to prove the $c$-theorem from the
strong subadditivity condition for this setup. Indeed, strong subadditivity
leads to \begin{equation} S(l_1)+S(l_2)\geq S(l_3)+S(l_4), \end{equation} where we assume
$l_1l_2=l_3l_4$ and $l_4\leq l_1, l_2\leq l_3$. This is equivalent
to the concavity of the entropy as a function of $\log l$ i.e. \begin{equation}
l\frac{d}{dl}\left(l\frac{dS(l)}{dl}\right)\leq 0.\label{entc}\end{equation} Noting
that $c(l)=3l\frac{dS(l)}{dl}$, it is clear from (\ref{bulk}) that the inequality
(\ref{entc}) is precisely the entropic $c$-theorem.
Now we return to the relation to the $g$-theorem and thus we assume
that the bulk region is conformal. If we again employ the choice of
subsystems and the defect line as described in fig.\ref{gboost}, the
bulk log terms are completely canceled out. Since the other part of
the entanglement entropy can be regarded as an entropic g-function,
we simply obtain \begin{equation} \log g (A)\geq \log g (A\cup B). \label{ddd}
\end{equation} This indeed agrees with the g-theorem in a particular case. In
this way, we have learned that strong subadditivity for the
entanglement entropy is closely related to the $g$-theorem for
DCFTs. We leave a further study of this issue as a future problem.
\begin{figure}[htbp]
\begin{center}
\hspace*{0.5cm}
\includegraphics[height=3cm]{gtheorem.eps}
\caption{The simple setup of $A$ and $B$ at a common fixed time. Notice that
both $A$ and $B$ live in the same one dimensional space.}\label{gth}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\hspace*{0.5cm}
\includegraphics[height=3cm]{gboost.eps}
\caption{The relativistic setup of $A$ and $B$. The vertical and
horizontal directions represent the time
and space coordinates, respectively. The dotted light-like triangle
delimits the causal future.}\label{gboost}
\end{center}
\end{figure}
\section{Holographic Boundary Entropy of Defect}
\label{holographicBE} \setcounter{equation}{0} An interesting class
of 2d CFTs, in which the boundary entropy can be calculated using
its relation to the entanglement entropy, is those 2d CFTs which
have a dual description in terms of a higher dimensional
gravitational theory on an asymptotically AdS$_3$ background. A
method to calculate entanglement entropies in theories with
gravitational duals has been found in \cite{Ryu:2006bv,Ryu:2006ef}.
This equates the entanglement entropy of a spatial region ${\cal M}$
in the field theory bounded by $\partial {\cal M}$ with $\frac{A}{4
G_N^{(3)}}$, where $A$ is the area of a minimal area surface ending on $\partial {\cal M}$ and in a constant time slice of the 3d bulk . $G_N^{(3)}$
is the 3d Newton constant. We will apply this formula here to
known examples of DCFTs with a gravitational dual.
In particular, we will look at three different systems. The first system
we study is a Randall-Sundrum (RS) like toy model
\cite{Randall:1999vf,Karch:2000ct} of a brane coupled to gravity,
which for a certain range of tensions has a dual description in
terms of a DCFT. Since this model, in its simplest form, has not
been embedded in string theory or any other consitent
theory of quantum gravity, we don't know precisely what the
DCFT is (and whether it exists at all).
But the advantage is that in this case we can calculate
both the entanglement entropy as well as (in the ``probe" limit of
small tension) the high temperature free energy, confirming that the
two alternative definitions of the boundary entropy do indeed agree.
The second model we look at is the Janus solution
\cite{Bak:2003jk,Bak:2007jm}. In this example, one once more knows
the full bulk geometry and can calculate the entanglement entropy.
For Janus the dual DCFT is known and is of the interface type. We
can calculate the boundary entropy also in the limit of weak
coupling, where the calculation is tractable on the field theory
side. To leading order in the parameter controlling the jump across
the interface, we observe agreement between weak and strong
coupling.
Last, but not least, we look at defects with localized matter. These
systems often have a dual description in terms of a probe brane
embedded in the AdS$_3$ space. In these scenarios we don't have
access to the entanglement entropy without controlling the backreaction.
However, we can calculate the boundary entropy via the high
temperature free energy, as was already pointed out in
\cite{Yamaguchi:2002pa}. The dual field theory is once more well
understood and we can compare weak and strong coupling answers.
\subsection{Defect in a Toy Model}
Our first example of a 3-dimensional geometry with a dual
description in terms of a DCFT arises as a solution to the RS
\cite{Randall:1999vf} action of a 2d brane with tension $\lambda$ \begin{equation} S =\frac{1}{16 \pi G_N^{(3)}} \int
d^3 x \sqrt{-g} (R +\frac{2}{R_{AdS}^2}) - \lambda \int d^2 x
\sqrt{-g_I}, \end{equation} where $g_I$ is the determinant of the induced metric
on the 2d slice spanned by the brane. Without the brane the solution
to this system would be AdS$_3$ with curvature radius $R_{AdS}$. For
branes with a tension $\lambda$ less than a critical value $
\lambda_* = \frac{1}{4 \pi G_N^{(3)} R_{AdS}}$ one can find
solutions which have a brane with an AdS$_2$ geometry and hence
precisely preserve the symmetries expected from a dual DCFT. No
embedding of the system in this simple form into string theory is
known. Assuming that it makes
sense as a quantum theory, the observables in this theory have the
interpretation of correlation functions in some DCFT
\cite{Karch:2000ct}. For this toy model we do not have an
alternative definition of the DCFT.
\subsubsection{The Background Solution}
To construct the solution, consider the $d+1$ dimensional asymptotic
AdS background \begin{equation} \label{adsads} ds^2= R_{AdS}^2(dy^2+e^{2A(y)}
(ds_{AdS_{d}})^2). \end{equation} The pure AdS corresponds to $e^{A(y)}=\cosh
y$.
We are interested in the dual of a 2d CFT so we set $d=2$. Then we
can write $(ds_{AdS_{d=2}})^2=-\cosh ^2 r dt^2+dr^2$. In the
ordinary global coordinate we can rewrite as follows \begin{equation}
\label{globalads} ds^2=R_{AdS}^2(-\cosh^2 \rho
dt^2+d\rho^2+\sinh^2\rho d \theta ^2). \end{equation} For pure AdS$_3$, the coordinates are
related to each other via \begin{equation} \cosh y \cosh r=\cosh\rho,\ \ \ \
\sinh y=\sinh\rho\sin\theta. \end{equation} Using these global coordinates the
geometry on which the dual CFT lives is actually a circle and not
just a line. There are two defects at $\theta=0$ and
$\theta=\pi$, corresponding to the boundary points at fixed $y$ but
infinite $r$. In the presence of a codimension one defect with
tension $\lambda$, the equation of motion becomes
\begin{equation} -1+(A')^2+A''=8\pi G^{(3)}_N R_{AdS} \lambda \delta(y),
\end{equation} where we assumed that the brane is situated at $y=0$. This can
be solved by \begin{equation} e^{A(y)}=\cosh(|y|-y_*). \end{equation} The constant $y_*$ is
defined by \begin{equation} \tanh y_*=4\pi G^{(3)}_N R_{AdS} \lambda. \end{equation}
The spacetime with the backreaction due to the brane becomes two
copies of the partial AdS spacetime defined by $-y_*< y <\infty$ in
the AdS sliced coordinates (\ref{adsads}).
\subsubsection{Boundary Entropy from Entanglement Entropy}
As mentioned in the beginning of this section, the holographic
recipe for calculating entanglement entropies is to find at a fixed
time $t$ the minimal area surface in the bulk that ends on the
boundary of the region whose entropy we want to calculate. In the
case of a 3-dimensional bulk spacetime this minimal spatial
area is simply a geodesic. If we concentrate on the largest
region in the field theory that is symmetric around the defect we
are looking for a geodesic that connects the boundary points $\theta
= - \pi/2$ and $\theta=\pi/2$.
That is, in the coordinate system of eq. (\ref{adsads}) we
want to connect the point $r=0$ at $y = + \infty$ with the point
$r=0$ at $y=-\infty$. By symmetry it is easy to see that the
geodesic is $r=0$. We will return to this in more detail
later when we look at asymmetric regions.
For this longest geodesic we can easily calculate the extra length
$\Delta L$ induced by the defect brane as follows \begin{equation} \Delta
L=2R_{AdS}y_*. \end{equation}
Thus the extra contribution to the entanglement entropy becomes \begin{equation}
\Delta S_A=\frac{R_{AdS}y_*}{2G^{(3)}_N}. \end{equation} In the probe limit,
the brane tension is very small ($y_*\ll1$) and we can
approximately obtain \begin{equation} \label{rsee} \Delta S_A=2\pi R_{AdS}^2
\lambda. \end{equation} $\log g=\Delta S_A$ can directly be identified as the
boundary entropy of the dual DCFT.
\subsubsection{Boundary Entropy from Free Energy}
\label{btzrs}
Without the brane, turning on a finite temperature corresponds to
replacing the AdS$_3$ solution in the bulk with a BTZ black hole,
\begin{equation} \label{BTZmetric} ds^2 = - h(r_{BTZ}) dt^2 +
\frac{dr_{BTZ}^2}{h(r_{BTZ})} + r_{BTZ}^2 d \theta^2 \end{equation} with
$h(r_{BTZ}) = r_{BTZ}^2 - \mu +1$. For simplicity we switched to
units in which the curvature radius $R_{AdS}=1$. For $\mu=0$ this
is simply global AdS and reduces to eq. (\ref{globalads}) by a
change of coordinates $\sinh(\rho) = r_{BTZ}$. The BTZ black hole
has a horizon at $r_H$ such that $r_H^2 = \mu-1$. The temperature
of the black hole is given by $T=\frac{h'(r_H)}{4 \pi} =
\frac{r_H}{2 \pi}$.
In order to study the free energy of the DCFT at finite temperature
we need to find the generalization of the BTZ black hole with the
backreaction of the brane included. This is a difficult problem and
no solutions are known. However, in the small tension limit the
change of the geometry due to the brane can be neglected. As a power
series expansion in the tension of the brane, the leading
contribution comes from the on-shell action of the brane probe which
minimizes its worldvolume in the fixed BTZ black hole background
geometry. This is in complete analogy to the calculation that allows
one to calculate order $N_f N_c$ corrections to the order $N_c^2$
free energies in a theory with a large number of colors $N_c$ and a
finite number of flavors $N_f$ using probe branes
\cite{Karch:2002sh}. This technique has been first used for a free
energy calculation in \cite{Babington:2003vm} and has been confirmed
by many calculations since.
The action describing the embedding of the brane is
proportional to the worldvolume of the brane, \begin{equation} S_{probe} = -
\lambda \int d^2 x \sqrt{-g_I}. \end{equation} A simple embedding is given by
the union of $\theta=0$ and $\theta = \pi$. This is the finite
temperature generalization of the probe stretching straight across
the AdS$_3$ space along the central $y=0$ slice in the coordinate
system of eq. (\ref{adsads}). It is the minimal action configuration
which satisfies the boundary conditions that the probe ends on the
defects\footnote{Alternatively we can work in the analog of Poincare
Patch coordinates where we drop the 1 from $h(r)$ and think of
$\theta$ as living on the real line as opposed to on a circle. In
this case we describe a single defect at $\theta=0$.}, which are
located at $\theta=0$ and $\theta=\pi$. The induced metric on this
brane is $ds^2 = - h dt^2 + \frac{dr_{BTZ}^2}{h}$ and so the
determinant of the induced metric is 1. Wick rotating to
Euclidean signature and regulating the on-shell action by simply
subtracting the zero temperature answer we get for the free energy
associated with a single defect \begin{equation} F = -T S_{on-shell} =
\lambda \lim_{r_c \rightarrow \infty} \left (
\int_{r_H}^{r_c}dr - \int_0^{r_c} dr \right ) = -\lambda r_H = -2
\pi T \lambda. \end{equation} The entropy now can be calculated using the
standard relation \begin{equation} \label{rsfe} S = - \frac{\partial F}{\partial
T} = 2 \pi \lambda.\end{equation} Restoring the curvature radius $R_{AdS}$,
this is in perfect agreement with the answer for the boundary
entropy we got from the entanglement entropy, eq. (\ref{rsee}).
\subsection{Boundary Entropy and Janus Solution}
It is well-known that the near horizon limit of the D1-D5 system is
type IIB string theory on AdS$_3\times S^3\times T^4$. We assume
that there are $Q_1$ D1-branes and $Q_5$ D5-branes in this system.
The AdS$_3$/CFT$_2$ correspondence claims that the string theory in
this background is dual to the $(4,4)$ superconformal sigma model
whose target space is the symmetric product
$(T^4)^{Q_1Q_5}/S_{Q_1Q_5}$. We would like to deform this CFT so
that it includes a conformal defect. In particular, we are
interested in an interface which separates two regions with $T^4$ of
different radii. We assume that the radius of $T^4$ changes from
$R_+$ to $R_-$.
Recently, a 3 dimensional gravity background has been constructed \cite{Bak:2007jm} that is a
particular example of the Janus solutions.
The supergravity metric in the Einstein frame is
\begin{equation} ds^2_{IIB}=e^{\frac{\phi}{2}}(ds^2_{(3)}+d\Omega_3^2)
+e^{-\frac{\phi}{2}}ds^2_{T^4}.\label{sugrab} \end{equation}
The $(2+1)$ dimensional part $ds^2_{(3)}$ is described by the
Einstein-Hilbert action plus a scalar field $\phi$ (i.e. in Einstein
frame). This is because $\sqrt{-g^{(10)}}R^{(10)}=\sqrt{-g^{(3)}}R^{(3)}$.
For the Janus solution, the 3D metric is explicitly given by \begin{equation}
ds^2_{(3)}=R^2_{AdS}(dy^2+f(y)ds^2_{AdS_2}),\label{adsja} \end{equation} where
the function $f(y)$ is found to be \begin{equation}
f(y)=\frac{1}{2}(1+\sqrt{1-2\gamma^2}\cosh 2y). \end{equation} Also, the two
asymptotic values of the dilaton $\phi_{\pm}\equiv\phi(\pm \infty)$
are found to be \begin{equation} \phi_\pm=\phi_0
\pm\frac{1}{2\sqrt{2}}\log\left(\frac{1+\sqrt{2}\gamma}{1-\sqrt{2}\gamma}\right).
\end{equation}
For $\gamma=0$ the dilaton is constant, $\phi=\phi_0$, and the
metric reduces to pure AdS in the coordinate system of eq.
(\ref{adsads}).
The geodesic distance $L$ is needed in order to compute the
holographic entanglement entropy. To obtain it, we have to be careful about the
regularization of the UV divergence. This can be done by expressing
the asymptotically AdS metric always in the form $ds^2_{(3)}\simeq
R^2_{AdS}\frac{dz^2+dx^2-dt^2}{z^2}.$ Then the UV cutoff is always
given by $z=\epsilon$. In our case of (\ref{adsja}) we obtain \begin{equation}
\epsilon=e^{-y_{\infty}}\frac{2}{(1-2\gamma^2)^\frac{1}{4}}. \end{equation} In this way, we can
find the additional contribution to the entanglement entropy when we put a
non-zero value of the Janus deformation $\gamma$ to be \begin{equation}
\Delta L=L-L_{\infty}=2R_{AdS}(y_{\infty}(\gamma)-y_{\infty}(0))
=R_{AdS}\log\frac{1}{\sqrt{1-2\gamma^2}}. \end{equation}
The radius and the 3D Newton constant expressed in terms of the dual
2d CFT quantities are given by \begin{equation} \label{d1d5map}
R_{AdS}=\sqrt{g_6}(Q_1Q_5)^{1/4}l_s,\ \ \ \
4G^{(3)}_N=\sqrt{g_6}(Q_1Q_5)^{-3/4}l_s, \ \ \
g_6=g_s\sqrt{\frac{Q_5}{Q_1}}.\end{equation}
Thus, in the end we obtain the shift of the entanglement entropy as
follows: \begin{equation} \Delta S_A=\frac{\Delta
L}{4G^{(3)}_N}=\frac{Q_1Q_5}{2}\log\frac{1}{1-2\gamma^2}
=Q_1Q_5(\gamma^2+\gamma^4+\cdot\cdot\cdot).\label{resultads} \end{equation} We can claim
that this finite part which appears in the Janus background actually
corresponds to the boundary entropy (or the logarithm of the
$g$-function) by applying the relation (\ref{entg}).
Now we want to perform the direct computation of the boundary
entropy from the CFT side in order to compare with the above
result.
To treat the defect CFT we need the doubling trick discussed in
\cite{Bachas:2001vj}. Consider again a single compactified scalar
$\phi$ in the presence of the interface where the radius of the
scalar jumps from $R_+$ to $R_-$. This theory is equivalent to a
BCFT with two scalar fields whose radii are $R_+$ and $R_-$. The
boundary condition is the Neumann-Dirichlet type (i.e. there is a `D1-brane'
which wraps the diagonal $S^1$ in $T^2$) as we will review in
appendix A. Since the $g$-function is proportional to the tension of
the D-brane and is T-dual invariant, we obtain (see
\cite{Elitzur:1998va}) the following results for a single boson
compactified with the radius $\tilde{R}$ \begin{equation}
g_{N}=\sqrt{\frac{\tilde{R}}{\sqrt{2\alpha'}}},\ \ \ \ \
g_{D}=\sqrt{\frac{\sqrt{\alpha'}}{\sqrt{2}\tilde{R}}}, \end{equation} for Neumann and Dirichlet
boundary conditions, respectively. For the Dp-brane wrapped on $T^p$ with the
B-field (i.e. the gauge flux) we obtain \begin{equation} g=2^{-p/4}\cdot\det
(G-BG^{-1}B), \label{gdp} \end{equation} where we assume that all torus
coordinates have the periodicity $x_{i}\sim x_i+2\pi\sqrt{\alpha'}$.
Our system is described by a D1-brane stretching in the diagonal
direction of $T^2$. This is T-dual to a D2-brane with a gauge flux
$B_{12}=1$, which corresponds to a single D0 charge.
Plugging in $g_{11}=\frac{1}{R_1^2}$ and $g_{22}=R_2^2$, the formula
(\ref{gdp}) leads to \begin{equation}
g=\frac{1}{\sqrt{2}}\sqrt{\frac{R_+}{R_-}+\frac{R_-}{R_+}}. \end{equation} Indeed we can
confirm $g=1$ at $R_+=R_-$, which corresponds to the absence of the
defect. Thus we
get \footnote{The boundary entropy of this DCFT very
recently has also been studied in \cite{Bachas:2007td}.
Interestingly, the authors find that such an interface,
where the radius of a compact scalar jumps by a
finite amount, can increase the entropy by splitting into 2 defects
with smaller jumps. This process repeats and
ultimately one should obtain infinitely many
defects with infinitessimally small jumps.
In \cite{Bachas:2007td} this property is identified as an
instability in the sense of the renormalization group flow. The
CFT has relevant operators that drive the RG flow away from the
fixed point with a single defect. This only turns
into a dynamical instability if we promote
the radius of the scalar into a dynamical field. We thus should not expect to see
this as an instability in the spectrum of normalizable fluctuations
around the Janus geometry with fixed asymptotic behavior,
consistent with the positive energy theorem
proven for Janus-type solutions in \cite{Freedman:2003ax}.
}
\begin{equation} \Delta S_{bdy}=\log g=
\log\frac{\sqrt{\frac{R_+}{R_-}+\frac{R_-}{R_+}}}{\sqrt{2}}. \end{equation}
Then we need to estimate the value of $\frac{R_+}{R_-}$ dual to the
Janus solution. First, we notice that the warp factor of the $T^4$
part becomes the constant $1$ in the {\it string} frame because
$G^{Einstein}_{\mu\nu}=e^{-\frac{1}{2}\phi}G^{string}_{\mu\nu}$. Thus
the kinetic term of the $(T^4)^{Q_1Q_5}/S_{Q_1Q_5}$ sigma model
should be proportional to $\frac{1}{g_s}=e^{-\phi}$. Explicitly, this
term goes like $\sim e^{-\phi}\int dz^2 G^{string}_{\mu\nu}\partial X^\mu
\bar{\partial} X^\nu$. Thus the radius is proportional to $e^{-\phi/2}$.
The ratio $R_+/R_-$ in the Janus solution becomes \begin{equation}
\frac{R_+}{R_-}=\left(\frac{1+\sqrt{2}\gamma}{1-\sqrt{2}\gamma}\right)^{\frac{1}{2\sqrt{2}}}.\label{ratior}
\end{equation} We can estimate the boundary entropy as follows \begin{equation}
S_{bdy}=4Q_1Q_5\log\frac{1}{\sqrt{2}}
\sqrt{\left(\frac{1+\sqrt{2}\gamma}{1-\sqrt{2}\gamma}\right)^{\frac{1}{2\sqrt{2}}}+
\left(\frac{1-\sqrt{2}\gamma}{1+\sqrt{2}\gamma}\right)^{\frac{1}{2\sqrt{2}}}}=
Q_1Q_5(\gamma^2+\frac{7}{6}\gamma^4+\cdot\cdot\cdot). \label{resultcft}\end{equation} As
expected, the boundary entropy in the free theory can also be
calculated via the free energy yielding identical results. We
present that calculation in appendix \ref{appendix2point}.
Thus the leading term ($\sim \gamma^2$) from AdS (\ref{resultads})
agrees with the one from CFT (\ref{resultcft}). Thinking of the
Janus field theory in the framework of conformal perturbation
theory, as in \cite{Clark:2004sb}, this agreement hints at a
non-renormalization of some correlation functions of the Lagrangian.
The relevant correlation functions are those of the Lagrangian with the twist fields that produce the $n-$sheeted
Riemann surface, corresponding to $Z_n$ in (\ref{formulaz}); refer
to \cite{Calabrese:2004eu,Ryu:2006ef} for general discussion. Also,
as shown in fig.\ref{comp}, the deviations of (\ref{resultads}) from (\ref{resultcft}) are
very small for any value of $\gamma$. We may notice that the
boundary entropy in the free field theory is always larger than that
in AdS (i.e. at strong coupling), which is natural.
\begin{figure}[htbp]
\begin{center}
\hspace*{0.5cm}
\includegraphics[height=4cm]{compare.eps}
\caption{The plot of the boundary entropy from both the AdS and free CFT calculation.
We plotted the values of $\frac{S_{bdy}}{Q_1Q_5}=\frac{\log g}{Q_1Q_5}$ as
a function of $\gamma$. Notice that $\gamma$ can take the values
$0\leq \gamma \leq \frac{1}{\sqrt{2}}.$ The upper and lower curve
correspond to the free field CFT result (\ref{resultcft}) and AdS
result (\ref{resultads}), respectively. They almost coincide with
each other, but there is a small deviation. }\label{comp}
\end{center}
\end{figure}
\subsection{Probe Computations for D1 D5 System}
Last, but not least, we want to study another set of DCFTs. A whole
class of DCFTs can be realized via probe\footnote{ Probe brane here
refers to the limit where the backreaction of the brane on the
geometry is negligible, so that the brane simply minimizes its
worldvolume action in a given fixed background. In the field theory
this corresponds to the quenched approximation which is justified by
a large number of colors limit.} D-branes in known AdS backgrounds
\cite{Karch:2000gx}. For AdS$_3$ such probe branes
were first discussed in \cite{Bachas:2000fr}.
In the dual field theory, the probe brane
corresponds to adding a finite number of localized matter fields
into a CFT with a large number of degrees of freedom, for example
$N_f$ fundamental hypermultiplets into a large $N_c$ gauge theory.
Focusing on 2d field theories, we can start with the AdS$_3\times
S^3\times T^4$ spacetime considered in the last subsection and add a
probe F1-string on AdS$_2$ or a probe D3 brane
on AdS$_2 \times S^2$ \cite{Raeymaekers:2006np, Couchoud:2003jw}. Without solving for the backreaction of these
branes we cannot extract the entanglement entropy. It is however
straightforward to obtain the free energy at high temperatures
directly from the probe action, just as we did in section \ref{btzrs}
for the RS brane in the small tension limit.
Consider $M$ F1-strings or $M$ D3 branes in the BTZ black hole
background eq. (\ref{BTZmetric}), since these are the potential
supersymmetric probes. These branes have an action which is given by
their area just as in the RS toy model. In addition there are also
couplings to the background form fields, in particular in the WZ
term in the D3 action. We must consider how these terms contribute to the action. In the
string theory setup the background is supported by a 3-form RR flux
$H$. A convenient gauge choice for the RR 2-form is to take it to be
of the form $B_{RR} = B(r) dt \wedge d\theta$. The probe embedding
we are looking for is the same $\theta=0$ one we discussed in the RS
toy model. Since the pullback of $B_{RR}$ for this
embedding is zero it does not contribute to the on-shell action and,
as before, we get the free energy (and hence the boundary entropy
associated with the defect) simply from the volume of the brane. The
result, as in eq. (\ref{rsfe}), is that for a brane of tension
$\lambda$ the boundary entropy of the defect dual to the probe brane
is \begin{equation} S = 2 \pi R_{AdS}^2 \lambda. \end{equation} For the F1 and D3 branes all
we need is to plug in the relevant values of the tension
$\lambda_{F1,3}$ using eq. (\ref{d1d5map}). The general formula is
that the tension of a Dp brane is $\lambda_p = \frac{1}{(2 \pi)^p}
\frac{1}{gs} \frac{1}{l_s^{p+1}} $ and $\lambda_{F1}=\frac{1}{2\pi
l_s^2}$ for the fundamental string. With this we get for a single
probe brane
\begin{eqnarray}
\nonumber \lambda_{F1} = \frac{1}{2 \pi l_s^2} = \frac{g_s Q_5}{2
\pi R_{AdS}^2} \,\,\, &\Rightarrow \,\,\,& S_{F1} = g_s Q_5, \\
R_{AdS}^2 \lambda_3 = \frac{R_{AdS}^2}{(2 \pi)^3 g_s l_s^4} =
\frac{g_s Q_5^2}{(2 \pi)^3 R_{AdS}^2} \,\,\, &\Rightarrow \, \, \,&
S_3 = \frac{g_s Q_5^2}{(2 \pi)^2}.
\end{eqnarray}
For $M$ probe branes we get $M$ times these expressions. A
boundary entropy scaling as $M Q_5$ is expected for the D3 brane
from the weak coupling consideration. This gets enhanced by a power
of the 't Hooft couplings $g_s Q_5$. Such a strong coupling
enhancement of the free energy has been seen in other probe systems
before, such as the D7 probe that adds flavor to ${\cal N} =4$
super-Yang-Mills, where the free energy scales as $\lambda N_f N_c$
instead of the naive $N_f N_c$ (see e.g.
\cite{Mateos:2006nu,Albash:2006ew,Karch:2006bv}). For the F1 string
we see a very similar effect. This determination of the boundary
entropy from the free energy contribution due to a probe brane can
easily be generalized to higher dimensional systems. What is unclear
to us at the moment is whether, as in 2d, in these higher
dimensional examples an equivalent definition of the boundary
entropy can also be given via the entanglement entropy. We hope to
return to this issue in the future.
\subsection{Size and Shape (In)dependence}
Given our understanding of the meaning of the boundary entropy, we
would expect that the contribution to the boundary entropy of a
defect should be independent of the size of the subsystem enclosing
the defect as long as the defect is in the center of the interval.
For a defect that is off-center the folding trick can not be used to
reduce the entanglement entropy calculation in the DCFT to the well
known case of a BCFT, as we pointed out in section \ref{befromee}.
It appears that in a DCFT the entanglement entropy of such an
asymmetrically shaped region depends explicitly on the microscopic
details and not just on the two universal numbers $c$ and $g$. In
this subsection we want to reanalyze this issue in the context of
the holographic calculation in the Janus framework.
We calculate the entanglement entropy of a spatial interval of
length $l$ containing the defect on the field theory side, but
potentially off-center. From the three-dimensional point of view, we
must find the geodesic length of a spacelike segment $r(y)$ in the
metric (\ref{adsads}) with one endpoint at $y = -\infty$, $r = r_0$
and one endpoint at $y = \infty$, $r = r_0 + \Delta r$. In line with
what we said above, we expect the boundary entropy to be independent
of the length $r_0$ but to depend on the asymmetry of the interval
about the defect parameterized by $\Delta r$.
The geodesic action is independent of $r$ and leads to the
conservation equation
\begin{equation}
\frac{f(y) r'}{\sqrt{f(y) r'^2 +1}} = \alpha
\end{equation}
where $\alpha$ is some constant that sets the asymmetry of the
interval that is nontrivially related to $\Delta r$ (this can be seen by integrating $r'$ from $y = -\infty$ to
$y = \infty$).
The fact that the geodesic length depends only on $r'$ tells us
immediately that the boundary entropy is independent of $r_0$ as
expected. In order to establish the dependence on $\alpha$, we must
calculate the geodesic length in the Janus system for some nonzero
$\alpha$ and subtract from it the geodesic length in the pure AdS
system, being careful that in both calculations the boundary
interval has the same length. It is very easy to see that in this
case the difference in entanglement entropies between Janus and pure
AdS gets a contribution from the detailed shape of the warpfactor
around the center of AdS (that is, around $y=0$).
\section{Conclusion}
\label{conclusion}
In this paper we have calculated the boundary entropy in several
strongly coupled 2d defect conformal field theories which have a
holographic dual. We confirmed that the definition of the boundary
entropy in terms of the entanglement entropy gives identical answers
to the definition in terms of a free energy at large temperature.
Perhaps most interestingly, we found that this equivalence only holds
in the case that one calculates the entanglement entropy for an
interval that has the defect at the center, so that the DCFT can be
mapped to a BCFT via the folding trick and the entanglement entropy
is completely specified by two universal numbers, the boundary
entropy and the central charge. In a DCFT, the entanglement entropy
of an asymmetric interval captures detailed information about the
microscopic details of the theory. In particular, from the knowledge
of the entanglement entropy for arbitrarily shaped intervals one can
reconstruct the length of all geodesics in the bulk and hence
presumably the bulk metric.
Our methods employed in the bulk can readily be generalized to
higher dimensions. In this case it is not clear if there is a
similar universal definition of a boundary entropy as in 2d, though
we may speculate that a coefficient of subleading divergent parts in
the entanglement entropy will be a counterpart of the boundary
entropy. However, it should still be interesting to calculate free
energies and entanglement entropies associated with defects in
strongly coupled theories in more than 2 dimensions.
\vskip2mm
\noindent {\bf Acknowledgments}
AK would like to thank the Yukawa Institute for Theoretical Physics
in Kyoto for their hospitality while this work was initiated. The
work of AK and ET was supported in part by the U.S. Department of
Energy under Grant No. DE-FG02-96ER40956. The work of TT is
supported in part by JSPS Grant-in-Aid for Scientific Research
No.18840027 and by JSPS Grant-in-Aid for Creative Scientific
Research No. 19GS0219.
|
1,108,101,565,502 | arxiv | \section{Introduction}
Similarly to light, gravitational waves can be gravitationally lensed by massive astrophysical objects, e.g., galaxies and galaxy clusters~\citep{Ohanian:1974ys,Thorne:1982cv,Deguchi:1986zz,Wang:1996as,Nakamura:1997sw,Takahashi:2003ix}.
Lensing changes the gravitational-wave amplitude without changing its frequency evolution~\citep{Deguchi:1986zz,Wang:1996as,Nakamura:1997sw,Takahashi:2003ix, Dai:2017huk,Ezquiaga:2020gdt}.
Moreover, strong lensing produces multiple images observable at the detectors as repeated events separated by minutes to months when lensed by galaxies~\citep{Ng:2017yiu,Li:2018prc,Oguri:2018muv}, and up to years
when lensed by galaxy clusters~\citep{Smith:2017mqu,Smith:2018gle,Smith:2019dis,Robertson:2020mfh,Ryczanowski:2020mlt}.
While much of the gravitational-wave lensing theory is similar to electromagnetic lensing, the detection methodologies and the science case are different. For example, in light lensing, one can observe strong lensing by discerning multiple images with telescope imaging. In GW lensing, we observe strongly lensed GWs as repeated events that can be identified with GW templates inaccessible to electromagnetic searches~\citep{Haris:2018vmn,Hannuksela:2019kle,Dai:2020tpj,Liu:2020par,Lo:2021nae,Janquart:2021qov}. The principal methodologies to detect gravitational-wave lensing with ground-based detectors have been developed in recent years~\citep{Cao:2014oaa,Lai:2018rto,Haris:2018vmn,Hannuksela:2019kle,Pang:2020qow, Pagano:2020rwj, Hannuksela:2020xor,Dai:2020tpj,Liu:2020par,Lo:2021nae,Janquart:2021qov}. Moreover, the LIGO-Vigro Collaboration (LVC) performed the first comprehensive search for gravitational-wave lensing signatures in the first half of the third LIGO-Virgo observing run recently~\citep{Abbott:2021iab}.
If detected, gravitational-wave lensing may enable several exciting scientific frontiers such as localisation of merging black holes to sub-arcsecond precision~\citep{Hannuksela:2020xor}, precision cosmography studies~\citep{Sereno:2011ty,Liao:2017ioi,Cao:2019kgn,Li:2019rns,Hannuksela:2020xor}, precise tests of the speed of gravitational-wave propagation~\citep{Baker:2016reh,Fan:2016swi,Mukherjee:2019wfw,Mukherjee:2019wcg}, tests of the gravitational-wave polarization content~\citep{Goyal:2020bkm}, and detecting intermediate-mass or primordial black holes~\citep{Lai:2018rto,Diego:2019rzc,Oguri:2020ldf}. They may also be useful in lens modelling by allowing one to break the mass-sheet degeneracy~\citep{Cremonese:2021puh}.
Recent strongly lensed gravitational-wave forecasts have predicted gravitational-wave lensing at a reasonable rate at design sensitivity of the Advanced LIGO and Advanced Virgo detectors~\citep{Ng:2017yiu,Li:2018prc,Oguri:2018muv,Xu:2021bfn,Mukherjee:2021}
(see also~\citet{Smith:2017mqu,Smith:2018gle,Smith:2019dis,Robertson:2020mfh,Ryczanowski:2020mlt} for estimates for galaxy clusters).
In addition, \citet{Xu:2021bfn} studied lensing forecasts in the context of probing the black hole and lens populations, while \citet{Mukherjee:2021} studied the impact of the binary coalescence times on the rate of lensing.
\citet{Haris:2018vmn} characterized the distribution of lensed events.
Here we further investigate strong lensing forecasts with a focus on the lensing science case and searches.
The science targets depend on the number of identified pairs. Suppose we have access to four lensed images of a gravitational-wave event. In that case, we might localise the gravitational-wave event to its host galaxy by comparing the image properties of the lensed wave with those produced by galaxies independently observed in the electromagnetic bands~\citep{Hannuksela:2020xor}. Two images might still allow us to constrain the number of candidates~\citep{Sereno:2011ty,Yu:2020agu}, but to a lesser degree as we will need to rely mainly on the magnification ratios to pinpoint the source location.\footnote{A search for a system lensed by a galaxy cluster might also be promising, even with two images~\citep{Smith:2017mqu,Smith:2018gle,Smith:2019dis,Robertson:2020mfh,Ryczanowski:2020mlt}.} More images also allow for better cosmography~\citep{Sereno:2011ty,Liao:2017ioi,Cao:2019kgn,Li:2019rns,Hannuksela:2020xor} and polarization tests~\citep{Goyal:2020bkm}.
Therefore, in Sec.~\ref{sec:lensed_rates}, we investigate the number of images discoverable in LIGO~\citep{Harry:2010zz,TheLIGOScientific:2014jea,TheVirgo:2014hva,Martynov:2016fzi,TheLIGOScientific:2016agk}, Virgo~\citep{TheVirgo:2014hva}, KAGRA~\citep{Somiya:2011np,Aso:2013eba,Akutsu:2020his}, A+~\citep{Abbott:2020qfu}, and LIGO Voyager~\citep{Adhikari:2020gft}.
In particular, we might identify two or more super-threshold triggers when we search for multiply imaged, strongly lensed gravitational waves~\citep{Li:2018prc}. However, it is also entirely plausible to observe some of these multiple images below the usual noise threshold as sub-threshold triggers~\citep{Li:2019osa,McIsaac:2019use,Mukherjee:2021}.
Thus, we also comment on sub-threshold triggers.
Another important question to address is how lensing forecasts can help the strong lensing parameter estimation (see~\citet{Haris:2018vmn,Hannuksela:2019kle,Liu:2020par,Lo:2021nae,Janquart:2021qov}). In particular, unlensed events can mimic a strongly lensed event by chance, resulting in a false alarm. The probability of a false alarm increases as we detect more events ($\propto N^2$, number of events squared) until the likelihood of a false alarm occurring becomes inevitable. However, we show how incorporating knowledge of the galaxy lensing time delay can significantly improve searches so that the false alarm increases at the same rate as it does for usual searches (Sec.~\ref{sec:FAP}).
While the time delay effect has been investigated, e.g., in \citet{Haris:2018vmn}, it has usually been discussed in the context of an additional improvement upon the usual searches.
Here we point out how, without the information of the lensing time-delay distribution, strong lensing searches may rapidly become intractable due to the growing number of candidate pairs.
Finally, we report the redshift distribution of lensed events and the Einstein radii of the systems that lens them and briefly comment on the science case (Sec.~\ref{sec:redshift}).
We conclude in Sec.~\ref{sec:conclusions}.
Throughout this paper, we assume a flat $\Lambda$CDM cosmology with $H_0=70 \, \rm km \, s^{-1} \, Mpc^{-1}$ and $\Omega_m = 0.31$, and all uncertainties quoted are at the 90 \% confidence level.
\section{Catalogue of lensed events}
\label{sec:lensed_catalogue}
We model the mass distribution of binary black holes following the observational results for the \textsc{Power Law + Peak model} of~\citet{GWTC2:rates}, setting the mass power-law index $\alpha=2.63$, mass ratio power-law index $\beta_q = 1.26$, low-mass tapering at $\delta_m = 4.82$~$\rm M_\odot$, minimum and maximum masses $m_{\rm min}=4.59$~$\rm M_\odot$ and $m_{\rm max}=86.22$~$\rm M_\odot$, and a Gaussian peak at $\mu_{\rm m}=33.07$~$\rm M_\odot$ with a width $\sigma_m=5.69$~$\rm M_\odot$, for a fraction of the population $\lambda_{\rm peak} = 0.10$. These values are consistent with the LIGO--Virgo population studies~\citep{GWTC2:rates}. We adopt a fit to the Population I/II star merger-rate density normalized to the local merger-rate density following~\citet{Oguri:2018muv},
\begin{equation}
\label{eqn:mergerrate}
\mathcal{R}_\textrm{m}(z_s) = \frac{\mathcal{R}_0(b_4+1)e^{b_2 z_s}}{b_4+e^{b_3 z_s}} {\rm Gpc^{-3}}\, {\rm yr^{-1}} \,,
\end{equation}
where $\mathcal{R}_0 $ is the local merger-rate density, and $b_2=1.6$, $b_3=2.1$, and $b_4=30$ are fitting parameters. We take the local merger-rate density to be consistent with the local merger-rate observations $\mathcal{R}_0 = 23.9^{+14.3}_{-8.6} \, \rm Gpc^{-3} \, yr^{-1}$~\citep{GWTC2:rates}, where we take the uncertainty to be constant with redshift.
The galaxy lens population follows the SDSS galaxy catalogue~\citep{Collett:2015roa}, and we loosely follow \citet{Haris:2018vmn} in the derivation of the lens population and our sampling procedure.
The strong lensing optical depth~\citep{Haris:2018vmn}
\begin{equation}
\label{eqn:opticaldepth}
\tau(z_s) = 4.17 \times 10^{-6} \left ( \frac{D_\textrm{c}(z_s)}{\rm Gpc} \right )^3\,,
\end{equation}
where $D_\textrm{c}(z_s)$ is the comoving distance\footnote{Note that the optical depth definition here refers to the probability that a given event is lensed irrespective of whether it is detected; the information about the binary black hole population and the selection bias is included separately in the rate computations (Appendix~\ref{app:lensed_rate_derivation}).}.
Note that here we have approximated the optical depth using the singular isothermal sphere (SIS) lens model; including ellipticity may yield a $\sim 5-10\, \%$ correction~\citep{More:2011rn, Xu:2021bfn}.
To facilitate quadruply imaged sources (lensed events that are split into four images) and realistic lens models, we adopt a power-law ellipsoidal mass distribution with external shear to approximate our lensing galaxies, available in~\textsc{lenstronomy}~\citep{Birrer:2018xgm}.
Specifically, we assume SDSS velocity dispersion and axis ratio profiles of elliptical galaxies in the local Universe~\citep{Collett:2015roa}, a 0.05 spread (one standard deviation) on the measurement of each shear component, and a typical power-law density slope with a mean slope $\gamma=2$ with $0.2$ spread~\citep{Koopmans:2009} (see Appendices \ref{app:bbh} \& \ref{app:lenses}, for the full population details).
To model the rate of \emph{detectable} events, we employ Monte Carlo importance sampling to sample the binary and the lens population (see Appendix \ref{app:lensed_rate_derivation} for the full details), selecting only events that pass the detection threshold on the signal-to-noise ratio (SNR). We assume spin-less binary black holes and adopt the \textsc{IMRPhenomD}~\citep{Husa:2015iqa, Khan:2015jqa} waveform model.
Our results partially extend previous forecast studies~\citep[e.g.,][]{Haris:2018vmn,Li:2018prc}, by considering an updated mass-population model, a network of detectors, and a power-law ellipsoidal mass distribution with external shear.
\section{Lensed rates}
\label{sec:lensed_rates}
\begin{table*}[t]
\centering
\begin{tabular}{r r l l l l l}
\hline \noalign{\smallskip}
\multicolumn{2}{r}{Observed rates} & L & L/H & L/H/V/K & L/H/V/K (A+) & L/H/V/K (Voyager) \\
\noalign{\smallskip}\hline \hline \noalign{\smallskip}
Lensed events: & total & $0.21^{+0.10}_{-0.07}$ $\text{yr}^{-1}$ & $0.65^{+0.32}_{-0.22}$ $\text{yr}^{-1}$ & $1.3^{+0.6}_{-0.4}$ $\text{yr}^{-1}$& $3.3^{+1.7}_{-1.1}$ $\text{yr}^{-1}$ & $16.8^{+8.4}_{-5.6}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{double} & $0.17^{+0.08}_{-0.06}$ $\text{yr}^{-1}$ & $0.50^{+0.25}_{-0.17}$ $\text{yr}^{-1}$ & $0.92^{+0.46}_{-0.31}$ $\text{yr}^{-1}$ & $2.5^{+1.2}_{-0.8}$ $\text{yr}^{-1}$ & $13.1^{+6.5}_{-4.4}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{triple} & $0.032^{+0.016}_{-0.011}$ $\text{yr}^{-1}$ & $0.11^{+0.06}_{-0.04}$ $\text{yr}^{-1}$ & $0.23^{+0.12}_{-0.08}$ $\text{yr}^{-1}$ & $0.55^{+0.28}_{-0.19}$ $\text{yr}^{-1}$ & $2.0^{+1.0}_{-0.7}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{quadruple} & $0.011^{+0.005}_{-0.004}$ $\text{yr}^{-1}$ & $0.038^{+0.019}_{-0.013}$ $\text{yr}^{-1}$ & $0.12^{+0.06}_{-0.04}$ $\text{yr}^{-1}$ & $0.30^{+0.15}_{-0.10}$ $\text{yr}^{-1}$ & $1.6^{+0.8}_{-0.6}$ $\text{yr}^{-1}$\\
\noalign{\smallskip} \hline \noalign{\smallskip}
\multicolumn{2}{r}{Unlensed events} & $370$ $\text{yr}^{-1}$ & $1.1 \times 10^3$ $\text{yr}^{-1}$ & $1.9 \times 10^3$ $\text{yr}^{-1}$ & $5.8 \times 10^3$ $\text{yr}^{-1}$ & $31 \times 10^3$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{Relative occurrence} & 1 : 1760 & 1 : 1650 & 1 : 1500 & 1 : 1740 & 1 : 1830 \\
\noalign{\smallskip} \hline
\end{tabular}
\caption{
The observed lensed event rates for different detector networks and detector sensitivities (LIGO Livingston [L] and Hanford [H] at their design, A+, and Voyager sensitivities; Virgo [V] and KAGRA [K] are always at their design sensitivities), categorised according to the \textit{observed} number of super-threshold images.
All uncertainties are at the 90 \% confidence level and are a direct consequence of the uncertainty in the local merger-rate density.
Unless otherwise specified, design sensitivity is assumed.
The rates can be subject to some uncertainties introduced by different merger-rate density models, the choice of the detection threshold, and detector down-time.
However, we also report the relative rate of occurrences, which we expect to be subject to less uncertainty.
}
\label{tab:rates}
\end{table*}
\begin{table*}[t]
\centering
\begin{tabular}{r r l l l l l}
\hline \noalign{\smallskip}
\multicolumn{2}{r}{Observed rates} & L & L/H & L/H/V/K & L/H/V/K (A+) & L/H/V/K (Voyager) \\
\noalign{\smallskip}\hline \hline \noalign{\smallskip}
Lensed events: & total & $0.30^{+0.15}_{-0.10}$ $\text{yr}^{-1}$ & $0.90^{+0.45}_{-0.30}$ $\text{yr}^{-1}$ & $1.7^{+0.9}_{-0.6}$ $\text{yr}^{-1}$& $4.3^{+2.1}_{-1.5}$ $\text{yr}^{-1}$ & $19.9^{+9.9}_{-6.7}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{double} & $0.23^{+0.12}_{-0.08}$ $\text{yr}^{-1}$ & $0.67^{+0.33}_{-0.22}$ $\text{yr}^{-1}$& $1.2^{+0.6}_{-0.4}$ $\text{yr}^{-1}$ & $3.2^{+1.6}_{-1.1}$ $\text{yr}^{-1}$ & $15.6^{+7.8}_{-5.2}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{triple} & $0.054^{+0.027}_{-0.018}$ $\text{yr}^{-1}$ & $0.17^{+0.08}_{-0.06}$ $\text{yr}^{-1}$& $0.32^{+0.16}_{-0.11}$ $\text{yr}^{-1}$ & $0.71^{+0.35}_{-0.24}$ $\text{yr}^{-1}$ & $2.3^{+1.1}_{-0.8}$ $\text{yr}^{-1}$ \\[2 pt]
\multicolumn{2}{r}{quadruple} & $0.015^{+0.008}_{-0.005}$ $\text{yr}^{-1}$ & $0.061^{+0.031}_{-0.021}$ $\text{yr}^{-1}$ & $0.18^{+0.09}_{-0.06}$ $\text{yr}^{-1}$ & $0.43^{+0.21}_{-0.14}$ $\text{yr}^{-1}$ & $2.0^{+1.0}_{-0.7}$ $\text{yr}^{-1}$\\
\noalign{\smallskip} \hline \noalign{\smallskip}
\multicolumn{2}{r}{Relative occurrence} & 1 : 1210 & 1 : 1180 & 1 : 1100 & 1 : 1350 & 1 : 1540 \\[2 pt]
\multicolumn{2}{r}{Overall increase} & 45 \% & 39 \% & 36 \% & 29 \% & 19\% \\
\noalign{\smallskip} \hline
\end{tabular}
\caption{
The observed lensed event rates for different detector networks and detector sensitivities (LIGO Livingston [L] and Hanford [H] at their design, A+, and Voyager sensitivities; Virgo [V] and KAGRA [K] are always at their design sensitivities), including events with an SNR $>7$.
Note that here we presume that SNR $>7$ is an indicative proxy for a detection using sub-threshold searches~\citep{Li:2019osa,McIsaac:2019use,Abbott:2021iab}, but a more comprehensive study inspecting selection and estimates based on the false alarm probability will be required to quantify the precise improvement.
Unless otherwise specified, design sensitivity is assumed.
}
\label{tab:subrates}
\end{table*}
We classify a super-threshold event as an event trigger with a network SNR $\geq 8$ (for a discussion on the suitability of this SNR limit, see, e.g.,~\citet{Abbott:2020qfu}).
Assuming the two LIGO, the Virgo, and the KAGRA detectors operating 100 \% of the time at design sensitivity, we find that the total \emph{observed} rate of lensed events is $1.3^{+0.6}_{-0.4}$~$\rm yr^{-1}$. The observed rate of unlensed events is $\sim 1900$ $\rm yr^{-1}$, which gives us a relative rate of 1 lensed event for every 1500 unlensed event detections.
The relative rate of lensed-to-unlensed detections is broadly consistent with findings from, e.g.,~\citet{Li:2018prc, Oguri:2018muv}.
The expected event rates for variable numbers of super-threshold images and different sensitivities (design, the A+ detector upgrade, and the planned LIGO-Voyager detector) are given in Table~\ref{tab:rates}.
The uncertainties in the observed rate here are a direct consequence of the uncertainty in the local merger-rate density.
Note that the rate of observed events here is increased by the network of detectors; the single-detector (LIGO Livingston) estimate for the rate of unlensed events is around $\sim 370$ events per year (consistent with, e.g.,~\citet{Xu:2021bfn}).
Note that once detector down-time is included, the observed rate can drop by a factor of two or more.
Moreover, there is some uncertainty in the choice of the detection threshold, in the sense that the usual templated searches classify the detection threshold based on the false alarm rate, and not the SNR~\cite[e.g.,][]{Abbott:2020qfu}.
We expect that such uncertainties can shift the total observed rates by perhaps an additional factor of a few.
However, the results can be re-scaled based on the fractional rate of lensed to unlensed events, which we expect to be less sensitive to detector down-time, the precise detection threshold, or the local merger-rate density.
\begin{figure}[!b]
\centering
\includegraphics[width=\linewidth]{ObservedRates_fit.pdf}
\caption{The observed lensed event rate as a function of detection threshold SNR $\rho_{th}$ for double (blue), triple (yellow) and quadruple (magenta) lensed event detections and the total rate (black). The observed rates, most notably the quadruple image detection rates, increase by several factors as the threshold SNR decreases.
}
\label{fig:supervssub}
\end{figure}
Targeted lensed searches, when at least one super-threshold counterpart image is available, may allow one to uncover so-called sub-threshold triggers below the usual noise threshold by reducing the background noise and glitch contribution~\citep{Li:2019osa, McIsaac:2019use}. We classify a sub-threshold event as an event trigger observed below a network SNR of 8, but above a network SNR of $\rho_{th}$, when at least one counter image with SNR $> 8$ is present. Since $\rho \propto d_L^{-1}$, \citet{Li:2019osa} provides an indicative increase in the effective distance of $\sim 15\%$ corresponding to $\rho_{th} = 7$. The expected event rates for variable numbers of detected images and detector sensitivities are given in Table~\ref{tab:subrates}. We find that the total number of observed quadruply lensed events, increases from $0.12^{+0.06}_{-0.04}$ $\text{yr}^{-1}$ to $0.18^{+0.09}_{-0.06}$ $\text{yr}^{-1}$, an increase of $51 \%$, when considering sub-threshold triggers. Furthermore, the total number of observed triply lensed events increases with $40 \%$ from $0.23^{+0.12}_{-0.08}$ $\text{yr}^{-1}$ to $0.32^{+0.16}_{-0.11}$ $\text{yr}^{-1}$ and for doubly lensed events there is an increase of $33 \%$ from $0.92^{+0.46}_{-0.31}$ $\text{yr}^{-1}$ to $1.2^{+0.6}_{-0.4}$ $\text{yr}^{-1}$. The ``double", ``triple" and ``quadruple" nomenclatures refer to the number of detected images, and not the number of images produced by the lens. The increase in detectable images further motivates follow-up sub-threshold searches~\citep{Li:2019osa,McIsaac:2019use}.
However, because the sub-threshold searches vary in their sensitivity and further improvements may still be possible, the SNR threshold choice may vary. Thus, a threshold of SNR $>7$ is not a flawless proxy for detection.
For this reason, we also show the detectable rates for variable SNR thresholds (Fig.~\ref{fig:supervssub}).
\begin{figure*}[!th]
\centering
\includegraphics[width = \linewidth]{TimeDelay.pdf}
\caption{\textbf{(a)} The time-delay distribution for observed double images. The shaded regions give the 90 \% confidence intervals for $\Delta t_{12} \sim 1.5 \text{ hr} - 133 \text{ days}$. \textbf{(b)} The time-delay distributions (with confidence intervals) for observed triply lensed sources between the first two images (blue), between the second and third images (magenta) and the sum of those two (black). \textbf{(c)} The time-delay distributions (with confidence intervals) for observed quadruply lensed sources between the first two images (blue; $\sim 4.8 \text{ hr} - 52 \text{ days}$), the second and the third images (magenta; $\sim 0.8 \text{ hr} - 12 \text{ days}$), the last two images (yellow; $\sim 1.7 \text{ hr} - 30 \text{ days}$), and the total of the three (black; $\sim 2.3 \text{ hr} - 28 \text{ days}$). Generally, the time delay between the lensed pairs is $\lesssim 93 \, {\rm days}$. Knowledge of the time-delay distribution is particularly useful in improving strong lensing parameter estimation.
}
\label{fig:time_delays}
\end{figure*}
\begin{figure}[!b]
\centering
\includegraphics[width = 0.95\linewidth]{FalseAlarmProbabilityPlaatje.pdf}
\caption{\emph{Graphical illustration of an unlensed event mimicking a strongly lensed event:} Two images of a single gravitationally lensed gravitational-wave event (red) and an unlensed event (blue). In this example, the unlensed gravitational wave signal (blue) is indistinguishable from the strongly lensed gravitational-wave images (right bottom panel).
Indeed, unlensed events can, in principle, resemble strongly lensed events, giving rise to strong lensing mimickers or "false alarms."
}
\label{fig:waveforms}
\end{figure}
We note that the rate estimates are subject to further uncertainties due to a largely (observationally) unconstrained high-redshift merger-rate density. The merger-rate density can be modeled, for example, by presuming that the observed binary black hole population originates from Population-I/II stars, as we have done here.
Still, there are variations to the specific predictions in the different models and population-synthesis simulations~\cite[e.g.,][]{Eldridge:2018nop,Neijssel:2019irh,Boco:2019teq,Santoliquido:2020axb,Abbott:2021iab,Mukherjee:2021}. Here we postpone the investigation of different model predictions and instead note that the rate of lensing will be constrained by direct observations of gravitational-wave lensing~\citep{Mukherjee:2021},
and to a degree by the stochastic gravitational-wave background~\citep{Buscicchio:2020bdq,Mukherjee:2020tvr,Buscicchio:2020cij,Abbott:2021iab}. This work focuses on the science case for gravitational-wave lensing, the strong lensing searches, and the relative improvement in the multiple-image detections due to detector upgrades.
\section{The lensing time-delay distribution and its effect on strong lensing searches}
\label{sec:FAP}
The expected observed time-delay distribution is a direct output of our mock catalogue of lensed events (Fig.~\ref{fig:time_delays}).
To test whether two gravitational-wave events are lensed, one must show that the waves are identical within detector accuracy (save for an overall difference in the complex phase, arrival time, and amplitude), as expected of the lensing hypothesis~\citep{Haris:2018vmn,Hannuksela:2019kle,Dai:2020tpj,Liu:2020par,Lo:2021nae,Janquart:2021qov,Abbott:2021iab}.
However, it is also possible for two waveforms to be near-identical within detector accuracy by chance, giving rise to strong lensing "mimickers" (see Fig.~\ref{fig:waveforms}, for an illustration).
Here we demonstrate how the galaxy-lensing time-delay prior allows us to keep the strong lensing searches tractable.
The time-delay distribution of the unlensed events follows a Poissonian process~\citep{Haris:2018vmn}. The distributions for lensed events are an output of our simulation (see Fig. \ref{fig:time_delays}).
Given an expected lensing time-delay distribution, this allows us to calculate a ranking statistic
\begin{figure}[!th]
\centering
\includegraphics[width = \linewidth]{RLUDistributions.pdf}
\caption{The $\mathcal{R}^L_U$ distributions of simulated unlensed (orange) and lensed (purple) event pairs. For unlensed event pairs, the survival function (SF) is shown, which is $1 - \text{CDF}$ (cumulative distribution function). Only a small fraction of unlensed events have an $\mathcal{R}^L_U$ similar or higher than lensed events. }
\label{fig:RLU}
\end{figure}
\begin{equation}
\mathcal{R}^L_U = \frac{p(\Delta t | \text{Lensed})}{p(\Delta t | \text{Unlensed})}\,,
\end{equation}
which quantifies how much more likely, \emph{a priori}, a certain arrival time difference between event pairs is under the lensed hypothesis than under the unlensed one.
The time-delay $\Delta t$ can, in principle, refer to the expected time-delay between any permutation of the image combinations from a single event. Time delays from triple- or quadruple-image systems are expected to be correlated, and including these correlations would further improve the discriminatory power of strong lensing searches. However, we will neglect the correlations between time delays in the following, as we only aim to demonstrate the basic principle here.
As a practical example, we take the time-delay distribution to be for the difference in arrival time between any two consecutive images from quadruply lensed systems (Fig.~\ref{fig:time_delays}, right panel, gray shaded region). This equates to the hypothesis that two triggers come from a quadruply lensed event, but it is unknown where they place in the chronological order.
Let us first inspect the improvement in the significance of lensed detections due to the inclusion of the lensed time-delay prior.
We simulate unlensed and lensed populations of events and compute the $\mathcal{R}^L_U$ for all event pairs.
Based on the survival function (Fig.~\ref{fig:RLU}), we find a decrease of a factor of $3.1 \times 10^{-2}$, on average, in the false alarm probability per event pair produced by a randomly chosen lensed event, due to the inclusion of lensing time-delay information.
That is, by incorporating the expected lensing time-delay distribution, the significance of lensed detections has improved, on average, by a factor of $\sim 32$.
However, the benefit of incorporating the time-delay distribution becomes even more apparent when inspecting a catalogue of events.
The total catalogue false alarm probability (the probability of finding at least one false alarm in a set of $N_{\text{pairs}}$ signal pairs)
\begin{equation}
\label{eqn:FAP}
\text{FAP} = 1 - \prod_{i = 0}^{N_{\text{pairs}}} (1 - p_i),
\end{equation}
where the false alarm per given event pair $p_i$ consists of an "intrinsic" false alarm probability, the probability that two events share a similar frequency evolution and thus mimic lensing by chance, and the probability that a lensed event produces a similar time-delay $\Delta t_i$ as the two unlensed events.
\begin{figure}[!t]
\centering
\includegraphics[width = \linewidth]{FalseAlarmProbability.pdf}
\caption{The catalogue false alarm probabilities (FAP) as a function of the observation run times $t_{obs}$, assuming a constant event rate $N = 510$ {events/yr}, time-delay distribution of all quadruples, time window $\Delta t_{\rm cluster} = 1$ {yr} and a FAP per event pair of $= 10^{-6}$. Shown are the FAP without windowing (black), the windowed FAP for galaxy cluster lensing (purple), and the ranked FAP for galaxy lensing (orange). Including galaxy lensing statistics changes the functional dependency in the exponential from $\propto t_{obs}^2$ to $\propto t_{obs}$. This reduces the FAP significantly for galaxy lensing when $t_{obs} \sim 1$ {yr}. For galaxy cluster lensing, the improvement is less significant owing to the longer lensing time delays. }
\label{fig:fAP}
\end{figure}
Without incorporating knowledge of the lensing time delays, all events $N$ from the observing run need to be taken into account with equal weight, giving $N_{\text{pairs}} = N(N - 1)/2$, where $N$ is the total number of single events.
This makes the likelihood of finding a false alarm inevitable as we obtain more gravitational wave detections (Fig.~\ref{fig:fAP}, black line).
\begin{figure}[!t]
\centering
\includegraphics[width = \linewidth]{RedshiftDistribution.pdf}
\caption{The observed redshift distributions for the galaxy lenses (purple) and lensed sources (orange), plotted on different scales. The unlensed source distribution (black) is shown for comparison.
Additionally, we show the distributions specifically for events that have been quadruply lensed (dashed), as an example of the versatility of the data.
The 90\% confidence interval for the unlensed sources is $z_s \sim 0.7 - 2.1$, while for the lensed sources $z_s \sim 1.0 - 3.9$.
Lensed events can thus allow us to probe events beyond the regular detector horizon.}
\label{fig:redshift_distributions}
\end{figure}
However, when including galaxy lensing statistics, we find that the catalogue false alarm probability increases linearly with time, similar to typical single-event false alarms (Fig.~\ref{fig:fAP}, orange line).
Indeed, we argue that prior knowledge of the lensing time delays not only offers an advantage in the strong lensing searches, but that it is necessary to enable the searches.
Without prior knowledge of the time delays, the searches will inevitably run into false alarms.
The implications are particularly important when considering events with large time delays, such as the GW170104--GW170814 event pair investigated in~\citet{Dai:2020tpj,Liu:2020par,Abbott:2021iab}.
Such events, if lensed, would be lensed by galaxy clusters, for which the lensing time-delay distribution is less well understood and the probability of a false alarm is significantly higher (Fig.~\ref{fig:fAP}, purple line).
Here we assume a simple uniform prior between $0$ and $1 \, \rm yr$ for the time-delay of galaxy clusters, mostly for illustrative purposes.
Therefore, we should be particularly careful in understanding the time-delay distribution and interpreting the results in light of the entire gravitational-wave catalogue for such events.
Unfortunately, the time-delay distribution is subject to astrophysical uncertainties in lens modeling and the modeling of the binary population.
Thus, we argue that careful follow-up investigations to understand the astrophysical uncertainties in modeling the statistical distribution of lensed events are vital to strong lensing searches.
Detailed investigation of the false alarm probability in gravitational-wave catalogues will be given in (\c{C}al{\i}\c{s}kan et al., in preparation).
Finally, we note that the inclusion of expected image types~\citep{Dai:2017huk} and relative magnifications~\citep{Lo:2021nae} may also improve the discriminatory power of strong lensing searches.
In our simulation, quadruple images typically consist of two subsequent type-I and two subsequent type-II images; the second and third images are type-I and type-II.
\section{Redshift and lens distribution}
\label{sec:redshift}
\begin{figure}[!t]
\centering
\includegraphics[width = \linewidth]{EinsteinRadii.pdf}
\caption{The distributions for Einstein radii of detected lensed events (purple) and the underlying population (black). The 90\% confidence interval for the detected Einstein radii is $\theta_E \sim 0.2 - 1.8$ arcsec, while the prior population generally has Einstein radii $\theta_E < 1.0$ arcsec. }
\label{fig:einsteinradii}
\end{figure}
Strongly lensed gravitational waves originate from higher redshifts than unlensed gravitational waves. Particularly, lensed events originate from redshifts $z_s \sim 1.0 - 3.9$, above the usual detector horizon (Fig.~\ref{fig:redshift_distributions}).
We note that strongly lensed gravitational-wave events can, in principle, be localised by combining gravitational-wave and electromagnetic measurements~\cite[e.g.,][]{Hannuksela:2020xor}.
Thus, when localised, they may allow for high-redshift luminosity distance measurements.
This may be particularly interesting for cosmology, where it has been suggested that some of the existing high-redshift luminosity distance measurements could be at odds with the standard $\rm \Lambda$CDM model~\cite[e.g.,][]{Risaliti:2018reu,Wong:2019kwg,DiValentino:2021izs}.
However, we note that the localisation itself depends on the redshift distribution and the lens properties; only some fraction of host galaxies can be located in electromagnetic lensing surveys if they are near enough and their Einstein radii are large enough to be resolvable.
We show the distribution of Einstein radii in Fig.~\ref{fig:einsteinradii}, which may be informative for such localisation studies.
Besides the fundamental interest, the characterization of the lensed events is important in understanding the lensing science case.
\section{Conclusions}
\label{sec:conclusions}
Here we have reported 1) the expected number of double, triple, and quadruple gravitational-wave image detections in upcoming observing runs, 2) the positive impact of incorporating the lensing time-delay distribution on the false alarm probability for multi-image searches, 3) the expected source redshift and Einstein radius distribution of lensed gravitational-wave events. We have also demonstrated how using a galaxy (or galaxy cluster) lensing time-delay prior in our searches allows us to reduce the complexity of double-image searches. By including a prior, the false alarm probability increases linearly with time (similar to non-lensed searches) rather than exhibiting quadratic growth with time.
However, more work is needed in modeling the merger-rate density, which is largely observationally unconstrained, in studying the precise improvement in the detection rates from sub-threshold searches and understanding the lensing time-delay distribution of events lensed by galaxy clusters.
A lot of work on the forecasts has now been done and, besides our work, many groups have found reasonable rates of gravitational-wave lensing at the design sensitivity and beyond~\citep{Ng:2017yiu,Li:2018prc,Oguri:2018muv,Xu:2021bfn,Mukherjee:2021}.
Further progress in estimating the precise rate will likely be impeded by the lack of binary black hole observations at high redshifts, where lensed gravitational waves originate from, although studies of the stochastic gravitational-wave background seem like a promising avenue~\citep{Buscicchio:2020bdq,Mukherjee:2020tvr,Buscicchio:2020cij,Abbott:2021iab}.
Nevertheless, we expect that direct gravitational-wave lensing observations will give the final verdict on the rates.
In the meantime, statistical forecasts can inform us of our tentative expectations, allow us to efficiently investigate the science case and potential improvements in search methodologies, and offer mock data simulations to stress-test our tools.
To facilitate such follow-up research, we have published our catalogue of simulated lensed gravitational-wave events in~\citet{wierda_a_renske_a_c_2021_4905030}.
We also hope that our work gives further motivation to include lensing statistics results in strong lensing searches. \\
\acknowledgements
\noindent
\textbf{Acknowledgements}
The authors thank Alvin Li, Chun-Lung Chan, Manchun Yeung, and Tjonnie Li for useful comments and feedback.
The authors also thank Jose Ezquiaga for detailed comments on the manuscript.
We also thank Thomas E. Collett, Mesut \c{C}al{\i}\c{s}kan, Daniel Holz, Anupreeta More, Haris K, Riccardo Buscicchio, and Jolien Creighton for discussion on related projects.
A.R.A.C.W., O.A.H, and C.V.D.B. are supported by the research program of the Netherlands Organisation for Scientific Research (NWO).
The authors are grateful for computational resources provided by the LIGO Laboratory and supported by the National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459.
|
1,108,101,565,503 | arxiv | \section{introduction}
\textcolor{black}{While the existence of quasicrystals \cite{Shechtman1984,Levine1984}
in nature is no longer debatable, it remains an open question if
materials can have a quasicrystalline ground state, and what the finite-temperature properties of this phase are \cite{DeBoissieu2012}. In addition, the characteristics of the phasonic degrees of freedom in quasicrystals are still the focus of much interest \cite{Widom2008, Kromer2012}. It is therefore
of great value to investigate the finite temperature physical properties of simple models with a quasicrystalline ground state. Such models can easily be constructed using the mathematical theory of tilings \cite{Grunbaum1986}, and have been extensively used for the study of quasicrystallinity \cite{Stadnik1999,Steinhardt1999,Suck2002,Trebin2006}. In particular, some finite temperature properties were studied using tiling models.
For example, the elastic properties of a three-dimensional model were shown to change upon a finite temperature phase transition \cite{Jeong1993,Dotera1994}, and a two-dimensional (2D) tiling model was recently shown to undergo a series of phase transitions leading from the quasicrystalline phase to the liquid phase through a number of intermediate periodic phases \cite{Nikola2013}.\\}
\begin{figure}[t]
\noindent \centering{}\textcolor{black}{\includegraphics[scale=0.3]{tile_edge_1S}\caption{\label{fig:tiles} (Color online) The 16 Ammann tiles.}
}
\end{figure}
\textcolor{black}{The model studied here is based on the 16 Ammann
tiles, each of which being decorated with one label (out of possible six) on each of its four edges (Figure \ref{fig:tiles}).
Ammann \cite{Grunbaum1986} showed that
These tiles can perfectly tile the plane such that adjacent edges have matching labels. All such Domino-like tiling configurations are non-periodic and share a quasicrystalline order: well-defined Bragg peaks are observed in the Fourier transform of the densities of each given tile-type at frequencies incommensurate with the reciprocal lattice vectors. For an infinite system,
there is an uncountable number of different perfect tiling configurations, parameterized by two continuous phases, $\chi_{1},\chi_{2}\in [0,1)$ (see Appendix \ref{App:map}). These phases \cite{Mermin1992,Lifshitz2011}, are related to the amplitudes of the Bragg peaks (see equation (\ref{eq:fourier transform of tile density}) below). For any finite patch of a perfect tiling, these phases are not well defined, and can be described by fuzzy angles, whose
uncertainty is inversely proportional to the linear size $L$ \cite{Koch2009}. The number of different tilings of a finite system scales linearly with $N=L^2$. Accordingly, a finite change of $\chi_{1}$ and $\chi_{2}$ is required in order to induce any change in a finite patch tiling. However, a continuous change of $\chi_{1}$ and $\chi_{2}$ induces a continuous change in the {\it infinite} configuration: the fraction of tiles modified by an infinitesimal change of these phases is linear in this change (Figure \ref{fig:domino}).
This hidden continuous symmetry of the perfect tilings is therefore manifestly non-local. }\
\begin{figure*}[t]
\centering
(a){\includegraphics[width=0.26\textwidth]{conf1}}
(b){
\includegraphics[width=0.26\textwidth]{conf2}}
{
\includegraphics[width=0.36\textwidth]{change_of_tiles_final2}}
\caption{(Color online) (a) A perfect tiling configuration on a $5\times5$ lattice generate with $\chi_{1}=0.35$ and $\chi_{2}=0.6$.
(b) The minimal change of $\chi_{2}$ required to change the $5\times5$ configuration shown in (a) is $\sim0.037$ leading to this tiling. The black dots at the center of the tiles denote tiles
that are changed in comparison with the configuration shown in (a).
(c) $\Delta N$, the fraction of changed tiles as a function of the change $\Delta\chi$ of $\chi_{2}$, with respect
to the configuration $\chi_{1}=0.35,\chi_{2}=0.3$.}
\label{fig:domino}
\end{figure*}
In what follows, we show that this global continuous symmetry has a major
impact on the finite temperature behavior of the model studied
here. Namely, like a truly local continuous symmetry, it does not allow the system to be ordered at any positive temperature.
In order to study the model at a finite temperature, one needs to define the Hamiltonian. A natural choice, introduced by Leuzzi and Parisi \cite{Leuzzi2000}, is to identify the energy of a configuration with the number of mismatching edges. Thus, the (uncountably degenerate) ground states of the model are the perfect tilings exhibiting quasicrystalline order. We wish to study the stability of this order to thermal fluctuations. In order to write the Hamiltonian in a convenient form, we define the 16-dimensional density vector $\overrightarrow{\rho}$, containing the 16 tile densities $\rho_{i}(\mathbf{r})$,
each of which is a unity if the tile at $\mathbf{r}$ is of type $i$, and zero otherwise. In terms of these, the Hamiltonian takes the form
\begin{equation}
H=\sum_{\mathbf{r}}\left[\overrightarrow{\rho}{}^{\dagger}(\mathbf{r})Y\overrightarrow{\rho}(\mathbf{r}+\mathbf{\hat{y}})+\overrightarrow{\rho}{}^{\dagger}(\mathbf{r})X\overrightarrow{\rho}(\mathbf{r}+\hat{\mathbf{x}})\right],\label{eq:energy}
\end{equation}
where $X$ and $Y$ are known interaction matrices, dictated by the above edge-matching rule, whose explicit form can be found in Appendix \ref{App:XY}. The unit vectors $\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ connect each site to (two of) its nearest neighbors. Note that the lattice constant is chosen as the length unit.
Note that in a general quasicrystalline system two kinds of gapless collective excitations exist: phonons and phasons
\cite{Socolar1986, Lifshitz2011}. Phonons describe locally uniform translations, while phasons describe correlated rearrangements of atoms. Our model is defined on a fixed lattice and therefore the low-energy excitations described by this model are the phasonic degrees of freedom.
\textcolor{black}{Previous works \cite{Leuzzi2000,Koch2009,Rotman2011}
have provided numerical evidence that the model undergoes a symmetry
breaking phase transition from a quasicrystalline low temperature
phase into a high temperature disordered phase. This seems to contradict
the well known Mermin-Wagner theorem, stating that continuous symmetries cannot be spontaneously broken in 2D (or one-dimensional)
systems \cite{Mermin1966,Hohenberg1967,Mermin1968}. However, the theorem relies on the existence of a {\it local}
order parameter field that can be changed continuously. In our case, each tile, and even the ground state of each finite patch, has a finite degeneracy and cannot be changed continuously. A slow gradient of $\chi_{1}$ and $\chi_{2}$ will not make any change in most finite patches of the system, and will be manifested by a discrete jump in the energy for some isolated patches. It is therefore not clear whether the Mermin-Wagner theorem applies here.}
\section{Absence of quasicrystalline order}
\textcolor{black}{We first provide an argument that quasicrystallinity is broken at any finite temperature, in a fashion similar to the case of a truly continuous local symmetry. For this purpose, we assume that the system is ordered at low temperatures, and self consistently calculate its finite temperature properties. It is then shown that thermal excitations destroy the order.}\
\textcolor{black}{The global symmetry is reflected in the
Fourier transform of the tile densities.
At a ground state characterized by the two phases $\chi_1,\chi_2$, the Fourier transform of $\overrightarrow{\rho}(\mathbf{r})$, $\overrightarrow{\psi}(\mathbf{q})$, takes the form
\begin{eqnarray}
\overrightarrow{\psi}(\mathbf{q}) & = & N\sum_{m,n,i,j}\delta(\mathbf{q}-2\pi(n\tau\mathbf{\hat{x}}+m\tau\mathbf{\hat{y}}+i\mathbf{\hat{x}}+j\mathbf{\hat{y}}))\nonumber\\
& & \times e^{2\pi i\left(n\chi_{1}+m\chi_{2}\right)}\overrightarrow{\psi_{0}}(n,m), \label{eq:fourier transform of tile density}
\end{eqnarray}
where $\chi_{1}$ and $\chi_{2}$ are
the continuous phases discussed above, $\tau=\frac{\sqrt{5}-1}{2}$ is the
inverse golden ratio, and $\overrightarrow{\psi_{0}}(n,m)$ are analytically calculated constant
amplitudes (see Appendix \ref{App:fourier components}). Note that $\mathbf{q}$ is defined modulo reciprocal lattice vectors
$G=2\pi(i\mathbf{\hat{x}}+j\mathbf{\hat{y}})$. Bragg peaks are thus spanned by 4 independent basis reciprocal vectors (like the closely related square Fibonacci tiling \cite{Lifshitz2002}), consistent with the quasicrystalline nature of the model.}
Assuming low temperature quasicrystalline order, only long wavelength
excitations should be considered. At scales smaller
than the typical wavelength of the contributing excitations, the system
appears ordered, slowly passing from one local ground
state to another. To express this idea formally, we define the local
Fourier transform of a function $f(\mathbf{x})$ as
\begin{equation}
f(\mathbf{x},\mathbf{k})=\frac{1}{A}\sum_{\mathbf{x'}}f(\mathbf{x'})e^{-i\mathbf{k}\cdot\mathbf{x}'}w_{\sigma}(\mathbf{x-x'}),\label{eq:local fourier transform}
\end{equation}
where $w_{\sigma}$ is a weight function with a finite length scale
$\sigma$ and $A=\sum_{\mathbf{x}}w_{\sigma}(\mathbf{x})$. This weight
function makes sure that we take only contributions around the point
$\mathbf{x}$. In order to simplify the analysis we take $w_{\sigma}$ to be
unity in some region with a length scale $\sigma\gg1$ around the origin,
and zero otherwise. As long as $\sigma$ is large
enough, the shape of this region is not important.
We now consider long wavelength excitations where $\chi_{1}(\mathbf{r})$ and $\chi_{2}(\mathbf{r})$ change slowly with $\mathbf{r}$, being approximately constant on length scale $\sigma$. As the system appears locally ordered, its local Fourier transform is
\begin{eqnarray}
\overrightarrow{\psi}(\mathbf{q}) & = & N\sum_{m,n,i,j}\delta(\mathbf{q}-2\pi(n\tau\mathbf{\hat{x}}+m\tau\mathbf{\hat{y}}+i\mathbf{\hat{x}}+j\mathbf{\hat{y}}))\nonumber\\
& & \times e^{2\pi i\left(n\chi_{1}(\mathbf{r})+m\chi_{2}(\mathbf{r})\right)}\overrightarrow{\psi_{0}}(n,m),
\label{eq:local fourier transform of small excitation}
\end{eqnarray}
where $\chi_{1}(\mathbf{r})$ and $\chi_{2}(\mathbf{r})$ are the
phases corresponding to the local ground state.
In terms of the local Fourier transform, the
Hamiltonian (Eq. \ref{eq:energy}) then takes the form
\begin{eqnarray}
H & = & \sum_{\mathbf{r},\mathbf{k}}\left[\overrightarrow{\psi}{}^{\dagger}(\mathbf{r},\mathbf{k})Y\overrightarrow{\psi}(\mathbf{r}+\mathbf{\hat{y}},\mathbf{k})e^{ik_{y}}+\right.\nonumber \\
& & \left.\overrightarrow{\psi}{}^{\dagger}(\mathbf{r},\mathbf{k})X\overrightarrow{\psi}(\mathbf{r}+\hat{\mathbf{x}},\mathbf{k})e^{ik_{x}}\right].
\end{eqnarray}
For a locally ordered configuration, one can plug in the local ground state approximation, equation
(\ref{eq:local fourier transform of small excitation}), and get the effective long wavelength Hamiltonian,
$E[\chi_{1}(\mathbf{r}),\chi_{2}(\mathbf{r})]$
\begin{eqnarray}
E & = & \sum_{\mathbf{r}}\sum_{m,n}\left(e^{2\pi i\left(n\partial_{x}\chi_{1}+m\partial_{x}\chi_{2}\right)}\overrightarrow{\psi_{0}^{\dagger}}(n,m)X\overrightarrow{\psi_{0}}(n,m)e^{i2\pi n\tau}+\right.\nonumber \\
& & \left.\text{ }e^{2\pi i\left(n\partial_{y}\chi_{1}+m\partial_{y}\chi_{2}\right)}\overrightarrow{\psi_{0}^{\dagger}}(n,m)Y\overrightarrow{\psi_{0}}(n,m)e^{i2\pi m\tau}\right),
\end{eqnarray}
where $\partial_{x}\chi_i=\chi_i(\mathbf{r}+\hat{\mathbf{x}})-\chi_i(\mathbf{r})$
and $\partial_{y}\chi_i=\chi_i(\mathbf{r}+\hat{\mathbf{y}})-\chi_i(\mathbf{r})$
are the discrete derivatives of $\chi_i$. The sums over $m$ and $n$ can be performed
numerically, and the final result, to lowest order in the derivatives,
is
\begin{eqnarray}
E & =\sum_{\mathbf{r}} & A\left(\left|\partial_{x}\chi_{1}\right|+\left|\partial_{y}\chi_{2}\right|\right)+B\left(\left|\partial_{x}\chi_{2}\right|+\left|\partial_{y}\chi_{1}\right|\right)\label{effective field theory}\\
& & +C\left(\left|\partial_{x}\chi_{1}+\tau\partial_{x}\chi_{2}\right|+\left|\partial_{y}\chi_{2}+\tau\partial_{y}\chi_{1}\right|\right)\nonumber \\
& & +D\left(\left|\partial_{y}\chi_{1}+\tau\partial_{y}\chi_{2}\right|+\left|\partial_{x}\chi_{2}+\tau\partial_{x}\chi_{1}\right|\right),\nonumber
\end{eqnarray}
where $A\approx1.00,B\approx1.94,C\approx1.57,D\approx0.61$.
We now investigate this effective Hamiltonian at finite temperatures.
Once we rephrased the low-T physics of the model in terms of truly continuous fields,
it is rather obvious that the Mermin-Wagner theorem applies, and thermal excitations
must destroy the order in any finite temperature. However, three notes are in order. First, note that the Mermin Wagner theorem holds even though the effective field theory is non-analytic, as long as it is continuous \cite{Ioffe2002}. Second, the transformation from the tiles degrees of freedom to the continuous phases involves a non trivial, singular, Jacobian, which at finite temperature translates into a complicated entropic term. While this entropic term remains unspecified, it must preserve the continuous symmetry in the local ground state approximation, and therefore should not affect our argument. Third, as discussed above, the local phases are never truly continuous. Each finite patch of the system has a finite ground state degeneracy, and thus the number of distinct values that any of the phases can take is finite and scales like the patch size $\sigma$, which can be taken to be the largest scale over which the system is in an approximate ground state. However, here one can invoke the discrete tile picture of the system: as any mismatch in a tiling costs at least one unit of energy, the density of mismatches at low temperatures is at most $O(\exp(-\Delta/T))$ with $\Delta$ of order unity. Thus, $\sigma$, the scale upon which the system is at a local ground state, can be made exponentially large as temperature drops down, and the system can
be effectively described by continuous phases.
The assumption of low temperature order leads to a contradiction, and the system is therefore not ordered at any finite temperature.
The transition found in \cite{Leuzzi2000,Koch2009,Rotman2011}
is therefore not a symmetry breaking transition. We now turn to investigate its
true nature.
\begin{figure}[t]
\noindent \centering{} \textcolor{black}{\includegraphics[scale=0.31]{m2_inset_final}
\caption{\label{fig:tiling_m2} (Color online) $\left|Q\right|^{2}=\frac{1}{2}\left(\left|Q_{x}\right|^{2}+\left|Q_{y}\right|^{2}\right)$
as a function of the system's size different temperatures (top to bottom: $T=0.34,0.36,0.38,0.4,0.41,0.415,0.43$). Note the log-log scale.
For $T$ very close but higher than $T_{c}\sim0.42$, an exponential decay
is evident. Inset: $\eta$ as a function of $T$ below the critical
temperature. }
}
\end{figure}
\section{Finite temperature behavior of the model}
\textcolor{black}{
A natural choice for the order parameters of the model, closely
related to the one defined in \cite{Rotman2011}, is the Fourier coefficients of the tile densities at
the basis reciprocal vectors:
\begin{equation}
q_{i}^{x}=\frac{1}{N}\sum_{\mathbf{r}}e^{-i2\pi\tau x}\rho_{i}(\mathbf{r})\text{, }q_{i}^{y}=\frac{1}{N}\sum_{\mathbf{r}}e^{-i2\pi\tau y}\rho_{i}(\mathbf{r}),\label{eq:naive order parameter-1}
\end{equation}
where $i$ is one of the 16 tile-types, and its choice is arbitrary. Note the need for
two order parameters, as the ground state manifold is parameterized
by two phases. While this form is correct, a more symmetric
and numerically preferable generalization is
\begin{equation}
Q_{x}=\frac{1}{N}\sum_{i,\mathbf{r}}e^{-i2\pi\tau x}e^{i\gamma_{i}^{x}}\rho_{i}(\mathbf{r})\text{, }Q_{y}=\frac{1}{N}\sum_{i,\mathbf{r}}e^{-i2\pi\tau y}e^{i\gamma_{i}^{y}}\rho_{i}(\mathbf{r}),\label{eq:order parameter-1}
\end{equation}
which sums the contribution from all tile-types $i$. The phases $\gamma_{i}^{x}$ and $\gamma_{i}^{y}$ are the relative phases between the Bragg peaks amplitudes observed for each tile-type (see Appendix \ref{App:fourier components}).
We measured these order parameters in the vicinity of the transition ($T_c\simeq0.418$) and below it, using Monte-Carlo simulations of the {\it{original tiling model}}.
Ground state configurations are nearly periodic with periodicities that are Fibonacci numbers (see Appendix \ref{App:map}).
We found that finite size effects are minimized using periodic boundary conditions, provided linear system size is a Fibonacci number.}
\textcolor{black}{
Well below the transition $\left|Q\right|^{2}\propto L^{-\eta(T)}$ (Figure \ref{fig:tiling_m2}), implying a power law decay of the correlation function with the
same exponent $\eta(T)$.
Above the transition $\left|Q\right|^{2}$ falls
exponentially, indicating short-range correlations.
This resembles the situation in the XY model, for example, which exhibits a quasi-long-range order (QLRO) at low-T and a topological Kosterlitz-Thouless (KT) transition \cite{Kosterlitz1972,Kosterlitz1974} to a disordered phase.}
The KT transition is associated with vortex
unbinding. It is therefore natural to ask whether single vortices
become stable at high temperatures in our system. A vortex in the
field $\chi_{1}$, for example, is given by a configuration associated with
\begin{equation}
\chi_{1}=\frac{1}{2\pi}\arctan\frac{y}{x},\chi_{2}=0.\label{eq:vortex solution}
\end{equation}
Using equation (\ref{effective field theory}), it is easy to see that the energy of the vortex diverges, $H_{vortex}\propto L$. The positional entropy of the vortex, on the other hand, grows with system's size only as $\log L$. A naive application of the standard KT argument, explaining the onset of vortex unbinding as a consequence of the positional entropy
overcoming the energy, would lead to the conclusion that in our case energy always
wins and vortices are
never stable. This conclusion is clearly wrong - our transition is associated with proliferation of vortices,
see figure \ref{fig:vortex}. Upon integrating-out the fast degrees of freedom, the field theory (\ref{effective field theory}) is likely to be renormalized into an effective gaussian free-energy functional, which results in a logarithmically-diverging vortex effective energy. This was shown to be the case for a similar tiling model at any finite temperature \cite{Tang1990}.
The standard KT argument for positional entropy overcoming the {\it effective} free-energy of the vortex at high temperatures does hold, and vortex unbinding will drive a topological phase transition. \\
\begin{figure}[t]
\noindent \centering{}\textcolor{black}{\includegraphics[scale=0.45]{vortex3}\caption{\label{fig:vortex} A typical configuration above the transition. The arrows represent the complex numbers $\psi_{1}(\mathbf{r},2\pi\tau\mathbf{\hat{x}})$ with $w_{\sigma}=\exp(-r^2/\sigma^2)$ and $\sigma=5$. We note that vortices similar to those shown in the figure were observed in all the configurations above $T_c$. Below $T_c$, no vortices were observed.}
}
\end{figure}
\textcolor{black}{In the usual KT scenario, where
the energy itself is quadratic to lowest order, $\eta\propto T$ at
low temperatures. In contrast, in our case $\eta$
shows a highly non-linear behavior (Figure \ref{fig:tiling_m2}, inset).
This too signals that the effective coupling constant is strongly renormalized, and becomes temperature dependent.
The very steep decay of $\eta$ as one moves away from
the transition implies that in finite lattices at low temperatures,
the system appears to be ordered, and the identification of the algebraic
correlations is very difficult in reasonably sized systems. This explains why previous works \cite{Leuzzi2000,Koch2009,Rotman2011} identified the low temperature phase as an ordered one.\
}
However, one feature of our transition deviates from the KT scenario. At the KT transition, the heat capacity $C_v$ exhibits a weak (numerically undetectable) $C_\infty$ essential singularity. As shown in \cite{Leuzzi2000,Koch2009,Rotman2011}, the tiling model exhibits a distinctive sharp peak in $C_v$ at $T_c$. In particular, we observed (for $L$ up to $89$) a clear power-law divergence $\left.\frac{dC_{V}}{dT}\right|_{T_{C}}\propto L^{-\epsilon}$ with $\epsilon$ (very roughly) $\sim 0.5(3)$, indicating a finite-order transition with $\nu\simeq1.25(20)$. Bearing in mind the large uncertainties in these numerical estimates, and the limited system sizes, this seems to suggest our transition may be of a different universality class than the standard KT transition. A qualitative change in critical behavior due to interaction between two XY fields was pointed out in the context of a double-layer XY model \cite{Parga1980}.
\\
It is worth saying a few words about the form of
the long wavelength Hamiltonian, equation (\ref{effective field theory}).
Usually, only analytic terms are considered when one constructs an effective field theory. The tiling model provides
an example where non-analytic terms arise naturally from first principles. In fact, it was already suggested that terms of the form $\sim\left|\partial\chi\right|$ describe the energy of phasons in general systems with a quasicrystalline
ground state \cite{Socolar1986}. A phase in which the {\it{free energy}} is characterized by such a non-analytic form is usually referred to as a locked phase \cite{Steinhardt1999}. In 3D, one expects to find a finite temperature transition from this phase to an unlocked phase, characterized by a quadratic free energy \cite{Jeong1993,Dotera1994}. However, in 2D systems, such as the one studied in this work, the transition occurs at zero temperature, as was shown in \cite{Tang1990}. A similar derivation of the energy can be made for an
analog three-dimensional system, where we expect that the equivalent of (\ref{effective field theory}) would be the
relevant low-temperature effective theory.\\
\section{Conclusions}
\textcolor{black}{We described here a topological phase transition in a system with discrete degrees of freedom. It is constructive to juxtapose this behavior with a similar scenario.
The clock model, where each spin can take one of $q$ possible planar directions \cite{Jose1977, Elitzur1979,Ortiz2012}, exhibits a KT transition for $q>4$. In this case, as long as $k_BT$ exceeds the energy of rotating a single spin between two neighboring directions, thermal fluctuations restore the continuous U(1) symmetry and one effectively gets back an XY model with
algebraically decaying correlations. Indeed, as temperature lowers the discrete nature is revealed, and a second phase transition occurs below which the system is ordered.
In comparison, in our tiling model the hidden continuous U(1) symmetry is restored not by temperature but rather by going into larger and larger finite ordered patches. The lower the temperature, the larger are the ordered patches in the system, and thus the QLRO phase survives for arbitrarily low temperatures. Given that the continuous symmetry discussed here is a general property of quasicrystals, similar arguments may lead to the conclusion that any 2D model with a quasicrystalline ground state (with either discrete or continuous degrees of freedom), cannot be ordered at any positive temperature. Indeed, algebraic correlations (but not a KT transition) were observed in a Penrose tiling model \cite{Tang1990} and various random tiling models \cite{Steinhardt1999}, and we expect the tiling model recently studied by Nikola {\it et al} \cite{Nikola2013} to exhibit QLRO at low temperatures as well. \textcolor{black}{ Furthermore, in our model rotational symmetry is explicitly broken by the underlying real-space lattice. However, the above formulation of the configuration in terms of the local phases enables one to study, in off-lattice models, the orientational QLRO of these fields. One expects a two-step melting of the QLRO quasicrystal through an intermediate "Hexatic" (or, rather, "Pentatic" for a five-fold symmetric quasicrystal) phase, as was predicted in \cite{De1989}.}}
\begin{acknowledgments}
We would like to acknowledge Ron Lifshitz, Giorgio Parisi, and Moshe Schwartz for many helpful discussions, as well as Ziv Rotman for helping us with the numerical simulations.
\end{acknowledgments}
|
1,108,101,565,504 | arxiv |
\section{Introduction}
\label{sec:intro}
The nonlinear Schr\"odinger equation (NLS) in $\mathbb R \times \mathbb R^d$,
\begin{equation}
\label{eqn:nls}
i \psi_t + \Delta \psi + g (|\psi|^2) \psi = 0, \quad \psi (0,\mathbf{x}) = \psi_0 (\mathbf{x}),
\end{equation}
appears in many different contexts. In applications, it appears as a
leading order approximation in nonlinear optics, many body quantum
systems, and hydrodynamics. It is also intrinsically interesting as
a canonical example of the competition between nonlinearity and
dispersion.
For appropriate choices of the nonlinearity $g: \mathbb R \to \mathbb R$, the
equation is known to possess \emph{soliton} solutions, nonlinear bound
states satisfying \eqref{eqn:nls} with the \emph{ansatz}
\begin{equation*}
\psi(t,\mathbf{x}) = e^{i \lambda t} R(\mathbf{x};\lambda),
\end{equation*}
where $\lambda>0$ is the \emph{soliton} parameter. It is conjectured
that any solution of \eqref{eqn:nls} with appropriate nonlinearity
that does not disperse as $t \rightarrow \infty$ must eventually
converge to a finite sum of stable solitons. This is referred to as the
``soliton resolution'' conjecture, a notoriously difficult problem to
formulate, see \cite{Tao2008}.
A natural property to investigate is the stability of the solitons.
In \cite{W1,W2,GSS}, a criterion for \emph{orbital} stability is
established. Briefly, it says that if the derivative of the $L^2$ norm of
$R(\mathbf{x}; \lambda)$ taken with respect to $\lambda$ is positive, then
the soliton is orbitally stable. The perturbation
remains small in a particular norm, $H^1$ in the case of NLS. If this
derivative is negative, the soliton is unstable.
Though these results on the orbital stability of solitons are very
powerful, relying on much of the variational structure of the
equations, they have three weaknesses. The first is that they do not
say if a perturbed soliton reaches an asymptotically constant state;
orbital stability only assures us that the perturbation remains small. The second
is that this approach provides no information if the
derivative of the $L^2$ norm vanishes. This is the case of the $L^2$ critical focusing NLS
equation, with $g(s) = s^{d/2}$ and for \emph{saturated}
nonlinearities which possess \emph{minimal mass} solitons. Finally,
the orbital stability fundamentally depends on the underlying equation
possessing a known variational structure. Though this is not a valid
criticism for \eqref{eqn:nls}, it is a problem for other equations,
such as those studied in \cite{simpson08as}.
Alternatively, results such as
\cite{BusPer,BusSul,Cuc,Schlag1,KS1,RodSchSof}, and many others, prove
{\it asymptotic} stability of a soliton or a collection of solitons;
the system converges to specific solitons to as $t \to \infty$, and
the rest of the mass disperses. Asymptotic stability is usually
proven perturbatively. The leading order behavior of the perturbation
to the soliton is governed by the linearized operator. First, linear
stability is proven by assesing the spectrum of the linearized
operator. Then the nonlinearity is shown to be dominated by the
linear flow. The spectrum of the linearized operator of \eqref{eqn:nls} with monomial
nonlinearity was studied in \cite{CGNT2007}.
Embedded eigenvalues of the linear operator are detrimental to
proving the necessary linear estimates. Indeed, they obstruct
the needed dispersive estimates, as
demonstrated in \cite{CucPel}. Thus, it is standard to make the
assumption that there are no eigenvalues
embedded in the essential spectrum. It is known that such a condition
cannot be proven directly using abstract properties of the linearized
operator, but must in fact be directly related to algebraic properties
of the soliton itself. In this work we develop an algorithm
for studying the spectral properties of the operator appearing when
one linearizes \eqref{eqn:nls} about a soliton solution. Furthermore,
we use this algorithm to prove the absence of \emph{embedded}
eigenvalues or resonances for four NLS problems. We collect these
results in the following section.
\begin{rem}
Though we only prove Theorem
\ref{thm:speccond} for a small number of cases, our objective in
this work is to present an approach for
verfiying the spectral hypotheses required for soliton stability theory.
\end{rem}
\subsection{Main Results}
\label{int:main}
Our results hinge on a so called {\it spectral property} based on
linearized matrix Schr\"odinger operators. The specific form, and its
motivations, are developed Section \ref{sec:normal}. In general, this
property can be formulated as:
\begin{defn}[The Generalized Spectral Property]
\label{def:specprop}
Let $d \geq 1$. Given $L_\pm$ and a skew adjoint operator
$\Lambda$, consider the two real Schr\"odinger operators
\begin{equation*}
\mathcal{L}_+ = -\Delta + \mathcal{V}_+, \ \mathcal{L}_- = - \Delta + \mathcal{V}_-,
\end{equation*}
defined by
\begin{align*}
\mathcal{L}_+ f & =\frac{1}{2}[L_+,\Lambda]f= \frac{1}{2} \left[
{L}_+ \Lambda f -
\Lambda {L}_+ f \right], \\
\mathcal{L}_- f & =\frac{1}{2}[L_-,\Lambda]f= \frac{1}{2} \left[
{L}_- \Lambda f - \Lambda {L}_- f \right]
\end{align*}
and
\begin{equation*}
\mathcal{V}_\pm = \frac{1}{2}\mathbf{x} \cdot \nabla V_\pm.
\end{equation*}
Let the real quadratic form for $\mathbf{z} = (u,v)^T \in H^1\times
H^1$ be
\begin{align*}
\mathcal{B}(\mathbf{z},\mathbf{z}) & = \mathcal{B}_+ (u,u) + \mathcal{B}_- (v,v) \\
& = \inner{\mathcal{L}_+ u}{ u} + \inner{\mathcal{L}_- v}{ v}.
\end{align*}
The system is said to satisfy a spectral property on the subspace $\mathcal{U}
\subseteq H^1\times H^1$ if there exists a universal constant
$\delta_0 > 0$ such that $\forall \mathbf{z} \in \mathcal{U}$,
\begin{equation*}
\mathcal{B}(\mathbf{z},\mathbf{z}) > \delta_0 \int \paren{ | \nabla \mathbf{z} |^2 + e^{-|\bold{y}|} |\mathbf{z}|^2 } d\bold{y}.
\end{equation*}
\end{defn}
In this work, the skew adjoint operator is
\begin{equation*}
\Lambda f \equiv \frac{d}{2} f + \mathbf{x} \cdot \nabla f = \frac{d}{d \lambda} \left[ \lambda^{\frac{d}{2}} f( \lambda \mathbf{x}) \right].
\end{equation*}
This has particular significance for the $L^2$ critical equation,
though we employ it in supercritcal problems. This (mis)application
is discussed in Section \ref{s:discussion}.
Our results rely on the key observation of G. Perelman \cite{GPer} that
\begin{thm}
\label{thm:coercive}
Given the $JL$ operator, arising from the linearization of NLS about a
soliton, where
\begin{equation*}
JL = \begin{pmatrix} 0 & L_- \\ -L_+ & 0 \end{pmatrix}
\end{equation*}
assume the $L_\pm$ operators satisfies the Spectral Property in the
sense of Definition \ref{def:specprop}. Then $JL$ has no
embedded eigenvalues on the subspace $\mathcal{U}$.
\end{thm}
\begin{proof}
Let us assume we have an embedded eigenstate
$\mathbf{z}_{{\mathrm{em}}} = (u_{{\mathrm{em}}}, v_{{\mathrm{em}}})^T \in \mathcal{U}$ corresponding to
eigenvalue $ i \tau_{{\mathrm{em}}}$, $\tau_{\mathrm{em}} > \lambda_0$. Then,
\begin{align*}
L_{-} v_{{\mathrm{em}}} & = i \tau_{{\mathrm{em}}} u_{{\mathrm{em}}} , \\
L_{+} u_{{\mathrm{em}}} & = - i \tau_{{\mathrm{em}}} v_{{\mathrm{em}}}.
\end{align*}
Plugging directly into the form,
\begin{equation*}
\begin{split}
\mathcal{B}(\mathbf{z}_{{\mathrm{em}}},\mathbf{z}_{{\mathrm{em}}}) & = \inner{\mathcal{L}_+
u_{\mathrm{em}}}{u_{\mathrm{em}}} + \inner{\mathcal{L}_- v_{\mathrm{em}}}{v_{\mathrm{em}}}\\
& = \frac{1}{2}\set{ \inner{\Lambda u_{\mathrm{em}}}{L_+ u_{\mathrm{em}}} +
\inner{L_+ u_{\mathrm{em}}}{\Lambda u_{\mathrm{em}}}} \\
&\quad+ \frac{1}{2} \set{\inner{\Lambda v_{\mathrm{em}}}{L_- v_{\mathrm{em}}} +
\inner{L_- v_{\mathrm{em}}}{\Lambda v_{\mathrm{em}}}}\\
& = \frac{1}{2}\set{ i \tau_{\mathrm{em}}\inner{\Lambda u_{\mathrm{em}}}{ v_{\mathrm{em}}}
-i \tau_{\mathrm{em}}
\inner{ v_{\mathrm{em}}}{\Lambda u_{\mathrm{em}}}} \\
&\quad+ \frac{1}{2} \set{-i \tau_{\mathrm{em}}\inner{\Lambda v_{\mathrm{em}}}{
u_{\mathrm{em}}} +
i \tau_{\mathrm{em}} \inner{ u_{\mathrm{em}}}{\Lambda v_{\mathrm{em}}}}\\
& = \frac{i \tau_{\mathrm{em}}}{2} \set{\inner{\Lambda u_{\mathrm{em}}}{ v_{\mathrm{em}}}
-\inner{ v_{\mathrm{em}}}{\Lambda u_{\mathrm{em}}} + \inner{v_{\mathrm{em}}}{\Lambda
u_{\mathrm{em}}} - \inner{\Lambda u_{\mathrm{em}}}{v_{\mathrm{em}}} }\\
&=0.
\end{split}
\end{equation*}
\end{proof}
We remark that this holds not just for embedded eigenvalues, but for
any purely imaginary eigenvalue. Thus, if the spectral property
holds, we are assured that there are no imaginary eigenvalues \emph{on
the designated subspace}. The subspace $\mathcal{U}$ will be set by our analysis
of the spectrum in Section \ref{spec:lin}.
\begin{defn}
\label{def:speccond}
Separately, we say that a linearized NLS problem satisfies of a {\it
spectral condition} if it lacks both:
\begin{itemize}
\item Embedded eigenvalues,
\item Endpoint resonances.
\end{itemize}
\end{defn}
Our second theorem, which relies on the first is:
\begin{thm}
\label{thm:speccond}
The spectral condition holds for the 3d cubic equation,
\eqref{eqn:nls} with $d=3$ and $g(s)=s$, linearized about the
ground state soliton $R$.
\end{thm}
We adapt the methods of \cite{FMR} to give a
numerically assisted proof of this result.
\begin{rem}
Though the main result of this paper will be to establish Theorem
\ref{thm:speccond}, our algorithm can also be used to
establish the spectral condition for the one dimensional equation
with $g(s) = s^{2.5}$ and $g(s)=s^3$. Again, this is for the
problem linearized about the ground state soliton.
\end{rem}
Separately, we establish that for 3d problems with nonlinearities
satisfying the conditions necessary for the existence of a soliton, as
discussed in \cite{BeLi}, one need only test for
embedded eigenvalues that are:
\begin{itemize}
\item Near the endpoints of the essential spectrum,
\item On a sufficiently low spherical harmonic.
\end{itemize}
In Appendix \ref{sec:mourre} we give a proof of the
following result using positive commutator arguments otherwise known
as Mourre estimates:
\begin{thm}
\label{thm:spec}
Given a Hamiltonian $\mathcal{H}$, there exists some $M > 0$ such
that for $|\mu| > M$, there are no solutions $u_{\mu}$ such that
\begin{eqnarray*}
\mathcal{H} u_\mu = \mu u_\mu.
\end{eqnarray*}
Similarly, if $d \geq 2$, for any embedded eigenvalue, $u_{\mu}$,
there exists a $K>0$ such that the spherical harmonic decomposition
\begin{eqnarray*}
u_\mu = \sum_{k = 0}^\infty \alpha_k (r,\mu) \phi_k (\phi,\theta)
\end{eqnarray*}
consists only of harmonics with $k <K$.
\end{thm}
\begin{rem}
Such results are well-known using resolvent estimate techniques;
however, our approach provides easily computable limits on $M$ and
$K$ in terms of the soliton solution.
\end{rem}
\subsection{Organization of Results}
\label{sec:org}
In Sections \ref{sec:nlsprops}, \ref{sec:sollin} and
\ref{spec:lin}, we review fundamental properties of
\eqref{eqn:nls} and the associated linearized operator.
In Sections \ref{spec:embres} and \ref{specnum:disc}, we collect
results and adapt the techniques of \cite{ES1} to prove properties of
the discrete spectrum and the absence of embedded resonances.
Finally, in Section \ref{sec:normal}, we prove an appropriate spectral property as
in Definition \ref{def:specprop}, based on the work in \cite{FMR}.
G.~Perelman's observation then rules out embedded eigenvalues. This
proves Theorem \ref{thm:speccond}.
In Appendices \ref{spec:eig} and \ref{spec:sphhar},
we use Mourre multipliers to eliminate large embedded eigenvalues and
large spherical harmonics from the expansion of an embedded
eigenvalue. Though these results have been known via resolvent
methods for some time, we aim to collect as much analytic
information about the spectrum as possible, providing bounds for future
estimates and computations. An overview of our numerical methods with
benchmarks is then presented in Appendix \ref{sec:numerics}.
{\sc Acknowledgments.} This paper is an extension of a result of a
thesis done by the first author under the direction of Daniel Tataru
at the University of California, Berkeley that arose from a discussion
with Wilhelm Schlag and Galina Perelman at the Mathematisches
Forschungsinstitut Oberwolfach. The first author is supported by an
NSF Postdoctoral Fellowship. The second author is supported in part
by NSERC. In addition, the authors wish to thank Gadi Fibich, Michael
Weinstein, Ian Zwiers and especially Wilhelm Schlag for many helpful
conversations throughout the development of the paper.
\section{Properties of the Nonlinear Schr\"odinger Equation}
\label{sec:nlsprops}
In this section we briefly review some important properties of
\eqref{eqn:nls}. For additional details, we refer the reader to the
texts \cite{sulem1999nse, Caz}.
In general, for nonlinearity $g: \mathbb R \to \mathbb R$, \eqref{eqn:nls} possesses
the following invariants for data $\psi_0 \in H^1$ and $|\mathbf{x}|\psi_0 \in
L^2$:
\begin{description}
\item[Conservation of Mass (or Charge)]
\begin{equation*}
Q (\psi) = \frac{1}{2} \int_{\mathbb R^d} |\psi|^2 d\mathbf{x} = \frac{1}{2} \int_{\mathbb R^d} |\psi_0|^2 d\mathbf{x},
\end{equation*}
\item[Conservation of Energy]
\begin{align*}
E(\psi) & = \int_{\mathbb R^d} | \nabla \psi |^2 d\mathbf{x} - \int_{\mathbb R^d} G(|\psi|^2) d\mathbf{x} = \int_{\mathbb R^d} | \nabla \psi_0 |^2 d\mathbf{x} - \int_{\mathbb R^d} G(|\psi_0|^2)
d\mathbf{x},
\end{align*}
where
\begin{equation*}
G(t) = \int_0^t g(s) ds.
\end{equation*}
\item[Pseudo-Conformal Conservation Law]
\begin{equation*}
\| (\mathbf{x} + 2 i t \nabla ) \psi \|^2_{L^2} - 4 t^2 \int_{\mathbb R^d} G(|u|^2) d\mathbf{x} = \|\b x \psi \|^2_{L^2} - \int_0^t \theta (s) ds,
\end{equation*}
where
\begin{equation*}
\theta (s) = \int_{\mathbb R^d} (4 (d+2) G(|\psi|^2) - 4 d g(|\psi|^2) |\psi|^2) d\mathbf{x}.
\end{equation*}
Note that $(\mathbf{x} + 2 i t \nabla )$ is the Hamilton flow of the linear
Schr\"odinger equation, so the above identity relates how the
solution to the nonlinear equation is effected by the linear flow.
\end{description}
Detailed proofs of these conservation laws can be arrived at easily
using energy estimates or Noether's Theorem, which relates
conservation laws to symmetries of an equation.
In this work, we restrict our attention to \emph{focusing}
nonlinearities, such that $g(s)\geq 0$ for all $ s\in \mathbb R$. These are
the nonlinearities that can yield soliton solutions. Often, $g(s) =
s^\sigma$ for some $\sigma>0$. We examine one instance of the power
nonlinearity, the three dimensional cubic problem ($\sigma = 1$).
As noted, a soliton solution takes the form
\begin{equation*}
\psi (t,\mathbf{x}) = e^{i \lambda t} R(\mathbf{x};\lambda),
\end{equation*}
where $\lambda > 0$ and $R (\mathbf{x};\lambda)$ is a positive, radially
symmetric, exponentially decaying solution of the equation:
\begin{equation}
\label{eqn:sol}
\Delta R - \lambda R + g(\abs{R}^2) R= 0.
\end{equation}
For power nonlinearities, the existence and uniqueness of the
ground state soliton is well known. Additionally, the scaling
properties of this case permit us to take
$\lambda = 1$. We do this in all that follows.
Existence of the soliton is proved by in \cite{BeLi} by minimizing the functional
\[
T(\psi) = \int | \nabla \psi |^2 d\mathbf{x}
\]
with respect to the constraint of fixed
\[
V(\psi) = \int [ G(|\psi|^2) - \frac{\lambda}{2} |\psi|^2 ] d\mathbf{x}.
\]
Then, using the minimizing sequence and Schwarz symmetrization, one obtains the
existence of the nonnegative, spherically symmetric, decreasing
soliton solution. Uniqueness is established in \cite{Mc} by ODE
methods.
An important relation is that $Q({\lambda}) = Q(R(\cdot;\lambda))$ and
$E({\lambda}) = E(R(\cdot;\lambda))$ are differentiable with respect
to $\lambda$. This fact can be determined from the early works of
Shatah, namely \cite{Sh1}, \cite{Sh2}. By differentiating Equation
\eqref{eqn:sol}, $Q$ and $E$ with respect to $\lambda$, we have
\begin{equation*}
\partial_{\lambda} E = - \lambda \partial_{\lambda} Q.
\end{equation*}
Variational techniques developed in \cite{GSS} and \cite{ShSt} tell us
that when $\delta ( \lambda ) = E({\lambda}) + \lambda Q({\lambda})$
is convex, or $\delta '' (\lambda) > 0$, the soliton is orbitally stable.
For $\delta '' (\lambda) < 0$ the soliton is unstable to small perturbations.
This stability (instability) directly is closely related to the
eigenvalues of the matrix Hamiltonians resulting from linearizing NLS about a soliton. For a brief
reference on this subject, see \cite{SS}, Chapter 4.
\section{Linearization about a Soliton}
\label{sec:sollin}
Let us write down the form of NLS linearized about a soliton
solution. First, we assume we have a solution $\psi = e^{i
\lambda t}(R + \phi(\mathbf{x},t))$. Inserting this into the equation, we have
\begin{equation}
i (\phi)_t + \Delta (\phi) =-g( R^2) \phi - 2 g'( R^2 ) R^2 \text{Re}(\phi) + \mathrm{O} (\phi^2),
\end{equation}
by splitting $\phi$ up into its real and imaginary parts, then doing a
Taylor Expansion. Hence, if $\phi = u + iv$, we get
\begin{equation}
\partial_t \begin{pmatrix}
u \\
v
\end{pmatrix} =J L \begin{pmatrix}
u \\
v
\end{pmatrix},
\end{equation}
where
\begin{equation}
JL = \begin{pmatrix}
0 & L_{-} \\
-L_{+} & 0
\end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} L_+ & 0 \\ 0 & L_- \end{pmatrix}
\end{equation}
and
\begin{align}
L_{-} &= - \Delta + \lambda -V_-,&& V_- =g( R ),\\
L_{+} &= - \Delta + \lambda - V_+,&& V_+ = g( R ) + 2 g' (R^2) R^2.
\end{align}
Alternatively, if we formulate the problem in terms of $\phi$ and
$\phi^*$,
\begin{equation}
\partial_t \begin{pmatrix}\phi \\ \phi^* \end{pmatrix} = i \mathcal{H} \begin{pmatrix}\phi \\ \phi^* \end{pmatrix},
\end{equation}
where
\begin{align}
\label{eqn:mathcalH}
\mathcal{H} = \begin{pmatrix} -\Delta + \lambda - V_1 & -V_2 \\
V_2 &\Delta - \lambda + V_1\end{pmatrix}
\end{align}
and
\begin{equation}
V_1 = g(R^2) + g'(R^2)R^2, \quad V_2 = g'(R^2)R^2.
\end{equation}
The potentials in the two formulations are related by $V_+ = V_1 +
V_2$ and $V_- = V_1 - V_2$.
There are many things we can immediately say about $L_{-}$, $L_{+}$,
$JL$ and $\mathcal{H}$. For a reference on the spectral theory
involved, see Hislop-Sigal \cite{HS} or Reed-Simon \cite{RSv4}. First
of all, both $L_{-}$ and $L_{+}$ are self-adjoint operators. Also,
$L_{-}$ is a non-negative definite operator and its null space is
$\text{span}\{ R \}$. Note also that the functions $\frac{\partial
R}{\partial x_j}$ for $j = 1,2,...,d$ are in the null space of
$L_{+}$. By comparison with the operator $\Delta + \lambda$ and using
the fact that $R$ decays exponentially, we see that the essential
spectrum of $\mathcal{H}$ is the set $(-\infty,\lambda] \cup
[\lambda,\infty)$ from Weyl's Theorem, see \cite{ES1} and \cite{RSv4}.
Equivalently, the essential spectrum of $JL$ is $(-i \infty, i
\lambda] \cup [i \lambda, i \infty)$. Indeed, $\sigma(JL) = i
\sigma(\mathcal{H})$. Finally, using the fact that $L_{-}$ is
non-negative definite and looking at eigenvalues $\mathcal{H}^2$, we
see
\begin{eqnarray}
L_{-} L_{+} u = \nu^2 u.
\end{eqnarray}
However, this can be rewritten as
\begin{eqnarray*}
T v = L_{-}^{\frac{1}{2}} L_{+} L_{-}^{\frac{1}{2}} v = \nu^2 v
\end{eqnarray*}
for $v = L_{-}^{\frac{1}{2}} u$. Since the operator $T$ is
self-adjoint, we must have $\nu \in \mathbb R \cup i \mathbb R$.
Typically, asymptotic stability is studied with the following
assumptions made on the matrix Hamiltonian:
\begin{defn}
\label{spec:defn1}
A Hamiltonian, $\mathcal{H}$ is called admissible if the following
hold:
\begin{enumerate}
\item There are no embedded eigenvalues in the essential spectrum.
\item The only real eigenvalue in $[-\lambda, \lambda ]$ is $0$.
\item The values $\pm \lambda$ are not resonances.
\end{enumerate}
\end{defn}
\begin{defn}
\label{spec:defn2}
Let NLS have nonlinearity $g$. We call $g$ admissible at $\lambda$ if there
exists a soliton, $R_\lambda$, for NLS and the
Hamiltonian, $\mathcal{H}$, resulting from linearization about
$R_\lambda$ is admissible in terms of Definition \ref{spec:defn1}.
\end{defn}
\begin{rem}
For simplicity in exposition, a matrix
Hamiltonian, $\mathcal{H}$, is said to be {\it admissible} if it
satisfies several {\it spectral conditions}. One of these properties
is a lack of embedded eigenvalues in the essential spectrum, which to establish we employ a {\it spectral property} as in \cite{FMR}.
\end{rem}
The spectral conditions of $\mathcal{H}$ from Definition \ref{spec:defn1} are generally required to
prove dispersive estimates for the evolution operator associated with
the linearized Hamiltonian equation, which in turn are required to
prove asymptotic stability results for solitons. See
\cite{Schlag1,RodSchSof} for further discussion. Let $P_d$ and $P_c$
be the projections onto the discrete and continuous spectrum of
$\mathcal{H}$.
See Figure \ref{fig:spec_decomp} for a description of the spectral
decomposition for $\mathcal{H}$ resulting from linearizing about
solitons with subcritical, critical and supercritical stability
properties.
\begin{figure}
\centering \scalebox{0.75}{\includegraphics{spec_decomp_sup}}
\caption{Plots of the spectral decomposition for $\mathcal{H}$ when
$R_\lambda$ is exponentially unstable (supercritical behavior).}
\label{fig:spec_decomp}
\end{figure}
\section{Spectral Properties of the Linearized Hamiltonian}
\label{spec:lin}
We now give more detailed and formal statements on the spectral
properties of the operator under investigation.
\subsection{A Survey of Results on the Spectrum of $\mathcal{H}$}
\label{sec:survey}
\subsubsection{An Analytic Result on the Spectrum of $\mathcal{H}$}
\label{sec:Hanalytic}
We formalize the heuristic discussion from Section \ref{sec:sollin}
with the following theorem from \cite{ES1}. Let us write the operator as
\begin{eqnarray*}
\mathcal{H} = \mathcal{H}_0 + V = \left[ \begin{array}{cc}
-\Delta + \lambda & 0 \\
0 & \Delta + \lambda
\end{array} \right] + \left[ \begin{array}{cc}
-V_1 & -V_2 \\
V_2 & V_1
\end{array} \right].
\end{eqnarray*}
In \cite{ES1}, the authors proved the following properties of the spectrum:
\begin{thm}[Erdogan-Schlag]
Assume there are no embedded eigenvalues in the continuous spectrum
of $\sigma ( \mathcal{H} )$. The essential spectrum of
$\mathcal{H}$ equals $(-\infty, -\lambda] \cup [\lambda, \infty)$.
Moreover, $\sigma ( \mathcal{H} ) = - \sigma( \mathcal{H}) =
\overline{\sigma( \mathcal{H} )} = \sigma( \mathcal{H}^* )$ and
$\sigma( \mathcal{H} ) \subset {\mathbb R} \cup i {\mathbb R}$. The discrete
spectrum consists of eigenvalues $\{ z_j \}_{j=1}^N$, $0 \leq N \leq
\infty$, of finite multiplicity. For each $z_j \neq 0$, the
algebraic and geometric multiplicities coincide and $\mathrm{Ran}
(\mathcal{H} - z_j)$ is closed. The zero eigenvalue has finite
multiplicity.
\end{thm}
\subsubsection{Absence of Embedded Resonances}
\label{spec:embres}
This result is developed in the earlier work of Erdogan-Schlag
\cite{ES1} and Agmon \cite{Ag}.
Define the space
\begin{eqnarray*}
X_{\sigma} = L^{2,\sigma} \times L^{2,\sigma},
\end{eqnarray*}
where
\begin{eqnarray*}
L^{2,\sigma} = \{ f | |\mathbf{x}|^\sigma f \in L^2\}.
\end{eqnarray*}
Then, we have the following Theorem, proved in \cite{ES1}:
\begin{thm}[Erdogan-Schlag]
Let $V_1$, $V_2$ have sufficient decay at $\infty$. Then for any
$\mu$ such that $|\mu| > \lambda$, $(\mathcal{H}_0 - (\mu \pm
i0))^{-1} V:X_{-\frac{1}{2}-} \to X_{-\frac{1}{2}-}$ is a compact
operator, and
\begin{eqnarray*}
I + (\mathcal{H}_0 - (\mu \pm i0))^{-1} V
\end{eqnarray*}
is invertible on these spaces.
\end{thm}
The proof relies on a similar argument to a restriction theorem from
harmonic analysis, which follows from a calculation using the specific
structure of $R_0 (z) = (-\Delta-z)^{-1}$. This strategy emulates
closely that of the bootstrapping argument of Agmon, \cite{Ag}, for scalar operators.
\subsubsection{Discrete Spectrum}
\label{specnum:disc}
We wish to show that the spectrum of $\mathcal{H}$ when linearized
about the ground state has the discrete spectral
decomposition Figure \ref{fig:spec_decomp}.
\begin{rem}
For the $3d$ cubic
nonlinearity, the structure of the discrete spectrum away from the
essential spectrum has been verified numerically in \cite{DeSc}, whose
methods we recall briefly here.
\end{rem}
In \cite{Schlag1}, using arguments derived from \cite{Per1}, it is
shown that the discrete spectrum for supercritical exponents is
determined by the discrete spectrum of $L_{\pm}$. We present here a
slightly stronger version that works for linearizations about a
minimal mass soliton, $R=R_{\min}$, in saturated nonlinearities.
Though we will not numerically analyze any saturated nonlinearities in
the current work, we present the generalized argument saturated
nonlinearities are also of interest.
The verification of the discrete spectrum heavily relies on the
following result:
following
\begin{thm}[Schlag]
Assume that $L_{-}$ has no discrete eigenvalues on the interval
$(0,\lambda]$ and $\mathcal{H}$ is a Hamiltonian as in
\eqref{eqn:mathcalH} resulting from linearizing about a minimal mass
soliton. Then, the only discrete eigenvalue for $\mathcal{H}$ in
the interval $[-\lambda,\lambda]$ is $0$.
\end{thm}
\begin{rem}
A very similar theorem appeared in \cite{Schlag1} proving the same
result for the Hamiltonian $\mathcal{H}$ of the form
\eqref{eqn:mathcalH} formed from $g(s) = s$ with $\lambda = 1$ and
$\mathbf{x} \in {\mathbb R}^3$. The proof follows with minimal changes and is
adaptable to many cases, hence we include it below for completeness.
\end{rem}
\begin{proof}
We argue by contradiction. To this end, assume $\mathcal{H}$ has an
eigenvalue away from $0$, say at $E$. Let $\lambda = 1$ for
simplicity. Then $\mathcal{H}^2$ has an eigenvalue at some value
$E^2 \in (0,1]$. Hence, we have
\begin{eqnarray*}
L_{-} L_{+} u_E = E^2 u_E,
\end{eqnarray*}
for $E^2<1$. Since $L_{-}$ is self-adjoint, we see that $u_E \perp
\phi$. By elliptic regularity, we have that $u_E \in H^4_{loc}$.
Let $P$ be the projection orthogonal to $\phi$. Let $A = P L_{+}P$.
Using that
\begin{equation*}
\mathrm{ker}(L_{+}) = \mathrm{span}\{ \partial_j \phi | 1 \leq j \leq d \}
\end{equation*}
and
\begin{eqnarray}
\label{eqn:es2}
(\mathcal{H} - z)^{-1} & = & (\mathcal{H}_0 -z)^{-1} [I - U_1 [ I
- U_2 J (\mathcal{H}_0 -z)^{-1} U_1]^{-1} \\
&& \times U_2 J (\mathcal{H}_0 -z)^{-1} U_1]^{-1}] , \notag
\end{eqnarray}
we have
\begin{equation*}
\mathrm{ker} (L_{+}) = \mathrm{span}\{ \partial_\lambda R, \ \partial_j R | 1 \leq j \leq d \}
\end{equation*}
since
\begin{eqnarray*}
\langle R, \partial_\lambda R \rangle = 0.
\end{eqnarray*}
Take $E_0$ to be the unique negative eigenvalue for $L_{+}$. Then,
define
\begin{eqnarray*}
g(\alpha) = \langle (L_{+} - \alpha)^{-1} R, R \rangle,
\end{eqnarray*}
which is well-defined and differentiable on $(E_0,1)$ since $\phi$
is orthogonal to the kernel of $L_{+}$. We have
\begin{eqnarray*}
g'(\alpha) & = & \langle (L_{+} - \alpha)^{-1} R,R \rangle > 0
\end{eqnarray*}
and
\begin{eqnarray*}
g(0) & = & \frac12 \langle R, \partial_\lambda R \rangle = 0.
\end{eqnarray*}
Hence, $g(0) = 0$ is the only $0$ for $g$ in the interval $(E_0,1)$
since
\begin{eqnarray*}
\lim_{\alpha \to E_0} g(\alpha) \to -\infty .
\end{eqnarray*}
Conversely, if $A f = \alpha f$ for some $-\infty < \alpha < 1$,
$\alpha \neq 0$ and $f \in L^2$, then $f \perp R$ and
\begin{eqnarray*}
(PL_{+}P-\lambda)f = (A-\lambda) f = 0.
\end{eqnarray*}
Since
\begin{eqnarray*}
E_0 \langle f, f \rangle \leq \langle L_{+} f, f \rangle = \lambda \langle f, f \rangle,
\end{eqnarray*}
we know that $\lambda \geq E_0$. If $\lambda = E_0$, then $f$ is a
ground state of $L_{+}$ and hence not orthogonal to $R$. However,
$g(\lambda) = 0$, hence $\lambda = 0$. So, $A$ has a collection of
eigenvalues at $0$. Define
\begin{eqnarray*}
\mathcal{G} = \mathrm{span}\{ R, R_j, R_\lambda , u_E \}.
\end{eqnarray*}
We would like to show that $\dim (\mathcal{G}) = d+3$. Since $\phi$
is orthogonal to all the other functions, we need only show that the
equation
\begin{eqnarray}
\label{eqn:lin}
c_1 u_E + c_2 R_\lambda + \sum_{j =1}^d c_{j+2} R_j = 0
\end{eqnarray}
has only the trivial solution $c_j = 0$ for all $j$. By applying
$L_{+}$ to \eqref{eqn:lin}, we see
\begin{eqnarray*}
c_1 L_{+} u_E + c_2 R = 0.
\end{eqnarray*}
Taking the inner product with $u_E$, we conclude $c_1 = 0$.
This implies that $c_2 = 0$. As a result, $c_j = 0$ for $j =
2,\dots,d+2$. Now, if we can show that
\begin{eqnarray}
\label{eqn:max}
\sup_{\| f \|_{L^2}, f \in \mathcal{G} } \langle A f, f \rangle < 1,
\end{eqnarray}
then by the Courant minimax principle, there would be at least $d+3$
eigenvalues less than $1$ for $A$. However, we have shown there are
exactly $d+2$ of them. Note that neither the minimal eigenfunction
for $L_{+}$ nor $\phi$ itself are eigenvalues of $A$ due to
orthogonality arguments. Hence, if we can prove \eqref{eqn:max}, we
have proved the result.
By our assumption on the spectrum of $L_{-}$,
\begin{eqnarray*}
\langle P L_{-}^{-1} P f, f \rangle < \langle f, f \rangle
\end{eqnarray*}
for $f \neq 0$. Since $E \leq 1$ by assumption, we can prove the stronger result that
\begin{eqnarray*}
\langle A f, f \rangle \leq E^2 \langle P L_{-}^{-1} P f, f \rangle
\end{eqnarray*}
for all $f = a u_E + b \phi + \bold{c} \cdot \nabla R + d R_\lambda.$
To this end, we have
\begin{equation*}
\begin{split}
\langle A f, f \rangle & = \langle L_{+} (a u_E), a u_E + \bold{c} \cdot \nabla \phi + d \phi_\lambda \rangle \\
&\quad + \langle L_{+} \bold{c} \cdot \nabla R, a u_E + \bold{c} \cdot \nabla R + d R_\lambda \rangle \\
& = E^2 \langle L_{-}^{-1} (a u_E), a u_E + \bold{c} \cdot \nabla R + d R_\lambda \rangle \\
&\quad + E^2 \langle d R_\lambda, L_{-}^{-1} (a u_E) \rangle \\
& \leq E^2 \langle L_{-}^{-1} (a u_E), a u_E + \bold{c} \cdot \nabla R + d R_\lambda \rangle + E^2 \langle d R_\lambda, L_{-}^{-1} (a u_E) \rangle \\
&\quad + E^2 \langle L_{-}^{-1} (\bold{c} \cdot \nabla R + d R_\lambda), (\bold{c} \cdot \nabla R + d R_\lambda) \rangle \\
& \leq E^2 \langle P L_{-}^{-1} P f, f \rangle
\end{split}
\end{equation*}
since $L_{-}^{-1} {u_E} \perp \nabla R$ and $L_{-}$ is positive
definite on $\mathcal{G} \setminus R$.
\end{proof}
In order to test the discrete spectral assumptions, we
briefly recall the work of \cite{DeSc}, which requires some numerical computation.
First, let review the Birman-Schwinger method. let $H = L_{-} - \lambda = -\Delta - V$ for $V > 0$.
Since we are looking for small, positive eigenvalues of $L_{-}$, so
take $H f = -\alpha^2 f$ for $0< \alpha < \lambda$ so we have $L_{-} f
= (\lambda - \alpha^2) f$. Set $U = \sqrt{V}$ and $g = Uf$, then
\begin{eqnarray*}
g = U (-\Delta + \alpha^2)^{-1} U g.
\end{eqnarray*}
In other words, $g \in L^2$ is an eigenfunction for
\begin{eqnarray*}
K(\alpha) = U (-\Delta + \alpha^2)^{-1} U g
\end{eqnarray*}
with eigenvalue $1$ where $K > 0$, compact. Conversely, if $g
\in L^2$ satisfies $K(\alpha) g = g$, then
\begin{eqnarray*}
f = U^{-1} g = (-\Delta + \alpha^2)^{-1} Ug \in L^2
\end{eqnarray*}
and $Hf = -\alpha^2 f$. The eigenvalues of $K(\alpha)$ are seen to be
strictly increasing as $\alpha \to 0$ since
\begin{eqnarray*}
K'(\alpha) = -2 \lambda U (-\Delta + \alpha^2)^{-2} U.
\end{eqnarray*}
This implies that
\begin{eqnarray*}
\# \left\{ \alpha : \text{Ker} (H-\alpha^2) \neq \{0\} \right\} = \# \left\{ E>1 : \text{Ker} (K(0)-E) \neq \{0\} \right\}
\end{eqnarray*}
counted with multiplicity.
Finally, use the symmetric resolvent identity to see
\begin{eqnarray*}
(H-z)^{-1} & = & (-\Delta -z)^{-1} + (-\Delta -z)^{-1} U \\
&& \times \left[ I - U (-\Delta -z)^{-1} U \right]^{-1} U (-\Delta -z)^{-1}.
\end{eqnarray*}
Hence, the Laurent expansion about $z=0$ does not require negative
powers for $z$ iff $I + U(-\Delta -z)^{-1} U$ is invertible at $z =0$,
i.e.
\begin{equation*}
\mathrm{ker} (I - U (-\Delta)^{-1} U) = \{ 0 \}
\end{equation*}
by the Fredholm alternative since $V$ has exponential decay. Hence,
if $H$ has no resonance or eigenvalue at $0$, then $K(0)$ will not
have an eigenvalue at $1$. If we then count the eigenvalues
$\alpha_j$ in decreasing order for $K(0)$, then $H$ has exactly $N$
negative eigenvalues and neither an eigenvalue or resonance at $0$ iff
$\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_N > 1$ and
$\alpha_{N+1} < 1$. Hence, we can study numerically study $K$ for a
soliton of the saturated nonlinear Schr\"odinger equation. To do so,
we must accurately find a soliton, then use it as potential in the
truncation and discretization scheme presented in \cite{DeSc}, where
the gap condition is verified for the Hamiltonian resulting from
linearization about the $3d$ cubic ground state soliton.
\subsection{Generalized Kernel}
\label{s:kerg}
Let us review the generalized kernel of a Hamiltonian resulting
from linearizing about a soliton. Following \cite{W1},
we see by direct calculation that the vectors
\begin{eqnarray*}
\left[ \begin{array}{c}
0 \\
R
\end{array} \right], \left[ \begin{array}{c}
R_j \\
0
\end{array} \right]
\end{eqnarray*}
for all $j = 1, \dots, d$ are contained in $\mathrm{ker}(JL)$. Now, as
$Q(R_\lambda)$ is differentiable with respect to $\lambda$, we have by
a simple calculation that $L_{+} \partial_\lambda R = - R$ and $L_{-}
(x \phi) = -2 \nabla R$. Hence, the vectors
\begin{eqnarray*}
\left[ \begin{array}{c}
0 \\
x_j R
\end{array} \right], \left[ \begin{array}{c}
(\partial_\lambda R)_{\lambda_0} \\
0
\end{array} \right]
\end{eqnarray*}
in the generalized null space of order $2$. Notice that so far we
have constructed at $2d+2$ dimensional null space. Since we know the
null spaces of $L_{-}$ and $L_{+}$ exactly, these are unique. For
power nonlinearities, $g(s) = s^{\sigma}$,
\begin{equation}
(\partial_\lambda R)_{\lambda_0=1} = \frac12 (\frac{1}{\sigma} R + \mathbf{x} \cdot
\nabla R).
\end{equation}
We use this explicit form in our calculations.
As a result, we have the following
\begin{thm}
Let $g$ be the $L^2$ supercritical monomial nonlinearity. There exists a
$2d+2$ dimensional null space for $\mathcal{H}$, the matrix
Hamiltonian resulting from linearization about the ground state
soliton ($\lambda = 1$), consisting of the span of the vectors
\begin{eqnarray*}
\left\{ \left[ \begin{array}{c}
0 \\
R
\end{array} \right], \left[ \begin{array}{c}
R_j \\
0
\end{array} \right],
\left[ \begin{array}{c}
0 \\
x_j R
\end{array} \right], \left[ \begin{array}{c}
(\partial_\lambda R)_{\lambda_0} \\
0
\end{array} \right] \right\}.
\end{eqnarray*}
\end{thm}
\begin{proof}
The generalized null space of the adjoint can be found by reversing
the location of the non-zero elements in the above vectors.
Suppose that there exists a generalized eigenspace for the eigenvalue
$E \neq 0$. Then, there exists $\chi \neq 0$ and $\psi \neq 0$ such
that $(\mathcal{H} - E) \psi = \chi$ and $(\mathcal{H} - E) \chi = 0$.
Then, note that
\begin{eqnarray*}
(\mathcal{H}^2 - E^2) \psi & = & (A + E) \chi = 2 E \chi, \\
(\mathcal{H}^2 - E^2) \chi & = & 0.
\end{eqnarray*}
Hence, $\mathcal{H}^2$ has a generalized eigenspace at $E^2$. As a
result, we see that $T = L_{+} L_{-}$ has a generalized eigenspace at
$E^2$. Let $T \chi = E^2 \chi$ and $(T-E^2) \psi = c \chi$, for some
$c \neq 0$. Hence,
\begin{eqnarray*}
(L_{-}^{\frac{1}{2}} L_{+} L_{-}^{\frac{1}{2}} - E^2) L_{-}^\frac{1}{2} \psi_1 & = & c \chi_1 , \\
(L_{-}^{\frac{1}{2}} L_{+} L_{-}^{\frac{1}{2}} - E^2)^2
L_{-}^\frac{1}{2} \psi_1 & = & c L_{-}^{\frac{1}{2}} ( L_{+} L_{-} -
E^2) \chi_1 = 0,
\end{eqnarray*}
where given $P^c_R = I - P_R$, we have $\chi_1 = P^c_R \chi \neq 0$
since $T \chi = E^2 \chi$ and $\psi_1 = P^c_R \psi \neq 0$ since
$(T-E^2) \psi = c \chi$. However, this means that the self-adjoint
operator $L_{-}^{\frac{1}{2}} L_{+} L_{-}^{\frac{1}{2}}$ has a
generalized eigenvalue, which is impossible by an orthogonality
argument.
Since we have assumed there are no eigenvalues at the endpoints of the
continuous spectrum, there can be no accumulation and the number of
discrete eigenvalues is finite.
\end{proof}
\subsection{Natural Orthogonality Conditions}
\label{sec:verification}
As noted in Definition \ref{def:specprop}, even if the spectral
property holds, it only implies the absence of embedded eigenvalues on
a subspace of $\mathcal{U} \subset L^2\times L^2$. That
we must limit ourselves to a subspace will become clear
in Section \ref{sec:index_computations}, where we demonstrate that the operators $\mathcal{L}_\pm$
have negative eigenvalues.
This subspace will be defined as the
orthogonal complement to the span of a set of vectors. If this
collection of vectors is not chosen properly, we may find that the
spectral property holds though the operator still has embedded
eigenvalues. Thus the constraints on the set of vectors whose
orthogonal complement will define $\mathcal{U}$ are:
\begin{enumerate}
\item They must be orthogonal to any embedded eigenvalues,
\item Orthogonality with respect to them should induce positivity of
$\mathcal{L}$ on $\mathcal{U}$.
\end{enumerate}
A way of meeting both of these requirements is to use the discrete spectrum of the
adjoint matrix Hamiltonian, $\mathcal{H}^*$. To that end, we rely on the following simple results.
\begin{lem}
If $(\lambda, \vec{u})$ is an eigenvalue, eigenvector pair for $JL$
and $(\sigma, \vec{v})$ is an eigenvalue, eigenvector pair for
$(JL)^\ast$, then
\[
(\lambda - \sigma^\ast) \inner{\vec{u}}{\vec{v}} = 0 .
\]
Thus, if $\lambda - \sigma^\ast \neq 0$, the states are orthogonal.
\end{lem}
\begin{cor}
An eigenstate of $JL$ associated with an imaginary (possibly
embedded) eigenvalue, $i \tau \neq 0$, is orthogonal to
$\mathrm{ker}_{\mathrm{g}}((JL)^\ast)$.
\end{cor}
\begin{cor}
Let $(i \tau \neq 0, \vec{\psi})$ and $(\lambda >0, \vec{\phi})$ be
eigenvalue, eigenvector pairs of $JL$. Then
\begin{align*}
\inner{\psi_1}{\phi_2} & = 0,\\
\inner{\psi_2}{\phi_1} & = 0.
\end{align*}
\end{cor}
\begin{proof}
By the Hamiltonian symmetry of the problem, $-\lambda$ $(\phi_2,
\phi_1)^T$ and $\lambda$, $(-\phi_2,-\phi_1)^T$ are eigenvalue pairs
of the adjoint, $(JL)^\ast$. Therefore,
\begin{align*}
\inner{\psi_1}{\phi_2} - \inner{\psi_2}{\phi_1} & = 0, \\
-\inner{\psi_1}{\phi_2} - \inner{\psi_2}{\phi_1} & = 0.
\end{align*}
Adding and subtracting these equations gives the result.
\end{proof}
These trivial observations motivate using the known spectrum of the
adjoint system in constructing the orthogonal subspace. For the 3D
cubic problem, we can thus use make use of eigenstates coming from the
origin and the two off axis, real eigenvalues.
\section{Bilinear Forms and the Spectral Property}
\label{sec:normal}
We show here how the Spectral
Property \ref{def:specprop} is a condition sufficient
for showing there are no embedded eigenvalues, thus proving Theorems
\ref{thm:coercive} and \ref{thm:speccond}.
For general nonlinearities, we assume the operator resulting from
linearizing about a soliton $R$ has as discrete spectrum that is one
of the following:
\begin{itemize}
\item $(i)$ a $2d+2$ dimensional null space given by $R, \nabla
R, \partial_\lambda R, \mathbf{x} R$ plus $2$ eigenfunctions with symmetric
discrete eigenvalues $\lambda_0$, $-\lambda_0$ such that
$\lambda_0^2 \in {\mathbb R}$,
\item $(ii)$ a $2d+4$ dimensional null space given by $R, \nabla
R, \partial_\lambda R, \mathbf{x} R, \alpha, \beta$.
\end{itemize}
In the following subsections, we establish the following:
\begin{thm}
The generalized spectral property holds for the 3d cubic problem for
$( f, g)^T \in \mathcal{U} \subset L^2 \times L^2 $ specified by the following
orthogonality conditions:
\begin{equation*}
\inner{f}{R} =0,\quad \inner{g}{R + \mathbf{x} \cdot \nabla R}=0,
\quad\inner{f}{\phi_2} =0,\quad\inner{f}{ x_j R}
=0\quad\text{for $j=1,\ldots d$}, \end{equation*}
where $\vec{\phi} = ( \phi_1, \phi_2)^T$ is the
eigenstate associated with the positive eigenvalue $\sigma>0$.
\end{thm}
This subspace is motivated by the observations in Section
\ref{sec:verification}, as all of the elements we are orthogonal to
arise from the spectrum of the adjoint problem.
\begin{rem}
Note that henceforward we assume here the unstable eigenfunction for the
$3d$ cubic problem is radial. This claim is
substantiated by direct integration in our numerical results section,
as well as the fact that the spectral decomposition for the critical
problem remains valid under the assumption of radial symmetry, hence
the multiplicity of radial eigenfunctions must be at least $4$. Since
$R$, $R_\lambda$ are the only radial components of the kernel, the
unstable eigenmode must also be radial.
\end{rem}
Following \cite{FMR}, we computationally verify the spectral property
in the following steps. First, the bilinear form is decomposed by
spherical harmonics into
\begin{equation}
\begin{split}
\mathcal{B}(\mathbf{z},\mathbf{z}) & = \mathcal{B}_+(f,f) + \mathcal{B}_-(g,g) \\
&=\sum_{k=0}^\infty \mathcal{B}_+^{(k)}(f^{(k)},f^{(k)}) + \sum_{k=0}^\infty
\mathcal{B}_-^{(k)}(g^{(k)},g^{(k)}) ,
\end{split}
\end{equation}
where $\mathbf{z} = (f,g)^T$, and $f^{(k)}$ and $g^{(k)}$ are the
components of $f$ and $g$ in the $k$-th spherical harmonic.
We then identify the dimension of the subspace of negative eigenvalues for
$\mathcal{L}_\pm^{(k)}$. Though at first this would appear to require an
infinite number of computations, a monotonicity property of
these operators with respect to $k$ limits this to a finite number of
harmonics. We then show that our orthogonality conditions are
sufficient to point us away from the negative directions, allowing us
to prove our result.
\subsection{The Index of an Operator}
For a bilinear form $B$ on a vector space $V$, the index of $B$ with
respect to $V$ is given by
\begin{equation*}
\begin{split}
\text{ind}_V (B) \equiv \max \{ k \in \mathbb{N} \mid& \text{there exists a subspace $P$ of codimension $k$} \\
& \text{such that $B|_{P}$ is positive} \} .
\end{split}
\end{equation*}
Our results rely on the following generalization of Theorem XIII.8 of
\cite{RSv4}, which is in turn an extension of the Sturm
Oscillation Theorem (Section XIII.7 of \cite{RSv4}):
\begin{thm}
\label{thm:rs_idx}
Let $U^{(k)}$ be the solution to
\begin{equation*}
L^{(k)} U^{(k)} = - \frac{d^2}{dr^2}U^{(k)} - \frac{d-1}{r} \frac{d}{dr} U^{(k)} + V(r) U^{(k)} + \frac{k(k +
d-1)}{r^2} U^{(k)}= 0
\end{equation*}
with initial conditions given by the limits
\begin{equation*}
\lim_{r\to 0} \frac{U^{(k)}(r)}{r^k} = 1, \quad\lim_{r\to 0} \frac{d}{dr} \frac{U^{(k)}(r)}{r^k} = 0,
\end{equation*}
where $V$ is sufficiently smooth and decaying at $\infty$. Then,
the number $N(U^{(k)})$ of zeros of $U^{(k)}$ is finite and
\begin{align*}
{\mathrm{ind}}_{H^1_{\mathrm{rad}}} (B^{(0)}) &= N(U^{(0)}), \\
{\mathrm{ind}}_{H^1_{{\mathrm{rad}}+}} (B^{(k)}) &= N(U^{(k)}),\quad k \geq 1,
\end{align*}
where $B^{(k)}$ is the bilinear form associated to $L^{(k)}$.
\end{thm}
The space $H^1_{\mathrm{rad}}$ is the set of radially symmetric $H^1(\mathbb R^d)$
functions. The space $H^1_{{\mathrm{rad}}+}$ is the subset of $H^1_{\mathrm{rad}}$ for
which
\[
\int \frac{\abs{f}^2}{\abs{\mathbf{x}}^2} d\mathbf{x} < \infty.
\]
We will omit the subscript notation in our subsequent
index computations. It will be $H^1_{\mathrm{rad}}$ for $k=0$ and $H^1_{{\mathrm{rad}}+}$ for $k\geq 1$.
If one wishes to remove the limits from the statement of the initial
conditions, let $U^{(k)}(r)= r^k \widetilde{U}^{(k)}(r)$. Then the operator becomes
\[
\widetilde{L}^{(k)} = - \frac{d^2}{dr^2} - \frac{d-1 + 2k}{r} \frac{d}{dr} + V
\]
and the initial conditions become $\widetilde{U}^{(k)}(0) = 1$ and $\frac{d}{dr}
\widetilde{U}^{(k)}(0) = 0$. Indeed, we use precisely this change of variables
when making our numerical computations; see Appendix
\ref{s:numerical_sing}. The proof can be adapted from the proof of
Theorem XIII.8 in \cite{RSv4}.
\begin{cor}
\label{c:idx_mono}
The index is monotonic with respect to $k$,
\[
{\mathrm{ind}}(B^{(k+1)})\leq {\mathrm{ind}}(B^{(k)}) .
\]
\end{cor}
This has the useful consequence that once we find an $k$ for which
${\mathrm{ind}}(B^{(k)}) = 0$, we can immediately conclude that $B^{(k')}\geq 0$
for all $k' \geq k$.
Once we have computed the number of directions of each
$\mathcal{L}_\pm^{(k)}$ that prevents it from being positive, we can check
that we have a sufficient number of orthogonal conditions to point us
into the positive subspace.
\subsection{Numerical Estimates of the Index}
\label{sec:index_computations}
To compute the indexes of the operators, we proceed as follows.
In dimension three, we solve the initial value problems
\begin{align}
\mathcal{L}_+^{(0)} U^{(0)} &= 0, \quad U^{(0)} (0)= 1, \quad \frac{d}{dr}U^{(0)} (0)=0, \\
\mathcal{L}_-^{(0)} Z^{(0)} &= 0, \quad Z^{(0)} (0)= 1, \quad
\frac{d}{dr}Z^{(0)} (0)=0
\end{align}
for radially symmetric functions $U^{(0)}$ and $Z^{(0)}$. For higher
harmonics, $k>0$, we solve the initial value problems
\begin{align}
\mathcal{L}_+^{(k)} U^{(k)} &= 0, \quad U^{(k)} (0)= 0, \quad \lim_{r\to
0}
r^{-k} U^{(k)} (r)=1 ,\\
\mathcal{L}_-^{(k)} Z^{(k)} &= 0, \quad Z^{(k)} (0)= 0, \quad \lim_{r\to
0} r^{-k} Z^{(k)} (r)=1
\end{align}
for radially symmetric functions $U^{(k)}$ and $Z^{(k)}$.
\begin{prop}
\label{prop:index_computations}
The indexes of 3d Cubic NLS are:
\begin{gather*}
{\mathrm{ind}} \mathcal{L}_+^{(0)} = 1, \quad {\mathrm{ind}} \mathcal{L}_+^{(1)} = 1, \quad {\mathrm{ind}}
\mathcal{L}_+^{(2)} = 0,\\
{\mathrm{ind}} \mathcal{L}_-^{(0)} = 1,\quad {\mathrm{ind}} \mathcal{L}_-^{(1)} = 0.
\end{gather*}
\end{prop}
Once this proposition is established, Corollary \ref{c:idx_mono}
immediately gives us
\begin{cor}
\label{c:idx_mono_cubic}
For 3d Cubic NLS,
\begin{align*}
{\mathrm{ind}} \mathcal{B}_+^{(k)} =0&\quad\textrm{for $k>2$},\\
{\mathrm{ind}} \mathcal{B}_-^{(k)} =0&\quad\textrm{for $k>1$},
\end{align*}
where $\mathcal{B}_{\pm}^{(k)}$ is the bilinear form associated with $\mathcal{L}_{\pm}^{(k)}$.
\end{cor}
Using the method discussed in Section \ref{sec:numerics}, we compute
$U^{(\alpha)}$ and $Z^{(\alpha)}$ for each problem, $\alpha =0,1,2$.
The profiles appear in Figures
\ref{fig:3dcubic_index_k0},\ref{fig:3dcubic_index_k1},\ref{fig:3dcubic_index_k2}.
All were computed with a tolerance setting $10^{-13}$.
As a consistency check on the numerics, we note that asymptotically,
the potential vanishes, and
\begin{align}
\mathcal{L}_\pm^{(k)} &\approx -\frac{d^2}{dr^2}
-\frac{d-1}{r}U^{(k)}+\frac{k(k+d-2)}{r^2}.
\end{align}
In the region where $r \gg 1$, and the equations are essentially free
and the solutions must behave as:
\begin{subequations}
\begin{equation}
\label{eq:Uk_asympt}
U^{(k)}(r) \approx C_0^{(k)} r^k + C_1^{(k)} r^{2 - d - k}
\end{equation}
and
\begin{equation}
\label{eq:Zk_asympt}
Z^{(k)}(r) \approx D_0^{(k)} r^k + D_1^{(k)} r^{2 - d - k}.
\end{equation}
\end{subequations}
Estimating these constants from the numerics, we see that they have
the ``correct'' signs. For instance in Figure
\ref{fig:3dcubic_index_k0} (a), $U^{(0)}$ clearly has one zero
crossing. For sufficiently large $r$, the function appears to be
increasing past a local minimum. However, since the constants appear
to have stabilized, we contend we have entered the free region; the
signs and magnitudes of the constants thus forbid another zero.
\begin{figure}
\centering \subfigure[$k=0$ harmonic]{
\includegraphics[width=2.5in]{cubic_3d_idx0}
} \subfigure[$k=0$ harmonic asymptotics]{
\includegraphics[width=2.5in]{cubic_3d_idx0_consts}
}
\caption{Index computations for 3d Cubic NLS. The number of zero
crossings (other than $r=0$), determines the codimension of the
subspace on which the operator $\mathcal{L}_\pm^{(k)}$ is positive.}
\label{fig:3dcubic_index_k0}
\end{figure}
\begin{figure}
\centering \subfigure[$k=1$ harmonic]{
\includegraphics[width=2.5in]{cubic_3d_idx1}
} \subfigure[$k=1$ harmonic asymptotics]{
\includegraphics[width=2.5in]{cubic_3d_idx1_consts}
}
\caption{Index computations for 3d Cubic NLS. The number of zero
crossings (other than $r=0$), determines the codimension of the
subspace on which the operator $\mathcal{L}_\pm^{(k)}$ is positive.}
\label{fig:3dcubic_index_k1}
\end{figure}
\begin{figure}
\centering \subfigure[$k=2$ harmonic]{
\includegraphics[width=2.5in]{cubic_3d_idx2}
} \subfigure[$k=2$ harmonic asymptotics]{
\includegraphics[width=2.5in]{cubic_3d_idx2_consts}
}
\caption{Index computations for 3d Cubic NLS. The number of zero
crossings (other than $r=0$), determines the codimension of the
subspace on which the operator $\mathcal{L}_\pm^{(k)}$ is positive.}
\label{fig:3dcubic_index_k2}
\end{figure}
\begin{prop}
\label{prop:idx_perturbation}
For the operators in Proposition \ref{prop:index_computations} and
Corollary \ref{c:idx_mono_cubic} there exists a universal
$\delta_0>0$, sufficiently small, such that for the perturbed
operators
\[
\overline{\mathcal{L}}_\pm^{(k)} = {\mathcal{L}}_\pm^{(k)} - \delta_0 e^{-\abs{\mathbf{x}}^2}
\]
the associated bilinear forms have the property that
\[
{\mathrm{ind}}(\overline{\mathcal{B}}_\pm^{(k)}) = {\mathrm{ind}}(\mathcal{B}_\pm^{(k)}).
\]
\end{prop}
\begin{proof}
The proof here follows obviously from definition of the index of
$B$, namely the positivity of the quadratic form $B$ on the
subspace for individual operators. Let
$\delta_0$ be a sufficiently small value that it holds for
$\overline{\mathcal{L}}_+^{(k)}$ for $k = 0, 1, 2$ and
$\overline{\mathcal{L}}_-^{(k)}$ for $k = 0,1$. This $\delta_0$
now holds for all higher values of $k$, again by a monotonicity
argument. Indeed, for $k > 2$,
\[
\overline{\mathcal{B}}_+^{(k)}(f,f) = \overline{\mathcal{B}}_+^{(2)}(f,f) + \int
\frac{k^2 + 2k - 8 }{\abs{\mathbf{x}}^2}\abs{f}^2 d\mathbf{x} \geq 0.
\]
Because the form remains positive, this confirms that its index of zero is unperturbed.
\end{proof}
\subsection{Invertibility of Operators}
In conjunction with the results on the indexes of operators, we need
to compute a number of inner products of the form $\inner{\mathcal{L}
u}{u}$, where $\mathcal{L}$ is one of our operators and $u$ solves $\mathcal{L}
u = f$. These are computed numerically, but we can rigorously justify
the existence and unqiuess of these solutions, $u$, for the problems
under consideration.
\begin{prop}[Numerically Verified for 3d Problems]
\label{prop:eu_bvp_3d}
Let $f$ be a smooth, radially symmetric, localized function
satisfying the bound $\abs{f(r)} \leq C e^{-\kappa r}$ for some
positive constants $C$ and $\kappa$. There exists a unique radially
symmetric solution
\[
(1+r^{k+1})u \in L^\infty([0,\infty))\cap C^2([0,\infty))
\] to
\begin{equation}
\mathcal{L} u = f,
\end{equation}
where $\mathcal{L} = \mathcal{L}^{(k)}_\pm$ for one of the 3d problems.
\end{prop}
\begin{proof}
This is Proposition 2 and 4 of \cite{FMR}, along with our
computations of the indexes in Proposition
\ref{prop:index_computations}. See Appendix \ref{sec:inverse_proof}
for a proof in 1d.
\end{proof}
\begin{cor}
The solutions in Proposition \ref{prop:eu_bvp_3d} are smooth and
decay $\propto r^{-1 - k}$ as $r\to \infty$.
\end{cor}
\subsection{Estimates of Inner Products}
In order to prove the spectral property for each of these NLS
equations, we need to approximate the bilinear forms associated with
$\mathcal{L}_\pm^{(\alpha)}$ on certain functions. These particular
functions are, generically, of the form $\mathcal{L} u = f$, where $f$ is
from one of the orthogonality conditions.
\begin{prop}[Numerical approximation of inner products]
\label{prop:ip_estimates_3d}
For the $3d$ cubic problem, let $U_1^{(0}$, $U_2^{(0)}$,
$U_1^{(1)}$, and $Z_1^{(0)}$ solve
\begin{align}
\mathcal{L}_+^{(0)} U_1^{(0)} &= R, \quad (1+r)U_1^{(0)}\in L^\infty,\\
\mathcal{L}_+^{(0)} U_2^{(0)} &=\phi_2, \quad (1+r)U_2^{(0)}\in L^\infty,\\
\mathcal{L}_+^{(1)} U_1^{(1)} &= r R, \quad (1+r^2)U_1^{(0)}\in L^\infty,\\
\mathcal{L}_-^{(0)} Z_1^{(0)} &= R + r R', \quad (1+r)Z_1^{(0)}\in
L^\infty,
\end{align}
where $\vec{\phi}$ is the eigenstate associated with the positive
real eigenvalue of $JL$. Then,
\begin{align}
K_1^{(0)}&\equiv \inner{\mathcal{L}_+^{(0)}
U_1^{(0)}}{U_1^{(0)}}= 1.04846,\\
K_2^{(0)}&\equiv \inner{\mathcal{L}_+^{(0)} U_2^{(0)}}{U_2^{(0)}}=
0.00215981,\\
K_3^{(0)}&\equiv\inner{\mathcal{L}_+^{(0)}
U_1^{(0)}}{U_2^{(0)}}=-0.116369, \\
K_1^{(1)}&\equiv\inner{\mathcal{L}_+^{(1)} U_1^{(1)}}{U_1^{(1)}}=-0.581854,\\
J_1^{(0)}&\equiv\inner{\mathcal{L}_-^{(0)}
Z_1^{(0)}}{Z_1^{(0)}}= -0.662038.
\end{align}
\end{prop}
\begin{proof}
These result follows from direct computation.
\end{proof}
Finally, we state the following
\begin{prop}
\label{p:ip_est_3d_peturb}
For each case in Proposition \ref{prop:ip_estimates_3d}, there
exists a $\delta_0$ sufficiently small such that inner products
associated with $\overline{\mathcal{L}}$ can be made arbitrarily close to
$\mathcal{L}$. These values will be denoted with overlines.
\end{prop}
\begin{proof}
This follows immediately from the invertibility of the operator and continuity.
\end{proof}
\subsection{Proof of the Spectral Property}
\label{sec:spec_prop}
We are now ready to prove the spectral property. We prove positivity
of $\overline{\mathcal{B}}_+^{(0)}$, the other cases are similar. Our proof
closely follows Step 1 and Step 3 of Section 2.4 of \cite{FMR}.
Since $K_1^{(0)}$ and $K_2^{(0)}>0$, orthogonality to $R$ and $\phi_2$
will not give positivity. However, if $f$ is orthogonal to both of
these, then it is also orthogonal to
\[
q = R - \frac{K^{(0)}_3}{K^{(0)}_2} \phi_2
\]
and
\[
\begin{split}
\inner{\mathcal{L}_+^{(0)} q}{q}&= K_1^{(0)} - 2
\frac{K^{(0)}_3}{K^{(0)}_2} K_3^{(0)}
+ \paren{\frac{K^{(0)}_3}{K^{(0)}_2}}^2 K^{(0)}_2\\ &=
-\frac{1}{K^{(0)}_2}((K^{(0)}_3)^2 - K^{(0)}_1 K^{(0)}_2)\\ & =
-5.22138.
\end{split}
\]
By Proposition \ref{p:ip_est_3d_peturb}, we can take $\delta_0$
sufficiently small such that
\[
-\frac{1}{\overline{K}^{(0)}_2}\paren{(\overline{K}^{(0)}_3)^2 -
\overline{K}^{(0)}_1 \overline{K}^{(0)}_2}<0 .
\]
We proceed with this value of $\delta_0$.
Let $\overline{Q}$ solve
\[
\overline{\mathcal{L}}_+^{(0)} \overline{Q} = q.
\]
Obviously,
\[
Q = \overline{U}_1^{(0)} -
\frac{K^{(0)}_3}{K^{(0)}_2} \overline{U}_2^{(0)}
\]
and
\[
\overline{\mathcal{B}}_+^{(0)}(\overline{Q} ,\overline{Q} ) < 0.
\]
For a moment, suppose $\overline{Q} \in H^1_{\mathrm{rad}}$; it is not since it
decays too slowly to be in $L^2$. We could then imagine decomposing
$H^1_{\mathrm{rad}}$ into $\mathrm{span}\{\overline{Q}\}$ and its orthogonal complement,
where the orthgonalization is done with respect to the
$\overline{\mathcal{B}}_+^{(0)}$ quadratic form. Since
$\overline{\mathcal{B}}_+^{(0)}(\overline{Q},\overline{Q})<0$, the form is
{\it non-degenerate} and this decomposition is well defined. Since
${\mathrm{ind}}_{H^1_{\mathrm{rad}}} \overline{\mathcal{B}}_+^{(0)} = 1$,
$\overline{\mathcal{B}}_+^{(0)}\geq 0$ on $\mathrm{span}\{\overline{Q}_A\}^\perp$.
To prove this claim, we argue by contradiction. Suppose there were an
element, $Z \in \mathrm{span}\{\overline{Q}_A\}^\perp$ for which
$\overline{\mathcal{B}}_+^{(0)}(Z,Z)<0$. Then, because of our
decomposition, $\overline{\mathcal{B}}_+^{(0)}<0$ on $\mathrm{span}\{Z,
\overline{Q}_A\}$, a space of dimension 2. This contradicts our index
calculation, proving the claim.
Continuing, if $u\in H^1_{\mathrm{rad}}$, $u\perp q$ (with respect
to $L^2$), then using the
hypothetical orthogonal decomposition,
\[
u = c \overline{Q} + u^\perp.
\]
If $c =0$, then $u$ lies in a subspace of $H^1_{\mathrm{rad}}$ on which
$\overline{\mathcal{B}}_+^{(0)}\geq 0$, giving the desired positivity.
Indeed, the orthogonality condition, $u \perp q$, is sufficient to
ensure $u$ is orthogonal to $\overline{Q} $ with respect to the
$\overline{\mathcal{B}}_+^{(0)}$ quadratic form. Taking the inner product of
$u$ with $q$,
\begin{equation*}
\begin{split}
0 &= c \inner{\overline{Q}}{ q} +
\inner{u^\perp}{q} = c
\overline{\mathcal{B}}_+^{(0)}(\overline{Q},\overline{Q}) +
\inner{u^\perp}{\overline{\mathcal{L}}_+^{(0)}\overline{Q}}\\
&= c
\overline{\mathcal{B}}_+^{(0)}(\overline{Q},\overline{Q}) +
\overline{\mathcal{B}}_+^{(0)}(u^\perp,\overline{Q})= c
\overline{\mathcal{B}}_+^{(0)}(\overline{Q},\overline{Q}) + 0.
\end{split}
\end{equation*}
Since
$\overline{\mathcal{B}}_+^{(0)}(\overline{Q},\overline{Q})\neq 0$,
we have $c = 0$.
Unfortunately, the above argument does not work as stated because
$\overline{Q}$ is not in $L^2$! To get positivity of
$\overline{\mathcal{B}}_+^{(0)}$, we regularize the problem and follow the
above scheme. First, we introduce the smooth cutoff function
$\chi_A(r)=\chi(r/A)$, defined such that
\[
\chi(r)=\begin{cases}
1 & r < 1\\
0 & r\geq 2
\end{cases}
\]
and the norm
\[
\norm{f}_{\pm}^2 = \norm{\nabla f}_{L^2}^2 + \int \abs{\mathcal{V}_\pm} \abs{f}^2.
\]
Let $\overline{Q}_A(r) = \overline{Q}(r)\chi_A(r)$. Next, we
observe that
\begin{equation}
\lim_{A\to +\infty} \norm{\overline{Q}_A - \overline{Q}}_+ +
\abs{\overline{\mathcal{B}}_+^{(0)}(\overline{Q} ,\overline{Q} ) -
\overline{\mathcal{B}}_+^{(0)}(\overline{Q}_A ,\overline{Q}_A )}=0.
\end{equation}
Since $\overline{\mathcal{B}}_+^{(0)}(\overline{Q} ,\overline{Q} ) < 0
$, for sufficiently large $A$,
$\overline{\mathcal{B}}_+^{(0)}(\overline{Q}_A ,\overline{Q}_A )
<0$ too. Thus, we can legitimately decompose $H^1_{\mathrm{rad}}$ as
\begin{equation}
H^1_{\mathrm{rad}} = \mathrm{span}\{\overline{Q}_A\} \oplus \mathrm{span}\{\overline{Q}_A\}^\perp
\end{equation}
with the orthgonalization is done with respect to the quadratic form
$\overline{\mathcal{B}}_+^{(0)}$.
Finally, let $u \in H^1_{\mathrm{rad}}$, $u \perp R$ and $u \perp \phi_2$. Then
$u \perp q$. With $A$ sufficiently large to make the
above decomposition valid,
\begin{equation}
u = c(A) \overline{Q}_A + u^\perp_A.
\end{equation}
As in the heuristic argument $\overline{\mathcal{B}}_+^{(0)}(u^\perp_A,
u^\perp_A)\geq 0$. Thus,
\[
\overline{\mathcal{B}}_+^{(0)}(u,u) = c(A)^2
\overline{\mathcal{B}}_+^{(0)}(\overline{Q}_A, \overline{Q}_A) +
\overline{\mathcal{B}}_+^{(0)}(u^\perp_A, u^\perp_A) \geq c(A)^2
\overline{\mathcal{B}}_+^{(0)}(\overline{Q}_A, \overline{Q}_A) .
\]
We will have our result if $c(A)\to 0$ as $A\to +\infty$.
Since $\inner{u}{q}=0$,
\[
\begin{split}
c(A) \inner{\overline{U}_A^\ast} {q} &= -
\inner{u^\perp_A}{q}\\
& = - \inner{u^\perp_A}{\overline{\mathcal{L}}_+^{(0)}
\overline{Q}}\\
& = - \inner{u^\perp_A}{\overline{\mathcal{L}}_+^{(0)}\paren{
\overline{Q}- \overline{Q}_A }}.
\end{split}
\]
Therefore,
\[
\abs{c(A)}= \frac{\abs{\inner{u^\perp_A}{\overline{\mathcal{L}}_+^{(0)}\paren{
\overline{Q}- \overline{Q}_A
}}}}{\abs{\inner{\overline{U}_A^\ast} {q} }}\leq
\frac{\norm{u_A^\perp}_+\norm{\overline{Q} - \overline{Q}_A}_+}{\abs{\inner{\overline{U}_A^\ast} {q} }}.
\]
Also,
\[
\begin{split}
\abs{\inner{\overline{U}_A^\ast} {q} - \inner{\overline{Q}} {q}} &=
\abs{\inner{\overline{U}_A^\ast- \overline{Q}}
{\overline{\mathcal{L}}_+^{(0)} \overline{Q}} }\leq \norm{\overline{Q} - \overline{Q}_A}_+\norm{\overline{Q}}_+.
\end{split}
\]
Because this vanishes as $A\to +\infty$, we have that for all $A$
sufficiently large,
\[
\abs{c(A)}\leq C \norm{u_A^\perp}_+\norm{\overline{Q} - \overline{Q}_A}_+
\]
for a constant $C$ independent of $A$.
By construction,
\[
\norm{u_A^\perp}_+ \leq C\paren{\norm{u}_+ + c(A)
\norm{\overline{U}_A^\ast}_+}\leq C\paren{\norm{u}_+ + c(A)
\norm{\overline{U}_A^\ast - \overline{Q}}_+ + \norm{\overline{Q}}_+}.
\]
Substituting into our previous estimate on $c(A)$,
\[
\abs{c(A)} \leq C \paren{\norm{u}_+ + c(A)
\norm{\overline{U}_A^\ast - \overline{Q}}_+ + \norm{\overline{Q}}_+}\norm{\overline{Q} - \overline{Q}_A}_+.
\]
We can clearly see that as $A\to +\infty$, $c(A)\to + 0$. We
conclude,
\[
\overline{\mathcal{B}}_+^{(0)}(u,u) \geq 0
\]
for $u \in H^1_{\mathrm{rad}}$ and $u \perp R$ and $u\perp \phi_2$. This
yields the estimate
\[
\mathcal{B}_+^{(0)}(u,u) \geq \delta_0 \int e^{-\abs{\mathbf{x}}^2} \abs{u}^2dx.
\]
Following the same analysis for $\overline{\mathcal{L}}_-$, we conclude
\[
\mathcal{B}_-^{(0)}(g,g) \geq \delta_0 \int e^{-\abs{\mathbf{x}}^2} \abs{g}^2dx
\]
for $g\in H^1_{\mathrm{rad}}$ and $g \perp R + r R'$ since $J_1^{(0)}<0$. Repeating this again for
$f \in H^1_{{\mathrm{rad}}(1)}$, $f \perp r R$, we get
\[
\mathcal{B}_-^{(1)}(f,f) \geq \delta_0 \int e^{-\abs{\mathbf{x}}^2} \abs{f}^2dx
\]
because $K_1^{(1)}<0$.
Let us assume that $\delta_0$ has been taken sufficiently small such
that:
\begin{itemize}
\item The indexes of all operators are unperturbed, as in Proposition \ref{prop:idx_perturbation},
\item The above arguments on the positivity of
$\overline{\mathcal{B}}_+^{(k)}$ for $k=0,1$ and
$\overline{\mathcal{B}}_-^{(0)}$ hold.
\end{itemize}
Then for $\mathbf{z} = (f,g)^T$ satisfying the orthogonality conditions,
\begin{equation*}
\begin{split}
\overline{\mathcal{B}}(\mathbf{z},\mathbf{z}) &= \overline{\mathcal{B}}_+(f,f)+
\overline{\mathcal{B}}_-(g,g) \\
&=\sum_{k=0}^\infty \overline{\mathcal{B}}_+^{(k)}(f^{(k)},f^{(k)}) + \sum_{k=0}^\infty
\overline{\mathcal{B}}_-^{(k)}(g^{(k)},g^{(k)}) \\
& = \sum_{k=0}^\infty {\mathcal{B}}_+^{(k)}(f^{(k)},f^{(k)}) + \sum_{k=0}^\infty
{\mathcal{B}}_-^{(k)}(g^{(k)},g^{(k)}) - \delta_0\int
e^{-\abs{\mathbf{x}}^2} \paren{\abs{f}^2 + \abs{g}^2}d\mathbf{x}\\
& = \mathcal{B}(\mathbf{z},\mathbf{z}) - \delta_0\int
e^{-\abs{\mathbf{x}}^2} \abs{\mathbf{z}}^2 d\mathbf{x}\geq 0.
\end{split}
\end{equation*}
We almost have the expression in Definition \ref{def:specprop}. To complete the proof, note that for any $\theta \in (0,1)$,
\[
(1+\theta) \mathcal{B}(\bold{z},\bold{z}) \geq \theta\paren{\int \abs{\nabla \bold{z}}^2 d\mathbf{x} +
\int V_+\abs{f}^2 + \int V_- \abs{g}^2 } + \delta_0 \int e^{-\abs{\mathbf{x}}^2} \abs{\bold{z}}^2d\mathbf{x}.
\]
We can take $\theta=\theta_\star$ sufficiently small such that
\[
\theta_\star\paren{
\int V_+\abs{f}^2 + \int V_- \abs{g}^2 } + \delta_0 \int
e^{-\abs{\mathbf{x}}^2} \abs{\bold{z}}^2d\mathbf{x}\geq \frac{\delta_0}{2}\int
e^{-\abs{\mathbf{x}}^2} \abs{\bold{z}}^2d\mathbf{x}.
\]
Then,
\[
\mathcal{B}(\bold{z},\bold{z})\geq \frac{\theta_\star}{1+\theta_\star}\int \abs{\nabla \bold{z}}^2
d\mathbf{x} + \frac{\delta_0}{2(1+\theta_\star)}\int
e^{-\abs{\mathbf{x}}^2} \abs{\bold{z}}^2d\mathbf{x}.
\]
Shrinking $\delta_0$ again, so that it is smaller than
\[
\min\set{\frac{\theta_\star}{1+\theta_\star}, \frac{\delta_0}{2(1+\theta_\star)}}
\]
gives us the spectral property.\begin{flushright}{$\blacksquare$}\end{flushright}
\section{Other Problems}
In principle, this scheme can be applied to any linearized nonlinear
Schr\"odinger equation. One finds the indexes of the operators, picks
an appropriate subspace to project away from, and computes the
necessary inner products. However, our experiments show that the
algorithm is not as universal as might be hoped. In this section
we exhibit the computations for several 1d NLS equations,
\begin{equation}
\label{e:nls_1d}
i \psi_t + \psi_{xx} + \abs{\psi}^{2\sigma} \psi = 0.
\end{equation}
Sometimes our approach works, ruling out embedded eigenvalues in a range
of supercritical cases, while in others it fails, leaving a large
range of interesting problems unresolved.
\subsection{Numerical Estimates of the Index}
\label{s:1d_idx}
As in the 3d problem, we first compute the indexes of the operators
$\mathcal{L}_\pm$ to identify the number of ``bad'' directions. In contrast
to the multidimensional problems where there are an arbitrarily high,
but finite, number of harmonics which must be examined, 1D problems
only require us to study the operators restricted to even and odd
functions. This requires the following results, whose proofs are
quite similar to that of Theorem \ref{thm:rs_idx}:
\begin{cor}
Let $U$ be the even solution to
\begin{gather*}
\left\{ \begin{array}{c}
L U = - U'' + V(r) U = 0, \\
U(0) = 1, U' (0) = 0,
\end{array} \right.
\end{gather*}
where $V$ is sufficiently smooth and decaying at $\infty$. Then,
the number $N(U)$ of zeros of $U$ is finite and
\begin{equation*}
{\mathrm{ind}}_{H^1_e} (B) = N(U),
\end{equation*}
where $B$ is the bilinear form associated with $L$.
\end{cor}
\begin{cor}
Let $U$ be the odd solution to
\begin{gather*}
\left\{ \begin{array}{c}
L U = - U'' - \frac{d-1}{r} U' + V(r) U = 0, \\
U(0) = 0, U' (0) = 1,
\end{array} \right.
\end{gather*}
where $V$ is sufficiently smooth and decaying at $\infty$. Then,
the number $N(U)$ of zeros of $U$ is finite and
\begin{equation*}
{\mathrm{ind}}_{H^1_o} (B) = N(U),
\end{equation*}
where $B$ is the bilinear form associated with $L$.
\end{cor}
$H^1_e$ and $H^1_o$ are the subspaces of $H^1(\mathbb R)$ restricted to even
and odd functions. In what follows, we shall omit them in the
subscripts of the indexes.
To proceed, we numerically solve the initial value problems
\begin{align}
\mathcal{L}_+^{(e)} U^{(e)} &=0, \quad U^{(e)}(0)=1, \quad\frac{d}{dx}U^{(e)}(0)=0,\\
\mathcal{L}_-^{(e)} Z^{(e)} &=0, \quad Z^{(e)}(0)=1, \quad
\frac{d}{dx}Z^{(e)}(0)=0
\end{align}
for even functions $U^{(e)}$ and $Z^{(e)}$. We then solve
\begin{align}
\mathcal{L}_+^{(o)} U^{(o)} &=0, \quad U^{(o)}(0)=0, \quad\frac{d}{dx}U^{(o)}(0)=1,\\
\mathcal{L}_-^{(o)} Z^{(o)} &=0, \quad Z^{(o)}(0)=0, \quad
\frac{d}{dx}Z^{(o)}(0)=1
\end{align}
for odd functions $U^{(o)}$ and $Z^{(o)}$. $\mathcal{L}_\pm$ have the same
definitions as before; the Laplacian is now one dimensional. As in
the 3d case, we verify {\it a postiori} that, asymptotically, $U^{\alpha}$ and
$Z^{\alpha}$ fit
\begin{align}
U^{\alpha(x)} &\approx C^{(\alpha)}_0 + C^{(\alpha)}_1 x,\\
Z^{\alpha(x)} &\approx D^{(\alpha)}_0 + D^{(\alpha)}_1 x.
\end{align}
In addition, the constants have the right signs to ensure we have the
correct number of zero crossings.
\begin{prop}[Numerically Verified]
\label{prop:idx_1d}
The indexes for the 1d NLS equation with $\sigma = 2, 2.1, 2.5, 3$
are:
\begin{gather*}
{\mathrm{ind}} \mathcal{L}_+^{(e)} =1, \quad {\mathrm{ind}} \mathcal{L}_-^{(e)} =1,\\
{\mathrm{ind}} \mathcal{L}_+^{(o)} =1, \quad {\mathrm{ind}} \mathcal{L}_-^{(o)} =0.
\end{gather*}
\end{prop}
\begin{proof}
Using the method discussed in Section \ref{sec:numerics}, we compute
$U^{(\alpha)}$ and $Z^{(\alpha)}$ for each problem, $\alpha = e, o$.
The profiles appear in Figures \ref{fig:1d_sig21_idx_even},
\ref{fig:1d_sig21_idx_odd}. All are computed with a relative
tolerance of $10^{-10}$ and an absolute tolerance of $10^{-12}$
using \textsc{Matlab} .
\begin{figure}
\centering \subfigure[Even functions]{
\includegraphics[width=2.5in]{1dcrit_idxe}
} \subfigure[Even function asymptotics]{
\includegraphics[width=2.5in]{1dcrit_idxe_consts}
}
\caption{Index computations for critical 1d NLS. The number of zero
crossings (other than $x=0$), determines the codimension of the
subspace on which the operator $\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sigcrit_idx_even}
\end{figure}
\begin{figure}
\centering \subfigure[Odd functions]{
\includegraphics[width=2.5in]{1dcrit_idxo}
} \subfigure[Odd function asymptotics]{
\includegraphics[width=2.5in]{1dcrit_idxo_consts}
}
\caption{Index computations for critical 1d NLS. The number of zero
crossings (other than $x=0$), determines the codimension of the
subspace on which the operator $\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sigcrit_idx_odd}
\end{figure}
\begin{figure}
\centering \subfigure[Even functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig21_idxe}
} \subfigure[Even function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig21_idxe_consts}
}
\caption{Index computations for 1d NLS with $\sigma=2.1$. The
number of zero crossings (other than $x=0$), determines the
codimension of the subspace on which the operator
$\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sig21_idx_even}
\end{figure}
\begin{figure}
\centering \subfigure[Odd functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig21_idxo}
} \subfigure[Odd function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig21_idxo_consts}
}
\caption{Index computations for 1d NLS with $\sigma=2.1$. The
number of zero crossings (other than $x=0$), determines the
codimension of the subspace on which the operator
$\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sig21_idx_odd}
\end{figure}
\begin{figure}
\centering \subfigure[Even functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig25_idxe}
} \subfigure[Even function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig25_idxe_consts}
}
\caption{Index computations for 1d NLS with $\sigma=2.5$. The
number of zero crossings (other than $x=0$), determines the
codimension of the subspace on which the operator
$\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sig25_idx_even}
\end{figure}
\begin{figure}
\centering \subfigure[Odd functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig25_idxo}
} \subfigure[Odd function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig25_idxo_consts}
}
\caption{Index computations for 1d NLS with $\sigma=2.5$. The
number of zero crossings (other than $x=0$), determines the
codimension of the subspace on which the operator
$\mathcal{L}_\pm^{(e/o)}$ is positive.}
\label{fig:1d_sig25_idx_odd}
\end{figure}
\begin{figure}
\centering \subfigure[Even functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig30_idxe}
} \subfigure[Even function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig30_idxe_consts}
}
\caption{Index computations for 1d NLS with $\sigma=3$. The number
of zero crossings (other than $x=0$), determines the codimension
of the subspace on which the operator $\mathcal{L}_\pm^{(e/o)}$ is
positive.}
\label{fig:1d_sig30_idx_even}
\end{figure}
\begin{figure}
\centering \subfigure[Odd functions]{
\includegraphics[width=2.5in]{supercrit_1d_sig30_idxo}
} \subfigure[Odd function asymptotics]{
\includegraphics[width=2.5in]{supercrit_1d_sig30_idxo_consts}
}
\caption{Index computations for 1d NLS with $\sigma=3$. The number
of zero crossings (other than $x=0$), determines the codimension
of the subspace on which the operator $\mathcal{L}_\pm^{(e/o)}$ is
positive.}
\label{fig:1d_sig30_idx_odd}
\end{figure}
\end{proof}
Proposition \ref{prop:idx_perturbation} applies to these 1d problems
too. As in the 3d case, we ultimately use a perturbed bilinear
form in the proof of the spectral property.
\subsection{Estimates of the Inner Products}
We now compute a series of inner products and show that in some cases
the natural orthogonality conditions are sufficient to yield a
spectral property. Rigoursly, these results require the following
Proposition on the invertibility of the $\mathcal{L}_\pm^{(e/o)}$ operators:
\begin{prop}[Numerically Verified for 1d Saturated Problem]
\label{prop:eu_bvp_1d}
Let $f$ be a smooth, localized function satisfying the bound
$\abs{f(x)} \leq C e^{-\kappa \abs{x}}$ for some positive constants
$C$ and $\kappa$. If $f$ is even/odd, there exists a unique
even/odd solution $u \in L^\infty(\mathbb R)\cap C^2(\mathbb R)$ to
\begin{equation}
\mathcal{L} u = f,
\end{equation}
where $\mathcal{L} = \mathcal{L}^{(e/o)}_\pm$ for a 1d problem.
\end{prop}
\begin{proof}
See Appendix \ref{sec:inverse_proof}.
\end{proof}
\begin{prop}[Numerical]
\label{prop:ip_estimates_1d}
Let $U_1^{(e)}$, $Z_1^{(e)}$, and $U_1^{(o)}$, all elements of
$L^\infty(\mathbb R)$, solve the following boundary value problems:
\begin{align}
\mathcal{L}_+^{(e)} U_1^{(e)} & =R , \quad \frac{d}{dx} U_1^{(e)}(0) = 0,\\
\mathcal{L}_-^{(e)} Z_1^{(e)} & =\frac{1}{\sigma}R + x R' , \quad \frac{d}{dx} Z_1^{(e)}(0) = 0,\\
\mathcal{L}_+^{(o)} U_1^{(o)} & =R' , \quad U_1^{(o)}(0) = 0.
\end{align}
Let
\begin{align}
K_1^{(e)} & \equiv\mathcal{B}_+^{(e)}(U_1^{(e)}) =
\inner{\mathcal{L}_+^{(e)} U_1^{(e)}}{U_1^{(e)}} ,\\
J_1^{(e)}& \equiv\mathcal{B}_-^{(e)}(Z_1^{(e)}) =
\inner{\mathcal{L}_-^{(e)} Z_1^{(e)}}{Z_1^{(e)}} ,\\
K_1^{(o)} & \equiv\mathcal{B}_+^{(o)}(U_1^{(o)}) =
\inner{\mathcal{L}_+^{(o)} U_1^{(o)}}{U_1^{(o)}} .
\end{align}
Then,
\begin{center}
\begin{tabular}{llll}
$\sigma$ & $K_1^{(e)}$ & $J_1^{(e)}$ & $K_1^{(o)}$\\
\hline
$2.0$ & $-0.557768$
& $0.292551$
& $-1.30410$
\\
$2.1$ & $-0.496932$
& $0.216284$
& $-1.21364$
\\
$2.5$ & $-0.297841$
& $-0.0216292$
& $-0.924662$
\\
$3.0$ & $-0.122559$
& $-0.218499$
& $-0.671783$.
\end{tabular}
\end{center}
\end{prop}
\subsubsection{Proof of the Spectral Property for Certain Supercritical Cases}
We restrict our attention to the 1d supercritical problems $\sigma =
2.5$ and $\sigma = 3$. Repeating the procedure of Section
\ref{sec:spec_prop}, for $\mathbf{z} = (f, g)^T$, the orthogonality of $f$ to $R$
and $x R$ gives us $\overline{\mathcal{L}}_+\geq 0$ and the orthogonality of $g$ to
$\frac{1}{\sigma }R + x R'$ gives us $\overline{\mathcal{L}}_-\geq 0$. This proves the
spectral property on the restricted subspace. Since these
orthogonality conditions are consistent with those formulated in
Section \ref{sec:verification}, we conclude that there are no non zero
purely imaginary eigenvalues.
\subsubsection{An Inconclusive Supercritical Case}
In the case of $\sigma = 2.1$, we have that $J_1^{(e)}>0$, which means
that orthogonality of $g$ with respect to $\frac{1}{\sigma} R + x R'$,
is insufficient to guarantee positivity of $\mathcal{L}_-$. It is possible
that if we extend our scope, as in the 3D cubic problem, to include
orthogonality to the eigenstate associated with the unstable eigenvalue
we will be able to prove the spectral property for this problem.
However, we do not pursue that here; rather we wish to highlight the
failure of our algorithm at a seemingly arbitrary supercritical
nonlinearity.
\subsubsection{The Critical Case}
\label{s:1d_crit}
The critical 1D problem, with $\sigma = 2$, is also inconclusive. As
in the supercritical problems we will look at the inner products
against $R$ and $\frac{1}{\sigma} R + x R'$. We also employ
inner products arising from with the rest of the generalized kernel,
$x^2 R$ and $\beta$, where $\beta$ solves
\begin{equation*}
L_+ \beta = - x^2 R.
\end{equation*}
See \cite{W1} for details. This motivates the following numerical result:
\begin{prop}[Numerical]
Let $Z_2^{(e)}$ solve
\begin{equation}
\mathcal{L}_-^{(e)}Z_2^{(e)} = \rho, \quad \frac{d}{dx} Z_1^{(e)}(0) = 0
\end{equation}
and let
\begin{align}
J_2^{(e)} & \equiv\mathcal{B}_-^{(e)}(Z_2^{(e)},Z_2^{(e)}) =
\inner{\mathcal{L}_-^{(e)} Z_2^{(e)}}{Z_2^{(e)}} ,\\
J_3^{(e)} &\equiv\mathcal{B}_-^{(e)}(Z_1^{(e)},Z_2^{(e)})
=\inner{\mathcal{L}_-^{(e)} Z_1^{(e)}}{Z_2^{(e)}}.
\end{align}
Then,
\begin{align}
J_2^{(e)} & =3.77915 ,\\
J_3^{(e)} & =0.864273.
\end{align}
\end{prop}
Since $K_1^{(e)}<0$, we may conclude that $\mathcal{L}_+\geq 0$, when the
operator is restricted to even functions that are orthogonal to $R$.
However, orthogonality to neither $ \frac{1}{2} R + x R'$ nor $\rho$
is, individually, sufficient to gain positivity of $\mathcal{L}_-^{(e)}$.
We are thus motivated to consider orthogonality to the subspace
$\mathrm{span}\{\frac{1}{2} R + x R', \rho\}$, as in the proof of the 3D cubic
problem. We examine the quantity
\begin{equation}
-\frac{1}{J_2^{(e)}}\paren{(J_3^{(e)})^2 -J_1^{(e)}J_2^{(e)} } = 0.0948958.
\end{equation}
However, we need this to be negative. Thus, we have no set of natural
orthogonality conditions which yield a spectral
property. In this case, since $K_1^{(o)}<0$, the only
obstacle to the proof is the $\mathcal{L}_-^{(e)}$ operator.
\subsubsection{The Critical Case with Other Orthogonality Conditions}
If we had instead used the orthogonality condition, $g \perp R$, and
then solved the boundary value problem $\mathcal{L}_-^{(e)}\check{Z}_{3}^{(e)} = R$,
the inner product,
\[
\check{J}_1^{(e)} \equiv\inner{\mathcal{L}_-^{(e)}\check{Z}_1^{(e)}}{\check{Z}_1^{(e)}} = -3.770731.
\]
This would give us a spectral property, but it is not a convenient
subspace.
Suppose we use the orthogonality conditions of \cite{FMR}, and let $g
\perp \Lambda R$ and $g \perp \Lambda^2 R$. Then, we compute as
follows: Let $\hat{Z}_1^{(e)}$, and $\hat{Z}_2^{(e)}$ solve the
following problems:
\begin{align}
\mathcal{L}_-^{(e)} \hat{Z}_1^{(e)} & =\Lambda R , \quad \frac{d}{dx} \hat{Z}_1^{(e)}(0) = 0,\\
\mathcal{L}_-^{(e)} \hat{Z}_2^{(e)} & =\Lambda R , \quad \frac{d}{dx}
\hat{Z}_2^{(e)}(0) = 0,
\end{align}
and define the inner products
\begin{align}
\hat{J}_1^{(e)} &
\equiv\mathcal{B}_-^{(e)}(\hat{Z}_1^{(e)},\hat{Z}_1^{(e)}) =
\inner{\mathcal{L}_-^{(e)} \hat{Z}_1^{(e)}}{\hat{Z}_1^{(e)}} ,\\
\hat{J}_2^{(e)} &
\equiv\mathcal{B}_-^{(e)}(\hat{Z}_2^{(e)},\hat{Z}_2^{(e)}) =
\inner{\mathcal{L}_-^{(e)} \hat{Z}_2^{(e)}}{\hat{Z}_2^{(e)}} ,\\
\hat{J}_3^{(e)}
&\equiv\mathcal{B}_-^{(e)}(\hat{Z}_1^{(e)},\hat{Z}_2^{(e)})
=\inner{\mathcal{L}_-^{(e)} \hat{Z}_1^{(e)}}{\hat{Z}_2^{(e)}}.
\end{align}
We will find that
\begin{align}
\hat{J}_1^{(e)} & = 0.292551, \\
\hat{J}_2^{(e)} & =2.57656,\\
\hat{J}_3^{(e)} & = -1.27657.
\end{align}
As one would hope, given that the 1D spectral property was established
in \cite{MR-Annals},
\begin{equation}
-\frac{1}{\hat{J}_2^{(e)}}\paren{(\hat{J}_3^{(e)})^2
-\hat{J}_1^{(e)}\hat{J}_2^{(e)} } =-0.339932.
\end{equation}
This sign ensures that projection away from those two directions is
sufficient to point us away from the negative eigenvalue, rendering
$\overline{\mathcal{L}}_-^{(e)}\geq 0$.
\section{Discussion}
\label{s:discussion}
We have demonstrated a computer assisted algorithm for proving the
positivity of a bilinear form, $\mathcal{B}$, on a subspace $\mathcal{U}$.
Because of the relationship between $\mathcal{B}$, $\mathcal{L}$, and the
linearized operator, $JL$, we infer that there are no embedded
eigenvalues. We succeeded with this program in the case of the 3d
cubic equation, and a two supercritical 1d problems. C.~Sulem has
suggested t that is likely to also be successful for solitons (with
$\lambda = 1$) of the 3d cubic-quintic equation,
\[
i \psi_t + \Delta \psi + \abs{\psi}^2 \psi - \gamma \abs{\psi}^4 \psi = 0
\]
for $\gamma$ sufficiently close to zero. We also anticipate success
for other 1d supercritical problems with $\sigma$ sufficiently large.
These cases warrant further study.
For subcritical problems, a similar algorithm should apply, though it
will certainly require a additional orthogonality
conditions. Many subcritical problems contian eigenvalues inside the
spectral gap. Since our approach does not distinguish between
embedded eigenvalues and imaginary eigenvalues in the gap, it would be
essential to project away from those states.
It remains to be seen how to extend our technique to other NLS/GP
equations. Indeed, the failure in the 1D critical problem is
curious. The success or failure of the approach is likely related to
the choice of our operator $\Lambda = d/2 + \mathbf{x}
\cdot \nabla$. In \cite{FMR}, the authors proved the spectral property
using this $\Lambda$, as it is generated by the scaling invariance of
the mass critical problem. This results in the so-called
``pseudoconformal invariant'' for critical NLS and has great
implications for blow-up. See \cite{MR-Annals}
and \cite{sulem1999nse} for additional details.
Finally, recall that $\Lambda$ determines the operators $\mathcal{L}_\pm$. These
each have an index identifying the number of negative' directions. We
then choose orthogonality conditions that simultaneously must satisfy
the two properties:
\begin{enumerate}
\item They must be orthogonal to any embedded eigenvalues,
\item Orthogonality in $L^2$ with respect to these directions must
imply orthogonality to the negative directions of $\mathcal{L}_\pm$, with
respect to the $\mathcal{B}_\pm$ quadratic form.
\end{enumerate}
The first requirement is satisfied by the vectors from the adjoint
problem, as discussed in Section \ref{sec:verification}. We appear to
have little flexibility in altering these. Changing
$\Lambda$ will change $\mathcal{L}_\pm$; in turn this changes the negative
directions. Thus, a different skew adjoint operator may extend the
applicability of the algorithm.
|
1,108,101,565,505 | arxiv | \section{Introduction}
\label{sec:Intro}
Many important environmental and biological phenomena that occur in our daily lives involve fluids partly flowing freely and partly filtrating through a porous medium. Such processes are of interest in physiology when studying cancer growth and filtration of blood through arterial vessel walls. The free/porous flow plays an important role in numerous industrial applications of air and oil filter design, oil exploration, or chemical reactor simulations. There is also considerable environmental interest in studies of groundwater remediation, geologic CO$_2$ sequestration, and bacterial biofilms.
Due to its wide applicability, the modeling of free/porous flow has recently received considerable attention, and mathematical and numerical analyses have been done. The flow is often modeled by the incompressible Stokes equations in the free domain and the Darcy equations in the porous domain, coupled across the interface through suitable conditions. The numerical solutions of these flows have been almost exclusively based on the finite element method~\cite{sun21, li18, discacciati18, chid16, hanspal13, chen11, burman07, correa09, arbogast07, discacciati02, discacciati07, galvis06, guest06, hanspal06, layton03, mardal02, masud02}; a singularity method was used to obtain the solution in~\cite{elasmi01, sekhar00}, and a MAC scheme in~\cite{shiue18}. There are also recently developed partitioned methods for time-dependent Stokes-Biot problems, which utilize Robin-type coupling conditions at the interface~\cite{bukac15, ambart19}.
Despite the significant efforts in recent years to design numerical methods of coupled free/porous flows, these problems remain a challenge today, owing to the multiphysics nature of the system and the large differences in the physical parameters in the governing equations. The severe limitations of the direct solution necessitate the development of techniques that would enable one to split the system and solve the smaller Stokes and Darcy problems separately while exchanging information through the boundary conditions only. These techniques, called domain decomposition methods (DDM), have been used to solve various problems~\cite{quarteroni99}. DDMs are based on dividing the computational domain into several subdomains with or without overlap, and formulating the so-called transmission conditions along the subdomain interfaces. The coupling across the interface is then replaced by an iterative procedure, where the smaller problems are solved independently at each iteration. For the coupled flow problem, similar techniques have been developed and investigated in the context of the finite element method, e.g.~\cite{sun21, cao14, vassilev14, quarteroni99, discacciati02, discacciati07, galvis06}. In particular, we are interested in the Dirichlet-Neumann iterative methods~\cite{quarteroni99, discacciati02}, where the interface conditions are split between the Stokes and Darcy problems.
In this paper we develop a simple sequential iterative algorithm inspired by the Dirichlet-Neumann methods to decouple the free and porous problems, and formulate highly accurate second-kind boundary integral equations to solve for the fluid quantities. Boundary integral equation methods (BIEM) are extremely powerful in solving many differential equations~\cite{atkinson97, colton98, kress99, hsiao08}, and their importance is well recognized. They have been extensively applied to solve Stokes problems~\cite{pozrikidis92, cortez01, cortez05, hsiao08, greengard96}, as well as porous media flows~\cite{liggett83, rungamornrat06, lough98, pozrikidis03}. In two dimensions, a Robin-Robin DDM was designed for the Stokes-Darcy system and boundary integral equations were used to solve the local problems~\cite{boubendir-tlupova-13}. Non-local operators based on the integral formulations were used in the Robin transmission conditions, leading to a significant improvement in convergence of the DDM. This method is more efficient in overcoming the challenges of the direct solution of~\cite{tlupova09, boubendir09}. However, to the best of our knowledge, no constructive and efficient way of solving the coupled free/porous system in 3D has been developed using BIEM. Attempting a direct solution of the coupled Stokes-Darcy system with BIEM will lead to a formulation that is inevitably of mixed-kind, due to the incompatibility of the Stokes and Darcy differential operators. Once the system is split through the DDM procedure however, suitable formulations can be applied to each problem. The significant advantages of BIEM are the reduced dimensionality as the boundary alone needs to be discretized, high achievable accuracy at points on and off the boundary, the availability of well-conditioned formulations where the efficiency of the solution is maintained with grid refinement, and efficient solutions for problems where the free domain is large or infinite, with boundaries that have complicated geometry, as well as moving boundaries.
To address the singularities that develop in the integrands when solving the integral equations, we use a regularization technique developed in~\cite{beale01, beale04} for Laplace's equation, and recently developed for the Stokes equations in~\cite{tlupova18}. The smoothing can be chosen to have high order on the boundary, so that the accuracy in solving the integral equations generally approaches $O(h^5)$, where $h$ is the grid size in the coordinate planes. This approach is quite simple in that it does not require any special quadrature for evaluating the integrals on the boundary. To discretize the integrals a quadrature rule for closed surfaces from~\cite{wilson, beale16} is applied. This quadrature rule works well for general surfaces without requiring coordinate charts, using projections on coordinate planes instead. As the weights for the quadrature points are found from a partition of unity on the unit sphere, they do not depend on the particular surface. With this approach, the overall formulation is simple to implement, and the data structure needed to describe the boundary is minimal.
The paper is organized as follows. In Section~\ref{sec:ProblemFormulation}, we state the governing equations and suitable interface conditions for a Stokes-Darcy system. In Section~\ref{sec:DDM}, we outline a Dirichlet-Neumann type iterative procedure to split and sequentially solve the Darcy and Stokes problems. A convergence analysis of this algorithm for a spherical geometry based on spherical harmonics representations is conducted in Section~\ref{sec:Analysis}. In Section~\ref{sec:BIEM}, a boundary integral formulation is developed for both Darcy and Stokes equations, including a description of the treatment of singularities in the integrands in Section~\ref{sec:Regularization} and the quadrature in Section~\ref{sec:Quadrature}. The complete algorithm is presented in Section~\ref{sec:Algorithm} and the discrete iteration operator is computed in Section~\ref{sec:DiscIterOper}. Numerical results are presented in Section~\ref{sec:NumericalResults}, where we use a known analytical solution for a viscous flow around a porous sphere as a benchmark problem, as well as test the algorithm on other geometries.
\section{Problem formulation}
\label{sec:ProblemFormulation}
To describe the model equations, we denote the free flow quantities and the porous domain quantities by subscripts $S$ and $D$, respectively. Figure~\ref{Schematic} shows an example two-dimensional setting, where a free fluid flow in a bounded region $\Omega_S$ and a flow in a porous region $\Omega_D$ are coupled across the common interface $\Sigma$.
\begin{figure}[h]
\begin{centering}
\scalebox{0.75}{\includegraphics{figure1.pdf}}
\caption{The domain with the free fluid region $\Omega_S$ and the porous medium region $\Omega_D$, with the common interface $\Sigma$.}
\label{Schematic}
\end{centering}
\end{figure}
In the region of free fluid flow $\Omega_S$, we assume the low Reynolds number regime modeled by the incompressible Stokes equations,
\begin{equation}
\label{StokesEqs1}
\textrm{In} \ \Omega_{S} :
\left \{ \begin{array}{l}
\nabla p_S - \mu \Delta\bd{u}_S = 0, \\[2pt]
\nabla \cdot \bd{u}_S = 0,
\end{array}\right.
\end{equation}
where $\mu$ is the fluid viscosity, $p_S$ and $\bd{u}_S$ are the fluid pressure and velocity, respectively. Darcy's equations govern the flow in the porous medium,
\begin{equation}
\label{DarcyEqs1}
\textrm{In} \ \Omega_{D} :
\left \{ \begin{array}{l}
\nabla p_D + \mu K^{-1}\bd{u}_D = 0, \\[2pt]
\nabla \cdot \bd{u}_D = 0.
\end{array}\right.
\end{equation}
Here $\bd{u}_D$ and $p_D$ are respectively the (averaged) fluid velocity and the hydrostatic pressure, and $K$ is a symmetric and positive definite permeability tensor that depends on the microstructure of the porous medium. The medium is called homogeneous if the permeability $K(\bd{x})$ is constant throughout the domain, and isotropic when permeability is a scalar, $K = \kappa I$. In this work, we treat porous media that are homogeneous and isotropic.
The interface between these two systems of partial differential equations serves as a replacement of the boundary layer through which the velocity changes rapidly. A lot of attention has been devoted to the formulation and analysis of appropriate coupling conditions~\cite{beavers67,saffman71, jager96, jager00, ochoa-tapia95_1, ochoa-tapia95_2, sahraoui92}. The incompressibility condition leads to continuity of normal velocity components,
\begin{equation}
\label{Vel_contin}
\bd{u}_S \cdot \bd{n}_S = -\bd{u}_D \cdot \bd{n}_D,
\end{equation}
where $\bd{n}_S$ and $\bd{n}_D$ are the unit normal vectors that point out of the regions $\Omega_D$ and $\Omega_S$ respectively, so that $\bd{n}_D = -\bd{n}_S$ on $\Sigma$ (see Fig.~\ref{Schematic}). In addition, a suitable condition on the tangential velocity is the slip condition of Beavers-Joseph-Saffman~\cite{beavers67, saffman71},
\begin{equation}
\label{BeaversJoseph}
\bdg{\tau}_S\cdot\bdg{\sigma}_S\cdot\bd{n}_S = \frac{\gamma}{\sqrt{\kappa}}\bd{u}_S \cdot \bdg{\tau}_S,
\end{equation}
where $\bdg\sigma_S = -p_S I+\mu[\nabla\bd{u}_S + (\nabla\bd{u}_S)^T]$ is the Stokes stress tensor, $\bdg{\tau}_S$ is the tangent vector to the interface, and $\gamma$ is a dimensionless quantity that depends on the structure of the porous material. Well-posedness of the mathematical formulation with this condition has been analyzed for steady flows~\cite{burman07, discacciati02, galvis06, layton03} as well as unsteady flows~\cite{cao09}. The final condition represents the balance of normal forces across the interface,
\begin{equation}
\label{Press_contin}
\bd{n}_S\cdot\bdg{\sigma}_S\cdot\bd{n}_S = -p_D,
\end{equation}
which allows the pressure to be discontinuous across the interface.
A fundamental difficulty in the development of robust computational methods for the free/porous flow system~\eqref{StokesEqs1}-\eqref{Press_contin} is due to the fact that the PDEs that govern the fluid flow in the free region and the flow in the porous medium have different properties. The differential operators in the two subdomains are incompatible, inhibiting the simultaneous solution of the equations by a direct method. In addition, the equations include parameters, the viscosity $\mu$ and permeability $\kappa$, that could differ by several orders of magnitude in practice.
\section{The Dirichlet-Neumann iterative method}
\label{sec:DDM}
The severe limitations of the direct solution necessitate the development of techniques that would enable one to split the system and solve the smaller Stokes and Darcy problems separately while exchanging information through the boundary conditions only. We develop a numerical scheme based on the sequential Dirichlet-Neumann (D-N) iterative method~\cite{quarteroni99, discacciati02}. The outline of the algorithm is as follows. First, we define $q = -\bd{u}_S\cdot \bd{n}_S$ as the iteration variable on the interface $\Sigma$, and choose an initial guess $q^{(0)} = 0$. Then, for $k=1,2,3,...$ until convergence, perform the following steps:
\begin{enumerate}
\item Solve the Darcy problem with the interface condition~\eqref{Vel_contin},
\begin{equation}
\label{DarcyProblem1}
\left \{ \begin{array}{ll}
\nabla p^{(k)}_D + \mu \kappa^{-1}\bd{u}^{(k)}_D = 0 & \textrm{in} \ \Omega_{D}, \\[2pt]
\nabla \cdot \bd{u}^{(k)}_D = 0 & \textrm{in} \ \Omega_{D},\\[2pt]
\bd{u}^{(k)}_D \cdot \bd{n}_D = q^{(k-1)} & \textrm{on} \ \Sigma.
\end{array}\right.
\end{equation}
The interface condition is a Neumann condition for $p_D$.
\item Using $p^{(k)}_D$ from Step 1, solve the Stokes problem with the interface conditions~\eqref{BeaversJoseph}-\eqref{Press_contin},
\begin{equation}
\label{StokesProblem1}
\left \{ \begin{array}{ll}
\nabla p^{(k)}_S - \mu \Delta\bd{u}^{(k)}_S = \bd{0} & \textrm{in} \ \Omega_{S}, \\[2pt]
\nabla \cdot \bd{u}^{(k)}_S = 0 & \textrm{in} \ \Omega_{S},\\[2pt]
\bdg{\tau}_S\cdot\bdg{\sigma}^{(k)}_S\cdot\bd{n}_S = \frac{\gamma}{\sqrt{\kappa}}\bd{u}^{(k-1)}_S\cdot\bdg{\tau}_S & \textrm{on} \ \Sigma,\\[2pt]
\bd{n}_S\cdot\bdg{\sigma}^{(k)}_S\cdot\bd{n}_S = -p^{(k)}_D & \textrm{on} \ \Sigma.
\end{array}\right.
\end{equation}
\item Update the iteration variable
\begin{equation}
\label{q_update}
q^{(k)} = (1-\theta) \, q^{(k-1)} - \theta \, \bd{u}^{(k)}_S\cdot \bd{n}_S
\end{equation}
using $\bd{u}^{(k)}_S$ from Step 2, where $\theta \in (0,1)$ is a relaxation parameter.
\end{enumerate}
The convergence criterion is taken as the relative error between $q^{(k+1)}$ and $q^{(k)}$, which we call the DDM residual, falling below a prescribed tolerance. The solutions of the now separate Darcy \eqref{DarcyProblem1} and Stokes \eqref{StokesProblem1} problems are based on the boundary integral formulations which we describe in Section~\ref{sec:BIEM}.
\section{Convergence analysis}
\label{sec:Analysis}
We analyze the convergence properties of the Dirichlet-Neumann iterative scheme using the spherical harmonics representation of both problems, assuming as the interface a sphere of radius $R$ centered at the origin, with the Darcy problem inside and the Stokes problem outside the sphere.
\subsection{Darcy problem inside a sphere}
We consider the problem
\begin{equation}
\label{CA1}
\left \{ \begin{array}{ll}
\tilde{\kappa}\, \nabla p + \bd{u} = 0 & \textrm{in} \ \Omega, \\[2pt]
\nabla \cdot \bd{u} = 0 & \textrm{in} \ \Omega,\\[2pt]
\bd{u} \cdot \bd{n} = q & \textrm{on} \ \Sigma,
\end{array}\right.
\end{equation}
where $\Omega$ is the sphere of radius $R$ centered at the origin, $\Sigma$ is its boundary, $\bd{n}$ is the outward unit normal vector, and $\tilde{\kappa} = \kappa/\mu$.
In spherical coordinates $(r,\theta,\phi)$, the general solution to Laplace's equation $\Delta p=0$ in $\Omega$ can be written as
\begin{equation}
\label{CA2}
p = \sum_{n=0}^\infty p_n,
\end{equation}
with
\begin{equation}
\label{CA3}
p_n = \sum_{m=-n}^n a_n^m\, r^n\, Y_n^m(\theta,\phi),
\end{equation}
where $p_n$ is a solid harmonic of order $n$, $Y_n^m$ is the spherical harmonic function of degree $n$ and order $m$, and $a_n^m\in\mathbb{C}$ are constants.
With $\bd{n}=\bd{x}/r$ and
\begin{equation}
\label{CA4}
\nabla p = \sum_{n=0}^\infty \sum_{m=-n}^n \left[ \frac{\partial}{\partial r} (a_n^m\, r^n)\, \frac{\bd{x}}{r} \, Y_n^m(\theta,\phi) + a_n^m\, r^n\, \nabla Y_n^m(\theta,\phi)\right],
\end{equation}
for the normal velocity we get
\begin{equation}
\label{CA5}
\bd{u}\cdot\bd{n} = -\tilde{\kappa}\, \nabla p \cdot \bd{n} = -\tilde{\kappa} \sum_{n=0}^\infty \sum_{m=-n}^n \left[ n\, a_n^m\, r^{n-1}\, Y_n^m(\theta,\phi) \right].
\end{equation}
Using a similar representation for $q$ on $\Sigma$,
\begin{equation}
\label{CA6}
q = \sum_{n=0}^\infty \sum_{m=-n}^n q_n^m\, Y_n^m(\theta,\phi),
\end{equation}
and imposing the Neumann boundary condition in~\eqref{CA1} on $r=R$, we find for the coefficients $a_n^m$,
\begin{equation}
\label{CA7}
-\tilde{\kappa}\, n\, a_n^m\, R^{n-1} = q_n^m.
\end{equation}
\subsection{Stokes problem outside a sphere}
We now consider the solution of the Stokes problem with a no-slip boundary condition for simplicity,
\begin{equation}
\label{CA8}
\left \{ \begin{array}{ll}
\nabla p - \mu\, \Delta\bd{u} = 0 & \textrm{in} \ \Omega^c, \\[2pt]
\nabla \cdot \bd{u} = 0 & \textrm{in} \ \Omega^c,\\[2pt]
\bd{u}\cdot\bdg{\tau} = 0 & \textrm{on} \ \Sigma,\\[2pt]
\bd{n}\cdot \bdg{\sigma} \cdot \bd{n} = -p_D & \textrm{on} \ \Sigma,
\end{array}\right.
\end{equation}
where $p_D$ is the Darcy pressure on $\Sigma$ and $\Omega^c$ is the exterior of the sphere of radius $R$.
The solution to the Stokes equations in spherical harmonics is referred to as Lamb's general solution~\cite{lamb32}. Since the Stokes pressure satisfies Laplace's equation, it can be represented similar to~\eqref{CA2}-\eqref{CA3}, but for the exterior solution we keep only the negative harmonics,
\begin{equation}
\label{CA9}
p = \sum_{n=0}^\infty p_{-n-1},
\end{equation}
with
\begin{equation}
\label{CA10}
p_{-n-1} = \sum_{m=-n}^n b_n^m\, r^{-n-1}\, Y_n^m(\theta,\phi).
\end{equation}
The disturbance velocity field is represented by (see~\cite{kim-karrila05})
\begin{equation}
\label{CA11}
\bd{u} = \sum_{n=1}^\infty \left[ -\frac{(n-2)\, r^2\, \nabla p_{-n-1}}{2\mu\, n (2n-1)} + \frac{(n+1)\, \bd{x}\, p_{-n-1}}{\mu\, n (2n-1)} + \nabla \phi_{-n-1} + \nabla \times (\bd{x}\, \chi_{-n-1}) \right],
\end{equation}
where
\begin{equation}
\label{CA12}
\phi_{-n-1} = \sum_{m=-n}^n c_n^m\, r^{-n-1}\, Y_n^m(\theta,\phi),
\end{equation}
\begin{equation}
\label{CA13}
\chi_{-n-1} = \sum_{m=-n}^n d_n^m\, r^{-n-1}\, Y_n^m(\theta,\phi).
\end{equation}
The first two terms in~\eqref{CA11} correspond to a particular solution, while the rest corresponds to the homogeneous solution, constructed from a potential (the $\nabla\phi$ term) and a toroidal field (the $\nabla \times (\bd{x}\, \chi)$ term)~\cite{kim-karrila05}.
The stress vector acting across the sphere can be expressed in general as (see~\cite{happel-brenner83} equation (3-2.37) or~\cite{kim-karrila05} exercise 4.6, p.104),
\begin{eqnarray}
\label{CA14}
\bdg{\sigma}_n = \bdg{\sigma}\cdot \bd{n} = \frac{1}{r} \sum_{n=-\infty}^\infty \left[ \frac{n(n+2)\, r^2\, \nabla p_n}{(n+1) (2n+3)} - \frac{(2n^2+4n+3)\, \bd{x}\, p_n}{(n+1) (2n+3)} \right] \nonumber\\
+ \frac{\mu}{r} \sum_{n=-\infty}^\infty \left[ 2(n-1)\nabla \phi_n + (n-1)\nabla \times (\bd{x}\, \chi_n) \right].
\end{eqnarray}
For the exterior problem, we again only keep the negative harmonics and replace $n$ in~\eqref{CA14} with $-n-1$, to obtain
\begin{eqnarray}
\label{CA15}
\bdg{\sigma}_n = \bdg{\sigma}\cdot \bd{n} = \frac{1}{r} \sum_{n=1}^\infty \left[ \frac{(n-1)(n+1)\, r^2\, \nabla p_{-n-1}}{n (2n-1)} - \frac{(2n^2+1)\, \bd{x}\, p_{-n-1}}{n (2n-1)} \right] \nonumber\\
- \frac{\mu}{r}\, \sum_{n=1}^\infty \left[ 2(n+2)\nabla \phi_{-n-1} + (n+2)\nabla \times (\bd{x}\, \chi_{-n-1}) \right].
\end{eqnarray}
We now turn to the boundary conditions. The no-slip condition on $r=R$ implies (see~\cite{kim-karrila05} p.89) that $\chi$ is not needed and
\begin{equation}
\label{CA16}
\sum_{n=1}^\infty \left[ -\frac{n(n+1)\, R}{2\mu\, (2n-1)}\, p_{-n-1} |_{r=R} + \frac{(n+1)(n+2)}{R} \phi_{-n-1} |_{r=R} \right] = 0,
\end{equation}
which leads to
\begin{equation}
\label{CA17}
\phi_{-n-1} |_{r=R} = \frac{n\, R^2}{2\mu\, (n+2)(2n-1)} p_{-n-1} |_{r=R}.
\end{equation}
Note that for the radial component of velocity we have from~\eqref{CA11},
\begin{equation}
\label{CA18}
u_r = \sum_{n=1}^\infty \left[ -\frac{(n-2)\, r^2}{2\mu\, n (2n-1)}\, \frac{\partial p_{-n-1}}{\partial r} + \frac{(n+1)\, r\, p_{-n-1}}{\mu\, n (2n-1)} + \frac{\partial \phi_{-n-1}}{\partial r} \right].
\end{equation}
Using Euler's theorem for homogeneous polynomials,
\begin{equation}
\label{CA19}
r\, \frac{\partial h_n}{\partial r} = n\, h_n,
\end{equation}
where $h_n$ is any solid harmonic of order $n$, the radial velocity in~\eqref{CA18} simplifies to
\begin{equation}
\label{CA20}
u_r = \sum_{n=1}^\infty \left[ \frac{(n+1)\, r\, p_{-n-1}}{2\mu\, (2n-1)} - \frac{(n+1)}{r} \phi_{-n-1} \right].
\end{equation}
Similarly, for the radial component of the stress vector $\bdg{\sigma}_n$,
\begin{eqnarray}
\label{CA21}
\sigma_{nr} = \frac{1}{r} \sum_{n=1}^\infty \left[ \frac{(n-1)(n+1)\, r^2}{n (2n-1)}\, \frac{\partial p_{-n-1}}{\partial r} - \frac{(2n^2+1)\, r\, p_{-n-1}}{n (2n-1)} \right] \nonumber\\
- \frac{\mu}{r}\, \sum_{n=1}^\infty \left[ 2(n+2) \frac{\partial \phi_{-n-1}}{\partial r} \right],
\end{eqnarray}
which simplifies again using~\eqref{CA19},
\begin{equation}
\label{CA22}
\sigma_{nr} = \sum_{n=1}^\infty \left[ -\frac{(n^2+3n-1)}{(2n-1)} p_{-n-1} + \frac{2\mu}{r^2} (n+1)(n+2) \phi_{-n-1} \right].
\end{equation}
On the boundary $r=R$ using~\eqref{CA17}, this becomes
\begin{equation}
\label{CA23}
\sigma_{nr}|_{r=R} = \sum_{n=1}^\infty \left[ - p_{-n-1} \right]_{r=R},
\end{equation}
and therefore enforcing the boundary condition on the normal stress
\begin{equation}
\label{CA24}
\bdg{\sigma}_n\cdot\bd{n} = \sigma_{nr}|_{r=R} = -p_D
\end{equation}
leads to
\begin{equation}
\label{CA25}
\sum_{n=1}^\infty \sum_{m=-n}^n b_n^m\, R^{-n-1}\, Y_n^m(\theta,\phi) = \sum_{n=1}^\infty \sum_{m=-n}^n a_n^m\, R^n\, Y_n^m(\theta,\phi).
\end{equation}
Using~\eqref{CA7}, we write
\begin{equation}
\label{CA26}
b_n^m = -\frac{R^{n+2}}{\tilde{\kappa}\, n} q_n^m.
\end{equation}
\subsection{Spectrum of the Dirichlet-Neumann procedure}
We now diagonalize the iteration operator of the Dirichlet-Neumann method defined on the sphere. To derive the spectrum of the iteration operator, note that the iteration variable $q$ is updated at each iteration according to~\eqref{q_update}, so we need the normal component of Stokes velocity on the sphere. We evaluate the radial velocity~\eqref{CA20} on the boundary $r=R$ and use~\eqref{CA17} to get for the normal Stokes velocity,
\begin{equation}
\label{CA27}
u_r |_{r=R} = \sum_{n=1}^\infty \left[ \frac{(n+1)\, R}{\mu\, (n+2)(2n-1)} p_{-n-1}|_{r=R} \right].
\end{equation}
Using~\eqref{CA26}, this becomes
\begin{equation}
\label{CA28}
u_r |_{r=R} = -\frac{R^2}{\kappa} \sum_{n=1}^\infty \left[ \frac{(n+1)}{n(n+2)(2n-1)} \sum_{m=-n}^n q_n^m\, Y_n^m(\theta,\phi) \right].
\end{equation}
For the DDM update we get
\begin{equation}
\label{CA29}
q^{(k)} = (1-\theta) q^{(k-1)} + \theta u_r^{(k)} |_{r=R},
\end{equation}
which we write in the form of the diagonalized iteration operator $q^{(k)} = \mathcal{A} q^{(k-1)}$ as
\begin{equation}
\label{CA30}
q_n^{m(k)} = \mathcal{A}_n q_n^{m(k-1)},
\end{equation}
where
\begin{equation}
\label{CA31}
\mathcal{A}_n = (1-\theta) - \theta\frac{R^2}{\kappa} \frac{(n+1)}{n(n+2)(2n-1)}, \qquad n=1,2,3,...
\end{equation}
Figure~\ref{Sphere_modes} shows these coefficients for $n=1,...,50$ for different values of permeability $\kappa$ as well as different $\theta$. We can observe that for $\kappa=1$, a larger value of $\theta$ leads to smaller coefficients $\mathcal{A}_n$ and therefore the iterations are expected to converge faster, as we will show in the numerical tests. For smaller values of $\kappa$ however, the coefficients $\mathcal{A}_n$ get close to 1 as $n$ increases, independent of the value of $\theta$. This can hinder the convergence rate of the algorithm when using a successive approximation technique as in~\eqref{q_update}. We will show in our numerical tests that this indeed is what happens, and switch to using a more efficient method, and one that does not require all eigenvalues to be less than 1, such as GMRES to help mitigate this issue. Interesting approaches that will be investigated in future work are (i) different decoupling, e.g. Robin-Robin transmission conditions, and (ii) using in place of $\theta$ non-local operators based on the boundary integral formulation to speed up the convergence rates. Similar techniques were shown to dramatically speed up the iterations in two dimensions~\cite{boubendir-tlupova-13}.
\begin{figure}[htb]
\begin{centering}
\scalebox{0.27}{\includegraphics{rad1_modes_k1.eps}}
\scalebox{0.27}{\includegraphics{rad1_modes_k1e-2.eps}}
\scalebox{0.27}{\includegraphics{rad1_modes_k1e-4.eps}}
\caption{Harmonic coefficients $\mathcal{A}_n$ from~\eqref{CA31} of the iteration operator on the unit sphere, for permeability values $\kappa=1, 10^{-2}, 10^{-4}$.}
\label{Sphere_modes}
\end{centering}
\end{figure}
\section{Boundary integral formulations}
\label{sec:BIEM}
There are two main approaches to formulating boundary integral equations, usually referred to as the 'direct' and 'indirect' approaches. The direct approach is based on the Green's representation formula and is the approach we take in this work. While the analytic properties of the integral operators are the same with both approaches, our primary interest in the direct approach is that the resulting equations are for the physically relevant Dirichlet or Neumann data.
\subsection{Darcy problem}
The Green's representation formula,
\begin{equation}
\label{Greens}
p_D(\bd{y}) = - \int_{\partial\Omega_D} \frac{\partial p_D(\bd{x})}{\partial n(\bd{x})} \,G(\bd{y,x})\, dS(\bd{x}) + \int_{\partial\Omega_D} p_D(\bd{x}) \, \frac{\partial G(\bd{y,x})}{\partial n(\bd{x})} \, dS(\bd{x}) , \qquad \bd{y}\in\Omega_D,
\end{equation}
represents any harmonic function $p_D$ in terms of its Cauchy data, that is, its boundary values and its normal derivative on the boundary. The first integral in~\eqref{Greens} is the single-layer potential and the second integral is the double-layer potential, where the kernels are $G(\bd{y,x}) = -1/4\pi |\bd{y} - \bd{x}|$ and its derivative in the outward normal $\frac{\partial G(\bd{y,x})}{\partial n(\bd{x})} = \nabla_\bd{x} G(\bd{y},\bd{x})\cdot \bd{n}(\bd{x}) = -(\bd{y}-\bd{x})\cdot \bd{n}(\bd{x})/4\pi |\bd{y}-\bd{x}|^3$. The double layer potential undergoes a discontinuity across the boundary, and the jump relations
\begin{equation}
\label{Jump}
\int_{\partial\Omega_D} \frac{\partial G(\bd{y,x})}{\partial n(\bd{x})} \, dS(\bd{x})
= \left\{ \begin{array}{ll}
1, & \bd{y}\in\Omega_D\\
1/2, & \bd{y}\in\partial\Omega_D\\
0, & \bd{y}\in\mathbb{R}^3 \setminus \bar{\Omega}_D
\end{array}\right.
\end{equation}
are used to extend~\eqref{Greens} to the boundary,
\begin{equation}
\label{Greens_bdry}
\frac12 p_D(\bd{y}) = - \int_{\partial\Omega_D} \frac{\partial p_D(\bd{x})}{\partial n(\bd{x})} \,G(\bd{y,x})\, dS(\bd{x}) + \int_{\partial\Omega_D} p_D(\bd{x}) \, \frac{\partial G(\bd{y,x})}{\partial n(\bd{x})} \, dS(\bd{x}) , \qquad \bd{y}\in\partial\Omega_D.
\end{equation}
We write~\eqref{Greens_bdry} as a second-kind integral equation assuming known Neumann data $\partial p_D/\partial n$ on the boundary,
\begin{equation}
\label{Darcy_BIEM}
\frac12 p_D = H p_D + b_D, \qquad \qquad b_D = -L \frac{\partial p_D}{\partial n},
\end{equation}
where by $L$ and $H$ we denote the integral operators corresponding to the single- and double-layer potentials, respectively. The integral equation~\eqref{Darcy_BIEM} can be solved by successive approximations, otherwise known as a Neumann's iteration scheme (see, for example,~\cite{kress99} or~\cite{hsiao08}, Sec. 5.6.7),
\begin{equation}
\label{Darcy_BIEM_iter}
p_D^{(n+1)} = \frac12 p_D^{(n)} + H p_D^{(n)} + b_D, \qquad n=0,1,2,...
\end{equation}
\subsection{Stokes problem}
We follow a similar 'direct' approach for the Stokes problem and use the following integral representation for the velocity,
\begin{equation}
\label{Stokes}
\bd{u}_S(\bd{y}) = \bd{u}^\infty -\frac{1}{8\pi}\int_{\partial\Omega_S} S(\bd{y,x})\, \bd{f}_S(\bd{x}) \, dS(\bd{x}) + \frac{1}{8\pi}\int_{\partial\Omega_S} \bd{u}_S(\bd{x}) \cdot T(\bd{y,x}) \cdot \bd{n}_S(\bd{x}) \, dS(\bd{x}) , \quad \bd{y}\in\Omega_S,
\end{equation}
where $\bd{n}_S$ is the unit outward normal vector to the Stokes domain boundary, $\bd{f}_S = \bdg{\sigma}_S\cdot\bd{n}_S$ is the surface force, or traction, on the boundary $\partial\Omega_S$, $\bdg{\sigma}_S$ is the stress tensor, and
\begin{equation}
S_{ij}(\bd{y,x}) = \frac{\delta_{ij}}{|\bd{\hat{y}}|} + \frac{\hat{y}_i \ \hat{y}_j}{|\bd{\hat{y}}|^3}, \qquad \qquad
T_{ijk} (\bd{y,x}) = -6 \frac{ \hat{y}_i\hat{y}_j\hat{y}_k}{|\bd{\hat{y}}|^5},
\end{equation}
are the Stokeslet and stresslet fundamental solutions of the Stokes equations, where $\bd{\hat{y}} = \bd{y}- \bd{x}$, $\delta_{ij}$ is the Kronecker delta, and $i,j,k = 1,2,3$ are Cartesian coordinates. A background flow $\bd{u}^\infty$ is added for external flows.
The representation~\eqref{Stokes} is based on the Lorentz reciprocal theorem, and again, the first integral is called the single-layer potential and the second integral is the double-layer potential of Stokes flow. The double-layer undergoes a discontinuity across the boundary much like~\eqref{Jump}, and we again use jump conditions to continuously extend the representation~\eqref{Stokes} to the boundary:
\begin{equation}
\label{Stokes_bdry}
\frac12 \bd{u}_S(\bd{y}) = \bd{u}^\infty -\frac{1}{8\pi}\int_{\partial\Omega_S} S(\bd{y,x})\, \bd{f}_S(\bd{x}) \, dS(\bd{x}) + \frac{1}{8\pi}\int_{\partial\Omega_S} \bd{u}_S(\bd{x}) \cdot T(\bd{y,x}) \cdot \bd{n}_S(\bd{x}) \, dS(\bd{x}) , \quad \bd{y}\in\partial\Omega_S.
\end{equation}
We write~\eqref{Stokes_bdry} as
\begin{equation}
\label{Stokes_BIEM}
\frac12 \bd{u}_S = K \bd{u}_S + b_S, \qquad \qquad b_S = -M \bd{f}_S + \bd{u}^\infty,
\end{equation}
where by $M$ and $K$ we denote the integral operators corresponding to the Stokes single- and double-layer potentials, respectively. Assuming a known surface force, equation~\eqref{Stokes_BIEM} is a second-kind integral equation for $\bd{u}_S$, and we solve it by successive approximations
\begin{equation}
\label{Stokes_BIEM_iter}
\bd{u}_S^{(n+1)} = \frac12 \bd{u}_S^{(n)} + K \bd{u}_S^{(n)} + b_S, \qquad n=0,1,2,...
\end{equation}
It is a well-known challenge in the numerical solution of the integral equations~\eqref{Darcy_BIEM_iter} and~\eqref{Stokes_BIEM_iter} that singularities develop in the integrands $H, L, K, M$, and we address this issue by using a high accuracy regularization technique which we describe next.
\subsection{High-order regularization of kernels on the surface}
\label{sec:Regularization}
The integrands in~\eqref{Greens_bdry} and~\eqref{Stokes_bdry} exhibit singularities for $\bd{y}=\bd{x}$. To address this issue, we first use subtraction in the double layer integrals to reduce the singularities,
\begin{subequations}
\begin{align}
\label{H_subtr}
(H p_D) (\bd{y}) &= \int_{\partial\Omega_D} [p_D(\bd{x}) - p_D(\bd{y})] \, \frac{\partial G(\bd{y,x})}{\partial n(\bd{x})} \, dS(\bd{x}) + \frac12 p_D(\bd{y}), \quad \bd{y}\in\partial\Omega_D, \\[6pt]
\label{K_subtr}
(K \bd{u}_S) (\bd{y}) &= - \frac{1}{8\pi}\int_{\partial\Omega_S} [\bd{u}_S(\bd{x}) - \bd{u}_S(\bd{y})] \cdot T(\bd{y,x}) \cdot \bd{n}(\bd{x})dS(\bd{x}) - \frac12 \bd{u}_S(\bd{y}), \quad \bd{y}\in\partial\Omega_S.
\end{align}
\end{subequations}
Next, we apply the regularization method of~\cite{beale01, beale04} for Laplace's equation, and recently developed for the Stokes equations in~\cite{tlupova18}. The approach is to replace $1/r^p$, where $r=|\bd{\hat{y}}|$, by a smooth version $s(r/\delta)/r^p$, where $s(r/\delta)$ is a radial smoothing function and $\delta>0$ is a smoothing, or regularization, parameter. As we are interested in evaluating the integrals on the boundary, the smoothing factor $s$ can be chosen to have high order, so that the accuracy in solving the integral equations approaches $O(h^5)$, where $h$ is the grid size in the coordinate planes. The derivation of these smoothing functions can be found in~\cite{beale01, beale04, tlupova18}. For completeness, we state the regularization formulas for both the Darcy and Stokes problems. The kernels in the Darcy representation~\eqref{Greens_bdry} are replaced with their regularized versions,
\begin{equation}
G^\delta(\bd{y,x}) = -\frac{s_1(r/\delta)}{4\pi r}, \qquad \qquad \nabla G^\delta(\bd{y,x}) = \nabla G^\delta(\bd{y,x}) s_2(r/\delta) = \frac{\bd{\hat{y}}}{4\pi r^3} s_2(r/\delta).
\end{equation}
The regularized single and double layer Stokes kernels in the representation~\eqref{Stokes_bdry} are
\begin{equation}
S^\delta_{ij}(\bd{y,x}) = \delta_{ij} \frac{s_1(r/\delta)}{r} + \hat{y}_i \ \hat{y}_j \frac{s_3(r/\delta)}{r^3}, \qquad \qquad
T^\delta_{ijk} (\bd{y,x}) = -6 \hat{y}_i\hat{y}_j\hat{y}_k \frac{ s_4(r/\delta)}{r^5}.
\end{equation}
The radial smoothing functions $s_i(r/\delta)$ are chosen to yield high order accuracy when the integrals are evaluated on the surface, with the error approaching $O(h^5)$ in discretization size $h$ as shown in~\cite{beale01, beale04, tlupova18},
\begin{subequations}
\begin{align}
s_1(r) &= \erf(r) - \frac{2}{3\sqrt{\pi}} r (2 r^2 - 5) e^{-r^2},\\
s_2(r) &= \erf(r) + \frac{2}{3\sqrt{\pi}} r (2 r^2 - 3) e^{-r^2},\\
s_3(r) &= \erf(r) - \frac{2}{3\sqrt{\pi}} r (4 r^4 - 14 r^2 + 3) e^{-r^2}, \\
s_4(r) &= \erf(r) - \frac{2}{9\sqrt{\pi}} r (8 r^6 - 36 r^4 + 6 r^2 + 9) e^{-r^2},
\end{align}
\end{subequations}
where $\erf$ is the error function. This approach is quite simple in that it does not require any special quadrature for evaluating the integrals on the boundary.
\subsection{Quadrature}
\label{sec:Quadrature}
With the integrands smoothed out, to discretize the integrals a quadrature rule for closed surfaces from~\cite{wilson, beale16} is applied, and we briefly describe it here. First, an angle $\theta$ is chosen and a partition of unity on the unit sphere is defined,
consisting of functions $\psi_1, \psi_2, \psi_3$ with $\Sigma_i \psi_i \equiv 1$ such that
$\psi_i(\bd{n}) = 0$ if $|\bd{n}\cdot\bd{e}_i| \leq \cos{\theta}$, where $\bd{e}_i$ is the
$i$th coordinate vector. Here we use $\theta = 70^o$. For mesh size $h$,
a set $R_3$ of quadrature points consists of points $\bf{x}$ on the surface of the form
$(j_1h,j_2h,x_3)$ such that $|\bd{n(x}) \cdot \bd{e}_3| \geq \cos{\theta}$, where
$\bd{n}(\bd{x})$ is the unit normal at $\bd{x}${, see Fig.~\ref{Surfaces}}. Sets $R_1$ and $R_2$ are
defined similarly. For a function $f$ on the surface the integral is computed as
\begin{equation}
\label{Quadrature}
\int_S f(\bd{x}) \,dS(\bd{x}) \approx \sum_{i=1}^3 \sum_{\bd{x} \in R_i}
\frac{\psi_i(\bd{n}(\bd{x}))\,f(\bd{x}) }{| \bd{n}(\bd{x})\cdot\bd{e}_i |} \,h^2.
\end{equation}
{The partition of unity functions $\psi_i$ are constructed from the $C^\infty$ bump function $b(r) = e^{r^2/(r^2 - 1)}$ for $|r| < 1$ and zero otherwise. The quadrature is effectively reduced to the trapezoidal rule without boundary. Thus for regular integrands the quadrature has arbitrarily high order accuracy, limited only by the degree of smoothness of the integrand and surface.} The points in $R_i$
can be found by a line search since they are well separated; see~\cite{wilson} and~\cite{beale16}.
Suppose the surface is given by $\phi(x_1, x_2, x_3) = 0$, with $\phi > 0$ outside, and the normal vector $\nabla\phi$ is predominantly in the $x_3$ direction. Then the parameterization is given by $x_3 = z(x_1, x_2)$, where $z$ is the vertical coordinate on the surface. Derivatives of this parameterization are computed implicitly by $z_i \equiv \partial z/ \partial x_i = -\phi_i/\phi_3$, the tangent vectors are defined as $\bd{T}_1 = (1, 0, z_1)$ and $\bd{T}_2 = (0, 1, z_2)$. The outward normal is $\bd{n} = \nabla\phi/|\nabla\phi |$ or $\bd{n} = \pm (-z_1,-z_2, 1)/\sqrt{1+z_1^2+z_2^2}$, where "+" is used if $x_3 > z(x_1, x_2)$ outside and "-" is used otherwise.
In Figure~\ref{Surfaces}, we show the set $R_3$ of quadrature points, given by the top and bottom coordinate grids $(j_1h,j_2h,x_3)$, for the following three surfaces:
\begin{subequations}
\begin{align}
\label{Sphere}
\phi(x_1,x_2,x_3) &= x_1^2 + x_2^2 + x_3^2 - 1, \\[6pt]
\label{Ellipsoid}
\phi(x_1,x_2,x_3) &= \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} - 1, \\[6pt]
\label{Molecule}
\phi(x_1,x_2,x_3) &= \sum_{k=1}^4 \exp(- |{\bf x} - {\bf x}_k|^2/r^2) - c.
\end{align}
\end{subequations}
For the ellipsoid~\eqref{Ellipsoid} we set $a=1,b=0.6,c=0.4$, and for the 'four-atom molecule' surface~\eqref{Molecule} we use centers $\bd{x}_1 = (\sqrt{3}/3,0,-\sqrt{6}/12)$, $\bd{x}_{2,3} = (-\sqrt{3}/6,\pm .5,-\sqrt{6}/12)$, $\bd{x}_4 = (0,0,\sqrt{6}/4)$ and $r = .5$, $c = .6$, as in~\cite{beale16}.
\begin{figure}[htb]
\begin{centering}
\scalebox{0.4}{\includegraphics{surfaces.eps}}
\caption{Quadrature points generated by $\psi_3$ and $h=1/16$ on the surface of the unit sphere~\eqref{Sphere} (top left), the ellipsoid with semiaxes $a=1,b=0.6,c=0.4$~\eqref{Ellipsoid} (top right), and the 'molecule'~\eqref{Molecule} (bottom row). The quadrature points are at the intersections of the lines.}
\label{Surfaces}
\end{centering}
\end{figure}
This quadrature rule works well for general surfaces without requiring coordinate charts, using projections on coordinate planes instead. As the weights for the quadrature points are found from a partition of unity on the unit sphere, they do not depend on the particular surface. The quadrature points can be found efficiently if, for example, the surface is given analytically or numerically as the level set of a function. With this approach, the formulation is very simple to implement, the data structure and information needed to describe the boundary are minimal, and there are no parameters to fine-tune except the regularization parameter $\delta$ (which is usually taken as a multiple of grid size $h$, see the discussion in~\cite{ tlupova18}).
\section{Algorithm}
\label{sec:Algorithm}
Now we summarize the algorithm for solving the Stokes-Darcy system. First, we note that
\begin{equation}
\frac{\partial p_D}{\partial n} = -\frac{\mu}{\kappa} \bd{u}_D \cdot \bd{n}_D,
\end{equation}
and by combining the interface conditions \eqref{BeaversJoseph} and \eqref{Press_contin},
\begin{equation}
\label{f}
\bd{f}_S = \bdg{\sigma}_S\cdot\bd{n}_S = -p_D \bd{n}_S - \frac{\gamma}{\sqrt{\kappa}} [\bd{u}_S - (\bd{u}_S\cdot \bd{n}_S)\bd{n}_S],
\end{equation}
so we rewrite the Darcy~\eqref{Darcy_BIEM} and Stokes~\eqref{Stokes_BIEM} integral equations as
\begin{subequations}
\begin{align}
\label{Darcy_final}
\frac12 p_D &= H p_D + b_D, \qquad \qquad b_D = \frac{\mu}{\kappa} L (\bd{u}_D \cdot \bd{n}_D),\\[6pt]
\label{Stokes_final}
\frac12 \bd{u}_S &= K \bd{u}_S + b_S, \qquad \qquad b_S = -M \bd{f}_S + \bd{u}^\infty,
\end{align}
\end{subequations}
We outline the algorithm of the iterative Stokes and Darcy solutions using successive approximations approach, keeping in mind that the successive approximations are eventually replaced by GMRES for a more efficient solution.
\begin{enumerate}
\item Let $k=0$, $\bd{u}_S^{(0)} = \bd{0}$ and $q^{(0)} = -\bd{u}_S^{(0)}\cdot \bd{n}_S = 0$.
\item For $k=1,2,...$ until convergence, do:
\begin{enumerate}
\item Compute the Laplace single layer potential $b_D$ in~\eqref{Darcy_final} using $\bd{u}_D\cdot \bd{n}_D = q^{(k-1)}$.
\item Solve the Darcy problem \eqref{Darcy_final} for $p_D^{(k)}$ via (i) successive approximations \eqref{Darcy_BIEM_iter} or (ii) GMRES.
\item Compute $\bd{f}_S$ as in~\eqref{f}, using $p_D^{(k)}$ and $\bd{u}^{(k-1)}_S$.
\item Compute the Stokes single layer potential $b_S$ in~\eqref{Stokes_final}.
\item Solve the Stokes problem \eqref{Stokes_final} for $\bd{u}_S^{(k)}$ via (i) successive approximations \eqref{Stokes_BIEM_iter} or (ii) GMRES.
\item Set $q^{(k)} = (1-\theta)q^{(k-1)} - \theta \bd{u}_S^{(k)}\cdot \bd{n}_S$.
\end{enumerate}
\end{enumerate}
Here $\theta \in (0,1)$ is a relaxation parameter. We assume convergence when the relative error between $q^{(k)}$ and $q^{(k-1)}$, which we call the residual,
\begin{equation}
\label{DDM_residual}
\left( \frac{\sum_{i=1}^N \left(q^{(k)}_i - q^{(k-1)}_i\right)^2}{\sum_{i=1}^N \left(q^{(k)}_i\right)^2} \right)^{1/2},
\end{equation}
falls below a prescribed tolerance.
\subsection{Discrete iteration operator}
\label{sec:DiscIterOper}
To analyze the convergence properties of DDM in the discrete solution, we look at
the iteration matrix obtained from the integral formulation and its eigenvalues. For simplicity, we assume $\bd{u}^\infty=0$ and the slip coefficient $\gamma=0$. With the iteration variable defined as $q = -\bd{u}_S\cdot \bd{n}_S$, each Dirichlet-Neumann iteration includes solving the following two problems:
\begin{equation}
\frac12 p_D^{(k)} = \tilde{H}p_D^{(k)} + \frac{\mu}{\kappa} \tilde{L} q^{(k-1)},
\end{equation}
\begin{equation}
\frac12 \bd{u}_S^{(k)} = \tilde{K} \bd{u}_S^{(k)} + \tilde{M}_n p_D^{(k)},
\end{equation}
where $\tilde{H}, \tilde{K}$ are the matrices corresponding to the double layer potentials for Laplace and Stokes, respectively, and $\tilde{L}, \tilde{M}_n$ are the matrices corresponding to the single layer for Laplace and Stokes (the Stokes single layer $\tilde{M}_n$ including an extra multiplication by the normal). The 'tilde' symbol denotes the discrete version of the operators, with the kernels smoothed out as in Section~\ref{sec:Regularization} and the quadrature in Section~\ref{sec:Quadrature} applied. All these are dense matrices.
We can then write the discrete representation of the iteration operator as follows:
\begin{eqnarray}
q^{(k)} &= & (1-\theta)q^{(k-1)} - \theta \bd{u}_S^{(k)}\cdot \bd{n}_S \nonumber \\
&=& (1-\theta)q^{(k-1)} - \theta \left[(I/2 - \tilde{K} )^{-1} \tilde{M}_n p_D^{(k)} \right]\cdot \bd{n}_S \nonumber \\
&=& (1-\theta)q^{(k-1)} - \theta \left[(I/2 - \tilde{K} )^{-1} \tilde{M}_n\right] \cdot \bd{n}_S \, p_D^{(k)} \nonumber \\
&=& (1-\theta)q^{(k-1)} - \theta \left[(I/2 - \tilde{K} )^{-1} \tilde{M}_n\right] \cdot \bd{n}_S \, (I/2-\tilde{H})^{-1} \, \frac{\mu}{\kappa} \, \tilde{L} q^{(k-1)}.
\label{Discrete_iter_oper}
\end{eqnarray}
We write~\eqref{Discrete_iter_oper} as
\begin{equation}
q^{(k)} = \tilde{\mathcal{A}} q^{(k-1)},
\end{equation}
with the iteration matrix
\begin{equation}
\label{A_mat}
\tilde{\mathcal{A}} = I - \theta \left\{ I + \frac{\mu}{\kappa}\big[(I/2 - \tilde{K} )^{-1} \tilde{M}_n\big] \cdot \bd{n}_S \ \big[(I/2-\tilde{H})^{-1} \tilde{L}\big] \right\}.
\end{equation}
The operator in the first square brackets corresponds to solving the Stokes problem for $\bd{u}_S$, while the operator in the second square brackets corresponds to solving the Darcy problem for $p_D$.
\section{Numerical Results}
\label{sec:NumericalResults}
Here we analyze the algorithm in terms of accuracy, convergence, and dependence on physical parameter values for several surfaces. The code was written in C++, compiled with the clang compiler, and all computations were performed on a MacBook Pro 2.3 GHz and 32 GB RAM, running the Big Sur OS.
\subsection{BIEM accuracy}
We first demonstrate the high achievable accuracy of the integral formulations on the boundary. We solve the Darcy and Stokes integral equations independently of each other, using successive approximations~\eqref{Darcy_BIEM_iter} and~\eqref{Stokes_BIEM_iter}, each with a known exact solution. For the Darcy problem test, we used $\kappa=1$ and the harmonic function $p_D = e^x\sin y$ on the sphere of radius $R=0.8$. For the Stokes problem test, we used the solution similar to Sec. 4.4. of~\cite{tlupova18}. On the outside of a sphere of radius 0.8, we assume the velocity given by a point force singularity of strength $(1,0,0)$, placed at $(0.2, 0,0)$. We then take the solution on the inside of the sphere to be 0, so that the strengths in the Stokes single and double layer potentials are the jumps in velocity and surface force across the surface.
\begin{figure}[h]
\begin{centering}
\scalebox{0.4}{\includegraphics{err_sphere_on.eps}}
\caption{Convergence in grid size $h$ of separate Darcy and Stokes problems solved on the surface of a unit sphere by successive approximations \eqref{Darcy_BIEM_iter} and \eqref{Stokes_BIEM_iter}, with $\delta=3h$.}
\label{Errors}
\end{centering}
\end{figure}
Figure~\ref{Errors} shows the accuracy of this solution, where error is defined as the $L_2$ norm of the difference in the computed and exact solutions, $p_D$ for Darcy and $\bd{u}_S$ for Stokes. We used a smoothing parameter $\delta = 3h$ here and throughout this paper, where $h$ is the grid size in the coordinate planes. A larger choice of $\delta$ is needed to ensure the regularization error is dominant over the discretization error, so that the total error approaches $O(h^5)$ (see~\cite{tlupova18}). As expected, the convergence rate observed in Fig.~\ref{Errors} is $O(h^5)$. Other surfaces, such as a thin ellipsoid, are generally expected to give somewhat larger errors due to the larger curvature and varied spacing~\cite{tlupova18}.
\subsection{Discrete iteration operator}
Here we validate the numerical implementation of the Dirichlet-Neumann iterations for the spherical geometry. Similar to the setup used in Section~\ref{sec:Analysis}, consider the Darcy problem inside the unit sphere centered at the origin, coupled with the Stokes problem outside the sphere. Parameters are set to $\mu=1$, $\gamma=0$, and the unit sphere was discretized using $h=1/16$ leading to $N=4302$ quadrature points. In Figure~\ref{Sphere_eigenvalues}, the eigenvalues of the discrete iteration matrix $\tilde{\mathcal{A}}$ in~\eqref{A_mat} are computed and the first 50 are shown, to compare with the harmonic coefficients $\mathcal{A}_n$ of the iteration operator in Figure~\ref{Sphere_modes}.
\begin{itemize}
\item $\kappa=1$. In the left figure, the eigenvalues are shown for permeability $\kappa=1$ and varying relaxation parameter $\theta$ values. Again we see that a larger value of $\theta \in (0,1)$ leads to smaller eigenvalues, where most cluster around the value $1-\theta$. We then expect the successive approximations to converge faster for larger $\theta$.
\item $\kappa \ll 1$. Next, keeping $\mu=1$ fixed, we compute the eigenvalues for smaller values of permeability, $\kappa=10^{-2}$ (center figure), and $\kappa=10^{-4}$ (right figure). For these, we set the relaxation parameter $\theta=c\kappa$, and vary the value of coefficient $c=0.5, 0.75, 0.9$ as before. In all cases, the vast majority of the eigenvalues cluster around 1 (we saw this with the harmonic coefficients as well). As we will show, this is going to severely inhibit convergence speeds of DDM by successive approximations, and we will perform the DDM iterations using GMRES instead.
\end{itemize}
\begin{figure}[htb]
\begin{centering}
\scalebox{0.27}{\includegraphics{rad1_eig_k1.eps}}
\scalebox{0.27}{\includegraphics{rad1_eig_k1e-2.eps}}
\scalebox{0.27}{\includegraphics{rad1_eig_k1e-4.eps}}
\caption{Eigenvalues of the iteration matrix $\tilde{\mathcal{A}}$ in~\eqref{A_mat} on the unit sphere for $h=1/16$, for permeability values $\kappa=1, 10^{-2}, 10^{-4}$.}
\label{Sphere_eigenvalues}
\end{centering}
\end{figure}
\subsection{Benchmark problem: Stokes flow past a porous sphere}
In this section we test our algorithm using the problem of uniform flow of a viscous fluid past a porous spherical shell. Assuming that the fluid in the permeable sphere $0\leq r \leq R$ obeys Darcy's law and the fluid outside the sphere satisfies the Stokes equations, the solution was obtained by Joseph and Tao~\cite{joseph-tao64}. The matching conditions enforced on the sphere are continuity of pressure and normal velocity, as well as the no-slip condition for the tangential velocity of the exterior flow. Considering a stationary porous sphere of radius $R$ in a uniform streaming viscous fluid $\bd{U} = (0,0,U)$, the solution is given in spherical coordinates $(r,\theta,\phi)$,
\begin{eqnarray}
p^D &= &-3\mu\, U\, r\, \cos\theta / (2R^2+\kappa),\\
u^D_r &=& 3 U\, \kappa\, \cos\theta / (2R^2+\kappa),\\
u^D_\theta &=& -3 U\, \kappa\, \sin\theta / (2R^2+\kappa),
\end{eqnarray}
for the Darcy problem, and
\begin{eqnarray}
p^S &= &-3\mu\, R\, U\, \cos\theta / r^2(2+\kappa/R^2),\\
u^S_r &=& \left\{ \frac{-3 R U}{(2+\kappa/R^2)\, r} \left( 1-\frac{R^2}{3r^2} \left(1+\frac{2\kappa}{R^2} \right) \right) + U \right\} \cos\theta,\\
u^S_\theta &=& \left\{ \frac{3 R U}{(2+\kappa/R^2)\, 2\, r} \left( 1+\frac{R^2}{3r^2} \left(1+\frac{2\kappa}{R^2} \right) \right) - U \right\} \sin\theta,
\end{eqnarray}
for the Stokes problem, where $u_\phi=0$ due to axial symmetry. The drag on the porous sphere, defined as the hydrodynamic force exerted on the sphere,
\begin{equation}
\label{drag_int}
\bd{D} = \int_{\partial\Omega} \bd{f}\, dS(\bd{x}),
\end{equation}
where $\bd{f} = \bdg{\sigma}_S\cdot\bd{n}$, $\bd{n}=\bd{e}_r$, was found to equal
\begin{equation}
\label{drag_value}
\bd{D} = \frac{6\pi\, \mu\, R\, U}{1+\kappa/(2R^2)}.
\end{equation}
The usual solution and drag for the flow past a solid sphere are recovered from these formulas by setting $\kappa=0$.
To test our algorithm against this solution, we compute the Stokes stress vector as $\bd{f} = f_r \bd{e}_r + f_\theta \bd{e}_\theta$ with
\begin{equation}
\label{f_r}
f_r = -p_D + 2\mu \frac{\partial u_r}{\partial r},
\end{equation}
\begin{equation}
\label{f_theta}
f_\theta = \mu \left( r \frac{\partial}{\partial r} \left(\frac{u_\theta}{r}\right) + \frac{1}{r} \frac{\partial u_r}{\partial\theta} \right),
\end{equation}
being the normal and tangential components. Note that we replaced $p_S$ with $p_D$ in~\eqref{f_r} since the exact solution was derived assuming continuous pressure $p_D=p_S$ on the boundary. To compute the drag numerically, we integrate~\eqref{drag_int} using our usual quadrature, and compare to the exact value given by~\eqref{drag_value}.
In our tests, we use a unit sphere $R=1$, $U=1, \mu=1$ and no-slip $\gamma=0$. We use a tolerance of $10^{-9}$ for the residual~\eqref{DDM_residual} in the Dirichlet-Neumann iterations, and for the solution of the integral equations, we use successive approximations~\eqref{Darcy_BIEM_iter} and~\eqref{Stokes_BIEM_iter}, as well as GMRES, both with tolerance $10^{-9}$. Table~\ref{table:Sphere_k1} shows the results for $\kappa=1$. For different grid sizes $h$, displayed are: the number of quadrature points $N$ on the sphere surface, the number of Dirichlet-Neumann iterations via successive approximations (D-N SA) for two values of $\theta=0.5$ and $\theta=0.75$, the number of successive approximations (local SA) needed to solve the Darcy and Stokes problems at each D-N step to reach the set tolerance, the number of GMRES iterations (local GMRES) for each problem at each D-N step for the same tolerance, and finally the errors in the Darcy and Stokes solutions computed in $L_2$ norm as
\begin{equation}
\label{error_Darcy}
||p^D_h-p^D||_2 = \left(\sum_{i=1}^N \left(p_D(\bd{x}_i)- p_D^{ex}(\bd{x}_i)\right)^2 \Big/ N \right)^{1/2},
\end{equation}
\begin{equation}
\label{error_Stokes}
||\bd{u}^S_h-\bd{u}^S||_2 = \left(\sum_{i=1}^N \left|\bd{u}_S(\bd{x}_i)- \bd{u}_S^{ex}(\bd{x}_i)\right|^2 \Big/ N \right)^{1/2},
\end{equation}
where $p_D^{ex}$ and $\bd{u}_S^{ex}$ are the exact values and $|\bd{u}|$ is the Euclidean norm. For a more detailed view, Figure~\ref{Sphere_k1_resi_err} shows the DDM residual and the errors in drag, $p_D$ and $\bd{u}_S$, by comparing the numerical solution to the exact solution at every D-N iteration for grid sizes $h=1/16$ and $h=1/32$ and varying $\theta$.
We make the following observations:
\begin{itemize}
\item \underline{D-N convergence and dependence on the relaxation parameter $\theta$}. For $\theta=0.5$, about 19 D-N iterations are required to reach the tolerance, independent of the grid size, while for $\theta=0.75$ only 7 iterations are needed for $h=1/16$, and this number reduces quickly to 2 iterations for smaller grid sizes. Figure~\ref{Sphere_k1_resi_err} (top left) shows the gradual decrease in the residual for $\theta=0.5$ and a more rapid decrease for $\theta=0.75$. This faster convergence can be attributed to the smaller spectral radius of the iteration operator, as was observed for $\kappa=1$ in Figures~\ref{Sphere_modes} and~\ref{Sphere_eigenvalues}. This is remarkable given that with smaller $h$ the local Stokes and Darcy problems become larger.
\item \underline{Solution errors}. Further, we see that the errors in drag, $p_D$ and $\bd{u}_S$ in Figure~\ref{Sphere_k1_resi_err} all level off at the BIEM accuracy shown in Table~\ref{table:Sphere_k1} (again, much faster with $\theta=0.75$). From the error values in Table~\ref{table:Sphere_k1} we see that the convergence rate is $O(h^5)$ or higher with grid refinement.
\item \underline{Local problems: SA vs. GMRES}. Table~\ref{table:Sphere_k1} shows the number of successive approximations (local SA) performed at each D-N iteration, as well as the number of GMRES iterations (local GMRES) as an alternative. For SA, a range is shown because the number decreases slowly with each D-N iteration. It is clear that successive approximations do not converge quickly, and GMRES is far superior in solving the local Stokes and Darcy problems at each iteration. Both methods require evaluating the double layer potential at each iteration, and since GMRES performs far fewer double layer evaluations, we use GMRES in all our remaining tests.
\item \underline{Local problems: CPU time}. Both SA and GMRES require evaluating the double layer potential at each iteration, which in discretized form is a dense linear system. The Darcy system is $N\times N$, while the Stokes system is $3N\times 3N$, where $N$ is the number of quadrature points. The computational cost is then $O(N^2)$ for each problem. The actual CPU times for each double layer integral evaluation are shown in Figure~\ref{Sphere_k1_CPU}. Note that the discretized integrals were implemented to take advantage of the target-source symmetry in all kernels, reducing the computational time in half. If very large systems are considered, the computational time could be reduced to $O(N\log N)$ or $O(N)$ using a fast summation algorithm such as a treecode~\cite{wang-krasny-tlupova2} or a fast multipole method~\cite{tornberg-greengard-08}.
\item \underline{Dependence on $\kappa$}. We next proceed to study the convergence of D-N iterations for different values of permeability. For $\kappa=1$, we see a strong dependence on the values of $\theta$ when successive approximations are used, because the spectral radius of the iteration operator is smaller for larger $\theta$. Table~\ref{table:Sphere_diff_k} reiterates some of these results and also shows the results for $\kappa=10^{-2}$ and $\kappa=10^{-4}$. For small values of $\kappa$ such as these, we saw that the spectral radius of the iteration operator is very close to 1, and this indeed inhibits convergence. The number of successive approximations needed for D-N convergence (D-N by SA in Table~\ref{table:Sphere_diff_k}) exceeds 100 because the residual decreases very slowly. This is not improved by using a larger $\theta$ as it did for $\kappa=1$ (in all cases, we set $\theta=c\kappa$ with $c\in (0,1)$). Because of this, we perform the D-N iterations using GMRES, which converges much faster. The D-N by GMRES column in Table~\ref{table:Sphere_diff_k} shows the number of iterations for the same tolerance $10^{-9}$ for the residual. The number of iterations is 7 for $\kappa=10^{-2}$ and 8 for $\kappa=10^{-4}$ for the coarsest grid $h=1/16$. As the grid size is reduced and the solution becomes more accurate, the number of D-N iterations by GMRES reduces. For the solution errors, which are also shown in the left graph in Figure~\ref{Errors_All}, we observe convergence of the order $O(h^5)$.
\end{itemize}
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c||c|c||c|c||c|c||c|c|}
\multirow{2}{*}{$h$} & \multirow{2}{*}{$N$} & \multicolumn{2}{c||}{D-N SA} & \multicolumn{2}{c||}{local SA} & \multicolumn{2}{c||}{local GMRES} & \multirow{2}{*}{$||p^D_h-p^D||_2$} & \multirow{2}{*}{$||\bd{u}^S_h-\bd{u}^S||_2$}\\
& & $\theta=0.5$ & $\theta=0.75$ & Darcy & Stokes & Darcy & Stokes & & \\
\hline
1/16 & 4302 & 19 & 7 & 37-2 & 26-2 & 5 & 8 & 3.450e-05 & 1.053e-04 \\
1/32 & 17070 & 19 & 5 & 37-1 & 26-1 & 4 & 7 & 1.442e-06 & 5.525e-06 \\
1/64 & 68166 & - & 2 & - & - & 3 & 5 & 2.605e-08 & 1.716e-07 \\
1/128 & 272718 & - & 1 & - & - & 2 & 3 & 3.236e-10 & 4.735e-09 \\
\end{tabular}
\caption{Flow past a porous sphere with $\kappa=1$, DDM residual tolerance = $10^{-9}$, local solution tolerance = $10^{-9}$.}
\label{table:Sphere_k1}
\end{center}
\end{table}
\begin{figure}[!htb]
\begin{centering}
\scalebox{0.4}{\includegraphics{rad1_k1_gmres_resi.eps}}
\scalebox{0.4}{\includegraphics{rad1_k1_gmres_drag.eps}} \\
\scalebox{0.4}{\includegraphics{rad1_k1_gmres_pd.eps}}
\scalebox{0.4}{\includegraphics{rad1_k1_gmres_us.eps}}
\caption{Flow past a porous sphere with $\kappa=1$, tolerance $10^{-9}$ for residual/local solutions. Top left: DDM residual, top right: error in drag, bottom left: error in $p_D$, bottom right: error in $\bd{u}_S$.}
\label{Sphere_k1_resi_err}
\end{centering}
\end{figure}
\begin{figure}[!htb]
\begin{centering}
\scalebox{0.4}{\includegraphics{rad1_k1_gmres_CPU.eps}}
\caption{Flow past a porous sphere with $\kappa=1$. CPU time in seconds on each evaluation of the double layer potential (one successive approximation or one GMRES iteration).}
\label{Sphere_k1_CPU}
\end{centering}
\end{figure}
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c||c|c||c|c||c|c||c|c|}
\multirow{2}{*}{$\kappa$} & \multirow{2}{*}{$h$} & \multicolumn{2}{c||}{D-N by SA} & \multicolumn{2}{c||}{D-N by GMRES} & \multicolumn{2}{c||}{local GMRES} & \multirow{2}{*}{$||p^D_h-p^D||_2$} & \multirow{2}{*}{$||\bd{u}^S_h-\bd{u}^S||_2$} \\
& & $\theta=0.5\kappa$ & $\theta=0.75\kappa$ & $\theta=0.5\kappa$ & $\theta=0.75\kappa$ & Darcy & Stokes & & \\
\hline
$1$ & 1/16 & 19 & 7 & 4 & 4 & 5 & 8 & 3.450e-05 & 1.053e-04 \\
& 1/32 & 19 & 5 & 4 & 4 & 6 & 6 & 1.442e-06 & 5.525e-06 \\
&1/64 & - & 2 & 3 & 3 & 6 & 6 & 2.605e-08 & 1.716e-07 \\
\hline
$10^{-2}$ & 1/16 & $>100$ & $>100$ & 7 & 7 & 6 & 9 & 9.484e-05 & 4.500e-05 \\
& 1/32 & $>100$ & $>100$ & 6 & 7 & 7 & 7 & 3.498e-06 & 2.576e-06 \\
&1/64 & - & - & 4 & 4 & 8 & 7 & 3.084e-08 & 8.546e-08 \\
\hline
$10^{-4}$ & 1/16 & $>100$ & $>100$ & 8 & 8 & 8 & 11 & 5.474e-04 & 4.243e-05 \\
& 1/32 & $>100$ & $>100$ & 2 & 2 & 8 & 9 & 3.490e-06 & 2.554e-06 \\
&1/64 & - & - & 1 & 1 & 5 & 5 & 3.899e-08 & 8.451e-08 \\
\end{tabular}
\caption{Flow past a porous sphere with varying $\kappa$, tolerance $10^{-9}$ for residual/local solutions.}
\label{table:Sphere_diff_k}
\end{center}
\end{table}
\subsection{Thin porous ellipsoid}
Next we test the method on a geometry that has a larger curvature, a thin ellipsoid defined in~\eqref{Ellipsoid}
with $a=1,b=0.6,c=0.4$. In this case the exact solution is not known, and we compute the error empirically as $||p_h - p_{h/2}||_2$ and $||\bd{u}_h - \bd{u}_{h/2}||_2$, with similar definitions of the $L_2$ norm as in~\eqref{error_Darcy}-\eqref{error_Stokes}. Table~\ref{table:Ellipsoid_diff_k} shows the results for permeability values $\kappa = 1, 10^{-2}, 10^{-4},10^{-7}$. As we saw in the sphere example, the number of DDM iterations increases slightly when we go from $\kappa=1$ to $\kappa<1$. Unlike the sphere, however, as the grid is refined, the number of DDM iterations does not decrease but rather increases slightly. Furthermore, for the small value of the permeability $\kappa=10^{-4}$, the DDM iterations do not converge to the prescribed tolerance $10^{-9}$ quickly. In the number of iterations shown in the table (denoted by *), the relative residual reaches a tolerance of about $10^{-6}$, with the consecutive residuals differing by less than $10^{-9}$. As expected for the thin ellipsoid, the solution accuracy is lower than what we observed with the sphere.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c||c|c||c|c||c|c|}
\multirow{2}{*}{$\kappa$} & \multirow{2}{*}{$h$} & \multirow{2}{*}{$N$} & \multicolumn{2}{c||}{D-N by GMRES} & \multicolumn{2}{c||}{local GMRES} & \multirow{2}{*}{$||p^D_h-p^D_{h/2}||_2$} & \multirow{2}{*}{$||\bd{u}^S_h-\bd{u}^S_{h/2}||_2$} \\
& & & $\theta=0.5\kappa$ & $\theta=0.75\kappa$ & Darcy & Stokes & & \\
\hline
$1$ & 1/16 & 1742 & 5 & 5 & 7 & 11 & 3.731e-04 & 6.052e-04 \\
& 1/32 & 6902 & 5 & 5 & 6 & 10 & 3.486e-05 & 4.399e-05 \\
&1/64 & 27566 & 5 & 5 & 5 & 9 & 3.300e-06 & 3.589e-06 \\
&1/128 & 110250 & 5 & 5 & 5 & 8 & - & - \\
\hline
$10^{-2}$ & 1/16 & 1742 & 9 & 9 & 9 & 14 & 2.378e-02 & 5.744e-03 \\
& 1/32 & 6902 & 12 & 12 & 9 & 14 & 1.772e-03 & 1.797e-03 \\
&1/64 & 27566 & 12 & 12 & 8 & 11 & 8.347e-05 & 8.423e-05 \\
&1/128 & 110250 & 12 & 12 & 8 & 11 & - & - \\
\hline
$10^{-4}$ & 1/16 & 1742 & 9* & 8* & 11 & 16 & 1.689e-01 & 1.873e-02 \\
& 1/32 & 6902 & 11* & 11* & 11 & 17 & 2.913e-02 & 2.914e-02 \\
&1/64 & 27566 & 13* & 13* & 11 & 17 & 1.435e-03 & 1.436e-03 \\
&1/128 & 110250 & 13* & 13* & 11 & 18 & - & - \\
\hline
$10^{-7}$ & 1/16 & 1742 &
6 & 6 & 14 & 20 &
1.900e-01 & 1.865e-02 \\
& 1/32 & 6902 &
8 & 8 & 11 & 20 &
3.406e-02 & 3.408e-02 \\
& 1/64 & 27566 &
10 & 10 & 11 & 18 &
1.448e-03 & 1.449e-03 \\
& 1/128 & 110250 &
10 & 10 & 12 & 18 & - & - \\
\end{tabular}
\caption{Flow past a porous ellipsoid with varying $\kappa$, tolerance $10^{-9}$ for residual/local solutions.}
\label{table:Ellipsoid_diff_k}
\end{center}
\end{table}
\subsection{Porous molecule}
The four-atom molecule surface used in these tests is defined in~\eqref{Molecule}. Again, we compute the errors empirically similar to the ellipsoid. The results for different permeability values and grid sizes are shown in Table~\ref{table:Molecule_diff_k}. The method performs well for this surface, converging to the tolerance in just a few iterations, with the speed of convergence improving when the discretization of the surface is refined. For $h=1/128$ for example, which results in $N=272718$ quadrature points, only 2 DDM iterations are needed for $\kappa=1$ and $\kappa=10^{-2}$, while the $\kappa=10^{-4}$ { and $\kappa=10^{-7}$ cases only require} one iteration for the tolerance of $10^{-9}$. The solution accuracy is higher than for the ellipsoid and similar to the sphere accuracy, with the order of convergence as predicted $O(h^5)$ - this is shown in Figure~\ref{Errors_All}.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c||c|c||c|c||c|c|}
\multirow{2}{*}{$\kappa$} & \multirow{2}{*}{$h$} & \multirow{2}{*}{$N$} & \multicolumn{2}{c||}{D-N by GMRES} & \multicolumn{2}{c||}{local GMRES} & \multirow{2}{*}{$||p^D_h-p^D_{h/2}||_2$} & \multirow{2}{*}{$||\bd{u}^S_h-\bd{u}^S_{h/2}||_2$} \\
& & & $\theta=0.5\kappa$ & $\theta=0.75\kappa$ & Darcy & Stokes & & \\
\hline
$1$ & 1/16 & 4302 & 4 & 4 & 5 & 7 & 1.793e-05 & 1.221e-05 \\
& 1/32 & 17070 & 4 & 4 & 6 & 6 & 8.947e-07 & 9.358e-07 \\
&1/64 & 68166 & 3 & 3 & 6 & 6 & 1.892e-08 & 1.974e-08 \\
&1/128 & 272718 & 2 & 2 & 6 & 6 & - & - \\
\hline
$10^{-2}$ & 1/16 & 4302 & 8 & 8 & 6 & 9 & 1.689e-04 & 3.228e-05 \\
& 1/32 & 17070 & 7 & 7 & 7 & 7 & 6.102e-06 & 6.174e-06 \\
&1/64 & 68166 & 4 & 5 & 7 & 7 & 5.877e-08 & 6.263e-08 \\
&1/128 & 272718 & 2 & 2 & 7 & 7 & - & - \\
\hline
$10^{-4}$ & 1/16 & 4302 & 8 & 11 & 8 & 10 & 1.052e-03 & 2.158e-05 \\
& 1/32 & 17070 & 2 & 8 & 9 & 9 & 5.883e-06 & 5.981e-06 \\
&1/64 & 68166 & 1 & 1 & 5 & 5 & 7.676e-08 & 8.000e-08 \\
&1/128 & 272718 & 1 & 1 & 3 & 3 & - & - \\
\hline
$10^{-7}$ & 1/16 & 4302 &
1 & 1 & 8 & 12 &
1.330e-04 & 4.356e-05 \\
& 1/32 & 17070 &
1 & 1 & 8 & 10 &
5.710e-06 & 5.797e-06 \\
& 1/64 & 68166 &
1 & 1 & 7 & 8 &
7.679e-08 & 8.002e-08 \\
& 1/128 & 272718 & 1 & 1 & 3 & 3 & - & - \\
\end{tabular}
\caption{Flow past a porous molecule with varying $\kappa$, tolerance $10^{-9}$ for residual/local solutions.}
\label{table:Molecule_diff_k}
\end{center}
\end{table}
\begin{figure}[!htb]
\hspace{-1.2cm} \scalebox{0.4}{\includegraphics{Errors_All.eps}}
\caption{Solution errors for three surfaces: unit sphere, thin ellipsoid, four-atom molecule.}
\label{Errors_All}
\end{figure}
\section{Conclusions and future work}
We have presented a domain decomposition method of a sequential Dirichlet-Neumann type to solve the Stokes-Darcy system of PDEs. The method is based on splitting the interface conditions so that first, the Darcy problem is solved with one of the conditions, and then the Stokes problem is solved with the other two. The information is exchanged at each step of this iterative procedure. The local Stokes and Darcy problems are solved by second kind boundary integral formulations. A regularization technique specifically designed to achieve high accuracy on the boundary is applied, and the quadrature method of \cite{wilson, beale16} is used, which does not use coordinate charts and is simple to implement. Numerical results validate the algorithm against a problem with a known analytical solution of viscous flow past a porous sphere, as well as demonstrate the applicability of the method to more general geometries.
Our convergence analysis of the method shows strong dependence of the spectral radius of the iteration operator on the permeability $\kappa$. For small values of $\kappa$, the eigenvalues cluster close to 1, inhibiting convergence of successive approximations. While we demonstrate that using a Krylov subspace method such as GMRES mitigates this issue, more robust DDMs that are less dependent on the physical parameters are desirable and will be investigated. It was shown in the two-dimensional case~\cite{boubendir-tlupova-13} that a Robin-Robin type domain decomposition method with non-local operators (based on the boundary integral formulations) can be chosen in the de-coupling of transmission conditions to dramatically improve the convergence properties of the solution. The dependence of convergence rates on the physical parameters of viscosity $\mu$ and permeability $\kappa$ can be reduced as well. Similar approaches will be investigated in the three-dimensional case.
In many relevant applications, the porous medium has a heterogeneous structure, and the BIEM is not a suitable choice to model the flow there. This work can be extended to model such porous media, by coupling the BIEM in the domain of free flow with more appropriate techniques such as the finite element method (FEM) in the porous domain with variable properties. With this approach, the FEM can handle the porous domain where the heterogeneities occur, while the BIEM will efficiently deal with the Stokes domain, which can be large or even infinite.
To limit the scope of this paper, only the steady state problem was considered with a single interface. In future work, time-dependent problems and multi-surface cases will be considered. In such a simulation, several surfaces can be close to each other, and it is well-known that the integrands become nearly singular and therefore numerically non-trivial to integrate with sufficient accuracy. The correction methods of~\cite{beale01, beale04, tlupova18} will be used to improve the accuracy in the nearly singular case. Another challenge in such a simulation is the high computational cost of the direct evaluation of the resulting dense linear systems, which is $O(N^2)$, where $N$ is the number of unknowns. To be able to simulate large enough systems, the computational cost will be reduced using fast summation techniques such as the kernel-independent treecode~\cite{wang-krasny-tlupova2} or a fast multipole method~\cite{tornberg-greengard-08}.
\section*{Acknowledgments}
The author gratefully acknowledges support by the National Science Foundation under grant DMS-2012371.
\bibliographystyle{plain}
|
1,108,101,565,506 | arxiv | \section{Introduction}
Positron annihilation is one of the key tools in modern investigations
of the Fermi surface (FS) of solids \cite{west1995}, alongside quantum
oscillatory techniques, Compton scattering and angle-resolved photoemission.
Unlike other methods, however, the positron probe itself plays a crucial
role in the measured distribution, preferentially annihilating with those
electrons that are most able to screen its charge. In a typical metal,
free from vacancy-type defects, the electrons that are most readily able
to screen are, of course, those at the FS, advantageously leading to an
enhancement of the signal contributed from electrons at the FS. Attempts
to account for this enhanced contribution have, for the most part, relied
on detailed studies of the electron-positron interaction within the jellium
model \cite{kahana1963,arponen1979}, which is now essentially well-understood
\cite{stachowiak1993}. However, such schemes are yet to achieve good agreement
with experiment when applied to a wide range of metallic systems. Here,
we consider this problem from an experimental perspective, {\em measuring}
the state-dependent enhancement factor for some simple elemental metals,
and present a phenomenological (and empirical) correction to the work of
Barbiellini, Alatalo and their co-workers \cite{barbiellini1997,alatalo1996}
that offers excellent agreement with experiment.
The complex many-body interaction between the positron and the electron gas has
been intensively studied for many years \cite{sormann1996}. When the positron
enters a homogeneous electron gas, the attractive Coulomb interaction polarizes
the electron gas in the vicinity of the positron, leading to a cusp in the
unscreened electron density at the positron's position and the
associated enhancement of
the partial annihilation rate of those electrons that screen the charge.
The theory
of Kahana \cite{kahana1963} predicted a momentum-dependent enhancement,
in which the enhancement increases towards the Fermi momentum, $k_{\rm F}$,
and corresponds to the increased capability of electrons near the Fermi level
to screen the positron's charge, compared with lower-lying electron states.
However, the inhomogeneity of the electron gas in real lattices can have a
strong influence on the enhancement, even hiding the Kahana-like momentum
dependence \cite{barbiellini1997,sormann1996}.
It is worth pointing out that when considering enhancement there are
actually two separate, but related issues. Firstly, in the context of
calculating the correct positron lifetimes in solids, the enhancement of
the total electron density needs to be properly described in order to
calculate the positron annihilation rate. Secondly, a description of the
enhancement is needed when calculating the two-photon momentum densities
(which are the focus of the current paper). The former problem is easier because
the contact density can be parameterized in terms of the local electron and
positron densities (using the many-body results for jellium), but the latter
is a more difficult problem since in the framework of density functional
theory there is no formally exact way to calculate the two-photon
momentum density \cite{singh1985,barbiellini2003} (and as such all
models in the literature are, in practice, empirical).
Local density parameterizations, in which the enhancement is parameterized as
a function of the unscreened electron density, $n$, at the positron, have
been introduced
to account for the inhomogeneity of real systems. In these approaches, the
enhancement is usually expressed in terms of the electron gas parameter,
$r_s = (3/4\pi n)^{1/3}$, of which it
is a monotonically increasing function for typical crystallographic electron
densities. Some popular choices are the expressions of Arponen and
Pajanne \cite{arponen1979}, based on boson formalism and parameterized by
Barbiellini and co-workers \cite{barbiellini1995}, and those of Boro\'{n}ski
and Nieminen (BN) which are based upon an interpolation of Fermi liquid
results due to Lantto \cite{boronski1986}. Jarlborg and Singh (JS)
have used a local-density approach to solve a two-body electron-positron
Schr\"{o}dinger equation inside a spherical correlation cell that yields good
agreement with transition metals and their alloys for both momentum densities
\cite{jarlborg1987} and positron lifetimes \cite{barbiellini1991}, and is a
common choice to describe the enhancement of the momentum distribution in
metals \cite{barbiellini2003,major2004b}. More general parameterizations
have been proposed \cite{mijnarends1979,daniuk1987,jarlborg1991,sob1982}
that include Kahana-like momentum or energy dependence to describe the
results of positron measurements. More recently, theoretical prescriptions for
the enhancement have been developed that represent a significant departure from
the homogeneous electron gas or local-density approaches, based on, for example,
the generalized gradient approximation \cite{barbiellini1997}, Bloch-modified
ladder \cite{sormann1996} or weighted-density approximation \cite{rubaszek1998}.
Owing to the different screening properties of $d$ and $s$-$p$ electrons,
efforts to include a character, or state-dependent enhancement function
have been applied to several transition metals and their alloys
\cite{sob1982,svoboda1983,matsumoto1987,genoud1990}. \v{S}ob applied
such a scheme to data measured on a polycrystalline FeAl alloy, finding a
de-enhancement of the $d$ states by a factor of $\sim 2.2$ compared with the
$s$-$p$ states \cite{sob1982}, whereas the application of the same procedure by
Svoboda and \v{S}ob \cite{svoboda1983} to CuZn was found to favor a reduction
by a factor of $\sim 1.5$. Theoretically, such explicit state-dependence is
rarely included, although for flat $d$-bands it is implicitly present in any
energy-dependent model. Recently, Barbiellini and co-workers have developed a
theoretical and {\em ab initio} state-dependent prescription for calculating
the enhancement in a general system \cite{barbiellini1997,alatalo1996}, which
is based on the state-dependent annihilation rates calculated within the
generalized gradient approximation (GGA). Although it has been demonstrated
that the effects of enhancement do not shift the location in {\bf k}-space of
the Fermi breaks in positron measurements \cite{majumdar1965}, the influence
of the theoretical treatment of the enhancement, when rigid-band like shifts
are applied to the electronic structure and compared with experiment, has
not yet been investigated.
Here, we tackle the problem of describing the enhancement of the
positron annihilation rates from an experimental perspective. Employing
a state-dependent (SD) model for the enhancement similar to that of Ref.\
\cite{barbiellini1997}, we simultaneously fit both the FS and the enhancement
from {\em ab initio} electronic structure calculations to positron data
directly in {\bf k}-space in order to obtain a quantitative {\em measurement}
of the enhancement in metals. Additional comparisons with the calculational
scheme of Ref.\ \cite{barbiellini1997} are used to quantitatively assess the
applicability of such an SD enhancement model for electron-positron momentum
distributions. In particular, the accuracy of rigid-band-like approaches in
obtaining more realistic representations of the experimental FS are found
to be sensitively dependent on the particular enhancement employed in the
calculation.
The organisation of this paper is as follows. In Section \ref{s:method},
we introduce the method employed in this paper. In Section \ref{s:dmetals},
we apply this fitting technique to some $3d$ transition and noble metals,
namely V, Cr and Ag, and in Section \ref{s:alkali} we address the simple metal
Al. Finally, in Section \ref{s:model} we apply and investigate a
correction to the existing theory of Ref.~\cite{barbiellini1997} that provides
useful predictive power as a general model of enhancement in both transition
metals as well as simple metals. The application of this correction to Mo is
shown to quantitatively explain the difference in the momentum distributions of
Cr and iso-electronic Mo that is observed despite the similarity in their FS.
\section{Method}
\label{s:method}
\subsection{State-dependent enhancement}
The quantity measured by two dimensional angular correlation of
(electron-positron) annihilation radiation (2D-ACAR) experiments is a
once-integrated projection
(along a suitable crystallographic direction) of the so-called two-photon
momentum density, $\rho^{2\gamma}({\bf p})$,
\begin{equation}
\rho^{2\gamma}({\bf p}) = \sum_{i,{\bf G}} n_i
\vert C_{i,{\bf G}} \vert^2 \delta({\bf p}-{\bf k}-{\bf G}),
\label{e:rhop}
\end{equation}
where $n_i$ is the electron occupation density of state $i = \{j, {\bf k}\}$
($j$ is the band index), $C_{i,{\bf G}}$ are the coefficients of a plane-wave
expansion of the product of the electron and positron wavefunctions, in which
${\bf G}$ is a vector of the reciprocal lattice, and the $\delta$-function
expresses the conservation of crystal momentum. In a 2D-ACAR measurement, the 3D
quantity expressed in Eq.~\ref{e:rhop} is integrated along a particular
direction to yield a 2D projection of $\rho^{2\gamma}({\bf p})$, and the
projected axis is usually chosen to be a suitable high-symmetry crystallographic
axis. The FS enters Eq.~\ref{e:rhop} as discontinuous breaks in the momentum
density when {\bf p} traverses $i$ occupied ($n_i = N$) to $i$ unoccupied ($n_i
= N-1$) (i.e. when the band crosses the Fermi energy). The folding of
crystallographically equivalent {\bf p}-points of momentum using the so-called
Lock-Crisp-West procedure \cite{lock1973} yields the `reduced momentum density'
(RMD),
$\rho^{2\gamma}({\bf k})$,
\begin{equation}
\rho^{2\gamma}({\bf k}) = \sum_{i,{\bf G}}
\vert C_{i,{\bf G}} \vert^2.
\label{e:rmd}
\end{equation}
The $C_{i,{\bf G}}$ of Eq.~\ref{e:rhop} can be written in terms of the
single-particle electron and positron wavefunctions, $\psi_i({\bf r})$ and
$\psi^+({\bf r})$ as,
\begin{equation}
C_{i,{\bf G}} = \int {\rm d}^3{\bf r}
\exp[-{\rm i}({\bf k}+{\bf G})\cdot{\bf r}] \psi_i^{\rm ep}({\bf r}, {\bf r}).
\label{e:cig}
\end{equation}
Here, $\psi_i^{\rm ep}({\bf r}, {\bf r'})$ is the electron-positron pair
wavefunction for state $i$,
\begin{equation}
\psi_i^{\rm ep}({\bf r}, {\bf r'}) = \psi_i({\bf r'}) \psi^+({\bf r})
\sqrt{\gamma_i({\bf r})},
\label{e:psiep}
\end{equation}
where $\gamma_i({\bf r})$ is the state-dependent positron enhancement
factor (for the state $i$). Setting $\gamma=1$ in Eq.~\ref{e:psiep} is
equivalent to the independent particle model (IPM), although it should be
noted that the effects of the positron wavefunction are still included in that
case. The usual parameterizations of the enhancement, for example the BN or
JS models, are based on local-density parameterizations,
in which $\gamma_i({\bf r})
= \gamma({\bf r})$ is a function only of the unscreened local electron
density at the
location of the positron. Other state-dependent prescriptions exist
(e.g.\ \cite{sob1982,jarlborg1991}), although these have relied on the {\em
empirical} determination of the state dependence of the enhancement.
Barbiellini and co-workers \cite{barbiellini1997} have proposed a theoretical
prescription for applying a state-dependent positron enhancement factor to {\em
ab initio} calculations of the electronic structure and momentum density. In
their scheme, $\gamma_i$ is obtained through the partial annihilation rates,
such that,
\begin{equation}
\gamma_i = \lambda_i / \lambda_i^{\rm IPM},
\label{e:lambdai}
\end{equation}
where $\lambda_i$ is the partial annihilation rate of the state $i$ including
correlation effects, and $\lambda_i^{\rm IPM}$ is the partial annihilation rate
due to the IPM. The total annihilation rate, $\lambda$, may be
calculated from (here shown for the local density approximation, LDA),
\begin{equation}
\lambda = {\pi}r_e^2c \int {\rm d}^3 {\bf r}\;n^+({\bf r})n({\bf r})
\gamma({\bf r}),
\label{e:lambdalda}
\end{equation}
where $r_e$ is the classical electron radius, $c$ is the speed of light and
$n^+({\bf r})$ is the positron density. In
their calculations, the GGA was used for the calculation of $\lambda_i$,
which successfully reproduces the experimental annihilation rates rather well
\cite{barbiellini1995,barbiellini1996}.
\subsection{Practical approach}
We begin by computing the {\em ab initio}
electronic structure using the linearized muffin-tin orbital (LMTO) method,
within the atomic sphere approximation and including combined correction terms
\cite{barbiellini2003}. The $C_{i,{\bf G}}$ coefficients of Eq.~\ref{e:cig} are
then computed within the IPM (equivalent to setting $\gamma=1$ in
Eq.~\ref{e:psiep}), unfolded in such a way as to resolve the individual
contribution owing to the atom index ($n$), and orbital angular momentum
quantum number ($l$),
\begin{equation}
C^{\rm IPM}_{i,{\bf G}} = \sum_{n,l} C^{\rm IPM}_{i,{\bf G},n,l}.
\end{equation}
The momentum density in the first Brillouin zone (i.e.\ the RMD,
Eq.~\ref{e:rmd}) is computed for the IPM by,
\begin{equation}
\rho^{\rm IPM}_j ({\bf k}) = {\rm constant} \times \sum_{\bf G} \vert
(\sum_{n,l} C^{\rm IPM}_{i,{\bf G},n,l}) \vert^2.
\end{equation}
For the enhancement, we introduce the quantities $\gamma_{n,l}$ that describe
the enhancement of a state of atomic species $n$ and of orbital angular momentum
$l$ ($l = s, p, d, f$). These can then be incorporated into the calculation of
the RMD by,
\begin{equation}
\rho^{\rm SD}_j({\bf k}) = {\rm constant} \times \sum_{\bf G} \vert (\sum_{n,l}
\sqrt{\gamma_{n,l}}\;
C^{\rm IPM}_{i,{\bf G},n,l}) \vert^2.
\label{e:rhok}
\end{equation}
Note that in the above equation, the $\gamma_l$ multiply the $C_{i,{\bf G},n,l}$
coefficients, which are inside the sum over included {\bf G}-vectors, and so
the RMD must be re-computed for each $\gamma_l$ and cannot be expanded into a
sum of contributions to the momentum density from different $l$-orbitals.
In this way, $\gamma_l$ is a universal quantity, representing the partial
enhancement of a state with character $l$. The degree to which it is enhanced
depends on the coefficients of the wavefunctions in the LMTO calculation. The
enhancement, then, of a pure state of atomic species $n$ and orbital character
$l$ is given by $\gamma_{n,l} = \gamma_n \cdot \gamma_l$. Note that the band
characters (atomic species and orbital character) are strongly ${\bf
k}$-dependent, and of course vary from band to band due to hybridization with
other states, so our enhancement model is a general state-dependent model for
the enhancement (see, for example, Fig.\ \ref{f:gammak}),
but has its origins in the convenient properties of the LMTO wavefunctions.
The contribution due to core annihilations is an important consideration for
positron lifetime calculations \cite{puska1986}. However, the core contribution
is small and relatively independent of $k$ across the first Brillouin zone (BZ),
and can safely be omitted from this
calculation. Instead, the contribution from core states in the data is
described by a uniform background in the subsequent fitting procedure.
\subsection{Minimization procedure}
In the rigid-band
approach, the agreement between experiment and theory is iteratively
maximized with respect to a rigid shift of one or more of the energy
bands (typically those that constitute the FS), until convergence at the
minimum of the goodness-of-fit parameter is achieved.
This is similar to the method of Ref.~\cite{major2004b}, however, there
are some important differences.
In Ref.~\cite{major2004b}, the radial anisotropy of the
two-photon momentum density in {\bf p}-space served as the comparative
quantity, and the enhancement
was fixed to that chosen in its initial calculation (in that case, the JS model
was used). Here, we perform our comparison in {\bf k}-space, corresponding to
the Lock-Crisp-West-folded data, and explicitly include enhancement of the form
outlined above (SD model) in the fitting.
The advantage of operating in {\bf k}-space (aside from the
smaller array sizes involved) is
principally that we are sensitive directly to the projected Fermi breaks, rather
than the many weaker FS signatures that are distributed throughout the {\bf
p}-space spectrum. An additional consideration, however, is the contribution
from higher momentum components (Umklapp processes), whose enhancement has
presented a challenge for theoretical models (see, for example,
Ref.~\cite{sormann1996}). It is noted that operating in {\bf k}-space
involves the
folding of the Umklapp contributions into the first BZ, both experimentally and
theoretically, and that any non-trivial behavior of these contributions is
subsequently lost.
However, we have checked our results with the equivalent {\bf p}-space spectra,
and in particular near the Umklapp regions (as well as its integral, which
represents an analogue of the coincidence Doppler broadening spectra).
Crucially, we find the data are equally well-reproduced using such a
{\bf k}-space approach as they are with the traditional JS model.
The fitting parameters constitute the energy shift, $\delta_j$, for each band in
the fit (typically those that cross $E_{\rm F}$), two scaling parameters for
each experimental projection, $\Delta_m$ and $S_m$, that approximately relate to
the ${\bf k}$-independent core contribution and the number of counts in the
2D-ACAR spectra respectively, and the enhancement parameters, $\gamma_{n,l}$, of
which there are typically three for simple systems.
These are simultaneously adjusted using the {\tt MINUIT} package
\cite{minuit} and the computed {\bf k}-space density is compared with the data
until convergence is reached.
Note that we fit the ratios of the enhancement
parameters, absorbing their magnitude into the scaling parameters (the absolute
magnitude of the enhancement parameters is indistinguishable from the scaling
parameters in the data).
Our definition and treatment of the scaling parameters have an important
consequence. As mentioned above, we do not treat the enhancement of the core
electrons, preferring to concentrate on the description of the shape of the RMD.
Such an approach means that good agreement can be obtained with the IPM if we
consider a {\em negative} contribution from core annihilations. Whilst this is
clearly unphysical, it stems from the strong overestimation of the enhancement
of deeply-bound electrons within the IPM. Here, we are most interested in the
band properties of the momentum distribution (i.e.\ its {\em shape}),
and in particular its FS signatures. We point out that in the following
discussion, even when the IPM appears to give reasonable agreement with our
data, the agreement with positron lifetime measurements (see, for example,
Refs.~\cite{barbiellini1991,puska1986,takenaka2008}) would be very poor,
in contrast to the other enhancement models that are addressed here.
\section{Transition and noble metals}
\label{s:dmetals}
The transition metals and their alloys have traditionally been the subject of
the bulk of experimental investigations of the FS, and a good description of the
electron-positron momentum density and enhancement of such systems has been
vital in understanding 2D-ACAR, and indeed coincidence Doppler broadening
\cite{alatalo1996,asokakumar1996,makkonen2006} data. The JS model was
specifically developed with transition metals in mind, and is generally thought
to provide a good description of the enhancement for $d$-electron densities
(with $r_s \sim 1.8$) \cite{jarlborg1987,barbiellini1991,major2004b}. Here, we
begin by applying the SD enhancement model described above to
some metals whose FSs have been accurately determined via quantum-oscillatory
methods (Ag, V) and one whose FS is inaccessible to conventional FS
probes (paramagnetic Cr), first concentrating on the ``raw'' LMTO
calculations of the RMD. Comparisons are made with both the IPM and the JS model
for enhancement, as well as a simplified version of the Barbiellini-Alatalo
\cite{barbiellini1997,alatalo1996} enhancement scheme. Following this, we
rigidly fit the bands to the experimental data to obtain an experimental
measurement of the FS, in order to assess the sensitivity of this approach to
the FS.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{gammak.pdf}
\end{center}
\caption{(color online) The state-dependent enhancement of Ag from our model,
shown for two
energy bands along the path $\Gamma$-$X$ in the BZ.
(a) The dispersion of bands 1 and 6; (b) the enhancement from our
fit to the experimental data; (c) and (d) the character of bands 6 and 1
respectively.
Note that band 6 crosses $E_{\rm F}$, shown by the dotted
line in (a); above which the enhancement is unphysical.}
\label{f:gammak}
\end{figure}
Three 2D-ACAR projections ([100], [110] and [111]) were obtained from a single
crystal of Ag at $\sim 70$~K, with a resolution full width at half maximum
of $0.71 \times 1.11$ mrad in the $p_x$
and $p_y$ data axes respectively (corresponding to $\sim 12 \% \times 19$ \% of
the BZ of Ag). For V, four projections were obtained along the [100],
[110], [210] and [211] directions at room temperature with a resolution of
$1.11 \times 1.33$ mrad (with the exception of the [110] direction, which was
collected at $\sim 24$ K with resolution $0.83 \times 1.11$ mrad). Paramagnetic
Cr was measured along the [100] and [110] directions at 353~K, well above
the N\'{e}el temperature (with
a resolution function the same as the room temperature V measurements).
For each sample presented in this manuscript, the 2D-ACAR spectra have been
carefully checked to confirm the absence of any defect or impurity signatures
in the spectra.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{rawdata.pdf}
\end{center}
\caption{(color online) Comparison between experimental {\bf k}-space momentum
density for (a) Ag [100] projection,
(b) V [110] projection and (c) Cr [110] projection
and the computed raw band calculation of the RMD for IPM, JS and SD models. High
symmetry points in projection have been labelled.
Note that in (a) the horizontal and vertical axes are $\left<110\right>$
crystallographic axes; the $\left<100\right>$ axes are along the diagonal.}
\label{f:rawdata}
\end{figure*}
LMTO calculations were performed over 1505 k-points in the irreducible wedge of
the face-centered cubic BZ for Ag, and over 6201 k-points in the irreducible
wedge of the body-centered cubic BZ for V and Cr. For each material, the RMD was
computed for both the IPM and the JS parameterizations of the enhancement,
and convoluted with the appropriate experimental resolution function. This was
compared to the experimental data with adjustments only to the scaling
parameters, $S_m$ and $\Delta_m$. The SD model of the enhancement was then
obtained by simultaneously fitting the orbital enhancement factors, $\gamma_l$.
In each
case, a goodness-of-fit parameter, $\chi_{\rm red}^2$, was computed as a
weighted average of that from each experimental and theoretical projection.
Comparisons were also made for the raw band calculation with the model of Refs.\
\cite{barbiellini1997,alatalo1996} by computing the annihilation rates
associated with each orbital both within the GGA \cite{barbiellini1995} and IPM,
where the enhancement of each orbital in this model is the ratio of these
annihilation rates. However, it should be noted that this scheme still invokes a
parameterization of the enhancement in terms of the electron-gas parameter. To
compare with our experimental values, we integrate over {\em all} ${\bf
k}$-states to obtain the annihilation rate from all electrons of orbital quantum
number $l$. In Eq.~\ref{e:lambdalda}, this corresponds to substituting the
(partial) electron density due to each $s$, $p$, $d$ and $f$ orbital, $n_l({\bf
r})$ for $n({\bf r})$ to obtain $\lambda_l$ in the GGA, rather than the method
of Ref.\ \cite{barbiellini1997} in which $n_{j, {\bf k}}({\bf r})$ is used. In
the following, we refer to this as the simplified Barbiellini-Alatalo (SBA)
model (in which the `simplified' reflects the integration over all {\bf
k}-states).
\subsection{Raw band calculations}
For the raw band calculations, the comparison between theory and experiment
depend on (i) a good {\em ab initio} description of the electronic structure,
and in particular the FS, and (ii) a reliable understanding of the
electron-positron enhancement factor. The well-known FS of Ag consists of just a
single sheet that is only slightly perturbed from the free-electron sphere, most
notably along the [111] direction where the FS intersects with the BZ boundary
to form a neck at the $L$-point of the BZ. Band structure calculations within
the LDA reproduce the precise measurements of quantum oscillations
\cite{halse1969,coleridge1972,coleridge1982} very well and it therefore provides
an excellent candidate in which to test models for the enhancement, for which
the JS model would be an obvious choice. As demonstrated by
Fig.~\ref{f:rawdata}a and \ref{f:rawline}a, the experimental data are
well-described by the raw band calculations of the RMD, in which even the IPM
(including only positron wavefunction effects) works reasonably well.
Quantitatively, as demonstrated in Table \ref{t:rawband}, the JS enhancement
model is found to improve the agreement between experiment and theory,
particularly near the projected $\Gamma{X}$ and $L$ points of the BZ. However,
the current SD model is able to bring the theoretical RMD into much closer
agreement with the data by de-enhancing the $s$ and $d$ states relative to the
$p$ states. For Ag, the bands below $E_{\rm F}$ are predominantly $d$-bands,
with some hybridization with the $5s$ state, but the band that crosses $E_{\rm
F}$ has substantial $p$ character. The de-enhancement of $d$ states relative to
the $sp$ bands is well-known \cite{jarlborg1987} and is attributed to the
relative localization of $d$ electrons, particularly near the top of the
$d$-bands. That the $p$ enhancement appears to be quite strong can be explained
by a Kahana-like momentum enhancement, in which those electron states nearest
${\bf k}_{\rm F}$ are most enhanced. In Fig.\ \ref{f:gammak} the measured SD
enhancement is plotted along $\Gamma$-$X$, accompanied by the band dispersion
and character. As can be seen from Fig.\ \ref{f:gammak}b, the enhancement of
band 6 grows substantially as the band approaches the Fermi level, replicating
the Kahana-like momentum dependence of the enhancement. This is captured in our
model by the enhancement of the $p$-like states; as demonstrated by Fig.\
\ref{f:gammak}c, the enhancement of both bands 1 and 6 closely follow their
respective $p$ character.
Note that, owing to the weak contribution from high-lying $f$ states, a good
quantification of their enhancement is not possible.
\begin{table}[tb]
\begin{center}
\begin{tabular}{||c|c||cccc||c||}
\hline
\multicolumn{2}{||c||}{} &
$\gamma_s$ & $\gamma_p$ & $\gamma_d$ & $\gamma_f$ & $\chi_{\rm red}^2$ \\
\hline
\multirow{4}{*}{\bf Ag} & IPM & - & - & - & - & 12.51 \\
& JS & - & - & - & - & 10.37 \\
& SD & 0.81 & 1.00 & 0.81 & (0.76) & 6.57 \\
& SBA & 0.92 & 1.00 & 0.64 & (1.08) & 8.98 \\
\hline
\multirow{4}{*}{\bf V} & IPM & - & - & - & - & 19.74 \\
& JS & - & - & - & - & 26.56 \\
& SD & 0.69 & 1.00 & 0.78 & (0.57) & 13.32 \\
& SBA & 0.97 & 1.00 & 0.83 & (1.02) & 32.86 \\
\hline
\multirow{4}{*}{\bf Cr} & IPM & - & - & - & - & 8.15 \\
& JS & - & - & - & - & 5.82 \\
& SD & 0.82 & 1.00 & 0.54 & (0.87) & 2.54 \\
& SBA & 0.97 & 1.00 & 0.80 & (1.03) & 5.09 \\
\hline
\end{tabular}
\end{center}
\caption{The results of the fit between the different parameterizations of the
enhancement and the data.
For the SD
model, the $\gamma_l$ for each state obtained from the fit is also given,
normalized to $\gamma_p = 1$. The $\gamma_l$ for the SBA model are determined
from the partial annihilation rates described in Eq.\ \ref{e:lambdai}. The
errors in the fit of the $\gamma_l$ of the SD model are $\sim \pm 0.01$. Note
that the higher statistical precision of the V data yields a relatively large
$\chi_{\rm red}^2$ parameter.}
\label{t:rawband}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{rawline.pdf}
\end{center}
\caption{(color online) The raw-band RMD of (a) Ag, (b) V and (c) Cr shown in
Fig.\ \ref{f:rawdata}, shown here along a path in the BZ.}
\label{f:rawline}
\end{figure}
The situation is more complicated for V and Cr, for which the details of the
near-$E_{\rm F}$ electronic structure, including the precise dimensions of the
FS, are either not well-reproduced by our band calculations (V), or have not
been accurately determined (paramagnetic Cr). Figs.\ \ref{f:rawdata}b and
\ref{f:rawdata}c show representative 2D-ACAR projections of V and Cr (along the
[110] direction) respectively, compared with the corresponding theoretical
quantities. The FS of V is composed of two sheets. The first sheet (originating
from band 2 of predominantly $3d$ character) forms a small $\Gamma$-centred hole
octahedron that encloses $\sim 0.12$ holes and has remained unobserved in
quantum oscillation data, although its presence has been confirmed by 2D-ACAR
measurements \cite{singh1985b,pecora1988,singh1984} (this sheet is visible in
the experimental data in Fig.~\ref{f:rawdata}b at the projected ${\Gamma}N$
point). Band 3, on the other hand,
experiences appreciable hybridization with the $4p$ states above $E_{\rm F}$ and
forms a $\Gamma$-centred jungle-gym hole FS as well as some hole ellipsoids that
are centred at $N$. These $N$-hole ellipsoids can be clearly seen in the data of
Fig.\ \ref{f:rawdata}b at the projected $N$-point of the BZ, where the density
experiences a local dip due to their presence. These features in the IPM and JS
calculations are predicted to be substantially too large and too strong (see
Fig.\ \ref{f:rawline}b), and the enhancement of the high-density surrounding
region is not well reproduced. Although, as pointed out by Jarlborg and Singh,
the enhancement is expected to be less important for a less-full $d$-band, the
JS enhancement is actually found to perform worse than the IPM for V (see Table
\ref{t:rawband}), at least in the shape of the distribution (positron lifetime
predictions are substantially better described by the JS model
\cite{barbiellini1991}).
V was used as a test material by Jarlborg and Singh in their
presentation of the JS enhancement model, in which they comment that the IPM
already provides a reasonable description of the momentum density
\cite{jarlborg1987}, and that their model offered only weak improvement.
However, their comparisons were made with electronic structure calculations
where the bands had been rigidly shifted to agree with de Haas-van Alphen (dHvA)
measurements of the $N$-hole ellipsoids \cite{parker1974}. Indeed, in Fig.\
\ref{f:rawline}b it is obvious that the dimensions of the dips in the momentum
density along $NH$-$N$-$HN$ are incorrectly placed with respect to the data. In
Section \ref{ss:rigidband}, we address such inconsistencies by rigidly shifting
the bands to improve agreement between experiment and theory.
Cr neighbors V in the periodic table, having an extra electron,
yet its paramagnetic FS has remained relatively unexplored experimentally,
principally owing
to the emergence of an ordered spin-density wave phase below $\sim 312$ K
\cite{fawcett1988}, where the high temperature precludes quantum oscillatory
measurements in the paramagnetic phase, and strong spin fluctuations appear to
suppress the measurement of the nested sheets in the ordered phase
\cite{ditusa2010}. Theoretically,
the FS is composed of three sheets, the first of which (band 3) contributes some
small electron `lenses' midway between $\Gamma$ and $H$. Band 4 forms some $N$
hole ellipsoids and $H$-centred octahedra, whereas in band 5 there is a
$\Gamma$-centred electron `jack'. Molybdenum, isoelectronic to Cr, shares
a similar FS topology, in which the $N$-hole ellipsoids and the electron jack
can be clearly visualized in the [110]-projected {\bf k}-space density of
2D-ACAR measurements \cite{fretwell1995,hughes2004}. For Cr, these features,
shown in Fig.\ \ref{f:rawdata}c near the projected $N$ points, are obscured
in the measurement, presumably owing to
enhancement effects \cite{dugdale1998}. Indeed, the $N$-hole ellipsoids are
more evident in the IPM and JS projected densities than they are in the data.
Overall, the agreement between the IPM and JS calculations of the RMD and the
data is reasonable (see Table \ref{t:rawband}), but is particularly poor near
the projected $N$-points of the BZ as well as midway along the $NH$-$\Gamma{N}$
path (see Fig.\ \ref{f:rawline}c). Here, the knobs of the electron jack project
on top of one another, and the IPM and JS do not predict the de-enhancement of
the momentum density very well in this part of the BZ.
The application of the SD model, however, considerably improves the agreement
between experiment and theory by substantially de-enhancing the $s$ and $d$
states. In Figs.\ \ref{f:rawline}b and \ref{f:rawline}c, this can be most
clearly seen at the projected $N$-points of the BZ, as well as the momentum
density near the ${\Gamma}N$ points. For both V and Cr, the hybridization of the
valence states with the unoccupied $4p$ states indicates the importance of the
$p$ electrons in deciding the topology of the FS, and their proximity to $E_{\rm
F}$ means they have a strong impact on the enhancement of the momentum density
in 2D-ACAR measurements. Previous non-iterative comparisons of orbital-weighted
band theory and 2D-ACAR data have been made in {\bf p}-space for V
\cite{genoud1990}, in which a de-enhancement of the $s$ and $d$ states by $\sim
0.8$ relative to the $p$ states was favored. Our results are close to these,
where we obtain 0.69 and 0.78 for $s$ and $d$ states respectively, corresponding
to a slightly greater de-enhancement of the lower-lying $s$ states. Similarly
for paramagnetic Cr, Matsumoto and Wakoh \cite{matsumoto1987} estimated
(also non-iteratively) that the Cr $d$ states were de-enhanced by $\sim 0.67$
relative to the $sp$ states (which were considered together). Our results
correspond well with their findings, where we obtain 0.82 and 0.54 for $s$ and
$d$ states, relative to the $p$ states. The stronger enhancement of the Cr $s$
states (compared with V) may be explained by the higher occupation of the $4s$
states in Cr (they are almost twice as occupied in Cr).
Finally, we comment on the predictions for state enhancement made by the SBA
model. Apart from Ag, which has a much higher $d$ electron density than either V
or Cr, the orbital enhancement ratios are predicted to be very similar (see
Table \ref{t:rawband}), with a weak de-enhancement of the $s$-states and a
modest de-enhancement of the $d$-states relative to the $p$-states. For Ag, a
rather more exaggerated de-enhancement is predicted for the $d$-states.
Qualitatively, these results are in agreement with our measured values, but
differ substantially in magnitude and lead to a slightly higher $\chi^2_{\rm
red}$ parameter than the current SD model. Nevertheless, the SBA model provides
better agreement with the data than either the IPM or the JS model for Ag and
Cr, supplying a more robust predictive scheme for computing the enhancement in
2D-ACAR momentum distributions.
That it does not fair so well for V is mostly accounted for by the rather
larger corrections to the LDA band structure that are required for V, a
topic that will be returned to in the next section. Here, we emphasize that
the SBA model is expected to improve the agreement with the data over the
IPM or JS models when extensions to the LDA, such as non-local potentials
\cite{barbiellini1989} or self-energy corrections \cite{barbiellini2005},
that improve the description of the FS are included.
The most probable origin for the discrepancy between the SBA
scheme and our measured SD enhancement is the omission of a Kahana-like
energy-dependence or momentum-dependence in the predictive scheme of
Ref.~\cite{barbiellini1997}. As already highlighted, some of the results for the
enhancement in the current SD model, particularly the apparent strong $p$
enhancement in V and Cr, reflect a Kahana-like enhancement of those electron
states near $E_{\rm F}$. In our model, where a state is not too dispersive in
energy, this is naturally captured by enhancement of that state. The authors of
Ref.~\cite{barbiellini1997} comment that the Kahana-like energy- or
momentum-dependence appears to be less important than the state-dependence from
their results, a conclusion that this work substantiates, but these results
suggest that including such enhancement could produce a good improvement in the
agreement between experiment and theory. In Section \ref{s:model}, we apply
just such an energy-dependent term to the SBA enhancement factor, and
demonstrate the improved predictive capacity of such a model.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.73\linewidth]{fitdata.pdf}
\end{center}
\caption{(color online) Comparison between experimental data for
(a) V [110] projection and (b) Cr [110] projection
and the rigid band fit to the RMD for the SD model, shown in the same
way as Fig.\ \ref{f:rawdata}.}
\label{f:fitdata}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.00\linewidth]{fitline.pdf}
\end{center}
\vspace*{-0.2in}
\caption{(color online) The rigid-band fit to the RMD of (a) Ag, (b) V and (c)
Cr shown in Fig.\ \ref{f:fitdata}, shown here along a path in the BZ.}
\label{f:fitline}
\end{figure}
\subsection{Rigid-band fit}
\label{ss:rigidband}
In 2D-ACAR investigations of the FS, the traditional method of extracting the FS
from experimental data is to contour the data at a level that corresponds to
extrema in the first derivative of the data, and it is well known that
enhancement effects do not shift the location of these breaks
\cite{majumdar1965}.
While first-principles calculations are often able to make excellent qualitative
predictions about the nature of the Fermi surface, when subject to detailed
scrutiny in light of precise experimental data it is often found that
quantitative differences exist. Shortcomings in the approximations used in the
calculations (e.g.\ exchange-correlation functional, neglect of relativistic
effects) mean that in reality it is difficult to get the Fermi surface correct.
These differences can often be reduced or eliminated by small shifts of the
relevant bands with respect to the Fermi level, and
it has recently become feasible, and indeed quite common, to `tune' a
band-theoretical calculation in this way
(e.g. see Refs.~\cite{major2004b,utfeld2009,utfeld2010}).
Such an approach requires an accurate
description of the positron enhancement, if conclusions regarding details of the
FS itself are to be drawn from such a fit, and we now turn our attention to
investigate the behavior of our SD model applied to such detailed FS studies.
For the rigid-band fit of the electronic structure, the Fermi level for
each band near $E_{\rm F}$ was fitted to the data. In the case of the SD
enhancement model, the orbital enhancement factors were fitted simultaneously.
The $\chi_{\rm red}^2$ was computed as before, and the number of electrons
enclosed by the fitted FS (i.e. the occupied fraction of the Brillouin zone)
was obtained.
The results of the rigid-band fit to the data are displayed in Fig.\
\ref{f:fitdata}, and demonstrate substantial improvement over the corresponding
raw band calculations of Fig.\ \ref{f:rawdata}. In Table \ref{t:rigidfit},
the $\chi^2_{\rm red}$ is shown for each fit, along with the fitted orbital
enhancement factors for the SD model. Beginning with some general comments,
we note that the orbital enhancement parameters obtained in the
SD model are only moderately adapted as a consequence of including the
bands in the fit, and the same general trends are observed. Additionally,
it is noteworthy that in almost every case the shift in the energy band
(see Table \ref{t:bandshifts}) is found to be smallest for the SD model
(with the exception of band 3 of V).
Fig.~\ref{f:fitline} shows the RMD along the same path through the BZ
as in Fig.~\ref{f:rawline}, and we will now concentrate in more detail on the
agreement between experiment and theory.
As indicated by the small change
in electron count of the shifted bands, the change in the FS itself is small,
owing to the appreciable dispersion of band 6 at $E_{\rm F}$, and the shift
in the Fermi wavevector is ${\Delta}k_{\rm F} \sim 0.02\,(2\pi/a)$
(just $\sim 15\%$ of the resolution function).
\begin{table}[tb]
\begin{center}
\begin{tabular}{||c|c||cccc||c||}
\hline
\multicolumn{2}{||c||}{} &
$\gamma_s$ & $\gamma_p$ & $\gamma_d$ & $\gamma_f$ & $\chi_{\rm red}^2$ \\
\hline
\multirow{3}{*}{\bf Ag} & IPM & - & - & - & - & 5.53 \\
& JS & - & - & - & - & 4.84 \\
& SD & 0.88 & 1.00 & 0.85 & (0.76) & 4.09 \\
\hline
\multirow{3}{*}{\bf V} & IPM & - & - & - & - & 7.77 \\
& JS & - & - & - & - & 8.40 \\
& SD & 0.61 & 1.00 & 0.63 & (0.49) & 3.79 \\
\hline
\multirow{3}{*}{\bf Cr} & IPM & - & - & - & - & 2.52 \\
& JS & - & - & - & - & 1.85 \\
& SD & 0.83 & 1.00 & 0.61 & (1.27) & 1.28 \\
\hline
\end{tabular}
\end{center}
\caption{The results of the rigid-band fit between the different
parameterizations of the enhancement and the data, presented in the same
way as Table \ref{t:rawband}. The shifts in the energy bands for each fit
are shown in Table \ref{t:bandshifts}.}
\label{t:rigidfit}
\end{table}
The improvement is much more dramatic for V (Fig.\ \ref{f:fitline}b),
in which the size of the $N$-hole features in the data is now well-described
by all of the enhancement models, stemming from opposite shifts in bands 2
and 3.
After rigidly shifting the bands, the IPM and JS demonstrate similar shifts of
the energy bands and a similar goodness-of-fit parameter, leading to an excess
in occupied volume of 0.14 and 0.10 of an electron
The improvement in the SD model is more pronounced, however.
The $d$ bands are well known
to be placed too low by the LDA with respect to $sp$ bands. As noted in
Ref.~\cite{major2004b}, band 2 (of predominantly $d$ character) is pushed up in
energy by the fit towards the higher $p$ character of band 3, which is pulled
down by the fit, correcting this tendency.
\begin{table}[t]
\begin{center}
\begin{tabular}{||c|c||ccc||c||}
\hline
\multicolumn{2}{||c||}{} & \multicolumn{3}{c||}{band shifts / mRy} &
electron +/- \\
\hline
\multirow{4}{*}{\bf Ag} & & band 6 & - & - & \\
\cline{2-6}
& IPM & -21.2 & - & - &+0.08 \\
& JS & -18.3 & - & - &+0.07 \\
& SD & -14.2 & - & - &+0.05 \\
\hline
\multirow{4}{*}{\bf V} & & band 2 & band 3 & - & \\
\cline{2-6}
& IPM & +22.3 & -15.8 & - &+0.14 \\
& JS & +26.5 & -15.4 & - &+0.10 \\
& SD & +18.8 & -16.4 & - &+0.17 \\
\hline
\multirow{4}{*}{\bf Cr} & & band 3 & band 4 & band 5 & \\
\cline{2-6}
& IPM & -25.1 & +15.6 & +13.2 &-0.02 \\
& JS & -22.9 & +12.2 & +13.2 &-0.01 \\
& SD & -18.5 & +5.0 & +13.2 & 0.02 \\
\hline
\end{tabular}
\end{center}
\caption{The shifts in the energy bands for each of the rigid-band fits. Also
shown is the change in electron count in the BZ due to the fit. Note that for
Cr band 5 (that just grazes $E_{\rm F}$)
is completely expelled by all of the fits. The errors in the shifts
of the bands are in each case $\lesssim 1$~mRy.}
\label{t:bandshifts}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{crchi.pdf}
\end{center}
\caption{(color online) The real (top)
and imaginary (bottom) parts
of the static susceptibility,
$\chi_0({\bf q})$, of paramagnetic Cr, calculated for the raw
band LMTO calculations as well as the results of the rigid-band fit to the data
with the IPM, JS and SD models of enhancement. The dashed vertical line
represents the peak in the real part of the susceptibility. The inset shows
a slice of the FS through the $(001)$ plane, with the arrow depicting the
nesting that gives rise to the peak in $\mathfrak{Im}\;\chi_0({\bf q})$
between the hole (outer, red) and electron (inner, blue) FS sheets.}
\label{f:crchi}
\end{figure}
The energy shifts of the bands are also in good agreement with quantum
oscillations. Comparing the semi-axis radii of the ellipsoids with high-quality
dHvA parameterizations (see Table \ref{t:vfs}), we find much improved
correspondence with experiment than the raw calculation, and indeed they compare
favorably with the shifts of Ref.\ \cite{major2004b}. The differences in the
results of Ref.\ \cite{major2004b} and the current {\bf k}-space approach
reflect the different sensitivity of the two techniques to specific features of
the data (for example, compare the $N$-$P$ radius with the $N$-$H$ radius). It
is also worth mentioning that the jungle-gym FS also originates from band 3, and
that this will also contribute to the shifts of this band, and so considering
the orbits about the $N$-ellipsoids alone may be misleading. Unfortunately,
there is a dearth of data for this sheet of FS, and comparisons are hard to
draw. As a final point, quantum oscillations appear to be relatively insensitive
to the FS of band 2, whereas we find a strong dependence of our fit to that
band, in agreement with other positron studies in V
\cite{singh1985b,pecora1988}.
Finally, for Cr (Fig.\ \ref{f:fitline}c), the data are reasonably well described
by the SD model throughout the BZ, whereas the IPM and JS models struggle near
the ${\Gamma}N$ points in both the raw band calculations and the rigid-band
fits.
In the absence of high-precision FS data for paramagnetic Cr, owing to the
ordering temperature ($T_{\rm N} \sim 312$K) of the spin-density wave, a robust comparison can instead
be made of the nesting vector of the paramagnetic FS that is widely believed to
determine the ordering (and has remained difficult to establish experimentally).
High-resolution neutron diffraction measurements have established the ordering
vector to be ${\bf Q} = (0, 0, 0.9516)\;(2\pi/a)$ (see Ref.\ \cite{fawcett1988}
and references therein). Our raw LMTO calculations predict (via a computation
of the static susceptibility, $\chi_0 ({\bf q})$, see for example
Ref.~\cite{laverock2009})
a nesting vector
${\bf q} \sim 0.930 \;(2\pi/{\bf a})$ (see Fig.\ \ref{f:crchi}),
which is rather smaller than the neutron measurements. Since quantum
oscillations are precluded in the paramagnetic phase (and have recently remained
unobserved from the relevant FS sheets in the ordered phase owing to strong
spin-fluctuation induced
scattering \cite{ditusa2010}), the only data on the FS
that has been capable of extracting this nesting vector have been some recent
angle-resolved photoemission measurements on Cr(110) thin films
\cite{schafer1999,rotenberg2005}, in which a nesting vector of ${\bf q} \sim
0.950 \pm 0.005\;(2\pi/{\bf a})$ is reported, in very good agreement with
neutron measurements. Our rigid-fit to the data (Fig.\ \ref{f:crchi})
culminates in a FS nesting vector of ${\bf q} \sim 0.950 \pm 0.002\;(2\pi/{\bf
a})$, where the error quoted is the combined error from shifting the two bands
to match the experimental results, representing the highest-precision
experimental confirmation of the relevant dimensions of the FS of paramagnetic
Cr from a bulk measurement. In contrast to this excellent agreement, the shifts
of the bands obtained using the IPM and JS models suggest nesting
vectors of ${\bf q} \sim 0.924 \pm 0.002\;(2\pi/{\bf a})$ and ${\bf q} \sim
0.932 \pm 0.002\;(2\pi/{\bf a})$ respectively.
The conclusions we draw from this section are the following. First,
our approach provides a robust empirical means of {\em measuring} the orbital
electron-positron enhancement factors, that are truly state-dependent (i.e.\
{\bf k}-dependent). Second, this measurement is not so strongly dependent on the
accuracy of the band calculation, being rather more sensitive to the overall
shape of the momentum distribution. Third, simultaneous fitting of the energy
bands {\em and} the orbital enhancement lead to a tuned FS that is in better
agreement with other FS data than is the raw band calculation, as well as
in good agreement with previous {\bf p}-space fitting approaches. Finally,
the band shifts that are required to reproduce experimental data (that is
in better agreement) are generally smaller for the SD model than the other
approaches investigated here, indicating that artificially large rigid shifts
in the bands can develop as a consequence of an inadequate description of
the enhancement.
\begin{table}[t]
\begin{center}
\begin{tabular}{||c||c|c|c|c||c||}
\hline
direction & LMTO & Ref.\ \cite{parker1974} dHvA & SD fit & Ref.\
\cite{major2004b} fit \\
\hline
$N$-$P$ & 0.257 & 0.223 & $0.224\pm0.002$ & 0.245 \\
$N$-$\Gamma$ & 0.254 & 0.212 & $0.204\pm0.002$ & 0.231 \\
$N$-$H$ & 0.168 & 0.176 & $0.146\pm0.001$ & 0.160 \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of the semi-axis radii (in units of $2\pi/a$)
of the $N$ hole ellipsoids from our
raw LMTO calculation and the fitted SD momentum density for V. Comparisons are
made with the high-precision parameterizations of dHvA data of Ref.\
\cite{parker1974} as well with the fitting technique (also applied to 2D-ACAR
data) employed by Ref.\ \cite{major2004b}. The errors reflect the error in
locating the minimum of the fit with respect to the shift in the bands. Note
that the dHvA radii rely on the assumption of perfect ellipsoids.}
\label{t:vfs}
\end{table}
\section{Simple metals}
\label{s:alkali}
We now turn to the other regime of enhancement, in which the bands, of $sp$
character, are closer to the nearly free electron model. Aluminium and the
alkali metals (and their alloys) provide a more stringent test for the SD
model. The electron-gas parameter of Al is $r_s = 2.65$, above the
point at which the JS (which does not conserve the low-density limit) and
the BN (which does) begin to diverge; as a consequence JS is not expected
to perform well here.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{al_data.pdf}
\end{center}
\caption{(color online) Comparison between experimental data for Al projected
down the [111]
axis and the computed RMD for IPM, JS and SD models. (a) The raw LMTO band
calculation, and (b) the results of the rigid fitting of the energy bands. The
high symmetry points in projection ($\Gamma{L}$, $X$ and $W$) are shown in
(a), and the boundary of the first BZ is marked by the dotted line.}
\label{f:aldat}
\end{figure}
The FS of Al is composed of two bands, one which forms a $\Gamma$-centred hole
sheet from band 2 that lies completely in the first BZ, and from band 3 a
so-called `dismembered monster', that consists of square electron rings that run
the length of the edges of the first BZ except at the corners ($W$-points)
\cite{cracknell1969}. Two 2D-ACAR projections along the [110] and [111]
directions were measured at room temperature and compared with LMTO calculations
performed over 1505 k-points in the irreducible BZ using the IPM, JS and SD
enhancement as before.
In Fig.\ \ref{f:aldat}, the data for the [111] projection is shown alongside the
LMTO calculations of the RMD. The FS structure can be clearly seen in the data
(shown in the bottom right panel), where the low density in the center reflects
the band 2 hole sheet, and the higher density at the edges of the projected BZ
come from the electron rings of band 3. At the corner of the projected BZ (near
the $W$-point), the particularly high region is due to the projection of the
rings in neighbouring zones along the $\left< 111 \right>$ directions. As can be
seen in the left panels of Fig.\ \ref{f:aldat}, the IPM and JS models are
particularly poor at describing the enhancement at the edges of the zone that
connect these strong features. Moreover, a small local peak at $\Gamma{L}$ that
is predicted by both IPM and JS is not observed at all in the data.
Quantitatively, as might be expected from the electron density of Al, the JS
model fairs poorly for the raw band calculation, and even worse than the IPM
(see Table \ref{t:alfit}). The SD model, however, does an excellent job of
describing the RMD of Al, correctly accounting for the absence of the local peak
at $\Gamma{L}$ and the connectivity of the strong features near $W$, and leading
to an almost order-of-magnitude improvement in the $\chi^2_{\rm red}$ parameter.
Here, the $s$-states are de-enhanced substantially, presumably owing to them
lying very low in energy. Unlike the previous $d$-electron systems, the
de-enhancement of the (unoccupied) $d$ states is not observed for Al.
\begin{table}[tb]
\begin{center}
\begin{tabular}{||c||cccc||c||}
\hline
& $\gamma_s$ & $\gamma_p$ & $\gamma_d$ & $\gamma_f$ & $\chi_{\rm red}^2$ \\
\hline
\multicolumn{6}{||c||}{raw band} \\
\hline
IPM& - & - & - & - & 15.40\\
JS & - & - & - & - & 17.20\\
SD & 0.53& 1.00&(1.29)&(0.60)& 2.29\\
\hline
\multicolumn{6}{||c||}{rigid fit} \\
\hline
IPM& - & - & - & - & 3.93\\
JS & - & - & - & - & 3.92\\
SD & 0.60& 1.00&(1.09)&(0.72)& 2.12\\
\hline
\end{tabular}
\end{center}
\caption{The results of the fit between the different parameterizations of the
enhancement and the data for Al, shown in the same way as Table
\ref{t:rigidfit}. The band-shifts that accompany the rigid-band fit are shown
in Table \ref{t:alshift}.}
\label{t:alfit}
\end{table}
\begin{table}[tb]
\begin{center}
\begin{tabular}{||c||cc||c||}
\hline
& \multicolumn{2}{c||}{band shifts / mRy} & electron +/- \\
\hline
& band 2 & band 3 & \\
\hline
IPM & -39.2 & -30.0 & +0.21 \\
JS & -40.0 & -33.7 & +0.22 \\
SD & -11.9 & -10.2 & +0.07 \\
\hline
\end{tabular}
\end{center}
\caption{The shifts in the energy bands for each of the rigid-band fits of Al.}
\label{t:alshift}
\end{table}
When the bands are fitted, the agreement between data and theory for each model
is very good. However, for the IPM and JS models the local peak at $\Gamma{L}$
persists, albeit at a much weaker amplitude. Moreover, consistent with the
previous conclusions, the shifts in the energy bands are substantially larger
for the IPM and JS models than the SD model (see Table \ref{t:alshift}),
leading to an electron excess of
$\sim 0.21$ (over a single FS sheet). For the SD model, this discrepancy is
much reduced, at just 0.07 electrons. Similarly to Ag, the FS of Al is already
well-described by the LMTO calculation, and comparisons with quantum oscillatory
data \cite{kamm1963,larson1967} agree with the raw band and SD rigid-band fit
to within ${\Delta}k_{\rm F} \sim 0.03\,(2\pi/a)$ ($\sim 15\%$ of the
resolution function).
\section{Phenomenological model}
\label{s:model}
Given the above results, we aim to find a phenomenological model that imparts
predictive capability on the calculation of the RMD. Taking the SD fitted
FS as a baseline,
we attempt to improve on the SBA model of the enhancement. The predictions
of the SBA enhancement are, in general, satisfactory, offering a similar
description of the experimental RMD (in some cases slightly better, in others
slightly worse) to the JS enhancement model. The predictions of the SBA model
can be understood largely from the perspective of the localization of the
states, in which $s$ and $p$ states experience similar enhancement over the IPM,
with the $s$ states in transition metals slightly less than $p$ due to their
slightly more localized nature in these systems. The $d$ states are enhanced
much less in the transition metals, associated with the greater localization,
and the increasing localization as the $d$-band becomes more filled is reflected
by the greater de-enhancement of the $d$-states in Ag when compared with
either Cr or V.
The Kahana model for enhancement, applied to a homogeneous electron gas and
parameterized in terms of $(k/k_{\rm F})^2$, is not expected to work
well for $d$-band systems, in which the effects of the crystal lattice can
completely hide the Kahana nature. For this reason, Mijnarends and Singru
(MS)
\cite{mijnarends1979} proposed a scheme parameterized by $\epsilon = (E-E_{\rm
bot})/(E_{\rm F}-E_{\rm bot})$, where $E_{\rm bot}$ is the energy at the bottom
of the conduction band,
\begin{equation}
\gamma = a + b \epsilon + c \epsilon^2,
\label{e:ms}
\end{equation}
where $a$, $b$ and $c$ are constants determined by the electron gas parameter
$r_s$. For a parabolic $s$ band this is identical to Kahana's formalism.
MS demonstrated the applicability of their prescription for the case of
Cu, in which substantial improvement was found (in {\bf p}-space) with this
description. The SBA model accounts for the variations in enhancement due to the
localization of a particular orbital, and its overlap with the positron
wavefunction, but does not consider the proximity of a state to the Fermi level,
leading to an over-estimation of the enhancement of more tightly-bound, filled
$s$ electron shells.
Adding such a scheme to the SBA model was not found to
universally explain the variations in enhancement for our experimental
data without different choices of the constants $a$, $b$ and $c$ (in fact,
following MS, we choose to set $a = 1$ in Eq.~\ref{e:ms} so that $b
\rightarrow b/a$ and $c \rightarrow c/a$).
\begin{table}[tb]
\begin{center}
\begin{tabular}{||c||c||c|c|c||}
\hline
\hspace*{0.3in} & \hspace*{0.1in} SBA \hspace*{0.1in} &
\multicolumn{3}{c||}{SBA-MS} \\
\hline
& $\chi_{\rm red}^2$ & \hspace*{0.1in}$N(E_{\rm F})\hspace*{0.1in}$ &
\hspace*{0.1in}$b/a$\hspace*{0.1in} & \hspace*{0.1in}$\chi_{\rm
red}^2$\hspace*{0.1in} \\
\hline
V & 20.41 & 23.80 & 0.700 & 4.30 \\
Cr & 2.50 & 9.52 & 0.202 & 1.39 \\
Ag & 4.71 & 3.60 & 0.042 & 4.50 \\
\hline
Al & 7.90 & 5.04 & 0.589 & 2.28 \\
\hline
\end{tabular}
\end{center}
\caption{The linear component of MS-type energy-dependent enhancement ($b/a$)
obtained
by fitting the SBA model
to the data. $N(E_{\rm F})$ is given in units of states
/ Ry / unit cell, and the quadratic term, $c/a$, in Eq.~\ref{e:ms} is set
to 0. The $\chi_{\rm red}^2$ parameter is given before (SBA) and after (SBA-MS)
the application of the MS-type enhancement.}
\label{t:msfit}
\end{table}
Of particular interest in this analysis is the enhancement for V and Cr, which
are neighbors in the periodic table and would therefore, from the perspective
of a homogeneous electron gas, be expected to follow similar trends in their
enhancement owing to their similar electron density. As can be seen in Table
\ref{t:rigidfit} the measured enhancement of V and Cr is quite different, and
yet V and Cr can each be well-approximated by a calculation of the other's
electronic band structure, with a simple extrapolation of $E_{\rm F}$ to account
for the different band-fillings (i.e. the rigid-band approximation works well).
The usual prescriptions for enhancement, in terms of the electron density, or
even a MS type energy-dependent enhancement, fail to predict such different
shell enhancements. Substituting the measured SD enhancement parameters for Cr
into the V calculation, and vice versa, is not found to describe the data well,
enforcing the idea that the enhancement is substantially and fundamentally
different for these two elements.
Since V and
Cr are electronically very similar, exhibiting the same body-centered cubic
structure, the largest difference between the two is in their band filling and
FS. In V, $E_{\rm F}$ lies close to a peak in the $d$ density of states
with appreciable
($\sim 20$ \%) $p$-character, leading to a total number of states at $E_{\rm F}$
of $N(E_{\rm F}) = 23.80$ states per Ry per atom. In paramagnetic Cr, on the
other hand, the additional electron places $E_{\rm F}$ in a valley between the
bonding and anti-bonding $d$-states with $N(E_{\rm F}) = 9.52$ states per Ry per
atom. It follows that the number of electrons that are capable of screening the
positron impurity (and thus lead to the enhancement of the annihilation rate) in
V and Cr is very different, and cannot be captured by considerations of the
electron density or energy alone. However, such a concept does provide a route
to understanding the different SD enhancement models in V and Cr, and the
different constants $b/a$ and $c/a$ in Eq.~\ref{e:ms} that are required to
explain the data.
In order to test such a correction to the enhancement, we apply a MS-type
enhancement to the SBA model, which is then fitted to the experimental
data. According to Kahana's theory, the quadratic part of the enhancement
parameterization is fairly weak, with $c/a \approx 0.138$ for metallic
densities, and can be well approximated by just a linear component
($b/a$). Here, we adopt just this linear energy enhancement,
and set $c/a = 0$ in Eq.~\ref{e:ms}, leaving just a single fitting
parameter that is capable of adjusting the {\em shape} of the computed RMD:
\begin{equation}
\gamma_{\rm SBA-MS} = \gamma_{\rm SBA} [1 + (b/a) \epsilon].
\end{equation}
The results of such a model, which we refer to as SBA-MS enhancement,
are found to enormously improve the agreement between data and theory
for all materials (see Table \ref{t:msfit}),
leading to $\chi_{\rm red}^2$ parameters that approach
the SD model investigated in Section \ref{s:dmetals}. Moreover, for the three
transition metal elements addressed in this manuscript, this linear component
of energy enhancement is found to scale with the density of states at the
Fermi level (Table \ref{t:msfit}), providing an empirical
model for the
enhancement of $d$-band elements and compounds. For $s$-$p$ electron metals,
the enhancement is found to more closely resemble Kahana's parameters, and
two regimes emerge -- Kahana's prediction for $s$-$p$ simple metals,
and a $N(E_{\rm F})$-dependent set of parameters for $d$ electron metals.
It is interesting to compare our results for an MS-type enhancement to those
applied by Matsumoto and Wakoh for Cr \cite{matsumoto1987}, in which they obtain
a factor $b/a \sim 0.15$ in a non-optimized approach, very close to our 0.20. On
the other hand, Genoud \cite{genoud1990}, employing an MS-type enhancement
independently for $s$, $p$ and $d$ electrons in V, obtained $b/a = 0.1 - 0.2$
for $s$ and $p$ electrons, also in a non-optimized way, which is somewhat
smaller than the optimum $b/a = 0.70$ that we find.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{crmo.pdf}
\end{center}
\caption{(color online) Comparison between experimental data and the
corresponding theoretical
quantities in the raw IPM and rigid-band fit SD models of enhancement for (a) Cr
and (b) Mo. The RMD is shown along a path in the BZ in (c) and (d) for Cr and Mo
respectively.}
\label{f:crmo}
\end{figure}
Armed with such a model for the enhancement, we can now assess its
validity for another metal, specifically the $d$ electron metal Mo, for which we
have just a single [110] projection available, insufficient to permit a reliable
fit of the $\gamma_l$ parameters of the SD model. Instead, we apply
the SBA-MS model to the data, in which the $\gamma_l$'s are computed from
the partial annihilation rates of state $l$ and the energy-dependent enhancement
is provided from $N(E_{\rm F}) = 7.6$ states / Ry / atom and our preceding fit.
For comparison, we also compute the IPM and JS RMD. Both the JS and SBA models
of enhancement, by themselves, offer negligible improvement over the IPM, which
already provides a reasonable description of the data, and the application of
the SBA-MS model improves the agreement only modestly by $\sim 4$~\%. However,
lending freedom to the linear component $b/a$ is not found to provide any
additional improvement, emphasizing that the original IPM calculation was
already satisfactory.
One of the unresolved questions of 2D-ACAR in transition metals is why the {\bf
k}-space density of Cr and Mo (projected along the [110] direction) appear so
different, despite the apparent similarity of their isoelectronic and
isostructural FS topology \cite{dugdale1998} (see Fig.\ \ref{f:crmo}a,b). In
Ref.\ \cite{dugdale1998}, maximum-entropy filtering techniques were employed to
assess the FS breaks in both distributions, ruling out the FS topology as an
explanation; the question of whether positron effects or consequences of the
proximity to magnetic structure in Cr are to blame were left open and have
remained so despite several efforts to resolve the issue, both experimentally
and theoretically \cite{dugdale2000,biasini2000,rubaszek2002}. Here, we are able
to solve this issue, which stems from a strong over-estimation of the
enhancement near $\Gamma{N}$ in Cr, previously highlighted in Fig.\
\ref{f:rawline}. In Fig.\ \ref{f:crmo}c,d, the RMD along a path in the BZ is
shown
for both Cr and Mo for the 2D-ACAR data, and the IPM and SBA-MS models of
enhancement. It is clear that the IPM (which resembles the JS model) looks
similar for both metals, eliminating positron wavefunction effects (which are
included in the IPM) as responsible for the strong difference in the data. On
the other hand, the SBA-MS prediction (which closely resembles our measured SD
model), accounts for the data very closely, unambiguously establishing
enhancement effects as the key.
\section{Conclusions}
We have presented a detailed investigation of the positron enhancement
factor for several metals, providing a quantitative {\em measurement} of the
state-dependence of the enhancement. By combining this with a rigid shift of
the energy bands, we demonstrate that, when the band structure is optimized to
2D-ACAR measurements, the precise location of the Fermi breaks in {\bf k}-space
are sensitively dependent on the accuracy of the enhancement model used in
the calculation. Furthermore, we show that, by employing a state-dependent
model for the enhancement, much improved agreement between the `tuned'
calculation and high-precision quantum oscillatory data can be obtained.
In particular, for Cr our positron measurements yield a nesting vector that
is in excellent agreement with neutron measurements of the spin-density wave
ordering vector, with an estimated accuracy better than 0.5\% of the BZ.
Although alloys have not been investigated here, this approach also allows
for the contribution from different atomic sites to be separated in the
experimental data, allowing a determination of the fraction of annihilations
from each individual element's hybridized wavefunctions (for example, see
Ref.~\cite{al3li}).
Comparisons of our (empirical) model with other popular models
of the enhancement have been made, particularly with the {\em ab
initio} state-dependent model of Barbiellini, Alatalo and co-workers
\cite{barbiellini1997}, for which a semi-empirical energy-dependent correction
is proposed that is found to bring the theory into much better agreement
with the data. Such a combined model therefore provides an accurate
model for the enhancement in momentum density measurements,
such as those of 2D-ACAR or coincidence Doppler broadening techniques.
\section*{Acknowledgments}
We acknowledge the financial support of the EPSRC (UK). We are indebted to the
late Maurizio Biasini for providing us with the 2D-ACAR data of Ag, and would
also like to thank Bernardo Barbiellini for stimulating discussions.
|
1,108,101,565,507 | arxiv | \section{Introduction}
Triangular matrix factorization is a main building block in computational linear
algebra.
Driven by a large range of applications in computational sciences, parallel
numerical dense LU factorization has been intensively studied since
several decades which results in software of great maturity (e.g., LINPACK is used for benchmarking the efficiency of the top
500 supercomputers~\cite{DLP03}).
More recently, efficient sequential exact linear algebra
routines were developed~\cite{DGP08}. They are used in algebraic cryptanalysis, computational number theory, or integer linear programming and they benefit from the experience in numerical linear
algebra. In particular, a key point there is to embed the finite field elements
in integers stored as floating point numbers, and then rely on the efficiency of the
floating point matrix multiplication \texttt{dgemm} of the BLAS. The conversion
back to the finite field, done by costly modular reductions, is delayed as much as possible.
Hence a natural ingredient in the design of efficient dense linear algebra routines is the
use of block algorithms that result in gathering arithmetic operations in
matrix-matrix multiplications. Those can take full advantage of vector instructions and has a high computation
per memory access rate, allowing to fully overlap the data accesses by
computations and hence delivers peak performance efficiency.
In order to exploit the power of multi-core and many-core architectures, we now
investigate the parallelization of the finite field linear algebra routines.
We report in this paper the conclusions of our experience in parallelizing
exact LU decomposition for shared memory parallel computers. We try to emphasize
which specificities of exact computation domains led us to use different
approaches than that of numerical linear algebra.
In short, we will illustrate that numerical and exact LU factorization mainly differ in the
following aspects:
\vspace{-3mm}
\begin{itemize}
\item the pivoting strategies,
\item the cost of the arithmetic (of scalars and matrices),
\item the treatment of rank deficiencies.
\end{itemize}
\vspace{-2mm}
Those have a direct impact on the shape and granularity of the block
decomposition of the matrix used in the computation.
\vspace{-1em}
\paragraph{Types of block algorithms.}
Several schemes are used to design block linear algebra algorithms: the splitting can occur on one
dimension only, producing row or column slabs~\cite{KlvdGe95}, or both
dimensions, producing tiles~\cite{BLKD07}. Note that, here, we denote by tiles a
partition of the matrix into sub-matrices in the mathematical sense regardless
what the underlying data storage is.
Algorithms processing blocks can be either iterative or recursive.
Figure~\ref{fig:blockalg} summarizes some of the various existing block
splitting obtained by combining these two aspects.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.18\textwidth]{slabiter.png}
\hfill
\includegraphics[width=.18\textwidth]{slabrec.png}
\hfill
\includegraphics[width=.18\textwidth]{tileiter.png}
\hfill
\includegraphics[width=.18\textwidth]{tilerec.png}\vspace{-5pt}
\end{center}
\caption{Main types of block splitting}
\label{fig:blockalg}\vspace{-5pt}
\end{figure}
Most numerical dense Gaussian elimination algorithms, like in \cite{BLKD07}, use
tiled iterative block algorithms. In~\cite{DFLL11} the classic tiled iterative
algorithm is combined with a slab recursive one for the panel elimination.
Over exact domains, recursive algorithms are preferred to benefit from fast
matrix arithmetic (see below).
Slab recursive exact algorithms can be found in~\cite{JPS13} and references therein
and~\cite{DPS13} presents a tiled recursive algorithm.
\vspace{-1em}
\paragraph{The granularity} is the block dimension (or the dimension of the
smallest blocks in recursive splittings).
Matrices with dimensions below this threshold are treated by a
base-case variant (often referred to as the panel factorization).
It is an important parameter for optimizing efficiency: a finer grain allows
more flexibility in the scheduling when running numerous cores, but it also
challenges the efficiency of the scheduler and can increase the bus traffic.
\vspace{-1em}
\paragraph{The cost of the arithmetic.} In numerical linear algebra, the cost of
arithmetic operations is more or less associative: with dimensions above a rather low
threshold (typically a few hundreds), sequential matrix multiplication of the
BLAS reaches the peak efficiency of the processor. Hence the granularity has
very little impact on the efficiency of a block algorithm run in sequential.
On the contrary, over a finite field, a small granularity can imply a larger
number of costly modular reductions, as we will show in Section~\ref{sec:modcount}.
Moreover, numerical stability is not an issue over a finite field, and asymptotically fast matrix
multiplication algorithms, like Winograd's variant of Strassen
algorithm~\cite[\S 12]{GG99} can be used on top of the BLAS.
Their speed-up increases with matrix dimension. The cost of
sequential matrix multiplication over finite field is therefore not associative:
a larger granularity delivers better sequential efficiency.
\paragraph{Pivoting strategies and rank deficiencies.}
In dense numerical linear algebra, pivoting is used to ensure good
numerical stability and good data locality~\cite{GoVa96}.
In the context of dense {\em exact} linear algebra, stability is no longer an
issue. Instead, only certain pivoting strategies will reveal
the echelon form or, equivalently, the rank profile of the matrix~\cite{JPS13,DPS13}. This is a key
invariant used in many applications using exact Gaussian elimination, such as
Gr\"obner basis computations~\cite{F99a} and computational number
theory~\cite{stein2007modular}.
Over exact domains, the rank deficiency of some blocks also leads to unpredictable
dimensions of the tiles or slabs, as will be illustrated in Section~\ref{sec:rankdef}.
This makes the block splitting necessarily dynamic contrarily to the case of
numerical LU factorization where all panel blocks usually have full rank
and the splitting is done statically according to a granularity parameter.
Consequently the design of a parallel exact matrix factorization
necessarily differs from the numerical algorithms as follows:
\vspace{-3mm}
\begin{itemize}
\item granularity should be as large as possible, to reduce modular reductions
and benefit from fast matrix multiplication;
\item algorithms should preferably be recursive, to group arithmetic operations
in matrix products as large as possible.
\item block splitting and pivoting strategies must preserve and reveal the rank
profile of the matrix
\end{itemize}
\vspace{-2mm}
It also implies several requirements on the parallel run-time being used:
\vspace{-3mm}
\begin{itemize}
\item the block splitting has to be dynamically computed;
\item the computing load for each task is not known in advance (some panel
blocks may have high rank deficiency), making the tasks very heterogeneous.
\end{itemize}
\vspace{-2mm}
This motivated us to look into parallel execution runtimes using tasks with work-stealing based
scheduling.
All experiments have been conducted on a 32 cores Intel Xeon E5-4620 2.2Ghz
(Sandy Bridge) with L3 cache(16384 KB). The numerical BLAS is ATLAS v3.11.4,
LAPACK v3.4.2 and PLASMA v2.5.0. We used X-KAAPI-2.1 version with last git commit:
xkaapi\_2.1-30-g263c19c638788249. The gcc compiler version used is gcc 4.8.2 that
supports OpenMP 3.1.
We introduce in Section~\ref{sec:prelim} the algorithmic building blocks on
which our algorithms will rely and the parallel programming models and runtimes
that we used in our experiments.
In order to handle each problem separately, we focus in
Section~\ref{sec:fullrank} on the simpler case where no rank deficiency
occur. In particular Section~\ref{sec:modcount} presents detailed analysis of the
number of modular reductions required by various block algorithms including the tiled
and slab recursive, the left-looking, right-looking and Crout variants of the
tiled iterative algorithm. Lastly Section~\ref{sec:rankdef} deals with elimination
with rank deficiencies. We there present and compare new slab iterative,
tiled iterative and tiled recursive parallel algorithms that preserve
rank profiles. We then show that the latter can match state of the art
numerical routines, even when taking rank deficiencies into account.
\section{Preliminaries}
\label{sec:prelim}
\subsection{Auxiliary sequential routines}
All block algorithms that we will describe rely on four type of operations that
we denote using the BLAS/LAPACK naming convention:
\vspace{-2mm}
\begin{description}
\item[\texttt{gemm}\xspace:] general matrix multiplication, computing $C\leftarrow \alpha
A\times B +\beta C$,
\item[\texttt{trsm}\xspace:] solving upper/lower triang. syst. with matrix right/left h.s $B\leftarrow B
U^{-1}$.
\item[\texttt{laswp}\xspace:] permuting rows or columns by sequence of swaps.
\item[\texttt{getrf}\xspace:]
computing $(P,L,U,Q)$, $L$ and $U$ stored in place
of $A$, s.t. $A=PLUQ$.
\end{description}
\vspace{-2mm}
A first prefix letter \texttt{d} or \texttt{f} is specifies if the routine
works over double precision floating point numbers or finite field coefficients
and an optional prefix \texttt{p} stands for parallel implementation.
Our implementations use the sequential routines of the \texttt{fflas-ffpack}\xspace
library\footnote{\url{http://linalg.org/projects/fflas-ffpack}}~\cite{DGP08}. There,
the elements of a finite $\ensuremath{\mathbb{Z}}\xspace/p\ensuremath{\mathbb{Z}}\xspace$ for a prime $p$ of size about 20 bits
are integers stored in a double precision floating point number. The sequential
\texttt{fgemm}\xspace routine combines recursive steps of Winograd's algorithm calls to
numerical BLAS \texttt{dgemm}\xspace and reductions modulo $p$ when necessary. The \texttt{ftrsm}\xspace and
\texttt{fgetrf}\xspace routines use block recursive algorithms to reduce most arithmetic
operations to \texttt{fgemm}\xspace. More precisely \texttt{fgetrf}\xspace is either done by a slab
recursive algorithm~\cite{DGP08} or a tile recursive algorithm~\cite{DPS13}.
\subsection{Parallel programming models}
We base our implementation on the task based parallel features of OpenMP
standard.
This is motivated by the use of recursive algorithms where tasks are
mandatory. Now in tile iterative algorithms, loop with tasks happen to perform
at least as good as parallel loops.
\texttt{libgomp}\xspace is the GNU implementation of the OpenMP API for multi-platform
shared-memory parallel programming in C/C++ and Fortran.
Alternatively, we also used
\texttt{libkomp}\xspace~\cite{BGD12}, an optimized version of \texttt{libgomp}\xspace, based on the \texttt{XKaapi}\xspace runtime,
that reduces the overhead of the OpenMP directives and handles more
efficiently threads creation, synchronization and management.
In the experiments of the next sections, we will compare efficiency of the same
code linked against each of these two libraries.
\subsection{Parallel matrix multiplication}
In the iterative block algorithms, all matrix product tasks are sequential, on
the contrary the recursive block algorithms must call parallel matrix products
\texttt{pfgemm}\xspace, which we describe here. Operation \texttt{pfgemm}\xspace is of the form
$C \leftarrow \alpha A \times B+\beta C$.
In order to split the computation into independent tasks, only the row
dimension of $A$ and the column dimension of $B$ only are split.
The granularity of the split can be chosen in two different ways: either as a
fixed value, or by a ratio of the input dimension (e.g. the total number of
cores).
We chose the second option that maximizes the size of the blocks while ensuring
a large enough number tasks to for the computing ressources. All our experiments
showed that this option performs better than the first one.
When used as a subroutine in a parallel factorization, it will create more tasks
than the number of available cores, but this heuristic happen to be a good
compromise in terms of efficiency.
Figure~\ref{fig:pfgemmtime} shows the computation time on 32 cores of various
matrix multiplications: the numerical \texttt{dgemm}\xspace implementation of \texttt{Plasma-Quark}\xspace,
the implementation of \texttt{pfgemm}\xspace of \texttt{fflas-ffpack}\xspace using OpenMP tasks, linked
against the \texttt{libkomp}\xspace library. This implementation is run over the finite field
$\ensuremath{\mathbb{Z}}\xspace/131071\ensuremath{\mathbb{Z}}\xspace$ or over field of real double floating point numbers, with or
without fast Strassen-Winograd's matrix product.
One first notices that most routine perform very similarly. More precisely,
\texttt{Plasma-Quark}\xspace \texttt{dgemm}\xspace is faster on small matrices but the effect of
Strassen-Winograd's algorithm makes \texttt{pfgemm}\xspace faster on larger matrices, even
on the finite field where additional modular reductions occur.
\begin{figure}[ht!]
\centering
\includegraphics[width=.8\textwidth,angle=0.]{Gflops-MM.pdf}\\
\caption{Comparison of execution time exact vs numeric}
\label{fig:pfgemmtime}\vspace{-5pt}
\end{figure}
In terms of speed-up, the \texttt{pfgemm}\xspace reaches a factor of approximately 27 (using
32 cores) whereas the numerical \texttt{dgemm}\xspace of \texttt{Plasma-Quark}\xspace reaches a factor of 29,
but this is mostly reflect the fact that \texttt{dgemm}\xspace has a less efficient sequential
reference timing since it does not use Strassen-Winograd's algorithm.
Similarly, other basic routines used in the recursive block algorithms, such as
\texttt{ftrsm}\xspace (solving matrix triangular systems) and \texttt{flaswp}\xspace (permuting rows or
columns), have been parallelized by splitting a dimension into a constant number
of blocks (typically the number of cores).
\section{Eliminations with no rank deficiency}
\label{sec:fullrank}
In this section, we make the assumption that no rank deficiency occur during the
elimination of any of the diagonal block. This hypothesis is satisfied by
matrices with generic rank profile (i.e. having all their leading principal
minor non zero). This assumption allows us to focus on the problem of reducing
the modular reduction count.
\subsection{Modular reductions}
\label{sec:modcount}
When computing over finite field, it is of paramount importance to
reduce the number of modular reductions in the course of linear algebra
algorithms. The classical technique is to accumulate several
multiplications before reducing, namely replacing
$\sum_{i=1}^n (a_ib_i \mod p)$ with $\left(\sum_{i=1}^n a_ib_i\right)$ while keeping
the result exact. If $a_i$ and $b_i$ are integers between $0$ and
$p-1$ this is possible with integer or floating point units if the
result does not overflow, or in other words if
$n(p-1)^2<2^\text{mantissa}$, see, e.g., \cite{DGP08} for more
details.
This induces a splitting of matrices in blocks of size the largest $n^*$ satisfying
the latter condition. Now the use of block algorithms in parallel, introduces a
second blocking parameter that interferes in the number of reductions.
We will therefore compare the number of modular reductions of three variants of
the tile iterative algorithm (left-looking, right-looking and Crout, see~\cite{DDSV98}), the slab
recursive algorithm of~\cite{DGP08}, and the tile recursive algorithm of~\cite{DPS13}.
For the sake of simplicity, we will always assume that the block dimensions in the
parallel algorithms are always below $n^*$. In other words operations are done with full
delayed reduction for a single multiplication and any number of
additions: operations of the form $\sum a_i b_i$ are reduced modulo
$p$ only once at the end of the addition, but $a \cdot b \cdot c$ requires two
reductions.
For instance, with this model, the number of reductions required by a
classic multiplication of matrices of size $m\times k$ by $k\times n$
is simply: $R_{\texttt{gemm}\xspace}(m,k,n)=mn$.
From \cite[Theorem~3]{DPS13}, this extends also for triangular solving $m\times
m$ to $m\times n$: with
unit diagonal, $R_\texttt{utrsm}\xspace(m,m,n)=mn$ (actually the computation of the lowest row of
the solution requires no modulo as it is just a division by $1$, we will
therefore rather use $R_\texttt{utrsm}\xspace(m,m,n)=(m-1)n$)
and $R_\texttt{trsm}\xspace(m,m,n)=2mn$ (with the previous refinement for
$R_\texttt{utrsm}\xspace(m,m,n)$, this also reduces to $R_\texttt{trsm}\xspace(m,m,n)=(2m-1)n$).
Table~\ref{tab:blockvariants} sketches the different shapes of the associated
routine calls in the main loop of each variant.
\begin{table}[ht]\center
\begin{tabular}{ccc}
\toprule
Left looking &
Crout &
Right looking \\
\midrule
\hspace{-5pt}\begin{minipage}{.33\textwidth}
\begin{algorithmic}
\For{i=1 to n/k}
\State \texttt{utrsm}\xspace((i-1)k,(i-1)k,k)
\State \texttt{gemm}\xspace(n-(i-1)k,(i-1)k,k)
\State \texttt{pluq}\xspace(k,k)
\State \texttt{trsm}\xspace(k,k,n-ik)
\EndFor
\end{algorithmic}
\end{minipage} &
\hspace{-5pt}\begin{minipage}{.33\textwidth}
\begin{algorithmic}
\For{i=1 to n/k}
\State \texttt{gemm}\xspace(n-(i-1)k,(i-1)k,k)
\State \texttt{gemm}\xspace(k,(i-1)k,n-ik)
\State \texttt{pluq}\xspace(k,k)
\State \texttt{utrsm}\xspace(k,k,n-ik)
\State \texttt{trsm}\xspace(k,k,n-ik)
\EndFor
\end{algorithmic}
\end{minipage} &
\begin{minipage}{.31\textwidth}
\begin{algorithmic}
\For{i=1 to n/k}
\State \texttt{pluq}\xspace(k,k)
\State \texttt{utrsm}\xspace(k,k,n-ik)
\State \texttt{trsm}\xspace(k,k,n-ik)
\State \texttt{gemm}\xspace(n-ik,k,n-ik)
\EndFor
\end{algorithmic}
\end{minipage}\\
\bottomrule
\end{tabular}
\vspace{5pt}
\caption{Main loops of the Left looking, Crout and Right looking tile iterative
block LU factorization (see~\cite{DDSV98})}\label{tab:blockvariants}
\vspace{-5pt}
\end{table}
Then the number of modular reductions required for these different LU
factorization strategies is given in Table~\ref{tab:modcount}.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\center
\begin{tabular}{clc}
\toprule
\multirow{3}{*}{\begin{sideways}$k= 1$\end{sideways}}
& Iterative Right looking & $\frac{1}{3}n^3-\frac{1}{3}n$ \\
& Iterative Left Looking & $\frac{3}{2}n^2-\frac{3}{2}n+1$\\
& Iterative Crout & $\frac{3}{2}n^2-\frac{7}{2}n+3$\\
\midrule
\multirow{3}{*}{\begin{sideways}$k\geq 1$\end{sideways}}
& Tile Iterative Right looking & $\frac{1}{3k}n^3+\left(1-\frac{1}{k}\right)n^2+\left(\frac{1}{6}k-\frac{5}{2}+\frac{3}{k}\right)n$ \\
& Tile Iterative Left looking & $\left(2-\frac{1}{2k}\right)n^2+\left(-\frac{5}{2}k-1+\frac{2}{k}\right)n+2k^2-2k+1$\\
& Tile Iterative Crout & $\left(\frac{5}{2}-\frac{1}{k}\right)n^2+\left(-2k-\frac{5}{2}+\frac{3}{k}\right)n+k^2$\\
\midrule
& Tiled Recursive & $2n^2-n\log_2 n-n$\\
\midrule
& Slab Recursive & $(1+\frac{1}{4}\log_2 n)n^2-\frac{1}{2}n\log_2 n-n$\\
\bottomrule
\end{tabular}
\vspace{5pt}
\caption{Counting modular reductions in full rank block LU
factorization of an $n\times n$ matrix modulo $p$ when $np(p-1)<2^\text{mantissa}$,
for a block size of $k$ dividing $n$.}\label{tab:modcount}
\vspace{-10pt}
\end{table}
The last two rows of the table corresponds
to~\cite[Theorem~4]{DPS13} where $R_\texttt{utrsm}\xspace$ has been refined to $(m-1)n$ as mentioned above.
The first three rows are obtained by setting $k=1$ in the following block
versions.
The next three rows are obtained via the following analysis
where the base case (i.e. the $k \times k$ factorization) always uses the best
unblocked version, that is the Crout variant described above.
Following Table~\ref{tab:blockvariants}, we thus have:
\begin{itemize}
\item The right looking variant performs $\frac{n}{k}$ such $k\times k$ base cases,
$\texttt{pluq}\xspace(k,k)$, then, at iteration $i$, $(\frac{n}{k}-i)(\texttt{utrsm}\xspace(k,k,k)+\texttt{trsm}\xspace(k,k,k))$,
and $(\frac{n}{k}-i)^2$ \texttt{gemm}\xspace(k,k,k), for a total of
$\frac{n}{k}(\frac{3}{2}n^2-\frac{7}{2}n+3)+\sum_{i=1}^{\frac{n}{k}}
(n-ik)\left((3k-2) + (\frac{n}{k}-i)k\right)=\frac{1}{3k}n^3+\left(1-\frac{1}{k}\right)n^2+\left(\frac{1}{6}k-\frac{5}{2}+\frac{3}{k}\right)n$.
\item The Crout variant requires:
at each step, except the first one, to compute $R_{\texttt{gemm}\xspace}(n-ik,ik,k)$
reductions for the pivot and the elements below and
$R_\texttt{gemm}\xspace(k,ik,n-(i-1)k)$;
at each step, to perform one base case for the pivot block, to
solve unitary triangular systems, to the left, below the pivot, using
$(\frac{n}{k}-i)R_\texttt{utrsm}\xspace(k,k,k)$ reductions and to solve triangular systems to the
right, using $(\frac{n}{k}-i)R_\texttt{trsm}\xspace(k,k,k) $ reductions.
\item Similarly, the Left looking variant requires $R_\texttt{gemm}\xspace(n-ik, ik,
k)+R_\texttt{pluq}\xspace(k)+R_\texttt{utrsm}\xspace(ik, ik,k)+R_\texttt{trsm}\xspace(k,k,n-ik)$ reductions in the main loop.
\end{itemize}
In table~\ref{tab:overhead} we see that the left looking variant always performs less modular reductions.
Then the tiled recursive performs less modular reductions than the Crout variant
as soon as $2\leq k \leq \frac{n}{4}$.
Finally the right looking variant is clearly costs more modular reductions.
But the best algorithms here may not perform well in parallel, as will be shown next.
\begin{table}[htbp]
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l|ccc|ccc|cc}
\toprule
& \multicolumn{3}{c}{$k=212$} & \multicolumn{3}{c}{$k=\frac{n}{3}$} & \multicolumn{2}{c}{Recursive}\\
\midrule
& Right & Crout & Left & Right & Crout & Left & Tile & Slab\\
\midrule
n=3000 & \multicolumn{1}{c}{3.02} & 2.10 & \textbf{2.05} & \multicolumn{1}{c}{2.97} & 2.15 & 2.10 & \multicolumn{1}{c}{2.16} &
2.26\\
\midrule
n=5000 & \multicolumn{1}{c}{11.37} & 8.55 & 8.43 & \multicolumn{1}{c}{9.24} & 8.35 & 8.21 &\multicolumn{1}{c}{\textbf{7.98}} &
8.36\\
\midrule
n=7000 & \multicolumn{1}{c}{29.06} & 22.19 & 21.82 & \multicolumn{1}{c}{22.56} & 22.02 & 21.73 & \multicolumn{1}{c}{\textbf{20.81}} & 21.66\\
\bottomrule
\end{tabular}
\end{center}
\caption{Timings (in seconds) of sequential LU factorization variants on one core}
\label{tab:overhead} \vspace{-20pt}
\end{table}
\subsection{Parallel experiments}
\begin{figure}[ht!]
\centering
\includegraphics[width=.8\textwidth,angle=0.]{fullrank_modulo.pdf}
\caption{Parallel LU factorization on full rank matrices with modular operations}
\label{fig:moduloBlockLU}
\end{figure}
In Figure~\ref{fig:moduloBlockLU} we compare the tiled iterative variants with
the tiled recursive algorithm. The latter uses as a base case an iterative Crout
algorithm too which performs fewer modular operations,
The tiled recursive algorithm performs better than all other tiled iterative
versions.
This can be explained by a finer and more adaptive granularity and a better locality.
The left looking variant performs poorly for it uses an expensive sequential
\texttt{trsm}\xspace task. Although Crout and right-looking variant perform about the same
number of matrix products, those of an iteration of the right-looking variant
are independent, contrarily to those of the Crout variant, which explains a
better performance despite a larger number of modular reductions.
\begin{figure}[ht!]
\centering
\includegraphics[width=.8\textwidth,angle=0.]{fullrank-double.pdf}
\caption{Parallel LU factorization on full rank matrices without modular operations}
\label{fig:doubleBlockLU}
\end{figure}
Figure~\ref{fig:doubleBlockLU} shows the performance without modular
reductions, of the tiled recursive parallel implementation on full rank matrices
compared to \texttt{Plasma-Quark}\xspace.
The best block size for the latter library was
determined by hand for each matrix size. The two possible data-storage for
\texttt{Plasma-Quark}\xspace are used: the collection of tiles or the row-major data-storage.
Our tiled recursive parallel PLUQ implementation without modular reductions
behaves better than the \texttt{Plasma-Quark}\xspace \texttt{getrf\_tile}\xspace. This is mainly due
to the bi-dimensional cutting which allows for a faster panel elimination, parallel \texttt{trsm}\xspace
computations, more balanced \texttt{gemm}\xspace computations and some use of
Strassen-Winograd's algorithm. This explains why performance join again on more
than 24 cores: the size of the sequential blocks get below the threshold where
this algorithm speeds up computations (typically 2400 on this machine).
\section{Elimination with rank deficiencies}
\label{sec:rankdef}
\subsection{Pivoting strategies}
We now consider the general case of matrices with arbitrary rank
profile, that can lead to rank deficiencies in the panel eliminations.
Algorithms computing the row rank profile (or equivalently the column
echelon form) used to share a common pivoting strategy:
to search for pivots in a row-major fashion and consider the next row only if no
non-zero pivot was found (see~\cite{JPS13} and references therein).
Such an iterative algorithm can be translated into a slab recursive algorithm
splitting the row dimension in halves (as implemented in sequential
in~\cite{DGP08}) or into a slab iterative algorithm.
More recently, we presented in~\cite{DPS13} a more flexible pivoting strategy
that results in a tile recursive algorithm, cutting both dimensions
simultaneously. As a by product, both row and column rank profiles are
also computed simultaneously.
\vspace{-1em}
\paragraph{A slab iterative algorithm.}
In the slab iterative algorithm shown in Figure~\ref{fig:TileCUP}, each panel factorization has to be run by a
sequential algorithm. This sequential task is costly and therefore imposes a
choice of a fine granularity, which, as we saw, on the other hand implies more
modular reductions and a lesser speed-up of Strassen-Winograd's algorithm.
\begin{figure}[ht!]
\centering
\includegraphics[width=.7\textwidth,angle=0.]{tile_CUP.png}
\caption{Slab iterative factorization of a matrix with rank deficiencies}
\label{fig:TileCUP}
\end{figure}
Another difficulty is the fact that the starting column position of
each panel is determined by the rank of the blocks computed so far.
It can only be determined dynamically upon the execution.
This implies in particular that no data-storage by tiles, that fit the
tiles of the algorithm is possible here.
Moreover, the workload of each block operation may strongly vary,
depending on the rank of the corresponding slab. Such heterogeneous
tasks lead us to opt for work-stealing based runtimes instead of
static thread management.
\vspace{-1em}
\paragraph{Tiled iterative elimination.}
In order to speed-up the panel computation, we can split it into column
tiles. Thanks to the pivoting strategy of~\cite{DPS13}, it is still possible to
recover the rank profiles afterwards. Now with this splitting, the operations
remain more local and updates can be parallelized.
This approach shares similarities with the recursive computation of the panel
described in~\cite{DFLL11}.
Figure~\ref{fig:panel-pluq} illustrates this tile iterative factorization
obtained by the combination of a row-slab iterative algorithm, and a column-slab
iterative panel factorization.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\textwidth,angle=0.]{panel-PLUQ.png}
\caption{Panel PLUQ factorization}
\label{fig:panel-pluq}
\end{figure}
This optimization used in the computation of the slab factorization
improved the computation speed by a factor of 2, to achieve a speed-up
of 6.5 on 32 cores with \texttt{libkomp}\xspace.
\vspace{-1em}
\paragraph{Tiled recursive elimination.}
Lastly, the tile recursive algorithm described in~\cite{DPS13} can be run in
parallel using recursive tasks and the \texttt{pfgemm}\xspace, \texttt{ftrsm}\xspace and \texttt{flaswp}\xspace routines.
Contrarily to most recursive factorization algorithms, the recursive splitting
is done in four quadrants.
It has the interesting feature that if the tile top-left tile is rank deficient,
then the elmination of the bottom-left and top-right tiles can be parallelized.
Figure~\ref{fig:PLUQ-Gflops} shows performance obtained for the tiled recursive
and the tiled iterative factorization. Both
versions are tested using \texttt{libgomp}\xspace and \texttt{libkomp}\xspace libraries. The input \texttt{S16K} is a
$16000\times 16000$ matrix with low rank deficiency (rank is 15500). Linearly
independant rows and columns of the generated matrix are uniformly distributed
on the dimension.
\begin{figure}[ht!]
\centering
\includegraphics[width=.8\textwidth,angle=0.]{PLUQ-Gflops.pdf}\\
\caption{Performance of tiled recursive and tiled iterative factorizations
using \texttt{libgomp}\xspace and \texttt{libkomp}\xspace. Matrix dimension $n=16000$ with rank $15500$}
\label{fig:PLUQ-Gflops}
\end{figure}
The implementation with OpenMP of the tiled recursive LU maintained high
efficiency in the case of rank deficient matrices. It attained a speed-up of
13.6 on 32 cores. Besides the fact that it benefits from Strassen-winograd
implementation, it is adapted to minimize memory accesses and optimize data
placement.
Using \texttt{libkomp}\xspace instead of
\texttt{libgomp}\xspace library and numactl, for round and robin interleave memory placement,
that helps reducing dependency on bus speed, we manage to obtain peak
performance for our tiled recursive LU factorization.
\section{Conclusion}
We analyzed five different algorithms for the computation of Gaussian
elimination over a finite field.
The granularity surely optimizes the parallelization of these
algorithms but at the cost of more modular operations.
Algorithm optimizing modular reductions are unfortunately not the most efficient
in parallel. The best compromise is obtained with our recursive tiled algorithm
that performs best in both aspects.
\vspace{-1em}
\paragraph{Perspective}
Our future work focuses on two main issues. First, the use of specific allocators
that can be used for a better mapping of data in memory and reduce distant accesses.
Second, parallel programming frameworks for multicore processors \cite{KLDB10}
could be more effective than binding threads on each NUMA node.
Dataflow based dependencies, like when using OpenMP 4.0 directives,
can ensure more parallelism for recursive implementation using \texttt{libkomp}\xspace\cite{BGD12} library.
\vspace{-1em}
{\scriptsize
\bibliographystyle{abbrvurl}
|
1,108,101,565,508 | arxiv | \section{Introduction}
Motivated by the problem in which two or more objectives must be considered at the same time, even though they may conflict with each other, in this work we are interested the design of stochastic algorithms for multi-objective optimization. This type of problem is commonly found in everyday life, for example, in physics, engineering, social sciences, economy, biology, and many others^^>\cite{deb2001multi,Hwang79,Pardalos2018,jahn2004vector,eichfelder2021twenty}. Investing in the financial market while maximizing profit and minimizing risk or building a vehicle while maximizing performance and minimizing fuel consumption and pollutant emissions are examples of multi-objective optimization problems.
From a mathematical viewpoint the problem can be formulated through a variable $x\in \mathbb R^d$ describing a possible decision and assuming that $g_i(x)$ is the $i$-th objective for $i=1, \dots, m$, with $m \in \mathbb{N}$ being the total number of objectives. A multi--objective problem then requires to solve for a decision $x$
\begin{equation}
\min_{x \in \mathbb R^d} g(x)
\label{eq:mop}
\end{equation}
where $g(x) = (g_1(x) , \dots, g_m(x))^\top$. A solution to \eqref{eq:mop} corresponds to several optimal decisions. Here, we consider optimality in the sense of Pareto \cite{jahn2004vector}, i.e., no objective can be improved without necessarily degrade another objective. Without additional information about subjective preferences, there may be a (possibly infinite) number of Pareto optimal solutions, all of which are considered equally good.
Therefore, the optimization tasks consist of providing a set of optimal decisions. To this end, it is also desirable to have a \textit{diverse} set, that is, addressing the problem not only by optimizing fitness, but also by aiming to cover a variety of user-defined features of interest, in order to best describe the (possibly) broad set of optimal decisions.
Several methods have been proposed to numerically solve \eqref{eq:mop} and, as for single-objective optimization, they typically belong to either the class of metaheuristic algorithms or mathematical programming methods \cite{Sergeyev2018}.
Among metaheuristics \cite{talbi2009meta}, multi-objective evolutionary algorithms \cite{deb2001multi}, such as NSGA-II \cite{deb2002nsga2} and MOEA/D \cite{zhang2008moead}, have gained popularity among practitioners due to their flexibility and ease of use. At the same time, they usually lack of convergence analysis compared to mathematical programming methods. For more details on mathematical programming methods and evolutionary algorithms in multi-objective optimization we refer to the recent surveys \cite{eichfelder2021twenty, coello2020survey}.
We are interested in a particular class of stochastic particle optimization methods, called consensus-based optimization (CBO), which has recently gained popularity due to the use of mean-field techniques that can provide them with a rigorous mathematical foundation. Such methods consider interacting particle systems described by stochastic differential equations (SDEs) that combine a drift towards the estimated minimum and random exploration of the search space^^>\cite{pinnau2017consensus,carrillo2018analytical,fornasier2021consensusbased,carrillo2019consensus,fornasier2022aniso,benfenati2021binary,totzeck2020}. These approaches have been extended also to optimization problems over hypersurfaces \cite{fhps20-2,fhps20-2b,fornasier2020hypersurfaces}, constrained optimization \cite{borghi2021constrained,carrillo2021constrained} and multi-objective optimization \cite{borghi2022multi}. From a mathematical viewpoint, this class of metaheuristic methods is inspired by the corresponding mean-field dynamics based on particle swarming and multi-agent social interactions, which have been widely used to study complex systems in life sciences, social sciences and economics \cite{pareschi13,MR3274797,defrli13,Prigogine1977self,Vicseck}. These techniques have proven fruitful to demonstrate convergence towards a global minimum for single-objective problems, not only in the case of CBO methods, but also for the popular Particle Swarm Optimization (PSO) algorithm \cite{grassi2021from,huang2022PSO}, thus paving the way to provide a mathematical foundation for other metaheuristics.
In the same spirit, the authors proposed in \cite{borghi2022multi} a multi-objective optimization algorithm (M-CBO) by prescribing a CBO-type dynamics among several particles making use of a scalarization strategy. Scalarization strategies are a common tool in multi-objective optimization \cite{jahn2004vector} as they allow to translate problem \eqref{eq:mop} into a set of parametrized single-objective problems, which can be solved simultaneously in the case of particle-based optimization methods.
In this paper, we provide a convergence analysis for the method based on the mean-field description of the M-CBO dynamics. Furthermore, we improve the method in order to capture with uniform accuracy the shape of the Pareto front. This is done by iteratively updating parameters of the method to minimize specific diversity measures. Mathematically, this last feature is achieved by enlarging the phase space of the particles. A detailed analysis of the extended model is also presented by studying a mean-field approximation of the particle dynamics which allows to recover convergence guarantees towards optimal points and also underline a gradient-flow structure in the space of the parameters.
Recently, energy-based diversity measures have gained popularity in the multi-objective evolutionary optimization community due to their flexibility, scalability and theoretical properties \cite{coello2020survey}. In this formulation, a set of decision is diverse if it corresponds to a minimal configuration of a suitable two-body energy potential, breaking the problem down into finding such configurations. In the proposed algorithm, we obtain this by inserting a Vlasov-type dynamics in the space of the parameters. We prove that the particle dynamics can be written in the more general framework of non-local interaction equations over bounded domains. The later topic has been recently investigated e.g. in \cite{carrillo2016nonlocal,fatecau2017swarm,fatecau2019diffusion,patacchini2022nonlocal}.
The rest of the paper is organized as follows. In \cref{sec:2} we formally introduce the concept of optimality for \eqref{eq:mop} and present the scalarization strategy. Next,
in \cref{sec:3} we illustrate the particle dynamics both in the search space and in the space of parameters. \cref{sec:4} is devoted to the mathematical analysis of the system evolution using a mean-field description. Finally, in \cref{sec:5}
numerical examples on convex and non-convex as well as disjoint Pareto fronts are presented which confirm the theoretical results as well as the performance of the new method. Some concluding remarks are discussed in the last section.
\section{Problem definition and scalarization}
\label{sec:2}
We will use the following notation. Let $a \in \mathbb R^n$, $|a|$ indicates its euclidean norm and $(a)_l$ its $l$-th component, while for $A$ Borel set $A \subset \mathbb R^n$, $|A|$ indicates its Lebesgue measure. The symbols $\prec$ and $\preceq$ indicates the partial ordering with respect to the cone $\mathbb R^m_{>0}$ and $\mathbb R^m_{\geq 0}$ respectively.
\subsection{Pareto optimality and diversity}
When dealing with a vector-valued objective function $g: \mathbb R^d \to \mathbb R^m$
\begin{equation}
g(x) = \left (g_1(x), \cdots, g_m(x) \right)
\end{equation}
with $m\geq 2$, the interpretation of the minimization problem \eqref{eq:mop} is not unique, as the image space $\mathbb R^m$ is not fully ordered. We consider the notions of strong and weak Edgeworth-Pareto optimality which rely on the natural, component-wise, partial ordering on $\mathbb R^m$ \cite{jahn2004vector}.
\begin{definition}[Edgeworth-Pareto optimality.]
A point $\bar x \in \mathbb{R}^d$ is (strong) Edgeworth-Pareto (EP) optimal, or simply optimal, if $g(\bar x)$ is a minimal element of the image set $g(\mathbb{R}^d)$ with respect to the natural partial ordering, that is if there is no $x \in \mathbb{R}^d$ such that
\[ g_i(x) \leq g_i(\bar x) \;\; \textup{for all}\;\; i=1, \dots, m\,, \quad g(x) \neq g(\bar x)\,.\]
Alike, $\bar x$ is weakly EP optimal, if there is no $x \in \mathbb{R}^d$ such that
\[ g_i(x) < g_i(\bar x)\; \; \textup{for all}\;\; i=1, \dots, m\,.
\]
\noindent The set
$
F_x = \{ \bar x \in \mathbb R^d \, |\, \bar x \; \textup{is EP optimal} \}
$
constitutes the set of optimal EP points, while
\[
F = \{ g(\bar x) \in \mathbb R^m \, |\, \bar x \; \textup{is EP optimal} \}
\]
is the Pareto front.
\end{definition}
The multi-objective optimization problem \eqref{eq:mop} consists of finding the set of EP optimal points. Unlike single-objective problems, the set is typically uncountable and the optimization task involves finding a finite subset of optimal points. Those should ideally cover $F$ and the concept of \textit{diversity} is introduced to distinguish between two approximations \cite{deb2001multi}. Intuitively, if points on the Pareto front are more distanced, the diversity is higher. In view of the minimization problem, having a diverse approximation is desirable as it provides at the same cost a broader variety of possible solutions.
The most diverse approximation possible is possibly given by a set of point which is uniformly distributed over the Pareto front. Quantifying the diversity of an optimal set is of paramount importance both, to assess the performance of optimization methods and to design them. Indeed, oftentimes the heuristic of a specific method is constructed to specifically minimize, or maximize, a specific measure \cite{coello2020survey}.
Without knowledge of the exact Pareto front, popular diversity measures are given by \textit{hypervolume contribution} \cite{zitzler1998multi}, \textit{crowding distance} \cite{deb2002nsga2} and, recently, by the \textit{Riesz s-energy} \cite{coello2021overview, vega2021towards}. Our proposed algorithm will aim to minimize the latter (or similar energy-based measures) as it can be embedded in a mean-field framework. The exact definitions are introduced later.
To sum up, the multi-objective optimization problem we consider is a two-objective task itself, as one needs to find a set of points which are both EP optimal and optimize a suitable diversity measure.
\subsection{Scalarization strategy}
A popular way to approach \eqref{eq:mop} is to use a scalarization strategy \cite{jahn2004vector,book2005mop} which reduces the multi-objective problem to a (finite) number of single-objective sub-problems. Among the possible scalarization strategies, we consider the approximation sub-problems with weighted Chebyschev semi-norms \cite{jahn2004vector} where the single objectives are given by
\begin{equation}
G (x,w) := \max_{k\in \{ 1, \dots, m\}}\, w_k \,|g_k(x)|\,. \notag
\end{equation}
and are parametrized by a vector of weights $w$ which belongs to the unitary, or probability, simplex
\begin{equation}
\Omega :=\left \{ w \in \mathbb{R}^m_{\geq0}\; | \;\sum_{i=1}^m w_i = 1 \right\}\,.
\notag
\end{equation}
For each $w \in \Omega$ the subproblems then read
\begin{equation}
\min_{x \in \mathbb{R}^d} G(x, w).
\label{eq:sub}
\end{equation}
The link between the scalarized problems and the original multi-objective problem is given by the following result.
\begin{theorem}[{\cite[Corollaries 5.25, 11.21]{jahn2004vector}}]
Assume $g$ is component-wise positive.
\begin{enumerate}
\item[a)] A point $\bar x$ is weakly EP optimal if and only if $\bar x$ is a solution to \eqref{eq:sub} for some $w \in \Omega$.
\item[b)] Assume all sub-problems \eqref{eq:sub} attains an unique minimum. Then, $\bar x$ is EP optimal if and only if $\bar x$ is the solution to \eqref{eq:sub} for some $w \in \Omega$.
\end{enumerate}
\label{t:pareto}
\end{theorem}
\cref{t:pareto} shows the strength of the Chebyschev scalarization strategy which allows to find all the weakly EP optimal points, contrary to other strategies like linear scalarization \cite{jahn2004vector}. We remark that the proposed algorithm can also be applied to solve any other scalarized problems of the form \eqref{eq:sub} where the parameters are take from the unitary simplex $\Omega$.
Even though solving $N$ sub-problems with corresponding weights vectors $\{ W^i\}_{i=1}^N \subset \Omega$ ensures to find $N$ optimal points, we note that there is no guarantee to obtain a diverse approximation. Therefore, scalarization targets only one of the two objectives of the problem, without addressing the diversity of the solution. In the following, we introduce an algorithm where the parameters $W^i$ are dynamically changed during the computation to obtain a set of EP points which is also diverse.
\section{Adaptive multi-objective consensus based optimization}
\label{sec:3}
We propose a dynamics where $N\in \mathbb{N}$ particles interact with each other to solve $N$ scalar sub-problems given in the form \eqref{eq:sub}. We introduce the dynamics as a continuous-in-time process and leave the definition of the actual discrete optimization method to \cref{sec:5}.
At a time $t\geq0$, every particle is described by its position $X_t^i \in \mathbb R^d$ and its vector of weights $W_t^i \in \Omega$ which determines the optimization sub-problem the particle aims to solve. As a result, particles are described by $N$ tuples
\begin{align*}
(X_t^i, W_t^i) \quad \textup{for}\quad i= 1, \dots, N\, \quad \textup{for all} \quad t>0.
\end{align*}
in the augmented space $\mathbb R^d \times \Omega$.
The initial configuration is generated by sampling the positions $X^i_0$ from a common distribution $\rho_0 \in \mathcal{P}(\mathbb R^d)$ and by taking uniformly distributed weights vectors $W^i_0$ over $\Omega$.
The dynamics is prescribed to solve the multi-objective optimization task. We recall that \eqref{eq:mop} not only requires to find optimal points, but also points that are diverse, that is, well-distributed over the Pareto front. To this end, the optimization process is made of two mechanisms which address these two objectives separately.
\subsection{A consensus based particle dynamics in the search space}
The first mechanism prescribes the update of the position $\{X_t^i\}_{i=1}^N$, such that they converge towards EP optimal points. As in \cite{borghi2022multi}, this is done by introducing a CBO-type dynamics between the particles. To illustrate the CBO update rule, let us consider for the moment a fixed single-objective sub-problem parametrized by $w \in \Omega$. Similar to Particle-Swarm Optimization methods, in CBO dynamics at time $t>0$, the particles instantaneously move towards an attracting point $Y_t^\alpha$ which is given by a weighted average of their position:
\begin{equation}
Y^\alpha_t(w) = \frac{ \sum_{j=1}^N X^j_t\, \exp\left(-\alpha G(X^j_t, w)\right)}{\sum_{j=1}^N \exp\left(-\alpha G(X^j_t, w)\right)}.
\label{eq:Ya}
\end{equation}
Due to the coefficients used in \eqref{eq:Ya}, if $\alpha\gg1$, $Y^\alpha_t(w)$ is closer to the particles with low values of the objective function $G(\cdot, w)$ and, in the limiting case, it holds
\begin{equation} \notag
Y^\alpha_t(w) \longrightarrow \underset{X^j_t,\,j=1,\dots, N}{\textup{argmin}} G(X_t^j, w)\quad \textup{as} \quad \alpha \to \infty\,
\end{equation}
if the above minimum uniquely exists. This promotes the concentration of the particles in areas of the search space where the objective function $G(\cdot,w)$ attains low values and hence, more likely, a global minimum. We remark that the exponential coefficients correspond to the Gibbs distribution associated with the objective function and, moreover, that this choice is justified by the Laplace principle \cite{Dembo2010}. The later is an essential result to study the convergence of CBO methods \cite{fornasier2021consensusbased} and it states that for any absolutely continuous probability density $ \rho \in \mathcal{P}(\mathbb R^d)$ we have
\begin{equation}\notag
\lim_{\alpha \to \infty}\left( - \frac 1 \alpha \log \left( \int e^{-\alpha G(x, w)} d\rho (x) \right) \right) = \inf_{x \in \text{supp}(\rho)} G(x, w)\,.
\label{eq:laplace}
\end{equation}
Since in the multi-objective optimization dynamics each particle addresses a different sub-problem, each of them moves towards a different attracting point given by $Y^\alpha_t(W^i_t)$.
The drift strength is given by $\lambda >0$, while another parameter $\sigma>0$ determines the strength of an additional stochastic component.
The time evolution of the particles positions is determined by a system of SDE
\begin{equation}
dX^{i}_{t} = \lambda\left(Y_t^\alpha(W^i_t) - X_t^i \right)dt + \sigma D^i_t dB_t^i \quad \textup{for all} \quad i = 1, \dots, N \, ,
\label{eq:sdex}
\end{equation}
where $B_t^i$ are $d$-dimensional independent Brownian processes. The matrices $D_t^i$ characterize the random exploration process which might be isotropic \cite{pinnau2017consensus}
\begin{equation}
D_{t,\text{iso}}^i = | X_t^i - Y^\alpha_t(W^i_t)|\, I_d\,,
\label{eq:iso}
\end{equation}
$I_d$ being the $d$-dimensional identity matrix, or anisotropic \cite{carrillo2019consensus}
\begin{equation}
D_{t,\text{aniso}}^i = \text{diag} \left( (X_t^i - Y^\alpha_t(W^i_t))_1 , \dots ,(X_t^i - Y^\alpha_t(W^i_t))_d \right)\,.
\label{eq:aniso}
\end{equation}
Both explorations depend on the distance between $X^i_t$ and the correspondent attracting point making the stochastic component larger if the particle is far from $Y_t^\alpha(W^i_t)$. The difference lays on the direction of the random component: while in the isotropic exploration all dimensions are equally explored, the anisotropic one explores each dimension with a different magnitude.
The expected outcome of the position update rule \eqref{eq:sdex} is that every particle will find a minimizer of a sub-problem and hence, by \cref{t:pareto}, a weak EP optimal point. We have already mentioned that if the weights vectors are fixed to the initial, uniform, distribution $\{W_0^i \}_{i=1}^N$, that is
\[ \frac{d W_t^i}{dt} = 0 \quad \textup{for all} \quad i = 1, \dots, N \, , \]
there is no guarantee to obtain equidistant points on the front. Since it is impossible to determine beforehand the optimal distribution on $\Omega$, we propose a heuristic strategy which updates the vector weights promoting diversity.
\subsection{Uniform approximation of the Pareto front}
A popular diversity metric in multi-objective optimization is the \textit{hypervolume} contribution metric \cite{zitzler1998multi}, which has the drawbacks of being computationally expensive \cite{fonseca2009hyp} and, by definition, dependent on an estimate of $g$. Motivated by this and by the objective of designing algorithms which perform well for any shape of the Pareto front \cite{vega2021towards}, new energy-based diversity measures have recently gained popularity \cite{coello2020survey,coello2021overview}. Such measures quantify the diversity of a given empirical distribution $\rho^N \in \mathcal{P}(\mathbb R^d)$ by considering the pairwise interaction given by a two-body potential $U: \mathbb R^m \to (-\infty, \infty]$ on the image space
\begin{equation}
\mathcal{U}[ g\# \rho^N ] := \iint U\left (g(x) - g(y) \right)\, d\rho^N(y)\, d\rho^N(x)\,,
\label{eq:U}
\end{equation}
$g \# \rho^N$ being the push-forward measure of $\rho^N$.
The problem of finding well-spread points over the Pareto front is then equivalent to finding a configuration which is minimal with respect to the given energy
$\mathcal{U}$ where we recall that $F_x$ is the set of EP optimal points:
$$ \underset{\rho^N \in \mathcal{P}(F_x)}{\textup{min}}\; \mathcal{U}\, [ g \# \rho^N] \to \min.$$
A distribution $\nu^N$ is called diverse, if and only if
\begin{equation*}
\nu^N \in \underset{\rho^N \in \mathcal{P}(F_x)}{\textup{argmin}}\; \mathcal{U}\, [ g \# \rho^N]\,.
\end{equation*}
Any energy $ \mathcal{U}$ describing short range repulsion between particles, like Monge energy or repulsive-attractive power-law energy, is in principle a candidate to be a diversity measure. The Riesz $s$-energy given by
\begin{equation}
U_R(z) = \frac{1}{|z|^s} \quad \textup{with} \quad s = m-1
\label{eq:riesz}
\end{equation}
is a popular choice \cite{coello2021overview} due to its theoretically guarantees of being a good measure of the uniformity of points. Indeed, if $F$ is a $(m-1)$-dimensional manifold, the minimal energy configuration $\nu^N$ converges to the uniform Hausdorff distribution over $F$ as $N\to \infty$. We refer to \cite{hardin2005minimal} for the precise statements of the result and more details.
Inspired by the electrostatic potential between charged particles, the authors in \cite{braun2015preference} used a Newtonian potential which is also empirically proven to be a suitable diversity measure \cite{braun2015preference,Braun2018thesis}. See \cite{coello2021overview} for a numerical comparison between two-body potentials as diversity measures in evolutionary algorithms.
We will also compare different energies in \cref{sec:5} and consider $U\in \mathcal{C}^1(\mathbb R^m \setminus \{ 0\})$ to be any of the above.
Exact computation of minimal energy configurations of a system of $N$ particles is a well-studied problem as it is connected to e.g. crystallization phenomenon \cite{blanc2015crystal}. We note that, in our settings, the configuration $\rho^N$ is additionally mapped to the image space in \eqref{eq:U}, making the task even harder. Therefore, we propose an heuristic strategy that is expected to find only suboptimal configurations.
To promote diversity, we let the particles follow a vector field associated with $\mathcal{U}$. The movement will be only in parameter space $\{ W_t^i\}_{i=1}^N$ in order not interfere with the CBO optimization dynamics acting on the positions $\{X_t^i\}_{i=1}^N$.
Intuitively, if two particles are close to each other in the image space $g(\mathbb R^d)$, their weights vectors are pulled apart. This resemble a short range repulsion of $U$. To ensure $W_t^i$ remains in the unitary simplex $\Omega$, a projection to the tangent cone $T(W_t^i,\Omega)$
\begin{equation*}
P_{W_t^i} (h):= P_{T(W_t^i, \Omega)} (h) = \left\{ z \in T(W_t^i,\Omega) \,:\, |z - h| = \inf_{\xi \in T(W_t^i,\Omega)} | \xi - W_t^i|\right \}
\end{equation*}
for all $h \in \mathbb R^m$ is required, see also \cite{carrillo2016nonlocal} for more details. A parameter $\tau\geq0$ determines the time scale of the weights adaptation process with respect to the CBO dynamics \eqref{eq:iterx}. The process can be turned off for $\tau =0$.
In case of bi-objective problems, where $m=2$, we therefore obtain a Vlasov-type dynamics
\begin{equation}
\frac{dW^{i}_{t}}{dt} = - P_{T(W^i_t,\Omega)}\left ( -\frac{\tau}N \sum_{j=1}^N\nabla U\left(g(X_t^i) - g(X_t^j)\right) \right) \quad \textup{for all} \quad i = 1, \dots, N \, ,
\label{eq:sdew}
\end{equation}
which is well-defined as the parameters space is embedded in the image space $\mathbb R^m$. If $U$ has singularity in $0$, we set $\nabla U(0) = 0$. We note that the additional minus sign in \eqref{eq:sdew}, is due to explicit form of the relation determined by \cref{t:pareto} between the Pareto front and $\Omega$. This will become clear in the next section, as this choice gives a gradient flow structure to the parameters dynamics.
For $m> 2$, the relation between a weight vector $w \in \Omega$ and the correspondent (weakly) EP optimal point is more involved. Nevertheless, we prescribe a suitable heuristic dynamics as follows: let $U$ be given as
\begin{equation*}
U (z) = r(|z|) \quad \textup{for some}\quad r \in \mathcal{C}^1(\mathbb R_{\geq0}),
\end{equation*}
then the parameters dynamics reads
\begin{equation}
W^{i}_{t} = P_{T(W^i_t,\Omega)} \left ( - \frac \tau {N}\sum_{j=1}^N \frac{W_t^i - W_t^j}{|W_t^i - W_t^j|}\; r' \left(|g(X_k^i) - g(X_k^j)|\right)\right)
\label{eq:iterw}
\end{equation}
for all $i = 1, \dots, N$. The term $r'(\cdot)$ determines the strength and the sign of the interaction , while $(W_t^i - W_t^j)/|W_t^i - W_t^j|$ the direction of movement. As before, the projection step is needed due to the boundedness of $\Omega$. Even though \eqref{eq:iterw} can also be used when $m=2$, we will consider in the next section \eqref{eq:sdew} only.
Up to our knowledge, energy-based diversity metrics have only been used a selection criterion between candidate approximation of the Pareto front \cite{vega2021towards,coello2020reference}, and this is the first time the vector field associated to $\mathcal{U}$ is used to guide the particle dynamics in a metaheuristic multi-objective optimization method.
\section{Mean-field analysis of the particle dynamics}
\label{sec:4}
In this section, we give a statistical description of the optimization dynamics by presenting the corresponding mean-field model, which allows us to analyze the convergence of the method towards a solution to the multi-objective optimization problem. We restrict ourselves to the case where $m=2$ and the dynamics in $\Omega$ is given by \eqref{eq:sdew}. The particle dynamics is given by \eqref{eq:iterx} and \eqref{eq:iterw2}, respectively.
Similar to \cite{pinnau2017consensus}, we formally derive the mean-field equation of the large system \eqref{eq:sdex}, \eqref{eq:sdew} by making the so-called \textit{propagation of chaos} assumption on the marginals. In particular, let $F^N(t)$ be the particles probability distribution over $(\mathbb R^d \times \Omega)^N$ at a time $t \geq0$. We assume that $F^N(t) \approx f(t)^{\otimes N}$ that is, that the particles $(X_t^i, W_t^i), i=1, \dots,N$ are independently distributed according to $f(t) \in \mathcal{P}(\mathbb R^d \times \Omega)$ for some large $N\gg 1$.
In the following, we indicate with $\rho(t) \in \mathcal{P}(\mathbb R^d)$ the first marginal of $f(t)$ and with $\mu(t) \in \mathcal{P}(\Omega)$ the second marginal on the parameters space $\Omega$. As a consequence of the propagation of chaos assumption, we obtain that
\[
Y_t^\alpha(W^i_t) =\frac{\frac1N\sum_{i=1}^N X_t^i e^{-\alpha G(X_t^i,W^i_t)}}{\frac1N\sum_{i=1}^N e^{-\alpha G(X_t^i,W^i_t)}} \; \approx\; \frac{\int x e^{-\alpha G(x, W^i_t)} d\rho(t)}{\int e^{-\alpha G(x, W^i_t)}d\rho(t)} =: y^\alpha (\rho(t), W^i_t)
\]
and that
\[
\sum_{i=1}^N \nabla U \left( g(X_t^i) - g(X_t^j) \right) \approx \int \nabla U \left(g(X_t^i) - g(x) \right) d\rho(t)\,.
\]
The dynamics \eqref{eq:sdex}, \eqref{eq:sdew} is now independent on the index $i$ and we obtain the process $(X_t, W_t), t~>~0$ as
\begin{equation}
\begin{cases}
dX_t &=\lambda (y_t^\alpha(\rho(t), W_t) - X_t)dt + D(\rho(t),W_t) dB_t \\
dW_t&= \tau P_{W_t} \left(\int \nabla U \left( g(X_t) - g(x)\right) d\rho(t)\right)\, dt
\end{cases}
\label{eq:mono}
\end{equation}
where, $D(\rho_t,W_t)$ is defined consistently with \eqref{eq:iso} and \eqref{eq:aniso}. Process \eqref{eq:mono} is reformulated as
\begin{multline}
\frac\partial{\partial t} f(t,x,w) = - \lambda\nabla_x\cdot \Big( ( y^\alpha(\rho(t),w) - x) f(t,x,w) \Big)
+ \frac{\sigma^2}2 \Delta_x \big( D(\rho(t),w) f(t,x,w) \big)
\\
- \tau \nabla_w \cdot \left( P_w\left(\int \nabla U\left(g(x) - g(y)\right) d\rho(t,y) \right) f(t,x,w) \right)\,,
\label{eq:mf}
\end{multline}
with initial conditions $f(0,x,w) = \rho_0 \otimes \mu_0$, $\mu_0$ being the uniform distribution over the unit simplex $\Omega$, $\mu_0 = \textup{Unif}(\Omega)$.
The nonlinear partial differential equation \eqref{eq:mf} is a mean-field description of the microscopic dynamics generated by the optimization dynamics described in \cref{sec:3}. We note that the rigorous mean-field limit for single-objective CBO dynamics, which are similar to \eqref{eq:sdex}, was proven in \cite{huang2021meanfield}.
Following previous works, see e.g. \cite{carrillo2018analytical, fornasier2021consensusbased}, we consider such an approximation and mathematically analyze the proposed optimization method by studying a solution $f$ to \eqref{eq:mf}.
\subsection{Convergence to the Pareto front}
\label{s:4.2}
In the following, we assume $f \in \mathcal{C}\left([0,\infty), \mathcal{P}_2(\mathbb R^d \times \Omega)\right)$ to be a solution to \eqref{eq:mf} with initial data given by $\rho_0 \in \mathcal{P}_2(\mathbb R^d)$, $\mu_0 \in \mathcal{P}(\Omega)$. We assess the performance of a multi-objective algorithm by the average distance form the Pareto front $F$ and we use the Generational Distance ($GD$) \cite{van1998evolutionary} given by
\begin{equation}
GD[\rho(t)] = \left( \int \textup{dist}(g(x),F)^2 d\rho(t,x) \right)^{\frac{1}2}
\label{eq:GD}
\end{equation}
where $\rho(t)$ is the first marginal of $f(t)$.
In the following, we state conditions such that $DG[\rho(t)]$ decays up to a given accuracy $\varepsilon>0$.
\begin{assumption}[Uniqueness]
Every sub-problem \eqref{eq:sub} $w\in \Omega$ admits a unique solution $\bar x (w) \in F$. Moreover $\bar x \in \mathcal{C}^1 (\Omega, \mathbb R^d)$.
\label{a:1}
\end{assumption}
The uniqueness requirement is common in the analysis of CBO methods \cite{fornasier2021consensusbased}. This is due to difficulty to control the attractive term $y^\alpha$, whenever there are two or more minimizers. For example, assume $\nu$ is a measure concentrated in two different global minimizes of a sub-problem $w$, $\nu = \left(\delta_{\bar X_1} + \delta_{\bar X_2}\right)$: the attractive term could be located in the middle between them
\[
y^\alpha(\nu,w) = \frac12 \left (\bar X_1 + \bar X_2 \right)\,
\]
being obviously not(!) minimizer. The regularity assumption on $\bar x$ follows form the transport term in \eqref{eq:mf} with respect to $w$, and may be dropped if the interaction in the weights space is not present.
The next assumption requires that all scalar objective functions $G(x,w), w \in \Omega$ have a common lower and upper bounds in a neighborhood of the minimizer. See also \cite{rosasco2017geometry} and the references therein for more details on the following conditions.
\begin{assumption}[Stability at the minimizer]
In a neighborhood of their minimizer, $G(\cdot,w), w \in \Omega$ are $p$-conditioned and satisfy a growth condition: there exists a radius $R>0$, exponents $p>2, q>1$ and constants $c_1, c_2>0$ such that for all $w \in \Omega$
\[
c_1 | x - \bar x (w) |^p \leq G(x,w) - \min_{y\in \mathbb R^d} G(y,w) \leq c_2| x - \bar x (w) |^{1/q} \quad \text{for all}\quad x: \; |x - \bar x (w) | \leq R \,.
\]
Moreover, outside such a neighborhood, the function cannot be arbitrary close to the minimum: there exists $c_3>0$ such that for all $w \in \Omega$
\[
c_3 \leq G(x,w) - \min_{y\in \mathbb R^d} G(y,w) \quad \text{for all}\quad x:\; |x - \bar x (w) | \geq R \,.
\]
\label{a:2}
\end{assumption}
Finally, we assume the optimal EP points to be bounded. As in other CBO methods we also prescribe a condition on the initial data $\rho_0$ and $\mu_0$.
\begin{assumption}[Boundedness and initial datum] The set $F$ of optimal points is contained by a bounded, open set $H \subset \mathbb R^d, |H|>0$. The initial distribution $f_0$ is given by $f_0 = \rho_0 \otimes \mu_0$ with $\rho_0 = \textup{Unif}(H)$ and some $\mu_0\in \mathcal{P}(\Omega)$.
\label{a:3}
\end{assumption}
Assumptions \ref{a:1}--\ref{a:3} ensure that the results on the Laplace principle \cite{fornasier2021consensusbased} are applicable to all the different sub-problems \eqref{eq:sub} with uniform choice of $\alpha$. Therefore, under such assumptions, it possible to prove the convergence of each in the following sense. Let $\mathbb{E}_{f(t)}[|x -\bar x(w)|^2], f(t) \in \mathcal{P}(\mathbb R^d\times \Omega)$ denote the average $\ell_2$-error
\begin{equation}
\mathbb{E}_{f(t)}\left[|x -\bar x(w)|^2\right] = \int | x - \bar x (w)|^2\, df(t,x,w)\,
\label{eq:V}
\end{equation}
then, it holds:
\begin{theorem}[ {\cite[Theorem 12]{fornasier2021consensusbased}, \cite[Theorem 2]{fornasier2022aniso}}]
Assume (\ref{a:1})--(\ref{a:3}), $\nabla U \in L^\infty(\mathbb R^m)$ and let $f \in \mathcal{C}\left([0,\infty), \mathcal{P}_2(\mathbb R^d \times \Omega)\right)$ be a solution to \eqref{eq:mf} with initial datum $f_0$. Let $\kappa=d$ if isotropic diffusion \eqref{eq:iso} is used and $\kappa =1$ for anisotropic diffusion \eqref{eq:aniso}.
For any accuracy $\varepsilon$, $0<\varepsilon<\mathbb{E}_{f_0}\left[|x -\bar x(w)|^2\right]$, if
\begin{equation}
\kappa \sigma^2 + C \frac{\tau}{\sqrt{\varepsilon}} < \lambda\,, \quad \text{where} \quad C := \sqrt{2} \| \nabla W \|_{L^\infty(\mathbb R^m)} \|\nabla_w \bar x \|_{L^\infty(\Omega, \mathbb R^d)}
\end{equation}
and if $\alpha$ is sufficiently large, there exists a time $T>0$ such that
\[
\mathbb{E}_{f(T)}\left[|x -\bar x(w)|^2\right] = \varepsilon\,.
\]
Moreover, for all $t \in [0,T]$ it holds
\begin{equation}
\mathbb{E}_{f(t)}\left[|x -\bar x(w)|^2\right] \leq \mathbb{E}_{f_0}\left[|x -\bar x(w)|^2\right] e^{ - \left( \lambda - \kappa\sigma^2 - C \tau/\sqrt{\varepsilon} \right) t } \,.
\label{eq:Vdecay}
\end{equation}
\label{t:CBO}
\end{theorem}
We remark that the choice of $\alpha$ depends on the estimates given in \cref{a:2} and in particular on the accuracy $\varepsilon$.
\begin{corollary}
Under the settings of \cref{t:CBO}, if $g$ is Lipschitz continuous it holds
\[
GD[\rho(T)] =\textup{Lip}(g) \sqrt{\varepsilon}
\]
and, for all $t \in [0,T]$,
\[
GD[\rho(t)] \leq \textup{Lip}(g) \sqrt{\mathbb{E}_{f_0}\left[|x -\bar x(w)|^2\right]}
\exp\left( - \frac{\lambda - \kappa \sigma^2 - C \tau/\sqrt{\varepsilon} }{2} t \right)\,,
\]
where $\rho(t) \in \mathcal{P}(\mathbb R^d)$ is the first marginal of $f(t)$.
\label{c:1}
\end{corollary}
\begin{proof}
Since every sub-problem admits a unique solution (Assumption \ref{a:1}), by \cref{t:pareto} every solution $\bar x(w)$ is EP optimal and therefore its image $g(\bar x(w))$ belongs to the Pareto front $F$. Therefore
\[
\textup{dist}(g(x),F) \leq | g(x) - g(\bar x (w))| \leq \textup{Lip}(g) | x- \bar x(w)|
\]
from which follows that the generational distance $GD$ is bounded by the average $\ell_2$-error.
\end{proof}
\cref{t:CBO,c:1} show that CBO mechanism is able to successfully solve all sub-problems \eqref{eq:sub} simultaneously. In the next section, we will analyze the dynamics in the parameters space $\Omega$ to investigate the diversity of the computed solution.
\begin{remark}
In \cref{t:CBO}, $\tau$ needs to be taken of order $o(\sqrt{\varepsilon})$ suggesting that the parameters should adapt at a much slower time scale with respect to the positions, in order not to interfere with the CBO dynamics. With no weights vectors interaction, $\tau=0$, the decay estimate \eqref{eq:Vdecay} is independent of $\varepsilon$ and, in particular, the particles converge faster towards EP optimal points.
\label{r:tau0}
\end{remark}
\subsection{Decay of diversity measure}
\label{s:4.3}
The aim of interaction \eqref{eq:iterw} is improve the distribution of the parameters $\{W_t^i\}_{i=1}^N$ so that, in view of \cref{t:pareto}, the corresponding (weak) EP optimal points are well-distributed in the image space. Under suitable assumptions, such dynamics corresponds to a gradient flow on the unitary simplex $\Omega$.
For any sub-problem \eqref{eq:sub} parametrized by $w \in \Omega$, let $\bar x(w)$ be one of its global minima, which we assume exists. As we are interested in the relation between $w$ and its correspondent point on the Pareto front $g\left(\bar x(w)\right) \in F$, let us formally insert in the mean-field model \eqref{eq:mf} solutions of the form
\begin{equation}
f(t,x,w) = \delta(x - \bar x(w) ) \mu(t,w)
\label{eq:f}
\end{equation}
where $\mu(t) \in \mathcal{P}(\Omega)$. In ansatz \eqref{eq:f}, the location $x$ of the particle $(x,w)$ corresponds exactly to a solution $\bar x(w)$ to its sub-problem $w\in \Omega$. This is justified by the convergence result (\cref{t:CBO}) and by the fact that the positions dynamics takes place at a faster time scale then the parameters adaptation, see \cref{r:tau0}.
The reduced mean field equation in strong form is then given by
\begin{multline}\notag
\frac\partial{\partial t} f(t,x,w) = - \lambda\nabla_x\cdot \Big( \left( y^\alpha(\rho(t),w) - \bar x(w)\right) f(t,x,w) \Big)
+ \frac{\sigma^2}2 \Delta_x \big( D(\rho(t),w) f(t,x,w) \big)
\\
- \tau \nabla_w \cdot \left( P_w\left(\int \nabla U\left(g(\bar x(w)) - g(\bar x(v))\right) df(t,y,v) \right) f(t,x,w) \right)\,
\label{eq:mfreduced}
\end{multline}
and, the marginal $\mu(t)$ over $\Omega$ fulfills
\begin{equation}
\frac{\partial}{\partial t} \mu(t,w) = - \tau \nabla_w \cdot \left( P_w\left(\int \nabla U\left(\bar g(w) -\bar g(v)\right) d\mu(t,v) \right) \mu(t,w) \right)
\label{eq:red1}
\end{equation}
where for simplicity we introduced $\bar g := g \circ \bar x$.
\begin{assumption} The Pareto front $F$ is exactly the unitary simplex $\Omega$ and the potential energy $U$ is radially symmetric.
\label{a:4}
\end{assumption}
\begin{lemma} Under \cref{a:4}, for all $w,v \in \Omega$ it holds
\begin{equation}
P_w\Big( \nabla U\big(\bar g(w) - \bar g(v)\big) \Big) = - P_w\big(\nabla U (w - v) \big)\,.
\label{eq:P}
\end{equation}
\label{l:1}
\end{lemma}
\begin{proof}
We note that when $F = \Omega$, all weakly EP optimal points are also EP optimal and hence by \cref{t:pareto} $\bar g(w) = g(\bar x(w)) \in F$ for all $w \in \Omega$. Then, there exists $s \in [0,1]$ such that $\bar g (w) = (s, 1-s)$ for all $w$. By definition of the sub-problem \eqref{eq:sub} with $w = (w_1,w_2) = (w_1, 1-w_1)$,
\[
\bar g(w) = \min_{y \in F} \max \left\{ y_1w_1, y_2 w_2 \right\} =
\min_{s \in [0,1]} \max \{ s w_1, (1-s) (1-w_1) \}\,.
\]
At the minimizer, it must hold $s w_1 = (1-s)(1-w_1)$ and hence $s = (1-w_1)$. It follows that
\begin{equation}
\bar g (w) = A w, \quad \text{where} \quad A = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.
\label{eq:gbar}
\end{equation}
Since $U$ is radially symmetric it holds $\nabla U ( Aw - Av) = A \nabla U(w-v)$.
Finally, let us consider the basis $n_1 = (1,1)^\top, n_2 = (1,-1)^\top$ and a vector $u \in \mathbb R^2, u = u_1 n_1 + u_2 n_2$. We note that $P_w$ always projects towards $n_2$. Together with the fact that $Au = u_1n_1 - u_2n_2$, this leads to
\[
P_w(Au) = P_w(-u_2 n_2) = P_w(-u)\,
\]
and the identity \eqref{eq:P} follows.
\end{proof}
Thanks to \cref{l:1}, under \cref{a:4} equation \eqref{eq:red1} can be simplified to
\begin{equation}
\frac{\partial}{\partial t} \mu(t,w) = - \tau \nabla_w \cdot \left( P_w\left( - \int \nabla U\left(w -v )\right) d\mu(t,v) \right) \mu(t,w) \right)
\label{eq:red2}
\end{equation}
and initial conditions $\mu(0) = \mu_0$.
Equation \eqref{eq:red2} describes the continuum dynamics of particles which binary interact and that are confined to the set $\Omega$. Such aggregation model on bounded domains has been subject of several works, see for instance \cite{fatecau2017swarm,patacchini2022nonlocal}. Particularly relevant to the present work is \cite{carrillo2016nonlocal} where general prox-regular sets, like $\Omega$, are considered.
\begin{theorem}[{\cite[Theorem 1.5 ]{carrillo2016nonlocal}}]
Assume $U \in \mathcal{C}^1(\mathbb R^2)$ to be $\tilde\lambda$-geodetically convex on $\textup{Conv}(\Omega - \Omega)$ for some $\tilde\lambda \in \mathbb R$. For any initial data $\mu_0 \in \mathcal{P}_2(\Omega)$ there exists a locally absolutely continuous curve $\mu(t) \in \mathcal{P}(\Omega), t>0,$ such that $\mu$ is a gradient flow with respect to $\mathcal{U}$. Also, $\mu$ is a weak measure solution to \eqref{eq:red2}.
Furthermore,
\begin{equation}
\frac{d}{dt}\,\mathcal{U}(\mu(t)) \leq - \int \left | P_w\left( \nabla U \ast \mu(t) (w) \right)\right|^2 \mu(t,w) \,,
\label{eq:decayU}
\end{equation}
\label{t:flow}
where $*$ denotes the convolution operator.
\end{theorem}
Under \cref{a:4} and thanks to relation \eqref{eq:gbar} between $\Omega$ and $F$, \cref{t:flow} states that the energy over the front is decreasing. We note that the flow may convergence to the stationary points of \eqref{eq:red1} that are not minimal configurations, as observed in \cite{fatecau2017swarm} for even simple domains.
Clearly, without ansatz \eqref{eq:f}, there is no guarantee that the potential decreases along the evolution of the algorithm.
Quite the opposite, by \cref{t:CBO} particles are expected to concentrate on the Pareto front leading to an increased potential $\mathcal{U}$. Nevertheless, by \cref{t:CBO} there exists a time $T>0$ where
\[
\int | x- \bar x(w) |^2\, df(T,x,w) < \varepsilon \quad \text{and hence} \quad x \approx \bar x(w)
\]
making ansatz \eqref{eq:f} valid. Therefore, we claim that the reduced model \eqref{eq:red2} describes the dynamics for $t>T$. We will numerically investigate two phases of the algorithm: the first one when concentration over the Pareto happens, and the second when the potential $\mathcal{U}$ decays leading the an improved diversity of the solution.
\section{Numerical experiments}
\label{sec:5}
In this section, we numerically investigate the performance of the proposed method by testing it against several benchmark multi-objective problems.
The adaptive multi-objective consensus based optimization (AM-CBO) algorithm is obtained from an Euler--Maruyama time-discretization of \eqref{eq:sdex} and \eqref{eq:sdew} (or \eqref{eq:iterw} if $m>2$).
Let $\Delta t>0$ be a fixed time-step. For $k=0,1,\dots$, the particles positions are iteratively updated according to
\begin{equation}
X^i_{k+1} = X_k^i + \lambda\left(Y_k^\alpha(W^i_k) - X_k^i \right)\Delta t + \sigma D^i_k \sqrt{\Delta t}B_k^i
\label{eq:iterx}
\end{equation}
for all $i = 1, \dots, N$ where $B_k^i$ are multivariate independent random vectors, $B_k^i \sim \mathcal{N}(0,I_d)$. The update rule \eqref{eq:iterx} is overparametrized and in CBO optimization schemes typically $\lambda=1$ is used.
Similar to the projected gradient flow scheme used in \cite{patacchini2022nonlocal}, we replace the instantaneous projection to the tangential space $T(w,\Omega)$ by the projection $\Pi_{\Omega}$ to $\Omega$,
\[
\Pi_{\Omega} (v) = \left \{w \in \Omega\;:\: |v-w| = \inf_{\xi \in \Omega} |v - \xi| \right \} \quad \textup{for} \quad v \in \mathbb R^m
\]
and discretize the dynamics in $\Omega$ as
\begin{equation}
\begin{cases}
V^{i}_{k+1} &= W^i_k + \frac \tau {N}\sum_{j=1}^N \nabla U\left(g(X_k^i) - g(X_k^j)\right) \Delta t\\
W^{i}_{k+1} &= \Pi_{\Omega}\left( V^i_{k+1} \right)
\end{cases}
\,,
\label{eq:iterw2}
\end{equation}
for $m=2$, while for $m>2$ it reads
\begin{equation}
\begin{cases}
V^{i}_{k+1} &= W^i_k - \frac \tau {N}\sum_{j=1}^N \frac{W_k^i - W_k^j}{|W_k^i - W_k^j|}\; r' \left(|g(X_k^j) - g(X_k^i)|\right)\Delta t\\
W^{i}_{k+1} &= \Pi_{\Omega}\left( V^i_{k+1} \right)
\label{eq:iterw_bis}
\end{cases}
\,.
\end{equation}
The complete optimization method is described by Algorithm \ref{alg:1}. A remark on the computational complexity follows.
\begin{algorithm
\caption{AM-CBO}
\label{alg:1}
\begin{algorithmic}
\STATE{Set parameters: $\alpha, \lambda, \sigma, \tau, \Delta t$}
\STATE{Initialize the positions: $X^i_0 \sim \rho_0\,, i=1, \dots,N$}
\STATE{Initialize the weights vectors $\{ W_0^i\}_{i=1}^N $ uniformly in $\Omega$}
\STATE{$k\gets 0$}
\WHILE{stopping criterion is NOT satisfied}
\STATE{Compute $g(X^i_k)\,, i=1, \dots,N$ }
\FOR{$i=1, \dots, N$}
\STATE{compute $Y_k^\alpha(W_k^i)$ according to \eqref{eq:Ya}}
\STATE{sample $B^{i}_{k}$ from $\mathcal{N}(0, I_d)$}
\STATE{update $X^i_{k+1}$ according to \eqref{eq:iterx}}
\STATE{update $W^i_{k+1}$ according to \eqref{eq:iterw2} (or \eqref{eq:iterw_bis}})
\ENDFOR
\STATE{$k \gets k+1 $ }
\ENDWHILE
\RETURN $\{ X_{k}^i \}_{i=1}^N$
\end{algorithmic}
\end{algorithm}
For the sake of reproducible research, in the GitHub repository
\url{https://github.com/borghig/AM-CBO}
an implementation in MATLAB code of the proposed algorithm is made available.
\begin{remark}
Even though in every iteration the objective function $g$ is evaluated only $N$ times, the overall computational complexity is $\mathcal{O}(N^2)$ because the computation of $Y^i_k(w)$ requires $\mathcal{O}(N)$ computations, as well as the parameters update \eqref{eq:iterw} which is particularly costly.
One can reduce the computation complexity by considering only a random subset $I_k^M \subset \{1, \dots, N \}$ of $M \ll N$ particles when computing \eqref{eq:Ya} and \eqref{eq:iterw2}, by substituting
\[
\frac1N \sum_{j=1}^N (\cdot)^j \quad \textup{with} \quad \frac1{M} \sum_{j \in I^M_k} (\cdot)^j\,,
\]
whenever a sum over the different particles is performed. Inspired by Monte-Carlo particle simulations \cite{AlPa, JLJ}, this mini-random batch technique allows to lower the complexity to $\mathcal{O}(NM)$. We also note that that Fast Multipole Methods (FMM) \cite{greengard1987fast} may additionally be used to speed up the computation of the potential field, Then, the computational complexity of \eqref{eq:iterw2}, \eqref{eq:iterw_bis} is further reduced.
\end{remark}
\subsection{Performance metrics}
Denote by $\{X_k^i\}_{i=1}^N$ the set of particle positions at the $k$-th algorithm iteration and their empirical distribution by $\rho_k^N \in \mathcal{P}(\mathbb R^d)$.
We employ three different energies, the Riesz $s$-energy \eqref{eq:riesz}, Newtonian and the Morse potentials, both to measure the solutions diversity and to determine the dynamics of the vector weights. The Newtonian binary potential is given by
\begin{equation}
U_N (z) =
\begin{cases}
\log(|z|) & \textup{if} \;\; m = 2 \\
|z|^{2-m} & \textup{if}\;\; m >2
\end{cases}
\;,
\label{eq:newt}
\end{equation}
while the Morse potential is given
\begin{equation}
U_{M}(z) = e^{-C|z|} \quad \textup{with}\quad C>0.
\label{eq:morse}
\end{equation}
All considered potentials describe short-range repulsion between the particles. While the Morse potential is $\tilde \lambda$-geodetically convex, the Newtonian and Riesz repulsion are not. Since we will also employ the corresponding energies $\mathcal{U}_R$, $\mathcal{U}_N$, $\mathcal{U}_M$ to define the interaction between parameters, the constant $C$ can be considered as an algorithm parameter when the Morse repulsion is used.
To show the validity of the energy-based diversity metrics, we additional consider the hypervolume contribution metric $\mathcal{S}$ \cite{zitzler1998multi}. Let $g^* \in \mathbb R^m$ be a maximal element with respect to the natural partial ordering
\[
y_i \prec g^*_j \quad \textup{for all}\quad y \in F\,,
\]
the hypervolume measure is given by the Lebesque measure of the set of points between the computed solution and the maximal point $g^*$, that is
\begin{equation}
\mathcal{S} [\rho_k^N] = \left | \bigcup_{i=1}^N \left \{y \in \mathbb R^m \; | \; g(X^i_k) \prec y \prec g^* \right\} \right|.
\label{eq:hyp}
\end{equation}
Maximizing $\mathcal{S}$ has been shown to lead to a diverse approximation of the Pareto front \cite{emmerich2005emo}.
In \cref{s:4.2}, the convergence of the mean-field dynamics towards the Pareto front is shown by studying the evolution of the Generation Distance $GD$ \eqref{eq:GD}. In the experiments, we approximate this quantity by considering a reference approximation $\{ y^j\}_{j=1}^M$ of the front with $M=100$ points $y^i \in F$, $i = 1, \dots, M$ for every test problem. More details on the reference solution are given in \cref{app:1}.
For simplicity, we indicate the numerical approximation of the Generational Distance again by $GD$, which is defined by
\begin{equation}
GD[\rho_k^N] = \left( \frac 1 N \sum_{i=1}^N \textup{dist}(g(X_k^i), F_M)^2
\right)^{\frac12}\,.
\label{eq:GDnum}
\end{equation}
The Inverted Generational Distance $IGD$ is also considered. It consists of the average distance between the points of the reference solution $\{ y^i\}_{j=1}^M$ and the computed front
\begin{equation}
IGD[\rho_k^N] = \left( \frac 1 M \sum_{j=1}^M \textup{dist}(y^j, G_k)^2
\right)^{\frac12} \quad \textup{with}\quad G_k := \{g(X_k^i)\,|\,i = 1,\dots,N\}\,.
\label{eq:IGDnum}
\end{equation}
Contrary to $GD$ which only measures the distance form the Pareto front, $IGD$ takes in account the diversity of the computed solution, too. Hence, $IGD$ is also a suitable indicator of the optimality of the solution.
\subsection{Test problems}
Test problems with diverse Pareto front geometries are selected to show the performance of the proposed method. In the Lamé problems \cite{emmerich2007lame} the parameter $\gamma$ controls the front curvature: we use $\gamma = 0.25, 1, 3$ to obtain convex, linear and concave fronts respectively. We also consider the DO2DK \cite{branke2004finding} problems with $k=2,s=1$ and $k=4, s=2$. Here, the Pareto fronts have more complex geometries as they are not symmetric and, in one case, discontinuous.
All above problems are scalable to any dimension of the search space $d$ and in the image space $m$. For presentation purposes, we restrict ourselves to bi-objective optimization problems by setting $m=2$, but consider possibly large $d.$ In this case, the fronts analytical description are known, allowing us to obtain reference solutions. The problems definitions are recalled in \cref{app:2} for completeness.
\newcommand\ww{0.96}
\begin{figure}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0cm , clip, width=\ww\linewidth]{SingleRun_lame025_hist}
\bigskip
\includegraphics[trim = 0cm 0cm 0cm 0.5cm, clip, width=\ww\linewidth]{SingleRun_lame1_hist}
\bigskip
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=\ww\linewidth]{SingleRun_lame3_hist}
\bigskip
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=\ww\linewidth]{SingleRun_DO2DK_1_hist}
\bigskip
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=\ww\linewidth]{SingleRun_DO2DK_2_hist}
\caption{In black, particles position in the image space after a single run. The reference
solution is displayed in red. Four different parameters interaction strategies are used: no
interaction, Riesz, Newtonian and Morse potential. Histograms show the final distribution
over $\Omega$ (blue) and the optimal one (red).}
\label{fig:1}
\end{figure}
In this section, we use Algorithm \ref{alg:1} in four different scenarios
\begin{enumerate}
\item No parameters interaction $\tau=0$;
\item Riesz potential \eqref{eq:riesz}, with $\tau=10^{-5}$\,;
\item Newtonian potential \eqref{eq:newt}, with $\tau=10^{-3}$\,;
\item Morse potential \eqref{eq:morse}, with $\tau=10^{-1}$, $C=20$;
\end{enumerate}
The first scenario clearly corresponds to the standard M-CBO approximation, while the others to different AM-CBO strategies.
To validate model \eqref{eq:red2} and \cref{t:flow}, we update the parameters according to \eqref{eq:iterw2}.
The initial weights vectors $\{ W_0^i\}_{i=1}^N$ are taken (deterministically) uniformly distributed over $\Omega$, while the particle positions are uniformly sampled over $[0,1]^d$, $d=10$. We employ $N=100$ particles, which evolve for a maximum of $k_{\textup{max}} = 5000$ steps. The remaining parameters are set to $\lambda = 1, \sigma = 4, \alpha = 10^6$. This parameter choice consists of a compromise between the optimal parameters of each problem. Anisotropic diffusion \eqref{eq:aniso} is used and a projection step ensures the particle positions remain in the search space $[0,1]^d$, which is the same for all considered problems.
\cref{fig:1} shows the computed solutions, in the image-space, in the four different scenarios. Regardless of the interaction on $\Omega$, the particles always converge towards $EP$ optimal points and hence to the Pareto front. By definition of the Chebyshev sub-problems \eqref{eq:sub}, a uniform distribution in $\Omega$ leads to an uniform distribution of the particles over the front only when $F$ is linear (as in the Lamé problem $\gamma=1$). Indeed, \cref{fig:1} shows that the particles are well distributed even when there is not weights interaction ($\tau=0$). If the front geometry differs from this straight segment, the optimal parameters distribution on $\Omega$ differs form the uniform one. In particular, subsets of the Pareto front which are almost parallel to the axis are difficult to approximate without any interaction in the parameter space, see for instance Lamé $\gamma = 0.25$ and the DO2DK problems in \cref{fig:1}. When using $\tau \neq 0$, the solutions improves as the particles are more distributed over the entire front.
\begin{table}
\centering
\begin{tabular}{|l|l|c|c|c|c|c|c|c|}
\hline
Problem& Interaction & $GD$ & $\mathcal{U}_R$ & $\mathcal{U}_N$&$ \mathcal{U}_M$ & $\mathcal{S}$ & $IGD$ \\
\hline
Lam\'e 0.25& $\tau = 0$ & \cellcolor{g}2.33e-02 & 1.00e+10 & 2.41e+00 & 4.86e-01 &\cellcolor{g} 9.69e-01 & 1.31e-01 \\
\cline{2-8}
& Riesz & 8.74e+00 &\cellcolor{g} 5.65e+00 & -1.94e-01 & 9.62e-02 & 7.77e-01 & 4.06e-02 \\
\cline{2-8}
& Newtonian & 1.11e+01 & 8.23e+00 & -3.53e-01 & 1.14e-01 & 8.38e-01 & 4.25e-02 \\
\cline{2-8}
& Morse & 1.49e+01 & 1.81e+04 &\cellcolor{g} -1.36e+00 & \cellcolor{g}3.40e-02 & 7.45e-01 & \cellcolor{g}2.64e-02 \\
\hline
\hline
Lam\'e 1& $\tau = 0$ & \cellcolor{g}9.88e-02 & 9.60e+09 & 9.96e-01 & 1.26e-01 & 3.74e-01 & 8.28e-02 \\
\cline{2-8}
& Riesz & 1.63e-01 & \cellcolor{g}6.54e+00 & 8.57e-01 & 1.23e-01 & 4.59e-01 & \cellcolor{g}1.56e-02 \\
\cline{2-8}
&Newtonian & 9.81e-01 & 8.39e+00 & 5.77e-01 & 9.47e-02 & \cellcolor{g}4.62e-01 & 1.91e-02 \\
\cline{2-8}
&Morse & 6.83e-01 & 8.97e+05 & \cellcolor{g}4.41e-01 &\cellcolor{g} 7.95e-02 & 4.48e-01 & 1.78e-02 \\
\hline
\hline
Lam\'e 3& $\tau = 0$ &\cellcolor{g}1.93e-02 & 8.40e+09 & 9.56e-01 & 1.30e-01 & 8.45e-02 & 2.18e-02 \\
\cline{2-8}
& Riesz & 5.64e-02 & 7.06e+00 & 7.68e-01 & 1.14e-01 & 1.01e-01 & 1.32e-02 \\
\cline{2-8}
& Newtonian & 2.34e-01 & \cellcolor{g}6.33e+00 & 5.74e-01 & 9.10e-02 & \cellcolor{g}1.03e-01 & \cellcolor{g}1.11e-02 \\
\cline{2-8}
&Morse & 3.02e-01 & 9.57e+06 & \cellcolor{g}5.04e-01 & \cellcolor{g}7.84e-02 & 1.02e-01 & 1.29e-02 \\
\hline
\hline
DO2DK & $\tau = 0$ & 1.80e-01 & 1.00e+10 & -3.30e-01 & 6.94e-02 & 8.84e+01 & 2.82e-01 \\
\cline{2-8}
k=2,s=1 & Riesz & \cellcolor{g}5.03e-02 &\cellcolor{g} 1.58e+00 & \cellcolor{g}-6.04e-01 & 4.18e-02 & \cellcolor{g}8.94e+01 & 1.18e-01 \\
\cline{2-8}
& Newtonian & 6.48e-02 & 1.77e+01 & -5.98e-01 & 3.85e-02 & 8.94e+01 & \cellcolor{g}1.07e-01 \\
\cline{2-8}
& Morse & 9.59e-02 & 7.67e+08 & -5.64e-01 &\cellcolor{g}3.75e-02 & 8.94e+01 & 9.33e-02 \\
\hline
\hline
DO2DK &$\tau = 0$ & \cellcolor{g}6.60e-02 & 1.00e+10 & 2.69e+00 & 2.64e-01 & \cellcolor{g}8.66e+01 & 1.36e-01 \\
\cline{2-8}
k=4, s=2& Riesz & 8.95e-01 & \cellcolor{g}3.34e+00 & -1.27e-01 & \cellcolor{g}7.08e-02 & 8.40e+01 & \cellcolor{g}2.61e-02 \\
\cline{2-8}
& Newtonian & 1.50e+00 & 2.18e+01 &\cellcolor{g} -2.26e-01 & 7.52e-02 & 8.44e+01 & 3.61e-02 \\
\cline{2-8}
&Morse & 9.85e+00 & 1.94e+09 & -1.56e-01 & 9.09e-02 & 7.63e+01 & 3.45e-02 \\
\hline
\end{tabular}
\caption{Algorithm performance for the different settings and problems. Results are averaged over 25 runs.}
\label{table:results}
\end{table}
\cref{table:results} reports the performance metrics for all the problems. For most problems, the strategy $\tau=0$, with no interaction in $\Omega$ allows to reach lower values of $GD$. This is consistent with the analytical results \cref{t:CBO} and \cref{r:tau0}, which suggested that the additional dynamics may interfere with the CBO mechanism and, as a consequence, slow down the convergence towards optimal EP points. If the diversity metrics $\mathcal{U}_R$, $\mathcal{U}_N$, $\mathcal{U}_M$ and $\mathcal{S}$ are considered, dynamics including interaction of parameters allow to obtain more diverse solutions. Interestingly, using Morse binary potential in the interaction leads to a final lower Newtonian energy in some cases. We will investigate the role of the potential choice and $\tau$ in the next section.
In \cref{fig:1} the $IGD$ performance shows that letting particles interact in parameter space improves the overall quality of the solution. While the improvement is more substantial in problems with complex Pareto fronts (see for instance Lamé $\gamma = 0.25$, or DO2DK $k=2$), we remark that the additional mechanism allows to obtain better solutions. This is even true, if the parameter distribution is already optimal form the beginning (see Lamé $\gamma = 1$). We conjecture that this due to the additional stochasticity introduced by the potential. We will also study this aspect in the next subsection.
\cref{fig:2a,fig:2b} show the time evolution of $GD$, $\mathcal{U}_R$, $\mathcal{U}_N$, $\mathcal{U}_M$ and $IGD$ for two of the considered test problems. As suggested by the analysis of the mean-field model, in particular \cref{t:CBO}, $GD$ exponentially decays up to a maximum accuracy within the first iterations of the algorithm. This is due to the $CBO$ dynamics driving the particles around EP optimal points. At the same time, the potential energies increase as the particles are concentrating towards the front in the image-space. Another consequence of \cref{t:CBO} is that assumption \eqref{eq:f} is fulfilled and consequently the gradient-flow description \eqref{eq:red2} is valid. This is also observed in \cref{fig:2a,fig:2b} where the potentials start decreasing provided that relatively low $GD$ values are attained.
\begin{figure}
\centering
\begin{subfigure}{0.91\linewidth}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0cm , clip, width=1\linewidth]{experiment1_1}
\caption{Problem Lamé $\gamma = 0.25$}
\label{fig:2a}
\end{subfigure}
\begin{subfigure}{0.91\linewidth}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=1\linewidth]{experiment1_4}
\caption{Problem DO2DK $k=2, s=1$}
\label{fig:2b}
\end{subfigure}
\caption{Performance metric evolution, results are averaged over 25 runs.}
\label{fig:2}
\end{figure}
\subsection{Effect of the parameter $\tau$ and scalability}
By looking at the computational results, it becomes clear that the two phases of the algorithm, the one characterized by the CBO dynamics and the one characterized by the gradient-flows dynamics, have different scales. Typically, the former dynamics is much slower compared with the second one. This was consistent with assumptions to \cref{t:CBO}, where $\tau$ needs to be taken of order $o(\sqrt{\varepsilon})$.
\begin{figure}
\centering
\begin{subfigure}{1\linewidth}
\includegraphics[trim = 0cm 0cm 0cm 0cm , clip, width=1\linewidth]{ParOpt_besttau_GD}
\caption{Generational Distance (GD) \eqref{eq:GDnum}}
\label{fig:3a}
\end{subfigure}
\begin{subfigure}{1\linewidth}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=1\linewidth]{ParOpt_besttau_IGD}
\caption{Inverted Generational Distance (IGD) \eqref{eq:IGDnum}}
\label{fig:3b}
\end{subfigure}
\caption{Performance metrics as a function of $\tau$ for all the problems considered. Results are averaged over 10 runs.}
\label{fig:3}
\end{figure}
To experimentally investigate the importance, we test the algorithm for various values of $\tau$, keeping the remaining parameters fixed. \cref{fig:3a,fig:3b} show the final $GD$ and $IGD$ metrics when different binary potential are used during the computation.
As expected, relatively large values of $\tau$ lead to a strong interaction in parameter space that interferes with the CBO mechanism. As a result, the $GD$ metric increases for large values of $\tau$. Interestingly, the lowest $GD$ values are not always attained for the smallest values of $\tau$, suggesting that the additional weights vectors dynamics might help the CBO mechanism in optimizing the sub-problems.
The $IGD$ metrics in \cref{fig:3b},shows that the optimal value of $\tau$ is different for each test case. In particular, DO2DK problems benefit from a strong interaction in parameter space. This might be explained by the front geometry (\cref{fig:1}): the front length is long and, as consequence, the particles tend to be further apart in the image space, making the binary potential interaction weaker. Larger values of $\tau$ mitigate this effect, leading to better algorithm performances. If the extrema of the Pareto front are known in advance, one could address this issue by estimating the front length and choosing the parameter $\tau$ accordingly. We also note that algorithm seems to perform better when the Morse potential is used during the computation.
As already mentioned, the dynamics in $\Omega$ adds stochasticity to the particles position evolution. Hence, the additional diffusive term $\sigma D_k^iB_k^i$ in \eqref{eq:iterx} might not be necessary. Yet, taking $\sigma = 0$ yields poor approximations of the Pareto front, see \cref{fig:4a}, suggesting that the diffusive term is still of paramount importance for the particles exploration behavior and their statistical independence.
From \cref{fig:4b}, it is obvious that the optimal diffusion parameter $\sigma$ is larger, the smaller $\tau$ is. In particular, if $\tau=0$ the particles diverge from the optimal EP points only when $\sigma>10$, which is consistent with other CBO methods for single-objective optimization, see for instance \cite{benfenati2021binary}. At the same time, for some problems, if $\sigma$ is too small, larger values of $\tau$ improve the convergence towards optimal points.
\begin{figure}
\centering
\begin{subfigure}{1.\linewidth}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0cm , clip, width=1\linewidth]{ParOpt_tausigma_GD_morse}
\caption{$GD$}
\label{fig:4b}
\end{subfigure}
\begin{subfigure}{1.\linewidth}
\includegraphics[trim = 0cm 0cm 0cm 0.5cm , clip, width=1\linewidth]{ParOpt_tausigma_IGD_morse}
\caption{$IGD$}
\label{fig:4a}
\end{subfigure}
\caption{Metrics as functions of $\sigma$, for different values of $\tau$. Morse interaction is used, results are averaged over 5 runs.}
\label{fig:4}
\end{figure}
Finally, we test the algorithm performance for different dimensions $d$ of the search space, keeping the same parameters choice. If the same number $N=100$ of particles are used, the $IGD$ of the computed solutions increases as the space dimension $d$ becomes larger, see \cref{fig:5}. This effect can be simply reduced by increasing the number of particles linearly with the space dimension, see \cref{fig:5}.
\begin{figure}
\centering
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, width=1\linewidth]{ParOpt_IGDdim}
\caption{$IGD$ metrics as functions of the search space dimension $d$. Morse interaction is used, results are averaged over 20 runs.}
\label{fig:5}
\end{figure}
\section{Conclusions}
In this work, we proposed an adaptive stochastic particle dynamics based on consensus to solve multi-objective optimization problems. The method makes use of a scalarization strategy the break down the original problem into $N$ parametrized single-objective sub-problems.
The proposed algorithm, AM-CBO, extends prior work on multi-objective consensus based optimization by an additional adaptive dynamics in the parameter space in order to ensure that the particles distribute uniformly over the Pareto front. This is achieved by exploiting energy-based diversity measures. A rigorous mathematical analysis and numerical evidence are provided to validate this behavior. We theoretically investigated the long time behavior of the particle dynamics under the propagation of chaos assumption and establish convergence towards optimal points. Indeed, under appropriate assumptions, the particles are capable of solving several single-objective problems at the same time, with a remarkable save of computational cost with the respect to a naive approach. The additional dynamics on the parameter space is also analyzed based on results on non-linear aggregation equations. Numerical experiments show that the proposed method is capable to solve multi-objective problems with very different Pareto fronts. The algorithm scales well with the problem dimension, even when using a relatively small number of particles.
|
1,108,101,565,509 | arxiv | \section{Introduction}
A {\em discrete statistical model} \ is a subset
$\mathcal{M}$ of the open probability simplex $\Delta_{n}$. Each point $p$ in $\Delta_n$
is a probability distribution on the finite state space $\{0,1,\ldots,n\}$,~i.e.~$p = (p_0,p_1,
\ldots,p_n),$ where the $p_i$ are positive real numbers that satisfy
$p_0+p_1 + \cdots + p_n=1$. The model $\mathcal{M}$ is the set of all distributions
$p \in \Delta_n$ that are relevant for the particular application of interest.
In data analysis we are given an empirical distribution
$u = (u_0,u_1,\ldots,u_n)$. This is the point in the simplex $\Delta_n$
whose $i$th coordinate $u_i$ is the fraction of samples
observed to be in state~$i$.
The {\em maximum likelihood estimator} (MLE) of $\mathcal{M}$ is a
function $\Phi\colon \Delta_n \rightarrow \mathcal{M}$ that
takes the empirical distribution $u$ to a distribution
$\hat p = (\hat p_0,\hat p_1,\ldots,\hat p_n)$ that
best explains the given observations.
Here ``best'' is understood in the sense of likelihood inference.
This means that $\hat p = \Phi(u)$ is the point in $\mathcal{M}$
that maximizes the {\em log-likelihood function}
$ p \mapsto \sum_{i=0}^n u_i \cdot {\rm log}(p_i)$.
By convention, for any vector $u$ in $\mathbb{R}^{n+1}_{>0}$,
we set $\Phi(u) := \Phi(u/|u|)$ where $|u| = u_0+\cdots+u_n$.
Likelihood inference is consistent. This means that $\Phi(u) = u$ for $u \in \mathcal{M}$.
This follows from the fact that the log-likelihood function is
strictly concave on $\Delta_n$ and its unique maximizer is $p = u$.
Therefore, the MLE
$\Phi$ is a retraction from the simplex onto the model.
This point is fundamental for two fields at the crossroads of mathematics
and data science. {\em Information Geometry} \cite{Ay} views
the MLE as the nearest point map of a Riemannian metric on $\Delta_n$,
given by the Kullback-Leibler divergence of probability distributions.
{\em Algebraic Statistics} \cite{DSS} is concerned with models $\mathcal{M}$
whose MLE $\Phi$ is an algebraic function of $u$. This happens precisely when
the constraints that define $\mathcal{M}$ can be expressed in
terms of polynomials in $p$.
In this article we address a question that is fundamental for both fields:
\\ {\em For which models $\mathcal{M}$ is the MLE $\,\Phi$ a
rational function in the empirical distribution $u$?} \\
The most basic example where this happens
is the independence model for two binary random variables $(n=3)$.
Here $\mathcal{M}$ is a surface in the tetrahedron $\Delta_3$. That surface is a
familiar picture that serves as a point of entry for both
Information Geometry and Algebraic Statistics.
Points in $\mathcal{M}$ are positive rank one $2\times 2$ matrices
\begin{small} $\begin{bmatrix} p_0 & p_1 \\
p_2 & p_3 \end{bmatrix}$\end{small} whose entries sum to one.
The data takes the form of
a nonnegative integer $2 \times 2$ matrix $u$ of counts of observed frequencies. Hence
$ \,|u| = u_0{+}u_1{+}u_2{+}u_3$ is the sample size, and $u/|u|$ is
the empirical distribution. The MLE $\hat p = \Phi(u)$
is evaluated by multiplying the row and column~sums of $u$:
$$
\hat p_0 = \frac{(u_0 {+} u_1)(u_0{+}u_2)}{|u|^2} , \,
\hat p_1 = \frac{(u_0 {+} u_1)(u_1{+}u_3)}{|u|^2} , \,
\hat p_2 = \frac{(u_2 {+} u_3)(u_0{+}u_2)}{|u|^2} , \,
\hat p_3 = \frac{(u_2 {+} u_3)(u_1{+}u_3)}{|u|^2} .
$$
These four expressions are rational, homogeneous of degree zero,
and their sum is equal to~$1$.
We refer to \cite[Example 2]{huh14}
for a discussion of these formulas from our present perspective.
The independence model belongs to the class of graphical models \cite{lauritzen}.
Fix an undirected graph $G$ whose nodes represent random variables
with finitely many states.
The undirected graphical model $\mathcal{M}_G$ is a subset
of $\Delta_n$, where $n{+}1$ is the number of states in the joint distribution.
The graphical model $\mathcal{M}_G$ is \emph{decomposable} if and only if the graph $G$
is chordal. In this case, each coordinate $\hat p_i$ of the MLE is
an alternating product of linear forms
given by maximal cliques and minimal separators of~$G$.
A similar formula exists for directed graphical models,
also known as Bayesian networks.
In both cases, the coordinates of the MLE are not only rational functions, but even alternating products
of linear forms in $u = (u_0,u_1,\ldots,u_n)$.
This is no coincidence. Huh \cite{huh14} proved that
if $\Phi$ is a rational function then each of its coordinates is an
alternating product of linear forms, with numerator and denominator
of the same degree. Huh further showed that this alternating product must take a very
specific shape. That shape was discovered by Kapranov~\cite{Kap91},
who named it the {\em Horn uniformization}.
The results by Kapranov and Huh are valid for arbitrary
complex algebraic varieties. They make no reference
to a context where the coordinates are real, positive, and add up to~$1$.
The present paper makes the
leap from complex varieties back to statistical models. Building on the remarkable constructions
found by Kapranov and Huh, we here work in the setting of
real algebraic geometry that is required for statistical applications.
Our main result (Theorem~\ref{thm:main}) characterizes all models $\mathcal{M}$ in
$\Delta_n$ whose MLE is a rational function. It is stated in Section~2 and all its ingredients are presented in a self-contained manner.
In Section~3 we examine models with rational MLE that are familiar to
statisticians, such as decomposable graphical models and
Bayesian networks. Our focus lies on {\em staged tree models},
a far-reaching generalization of discrete Bayesian networks, as described
in the book by Collazo, G\"orgen and Smith \cite{CGS}.
We explain how our main result applies to these models.
The proof of Theorem~\ref{thm:main} is presented in Section~4.
This is the technical heart of our paper, building on the
likelihood geometry of \cite[\S 3]{huh2014likelihood}.
We also discuss the connection to toric geometry and geometric modeling
that appeared in recent work of Clarke and Cox
\cite{clarke2018moment}.
In Section~5 we present our algorithm
for constructing models with rational MLE, and we discuss
its implementation and some experiments.
The input is an integer matrix representing a toric variety, and the output is
a list of models derived from that matrix.
Our results suggest that only a small fraction of Huh's varieties in \cite{huh14}
are statistical models.
\section{How to be Rational}
Let $\mathcal{M}$ be a discrete statistical model in the open simplex
$\Delta_n$ that has a well-defined maximum likelihood estimator
$\Phi : \Delta_n \rightarrow \mathcal{M}$. We also write
$\Phi : \mathbb{R}^{n+1}_{> 0} \rightarrow \mathcal{M}$ for the induced
map $u \mapsto \Phi(u/|u|)$ on all positive vectors.
If the $n+1$ coordinates of $\Phi$ are rational functions in $u$,
then we say that $\mathcal{M}$
{\em has rational MLE}.
The following is our main result in this~paper.
\begin{theorem} \label{thm:main}
The following are equivalent for a discrete statistical model
$\mathcal{M} $ with MLE~$\Phi$:
\begin{itemize}
\item[(1)] The model $\mathcal{M}$ has {\bf rational MLE}. \vspace{-0.1cm}
\item[(2)] There exists a {\bf Horn pair} $(H,\lambda)$ such that $\mathcal{M}$ is the image of the Horn map $\,\varphi_{(H,\lambda)}$. \vspace{-0.1cm}
\item[(3)] There exists a {\bf discriminantal triple}
$(A,\Delta,{\bf m})$ such that $\mathcal{M}$ is the image
under the monomial map $\,\phi_{(\Delta,{\bf m})}\,$ of precisely
one orthant (\ref{eq:orthantdef}) of the dual toric variety $\,Y_A^*$. \vspace{-0.1cm}
\end{itemize}
The MLE of the model satisfies the following relation on the open orthant $\mathbb{R}^{n+1}_{>0}$\rm{:}\begin{equation}
\label{eq:threemaps}
\Phi \,= \, \varphi_{(H,\lambda)} \,= \, \phi_{(\Delta,\bf m)} \circ H.
\end{equation}
\end{theorem}
The goal of this section is to define all the terms seen in parts (2) and (3) of this theorem.
\begin{example} \label{ex:smalltree}
We first discuss Theorem~\ref{thm:main} for a very simple experiment:
{\em Flip a biased coin. If it shows heads, flip it again}.
This is a statistical model with $n=2$ given by the tree diagram
\begin{center}
\begin{tikzpicture}
\renewcommand{1}{1.3}
\renewcommand{1}{0.7}
\node (r) at (0*1,1*1) {\stage{ProcessBlue}{1}};
\node (b0) at (2*1,2*1) {\stage{ProcessBlue}{1}};
\node (b1) at (2*1,0*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a0) at (4*1,3*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a1) at (4*1,1*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\draw[->] (r) -- node [above] {\ls{$s_0$}} (b0);
\draw[->] (r) -- node [below] {\ls{$s_1$}} (b1);
\draw[->] (b0) -- node [above] {\ls{$s_0$}} (a0);
\draw[->] (b0) -- node [below] {\ls{$s_1$}} (a1);
\node [right, xshift=5] at (a0) {$p_0$};
\node [right, xshift=5] at (a1) {$p_1$};
\node [right, xshift=5] at (b1) {$p_2.$};
\end{tikzpicture}
\end{center}
The model $\mathcal{M}$ is a curve in the probability triangle $\Delta_2$.
The tree shows its parametrization
$$ \qquad \Delta_1\to \Delta_2 \,, \,\,
(s_0,s_1)\mapsto(s_0^2,s_0s_1,s_1)
\qquad \hbox{where $s_0,s_1 > 0$ and $s_0+s_1=1$.} $$
The implicit representation of the curve $\mathcal{M}$ is the quadratic equation
$p_0p_2-(p_0+p_1)p_1=0$.
Let $(u_0,u_1,u_2)$ be the counts from repeated experiments.
A total of $2u_0 + 2u_1 + u_2$ coin tosses were made.
We estimate the parameters as the empirical frequency of heads resp.~tails:
$$
\hat s_0 \,=\, \frac{2u_0 + u_1}{2u_0 + 2u_1 + u_2} \quad \text{and}\quad
\hat s_1 \,=\, \frac{u_1+u_2}{2u_0 + 2u_1 + u_2}.
$$
The MLE is the retraction from the triangle $\Delta_2$ to the curve $\mathcal{M}$
given by the rational formula
$$ \Phi(u_0,u_1,u_2)\,\,=\,\, (\hat s_0^2, \hat s_0 \hat s_1, \hat s_1) \,\,=\,\,\,
\begin{small}
\biggl(
\frac{(2u_0 + u_1)^2}{(2u_0 {+} 2u_1 {+} u_2)^2} \,,\,
\frac{(2u_0 {+} u_1)(u_1{+}u_2)}{(2u_0 + 2u_1 + u_2)^2}\, ,\,
\frac{u_1+u_2}{2u_0 {+} 2u_1 {+} u_2} \biggr).
\end{small}
$$
Hence $\mathcal{M}$ has rational MLE.
The corresponding Horn pair from part (2) in Theorem~\ref{thm:main} has
$$ H \,= \,\begin{small} \begin{pmatrix}
\phantom{-}2 & \phantom{-}1 & \phantom{-}0 \,\,\, \\
\phantom{-}0 & \phantom{-}1 & \phantom{-}1 \, \,\,\\
-2& -2& -1 \,\,\,
\end{pmatrix}\end{small} \quad {\rm and} \quad \lambda \,= \,(1,1,-1). $$
We next exhibit the discriminantal triple
$(A,\Delta,\bf m)$ in
part (3) of Theorem~\ref{thm:main}.
The matrix $A = \begin{pmatrix}1 & 1 & 1\end{pmatrix}$
gives a basis of the left kernel of $H$. The second entry is the polynomial
\begin{equation}
\label{eq:DeltaFactors} \Delta \,\, =\,\, x_3^2 - x_1^2 - x_1x_2 + x_2x_3 \,\, =\,\,
(x_3-x_1)(x_1+x_2+x_3). \end{equation}
The third entry marks the leading term ${\bf m}= x_3^2$. These data define the monomial map
$$ \phi_{(\Delta,{\bf m})} \,\, :\,\,
(x_1,x_2,x_3) \,\mapsto \,
\biggl(\,\frac{x_1^2}{x_3^2} \,,\, \frac{x_1x_2}{x_3^2} \, \,,-\frac{x_2}{x_3} \biggr).$$
The toric variety of the matrix $A$ is the point
$Y_A = \{(1:1:1)\}$ in $\mathbb{P}^2$. Our polynomial $\Delta$
vanishes on the line
$Y_A^{*} = \{x_1+x_2+x_3 =0 \}$ that is dual to $Y_A$.
The relevant orthant is the open line segment
$Y^{*}_{A,\sigma} \coloneqq \{(x_1:x_2:x_3) \in Y_A^* \,: \, x_1,x_2 >0 \,\,{\rm and} \,\, x_3 <0 \}$.
Part (3) in Theorem \ref{thm:main} says that $\mathcal{M}$ is the image of
$Y^*_{A,\sigma}$ under $\phi_{(\Delta,{\bf m})}$.
The MLE is $\Phi = \phi_{(\Delta,\bf m)} \circ H$.
\end{example}
We now come to the definitions needed for Theorem \ref{thm:main}.
Let $H = (h_{ij})$ be an $m \times (n{+}1)$ integer matrix whose columns sum to zero,
i.e.~$\,\sum_{i=1}^m h_{ij} = 0$ for $j=0,\ldots,n$.
We call such a matrix a \emph{Horn matrix}.
The following alternating products of linear forms have degree zero:
$$ (Hu)^{h_j} \,\,:=\,\,\prod_{i=1}^m \bigl(h_{i0} u_0 + h_{i1} u_1 + \cdots + h_{in} u_n\bigr)^{h_{ij}}
\qquad {\rm for} \,\, j=0,1,\ldots,n. $$
The Horn matrix $H$ is {\em friendly} if there
exists a real vector $\,\lambda = (\lambda_0,\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\neq 0$ for all $i$ such that the following identity holds in the
rational function field $\mathbb{R}(u_0,u_1,\ldots,u_n)$:
\begin{equation}
\label{eq:friendly} \lambda_0 (Hu)^{h_0} + \lambda_1 (Hu)^{h_1}+ \cdots
+ \lambda_n (Hu)^{h_n} \,\, = \,\, 1 .
\end{equation}
If this holds, then we say that $(H,\lambda)$ is a {\em friendly pair},
and we consider the rational function
\begin{equation}
\label{eq:rationalmap}
\mathbb{R}^{n+1} \,\to \,\mathbb{R}^{n+1} ,\,\,
u \, \mapsto \, \bigl(
\lambda_0 (Hu)^{h_0} , \,\lambda_1 (Hu)^{h_1}, \,\ldots ,\,
\lambda_n (Hu)^{h_n} \bigr).
\end{equation}
The friendly pair $(H,\lambda)$ is called a {\em Horn pair}
if no row of $H$ is zero or is a multiple of another row,
the function (\ref{eq:rationalmap}) is
defined for all positive vectors, and it maps these to positive vectors.
If these conditions hold then we write
$\,\varphi_{(H,\lambda)} : \mathbb{R}^{n+1}_{>0} \,\to \,\mathbb{R}^{n+1}_{>0} \,$
for the restriction of (\ref{eq:rationalmap}) to the positive orthant. We
call $\,\varphi_{(H,\lambda)}\,$ the {\em Horn map} associated to the
Horn pair $(H,\lambda)$.
\begin{remark}\label{rem:makeminimal}
Let $(H,\lambda)$ be a friendly pair satisfying the positivity condition for
the function~\eqref{eq:rationalmap}. To it we associate a Horn pair $(\tilde H, \tilde \lambda)$ by aggregating its collinear rows by summing them together, deleting the zero rows of $H$, and defining $\tilde \lambda$ as in \cite[Proposition~6.11]{clarke2018moment}. The pairs $(H,\lambda)$ and $(\tilde H, \tilde \lambda)$ define the same rational function~\eqref{eq:rationalmap}. Furthermore,
every Horn pair $(\tilde H,\tilde \lambda)$ can be uniquely recovered, up to permutation of its rows, from its Horn map $\varphi_{(\tilde H,\tilde \lambda)}$.
\end{remark}
\begin{example} \label{ex:illustrate}
We illustrate the equivalence of (1) and (2) in Theorem \ref{thm:main}
for the model described in \cite[Example 3.11]{huh2014likelihood}. Here $n=3$ and $m=4$ and
the Horn matrix equals
\begin{equation}
\label{eq:matrix44} H \quad = \quad \begin{small} \begin{pmatrix}
-1 & -1 & -2 & -2 \,\, \\
\phantom{-} 1 & \phantom{-} 0 & \phantom{-} 3 & \phantom{-} 2 \,\, \\
\phantom{-} 1 &\phantom{-} 3 & \phantom{-} 0 & \phantom{-} 2 \,\, \\
-1 & -2 & -1 & -2 \, \,\end{pmatrix}.\end{small}
\end{equation}
This Horn matrix is friendly because the following vector satisfies the identity (\ref{eq:friendly}):
\begin{equation}\label{eq:lambda14} \lambda \,=\, (\lambda_0,\lambda_1,\lambda_2,\lambda_3)\,=\,
\begin{small}
\biggl( \frac{2}{3} \,, \,-\frac{4}{27} \,,\, -\frac{4}{27}\,,\,\frac{1}{27} \biggr). \end{small}
\end{equation}
The pair $(H,\lambda)$ is a Horn pair, with associated Horn map
\begin{equation}
\label{eq:hornmap}
\varphi_{(H,\lambda)} : \,\mathbb{R}^{4}_{>0} \,\to \,\mathbb{R}^{4}_{>0} \,,\,\,
\begin{pmatrix} u_0 \\
u_1 \\ u_2 \\ u_3 \end{pmatrix} \mapsto \begin{pmatrix}
\frac{2 (u_0 + 3u_2 + 2u_3)(u_0+3u_1+2u_3)}{3(u_0+u_1+2u_2 + 2 u_3)(u_0+2u_1+u_2+2u_3)} \smallskip \\
\frac{4(u_0 + 3u_2 + 2u_3)^3}{27(u_0+u_1+2u_2 + 2 u_3)^2(u_0+2u_1+u_2+2u_3)}
\smallskip \\
\frac{4(u_0+3u_1+2u_3)^3}{27(u_0+u_1+2u_2 + 2 u_3)(u_0+2u_1+u_2+2u_3)^2}
\smallskip \\\,
\frac{(u_0 + 3u_2 + 2u_3)^2(u_0+3u_1+2u_3)^2}{27(u_0+u_1+2u_2 + 2 u_3 )^2(u_0+2u_1+u_2+2u_3)^2}\,
\end{pmatrix}.
\end{equation}
Indeed, this rational function evidently takes positive vectors to positive vectors.
The image of the map $\varphi_{(H,\lambda)} $ is a subset $\mathcal{M}$ of the tetrahedron
$\Delta_3 = \{p \in \mathbb{R}^4_{>0}:p_0+p_1+p_2 + p_3 = 1 \}$. We regard the subset
$\mathcal{M}$ as a discrete statistical model on the state space $\{0,1,2,3\}$.
The model $\mathcal{M}$ is the rational space curve of degree $4$ defined by the two quadratic equations
$$ 9p_1p_2 - 8 p_0p_3 \,=\,p_0^2 - 12(p_0+p_1+p_2+p_3)p_3 \,=\,0. $$
As in \cite[Example 3.11]{huh2014likelihood},
one verifies that the curve $\mathcal{M}$ has rational MLE, namely
$\,\Phi = \varphi_{(H,\lambda)}$.
\end{example}
We next define all the terms that are used in part (3) of Theorem \ref{thm:main}.
Fix a matrix $A = (a_{ij}) \in \mathbb Z^{r\times m}$ of rank $r$ that has the vector
$(1,\ldots,1)$ in its row span. The connection to (2) will be that the rows
of $A$ span the left kernel of $H$.
We identify the columns of $A$ with Laurent monomials in $r$ unknowns
$t_1,\ldots,t_r$. The corresponding monomial map is
\begin{equation}
\label{eq:monomap}
\gamma_A \,\,:\, (\mathbb{R}^*)^{r} \to \mathbb \mathbb{R} \mathbb{P}^{m-1}\,,\,\,\,
(t_1,\ldots,t_r) \,\mapsto \,
\biggl( \,\prod_{i=1}^r t_i^{a_{i1}}:
\,\prod_{i=1}^r t_i^{a_{i2}}:\,\, \cdots \,\, :
\,\prod_{i=1}^r t_i^{a_{im}} \biggr).
\end{equation}
Here $\mathbb{R}^* = \mathbb{R} \backslash \{0\}$ and $\mathbb{R} \mathbb{P}^{m-1}$
denotes the real projective space of dimension $m-1$.
Let $Y_A$ be the closure of the image of $\gamma_A$. This is
the projective toric variety given by the matrix~$A$.
Every point $x = (x_1:\cdots:x_m)$ in the dual projective space $(\mathbb{R} \mathbb P^{m-1})^\vee$
corresponds to a hyperplane $H_x$ in $\mathbb{R} \mathbb{P}^{m-1}$.
The dual variety $Y_A^*$ to the toric variety $Y_A$ is the closure of the~set
\[
\bigl\{\,x\in \mathbb{R} \mathbb P^{m-1} \mid \gamma_A^{-1}(H_x\cap Y_A)
\, \text{\rm{ is singular}} \,\bigr\}.
\]
A general point $x$ in the {\em dual toric variety}
$Y_A^*$ corresponds to a hyperplane $H_x$ that is
tangent to the toric variety $Y_A$ at a point $\gamma_A(t)$ with nonzero coordinates.
We identify sign vectors $\sigma \in \{-1,+1\}^m$ with orthants in $\mathbb{R}^m$.
These map in a $2$-to-$1$ manner to
orthants in $\mathbb{R} \mathbb{P}^{m-1}$. If we intersect them with $Y_A^*$,
then we get the {\em orthants} of the dual toric variety:
\begin{equation}
\label{eq:orthantdef} Y_{A,\sigma}^* \,\, = \,\, \bigl\{ \,x \in Y_A^*\, :\, \sigma_i \cdot x_i > 0
\,\,\hbox{for} \,\, i=1,2,\ldots,m \,\bigr\} \quad \subset \,\,\,\mathbb{R} \mathbb{P}^{m-1}.
\end{equation}
One of these is the distinguished orthant in Theorem~\ref{thm:main}, part (3).
\begin{example} \label{ex:fourtwo}
Fix $m=4$, $r=2$, and the following matrix with
$(1,1,1,1)$ in its row span:
\begin{equation}
\label{eq:matrix24}
A \,=\, \begin{pmatrix} 3 & 2 & 1 & 0 \\
0 & 1 & 2 & 3 \end{pmatrix}.
\end{equation}
As in \cite[Example 3.9]{huh2014likelihood}, the toric variety of $A$
is the {\em twisted cubic curve} in projective $3$-space:
$$ Y_A \,=\,\overline{\bigl\{ (t_1^3 : t_1^2t_2: t_1 t_2^2 : t_2^3) \in \mathbb{R} \mathbb{P}^3
\,: \, t_1,t_2 \in {\mathbb{R}}^* \bigr\}}.$$
The dual toric variety $Y_A^*$ is a surface in
$( \mathbb{R} \mathbb{P}^3)^\vee$. Its points $x$ represent planes
in $\mathbb{R} \mathbb{P}^3$ that are tangent to
the curve $Y_A$. Such a tangent plane corresponds to a cubic
$\,x_1 t^3+ x_2 t^2 +x_3 t + x_4 \,$ with a double root.
Hence, $Y_A^*$ is the surface of degree $4$ in
$(\mathbb{R} \mathbb P^{3})^\vee$ defined by the discriminant
\begin{equation}
\label{eq:disc24}
\Delta_A \,\, = \,\,
\underline{27 x_1^2 x_4^2} - 18 x_1 x_2 x_3 x_4 + 4 x_1 x_3^3 + 4 x_2^3 x_4 - x_2^2 x_3^2.
\end{equation}
All eight orthants $Y_{A,\sigma}^*$ are non-empty.
Representatives $x$ for the orthants are the eight cubics
$$ \begin{small} \begin{matrix} (t+1)^2(t+3),\,
(t+5)^2(t-1),\,
(t-1)^2(t+3),\,
(t+5)^2(t-8), \\
(t-3)^2(t+1),\,
(t-1)^2(t-3),\,
\underline{ (t-2)^2(t+3)},\,
(t+1)^2(t-3) .\end{matrix} \end{small}
$$
The underlined cubic is the point $\,x=(1,-1,-8,12 ) $ in $Y_{A,\sigma}^*$,
where $\sigma =(1,-1,-1,1)$.
\end{example}
We now present the key definition that is needed for
part (3) of Theorem~\ref{thm:main}.
Let $\Delta$ be a homogeneous polynomial with $n+2$
monomials, and let ${\bf m}$ be one of these monomials.
If we divide $\Delta$ by ${\bf m}$, then we obtain
a homogeneous Laurent polynomial of degree zero:
$$ \frac{1}{{\bf m}} \Delta\, \,= \,\,\, 1 \, - \,
\lambda_0 x_1^{h_{10}}x_2^{h_{20}} \cdots x_m^{h_{m0}} \,- \,
\lambda_1 x_1^{h_{11}}x_2^{h_{21}} \cdots x_m^{h_{m1}} \,-\, \cdots \, - \,
\lambda_n x_1^{h_{1n}}x_2^{h_{2n}} \cdots x_m^{h_{mn}}.
$$
We write $H_{(\Delta,{\bf m})}$ for the
$m \times (n+1)$ integer matrix with entries $h_{ij}$. Its column vectors are
denoted $h_j = (h_{1j},h_{2j},\ldots,h_{mj})$ for $j=0,1,\ldots,n$.
These data define the monomial map
$$\phi_{(\Delta,{\bf m})} \, : \, (\mathbb{R}^*)^m \rightarrow \mathbb{R}^{n+1} , \,
\,x \, \mapsto \, \bigl(
\lambda_0 x^{h_0} , \,\lambda_1 x^{h_1}, \,\ldots ,\,
\lambda_n x^{h_n} \bigr) .$$
\begin{definition} \label{def:dq}
A {\em discriminantal triple} $(A,\Delta,{\bf m})$ consists of
\begin{enumerate}
\item an $r \times m$ integer matrix $A$ of rank $r$ having $(1,1,\ldots,1)$ in its row span,
\vspace{-0.18cm}
\item an $A$-homogeneous polynomial $\Delta$ that vanishes
on the dual toric variety $Y_A^*$,
\vspace{-0.18cm}
\item a distinguished term ${\bf m}$ among those that occur in
the polynomial $\Delta$,
\end{enumerate}
such that $H_{(\Delta,\mathbf m)} = \tilde H_{(\Delta,\mathbf m)}$, the sign vector $\sigma\coloneqq \operatorname{sign}(H_{(\Delta,\mathbf m)}\cdot u)$ is the same for all positive column
vectors $u\in \mathbb R_{>0}^{n+1}$, and it satisfies the condition
\begin{equation}\label{eq:positivity}
\, \lambda_i \cdot \sigma^{h_i} > 0 \text{ for all } i=1,2,\ldots,m.
\end{equation}
\begin{remark}\label{rem:discriminantminimal}
Let $(A,\Delta,\mathbf m)$ be a triple as in Definition~\ref{def:dq}, 1--3, such that for its associated Horn matrix $H_{(\Delta,\mathbf m)}$ we have that $\operatorname{sign}(\tilde H_{(\Delta,\mathbf m)}\cdot u)$ is the same for all positive $u$ and the pair $(\tilde H_{(\Delta, \mathbf m)}, \tilde \lambda)$ satisfies \eqref{eq:positivity}. As in Remark~\ref{rem:makeminimal} we associate to $(A,\Delta,\mathbf m)$ a discriminantal triple $(\tilde A, \tilde \Delta, \tilde {\mathbf m})$ with $\operatorname{Im} \phi_{(\Delta,\mathbf m)} = \operatorname{Im} \phi_{(\tilde \Delta,\tilde {\mathbf m})}$.
\end{remark}
All definitions are now complete.
We next illustrate Definition~\ref{def:dq} for our running example.
\end{definition}
\begin{example}
Let $A$ be the $2 \times 4$ matrix in (\ref{eq:matrix24}),
$\Delta = \Delta_A$ its discriminant in (\ref{eq:disc24}), and
${\bf m} = 27x_1^2x_4^2$ the underlined term.
Then $(A,\Delta,{\bf m})$ is a discriminantal triple with
associated sign vector $\sigma=(-1,+1,+1,-1)$.
The orthant $Y_{A,\sigma}^*$ was highlighted
in Example \ref{ex:fourtwo}. It is a semialgebraic surface
inside $Y_A^* \subset \mathbb{R} \mathbb{P}^3$. This surface is mapped into
the tetrahedron $\Delta_3$~by
\begin{equation}
\label{eq:othermap} \phi_{(\Delta,{\bf m})} \,: \, (x_1,x_2,x_3,x_4)\,\mapsto \,
\biggl( \frac{2}{3} \frac{x_2x_3}{x_1 x_4},
-\frac{4}{27} \frac{x_3^3}{x_1 x_4^2},
-\frac{4}{27} \frac{x_2^3}{x_1^2 x_4},
\frac{1}{27} \frac{x_2^2 x_3^2}{x_1^2 x_4^2}
\biggr).
\end{equation}
The image of this map is a curve in $\Delta_3$, namely
the model $\mathcal{M}$ in Example~\ref{ex:illustrate}.
We verify (\ref{eq:threemaps}) by comparing
(\ref{eq:hornmap}) with (\ref{eq:othermap}).
The former is obtained from the latter by setting $x=Hu$.
\end{example}
We close this section with two remarks on Horn matrices, Horn pairs and Horn maps.
\section{Staged Trees}
We consider contingency tables $u = (u_{i_1 i_2 \cdots i_m})$ of format
$r_1 \times r_2 \times \cdots \times r_m$. Following \cite{DSS, lauritzen},
these represent joint distributions of discrete statistical models with
$n+1 =r_1r_2 \cdots r_m$ states.
For any subset $C \subset \{1,\ldots,m\}$, one considers
the marginal table $u_C$ that is obtained by summing out all indices not in $C$.
The entries of the marginal table $u_C$ are sums
of entries in $u$. Namely, to obtain the entry $u_{I,C}$ of $u_C$ for any state $I = (i_1,i_2,\ldots,i_m),$
we fix the indices of the states in $C$ and sum over the indices not in $C$.
For example, if $m=4$, $C=\{1,3\}$,
$I=(i,j,k,l)$, then $u_C$ is the $r_1 \times r_3$ matrix with entries
$$ u_{I,C} \,\, = \,\,
u_{i + k + } \,\, =\,\,\,
\sum_{j=1}^{r_2} \sum_{l=1}^{r_4} u_{ i j k l} .$$
Such linear forms are the basic building blocks for
the familiar models with rational MLE.
Consider an undirected graph $G$ with vertex set $\{1,\ldots,m\}$
which is assumed to be {\em chordal}.
The associated {\em decomposable graphical model} $\mathcal{M}_G$
in $\Delta_n$ has the rational MLE
\begin{equation}
\label{eq:mle_dg} \hat p_I \,\, = \,\, \frac{\prod_{C} u_{I,C} }{\prod_{S} u_{I,S}},
\end{equation}
where the product in the numerator is over all maximal cliques $C$ of $G$,
and the product in the denominator is over all separators $S$ in
a junction tree for $G$. See \cite[\S 4.4.1]{lauritzen}.
In what follows we regard $G$ as a directed graph, with edge
directions given by a
perfect elimination ordering on the vertex set $\{1,\ldots,m\}$.
This turns $\mathcal{M}_G$ into a Bayesian network.
More generally, a {\em Bayesian network} $\mathcal{M}_G$
is given by a directed acyclic graph $G$. We write
${\rm pa}(j)$ for the set of parents of the node $j$. The model $\mathcal{M}_G$
in $ \Delta_n$
has the rational MLE
\begin{equation}
\label{eq:mle_bn} \hat p_I \,\, = \,\,
\prod_{j=1}^m \frac{u_{I,{\rm pa}(j) \cup \{j\}}}{ u_{I,{\rm pa}(j)}}.
\end{equation}
If $G$ comes from an undirected chordal graph then
(\ref{eq:mle_dg}) arises from
(\ref{eq:mle_bn}) by cancellations.
\begin{example}[$m=4$] We revisit two examples
that were discussed on page 36 in \cite[\S 2.1]{DSS}.
The {\em star graph} $G = [14][24][34]$ is chordal. The MLE for $\mathcal{M}_G$ is the map
$\Phi$ with coordinates
$$ \hat p_{ijkl} \,\, =\,\,\, \frac{ u_{i++l} \cdot u_{+j+l} \cdot u_{++kl}}
{ u_{++++} \cdot u_{+++l}^2} \,\, =\,\,\,
\frac{u_{i+++}}{u_{++++}} \cdot
\frac{u_{+j+l}}{u_{+++l}} \cdot
\frac{u_{++kl}}{u_{+++l}} \cdot
\frac{u_{i ++l}}{u_{i+++}}.
$$
The left expression is (\ref{eq:mle_dg}).
The right is (\ref{eq:mle_bn}) for the directed graph
$1 \rightarrow 4$, $4 \rightarrow 2$, $4 \rightarrow 3$.
The {\em chain graph} $G = [12][23][34]$ is chordal. Its MLE is the map
$\Phi$ with coordinates
$$
\hat p_{ijkl} \,\, =\,\,\, \frac{ u_{i++l} \cdot u_{+jk+} \cdot u_{++kl}}
{ u_{+j++} \cdot u_{++k+}} \,\, =\,\,\, \varphi_{(H,\lambda)}(u)_{ijkl}.
$$
This is the Horn map in Proposition~\ref{prop:mle-staged-trees},
given by the specific pair $(H,\lambda)$ in Example~\ref{ex:stree}.
\end{example}
The formulas (\ref{eq:mle_dg}) and (\ref{eq:mle_bn}) are familiar
to statisticians. Theorem~\ref{thm:main} places them into a larger context.
However, some readers may find
our approach too algebraic and too general. Our aim in this section is lay out a
useful middle ground:
models given by staged~trees.
Staged trees were introduced by Smith and Anderson \cite{andersonSmith} as a generalization
of discrete Bayesian networks. They furnish an intuitive representation of
many situations that the above graphs $G$ cannot capture. In spite of their wide scope,
staged tree models are appealing because of their intuitive formalism for
encoding events. We refer to the textbook~\cite{CGS} for an introduction.
In what follows we study parts (1) and (2) in Theorem~\ref{thm:main} for staged~trees.
To define a {\em staged tree model}, we start with a directed rooted tree $\mathcal T$ having at least two edges emanating from each non-leaf vertex, a label set $S = \{s_i\mid i\in I\}$, and a labeling $\theta\colon \operatorname E(\mathcal T) \to S$ of the edges of the tree.
Each vertex of $\mathcal T$ has a corresponding \emph{floret}, which is the multiset of edge labels emanating from it. The labeled tree $\mathcal T$ is a \emph{staged tree} if any two florets are either equal or disjoint. Two vertices in $\mathcal{T}$ are in the same stage if their corresponding florets are the same. From this point on, $F$ denotes
the set of florets of $\mathcal{T}$.
\begin{definition} \label{def:stm}
Let $J$ be the set of root-to-leaf paths in the tree $\mathcal{T}$. We set $|J| = n+1$. For $i\in I$ and $j\in J$, let $\mu_{ij}$
denote the number of times edge label $s_i$ appears in the $j$-th root-to-leaf path.
The \emph{staged tree model} $\mathcal M_\mathcal{T} $ is the image of the parametrization
$$
\phi_{\mathcal{T}} : \Theta\to \Delta_n \, , \,\,
(s_i)_{i\in I} \mapsto (p_j)_{j\in J},
$$
where $\Theta:= \left\{(s_i)_{i \in I} \in (0,1)^{|I|} : \sum_{s_i\in f} s_i = 1 \text{ for all florets $f\in F$}\right\}$ is the parameter space of $\mathcal{M}_{\mathcal{T}}$, and $p_j = \prod_{i\in I} s_i^{\mu_{ij}}$ is the product of the edge parameters on the $j$-th root-to-leaf path.
\end{definition}
In the model $\mathcal M_\mathcal{T} $, the tree $\mathcal{T}$ represents
possible sequences of events. The parameter $s_i$ associated to an edge $vv'$ is the transition
probability from $v$ to $v'$. All parameter labels in a floret sum to $1$. The fact that distinct
nodes in $\mathcal{T}$ can have the same floret
of parameter labels enables staged tree models to encode conditional independence
statements \cite{andersonSmith}. This property allows us to represent any discrete
Bayesian network or decomposable model as a staged tree model.
Our first staged tree was seen in Example \ref{ex:smalltree}.
Here is another specimen.
\begin{example}[$n=15$] \label{ex:stree}
Consider the decomposable model for binary variables
given by the $4$-chain $G=[12][23][34]$.
Figure~$1$ shows a realization of $\mathcal{M}_G$
as a staged tree model $\mathcal M_\mathcal{T} $.
The leaves of $\mathcal{T}$ represent the outcome space $\{0,1\}^4$.
Nodes with the same color have the same associated floret.
The blank nodes all have different florets. The seven florets of $\mathcal{T}$ are
$$ \begin{small}
f_1 {=}\{s_0,s_1\}, f_2 {=}\{s_2,s_3\}, f_3 {=}\{s_4,s_5\}, f_4 {=} \{s_6,s_7\},
f_5 {=}\{s_8,s_9\}, f_6 {=}\{s_{10},s_{11}\}, f_7 {=}\{s_{12},s_{13}\}.
\end{small} $$
\end{example}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\renewcommand{1}{1.3}
\renewcommand{1}{0.5}
\node at (1,13*1) {$\mathcal{T}:$};
\node (r) at (1*1,7.5*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (d1) at (2*1,11.5*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (d2) at (2*1,3.5*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (c1) at (3*1,13.5*1) {\stage{Dandelion}{1}};
\node (c2) at (3*1,9.5*1) {\stage{ProcessBlue}{1}};
\node (c3) at (3*1,5.5*1) {\stage{Dandelion}{1}};
\node (c4) at (3*1,1.5*1) {\stage{ProcessBlue}{1}};
\node (b1) at (4*1,14.5*1) {\stage{Orchid}{1}};
\node (b2) at (4*1,12.5*1) {\stage{SpringGreen}{1}};
\node (b3) at (4*1,10.5*1) {\stage{Orchid}{1}};
\node (b4) at (4*1,8.5*1) {\stage{SpringGreen}{1}};
\node (b5) at (4*1,6.5*1) {\stage{Orchid}{1}};
\node (b6) at (4*1,4.5*1) {\stage{SpringGreen}{1}};
\node (b7) at (4*1,2.5*1) {\stage{Orchid}{1}};
\node (b8) at (4*1,0.5*1) {\stage{SpringGreen}{1}};
\node (a1) at (5*1,15*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a2) at (5*1,14*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a3) at (5*1,13*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a4) at (5*1,12*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a5) at (5*1,11*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a6) at (5*1,10*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a7) at (5*1,9*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a8) at (5*1,8*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a9) at (5*1,7*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a10) at (5*1,6*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a11) at (5*1,5*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a12) at (5*1,4*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a13) at (5*1,3*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a14) at (5*1,2*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a15) at (5*1,1*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\node (a16) at (5*1,0*1) {\tikz{\node[shape=circle,draw,inner sep=2pt] {};}};
\draw[->] (r) -- node [above] {\ls{$s_0$}} (d1);
\draw[->] (r) -- node [below] {\ls{$s_1$}} (d2);
\draw[->] (d1) -- node [above] {\ls{$s_2$}} (c1);
\draw[->] (d1) -- node [above] {\ls{$s_3$}} (c2);
\draw[->] (d2) -- node [above] {\ls{$s_4$}} (c3);
\draw[->] (d2) -- node [above] {\ls{$s_5$}} (c4);
\draw[->] (c1) -- node [above] {} (b1);
\draw[->] (c1) -- node [below] {} (b2);
\draw[->] (c2) -- node [above] {} (b3);
\draw[->] (c2) -- node [below] {} (b4);
\draw[->] (c3) -- node [above] {\ls{$s_6$}} (b5);
\draw[->] (c3) -- node [below] {\ls{$s_7$}} (b6);
\draw[->] (c4) -- node [above] {\ls{$s_8$}} (b7);
\draw[->] (c4) -- node [below] {\ls{$s_9$}} (b8);
\draw[->] (b1) -- node [above] {\ls{$s_{10}$}} (a1);
\draw[->] (b1) -- node [below] {\ls{$s_{11}$}} (a2);
\draw[->] (b2) -- node [above] {} (a3);
\draw[->] (b2) -- node [below] {} (a4);
\draw[->] (b3) -- node [above] {} (a5);
\draw[->] (b3) -- node [above] {} (a6);
\draw[->] (b4) -- node [above] {\ls{$s_{12}$}} (a7);
\draw[->] (b4) -- node [below] {\ls{$s_{13}$}} (a8);
\draw[->] (b5) -- node [above] {} (a9);
\draw[->] (b5) -- node [above] {} (a10);
\draw[->] (b6) -- node [above] {} (a11);
\draw[->] (b6) -- node [above] {} (a12);
\draw[->] (b7) -- node [above] {} (a13);
\draw[->] (b7) -- node [above] {} (a14);
\draw[->] (b8) -- node [above] {} (a15);
\draw[->] (b8) -- node [above] {} (a16);
\node [right, xshift=5] at (a1) {$p_{0000}$};
\node [right, xshift=5] at (a2) {$p_{0001}$};
\node [right, xshift=5] at (a3) {$p_{0010}$};
\node [right, xshift=5] at (a4) {$p_{0011}$};
\node [right, xshift=5] at (a5) {$p_{0100}$};
\node [right, xshift=5] at (a6) {$p_{0101}$};
\node [right, xshift=5] at (a7) {$p_{0110}$};
\node [right, xshift=5] at (a8) {$p_{0111}$};
\node [right, xshift=5] at (a9) {$p_{1000}$};
\node [right, xshift=5] at (a10){$p_{1001}$};
\node [right, xshift=5] at (a11){$p_{1010}$};
\node [right, xshift=5] at (a12){$p_{1011}$};
\node [right, xshift=5] at (a13){$p_{1100}$};
\node [right, xshift=5] at (a14){$p_{1101}$};
\node [right, xshift=5] at (a15){$p_{1110}$};
\node [right, xshift=5] at (a16){$p_{1111}$};
\end{tikzpicture}
\begin{tikzpicture}
\node (m) at (1*1,10*1) {
\scalebox{0.6}{$H={\begin{array}{l}
s_0\\s_1\\f_1\\s_2\\s_3\\f_2\\s_4\\s_5\\f_3
\\s_6\\s_7\\f_4\\s_8\\s_9\\f_5\\s_{10}\\s_{11}\\f_6\\s_{12}\\s_{13}\\f_7
\end{array} \left({
\begin{array}{cccccccccccccccc}
1&1&1&1&1&1&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&1&1&1&1&1&1\\
{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}\\
1&1&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&1&1&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&1&1&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&1&1\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&{-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}\\
1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&\cdot&\cdot&\cdot&\cdot\\
{-}&{-}&{-}&{-}&\cdot&\cdot&\cdot&\cdot&{-}&{-}&{-}&{-}&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&1\\
\cdot&\cdot&\cdot&\cdot&{-}&{-}&{-}&{-}&\cdot&\cdot&\cdot&\cdot&{-}&{-}&{-}&{-}\\
1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot\\
\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot\\
{-}&{-}&\cdot&\cdot&{-}&{-}&\cdot&\cdot&{-}&{-}&\cdot&\cdot&{-}&{-}&\cdot&\cdot\\
\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot\\
\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&1\\
\cdot&\cdot&{-}&{-}&\cdot&\cdot&{-}&{-}&\cdot&\cdot&{-}&{-}&\cdot&\cdot&{-}&{-}\\
\end{array}}\right)}$}};
\end{tikzpicture}
\end{center}\vspace {-0.1in}
\caption{
A staged tree $\mathcal{T}$ and its Horn matrix $H$
in Proposition~\ref{prop:mle-staged-trees}. Entries $-$ indicate $-1$.}\label{fig:tH}
\end{figure}
Next we show that staged tree models have rational MLE, so they
satisfy part (1) of Theorem \ref{thm:main}.
Our formula for $\Phi$ uses the notation for
$I,J$ and $\mu_{ij}$ introduced in Definition~\ref{def:stm}.
This formula is known in the literature on
chain event graphs (see e.g.~\cite{Silander.Leong.2013}).
\begin{proposition} \label{prop:mle-staged-trees}
Let $\mathcal{M}_{\mathcal{T}}$ be a staged tree model, and let $u = (u_j)_{j\in J}$ be a
vector of counts. For $i\in I$, let $f$ be the floret containing
the label $s_i$, and define the estimates
\[
\hat s_i \,\, \coloneqq \,\, \frac{\sum_j \mu_{ij}u_j}{\sum_{s_{\ell }\in f}\sum_{j} \mu_{\ell j} u_j}
\quad {\rm and} \quad
\hat p_j \,\,\coloneqq \,\, \prod_{i \in I} (\hat s_i)^{\mu_{ij}}.
\]
The rational function $\,\Phi\,$ that sends $\,(u_j)_{j\in J}\,$
to $\,(\hat p_j)_{j\in J}\,$ is the MLE of the model $\mathcal M_\mathcal{T}$.
\end{proposition}
\begin{proof}
We prove that the likelihood function $L(p,u)$ has a unique maximum at
$p = (\hat p_j)_{j\in J}$.
For $f\in F$, we fix the vector of parameters $s_f = (s_i)_{s_i\in f}$.
Associated with the floret $f$ is the local
likelihood function $L_f(s_f,u) = \prod_{s_i\in f} s_i^{\alpha_i}$, where
$\alpha_i =\sum_j \mu_{ij} u_j$. We have
\[ L(p,u) \, = \, \prod_j p_j^{u_j} \, = \,\prod_j \prod_i s_i ^ {u_j\mu_{ij}}\,
= \,\prod_i s_i^{\alpha_i} \, = \, \prod_{f\in F} L_f(s_f,u). \]
Since the $L_f$ depend on disjoint sets of unknowns, maximizing
$L$ is achieved by maximizing the factors $L_f$ separately.
But $L_f$ is the likelihood function of the full model $\Delta_{|f|-1}$,
given the data vector $(\alpha_i)_{s_i\in f}$. The MLE of that model is given by
$\hat s_i = \alpha_i / \sum_{s_\ell \in f} \alpha_\ell$, where $s_i \in f$.
We conclude that $\,\mathrm{argmax}_{s_f}\bigl(L_f(s_f,u)\bigr) = (\hat s_i)_{s_i\in f}\,$
and $\,\mathrm{argmax}_p \bigl( L(p,u) \bigr) = (\hat p_j)_{j \in J}$.
\end{proof}
\begin{remark}\label{elementary}
Here is a method for evaluating the MLE in Proposition~\ref{prop:mle-staged-trees}.
Let $[v]\subset J$ be the set of root-to-leaf paths through a fixed node $v$
in the tree $\mathcal{T}$ and define $u_{[v]}=\sum_{j\in [v]}u_j$.
The quotient $\frac{u_{[v']}}{u_{[v]}}$ is the empirical transition probability from
$v$ to $v'$ given arrival at $v$.
To obtain $\hat s_i$ we first compute the quotients $\frac{u_{[v']}}{u_{[v]}}$ for all edges $vv'$ with parameter
label $s_i$. Then we aggregate them by adding
their numerators and denominators separately.
We obtain
$\, s_i\,=\, (\sum u_{[v']})/(\sum u_{[v]})$,
where both sums range over all edges $vv'$ with parameter
label $s_i$.
\end{remark}
Proposition~\ref{prop:mle-staged-trees} yields an explicit
description of the Horn pair $(\tilde H,\tilde \lambda)$ associated to $\mathcal{M}_{\mathcal{T}}$.
\begin{corollary} \label{cor:horn-staged-trees}
Fix a staged tree model $\mathcal{M}_{\mathcal{T}}$ as above.
Let $H$ be the $(|I|+|F|)\times |J|$ matrix whose rows are indexed
by the set $I\sqcup F$ and entries are given by
\begin{align*}
&h_{ij}\,=\, \mu_{ij} \text{ for } i \in I, \text{ and } \\
&h_{fj}\,= \, -\sum_{s_{\ell} \in f}\mu_{\ell j} \text{ for }f \in F.
\end{align*}
Define the vector $\lambda\in \{-1,+1\}^{|J|}$ by
$\lambda_j= (-1)^{\sum_{f}h_{fj}}$. Then $(\tilde H,\tilde \lambda)$ is the Horn pair of
$\mathcal{M}_{\mathcal{T}}$, using the mapping $H\mapsto \tilde H$ defined in Remark~\ref{rem:makeminimal}.
\end{corollary}
Given a staged tree $\mathcal T$, we call the matrix $H$
in Corollary~\ref{cor:horn-staged-trees} the \emph{Horn matrix} of~$\mathcal T$.
\begin{remark}
In Corollary~\ref{cor:horn-staged-trees}, for a floret
$f\in F$, let $H_{f}$ be the submatrix of $H$ with row indices
$\{i:s_i\in f\}\cup\{f\}$. Then $H$ is the vertical concatenation
of the matrices $H_f$ for $f\in F$.
The matrix $\tilde H$ is obtained from $H$ by the row operations
described in Remark \ref{rem:makeminimal}.
\end{remark}
\begin{example}\label{ex:minimalH}
For the tree $\mathcal{T}$ in Example~\ref{ex:stree}, the Horn matrix
$H$ of $\mathcal M_\mathcal T$ is given in Figure~1.
Its rows indices are $(s_0,s_1,f_1,s_2,s_3,f_2,s_4,s_5,f_3
,s_6,s_7,f_4,s_8,s_9,f_5,s_{10},s_{11}, f_6,s_{12},s_{13},f_7)$.
The vector $\lambda$ for the friendly Horn matrix $H$
is the vector of ones $(1,\ldots,1)\in\mathbb{R}^{16}$.
Note that $(H,\lambda)$ is not a Horn pair.
We can delete the rows ${s_0,s_1,f_2,f_3}$
of the matrix $H$ by summing the pairs $(s_0,f_2)$ and $(s_1,f_3)$ and deleting zero rows. The result is the Horn pair $(\tilde H,\tilde \lambda)$.
\end{example}
Following \cite{gorgenSmith}, two staged trees $\mathcal{T}$ and $\mathcal{T}'$ are
called \emph{statistically equivalent}
if there exists a bijection between the sets of root-to-leaf paths of $\mathcal{T}$ and
$\mathcal{T}'$ such that, after applying this bijection, $\mathcal{M}_{\mathcal{T}}=\mathcal{M}_{\mathcal{T}'}$ inside the open simplex $\Delta_n$. Any staged tree model may have different but statisticaly equivalent tree
representations. In \cite[Theorem~1]{gorgenSmith}, the authors show that statistical equivalence
of staged trees can be determined by doing a sequence of operations on the trees, named \emph{swap} and \emph{resize}. One of the advantages
of describing a staged tree model via its Horn pair is that it gives a new criterion to decide whether two staged trees are statistically equivalent. This is simpler to implement than the criterion formulated in \cite{gorgenSmith}.
\begin{corollary}
Two staged trees are statistically equivalent if and only if their
their Horn pairs $(\tilde H, \tilde \lambda)$ agree.
\end{corollary}
One natural operation on a staged tree $\mathcal{T}$ is identifying two florets of the same size.
This gives a new tree $\mathcal{T}'$ and model $\mathcal{M}_{\mathcal{T}'}$ whose Horn matrix is readily obtained from that of~$\mathcal{T}$.
\begin{corollary} \label{cor:staging}
Let $\mathcal{T}'$ be a staged tree arising from $\mathcal{T}$ by identifying two florets $f$ and $f'$,
say by the bijection $(-)'\colon f\to f'$.
Then the Horn matrix $H'$ of $\mathcal{M}_{\mathcal{T}'}$ arises from
the Horn matrix $H$ of $\mathcal{M}_{\mathcal{T}}$ by replacing the blocks
$H_f$ and $H_{f'}$ in $H$ by the block $H'_f$ defined by
\begin{align*}
h'_{ij}&= h_{ij} +h_{i'j} \;\;\text{ for } s_i \in f,\\
h'_{fj}&= h_{fj}+h_{f'j}.
\end{align*}
\end{corollary}
\begin{proof}
This follows from the definition of the Horn matrices for $\mathcal{M}_{\mathcal{T}}$ and
$\mathcal{M}_{\mathcal{T}'}$.
\end{proof}
\begin{example} Let $\mathcal{T}'$ be the tree obtained from $\mathcal{T}$ in Example~\ref{ex:stree} by
identifying the florets $f_4$ and $f_5$. Then $\mathcal{M}_{\mathcal{T}'}$ is the
independence model of two random variables with four states.
\end{example}
\smallskip
Now we turn to part (3) of Theorem~\ref{thm:main}. We
describe the triple $(A,\Delta,{\bf m})$
for a staged tree model $\mathcal{M}_\mathcal{T}$ giving rise to its discriminantal triple $(\tilde A,\tilde \Delta,\tilde {\mathbf m})$ as in Remark~\ref{rem:discriminantminimal}. The pair
$(H,\lambda)$ was given in Corollary~\ref{cor:horn-staged-trees}.
Let $A$ be any matrix whose rows span the
left kernel of $H$, set $m=|I|+|J|$, and write $s$ for the $m$-tuple of parameters
$(s_i,s_f)_{i\in I, f\in F}$. From the Horn matrix in Corollary~\ref{cor:horn-staged-trees} we see that
\[\Delta={\bf m}\cdot \left(1-\sum_j (-1)^{\epsilon_j}
\prod_i \left(\frac{s_i}{s_f}\right)^{\mu_{ij}}
\right), \]
where $f$ depends on $i$, $\,{\bf m}=\lcm(\prod_i s_f^{\mu_{ij}}:f\in F)\,$
and $\,\epsilon_j={\sum_i \mu_{ij}}$.
The sign vector $\sigma$ for the triple $(A,\Delta,\mathbf m)$ is given by
$\sigma_i=+1$ for $i\in I$
and $\sigma_f=-1$ for $f\in F$.
Then $Y_{A,\sigma}^{*}$ gets mapped to $\mathcal{M}_{\mathcal{T}}$ via $\phi_{(\Delta,{\bf m})}$. Moreover, the
map $\phi_{\mathcal{T}}$ from Definition~\ref{def:stm} factors through $\phi_{(\Delta,{\bf m})}$. Indeed, if we define
$\iota: \Theta \to Y_{A,\sigma}^{*}$ by $(s_i)_{i\in I}\mapsto (s_i,-1)_{i \in I, f\in F}$, then $\phi_{\mathcal{T}}=\phi_{(\Delta,{\bf m})}\circ \iota$.
The derivation in the following example is an extension of that in
\cite[Example 3.13]{huh2014likelihood}.
\begin{example} \label{ex:4chainhorn}
Let $\mathcal{M}_{\mathcal{T}}$ be the $4$-chain model in Example~\ref{ex:stree}. Its associated
discriminant is
$$
\begin{matrix}
\Delta \,=\,
f_1 f_2 f_3 f_4 f_5 f_6 f_7 \!\!\!\!\! & \!
- \,s_0 s_2 s_6 s_{10} f_3 f_5 f_7 - s_0 s_2 s_6 s_{11} f_3 f_5 f_7
- s_0 s_2 s_7 s_{12} f_3 f_5 f_6 - s_0 s_2 s_7 s_{13} f_3 f_5 f_6 \\ & -\, s_0 s_3 s_8 s_{10} f_3 f_4 f_7
- s_0 s_3 s_8 s_{11} f_3 f_4 f_7 - s_0 s_3 s_9 s_{12} f_3 f_4 f_6 - s_0 s_3 s_9 s_{13} f_3 f_4 f_6 \\ &
- \,s_1 s_4 s_6 s_{10} f_2 f_5 f_7 - s_1 s_4 s_6 s_{11} f_2 f_5 f_7 - s_1 s_4 s_7 s_{12} f_2 f_5 f_6
- s_1 s_4 s_7 s_{13} f_2 f_5 f_6 \\ & \,- \,s_1 s_5 s_8 s_{10} f_2 f_4 f_7 - s_1 s_5 s_8 s_{11} f_2 f_4 f_7
- s_1 s_5 s_9 s_{12} f_2 f_4 f_6 - s_1 s_5 s_9 s_{13} f_2 f_4 f_6.
\end{matrix}
$$
Our notation for the parameters matches the row labels of the Horn matrix $H$ in
Figure~\ref{fig:tH}. This polynomial of degree $7$ is irreducible, so it equals the $A$-discriminant:
$\,\Delta = \Delta_A$.
The underlying matrix $A$ has format $13 \times 21$, and we represent it by its associated
toric ideal
$$
\begin{matrix}
I_A & = &\!\!\! \! \bigl\langle\,
s_{10} - s_{11}\,, \,\,
s_1 s_5 f_2 - s_0 s_3 f_3\,, \,\,s_1 s_4 f_2 - s_0 s_2 f_3\,,\,\,
s_5 s_9 f_4 - s_4 s_7 f_5\,, \,\,s_3 s_9 f_4 - s_2 s_7 f_5, \\ & & \,\,\,
s_{12} - s_{13}, \,
s_5 s_8 f_4 - s_4 s_6 f_5, s_3 s_8 f_4 - s_2 s_6 f_5, \,
s_9 s_{13} f_6 - s_8 s_{11} f_7, \, s_7 s_{13} f_6 - s_6 s_{11} f_7, \\ & & \! \!
s_0 s_2 s_6 s_{11} - f_1 f_2 f_4 f_6,
s_0 s_2 s_7 s_{13} - f_1 f_2 f_4 f_7,
s_0 s_3 s_8 s_{11} - f_1 f_2 f_5 f_6,
s_0 s_3 s_9 s_{13} - f_1 f_2 f_5 f_7, \\ & &
s_1 s_4 s_6 s_{11} - f_1 f_3 f_4 f_6,
s_1 s_4 s_7 s_{13} - f_1 f_3 f_4 f_7,
s_1 s_5 s_9 s_{13} - f_1 f_3 f_5 f_7,
s_1 s_5 s_8 s_{11} - f_1 f_3 f_5 f_6
\bigr\rangle.
\end{matrix}
$$
The toric variety $Y_A = \mathcal{V}(I_A)$ has dimension $12 $ and degree $141$.
It lives in a linear space of codimension $2$ in $\mathbb{P}^{20}$, where it is defined by
eight cubics and eight quartics. The dual variety $Y_A^* = \mathcal{V}(\Delta_A)$
is the above hypersurface of degree seven. We have
$ {\bf m} = f_1 f_2 f_3 f_4 f_5 f_6 f_7$, and
$\sigma $ is the vector in $\{-1,+1\}^{21}$ that has entry $+1$ at the indices corresponding to the $s_i$ and entry $-1$ at the indices corresponding to the $f_i$.
To obtain the discriminant $\tilde \Delta$ associated
to the Horn pair in Example~\ref{ex:minimalH},
we substitute 1 for $s_0,s_1,f_2,f_3$ in the polynomial~$\Delta$ and change all the minus signs to plus signs. See also the discussion
in Remark \ref{rem:makeminimal}.
\end{example}
It would be interesting to study the combinatorics of
the discriminantal triples for staged tree models. Our computations suggest
that, for many such models, the polynomial $\Delta$
is irreducible and is equal to the $A$-discriminant
$\Delta_A$ of the underlying configuration~$A$. However,
this is not true for all staged trees, as seen in equation (\ref{eq:DeltaFactors})
of Example~\ref{ex:smalltree}. We close this section
with a familiar class of models with rational MLE whose associated $\Delta$ factor.
\begin{example}\label{ex:multinomial} The {\em multinomial distribution} encodes the
experiment of rolling a $k$-sided die $m$ times. The associated model $\mathcal{M}$ is the
independence model for $m$ identically distributed random variables on $k$ states. We have
$n+1 = \binom{k+m-1}{m}$. The Horn matrix
$H$ is the $(k+1) \times (n+1)$ matrix
whose columns are the vectors $(-m,i_1,i_2,\ldots,i_k)^T$
where $i_1,i_2,\ldots,i_k$ are nonnegative integers whose sum equals $m$.
Here, $\, A = (1 \,\, 1 \,\, 1 \,\cdots \, 1)$, so the $A$-discriminant
is the linear form $\,\Delta_A = x_0+x_1+\cdots+x_k$. The following
polynomial is a multiple of $\Delta_A$:
$$ \Delta \,\,= \,\, (-x_0)^m - (x_1 + x_2 + \cdots +x_k)^m. $$
This $\Delta$, with its marked term ${\bf m} = (-x_0)^m$, encodes
the MLE for the model~$\mathcal{M}$.
\end{example}
\section{Proof of the Main Theorem}
In this section we prove Theorem~\ref{thm:main}.
This involves making precise how the objects in the three parts correspond to each other. Namely, models with rational MLE correspond to Horn pairs $(H,\lambda)$, and these correspond to pairs $(\Delta,\mathbf m)$ in a discriminantal triple.
For a pair $(H,\lambda)$ consisting of a Horn matrix $H$ and a coefficient vector $\lambda$, we denote by $\varphi$ the rational map defined in~(\ref{eq:rationalmap}). We recall that its $i$-th coordinate is
\begin{equation}\label{eq:rationalmap-coord}
\varphi_i(v) \,\,= \,\,\lambda_i\, \prod_{j=1}^m \biggl(\sum_{k=0}^n h_{jk}v_k \biggr)^{h_{ji}}.
\end{equation}
We also define the likelihood function $L_u\colon \mathbb R^{n+1}\to \mathbb R$ for the image of $\varphi$:
\begin{equation}\label{eq:likelihood-fct}
L_u(v)\, \coloneqq \,\prod_{i=0}^{n} \varphi_i(v)^{u_i}.
\end{equation}
Here $u\in \mathbb N^{n+1}$ is an arbitrary fixed data vector.
We start with the following key lemma.
\begin{lemma} \label{prop:horn-map-estimates-likelihoods}
Let $H = (h_{ij})$ be a Horn matrix, $\lambda$ a vector satisfying (\ref{eq:friendly}),
and $u\in \mathbb N^{n+1}$. The vector $u$ is the unique critical point of
its own likelihood function $L_u$, up to scaling.
\end{lemma}
\begin{proof}
We compute the partial derivatives of $L_u$.
For $\ell = 0,\dotsc, n$ we find
\begin{align*}
\frac{\partial}{\partial v_\ell} L_u(v)
&\,\,=\,\, \sum_{i=0}^n u_i\, \frac{L_u(v)}{\varphi_i(v)}\, \frac{\partial}{\partial v_\ell} \varphi_i(v)
\\ &\,\,= \,\, \sum_{i=0}^n u_i\, \frac{L_u(v)}{\varphi_i(v)}\,
\sum_{j=1}^m h_{ji}\, \frac{\varphi_i(v)}{\sum_{k=0}^{n}h_{jk} v_k}\, h_{j\ell}
\\ & \,\,=\,\, L_u(v)\, \sum_{j=1}^m \sum_{i=0}^n \frac{u_i\, h_{ji}\, h_{j\ell}}{\sum_{k=0}^n h_{jk} v_k}
\quad = \quad L_u(v)\, \sum_{j=1}^m \frac{h_{j\ell}\,\sum_{i=0}^n h_{ji} u_i}{\sum_{k=0}^n h_{jk} v_k}.
\end{align*}
For $v=u$, this evaluates to zero, since the sums in the fraction cancel and the $\ell$-th column of $H$ sums to zero.
The uniqueness of the critical point up to scaling follows from the fact that the
projective variety given by the image of $\varphi$
has ML-degree one, by \cite[Theorem~1]{huh14}.
\end{proof}
We use \cite{huh14} to explain the
relation between models with rational MLE and Horn pairs.
\begin{proof}[Proof of Theorem~\ref{thm:main}, Equivalence of (1) and (2).]
Let $\mathcal M$ be a model with rational MLE $\Phi$. The Zariski closure of
$\mathcal M$ is a variety of ML-degree one. By \cite[Theorem~1]{huh14}, there exists a Horn matrix $H$ and a coefficient vector $\lambda$ such that $\varphi = \Phi$. Now, the required sum-to-one and positivity conditions for $\varphi$ are satisfied because they are satisfied by the MLE $\Phi$. Indeed, the MLE of any
discrete statistical model maps positive vectors
$u$ in $\mathbb{R}^{n+1}_{> 0}$ into the simplex~$\Delta_n$.
Conversely, we claim that every Horn pair $(H,\lambda)$ specifies a nonempty model $\mathcal M$ with rational MLE. Indeed, define $\mathcal{M}$ to be the image of $\varphi_{(H,\lambda)}$. By the defining properties of the Horn pair, we have $\mathcal M \subset \Delta_n$. Lemma \ref{prop:horn-map-estimates-likelihoods} shows that $\varphi_{(H,\lambda)}$ is the MLE of $\mathcal M$.
\end{proof}
Next, we relate Horn pairs to discriminantal triples $(A,\Delta,\mathbf m)$.
The pair $(\Delta,\mathbf m)$ is the data that defines
$\mathcal M$ as an algebraic variety. The matrix $A$ and the derived sign vector $\sigma$
are witnesses of special properties of $(\Delta,\mathbf m)$. Namely, the polynomial $\Delta$ is $A$-homogeneous and vanishes on some dual toric variety, $Y_A^*$, whose $\sigma$-orthant
maps onto the model $\mathcal M$ via the map $\phi_{(\Delta, \mathbf m)}$.
The positivity condition of a Horn pair is supposed to translate into
the positivity condition in \eqref{eq:positivity}.
This translation is a consequence of the following key lemma.
\begin{lemma}\label{prop:positivity-domino}
Let $(H,\lambda)$ be a friendly pair. If there exists a vector $u\in \mathbb R^{n+1}$ such that $\varphi(u)>0$, then we have $\varphi(v) > 0$ for all $v$
in $\mathbb{R}_{>0}^{n+1}$ where it is defined.
\end{lemma}
\begin{proof}
The function $\varphi$ is homogeneous of
degree zero. It suffices to prove each coordinate of $\varphi(v)$ is a positive real number,
for all vectors $v$ with
positive integer entries. Indeed, every positive $v$ in $\mathbb{R}^{n+1}$
can be approximated by rational vectors, which can be scaled to be integral.
The open subset $U = \varphi^{-1}(\Delta_n)$ of~$\mathbb R^{n+1}$ contains $u$. If $U=\mathbb R^{n+1}$, then we are done. Else, $U$ has a nonempty boundary $\partial U$.
By continuity, $\partial U\subseteq \varphi^{-1}(\partial \Delta_n)$.
The likelihood function $L_v$ for the data vector $v$ vanishes on~$\partial U$.
We claim that $L_v$ has a critical point in $U$. The closed subset $\overline U$ is homogeneous. After passing to projective space $\mathbb P^n$, it becomes compact.
The likelihood function $L_v$ is well defined on
this compact set in $\mathbb{P}^n$, since it is homogeneous of degree zero,
and $L_v$ vanishes on the boundary.
Hence the restriction $L_v|_U$ is either identically zero or it has a critical point in $U$. But,
since $u\in U$ is a point with $L_v(u)\neq 0$, the second statement must be true.
Since $U$ is an open subset of $\mathbb R^{n+1}$, a critical point of the restriction
$L_v|_U$ is also a critical point of the function $L_v$ itself.
By Lemma~\ref{prop:horn-map-estimates-likelihoods},
this critical point must be $v$. Hence $v\in U$.
\end{proof}
\begin{corollary} \label{cor:sigmaexists}
Let $(H,\lambda)$ be a friendly pair, with $H = \tilde H$ as in
Remark~\ref{rem:makeminimal}. Fix~any positive vector $u$ in $ \mathbb R^{n+1}_{>0}$.
Then $(H,\lambda)$ is a Horn pair if and only if
$\lambda_i (Hu)^{h_i} > 0$ for $i=0,1\ldots,n$.
If this holds then the nonzero entries in each row of $H$ have
the same sign. In particular, the sign vector $\sigma = {\rm sign}(Hu)$ is
independent of the choice of~$u$.
\end{corollary}
\begin{proof}
The coordinates of $Hv$ are the linear factors of
the numerators and denominators of $\varphi(v)$.
We have shown in Lemma \ref{prop:positivity-domino}
that none of these numerators or denominators vanish on
$\Delta_n$, and hence the same holds for the coordinates of $Hv$.
This implies that the rows of $H$ have the desired sign property.
The characterization of Horn pairs now follows from (\ref{eq:rationalmap}).
\end{proof}
We prove the rest of Theorem~\ref{thm:main} by first explaining how to turn
$(H,\lambda)$ into a pair $(\Delta, \mathbf m)$ and then examining how the
constraints on Horn pairs and discriminantal triples are related.
\begin{proof}[Proof of Theorem~\ref{thm:main}, Equivalence of (2) and (3).]
Let $(H,\lambda)$ be a pair consisting of a Horn matrix and a coefficient vector. We construct a pair $(\Delta,\mathbf m)$ consisting of a polynomial $\Delta$ and a monomial $\mathbf m$ appearing in $\Delta$ as follows. For $k = 0,\dotsc, n+1$ let $h_k$ denote the columns of $H$, and write $h_k^+$ resp.\ $h_k^-$ for the positive part resp.\ the negative part of $h_k$, so that $h_k = h_k^+ - h_k^-$. In addition, let $\mathrm{max}_k (h_k^-)$ be the entrywise maximum of the $h^-_k$.
We define
\begin{equation}
\label{eq:mDelta}
\mathbf{m}\, =\, x^{\mathrm{max}_k (h_k^-)}
\quad \text{ and } \quad \Delta \,= \,\mathbf{m}\cdot
\biggl(1-\sum_{k=0}^n \lambda_k x^{h_k} \biggr).
\end{equation}
Conversely, from any pair $(\Delta, \mathbf m)$ as above, we construct a pair $(H,\lambda)$ by
the equation on the right hand side.
This specifies $H = (h_k)_k$ and $\lambda = (\lambda_k)_k$ uniquely.
We next proceed with comparing the defining properties for
Horn pairs with those for discriminantal triples.
\begin{claim}
If $(H,\lambda)$ is friendly and if the $r$ columns of an integer matrix $A$ with $AH=0$
span $\mathbb Z^r$, then $\Delta$ is $A$-homogeneous and vanishes on the dual toric variety $Y_A^*$. Conversely, if $\Delta$ is $A$-homogeneous and vanishes on $Y_A^*$ for some integer matrix $A$, then $(H,\lambda)$ is friendly.
\end{claim}
\begin{subproof}[Proof of Claim]
Let $(H,\lambda)$ be friendly and $A$ a matrix as above.
The Laurent polynomial $\,q := \Delta /{\mathbf m}\,$ from (\ref{eq:mDelta})
is a rational function on $\mathbb{P}^{m-1}$ that vanishes on the dual toric variety $Y_A^*$. To see this,
consider the exponentiation map $\varphi_2\colon \mathbb P^{m-1}\to \mathbb R^{n+1}$
that is defined by $\varphi_2(x)=\lambda * x^H,$ where $*$ is the entrywise product.
Let $f = 1 - (p_0 + \cdots + p_n)$. We have $q=f\circ \varphi_2$.
By \cite[Theorems~1 and~2]{huh14}, the function $\varphi_2$ maps an open
dense subset of $Y_A^*$ dominantly to the closure $\mathcal M$ of the image of
$\varphi_{(H,\lambda)}$. Since $f = 0$ on $\mathcal M$, we have
$f\circ \varphi_2 = 0$ on an open dense subset of $Y_A^*$,
hence $q = 0$ on $Y_A^*$, so $\Delta = 0$ there as well.
Conversely, let $\Delta$ be $A$-homogeneous and vanish
on $Y_A^*$ for some $A$. We claim that $q(x)$ is zero for all $x=Hu$ in the image of the
linear map $H$. We may assume $\mathbf m (x) \neq 0$. We must prove that $x$ is in the dual toric variety $Y_A^*$, since $\Delta$ vanishes on it. So, let $x_i = \sum_{j=0}^n h_{ij}u_j$ for $i=0,\dotsc n+1$. We
claim that $t=(1,\dotsc,1)$ is a singular point of the hypersurface
\[
\gamma_A^{-1}(H_x\cap Y_A) \,\,=\,\,
\left\{t \in \mathbb C^r \mid \sum_{i=1}^m x_i t^{a_i} = 0\right\}.
\]
First, the point $t$ lies on that hypersurface
since the columns of $H$ sum to zero:
\begin{align*}
\sum_{i=1}^m x_i \,=\, \sum_{i=1}^m \sum_{j=0}^n h_{ij} u_j
\,= \,\sum_{j=0}^m u_j \sum_{i=1}^m h_{ij} \,=\, 0.
\end{align*}
For $s=1,\dotsc,r$ we have $ \frac{\partial}{\partial t_s} t^{a_i} = a_{si} t^{a_i - e_s}$,
with $e_s$ the $s$-th canonical basis vector of $\mathbb Z^r$, and
\begin{align*}
\frac{\partial}{\partial t_s}\sum_{i=1}^n x_i t^{a_i}
\,\,= \,\,\sum_{i=1}^m \sum_{j=0}^n h_{ij} u_j a_{si} t^{a_i-e_s}
\,\,= \,\,\sum_{j=0}^n u_j \sum_{i=1}^m a_{si}h_{ij}t^{a_i-e_s}.
\end{align*}
This is zero at $t=(1,\dotsc,1)$ because $AH = 0$.
\end{subproof}
Next comes the point where we incorporate positivity. If a friendly pair $(H,\lambda)$ with $H = \tilde H$
is a Horn pair then the sign vector $\sigma$ satisfies
\eqref{eq:positivity}.
But conversely, if $(A,\Delta,\mathbf m)$ is a discriminantal triple
then \eqref{eq:positivity} holds, and Corollary \ref{cor:sigmaexists}
tells us that $(H,\lambda)$ is a Horn pair.
To complete the proof, let $\phi_{(\Delta,\mathbf m)}(x) \coloneqq \lambda * x^H$. We have
$\,\overline{\varphi_{(H,\lambda)}(\mathbb R^{n+1}_{>0})} =
\overline{\phi_{(\Delta,\mathbf m)}(Y_{A,\sigma}^*)}\,$ by \cite[Theorems~1 and~2]{huh14}, and
we have $\,\varphi_{(H,\lambda)} = \phi_{(\Delta,\mathbf m)} \circ H$ by construction.
\end{proof}
We proved that every model with rational MLE {\bf arises from} a toric variety $Y_A$.
In some cases, the model {\bf is itself} a toric variety $Y_C$.
It is crucial to distinguish the two matrices $A$ and $C$.
The two toric structures are very different.
For instance, every undirected graphical model is toric \cite[Proposition 3.3.3]{DSS}.
The toric varieties $ Y_C$ among staged tree models $\mathcal{M}_\mathcal{T}$ were classified in \cite{DG}.
The $4$-chain model $\mathcal{M}_\mathcal{T} = Y_C$ {\bf is itself} a toric variety of
dimension~$7$ in $\mathbb{P}^{15}$. But it {\bf arises from}
a toric variety $Y_A$ of dimension $12$ in $\mathbb{P}^{20}$, as seen in Example \ref{ex:4chainhorn}.
Toric models with rational MLE play an important role
in {\em geometric modeling} \cite{clarke2018moment, garciaSottile}.
Given an integer matrix $C\in \mathbb{Z}^{r\times (n+1) }$ and a vector of weights
$w \in \mathbb{R}^{n+1}_{>0}$, one considers the
\emph{scaled projective toric variety} $Y_{C,w}$ in $\mathbb{R} \mathbb{P}^{n}$.
This is defined as the closure of the
image of
\begin{equation}
\label{eq:monomapweights}
\gamma_{C,w} \,\,:\, (\mathbb{R}^*)^{r} \to \mathbb \mathbb{R} \mathbb{P}^{n}\,,\,\,\,
(t_1,\ldots,t_r) \,\mapsto \,
\biggl( \,w_1\prod_{i=1}^r t_i^{c_{i1}},
\,w_2\prod_{i=1}^r t_i^{c_{i2}},\, \ldots \, ,
\,w_m\prod_{i=1}^r t_i^{c_{im}} \biggr).
\end{equation}
The set of positive points in this toric variety is a discrete statistical
model $\mathcal{M}_{C,w}$ in $\Delta_{n}$.
There is a natural homeomorphism
from the toric model $\mathcal{M}_{C,w}$
onto the polytope of $C$. This is known among geometers
as the {\em moment map}, and as {\em Birch's Theorem}
in Algebraic Statistics. In geometric modeling the pair $(C,w)$ is used to define
\emph{toric blending functions}~\cite{toricPatches}.
It is highly desirable for the toric blending functions to
have \emph{rational linear precision} \cite{clarke2018moment, toricPatches}.
The property is rare and it depends in a subtle way on $(C,w)$.
Garcia-Puente and Sottile \cite{garciaSottile} established the connection
to algebraic statistics. They showed that
rational linear precision holds for $(C,w)$ if and only if the
statistical model $\mathcal{M}_{C,w}$ has rational MLE.
\begin{example}\label{ex:multinomial2} The most classical
blending functions with rational linear precision live on
the triangle $\{x \in \mathbb{R}^3_{>0}: x_1{+}x_2{+}x_3 = 1\}$.
They are the {\em Bernstein basis polynomials} of degree~$m$:
\begin{equation}
\label{eq:bernstein}
\frac{m!}{i!j!(m-i-j)!}x_1^i x_2^j x_3^{m-i-j} \,\, \,\text{ for}\;\;\; i,j \geq 0, \,i+j \leq m.
\end{equation}
Here $C$ is the $3 \times \binom{m+1}{2}$ matrix
whose columns are the vectors $(i,j,m-i-j)$. The weights are $w_{(i,j)}=\frac{m!}{i!j!(m-i-j)!}$.
The associated toric model $\mathcal{M}_{C,w}$ is the multinomial family,
where (\ref{eq:bernstein}) is the probability of observing $i$ times $1$, $j$ times $2$ and $m-i-j$ times
$3$ in $m$ trials.
This model is seen in Example \ref{ex:multinomial} and it has rational MLE.
Again, notice the distinction between the two toric varieties.
Here, $Y_A$ is a point in $\mathbb{P}^m$, whereas
$ Y_C$ is a surface in $\mathbb{P}^{\binom{m}{2}-1}$.
\end{example}
Clarke and Cox \cite{clarke2018moment} raise the problem of characterizing
all pairs $(C,w)$ with rational linear precision. This was solved
by Duarte and G\"orgen \cite{DG} for pairs arising from staged trees.
While the problem remains open in general, our theory
in this paper offers new tools.
We may ask for a characterization of
discriminantal triples whose models are toric.
\section{Constructing Models with Rational MLE}
Part (3) in Theorem~\ref{thm:main} allows us to construct models with rational MLE starting from
a matrix $A$ that defines a projective toric variety $Y_A$. In most cases, the
dual variety $Y_A^*$ is a hypersurface, and we can compute
its defining polynomial $\Delta_A$, the \emph{discriminant} \cite{gkz}.
The polynomial $\Delta$ in a discriminantal triple can be any homogeneous multiple of $\Delta_A$,
but we just take $\Delta = \Delta_A$ in Algorithm~\ref{algo:toric-to-models}.
For all terms $\mathbf m$ in $\Delta_A$, we check whether
$(A,\Delta_A,\mathbf m)$ is a discriminantal triple and, if so, we identify $\sigma$.
We implemented this algorithm in {\tt Macaulay2}.
Lines 1 and 18 of Algorithm~\ref{algo:toric-to-models} are computations that
rely on Gr\"obner bases. The
execution of Line 18 can be very slow. It may be
omitted if one is satisfied with obtaining the parametric description and MLE $\Phi^{(\ell)}$ of the model $\mathcal M_{\ell}$. For the check in Line 17, one does not need to compute $\Phi_i(v)$ numerically. Instead, one can just examine the signs and parities of the entries of~$H$.
\begin{algorithm}[t]\label{algo:toric-to-models}
\caption{From toric varieties to statistical models}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{An integer matrix $A$ of size $r\times m$ with $(1,\dotsc, 1)$ in its row span}
\Output{An integer $n$ and a collection of statistical models
$\mathcal M^{(\ell)} = (\Phi^{(\ell)}, I^{(\ell)})$, \\ where
$\Phi^{(\ell)}\colon \mathbb R^{n+1}\to\mathbb R^{n+1}$ is a rational MLE for $\mathcal M^{(\ell)}$,
and \\
$I^{(\ell)}\subseteq \mathbb R[p_0,\dotsc, p_n]$ is the defining prime ideal of $\mathcal M^{(\ell)}$.
}
\nllabel{line:discriminant}Compute the $A$-discriminant $\Delta_A\in \mathbb{Z}[x_1,\ldots, x_m]$\;
$n\leftarrow \#\mathrm{terms}(\Delta_A)-2$\;
$\mathrm{models} \leftarrow \{\}$\;
\For{$0\leq \ell\leq n$}{
$\mathbf{m}\leftarrow \mathrm{terms}(\Delta_A)_\ell$\;
$q\leftarrow 1-{\Delta_A}/{\mathbf m}$\;
\For{$0\leq i \leq n$}{
$\lambda_i \leftarrow \mathrm{coefficients}(q)_i$\;
$h_i \leftarrow \mathrm{exponent\_vectors}(q)_i$\;
$\Phi_i^{(\ell)}\leftarrow \textbf{(}u\mapsto \lambda_i \prod_{j=1}^m (\sum_{k=0}^n h_{jk}u_k)^{h_{ji}}\textbf{)}$\;
}
$H \leftarrow (h_i)_i$\;
\If{ $\operatorname{sign}(\tilde H)$ is not well-defined}{ discard this instance and \textbf{continue loop}\;}
Choose any positive vector $v$ in $\mathbb{R}^{n+1}_{>0}$\;
\If{$\Phi_i^{(\ell)}(v)>0$ for $i=0,1,\ldots,n$}
{
Compute the ideal $I^{(\ell)}$ of the image of $\Phi^{(\ell)}$\;
$\mathrm{models} \leftarrow \mathrm{models}\cup \{(\Phi^{(\ell)}, I^{(\ell)})\}$\;
}
}
\textbf{return} $\mathrm{models}$\;
\end{algorithm}
\begin{example}[$r=2,m=4$]
For distinct positive integers $\alpha, \beta,\gamma$ with $\gcd(\alpha,\beta,\gamma) = 1$, let
\[
A_{\alpha,\beta,\gamma} \,=\, \begin{pmatrix}
1& 1& 1& 1\\
0& \alpha& \beta& \gamma\end{pmatrix}.
\]
We ran Algorithm~\ref{algo:toric-to-models} for all $613$ such matrices with $0<\alpha<\beta<\gamma\leq 17$. Line 1 computes the discriminant $\Delta_A$ of the univariate polynomial
$ f(t)= x_1 + x_2 t^\alpha + x_3 t^\beta + x_4 t^\gamma $.
The number $n+2$ of terms of these discriminants equals
$7927/613 = 12.93 $ on average. Thus a total of $7927$ candidate triples
$(A,\Delta_A,{\bf m})$ were tested in Lines 12 to 21.
Precisely 123 of these were found to be discriminantal triples.
This is a fraction of 1.55 \%.
In other words, only 1.55 \% of the resulting complex varieties permitted by
\cite{huh14} are actually statistical models.
Here is a typical model that was discovered. Take $\alpha=1,\beta=4,\gamma = 7$.
The discriminant
$$ \begin{matrix} \Delta_A &=&
729 x_2^4 x_3^6-6912 x_1^3 x_3^7-8748 x_2^5 x_3^4 x_4+84672 x_1^3 x_2 x_3^5 x_4+34992 x_2^6 x_3^2 x_4^2 \\ & & -351918 x_1^3 x_2^2 x_3^3 x_4^2
-46656 x_2^7 x_4^3+518616 x_1^3 x_2^3 x_3 x_4^3 \,\,\underline{- \,823543 x_1^6 x_4^4}
\end{matrix}
$$
has $9$ terms, so $n=7$. The special term ${\bf m}$ is underlined. The
associated model is a curve of degree ten in $\Delta_7$. Its prime
ideal $I^{(\ell)}$ is generated by $18$ quadrics. Among them are
$15$ binomials that define a toric surface of degree six:
$49 p_1 p_2{-}48 p_0 p_3,3 p_0 p_4{-}p_2^2, \ldots,
361 p_3 p_7{-}128 p_5^2$.
Inside that surface, our curve is cut out by three other quadrics, like
$ \,26068 p_2^2 + 73728 p_0 p_5 $ $ +703836 p_0 p_6+234612 p_2 p_6+
78204 p_4 p_6+612864 p_0 p_7+212268 p_2 p_7+78204 p_4 p_7-8379 p_7^2
$.
\end{example}
\begin{example}[$r=3,m=6$]
For any positive integers $\alpha, \beta, \gamma, \varepsilon$, we consider the matrix
$$ A \,\,=\,\, \begin{small} \begin{pmatrix}
0 & \alpha & \beta & 0 & \gamma & \varepsilon \\
0 & 0 & 0 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1
\end{pmatrix}. \end{small} $$
The discriminant $\Delta_A$ is the {\em resultant} of two trinomials
$\,x_1 + x_2 t^\alpha + x_3 t^\beta \,$ and $\, x_4 + x_5 t^\gamma + x_6 t^\varepsilon\,$
in one variable $t$.
We ran Algorithm~\ref{algo:toric-to-models} for all
138 such matrices with
$ 0<\alpha<\beta\leq 17,\, 0<\gamma<\varepsilon\leq 17, \,
\gcd(\alpha,\beta) = \gcd(\gamma,\varepsilon) = 1$.
The number $n+2$ of terms of these discriminants equals
2665/138 = 19.31 on average. Thus a total of 2665 candidate triples
$(A,\Delta_A,{\bf m})$ were tested in Line 13.
Precisely 93 of these are discriminantal triples.
This is a fraction of 3.49~\%.
\end{example}
We now shift gears by looking at polynomials
$\Delta$ that are multiples of the $A$-discriminant.
\begin{example}[$r=1,m=4$]\label{ex:mult}
We saw in Examples~\ref{ex:smalltree} and~\ref{ex:multinomial} that interesting models can arise
from the matrix $A = (1\ 1\ \cdots \ 1)$ whose toric variety is just one point.
Any homogeneous multiple $\Delta$ of the linear form $\Delta_A = x_1 + x_2 + \cdots+ x_m$
can be used as input in Line 1 of Algorithm~\ref{algo:toric-to-models}. Here, taking $\Delta = \Delta_A$ results in the model given by the full simplex $\Delta_{m-2}$.
Let $m=4$ and abbreviate $x^a=x_{1}^{a_1} x_{2}^{a_2} x_{3}^{a_3} x_{4}^{a_4}$
and $|a|=a_1{+}a_2{+}a_3{+}a_4$ for $a\in \mathbb{N}^4$.
We conducted experiments with two families of multiples.
The first uses binomial multipliers:
$$ \qquad \Delta \, =\,(x^{a}+x^{b})\Delta_A\,\,\hbox{or}\,\, (x^{a}-x^{b})\Delta_A, \quad
\hbox{where $|a|=|b| \in \{1,2,\dots,8\}$ and $\gcd(x^a,x^b)=1$.} $$
This gives $1028$ polynomials $\Delta$. The numbers of polynomials of
degree $2,3,4,6,7,8,9,10$ is $6,\,21,\,46,\,81,\,126,\,181,\,246,\,321$.
For the second family we use the trinomial multiples
$$\,\,\, \Delta = (x^{a}+x^{b}+x^c)\Delta_A
\,\,\hbox{or}\,\, (x^{a}+x^{b}-x^c)\Delta_A, \,
\hbox{where $|a|{=}|b| {=} |c| \! \in\! \{1,2,3\}$ and $\gcd(x^a\!,x^b\!,x^c)=1$.} $$
Each list contains $4$ quadrics, $104$ cubics and $684$ quartics.
We report our findings in a table:
\smallskip
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Family & Pairs $(\Delta,\mathbf m)$ & Horn pairs & Percentage \\
\hline
$(x^{a}-x^{b})\Delta_A$ & 8212 & 12 & 0.15\%\\
$(x^{a}+x^{b})\Delta_A$ & 8218 & 0 & 0\% \\
$(x^{a}+x^{b}-x^c)\Delta_A$ & 8678 & 8 & 0.01\% \\
$(x^{a}+x^{b}+x^c)\Delta_A$ & 8968 & 0& 0\%\\
\hline
\end{tabular}
\end{center}
\smallskip
All $12$ Horn pairs in the first family represent the same model, up to a permutation of coordinates. All are coming from the six quadrics of the family. The model is the surface in $\Delta_4$
defined by the $2 \times 2$ minors of the matrix
$ \begin{pmatrix} p_0 & p_1 & p_2 \,\,
\\ p_0{+}p_1{+}p_2 & p_3 & p_4 \,\, \end{pmatrix} $.
This is a staged tree model similar to Example 2, but now with three choices
at each blue node instead of two.
\end{example}
In our construction of models with rational MLE, we start with families where $r$ and $m$ are fixed.
However, as the entries of the matrix $A$ go up, the number $n+1$ of states increases.
This suggests the possibility of listing all models for fixed small values of $n$. Is this list finite?
\begin{problem} Suppose that $n$ is fixed. Are there only finitely many models with rational MLE
in the simplex $\Delta_n$? Can we find absolute bounds, depending only on $n$, for
the dimension, degree and number of ideal generators of the associated varieties in $\mathbb{P}^n$?
\end{problem}
Algorithm~\ref{algo:toric-to-models} is a tool for studying these questions experimentally.
At present, however, we do not have any clear answers, even for $n=3$, where the models
are curves in a triangle.
\begin{small}
|
1,108,101,565,510 | arxiv | \section{Introduction}\label{sec1}
Optical properties of low-dimensional semiconductor nanostructures originate from excitons (Coulomb-bound electron-hole pairs) and exciton complexes such as biexcitons (coupled states of two excitons) and trions (charged excitons). These have pronounced binding energies in nanostructures due to the quantum confinement effect.\cite{HaugKoch,Cardona,Apphysrev} The advantage of optoelectronic device applications with low-dimensional semiconductor nanostructures lies in the ability to tune their properties in a controllable way. Optical properties of semiconducting carbon nanotubes (CNs), in particular, are largely determined by excitons,\cite{Dresselhaus07,Louie09} and can be tuned by electrostatic doping,\cite{Steiner09,Spataru10,Mueller10,Kato} or by means of the quantum confined Stark effect.\cite{Bondarev09PRB,Bondarev12PRB,Bondarev14PRBbec,Pedersen15} Carbon nanotubes are graphene sheets rolled-up into cylinders of one to a few nanometers in diameter and up to hundreds of microns in length, which can be both metals and semiconductors depending on their diameters and chirality.\cite{Saito,Dresselhaus} Over the past decade, optical nanomaterials research has uncovered intriguing optical attributes of their physical properties, lending themselves to a variety of new optoelectronic device applications.\cite{Papanikolas,Imamoglu,Avouris08,McEuen09,Hertel10,Bondarev10jctn,Strano11,BondarevNova11,Kono12,Baughman13,ChemPhysSI,BondarevOE15}
Formation of biexcitons and trions, though not detectable in bulk materials at room temperature, play a significant role in quantum confined systems of reduced dimensionality such as quantum wells,\cite{Birkedal,Singh,Thilagam,Lozovik,Bracker} nanowires,\cite{Forchel,Crottini,Sidor,Gonzales,Schuetz} nanotubes,\cite{Pedersen03,Pedersen05,Kammerlander07,Ronnow10,Ronnow11,Matsunaga11,Santos11,Bondarev11PRB,Bondarev14PRB,Watanabe12,Ronnow12,Colombier12,Yuma13} and quantum dots.\cite{Woggon,JonFbiexc,JonFtrion} Biexciton and trion excitations open up routes for controllable nonlinear optics and spinoptronics applications, respectively. The trion, in particular, has both net charge and spin, and therefore can be controlled by electrical gates while being used for optical spin manipulation, or to investigate correlated carrier dynamics in low-dimensional materials.
For conventional semiconductor quantum wells, wires, and dots, the binding energies of negatively or positively charged trions are known to be typically lower than those of biexcitons in the same nanostructure, although the specific trion to biexciton binding energy ratios are strongly sample fabrication dependent.\cite{Lozovik,Forchel,Sidor,Woggon} First experimental evidence for the trion formation in carbon nanotubes was reported by Matsunaga et al.\cite{Matsunaga11} and by Santos et al.\cite{Santos11} on $p$-doped (7,5) and undoped (6,5) CNs, respectively. Theoretically, R{\o}nnow et al.\cite{Ronnow10} have predicted that the lowest energy trion states in all semiconducting CNs with diameters of the order of or less than 1~nm should be stable at room temperature. They have later developed the fractional dimension approach to simulate binding energies of trions and biexcitons in quasi-1D/2D semiconductors, including nanotubes as a particular case.\cite{Ronnow11,Ronnow12} Binding energies of $63$~meV and $92$~meV are reported for the lowest energy trions\cite{Ronnow11} and biexcitons,\cite{Ronnow12} respectively, in the (7,5) nanotube. However, the recent nonlinear optics experiments were able to resolve both trions and biexcitons in the same CN sample,\cite{Colombier12,Yuma13} to report on the opposite tendency where the binding energy of the trion \emph{exceeds} that of the biexciton rather significantly in small diameter ($\lesssim\!1$~nm) CNs. Figure~\ref{fig0} shows typical experimental data for conventional low-dimension semiconductors (left panel) and small diameter semicondicting CNs (right panel). In the left panel, the biexciton resonance is seen to appear at lower photon energy than the trion one, in contrast with the right panel where the biexciton resonance manifests itself at greater photon energy than the trion resonance does. This clearly indicates greater trion binding energies than those of biexcitons in small diameter semiconducting CNs as opposed to conventional low-dimension semiconductors.
\begin{figure}[t]
\epsfxsize=17.5cm\centering{\epsfbox{fig0.eps}}\caption{Left panel: An example of the time-integrated photoluminescence spectra of a single CdSe quantum dot at multiple excitation intensities ($I_0\!=\!250~\mu$W). The spectra are normalized to the peak at 2.287~eV and are displaced vertically for clarity. Exciton (X) and trion (T) emission is present at the lowest excitation intensities. The biexciton (XX) emission peak shows up as the excitation intensity increases at the photon energy \emph{lower} than that of the trion peak, indicating the \emph{greater} biexciton binding energy than that of the trion. Reprinted with permission from Ref.[51]. Right panel: (a) Differential transmission spectra obtained for different probe laser time-delays after the excitation of the second bright exciton transition ($S_{22}$ excitation) of the semiconducting (6,5) CN by the pump laser with fluence $6.3\times10^{13}$ photons/pulse/cm$^2$. Induced transmission (IT) at 1.26~eV is due to the fast second-to-first bright exciton ($S_{22}\!\rightarrow\!S_{11}$) relaxation. (b)~Magnification of the spectral region below the energy of the first bright exciton excitation energy ($S_{11}$) in the (6,5) nanotube. Induced absorption (IA) at 1.08~eV that appears at longer time-delays is attributed to trion ($X^*$) formation. The short time-delay IA (black line) is due to biexciton ($XX$) formation at the photon energy 1.13~eV, which is \emph{greater} than the trion IA energy, indicating the \emph{lower} biexciton binding energy than that of the trion. Reprinted with permission from Ref.[50].}\label{fig0}
\end{figure}
More specifically, Colombier et al.\cite{Colombier12} reported on the observation of the binding energies $150$~meV and $106$~meV for the trion and biexciton, respectively, in the (9,7) CN. Yuma et al.\cite{Yuma13} reported even greater binding energies of $190$~meV for the trion versus $130$~meV for the biexciton in the smaller diameter (6,5) CN. (Their spectra are reproduced in Fig.~\ref{fig0}, right panel.) In both cases, the trion-to-biexciton binding energy ratio is greater than unity, decreasing as the CN diameter increases [1.46 for the 0.75~nm diameter (6,5) CN versus 1.42 for the 1.09~nm diameter (9,7) CN]. Trion binding energies greater than those of biexcitons are theoretically reported by Watanabe and Asano,\cite{Watanabe12} due to the energy band nonparabolicity and the Coulomb screening effect that reduces the biexciton binding energy more than that of the trion. Watanabe and Asano have extended the first order ($\mathbf{k}\cdot\mathbf{p}$)-perturbation series expansion model originally developed by Ando for excitons (see Ref.\cite{Ando2005} for review) to the case of electron-hole complexes such as trions and biexcitons. Figure~\ref{fig00} compares the differences between the trion and biexciton binding energies delivered by "phenomenological" and "unscreened" models termed as such to refer to the cases where the energy band nonparabolicity, electron-hole complex form-factors, self-energies and the screening effect are all neglected, and where all of them but screening are taken into account, respectively, with the difference given by the "screened" model. The latter is the Watanabe--Asano model which includes \emph{all} of the factors mentioned within the first order ($\mathbf{k}\cdot\mathbf{p}$)-perturbation theory. One can see that the "screened" model does predict greater trion binding energies than those of biexcitons as opposed to the phenomenological and unscreened models. However, the most the trion binding energy can exceed that of the biexciton within this model is $0.012\,(2\pi\gamma/L)$ equal to $21$ and $14$~meV for the (6,5) and (9,7) CNs, respectively, which is obviously not enough to explain the experimental observations.
\begin{figure}[t]
\epsfxsize=10.0cm\centering{\epsfbox{fig00.eps}}\caption{Differences between the trion and biexciton binding energies (in units of $2\pi\gamma/L$, where $L$ is the CN circumference and $\gamma\!=\!6.46$~eV$\,$\AA) as functions of the Coulomb coupling parameter $v_c$, which are obtained within the phenomenological, unscreened and screened models. The parameter $v_c$ is approximately equal to the ratio of the Coulomb energy to the CN band gap (typically $\lesssim\!0.25$). Reprinted with permission from Ref.[47].}\label{fig00}
\end{figure}
This article reviews the capabilities of the configuration space (Landau-Herring) method for the binding energy calculations of the lowest energy exciton complexes in quasi-1D/2D semiconductors. The approach was originally pioneered by Landau,\cite{LandauQM} Gor'kov and Pitaevski,\cite{Pitaevski63} Holstein and Herring\cite{Herring} in the studies of molecular binding and magnetism. The method was recently shown to be especially advantageous in the case of quasi-1D semiconductors,\cite{Bondarev11PRB,Bondarev14PRB} allowing for easily tractable, complete analytical solutions to reveal universal asymptotic relations between the binding energy of the exciton complex of interest and the binding energy of the exciton in the same nanostructure. The Landau-Herring method of the complex bound state binding energy calculation is different from commonly used quantum mechanical approaches reviewed above. These either use advanced simulation techniques to solve the coordinate-space Schr\"{o}dinger equation numerically,\cite{Pedersen05,Kammerlander07,Ronnow10,Ronnow11,Ronnow12} or convert it into the reciprocal (momentum) space to follow up with the ($\mathbf{k}\cdot\mathbf{p}$)-perturbation series expansion calculations.\cite{Watanabe12,Ando2005} Obviously, this latter one, in particular, requires for perturbations to be small. If they are not, then the method brings up an underestimated binding energy value, especially for molecular complexes such as biexciton and trion where the kinematics of complex formation depends largely on the asymptotic behavior of the wave functions of the constituents. This is likely the cause for Watanabe--Asano theory of excitonic complexes\cite{Watanabe12} to significantly underestimate the measurements by Colombier et al.\cite{Colombier12} and Yuma et al.\cite{Yuma13} on semiconducting CNs.
\begin{figure}[t]
\epsfxsize=11.5cm\centering{\epsfbox{fig000.eps}}\caption{Comparison of the biexciton binding energy given by the fractional dimension model of Ref.[48] to other coordinate-space formulated models ("phenomenological" in terms of Fig.~\ref{fig00}) and the configuration space model. Dimensionless units are used (defined in Sec.~2). Inset shows the relation between the effective CN radius and the dimension $D$ calculated according to Ref.[41]. Reference number correspondence to this article is as follows: "Present work"$\rightarrow\,$Ref.[48], "Refs.17,19"$\rightarrow\,$Ref.[40], "Ref.16"$\rightarrow\,$Ref.[39] (all of these are the phenomenological models in terms of Fig.~\ref{fig00}), "Ref.21"$\,\rightarrow\,$Ref.[45] (the configuration space model used in this article). Reprinted with permission from Ref.[48].}\label{fig000}
\end{figure}
The Landau-Herring configuration space approach does not have this shortcoming. It works in the \emph{configuration space} of the \emph{two} relative electron-hole motion coordinates of the \emph{two} non-interacting quasi-1D excitons that are modeled by the effective one-dimensional cusp-type Coulomb potential as proposed by Ogawa and Takagahara for 1D semiconductors.\cite{Ogawa91} Since the configuration space is different from the ordinary coordinate (or its reciprocal momentum) space, the approach does not belong to any of the models summarized in Fig.~\ref{fig00}. In this approach, the biexciton or trion bound state forms due to the exchange under-barrier tunneling between the equivalent configurations of the electron-hole system in the configuration space. The strength of the binding is controlled by the exchange tunneling rate. The corresponding binding energy is given by the tunnel exchange integral determined through an appropriate variational procedure. As any variational approach, the method gives an upper bound for the \emph{ground} state binding energy of the exciton complex of interest. As an example, Fig.~\ref{fig000} compares the biexciton binding energies calculated within several different models, including those coordinate-space formulated that are referred to as phenomenological in Fig.~\ref{fig00}, as well as the configuration space model. It is quite remarkable that with obvious overall correspondence to the other methods as seen in Fig.~\ref{fig000}, the Landau-Herring configuration space approach is the only to have been able consistently explain the experimental observations discussed above and shown in Fig.~\ref{fig0}, both for conventional low-dimension semiconductors and for semiconducting CNs. Whether the trion or biexciton is more stable (has greater binding energy) in a particular quasi-1D system turns out to depend on the reduced electron-hole mass and on the characteristic transverse size of the system.\cite{Bondarev14PRB} Trions are generally more stable than biexcitons in strongly confined quasi-1D structures with small reduced electron-hole masses, while biexcitons are more stable than trions in less confined quasi-1D structures with large reduced electron-hole masses. As such, a crossover behavior is predicted,\cite{Bondarev14PRB} whereby trions get less stable than biexcitons as the transverse size of the quasi-1D nanostructure increases --- quite a general effect which could likely be observed through comparative measurements on semiconducting CNs of increasing diameter. The method captures the essential kinematics of exciton complex formation, thus helping understand in simple terms the general physical principles that underlie experimental observations on biexcitons and trions in a variety of quasi-1D semiconductor nano\-structures. For semiconducting CNs with diameters $\lesssim1\!$~nm, the model predicts the trion binding energy greater than that of the biexciton by a factor $\sim\!1.4$ that decreases with the CN diameter increase, in reasonable agreement with the measurements by Colombier et al.\cite{Colombier12} and Yuma et al.\cite{Yuma13}
The article is structured as follows. Section 2 formulates the general Hamiltonian for the biexciton complex of two electrons and two holes in quasi-1D semiconductor. Carbon nanotubes of varying diameter are used as a model example for definiteness. The theory and conclusions are valid for any quasi-1D semiconductor system in general. The exchange integral and the binding energy of the biexciton complex are derived and analyzed. Section 3 further develops the theory to include the trion case. In Section 4, the trion binding energy derived is compared to the biexciton binding energy for semiconducting quasi-1D nanostructures of varying transverse size and reduced exciton effective mass. Section 5 generalizes the method to include trion and biexciton complexes formed by indirect excitons in layered quasi-2D semiconductor structures such as coupled quantum wells (CQWs) and bilayer self-assembled transition metal dichalchogenide heterostructures. Section 6 summarizes and concludes the article.
\section{Biexciton in quasi-1D}\label{sec2}
The problem is initially formulated for two interacting ground-state 1D excitons in a semiconducting CN. The CN is taken as a model for definiteness. The theory and conclusions are valid for any quasi-1D semiconductor system in general. The excitons are modeled by effective one-dimensional cusp-type Coulomb potentials, shown in Fig.~\ref{fig1}~(a), as proposed by Ogawa and Takagahara for 1D semiconductors.\cite{Ogawa91} The intra-exciton motion can be legitimately treated as being much faster than the inter-exciton center-of-mass relative motion since the exciton itself is normally more stable than any of its compound complexes. Therefore, the adiabatic approximation can be employed to simplify the formulation of the problem. With this in mind, using the cylindrical coordinate system [\emph{z}-axis along the CN as in Fig.~\ref{fig1}~(a)] and separating out circumferential and longitudinal degrees of freedom for each of the excitons by transforming their longitudinal motion into their respective center-of-mass coordinates,\cite{Bondarev09PRB,Ogawa91} one arrives at the Hamiltonian of the form\cite{Bondarev14PRB}
\begin{eqnarray}
\hat{H}(z_1,z_2,\Delta Z)=-\frac{\partial^2}{\partial\,\!z_{1}^2}-\frac{\partial^2}{\partial\,\!z_{2}^2}\hskip2cm\label{biexcham}\\
-\frac{1}{|z_{1}|\!+\!z_0}-\!\frac{1}{|z_{1}\!-\!\Delta Z|\!+\!z_0}-\!\frac{1}{|z_{2}|\!+\!z_0}-\!\frac{1}{|z_{2}\!+\!\Delta Z|\!+\!z_0}\hskip0.3cm\nonumber\\
-\frac{2}{|(\sigma z_1+z_2)/\lambda+\Delta Z|\!+\!z_0}-\frac{2}{|(z_1+\sigma z_2)/\lambda-\Delta Z|\!+\!z_0}\nonumber\\
+\frac{2}{|\sigma(z_1-z_2)/\lambda+\Delta Z|\!+\!z_0}+\frac{2}{|(z_1-z_2)/\lambda-\Delta Z|\!+\!z_0}\;.\nonumber
\end{eqnarray}
Here, $z_{1,2}\!=\!z_{e1,2}-z_{h1,2}$ are the relative electron-hole motion coordinates of the two 1D excitons separated by the center-of-mass-to-center-of-mass distance $\Delta Z\!=\!Z_2-Z_1$, $z_0$ is the cut-off parameter of the effective (cusp-type) longitudinal electron-hole Coulomb potential, $\sigma\!=\!m_e/m_h$, $\lambda\!=\!1+\sigma$ with $m_e$ ($m_h$) representing the electron (hole) effective mass. The "atomic units"\space are used,\cite{LandauQM,Pitaevski63,Herring} whereby distance and energy are measured in units of the exciton Bohr radius $a^\ast_B\!=\!0.529\,\mbox{\AA}\,\varepsilon/\mu$ and the Rydberg energy $Ry^\ast\!=\hbar^2/(2\mu\,m_0a_B^{\ast2})\!=\!13.6\,\mbox{eV}\,\mu/\varepsilon^2$, respectively, $\mu\!=\!m_e/(\lambda\,m_0)$ is the exciton reduced mass (in units of the free electron mass $m_0$) and $\varepsilon$ is the static dielectric constant of the electron-hole Coulomb potential.
\begin{figure}[t]
\epsfxsize=11.0cm\centering{\epsfbox{fig1.eps}}\caption{(a)~Schematic of the exchange coupling of two ground-state 1D excitons to form a~biexcitonic~state (arb.~units). Two collinear axes, $z_1$ and $z_2$, representing independent relative electron-hole motions in the 1st and 2nd exciton, have their origins shifted by $\Delta Z$, the inter-exciton center-of-mass separation. (b)~The coupling occurs in the configuration space of the two independent longitudinal relative electron-hole motion coordinates, $z_1$ and $z_2$, of each of the excitons, due to the tunneling of the system through the potential barriers formed by the two single-exciton cusp-type potentials [bottom, also in (a)], between equivalent states represented by the isolated two-exciton wave functions shown on the top.}\label{fig1}
\end{figure}
The first two lines in Eq.~(\ref{biexcham}) represent two non-interacting 1D excitons. Their individual potentials are symmetrized to account for the presence of the neighbor a distance~$\Delta Z$ away, as seen from the $z_1$- and $z_2$-coordinate systems treated independently [Fig.~\ref{fig1}~(a)]. The last two lines are the inter-exciton exchange Coulomb interactions --- electron-hole (line next to last) and hole-hole + electron-electron (last line), respectively. The binding energy $E_{X\!X}$ of the biexciton is given by the difference $E_g\!-2E_X$, where $E_g$ is the lowest eigenvalue of the Hamiltonian~(\ref{biexcham}) and $E_X\!=-Ry^\ast/\nu_0^2$ is the single-exciton binding energy with $\nu_0$ being the lowest-bound-state quantum number of the 1D exciton.\cite{Ogawa91} Negative $E_{X\!X}$ indicates that the biexciton is stable with respect to the dissociation into two isolated excitons. The strong transverse confinement in reduced dimensionality semiconductors is known to result in the mass reversal effect,\cite{HaugKoch,Cardona} whereby the bulk heavy hole state, that forming the \emph{lowest} excitation energy exciton, acquires a longitudinal mass comparable to the bulk \emph{light} hole mass ($\approx\!m_e$). Therefore, $m_h\!\approx\!m_e$ in our case of interest here, which is also true for graphitic systems such as CNs, in particular,\cite{Jorio05} and so $\sigma\!=\!1$ is assumed in Eq.~(\ref{biexcham}) in what follows with no substantial loss of generality.
The Hamiltonian (\ref{biexcham}) is effectively two dimensional in the configuration space of the two \emph{independent} relative motion coordinates, $z_1$ and $z_2$. Figure~\ref{fig1}~(b), bottom, shows schematically the potential energy surface of the two closely spaced non-interacting 1D excitons [second line of Eq.~(\ref{biexcham})] in the $(z_1,z_2)$ space. The surface has four symmetrical minima representing isolated two-exciton states shown in Fig.~\ref{fig1}~(b), top. These minima are separated by the potential barriers responsible for the tunnel exchange coupling between the two-exciton states in the configuration space. The coordinate transformation
\begin{equation}
x=\frac{z_1-z_2-\Delta Z}{\sqrt{2}}\,,\hskip1cm y=\frac{z_1+z_2}{\sqrt{2}}
\label{transformation}
\end{equation}
places the origin of the new coordinate system into the intersection of the two tunnel channels between the respective potential minima [Fig.~\ref{fig1}~(b)], whereby the exchange splitting formula of Refs.\cite{LandauQM,Pitaevski63,Herring} takes the form
\begin{equation}
E_{g,u}(\Delta Z)-2E_X=\mp J(\Delta Z).
\label{Egu}
\end{equation}
Here $E_{g,u}(\Delta Z)$ are the ground-state and excited-state energies, eigenvalues of the Hamiltonian~(\ref{biexcham}), of the two coupled excitons as functions of their center-of-mass-to-center-of-mass separation, and $J(\Delta Z)$ is the tunnel exchange coupling integral responsible for the bound state formation of two excitons. For biexciton, this takes the form
\begin{equation}
J_{X\!X}(\Delta Z)=\frac{2}{3!}\int_{\!-\Delta Z/\!\sqrt{2}}^{\Delta Z/\!\sqrt{2}}\!dy\left|\psi_{X\!X}(x,y)\frac{\partial\psi_{X\!X}(x,y)}{\partial x}\right|_{x=0},
\label{JXX}
\end{equation}
where $\psi_{X\!X}(x,y)$ is the solution to the Schr\"{o}dinger equation with the Hamiltonian~(\ref{biexcham}) transformed to the $(x,y)$ coordinates. The factor $2/3!$ comes from the fact that there are two equivalent tunnel channels in the biexciton problem, mixing three equivalent indistinguishable two-exciton states in the configuration space --- one state is given by the two minima on the $x$-axis and two more are represented by each of the minima on the $y$-axis [cf. Fig.~\ref{fig1}~(a) and Fig.~\ref{fig1}~(b)].
The function $\psi_{X\!X}(x,y)$ in Eq.~(\ref{JXX}) is sought in the form
\begin{equation}
\psi_{X\!X}(x,y)=\psi_0(x,y)\exp[-S_{X\!X}(x,y)]\,,
\label{psiXXxy}
\end{equation}
where
\begin{equation}
\psi_0(x,y)=\frac{1}{\nu_0}\exp\!\left[-\frac{1}{\nu_0}\left(|z_1(x,y,\Delta Z)|+|z_2(x,y,\Delta Z)|\right)\right]
\label{psi0xy}
\end{equation}
is the product of two single-exciton wave functions (ground state) representing the isolated two-exciton state centered at the minimum $z_1\!=\!z_2\!=\!0$ (or $x\!=\!-\Delta Z/\sqrt{2}$, $y\!=\!0$) of the configuration space potential [Fig.~\ref{fig1}~(b)]. This is the approximate solution to the Shr\"{o}dinger equation with the Hamiltonin given by the first two lines in Eq.~(\ref{biexcham}), where the cut-off parameter $z_0$ is neglected.\cite{Ogawa91} This approximation greatly simplifies problem solving, while still remaining adequate as only the long-distance tail of $\psi_0$ is important for the tunnel exchange coupling. The function $S_{X\!X}(x,y)$, on the other hand, is a slowly varying function to account for the major deviation of $\psi_{X\!X}$ from $\psi_0$ in its "tail area" due to the tunnel exchange coupling to another equivalent isolated two-exciton state centered at $z_1\!=\Delta Z$, $z_2\!=\!-\Delta Z$ (or $x\!=\!\Delta Z/\sqrt{2}$, $y\!=\!0$).
Substituting Eq.~(\ref{psiXXxy}) into the Schr\"{o}dinger equation with the Hamiltonian (\ref{biexcham}) pre-transformed to the $(x,y)$ coordinates, one obtains in the region of interest ($z_0$ dropped for the reason above)
\[
\frac{\partial S_{X\!X}}{\partial x}=\nu_0\left(\frac{1}{x+3\Delta Z/\sqrt{2}}-\frac{1}{x-\Delta Z/\sqrt{2}}+\frac{1}{y-\sqrt{2}\Delta Z}-\frac{1}{y+\sqrt{2}\Delta Z}\right),
\]
up to negligible terms of the order of the inter-exciton~van der Waals energy and up to the second order derivatives of~$S_{X\!X}$. This equation is to be solved with the boundary condition $S_{X\!X}(-\Delta Z/\sqrt{2},y)\!=\!0$ originating from the natural requirement $\psi_{_{X\!X}}(-\Delta Z/\sqrt{2},y)\!=\!\psi_0(-\Delta Z/\sqrt{2},y)$, to result in
\begin{equation}
S_{X\!X}(x,y)=\nu_0\!\left(\!\ln\!\left|\frac{x\!+\!3\Delta Z/\!\sqrt{2}}{x-\Delta Z/\!\sqrt{2}}\right|+\frac{2\sqrt{2}\Delta Z(x\!+\!\Delta Z/\!\sqrt{2})}{y^2-2\Delta Z^2}\!\right)\!.
\label{sXXxy}
\end{equation}
After plugging this into Eq.~(\ref{psiXXxy}) one can calculate the tunnel exchange coupling integral~(\ref{JXX}). Retaining only the leading term of the integral series expansion in powers of $\nu_0$ subject to $\Delta Z>1$, one obtains
\begin{equation}
J_{X\!X}(\Delta Z)=\frac{2}{3\nu_0^3}\left(\frac{e}{3}\right)^{2\nu_0}\!\Delta Z\,e^{-2\Delta Z/\nu_0}.
\label{JXXfin}
\end{equation}
The ground state energy $E_g(\Delta Z)$ of the two coupled 1D excitons in Eq.~(\ref{Egu}) is now seen to go through the negative minimum (biexcitonic state) as $\Delta Z$ increases. The minimum occurs at $\Delta Z_0=\nu_0/2$, whereby the biexciton binding energy takes the form
\[
E_{X\!X}=-J_{X\!X}(\nu_0/2)=-\frac{1}{9\nu_0^2}\left(\frac{e}{3}\right)^{2\nu_0-1}
\]
in atomic units. Expressing $\nu_0$ in terms of $E_X$, one obtains in absolute units the equation as follows
\begin{equation}
E_{X\!X}=-\frac{1}{9}\;|E_X|\left(\frac{e}{3}\right)^{2\sqrt{Ry^\ast/|E_X|}\,-\,1}\!\!\!.
\label{Exx}
\end{equation}
\begin{figure}[t]
\epsfxsize=11.0cm\centering{\epsfbox{fig2.eps}}\caption{Schematic of the two ground-state 1D excitons sharing the same hole to form a negative trion state (arb.~units). Two collinear axes, $z_1$ and $z_2$, representing independent relative electron-hole motions in the 1st and 2nd exciton, have their origins shifted by $\Delta Z$, the inter-exciton center-of-mass distance.}\label{fig2}
\end{figure}
\section{Trion in quasi-1D}\label{sec3}
The trion binding energy can be found in the same way using a modification of the Hamiltonian (\ref{biexcham}), in which two same-sign particles share the third particle of an opposite sign to form the two equivalent 1D excitons as Fig.~\ref{fig2} shows for the negative trion complex consisting of the hole shared by the two electrons. The Hamiltonian modified to reflect this fact has the first two lines exactly the same as in Eq.~(\ref{biexcham}), no line next to last, and one of the two terms in the last line --- either the first or the second one for the positive (with $z_{1,2}\!=\!z_{e}-z_{h1,2}$) and negative (with $z_{1,2}\!=\!z_{e1,2}-z_h$) trion, respectively. Obviously, due to the additional mass factor $\sigma$ (typically less than one for bulk semiconductors) in the hole-hole interaction term in the last line, the positive trion might be expected to have a greater binding energy in this model, in agreement with the results reported earlier.\cite{Sidor,Ronnow10} However, as was already mentioned in Sec.~\ref{sec2}, the mass reversal effect in \emph{strongly} confined reduced dimensionality semiconductors is to result in $\sigma\!=\!1$ in the trion Hamiltonian. The positive-negative trion binding energy difference disappears then. The negative trion case illustrated in Fig.~\ref{fig2}, is addressed below.
Just like in the case of the biexciton, the treatment of the trion problem starts with the coordinate transformation~(\ref{transformation}) to bring the trion Hamiltonian from the original (configuration space) coordinate system $(z_1,z_2)$ into the new coordinate system $(x,y)$ with the origin positioned as shown in Fig.~\ref{fig1}~(b). The tunnel exchange splitting integral in Eq.~(\ref{Egu}) now takes the form
\begin{equation}
J_{X^{\ast}}(\Delta Z)=\int_{\!-\Delta Z/\!\sqrt{2}}^{\Delta Z/\!\sqrt{2}}\!dy\left|\psi_{X^\ast}(x,y)\frac{\partial\psi_{X^\ast}(x,y)}{\partial x}\right|_{x=0},
\label{JXast}
\end{equation}
where $\psi_{X^\ast}(x,y)$ is the ground-state wave function of the Schr\"{o}dinger equation with the Hamiltonian (\ref{biexcham}) modified to the negative trion case, as discussed above, and then transformed to the $(x,y)$ coordinates. The tunnel exchange current integral $J_{X^{\ast}}(\Delta Z)$ is due to the electron position exchange relative to the hole (see Fig.~\ref{fig2}). This corresponds to the tunneling of the entire three particle system between the two equivalent indistinguishable configurations of the two excitons sharing the same hole in the configuration space $(z_1,z_2)$, given by the pair of minima at $z_1\!=\!z_2\!=\!0$ and $z_1\!=\!-z_2\!=\!\Delta Z$ in Fig.~\ref{fig1}~(b). Such a tunnel exchange interaction is responsible for the coupling of the three particle system to form a stable trion state.
Like in the case of the biexciton, one seeks the function $\psi_{X^{\ast}}(x,y)$ in the form
\begin{equation}
\psi_{X^{\ast}}(x,y)=\psi_0(x,y)\exp[-S_{X^{\ast}}(x,y)]\,,
\label{psixy}
\end{equation}
with $\psi_0(x,y)$ given by Eq.~(\ref{psi0xy}), where $S_{X^{\ast}}(x,y)$ is assumed to be a \emph{slowly} varying function to take into account the deviation of $\psi_{X^{\ast}}$ from $\psi_0$ in the "tail area" of $\psi_0$ due to the tunnel exchange coupling to another equivalent isolated two-exciton state centered at $z_1\!=\Delta Z$, $z_2\!=\!-\Delta Z$ (or $x\!=\!\Delta Z/\sqrt{2}$, $y\!=\!0$). Substituting Eq.~(\ref{psixy}) into the Schr\"{o}dinger equation with the negative trion Hamiltonian pre-transformed to the $(x,y)$ coordinates, one obtains in the region of interest
\[
\frac{\partial S_{X^{\ast}}}{\partial x}=\frac{\nu_0}{\Delta Z/\sqrt{2}-x}
\]
($|x|\!<\!\Delta Z/\sqrt{2}$, cut-off $z_0$ dropped) up to terms of the order of the second derivatives of~$S_{X^{\ast}}$. This is to be solved with the boundary condition $S_{X^{\ast}}(-\Delta Z/\sqrt{2},y)\!=\!0$ coming from the requirement $\psi_{X^{\ast}}(-\Delta Z/\sqrt{2},y)\!=\!\psi_0(-\Delta Z/\sqrt{2},y)$, to result in
\begin{equation}
S_{X^{\ast}}(x,y)=\nu_0\ln\frac{\sqrt{2}\Delta Z}{\Delta Z/\sqrt{2}-x}.
\label{sxy}
\end{equation}
After plugging Eqs.~(\ref{sxy}) and (\ref{psixy}) into Eq.~(\ref{JXast}), and retaining only the leading term of the integral series expansion in powers of $\nu_0$ subject to $\Delta Z>1$, one obtains
\begin{equation}
J_{X^{\ast}}(\Delta Z)=\frac{2}{2^{2\nu_0}\nu_0^3}\Delta Z\,e^{-2\Delta Z/\nu_0}.
\label{JXastfin}
\end{equation}
Inserting this into the right-hand side of Eq.~(\ref{Egu}), one sees that the ground state energy $E_g$ of the three particle system goes through the negative minimum (the trion state) as $\Delta Z$ increases. The minimum occurs at $\Delta Z_0=\nu_0/2$, whereby the trion binding energy in atomic units takes the form
\[
E_{X^{\!\ast}}=-J_{X^{\ast}}(\nu_0/2)=-\frac{1}{e\,2^{2\nu_0}\nu_0^2}.
\]
In absolute units, expressing $\nu_0$ in terms of $E_X$, one obtains
\begin{equation}
E_{X^{\!\ast}}=-\frac{|E_X|}{e\,2^{2\sqrt{Ry^\ast/|E_X|}}}.
\label{Exstar}
\end{equation}
\begin{figure}[t]
\epsfxsize=12.0cm\centering{\epsfbox{fig3.eps}}\caption{Trion binding energy, biexciton binding energy and their ratio given by Eqs.~(\ref{Exstar}), (\ref{Exx}) and (\ref{ExstarExx}), respectively, with $|E_X|\!=\!Ry^\ast\!/r^{0.6}$ as functions of the dimensionless nanotube radius.}\label{fig3}
\end{figure}
\section{Comparative analysis of $E_{X\!X}$ and $E_{X^{\!\ast}}$ in quasi-1D}
From Eqs.~(\ref{Exx}) and (\ref{Exstar}), one has the trion-to-biexciton binding energy ratio as follows
\begin{equation}
\frac{E_{X^{\!\ast}}}{E_{X\!X}}=3\!\left(\frac{3}{2e}\right)^{\!2\sqrt{Ry^\ast/|E_X|}}\!\!.
\label{ExstarExx}
\end{equation}
If one now assumes $|E_X|\!=\!Ry^\ast\!/r^{0.6}$ ($r$ is the dimensionless CN radius, or transverse confinement size for quasi-1D nanostructure in general) as was demonstrated earlier by variational calculations\cite{Pedersen03} to be consistent with many quasi-1D models,\cite{Bondarev09PRB,Ogawa91,Loudon} then one obtains the $r$-dependences of $|E_{X^{\!\ast}}|$, $|E_{X\!X}|$ and $E_{X^{\!\ast}}\!/E_{X\!X}$ shown in Fig.~\ref{fig3}. The trion and biexciton binding energies both decrease with increasing $r$ --- in such a way that their ratio remains greater than unity for small enough $r$ --- in full agreement with the experiments by Colombier et al.\cite{Colombier12} and Yuma et al.\cite{Yuma13} However, since the factor $3/2e$ in Eq.~(\ref{ExstarExx}) is less than one, the ratio can also be less than unity for $r$ large enough (but not too large, so that our configuration space method still works).
As $r$ goes down, on the other hand, the biexciton-to-exciton binding energy ratio $|E_{X\!X}/E_X|$ in Eq.~(\ref{Exx}) slowly grows, approaching the pure 1D limit $1/3e\approx0.12$. Similar tendency can also be traced in the Monte-Carlo simulation data of Ref.\cite{Kammerlander07} The equilibrium inter-exciton center-of-mass distance in the biexciton complex goes down with decreasing $r$ as well, $\Delta Z_0=\nu_0/2=\!1/(2\sqrt{|E_X|}\,)$ (atomic units). This supports experimental evidence for enhanced exciton-exciton annihilation in small diameter CNs.\cite{THeinz,Valkunas,Kono} The trion-to-exciton binding energy ratio $|E_{X^\ast}/E_X|$ of Eq.~(\ref{Exstar}) increases with decreasing $r$ faster than $|E_{X\!X}/E_X|$ (Fig.~\ref{fig3}), to yield $E_{X^\ast}/E_{X\!X}\!\approx\!3$ as the pure 1D limit for the trion-to-biexciton binding energy ratio.
\begin{figure}[t]
\epsfxsize=11.5cm\centering{\epsfbox{fig4.eps}}\caption{Trion ($X^\ast$) and biexciton ($X\!X$) binding energies given by Eqs.~(\ref{Exstar}) and (\ref{Exx}) with $|E_X|\!=\!Ry^\ast\!/r^{0.6}$, as functions of the CN radius and $\mu$ with $\varepsilon\!=\!1$ (a), and as functions of the CN radius and $\varepsilon$ with $\mu\!=\!0.04$ (b). Vertical parallel planes indicate the radii of the (6,5) and (9,7) CNs studied experimentally.}\label{fig4}
\end{figure}
When $E_{X^{\!\ast}}\!/E_{X\!X}$ is known, one can use Eq.~(\ref{ExstarExx}) to estimate the effective Bohr radii $a^\ast_B$ for the excitons in the CNs of known radii. For example, substituting $1.46$ for the 0.75~nm diameter (6,5) CN and $1.42$ for the 1.09~nm diameter (9,7) CN, as reported by Yuma et al.\cite{Yuma13} and Colombier et al.,\cite{Colombier12} respectively, into the left hand side of the transcendental equation~(\ref{ExstarExx}) and solving it for $a^\ast_B$, one obtains the effective exciton Bohr radius $a^\ast_B\!=\!2$~nm and $2.5$~nm for the (6,5) CN and (9,7) CN, respectively. This agrees reasonably with previous estimates.\cite{Pedersen03,Yuma13}
In general, the binding energies in Eqs.~(\ref{Exstar}) and (\ref{Exx}) are functions of the CN radius (transverse confinement size for a quasi-1D semiconductor nanowire), $\mu$ and~$\varepsilon$. Figures~\ref{fig4}~(a) and~\ref{fig4}~(b) show their 3D plots at fixed $\varepsilon\;(=\!1)$ and at fixed $\mu\;(=\!0.04)$, respectively, as functions of two remaining variables. The reduced effective mass $\mu$ chosen is typical of large radius excitons in small-diameter CNs.\cite{Jorio05} The unit dielectric constant $\varepsilon$ assumes the CN placed in air and the fact that there is no screening in quasi-1D semiconductor systems both at short and at large electron-hole separations.\cite{Louie09} This latter assumption of the unit background dielectric constant remains legitimate for \emph{small} diameter ($\lesssim\!1$~nm) semiconducting CNs in dielectric screening environment, too, --- for the lowest excitation energy exciton in its ground state of interest here (not for its excited states though), in which case the environment screening effect is shown by Ando to be negligible,\cite{Ando2010} diminishing quickly with the increase of the effective distance between the CN and dielectric medium relative to the CN diameter.
Figure~\ref{fig4}~(a) can be used to evaluate the relative stability of the trion and biexciton complexes in quasi-1D semiconductors. One sees that whether the trion or the biexciton is more stable (has the greater binding energy) in a particular quasi-1D system depends on $\mu$ and on the characteristic transverse size of the nanostructure. In strongly confined quasi-1D systems with relatively small $\mu$, such as small-diameter CNs, the trion is generally more stable than the biexciton. In less confined quasi-1D structures with greater $\mu$ typical of semiconductors,\cite{Cardona} the biexciton is more stable than the trion. This is a generic peculiarity in the sense that it comes from the tunnel exchange in the quasi-1D electron-hole system in the configuration space. Greater $\mu$, while not affecting significantly the single charge tunnel exchange in the trion complex, makes the neutral biexciton complex generally more compact, facilitating the mixed charge tunnel exchange in it and thus increasing the stability of the complex. From Fig.~\ref{fig4}~(b) one sees that this generic feature is not affected by the variation of $\varepsilon$, although the increase of $\varepsilon$ decreases the binding energies of both excitonic complexes --- in agreement both with theoretical studies\cite{Ronnow10} and with experimental observations of lower binding energies (compared to those in CNs) of these complexes in conventional semiconductor nanowires.\cite{Forchel,Crottini,Sidor,Gonzales,Schuetz} The latter are self-assembled nano\-structures of one (transversely confined) semiconductor embedded in another (bulk) semiconductor with the characteristic transverse confinement size typically greater than that of small diameter CNs, and so both inside and outside material dielectric properties matter there.
\begin{figure}[t]
\epsfxsize=12.0cm\centering{\epsfbox{fig5.eps}}\caption{Cross-section of Fig.~\ref{fig4}(a) at $\mu\!=\!0.04$ showing the relative behavior of the trion and biexciton binding energies in semiconducting CNs of increasing radius.}\label{fig5}
\end{figure}
Figure~\ref{fig5} shows the cross-section of Fig.~\ref{fig4}~(a) taken at $\mu\!=\!0.04$ to present the relative behavior of $|E_{X^{\!\ast}}|$ and $|E_{X\!X}|$ in semiconducting CNs of increasing radius. Both $|E_{X^{\!\ast}}|$ and $|E_{X\!X}|$ decrease, and so does their ratio, as the CN radius increases. From the graph, $|E_{X^{\!\ast}}|\!\approx\!170$ and $125$~meV, $|E_{X\!X}|\!\approx\!120$ and $95$~meV, for the (6,5) and (9,7) CNs, respectively. This is to be compared with $190$ and $130$~meV for the (6,5) CN versus $150$ and $106$~meV for the (9,7) CN reported experimentally.\cite{Colombier12,Yuma13} One sees that, as opposed to perturbative theories,\cite{Watanabe12} the present configuration space theory underestimates experimental data just slightly, most likely due to the standard variational treatment limitations. It does explain well the trends observed, and so the graph in Fig.~\ref{fig5} can be used as a guide for trion and biexciton binding energy estimates in small diameter ($\lesssim\!1$~nm) nanotubes.
\section{Configuration space method as applied to quasi-2D systems}
Recently, there has been a considerable interest in studies of optical properties of coupled quantum wells (CQWs).\cite{Kezer14,Kezer14jmpb,Govorov13,Muljarov12,Jeremy11,Gossard11,Govorov11,Butov09,Butov09prb,Snoke09,Bloch06,Snoke06} The CQW semiconductor nanostructure (Fig.~\ref{fig6}) consists of two identical semiconductor quantum wells separated by a thin barrier layer of another semiconductor. The tunneling of carriers through the barrier makes two wells electronically coupled to each other. As a result, an electron (a hole) can either reside in one of the wells, or its wave function can be distributed between both wells. A Coulomb bound electron-hole pair residing in the same well forms a direct exciton [Fig.~\ref{fig6}~(a)]. If the electron and hole of a pair are located in different wells, then an indirect exciton is formed [Fig.~\ref{fig6}~(b)].
\begin{figure}[t]
\epsfxsize=12.0cm\centering{\epsfbox{fig6.eps}}\vskip-0.5cm\caption{Schematic view of the trion formed by an electron and a direct exciton (a) and that formed by an electron and an indirect exciton (b) in the coupled quantum well nanostructure. In (a), exciton configurations $(h-e_1)$ and $(h-e_2)$ are inequivalent. In (b), they are equivalent and so the configuration space method can be used to evaluate the binding energy of the trion complex.}\label{fig6}
\end{figure}
Physical properties of CQWs can be controlled by using external electro- and magnetostatic fields. (See, e.g., Refs.\cite{Govorov13,Muljarov12} and refs. therein.) For example, applying the electrostatic field perpendicular to the layers increases the exciton radiative lifetime due to a reduction in the spatial overlap (contact density) between the electron and hole wave functions. As this takes place, the exciton binding energy reduces due to an increased electron-hole separation to make the exciton less stable against ionization, in contrast with the exciton magnetostatic stabilization effect under the same geometry.\cite{Govorov13} The tunneling effect is also enhanced as the electric field allows the carriers to leak out of the system, resulting in a considerable shortening of the photoluminescence decay time.
CQWs embedded into Bragg-mirror microcavities show a special type of voltage-tuned exciton polaritons, which can be used for low-threshold power polariton lasing.\cite{Jeremy11} New non-linear phenomena are also reported for these CQW systems both theoretically and experimentally, such as Bose condensation\cite{Kezer14} and parametric oscillations\cite{Bloch06} of exciton polaritons. For laterally confined CQW structures, experimental evidence for controllable formation of multiexciton Wigner-like molecular complexes of indirect excitons (single exciton, biexciton, triexciton, etc.) was reported recently.\cite{Govorov13} Trion complexes formed both by direct and by indirect excitons, as sketched in Figs.~\ref{fig6}~(a) and (b), were observed in CQWs as well.\cite{Shields97} All these findings make CQWs a much richer system capable of new developments in fundamental quantum physics and nanotechnology as compared to single quantum wells.\cite{Jeremy11,Gossard11,Govorov11,Butov09,Bloch06} They open up new routes for non-linear coherent optical control and spinoptronics applications with quasi-2D semiconductor CQW nanostructures.
Very recently, the problem of the trion complex formation in CQWs was studied theoretically in great detail for trions composed of a \emph{direct} exciton and an electron (or a hole) located in the neighboring quantum well\cite{Kezer14jmpb} [as sketched in Fig.~\ref{fig6}~(a)]. Significant binding energies are predicted on the order of $10$~meV at interwell separations $d\sim\!10-20$~nm for the lowest energy positive and negative trion states, to allow one suggest a possibility for trion Wigner crystallization. Figure~\ref{fig6}~(b) shows another possible trion complex that can also be realized in CQWs. Here, the trion is composed of an \emph{indirect} exciton and an electron (or a hole) in such a manner as to keep two same-sign particles in the same quantum well with the opposite-sign particle being located in the neighboring well. This can be viewed as the two equivalent configurations of the three-particle system in the configuration space ($\rho_1$, $\rho_2$) of the two \emph{independent} in-plane projections of the relative electron-hole distances $r_1$ and $r_2$ in the two \emph{equivalent} indirect excitons sharing the same electron, or the same hole as shown in Fig.~\ref{fig6}~(b). Such a three-particle system in the quasi-2D semiconductor CQW nanostructure is quite analogous to the quasi-1D trion presented in Sec.~3 above [cf. Fig.~\ref{fig6}~(b) and Fig.~\ref{fig2}]. Therefore, the Landau-Herring configuration space approach can be used here as well to evaluate the binding energy for this special case of the quasi-2D trion state.
Following is a brief outline of how one could proceed with the configuration space method to obtain the ground state binding energy for the quasi-2D trion complex sketched in Fig~\ref{fig6}~(b). A complete analysis of the problem will be presented elsewhere. The method requires knowledge of the ground state characteristics of the indirect exciton (abbreviated as "$I\!X$" in what follows). Specifically, one needs to know the quasi-2D ground state energy $E_{I\!X}(d)$ and corresponding \emph{in-plane} relative electron-hole motion wave function $\psi_{I\!X}(\rho,d)$ for the indirect exciton in the CQW system with the interwell distance $d$. These can be found by solving the radial Scr\"{o}dinger equation that is obtained by decoupling radial relative electron-hole motion in the cylindrical coordinate system with the $z$-axis being perpendicular to the QW layers [see Fig~\ref{fig6}~(b)]. Such an equation was derived and analyzed previously by Leavitt and Little.\cite{Leavitt90} The energy and the wave function of interest are as follows
\begin{eqnarray}
E_{I\!X}(d)=\lambda^2-\frac{4\lambda+4\lambda^4d^2E_1(\lambda d)\exp(2\lambda d)}{1+2\lambda d}\,,\nonumber\\[-0.2cm]
\label{indirect}\\[-0.2cm]
\psi_{I\!X}(\rho,d)=N\exp[-\lambda(\sqrt{\rho^2+d^2}-d)]\,,\hskip0.5cm\nonumber
\end{eqnarray}
where $E_1(x)\!=\!\int_{x}^{\infty}(e^{-t}/t)\,dt$ is the exponential integral, $\lambda=\!\lambda(d)\!=\!2/(1+2\sqrt{d}\,)$, the normalization constant $N$ is determined from the condition
\[
\int_{0}^{\infty}\!|\psi_{I\!X}(\rho,d)|^2\rho\,d\rho=1,
\]
and all quantities are measured in atomic units as defined in Sec.~2.
\begin{figure}[t]
\epsfxsize=12.0cm\centering{\epsfbox{fig7.eps}}\caption{Schematic of the charge-neutral spin-aligned Wigner crystal structure formed by two trions. They are one composed of the indirect exciton $(h_1-e_1)$ and the hole $h_2$, and another one composed of the indirect exciton $(h_3-e_3)$ and the electron $e_2$ [cf. Fig.~\ref{fig6}~(b)]. The structure can also be viewed as a triexciton, a coupled state of three indirect singlet excitons.}\label{fig7}
\end{figure}
With Eq.~(\ref{indirect}) in place, one can work out strategies for tunneling current calculations in the configuration space ($\rho_1$, $\rho_2$) of the two \emph{independent} in-plane projections of the relative electron-hole distances $r_1$ and $r_2$ in the two \emph{equivalent} indirect excitons as shown in Fig.~\ref{fig6}~(b). Both tunneling current responsible for the trion complex formation and that responsible for the biexciton complex formation can be obtained in full analogy with how it was done above for the respective quasi-1D complexes. The only formal difference now is the change in the phase integration volume from $dz_1dz_2$ to $\rho_1d\rho_1\rho_2d\rho_2$. Minimizing the tunneling current with respect to the center-of-mass-to-center-of-mass distance of the two equivalent indirect excitons results in the binding energy of a few-particle complex of interest. Note that the method applies to the complexes formed by \emph{indirect} excitons only as they allow equivalent configurations for a few-particle system to tunnel throughout in the configuration space ($\rho_1$, $\rho_2$), thereby forming a respective (tunnel) coupled few-particle complex.
Binding energy calculations for indirect exciton complexes in semiconductor CQW nanostructures are important to understand the principles of the more complicated electron-hole structure formation such as that shown in Fig.~\ref{fig7}. This is a coupled charge-neutral spin-aligned Wigner-like structure formed by two trions, one positively charged and another one negatively charged. The entire structure is electrically neutral, and it has an interesting electron-hole spin alignment pattern. This structure can also be viewed as a triexciton, a coupled state of three indirect singlet excitons. One could also imagine a Wigner-like crystal structure formed by unequal number of electrons and holes, as opposed to that in Fig.~\ref{fig7}, whereby the entire coupled structure could possess net charge and spin at the same time to allow precise electro- and magnetostatic control and manipulation by its optical and spin properties. Such Wigner-like electron-hole crystal structures in CQWs might be of great interest for spinoptronics applications.
All in all, indirect excitons, biexcitons and trions formed by indirect excitons are those building blocks that control the formation of more complicated Wigner-like electron-hole crystal structures in CQWs. The configuration space method presented here allows one to study the binding energies for these building blocks as functions of CQW system parameters, and thus to understand how stable electron-hole Wigner crystallization could possibly be in these quasi-2D nanostructures. The method should also work well for biexciton and trion complexes in quasi-2D self-assembled transition metal dichalcogenide heterostructures, where electrons and holes accumulated in the opposite neighboring monolayers are recently reported to form indirect excitons with new exciting properties such as increased recombination time\cite{Ceballos14} and vanishing high-temperature viscosity.\cite{Fogler14}
\section{Conclusion}
Presented herein is a universal configuration space method for binding energy calculations of the lowest energy neutral (biexciton) and charged (trion) exciton complexes in spatially confined quasi-1D semiconductor nanostructures. The method works in the effective two-dimensional configuration space of the two relative electron-hole motion coordinates of the two non-interacting quasi-1D excitons. The biexciton or trion bound state forms due to under-barrier tunneling between equivalent configurations of the electron-hole system in the configuration space. Tunneling rate controls the binding strength and can be turned into the binding energy by means of an appropriate variational procedure. Quite generally, trions are shown to be more stable (have greater binding energy) than biexcitons in strongly confined quasi-1D structures with small reduced electron-hole masses. Biexcitons are more stable than trions in less confined structures with large reduced electron-hole masses. A universal crossover behavior is predicted whereby trions become less stable than biexcitons as the transverse size of the quasi-1D nanostructure increases.
An outline is given of how the method can be used for electron-hole complexes of indirect excitons in quasi-2D semiconductor systems such as coupled quantum wells and van der Waals bound transition metal dichalcogenide heterostructures. Here, indirect excitons, biexcitons, and trions formed by indirect excitons control the formation of more complicated Wigner-like electron-hole crystal structures. The configuration space method can help develop understanding of how stable Wigner crystallization could be in these quasi-2D nanostructures. Wigner-like electron-hole crystal structures are of great interest for future spinoptronics applications.
\section*{Acknowledgments}
This work is supported by the US Department of Energy (DE-SC0007117). Discussions with David Tomanek (Michigan State U.), Roman Kezerashvili (NY$\,\!$CityTech), and Masha Vladimirova (U. Montpellier, France) are acknowledged. I.V.B. thanks Tony Heinz (Stanford U.) for pointing out Ref.\cite{Louie09} of relevance to this work.
|
1,108,101,565,511 | arxiv | \section{Introduction}
Let $d\ge2$ and $\phi_s(r)=\sqrt{s^2+r^2}$ for $r\ge0$ and $s>0$. We write $D$ for the operator $\sqrt{-\Delta_x}$, that is
\[
\widehat{Df}(\xi)=|\xi|\widehat{f}(\xi)
\]
where $\widehat{\cdot}$ denotes the (spatial) Fourier transform defined by
\[
\widehat{f}(\xi)=\int_{\mathbb R^d}e^{-ix\cdot\xi}f(x)\,\mathrm{d} x
\]
for appropriate functions $f$ on $\mathbb R^d$. Additionally, we define $D_\pm$ by
\[
\widetilde{D_\pm f}(\tau,\xi)=||\tau|\pm|\xi||\widetilde{f}(\tau,\xi),
\]
where $\widetilde{\cdot}$ is the space-time Fourier transform of appropriate functions $f$ on $\mathbb R\times \mathbb R^d$. The d'Alembertian operator $\partial_t^2-\Delta_x$ will be denoted by $\square$, so that $\square=D_-D_+$. Our main object of interest is the Klein--Gordon propagator given by
\[
e^{it\phi_s(D)} f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb R^d} e^{i(x\cdot\xi+t\phi_s(|\xi|))}\widehat{f}(\xi)\,\mathrm{d}\xi
\]
for sufficiently nice initial data $f$.
As part of the study of sharp bilinear estimates for the Fourier extension operator and inspired by work of Ozawa--Tsutsumi \cite{OT98}, Beltran--Vega \cite{BV19} very recently presented the following sharp estimate associated to the Klein--Gordon propagator
\begin{align}\label{ineq:BV}
&\|D^{\frac{2-d}{2}}(e^{it\phi_s(D)}f\overline{e^{it\phi_s(D)}g})\|_{L^2(\mathbb R^d\times\mathbb R)}^2\\
&\quad\le
(2\pi)^{1-3d}
\int_{(\mathbb R^d)^2}
|\widehat{f}(\eta_1)|^2|g(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
K^\mathrm{BV}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2,\nonumber
\end{align}
where
\[
K^\mathrm{BV}(\eta_1,\eta_2)
=
\int_{\mathbb S^{d-1}} \frac{
\phi_s(|\eta_1|)+\phi_s(|\eta_2|)
}{
(\phi_s(|\eta_1|)+\phi_s(|\eta_2|))^2-((\eta_1+\eta_2)\cdot\theta)^2
}
\,\mathrm{d}\sigma(\theta).
\]
The estimate \eqref{ineq:BV} has some interesting connections to well-known results. For example, as we shall see in more detail later, \eqref{ineq:BV} leads to null-form type estimates by appropriately estimating the kernel. In particular, when $d=2$ the Strichartz estimate
\begin{equation}\label{ineq:Strichartz, d=2}
\|
e^{it\phi_1(D)}f\|_{L^4(\mathbb R^{2+1})}
\leq
2^{-\frac14}
\|f\|_{H^\frac12(\mathbb R^2)}
\end{equation}
with the optimal constant follows from \eqref{ineq:BV}, more generally, for $d\geq2$ so does the null-form estimate
\begin{align}\label{ineq:Strichartz non-wave}
\|
D^{\frac{2-d}{2}}
|e^{it\phi_s(D)}f|^2\|_{L^2(\mathbb{R}^{d+1})}^2
\le
\frac{\pi^{2-d}|\mathbb S^{d-2}|}{2^ds}
\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^d)}^4,
\end{align}
although the optimality of the constant may be no longer true when $d\geq3$.
Here, the inhomogeneous Sobolev norm of order $\alpha$ is defined by
\[
\|f\|_{H^\alpha(\mathbb R^d)}:=\|\phi_1(D)^\alpha f\|_{L^2(\mathbb R^d)}.
\]
The estimate \eqref{ineq:Strichartz, d=2} with the optimal constant was first obtained by Quilodr\'an \cite{Qu15}. Bilinear estimates which bear resemblance to \eqref{ineq:BV} for the Klein--Gordon equation, as well as the Schr\"odinger and wave equation, have often arisen in the pursuit of optimal constants for Strichartz estimates and closely related null-form type estimates. As well as the aforementioned work of Beltran--Vega \cite{BV19}, estimates of the form \eqref{ineq:BV} for the Klein--Gordon propagator can be found in work of Jeavons \cite{Jv14} (see also \cite{Jvthesis}). For the Schr\"odinger equation, in addition to the Ozawa--Tsutsumi estimates in \cite{OT98}, estimates resembling \eqref{ineq:BV} may be found in work of Carneiro \cite{Cr09} and Planchon--Vega \cite{PV09}, with a unification of each of these results by Bennett et al. in \cite{BBJP17}. For the wave equation, Bez--Rogers \cite{BR13} and Bez--Jeavons--Ozawa \cite{BJO16} have established estimates resembling \eqref{ineq:BV}. We also remark that the related literature on sharp Strichartz estimates is large. In addition to the papers already cited, this body of work includes, for example, \cite{BJ15, COS19, COSS18, Fs07, HZ06, Ku03}; the interested reader is referred to the survey article by Foschi--Oliveira e Silva \cite{FO17} for further information.
Let $\Gamma(z)$ be the gamma function of $z$ (with $\mathrm{Re}(z)>0$) and
\begin{equation}
\mathcal K_a^b(\eta_1,\eta_2)
:=
\frac{\left(\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2-s^2\right)^b}
{\left(\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2+s^2\right)^a}.\nonumber
\end{equation}
In the present paper, we establish the following new bilinear estimates for the Klein--Gordon propagator.
\begin{theorem}\label{thm:main}
For $d\ge2$ and $\beta>\frac{1-d}{4}$, we have the estimate
\begin{align}\label{ineq:main}
&\||\square|^\beta (e^{it\phi_s(D)}f\overline{e^{it\phi_s(D)}g})\|_{L^2(\mathbb{R}^{d+1})}^2\\
&\quad\le
\mathbf{KG}(\beta,d)
\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d} \eta_1\mathrm{d} \eta_2,\nonumber
\end{align}
with the optimal constant
\[
\mathbf{KG}(\beta,d)
:=
2^{\frac{-5d+1}{2}+2\beta}\pi^{\frac{-5d+1}{2}}\frac{\Gamma(\tfrac{d-1}{2}+2\beta)}{\Gamma(d-1+2\beta)}.
\]
\end{theorem}
In the case when $s\to0$, certain sharp bilinear estimates for solutions to the wave equation with the operator $|\square|^\beta$ have been deeply studied by Bez--Jeavons--Ozawa \cite{BJO16}. One may note that, when $d=2$, a slightly larger range of $\beta$ is valid in Theorem \ref{thm:main} than one for the corresponding result \eqref{ineq:biest sharp BJO} for the wave case in \cite{BJO16}.
In order to prove Theorem \ref{thm:main}, we employ their argument and adapt it into the context of the Klein--Gordon equation. As a consequence of Theorem \ref{thm:main}, we will generate null-form type estimates of the form
\begin{equation}\label{ineq:corollaries}
\||\square|^\beta |e^{it\phi_s(D)}f|^2\|_{L^2(\mathbb{R}^{d+1})}
\le
C
\|\phi_s(D)^\alpha f\|_{L^2(\mathbb R^d)}^2
\end{equation}
for certain pairs $(\alpha,\beta)$ with the optimal constant.
\subsection{Wave regime}
For $d\geq4$, the kernel $K^\mathrm{BV}$ can be estimated\footnote{
We observe that $d\geq4$ is important here. For $d=3$, the estimate \eqref{ineq:biest D} actually does not hold. The counterexample has been given by Foschi \cite{Fs00} for the wave equation, and the same argument appropriately adapted works for the Klein--Gordon propagator.
}
as
\begin{align*}
K^{\mathrm{BV}}(\eta_1,\eta_2)
&\leq
\frac{|\mathbb S^{d-1}|}{\phi_s(|\eta_1|)+\phi_s(|\eta_2|)}
\int_{-1}^1
\left(
1-\left|\frac{\eta_1+\eta_2}{\phi_s(|\eta_1|)+\phi_s(|\eta_2|)}\right|^2\lambda^2
\right)^{-1}
(1-\lambda^2)^{\frac{d-3}{2}}
\,\mathrm{d}\lambda\\
&\leq
\frac {C}{\phi_s(|\eta_1|)+\phi_s(|\eta_2|)}
\end{align*}
for some absolute constant $C$ since $|\eta_1+\eta_2|\leq\phi_s(|\eta_1|)+\phi_s(|\eta_2|)$. The integral in the first inequality is surely finite as long as $d\geq4$. Then, it follows from the arithmetic-geometric mean that
\[
\phi_s(|\eta_1|)\phi_s(|\eta_2|)K^{\mathrm{BV}}(\eta_1,\eta_2)
\leq
C\phi_s(|\eta_1|)^\frac12\phi_s(|\eta_2|)^\frac12,
\]
and hence the null-form type estimate
\begin{equation}\label{ineq:biest D}
\|D^{\frac{2-d}{2}}(e^{it\phi_s(D)}f\overline{e^{it\phi_s(D)}g})\|_{L^2(\mathbb R^d\times\mathbb R)}
\le
C
\|\phi_s(D)^\frac14f\|_{L^2(\mathbb R^d)}
\|\phi_s(D)^\frac14g\|_{L^2(\mathbb R^d)}
\end{equation}
holds. When $s\to0$, the estimate \eqref{ineq:biest D} yields
\begin{equation}\label{ineq:null +-}
\|D^{\beta_0}D_-^{\beta_-}D_+^{\beta_+}(e^{itD}f\overline{e^{itD}g})\|_{L^2(\mathbb R^{d+1})}
\leq
C\|f\|_{\dot{H}^{\alpha_1}}\|g\|_{\dot{H}^{\alpha_2}},
\end{equation}
in the case of $(\beta_0,\beta_-,\beta_+,\alpha_-,\alpha_+)=(\frac{2-d}{2},0,0,\frac14,\frac14)$ for the propagator $e^{itD}$ associated with the wave equation. The estimate \eqref{ineq:null +-}, as well as the corresponding $(++)$ case (while \eqref{ineq:null +-} is $(+-)$ case),
\begin{equation}\label{ineq:null ++}
\|D^{\beta_0}D_-^{\beta_-}D_+^{\beta_+}(e^{itD}fe^{itD}g)\|_{L^2(\mathbb R^{d+1})}
\leq
C\|f\|_{\dot{H}^{\alpha_1}}\|g\|_{\dot{H}^{\alpha_2}}
\end{equation}
has found important applications in study of nonlinear wave equations. This type of estimate has been studied back in work of Beals \cite{Be83} and Klainerman--Machedon \cite{KM93,KM96a,KM97a}. A complete characterization of the admissible exponents $(\beta_0,\beta_-,\beta_+,\alpha_-,\alpha_+)$ for \eqref{ineq:null +-} and \eqref{ineq:null ++} was eventually obtained by Foschi--Klainerman \cite{FK00}. Such a characterization when the $L_{t,x}^2$ norm on the left-hand side of \eqref{ineq:null +-} is replaced by $L_t^qL_x^r$ has also drawn great attention. Using bilinear Fourier restriction techniques, Bourgain \cite{Br95} made a breakthrough contribution, then Wolff \cite{Wo01} and Tao \cite{Tao01} (in the endpoint case; see also Lee \cite{Lee06} and Tataru \cite{Tt07}) completed the diagonal case $q=r$. For the non-diagonal case we refer readers to \cite{LV08} due to Lee--Vargas for a complete characterization when $d\geq4$ and partial results when $d=2$, $3$. Soon later Lee--Rogers--Vargas \cite{LRV08} completed $d=3$, but a gap between necessary and sufficient conditions still remains when $d=2$.\\
As a means of comparing our bilinear estimate \eqref{ineq:main} with \eqref{ineq:BV}, we note that using the trivial bound
\begin{equation}\label{ineq:keyest wave}
\frac{\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2-s^2}{\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2+s^2}\le1,
\end{equation}
we estimate our kernel as
\begin{equation}\label{ineq:kernelest wave}
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\le
\mathcal K_0^{\frac{d-3}{2}+2\beta}(\eta_1,\eta_2).
\end{equation}
For $\beta\geq\frac{3-d}{4}$, it follows that
\begin{align}\label{ineq:biest D_-D_+ }
\||\square|^{\beta}(e^{it\phi_s(D)}f \overline{e^{it\phi_s(D)}g})\|_{L^2(\mathbb{R}^{d+1})}
\le
C
\|\phi_s(D)^{\frac{d-1}{4}+\beta}f\|_{L^2(\mathbb{R}^d)}
\|\phi_s(D)^{\frac {d-1}{4}+\beta}g\|_{L^2(\mathbb{R}^d)}
\end{align}
for some absolute constant $C$ (for instance, use $\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot \eta_2-s^2\leq 2\phi_s(|\eta_1|)\phi_s(\eta_2|)$), which, as in the discussion for the Beltran--Vega bilinear estimate, places Theorem \ref{thm:main} in the framework of null-form type estimates. If we \textit{formally} set $\beta=\frac{2-d}{4}$ in \eqref{ineq:biest D_-D_+ } to get data with regularity whose order is $\frac14$ as in \eqref{ineq:biest D}, the order of ``smoothing" from $|\square|^\beta$ becomes $2\beta=\frac{2-d}{2}$, which is compatible with \eqref{ineq:biest D}. Unfortunately, $\frac{2-d}{4}$ is outside the range $\beta\geq\frac{3-d}{4}$ and, in fact, as we shall see in Proposition \ref{prop:necessary condition}, $\beta\geq \frac{3-d}{4}$ is \emph{a necessary condition} for \eqref{ineq:biest D_-D_+ }. Nevertheless, as an application of Theorem \ref{thm:main} one can widen the range to, at least, $\beta\geq\frac{2-d}{4}$ if one considers radially symmetric data. We shall state this result as part of the forthcoming Corollary \ref{cor:biest radial}. In addition, for a large range of $\beta$ we shall in fact obtain the \emph{optimal constant} for such null-form type estimates; to state our result, we introduce the constant
\[
\mathbf{F}(\beta,d)
:=
2^{d-3+4\beta}\pi^{-\frac d2}\frac{\Gamma(\tfrac d2)\Gamma(\tfrac{d-1}{2}+2\beta)}{(d-2+2\beta)\Gamma(\tfrac{3d-5}{2}+2\beta)}.
\]
\begin{corollary}\label{cor:biest radial}
Let $d\ge2$, $\beta\geq\frac{2-d}{4}$. Then, there exists a constant $C>0$ such that \eqref{ineq:biest D_-D_+ } holds whenever $f$ and $g$ are radially symmetric.
Moreover, for $\beta\in[\frac{2-d}{4},\tfrac{3-d}{4}]\cup[\tfrac{5-d}{4},\infty)$, the optimal constant in \eqref{ineq:biest D_-D_+ } for radially symmetric $f$ and $g$ is $\mathbf{F}(\beta,d)^\frac12$, but there does not exist a non-trivial pair of functions $(f,g)$ that attains equality.
\end{corollary}
In the case of the wave propagator when $s\to0$, the estimate \eqref{ineq:biest D_-D_+ } becomes
\begin{equation}\label{ineq:Strichartz sharp BJO}
\||\square|^\beta(e^{itD}f\overline{e^{itD}g})\|_{L^2(\mathbb R^{d+1})}
\leq
\mathbf{F}(\beta,d)^\frac12
\|f\|_{\dot{H}^{\frac{d-1}{4}+\beta}(\mathbb R^d)}
\|g\|_{\dot{H}^{\frac{d-1}{4}+\beta}(\mathbb R^d)}
\end{equation}
and, in certain situation, it is known that the constant $\mathbf{F}(\beta,d)^\frac12$ is optimal. In the case $\beta=0$ and $d=3$, pioneering work of Foschi \cite{Fs07} established the optimality of the constant $\mathbf{F}(0,3)^\frac12$. The constant $\mathbf{F}(\beta,d)^\frac12$ is also known to be optimal when $(\beta,d)=(0,4)$ and $(\beta,d)=(0,5)$; the latter was established by Bez--Rogers \cite{BR13} building on work of Foschi and obtained via the bilinear estimate
\begin{equation}\label{ineq:Strichartz sharp BJO beta=0}
\|e^{itD}f\overline{e^{itD}g}\|_{L^2(\mathbb R^{d+1})}^2
\le
\textbf{W}(0,d)
\int_{(\mathbb R^{d})^2}|
\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2|\eta_1||\eta_2|
K_0^\mathrm{BR}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2.
\end{equation}
Here, $\textbf{W}(\beta,d)$ turns out to be $\mathbf{KG}(\beta,d)$ and $K_{\beta}^\mathrm {BR}$ formally coincides with the special case of our kernel $\mathcal K_{\frac12}^{\frac{d-2}{2}+2\beta}$ when $s=0$. The optimality of $\mathbf{F}(0,4)^\frac12$ in \eqref{ineq:Strichartz sharp BJO} was proved by Bez--Jeavons \cite{BJ15} by making use of \eqref{ineq:Strichartz sharp BJO beta=0}, polar coordinates and techniques from the theory of spherical harmonics.
Soon later, imposing an additional radial symmetry on the initial data $f$ and $g$, Bez--Jeavons--Ozawa proved the optimality of $\mathbf{F}(\beta,d)^\frac12$ in \eqref{ineq:Strichartz sharp BJO} when $d\geq2$ and\footnote{It can also be seen from the form of the sharp constant that widening the range $\beta>\beta_d$ is impossible.} $\beta>\beta_d:=\max\{\frac{1-d}{4},\frac{2-d}{2}\}$ by establishing the null-form type bilinear estimate
\begin{equation}\label{ineq:biest sharp BJO}
\||\square|^\beta(e^{itD}f\overline{e^{itD}g})\|_{L^2(\mathbb R^{d+1})}^2
\le
\textbf{W}(\beta,d)
\int_{(\mathbb R^{d})^2}|
\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2|\eta_1||\eta_2|
K_\beta^\mathrm{BR}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2,
\end{equation}
which again coincides\footnote{Because of this fact, we expect that \eqref{ineq:biest D_-D_+ } is valid with $C=\mathbf{F}(\beta,d)^\frac12$ for $\beta\in(\beta_d,\frac{2-d}{4})$ as well, but we do not pursue this here.} with \eqref{ineq:main} formally substituted with $s=0$. They accomplished the result by taking advantage of an exceedingly nice structure of the homogeneity that the kernel $\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}$ ($=K_\beta^\mathrm{BR}$) possesses, specifically, when $s=0$;
\[
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(r_1\theta_2,r_2\theta_2)
=
(r_1r_2)^{\frac{d-3}{2}+2\beta}(1-\theta_1\cdot\theta_2)^{\frac{d-3}{2}+2\beta},\qquad r_1,r_2>0,\ \theta_1,\theta_2\in \mathbb S^{d-1},
\]
This property completely divides the right-hand side of \eqref{ineq:biest sharp BJO} into radial and angular components if the initial data are radial symmetric.
In contrast, our concern is the case $s>0$ and the lack of homogeneity in the kernel causes significant difficulty in this regard. This can be seen as responsible for the gap $(\frac{3-d}{4},\frac{5-d}{4})$ (as well as the range $(\beta_d,\frac{2-d}{4})$) in Corollary \ref{cor:biest radial}, for which we also expect \eqref{ineq:biest D_-D_+ } still holds with $C=\mathbf{F}(\beta,d)^\frac12$.\\
In the current paper, we prove Corollary \ref{cor:biest radial} by first making use of our bilinear estimate \eqref{ineq:main}. One can show, however, that it is impossible to obtain the optimality of $\mathbf{F}(\beta,d)^\frac12$ in \eqref{ineq:biest D_-D_+ } for radial data and any $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$ once one makes use of \eqref{ineq:main} as a first step; somewhat surprisingly given that \eqref{ineq:main} is sharp. This is a consequence of Lemma \ref{lem:B<B} in the specific setting.
There are some special cases of $\beta$; the endpoints $\frac{3-d}{4}$ and $\frac{5-d}{4}$ of the gap, at which we can remove the radial symmetry hypothesis on the initial data and still keep the optimal constants.
\begin{corollary}\label{cor:wave}
Let $d\ge2$. Then, the estimate \eqref{ineq:corollaries} holds with the optimal constant $C=\mathbf{F}(\beta,d)^\frac12$
for $(\alpha,\beta)=(\frac12,\frac{3-d}{4})$ and $(\alpha,\beta)=(1,\frac{5-d}{4})$, but there are no extremisers. Furthermore, when $(\alpha,\beta)=(1,\frac{5-d}{4})$, we have the refined Strichartz estimate
\begin{align}\label{ineq:Strichartz refined general J}
\||\square|^\frac{5-d}{4} |e^{it\phi_s(D)}f|^2\|_{L^2(\mathbb{R}^{d+1})}
\le
\mathbf{F}(\tfrac{5-d}{4},d)^\frac12
\left(
\|\phi_s(D)f\|_{L^2(\mathbb R^d)}^4
-s^2
\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^d)}^4
\right)^\frac12,
\end{align}
where the constant is optimal and there are no extremisers.
\end{corollary}
Corollary \ref{cor:wave} generalizes the following recent results. In the context of the Klein--Gordon equation, Quilodr\'an \cite{Qu15} appropriately developed Foschi's argument in \cite{Fs07} and proved the sharp Strichartz estimate
\begin{equation}\label{ineq:Strichartz sharp Q}
\|e^{it\phi_1(D)}f\|_{L^q(\mathbb R^{d+1})}
\leq
\mathbf{H}(d,q)
\|f\|_{H^\frac12(\mathbb R^d)}
\end{equation}
for $(d,q)=(2,4)$, $(2,6)$, $(3,4)$, which are the endpoint cases of the admissible range of exponent $q$, namely,
\[
\frac{2(d+2)}{d}\le q\le \frac{2(d+1)}{d-1}.
\]
The constant $\mathbf{H}(d,q)$ denotes the optimal constant so that \eqref{ineq:Strichartz sharp Q} in the case $(d,q)=(3,4)$ is recovered by Corollary \ref{cor:wave} in the case $(\alpha,\beta)=(\frac12,\frac{3-d}{4})$ and $\mathbf{F}(0,3)^\frac14=\mathbf{H}(4,3)$ holds. In \cite{Qu15}, Quilod\'an also proved that there is no extremiser which attains \eqref{ineq:Strichartz sharp Q} for $(d,q)=(2,4)$, $(2,6)$, $(3,4)$.
Later Carneiro--Oliveira e Silva--Sousa \cite{COS19} further revealed the nature of
\eqref{ineq:Strichartz sharp Q}
for $d=1$, $2$, by answering questions raised in \cite{Qu15}; in particular, they found the best constant in \eqref{ineq:Strichartz sharp Q} for $(d,q)=(1,6)$ and absence of the extremisers (the case $(d,q)=(1,6)$ is the endpoint of the admissible range of $6\leq q\leq \infty$ when $d=1$). Meanwhile, they also established there exist extremisers in the non-endpoint cases in low dimensions $d=1$, $2$. A subsequent study by the same authors in collaboration with Stovall \cite{COSS18} proved the analogous results in the non-endpoint cases for higher dimensions $d\geq3$ by using some tools from bilinear restriction theory.
In \cite{Jv14}, Jeavons obtained the following refined Strichartz estimate in five spatial dimensions
\begin{align}\label{ineq:Strichartz refined J}
\|e^{it\phi_s(D)}f\|_{L^4(\mathbb R^{5+1})}
\leq
\mathbf{F}(0,5)^\frac14
\left(
\|\phi_s(D)f\|_{L^2(\mathbb R^5)}^4
-
s^2\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^5)}^4
\right)^\frac14,
\end{align}
which recovers the inequality \eqref{ineq:Strichartz sharp BJO} when $(\beta,d)=(0,5)$ in the limit $s\to0$. Moreover, by simply omitting the negative second term, it follows that
\begin{align*}
\|e^{it\phi_1(D)}f\|_{L^4(\mathbb{R}^{5+1})}
\le
\mathbf{F}(0,5)^\frac14
\|f\|_{H^1(\mathbb R^5)},
\end{align*}
where the constant $\mathbf{F}(0,5)^\frac14=(24\pi^2)^{-\frac14}$ is still sharp. These results are recovered too by Corollary \ref{cor:wave} in the case $(\alpha,\beta)=(1,\frac{5-d}{4})$.
\subsection{Non-wave regime}
One may examine the Beltran--Vega bilinear estimate \eqref{ineq:BV} from a somewhat different perspective to that taken in our earlier discussion which led to \eqref{ineq:biest D}.
For $d\geq2$ the kernel $K^\mathrm{BV}$ can be transformed as
\begin{align*}
K^\mathrm{BV}(\eta_1,\eta_2)
&=
|\mathbb S^{d-2}|\int_0^{\pi}
\frac{
\phi_s(|\eta_1|)+\phi_s(|\eta_2|)
}{
(\phi_s(|\eta_1|)+\phi_s(|\eta_2|))^2-|\eta_1+\eta_2|^2\cos^2 \theta
}
(\sin\theta)^{d-2}
\,\mathrm{d}\theta\\
&=
|\mathbb S^{d-2}|\int_{-\frac\pi2}^{\frac\pi2}
\frac{
\tau(\cos\theta)^{d-2}
}{
(\tau^2-|\xi|^2)+|\xi|^2\cos^2\theta
}
\,\mathrm{d}\theta\\
&=
|\mathbb S^{d-2}|\int_{-\frac\pi2}^{\frac\pi2}
\frac{
\tau(\cos\theta)^{d-2}
}{
(\tau^2-|\xi|^2)\tan^2\theta+\tau^2
}
\frac{\mathrm{d}\theta}{\cos^2\theta},
\end{align*}
where we denote $\tau=\phi_s(|\eta_1|)+\phi_s(|\eta_2|)$ and $\xi=\eta_1+\eta_2$ for the sake of convenience. By applying the fact\footnote{Note that the equality holds when $d=2$} $(\cos \theta)^{d-2}\leq1$, the change of variables $\tan\theta\mapsto\frac{\tau}{\sqrt{\tau^2-|\xi|^2}}x$, it follows that
\[
K^\mathrm{BV}(\eta_1,\eta_2)
\leq
\frac{|\mathbb S^{d-2}|}{\sqrt{\tau^2-|\xi|^2}}\int_{-\infty}^\infty\frac{\mathrm{d} x}{x^2+1}
=
\frac{\pi|\mathbb S^{d-2}|}{\sqrt{\tau^2-|\xi|^2}}.
\]
Hence, another key relation (instead of \eqref{ineq:kernelest wave});
\begin{equation}\label{ineq:keyest non-wave}
\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2\geq s^2,
\end{equation}
implies
\[
K^\mathrm{BV}(\eta_1,\eta_2)
\leq
\frac{\pi|\mathbb S^{d-2}|}{2s},
\]
with which, as informed earlier, the inequality \eqref{ineq:BV} directly yields \eqref{ineq:Strichartz non-wave}.
By comparison with \eqref{ineq:biest D}, the regularity level on the initial data has increased to $H^\frac12$ but this has allowed for a wider range of $d$ which, in particular, includes $d=2$ in which case \eqref{ineq:Strichartz non-wave} coincides with the sharp $H^\frac12\to L_{x,t}^4$ Strichartz estimate \eqref{ineq:Strichartz, d=2} obtained by Quilodr\'an.
Note that, in the non-wave regime, we are not allowed to let $s\to0$ because of the factor $s^{-1}$ appearing in the constant.
On the other hand, Theorem \ref{thm:main} also yields \eqref{ineq:Strichartz, d=2} as a special case of the following family of sharp null-form type estimates valid in all dimensions $d\ge2$. Indeed, since we have another kernel estimate
\begin{equation}\label{ineq:kernelest non-wave}
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\leq
2^{-\frac12}
\mathcal K_0^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
s^{-1}
\end{equation}
via \eqref{ineq:keyest non-wave}, we immediately deduce the following from Theorem \ref{thm:main}.
\begin{corollary}\label{cor:non-wave}
Let $d\ge2$. Then the estimate \eqref{ineq:corollaries} holds with the optimal constant
$$
C
=
\left(\frac{2^{-d+1}\pi^{\frac{-d+2}{2}}}{s{\Gamma(\frac d2)}}\right)^\frac12
$$
for $(\alpha,\beta)=(\frac12,\frac{2-d}{4})$, but there are no extremisers. Furthermore, when $(\alpha,\beta)=(1,\frac{4-d}{4})$, we have the refined Strichartz estimate
\begin{align}\label{ineq:Strichartz refined non-wave}
\||\square|^\frac{4-d}{4} |e^{it\phi_s(D)}f|^2\|_{L^2(\mathbb{R}^{d+1})}^2
\le
\left(
\frac{2^{-d+1}\pi^{\frac{-d+2}{2}}}{s\Gamma(\frac{d+2}{2})}
\right)^\frac12
\left(
\|\phi_s(D)f\|_{L^2(\mathbb R^d)}^4
-s^2
\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^d)}^4
\right)^\frac12,
\end{align}
where the constant is optimal and there are no extremisers.
\end{corollary}
One may note that \eqref{ineq:Strichartz refined non-wave} provides a sharp form of the following refined Strichartz inequality in the analogous manner of \eqref{ineq:Strichartz refined J} when $d=4$:
\begin{equation}\label{i:new}
\|e^{it\phi_1(D)}f\|_{L^4(\mathbb{R}^{4+1})}
\le
(16\pi)^{-\frac14}
(
\|f\|_{H^1(\mathbb R^4)}^4
-
\|f\|_{H^\frac12(\mathbb R^4)}^4
)^\frac14,
\end{equation}
however we are unable to conclude whether the constant $(16\pi)^{-\frac14}$ continues to be optimal if we drop the second term on the right-hand side, which is discussed in Section \ref{s:sharpness non-wave}.
\\
For solutions $u$ of certain PDE, in addition to the null-form estimates \eqref{ineq:null +-}, estimates which control quantities like $|u|^2$ through its interplay with other types of operators have appeared numerous times in the literature. In particular, we note that the approach taken by Beltran--Vega \cite{BV19}, which in turn built on work of Planchon--Vega \cite{PV09}, rested on interplay with geometric operators such as the Radon transform or, more generally, the $k$-plane transform. For related work in this context of interaction with geometrically-defined operators, we also refer the reader to work of Bennett et. al \cite{BBFGI18} and Bennett--Nakamura \cite{BN20}.
Our approach to proving Theorem \ref{thm:main} more closely follows the argument in \cite{BJO16} and does not appear to fit into such a geometric perspective.\\
\subsection*{Summary of results}
Theorem \ref{thm:main}, a natural generalization of \eqref{ineq:biest sharp BJO} in the context of the Klein--Gordon, reproduces several known Strichartz-type inequalities with the sharp constant. The following is the summary of our results and remaining open problems.
\begin{itemize}
\item Corollary \ref{cor:wave} recovers \eqref{ineq:corollaries} when $(\beta,d)=(0,3)$ due to Quilodr\'an \cite{Qu15}.
\item Corollary \ref{cor:wave} recovers \eqref{ineq:corollaries} when $(\beta,d)=(0,5)$ due to Jeavons \cite{Jv14}.
\item Corollary \ref{cor:wave} recovers \eqref{ineq:Strichartz refined J} when $(\beta,d)=(0,5)$ with $\alpha=\frac34$ due to Jeavons \cite{Jv14}.
\item[$\circ$] For $(\beta,d)=(0,4)$, it remains open whether \eqref{ineq:corollaries} holds with $C=\mathbf{F}(0,4)^\frac12$ as the sharp constant.
\item Corollary \ref{cor:non-wave} recovers \eqref{ineq:corollaries} when $(\beta,d)=(0,2)$ due to Quilodr\'an \cite{Qu15}.
\item Corollary \ref{cor:non-wave} yields \eqref{i:new}, an analogous refined Strichartz inequality of \eqref{ineq:Strichartz refined J}, in the case $(\beta,d)=(0,4)$.
\item [$\circ$] Corollary \ref{cor:non-wave} recovers \eqref{ineq:corollaries} when $(\beta,d)=(0,4)$ with the constant $(16\pi)^{-\frac14}$, but we do not know whether the constant is sharp.
\end{itemize}
\subsection*{Notation/Useful formulae}
Throughout the paper, we denote $A\gtrsim B$ if $A\ge CB$, $A\lesssim B$ if $A\le CB$ and $A\sim B$ if $C^{-1}B\le A \le CB$ for some constant $C>0$. The gamma function and the beta function are defined by
\[
\Gamma(z):=\int_0^\infty x^{z-1}e^{-x}\,\mathrm{d} x\quad \text{and} \quad B(z,w):=\int_0^1 \lambda^{z-1}(1-\lambda)^{w-1}\,\mathrm{d}\lambda,\]
respectively, for $z$, $w\in\mathbb C$ satisfying $\mathrm{Re}(z)$, $\mathrm{Re}(w)>0$.
Regarding those, we use the following well-known formulae multiple times;
\begin{equation}\label{eq:volume of d-sphere}
|\mathbb{S}^{d-1}|=\frac{2\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})},
\end{equation}
and
\[
B(z,w)=\frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}.
\]
Also, it is worth to note here that the inverse Fourier transform of an appropriate function $g$ on $\mathbb R^d$ is given by $g^\vee(x)=(2\pi)^{-d}\int_{\mathbb R^d}e^{ix\cdot\xi}g(\xi)\,\mathrm{d}\xi$ so that the following hold:
\begin{itemize}
\item $\widehat{fg}(\xi)=(2\pi)^{-d}\widehat{f}*\widehat{g}(\xi),\quad \xi\in\mathbb R^d.$
\item $\|f\|_{L^2(\mathbb R^d)}^2=(2\pi)^{-d}\|\widehat{f}\|_{L^2(\mathbb R^d)}^2$ (Plancherel's theorem).
\item $\|\phi_s(D)^\alpha f\|_{L^2(\mathbb R^d)}
=
(2\pi)^{-d}\left(\int_{\mathbb R^d}\phi_s(|\xi|)^{2\alpha}|\widehat{f}(\xi)|^2\,\mathrm{d}\xi\right)^\frac12$.
\end{itemize}
\subsection*{Structure of the paper}
\begin{itemize}
\item Section \ref{sec:proof of Theorem main}: We first prove Theorem \ref{thm:main} by adapting the argument of \cite{BJO16}.
\item Section \ref{sec:some observations wave regime}: We prove \eqref{ineq:biest D_-D_+ } for radial data and then show that, specifically for $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]\cup[\frac{5-d}{4},\infty)$, the estimate \eqref{ineq:biest D_-D_+ } with $C=\mathbf{F}(\beta,d)^\frac12$ holds. We also make an observation that suggests it may be difficult to obtain the optimal constant in \eqref{ineq:biest D_-D_+ } for $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$ even for radially symmetric data (see Proposition \ref{prop:a counter example}). At the end of this section, we show $\beta\geq\frac{3-d}{4}$ is necessary for \eqref{ineq:biest D_-D_+ } to hold for general data.
\item Section \ref{sec:sharpness of constants}: The aim of this section is to complete the proof of the corollaries. We first introduce how to deduce the refined form of the Strichartz estimate, and then focus on the sharpness of the constants in Corollary \ref{cor:biest radial}, Corollary \ref{cor:wave} and Corollary \ref{cor:non-wave}. They all are proved by the same method, but it differs from that in \cite{Jv14} or \cite{BJO16}, as here we need to deal with the more delicate situation of the non-wave regime.
The non-existence of extremisers is also discussed.
\item Section \ref{sec:analogous results ++}: We end the paper with Section \ref{sec:analogous results ++} by discussing analogous results for the $(++)$ case. As \cite{BJO16} has already observed, the $(++)$ case is far easier than the $(+-)$ case, and this will become clear from our argument in this section. We employ the null-form $|\square-(2s)^2|$ instead of $|\square|$ in order to follow the ideas of the proof of Theorem \ref{thm:main} and obtain an analogous bilinear estimate (Theorem \ref{thm:main ++}).
\end{itemize}
\subsection*{Acknowledgment}
The first author was supported by JSPS Postdoctoral Research Fellowship (No. 18F18020), and the second author was supported by JSPS KAKENHI Grant-in-Aid for JSPS Fellows (No. 20J11851). Authors express their sincere gratitude to Neal Bez, second author's adviser, for introducing the problem, sharing his immense knowledge and continuous support. They also wish to thank the anonymous referee for a very careful reading of the manuscript and many valuable suggestions and comments.
\section{Proof of Theorem \ref{thm:main}}\label{sec:proof of Theorem main}
Although some steps require additional care due to the extra parameter $s$, broadly speaking Theorem \ref{thm:main} can be proved by adapting the argument for wave propagators presented in \cite{BJO16}, whose techniques originated in \cite{BBI15} (see also \cite{BBJP17}). The key tool here is the following Lorentz transform given by $L$; for fixed $(\tau,\xi)\in\mathbb R\times \mathbb R^d$ such that $\tau>|\xi|$,
\[
L{t\choose x}={\gamma(t-\zeta\cdot x)\choose x+(\frac{\gamma-1}{|\zeta|^2}\zeta\cdot x-\gamma t)\zeta},\quad (t,x)\in\mathbb{R}\times\mathbb{R}^d,
\]
where $\zeta:=-\frac{\xi}{\tau}$ and $\gamma:=\frac{\tau}{(\tau^2-|\xi|^2)^{\frac12}}$. It is well known that the measure $\frac{\delta(\sigma-\phi_s(|\eta|))}{\phi_s(|\eta|)}$ for $(\sigma,\eta)\in\mathbb R\times\mathbb R^d$ is invariant under the Lorentz transform $L$, $|\det L|=1$, and
\begin{equation}\label{Lorentz}
L{(\tau^2-|\xi|^2)^{\frac12}\choose 0}={\tau\choose \xi}.
\end{equation}
Let us first introduce two lemmas whose proof come later in this section.
\begin{lemma}\label{lem:reformation of J}
For $\eta_1$, $\eta_2\in\mathbb{R}^{d}$ and $\beta>\frac{1-d}{4}$, define
\begin{align}\label{eq:reformation of J}
J^{2\beta}(\eta_1,\eta_2)
:=
\int_{\mathbb{R}^{2d}}\frac{|\phi_s(|\eta_1|)\phi_s(|\eta_4|)-\eta_1\cdot\eta_4-s^2|^{2\beta}}{\phi_s(|\eta_3|)\phi_s(|\eta_4|)}\delta {\tau-\phi_s(|\eta_3|)-\phi_s(|\eta_4|)\choose \xi-\eta_3-\eta_4}\,\mathrm{d}\eta_3\mathrm{d}\eta_4,
\end{align}
where $\tau=\phi_s(|\eta_1|)+\phi_s(|\eta_2|)$ and $\xi=\eta_1+\eta_2$.
Then, we have
\[
J^{2\beta}(\eta_1,\eta_2)=(2\pi)^{\frac{d-1}{2}}\frac{\Gamma(\tfrac{d-1}{2}+2\beta)}{\Gamma(d-1+2\beta)}
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2).
\]
\end{lemma}
\begin{lemma}\label{lem:spherical rearrangement}
Let $\eta_1$, $\eta_2 \in\mathbb{R}^d$. Set
\[
\xi=\eta_1+\eta_2,\qquad\tau=\phi_s(|\eta_1|)+\phi_s(|\eta_2|)
\]
and $\eta\in\mathbb{R}^d$ satisfying
\[
2\phi_s(|\eta|)=(\tau^2-|\xi|^2)^{\frac12}.
\]
Then,
there exists $\omega_*\in\mathbb{S}^{d-1}$ depending only on $\eta_1$, $\eta_2$ and $|\eta|$ such that
\begin{align*}
{\phi_s(|\eta_1|)\choose-\eta_1}\cdot L{\phi_s(|\eta|)\choose\eta}-s^2&=|\eta|^2\left(1+\frac{\eta}{|\eta|}\cdot\omega_*\right).
\end{align*}
\end{lemma}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Let $u(t,x)=e^{it\phi_s(D)}f(x)$ and $v(t,x)=e^{it\phi_s(D)}g(x)$. By the expressions $\widetilde{u}(\tau,\xi)
=
2\pi\delta(\tau-\phi_s(|\xi|))\widehat{f}(\xi)$ and $\widetilde{\overline{v}}(\tau,\xi)
=
2\pi\delta(\tau+\phi_s(|\xi|))\overline{\widehat{g}}(-\xi)$, Plancherel's theorem, and appropriately relabeling the variables, one can deduce
\begin{align*}
&
(2\pi)^{3d-1}
\||\Box|^\beta (u\overline{v})\|_{L^2(\mathbb{R}^{d+1})}^2\\
&\quad=
(2\pi)^{-4}
\int_{\mathbb{R}^{d+1}}|\tau^2-|\xi|^2|^{2\beta}|\widetilde{u}*\widetilde{\overline{v}}(\xi,\tau)|^2\,\mathrm{d}\tau\mathrm{d}\xi\\
&\quad=
\int_{\mathbb R^{4d}}\int_{\mathbb R^{d+1}}
|\tau^2-|\xi|^2|^{2\beta}\widehat{f}(\eta_1)\overline{\widehat{g}(-\eta_2)}
\overline{\widehat{f}(\eta_3)}\widehat{g}(-\eta_4)\\
&\quad\qquad\times
\delta{\tau-\phi_s(|\eta_1|)+\phi_s(|\eta_2|)\choose \xi-\eta_1-\eta_2}
\delta{\tau-\phi_s(|\eta_3|)+\phi_s(|\eta_4|)\choose \xi-\eta_3-\eta_4}\,\mathrm{d}\tau\mathrm{d}\xi\mathrm{d}\eta_1\mathrm{d}\eta_2\mathrm{d}\eta_3\mathrm{d}\eta_4\\
&\quad=
2^{2\beta}
\int_{\mathbb{R}^{4d}}|\phi_s(|\eta_1|)\phi_s(|\eta_4|)-\eta_1\cdot\eta_4-s^2|^{2\beta}
\frac{F(\eta_1,\eta_2)\overline{F(\eta_3,\eta_4)}}{(\phi_s(|\eta_1|)\phi_s(|\eta_2|)\phi_s(|\eta_3|)\phi_s(|\eta_4|))^{\frac12}}\\
&\quad\qquad\times
\delta {\phi_s(|\eta_1|)+\phi_s(|\eta_2|)-\phi_s(|\eta_3|)-\phi_s(|\eta_4|)\choose\eta_1+\eta_2-\eta_3-\eta_4}
\,\mathrm{d}\eta_1\mathrm{d}\eta_2\mathrm{d}\eta_3\mathrm{d}\eta_4.
\end{align*}
Here, the change of variables; $(\eta_2,\eta_4)\mapsto(-\eta_4,-\eta_2)$ has been performed in the last step and
\[
F(\eta_1,\eta_2):=\widehat f(\eta_1)\widehat g(\eta_2)\phi_s(|\eta_1|)^\frac12\phi_s(|\eta_2|)^\frac12.
\]
If we define $\Psi=\Psi_s(\eta_1,\eta_2,\eta_3,\eta_4)=\left(\frac{\phi_s(|\eta_1|)\phi_s(|\eta_2|)}{\phi_s(|\eta_3|)\phi_s(|\eta_4|)}\right)^\frac12$, then by the arithmetic-geometric mean we have
\[
|F(\eta_1,\eta_2)F(\eta_3,\eta_4)|
\le\frac12\left(|F(\eta_1,\eta_2)|^2\Psi+|F(\eta_3,\eta_4)|^2\Psi^{-1}\right)
\]
so that
\begin{equation}\label{ineq:a/g mean}
\frac{|F(\eta_1,\eta_2)F(\eta_3,\eta_4)|}{(\phi_s(|\eta_1|)\phi_s(|\eta_2|)\phi_s(|\eta_3|)\phi_s(|\eta_4|))^{\frac12}}\le\frac12\left(\frac{|F(\eta_1,\eta_2)|^2}{\phi_s(|\eta_3|)\phi_s(|\eta_4|)}+\frac{|F(\eta_3,\eta_4)|^2}{\phi_s(|\eta_1|)\phi_s(|\eta_2|)}\right).
\end{equation}
The equality holds if and only if
\[
\phi_s(|\eta_1|)\phi_s(|\eta_2|)\widehat{f}(\eta_1)\widehat{g}(\eta_2)
=
\phi_s(|\eta_3|)\phi_s(|\eta_4|)\widehat{f}(\eta_3)\widehat{g}(\eta_4)
\]
almost everywhere on the support of the delta measure, which is satisfied by, for instance, $f=g=f_a$ with $a>0$ that is given by
\begin{equation}\label{extremisers}
\widehat{f_a}(\xi)
=
\frac{e^{-a\phi_s(|\xi|)}}{\phi_s(|\xi|)}.
\end{equation}
Therefore,
\begin{align*}
&\left(
(2\pi)^{-3d+1}2^{2\beta}
\right)^{-1}
\||\square|^\beta(u\overline{v})\|_{L^2(\mathbb R^{d+1})}^2\\
&\quad\leq
\frac12
\left[
\int_{\mathbb R^{2d}}F(\eta_1,\eta_2)J^{2\beta}(\eta_1,\eta_2)\,\mathrm{d}\eta_1\mathrm{d}\eta_2+\int_{\mathbb R^{2d}}F(\eta_2,\eta_1)J^{2\beta}(\eta_1,\eta_2)\,\mathrm{d}\eta_1\mathrm{d}\eta_2
\right],
\end{align*}
which implies \eqref{ineq:main} by applying Lemma \ref{lem:reformation of J}. One may note that the constant in \eqref{ineq:main} is sharp since we only apply the inequality \eqref{ineq:a/g mean} in the proof.
\end{proof}
We now prove the aforementioned lemmas.
\begin{proof}[Proof of Lemma \ref{lem:reformation of J}]
Let $\tau=\phi_s(|\eta_1|)+\phi_s(|\eta_2|)$ and $\xi=\eta_1+\eta_2$. Recall the Lorentz transform $L$.
The change of variables ${\sigma_j\choose\eta_j}\mapsto L{\sigma_j\choose\eta_j}$ for $j=3,4$ gives
\begin{align*}
J^{2\beta}(\eta_1,\eta_2)
&=
\int_{\mathbb{R}^{2(d+1)}}\left|{\phi_s(|\eta_1|)\choose-\eta_1}\cdot{\sigma_4\choose\eta_4}-s^2\right|^{2\beta}\\
&\qquad\times
\frac{\delta(\sigma_3-\phi_s(|\eta_3|))}{\phi_s(|\eta_3|)}\frac{\delta(\sigma_4-\phi_s(|\eta_4|))}{\phi_s(|\eta_4|)}\delta{\tau-\sigma_3-\sigma_4\choose\xi-\eta_3-\eta_4}\,\mathrm{d}\sigma_3\mathrm{d}\sigma_4\mathrm{d}\eta_3\mathrm{d}\eta_4\nonumber\\
&=
\int_{\mathbb{R}^{2(d+1)}}\left|{\phi_s(|\eta_1|)\choose-\eta_1}\cdot L{\sigma_4\choose\eta_4}-s^2\right|^{2\beta}\\
&\qquad\times
\frac{\delta(\sigma_3-\phi_s(|\eta_3|))}{\phi_s(|\eta_3|)}\frac{\delta(\sigma_4-\phi_s(|\eta_4|))}{\phi_s(|\eta_4|)}\delta{(\tau^2-|\xi|^2)^\frac12-\sigma_3-\sigma_4\choose \eta_3+\eta_4}\,\mathrm{d}\sigma_3\mathrm{d}\sigma_4\mathrm{d}\eta_3\mathrm{d}\eta_4\nonumber\\
&=
\int_{\mathbb{R}^{d}}\left|{\phi_s(|\eta_1|)\choose-\eta_1}\cdot L{\phi_s(|\eta|)\choose\eta}-s^2\right|^{2\beta}
\frac{1}{\phi_s(|\eta|)^2}\delta(2\phi_s(|\eta|)-(\tau^2-|\xi|^2)^{\frac12})\,\mathrm{d}\eta.
\end{align*}
By Lemma \ref{lem:spherical rearrangement} and switching to polar coordinates,
\begin{align*}
J^{2\beta}(\eta_1,\eta_2)
&=
\left(\int_{\mathbb{S}^{d-1}}(1+\theta\cdot\omega_*)^{2\beta}\,\mathrm{d}\sigma(\theta)\right)
\left(
\int_0^\infty
\frac{r^{4\beta}}{\phi_s(r)^2}\delta(2\phi_s(r)-(\tau^2-|\xi|^2)^{\frac12})r^{d-1}\,\mathrm{d} r
\right).
\end{align*}
The first integral can be further simplified as
\begin{align*}
\int_{\mathbb{S}^{d-1}}\left(1+\theta\cdot\omega_*\right)^{2\beta}\,\mathrm{d}\sigma(\theta)
&=
|\mathbb S^{d-2}|\int_{-1}^1(1+\lambda)^{2\beta}(1-\lambda^2)^{\frac{d-3}{2}}\,\mathrm{d}\lambda\\
&=
2^{d-2+2\beta} |\mathbb{S}^{d-2}|B \left(\tfrac{d-1}{2}+2\beta,\tfrac{d-1}{2}\right)
\end{align*}
by using the beta function $B$.
For the remaining radial integration, one can perform the change of variables $2\phi_s(r)\mapsto\nu$ in order to get
\begin{align*}
\int_0^\infty\frac{r^{4\beta}}{\phi_s(r)^2}\delta(2\phi_s(r)-(\tau^2-|\xi|^2)^{\frac12})r^{d-1}\,\mathrm{d} r
&=
2^{-d+2-4\beta}\int_{4s^2}^\infty\frac{(\nu^2-4s^2)^{\frac{d-2}{2}+2\beta}}{\nu}\delta(\nu-(\tau^2-|\xi|^2)^\frac12)\,\mathrm{d}\nu\\
&=
2^{\frac{-d+1}{2}-2\beta}
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\end{align*}
and hence
\begin{align*}
J^{2\beta}(\eta_1,\eta_2)
=
2^{\frac{d-3}{2}} |\mathbb{S}^{d-2}|B (\tfrac{d-1}{2}+2\beta,\tfrac{d-1}{2})
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2).
\end{align*}
Finally, simplifying the constant by the formula
\[
B (\tfrac{d-1}{2}+2\beta,\tfrac{d-1}{2})=\frac{\Gamma(\tfrac{d-1}{2}+2\beta)\Gamma(\tfrac{d-1}{2})}{\Gamma(d-1+2\beta)},
\]
we are done.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:spherical rearrangement}]
Observe first that
\[
L{\phi_s(|\eta|)\choose\eta}=\frac12{\tau+\frac{\xi\cdot\eta}{\phi_s(|\eta|)}\choose2\eta+\xi(1+\frac{\xi\cdot\eta}{(\tau+2\phi_s(|\eta|))\phi_s(|\eta|)})}.
\]
Then, a direct calculation gives
\begin{align*}
{\phi_s(|\eta_1|)\choose-\eta_1}\cdot L{\phi_s(|\eta|)\choose\eta}=
(\phi_s(|\eta|))^2\left(1+\frac{\eta}{|\eta|}\cdot|\eta|{z}\right),
\end{align*}
where
\[
z
=
\frac{[\phi_s(|\eta|)+\phi_s(|\eta_1|)]\eta_2-[\phi_s(|\eta|)+\phi_s(|\eta_2|)]\eta_1}{\phi_s(|\eta|)^2[\phi_s(|\eta_1|)+\phi_s(|\eta_2|)+2\phi_s(|\eta|)]}.
\]
Since we have the relation $2\phi_s(|\eta|)=\phi_s(|\eta_1|)\phi_s(|\eta_2|)-\eta_1\cdot\eta_2+s^2$, the numerator of $z$ can be simplified by
\begin{align*}
&|[\phi_s(|\eta|)+\phi_s(|\eta_1|)]\eta_2-[\phi_s(|\eta|)+\phi_s(|\eta_2|)]\eta_1|^2\\
&\quad=[\phi_s(|\eta|)+\phi_s(|\eta_1|)]^2|\eta_2|^2+[\phi_s(|\eta|)+\phi_s(|\eta_2|)]^2|\eta_1|^2-2[\phi_s(|\eta|)+\phi_s(|\eta_1|)][\phi_s(|\eta|)+\phi_s(|\eta_2|)]\eta_2\cdot\eta_1 \\
&\quad=[\phi_s(|\eta|)+\phi_s(|\eta_1|)]^2\phi_s(|\eta_2|)^2+[\phi_s(|\eta|)+\phi_s(|\eta_2|)]^2\phi_s(|\eta_1|)^2\\
&\quad\qquad-2[\phi_s(|\eta|)+\phi_s(|\eta_2|)][\phi_s(|\eta|)+\phi_s(|\eta_1|)](\phi_s(|\eta_1|)\phi_s(|\eta_2|)-2\phi_s(|\eta|)^2)\\
&\quad\qquad\qquad-s^2([\phi_s(|\eta|)+\phi_s(|\eta_1|)]^2+[\phi_s(|\eta|)+\phi_s(|\eta_2|)]^2+2[\phi_s(|\eta|)+\phi_s(|\eta_1|)][\phi_s(|\eta|)+\phi_s(|\eta_2|)])\\
&\quad=\left(\phi_s(|\eta|)^2-s^2\right)[\phi_s(|\eta_1|)+\phi_s(|\eta_2|)+2\phi_s(|\eta|)]^2,
\end{align*}
and so it follows that
\[
|z|=\frac{|\eta|}{\phi_s(|\eta|)^2}.
\]
Therefore,
\begin{align*}
{\phi_s(|\eta_1|)\choose-\eta_1}\cdot L{\phi_s(|\eta|)\choose\eta}-s^2
=|\eta|^2\left(1+\frac{\eta}{|\eta|}\cdot\omega_*\right),
\end{align*}
where we have set $\omega_*=\frac{z}{|z|}$.
\end{proof}
\section{On estimate \eqref{ineq:biest D_-D_+ }}\label{sec:some observations wave regime}
As announced in the introduction, we focus on Corollary \ref{cor:biest radial} by assuming the radial symmetry on initial data. Firstly, we prove the claimed inequality \eqref{ineq:biest D_-D_+ } with the constant $C=\mathbf{F}(\beta,d)$ by applying Theorem \ref{thm:main}. Here, to deal with the complexity raised in the context of the Klein--Gordon equation, deriving a monotonicity property of the corresponding kernel is useful (Lemma \ref{lem:monotonicity}). Secondly, we show that there is, unfortunately, no way to conclude the same as Corollary \ref{cor:biest radial}, via Theorem \ref{thm:main}, for $\beta$ in the gap $(\frac{3-d}{4},\frac{5-d}{4})$. Thirdly, we observe a phenomenon that the radial symmetry on the initial data allows the inequality to hold with a wider range of $\beta$, namely, its lower bound from $\beta>\frac{3-d}{4}$ to, at least, $\beta\geq\frac{2-d}{4}$.
\subsection{Estimate \eqref{ineq:biest D_-D_+ } with explicit constant}
Here, we prove \eqref{ineq:biest D_-D_+ } for radially symmetric data $f$ and $g$ for $\beta\geq\frac{2-d}{4}$ and an explicit constant $C<\infty$; for $\beta=[\frac{2-d}{4},\frac{3-d}{4}]\cup[\frac{5-d}{4},\infty)$, this explicit constant coincides with $\mathbf{F}(\beta,d)^\frac12$. In order to complete the proof of Corollary \ref{cor:biest radial}, we need to show the sharpness of $\mathbf{F}(\beta,d)^\frac12$ for $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]\cup[\frac{5-d}{4},\infty)$, and the non-existence of extremisers; for these arguments, we refer the reader to Section \ref{sec:sharpness of constants}.
\begin{lemma}\label{lem:monotonicity}
Let $a+b>-1$, $b>-1$ and $\kappa\in[0,1]$. Define
\[
h^{a,b}(\kappa)
:=
\int_{-1}^1(1-\kappa\lambda)^a(1-\lambda^2)^b\,\mathrm{d}\lambda.
\]
Then,
\[
\sup_{\kappa\in[0,1]}h^{a,b}(\kappa)<\infty.
\]
Moreover, for $a\in(-\infty,0]\cup[1,\infty)$
\[
\sup_{\kappa\in[0,1]}h^{a,b}(\kappa)
=
h^{a,b}(1)
=
2^{a+2b+1}B(a+b+1,b+1).
\]
\end{lemma}
\begin{proof
By the Lebesgue dominated convergence theorem,
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}\kappa}h^{a,b}(\kappa)&=-a\kappa\int_{-1}^1(1-\kappa\lambda)^{a-1}\lambda(1-\lambda^2)\,\mathrm{d}\lambda\\
&=a\kappa\int_0^1\left((1+\kappa\lambda)^{a-1}-(1-\kappa\lambda)^{a-1}\right)\lambda(1-\lambda^2)^b\,\mathrm{d}\lambda
\end{align*}
Thus,
\[
\begin{cases}
\frac{\mathrm{d}}{\mathrm{d}\kappa}h^{a,b}(\kappa)\ge0\qquad&\text{if $a\in(-\infty,0]\cup[1,\infty)$},\\
\frac{\mathrm{d}}{\mathrm{d}\kappa}h^{a,b}(\kappa)<0\qquad&\text{if $a\in(0,1)$}.
\end{cases}
\]
For $a\in(-\infty,0]\cup[1,\infty)$,
\[
\sup_{\kappa\in[0,1]}h^{a,b}(\kappa)
=
h^{a,b}(1)=\int_{-1}^1(1-\lambda)^a(1-\lambda^2)^b\,\mathrm{d}\lambda
\]
and the change of variables $1+\lambda\mapsto2\lambda$ gives
\[
\int_{-1}^1(1-\lambda)^a(1-\lambda^2)^b\,\mathrm{d}\lambda=2^{a+2b+1}B(a+b+1,b+1)<\infty
\]
if $a+b>0$ and $b>-1$. Similarly, for $a\in(0,1)$,
\[
h^{a,b}(\kappa)\le h^{a,b}(0)=2^{2b+1}B(b+1,b+1)<\infty
\]
if $b>-1$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:biest radial}]
Let $f$, $g$ be radially symmetric. By Theorem \ref{thm:main}, we have
\begin{align}\label{i:radial theorem}
&\||\square|^{\beta}(e^{it\phi_s(\sqrt{-\Delta})}f\overline{e^{it\phi_s(\sqrt{-\Delta})}g})\|_{L^2(\mathbb{R}^{d+1})}^2\nonumber\\
&\quad\le \mathbf{KG}(\beta,d)
\int_0^\infty\!\int_0^\infty
|\widehat{f}(r_1)|^2|\widehat{g}(r_2)|^2
\phi_s(r_1)^{\frac{d-1}{2}+2\beta}\phi_s(r_2)^{\frac{d-1}{2}+2\beta}\Theta_\frac12^{\frac{d-2}{2}+2\beta}(r_1,r_2)r_1^{d-1}r_2^{d-1}\,\mathrm{d} r_1\mathrm{d} r_2,
\end{align}
where
\[
\Theta_a^b(r_1,r_2)
:=
\int_{(\mathbb{S}^{d-1})^2}
\frac{
\left(1-\frac{r_1r_2\theta_1\cdot\theta_2}{\phi_s(r_1)\phi_s(r_2)}-\frac{s^2}{\phi_s(r_1)\phi_s(r_2)}\right)^b
}{
\left(1-\frac{r_1r_2\theta_1\cdot\theta_2}{\phi_s(r_1)\phi_s(r_2)}+\frac{s^2}{\phi_s(r_1)\phi_s(r_2)}\right)^a
}
\,\mathrm{d}\sigma(\theta_1)\mathrm{d}\sigma(\theta_2).
\]
We divide the range of $\beta$ into $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]$ and $\beta\in[\frac{5-d}{4},\infty)$ and treat these cases differently. First, let us consider $\beta\in[\frac{5-d}{4},\infty)$ as the easier case. By applying the fundamental kernel estimate \eqref{ineq:kernelest wave}, we have
\begin{align*}
\Theta_\frac12^{\frac{d-2}{2}+2\beta}(r_1,r_2)
&\leq
\Theta_0^{\frac{d-3}{2}+2\beta}(r_1,r_2)=|\mathbb S^{d-1}||\mathbb S^{d-2}|h^{\frac{d-3}{2}+2\beta,\frac{d-3}{2}}(\kappa)
\end{align*}
with $\kappa=\frac{r_1r_2}{\phi_s(r_1)\phi_s(r_2)}$. Since $d-3+2\beta\geq1$, Lemma \ref{lem:monotonicity} implies that
\[
\sup_{\kappa\in[0,1)}h^{\frac{d-3}{2}+2\beta,\frac{d-3}{2}}(\kappa)
=
h^{\frac{d-3}{2}+2\beta,\frac{d-3}{2}}(1),
\]
and hence
\[
\sup_{r_1,r_2>0}
\Theta_\frac12^{\frac{d-2}{2}+2\beta}(r_1,r_2)\le 2^{\frac{3d-7}{2}+2\beta}|\mathbb{S}^{d-1}||\mathbb{S}^{d-2}|B\left(d-2+2\beta,\tfrac{d-1}{2}\right),
\]
which yields \eqref{ineq:biest D_-D_+ } with $C=\mathbf{F}(\beta,d)^\frac12$.\\
For $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]$, in which case $ \frac{d-2}{2}+2\beta\in[0,\frac12]$, the basic idea of our argument is the same as above but it requires a few more steps. Let
\[
\Xi(\nu,\upsilon)
:=
\int_{-1}^1
\frac{
(
1-\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda
)^{\frac{d-2}{2}+2\beta}
}{
(
1+\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda
)^\frac12
}
\,\mathrm{d}\mu(\lambda)
\]
with $\nu$ and $\upsilon$ satisfying
\[
\nu\in[0,1],\qquad \upsilon^2\leq1-\nu^2,
\]
and $\mathrm{d}\mu(\lambda)=(1-\lambda^2)^\frac{d-3}{2}\,\mathrm{d}\lambda$. Then, from \eqref{i:radial theorem}, it suffices to show
\begin{equation}\label{i:Xi Xi Xi(0,0)}
\Xi(\nu,\upsilon)\leq\Xi(0,\upsilon)\leq\Xi(0,0).
\end{equation}
In order to show the first inequality of \eqref{i:Xi Xi Xi(0,0)}, we establish monotonicity in $\nu \in \big[0,\sqrt{\frac{1-\upsilon^2}{2}}\big]$, and calculate directly for $\nu\in\big[\sqrt{\tfrac{1-\upsilon^2}{2}},\sqrt{1-\upsilon^2}\big]$.
Indeed, it simply follows that
\begin{align*}
\partial_\nu\Xi(\nu,\upsilon)&\leq
-
\left(\frac{d-2}{2}+2\beta\right)
\int_0^1
\frac{(1-\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda)^{\frac{d-4}{2}+2\beta}}{(1+\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda)^\frac12}\left(1-\frac{\nu}{\sqrt{1-\nu^2-\upsilon^2}}\lambda\right)\,\mathrm{d}\mu(\lambda)\\
&\qquad
-
\frac12
\int_0^1
\frac{(1-\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda)^{\frac{d-2}{2}+2\beta}}{(1+\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda)^\frac32}\left(1+\frac{\nu}{\sqrt{1-\nu^2-\upsilon^2}}\lambda\right)\,\mathrm{d}\mu(\lambda)\\
&\qquad\quad
-
\left(\frac{d-2}{2}+2\beta\right)
\int_0^1
\frac{(1-\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda)^{\frac{d-4}{2}+2\beta}}{(1+\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda)^\frac12}\left(1+\frac{\nu}{\sqrt{1-\nu^2-\upsilon^2}}\lambda\right)\,\mathrm{d}\mu(\lambda)\\
&\qquad\qquad
-
\frac12
\int_0^1
\frac{(1-\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda)^{\frac{d-2}{2}+2\beta}}{(1+\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda)^\frac32}\left(1-\frac{\nu}{\sqrt{1-\nu^2-\upsilon^2}}\lambda\right)\,\mathrm{d}\mu(\lambda),
\end{align*}
which is non-positive since $\beta\geq\frac{2-d}{4}$ and
\[
1-\frac{\nu}{\sqrt{1-\nu^2-\upsilon^2}}\lambda\geq0
\]
for $\nu\in\big[0,\sqrt{\tfrac{1-\upsilon^2}{2}}\big]$. On the other hand, for $\nu\in\big[\sqrt{\tfrac{1-\upsilon^2}{2}},\sqrt{1-\upsilon^2}\big]$, which imposes $0\leq\sqrt{1-\nu^2-\upsilon^2}\leq\nu$, it follows that
\begin{align*}
\Xi(\nu,\upsilon)
&=
\int_0^1
\frac{
(
1-\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda
)^{\frac{d-2}{2}+2\beta}
}{
(
1+\nu-\sqrt{1-\nu^2-\upsilon^2}\lambda
)^\frac12
}
\,\mathrm{d}\mu(\lambda)
+
\int_0^1
\frac{
(
1-\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda
)^{\frac{d-2}{2}+2\beta}
}{
(
1+\nu+\sqrt{1-\nu^2-\upsilon^2}\lambda
)^\frac12
}
\,\mathrm{d}\mu(\lambda)\\
&\leq
\int_0^1 2 \,\mathrm{d}\mu(\lambda)\\
&
\leq
\int_0^1(1-\sqrt{1-\upsilon^2}\lambda)^{\frac{d-3}{2}+2\beta}\,\mathrm{d}\mu(\lambda)
+
\int_0^1(1+\sqrt{1-\upsilon^2}\lambda)^{\frac{d-3}{2}+2\beta}\,\mathrm{d}\mu(\lambda)\\
&=
\Xi(0,\upsilon).
\end{align*}
Here, the first inequality is justified as long as $\beta\geq\frac{2-d}{4}$ and the second inequality is given by the arithmetic-geometric mean:
\begin{align*}
\frac12
\left(
(1-\sqrt{1-\upsilon^2}\lambda)^{\frac{d-3}{2}+2\beta}
+
(1+\sqrt{1-\upsilon^2}\lambda)^{\frac{d-3}{2}+2\beta}
\right)
\geq
\left(
1-(1-\upsilon^2)\lambda^2
\right)^{\frac{d-3}{4}+\beta}
\geq1.
\end{align*}
Since the second inequality of \eqref{i:Xi Xi Xi(0,0)} can be readily proved by Lemma \ref{lem:monotonicity}, we have \eqref{ineq:biest D_-D_+ } with $C=\mathbf{F}(\beta,d)^\frac12$ for $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]$ as well.
\end{proof}
\subsection{Threshold of our argument for $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$}\label{sec:arg. on gap}
Although $C=\mathbf{F}(\beta,d)^\frac12$ will be shown to be optimal for $\beta\in[\frac{2-d}{4},\frac{3-d}{4}]\cup[\frac{5-d}{4},\infty)$ in the case of radial data, it remains unclear whether this continues to be true for $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$; here we establish that there is no way to obtain the constant $\mathbf{F}(\beta,d)^\frac12$ if one first makes use of Theorem \ref{thm:main}.
In order to show that, we shall invoke the following useful result for the beta function due to Agarwal--Barnett--Dragmir \cite{ABD00}:
\begin{lemma}[\cite{ABD00}]\label{lem:B<B}
Let $m$, $p$ and $k\in\mathbb R$ satisfy $m$, $p>0$, and $p>k>-m$. If we have
\begin{equation}\label{cond:B<B}
k(p-m-k)>0
\end{equation}
then
\[
B(p,m)>B(p-k,m+k)
\]
holds.
\end{lemma}
\begin{figure}[b]
\begin{center}
\begin{tikzpicture}[scale=1]
\draw [color=blue, fill=blue!30!,opacity=0.5, thick] (3,0.3)--(3.4,0.3)--(3.4,0.5)--(3,0.5)--(3,0.3);
\draw [->](-0.5,0)--(4,0);
\draw [->](0,-0.5)--(0,4);
\draw [color=red](0.4,4) .. controls (0.4,0.4) ..(4,0.4);
\draw [color=blue,dotted](3,0)--(3,4);
\draw [color=blue,dotted](3.4,0)--(3.4,4);
\draw [color=blue,dotted](0,0.5)--(4,0.5);
\draw [color=blue,dotted](0,0.3)--(4,0.3);
\node at (-0.4,-0.4) {$O$};
\node [right] at (4,0) {$r_1$};
\node [left] at (0,4) {$r_2$};
\node [below] at (3.2,0) {$\sim\frac 1\delta$};
\node [left] at (0,0.4) {$\sim\delta$};
\node [left] at (3,0.8) {$\textcolor{blue}{\mathcal O_\delta}$};
\end{tikzpicture}
\caption{The set $\mathcal O_\delta$ along the curve $r_1=r_2^{-1}$.} \label{fig:counter example}
\end{center}
\end{figure}
With this in hand, we establish the following.
\begin{proposition}\label{prop:a counter example}
Let $d\ge2$ and $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$. Then there exist radially symmetric $f$ and $g$ such that
\begin{align*}
&\mathbf{KG}(\beta,d)
\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2\\
&\quad>
\mathbf{F}(\beta,d)
\|\phi_s(D)^{\frac{d-1}{4}+\beta}f\|_{L^2(\mathbb{R}^d)}^2
\|\phi_s(D)^{\frac {d-1}{4}+\beta}g\|_{L^2(\mathbb{R}^d)}^2
\end{align*}
holds.
\end{proposition}
\begin{proof
Let $0<\delta\ll1$ and
\[
A=\left\{\xi\in\mathbb{R}^d\,:\, \frac12<|\xi|<2\right\}.
\]
Define $f_A$ and $g_A$ so that for $\xi\in\mathbb{R}^d$
\[
\widehat{f_A}(\xi)=\chi_A(\tfrac\xi\delta)\quad\text{and}\quad\widehat{g_A}(\xi)=\chi_A(\delta\xi),
\]
where $\chi_A$ is the characteristic function of $A$. By use of polar coordinates,
\begin{align*}
&\int_{\mathbb{R}^{2d}}
|\widehat{f_A}(\eta_1)|^2|\widehat{g_A}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2\\
&\qquad=
\int_{\mathcal{O}_\delta}|\widehat{f_A}(r_1)|^2|\widehat{g_A}(r_2)|^2(\phi_s(r_1)\phi_s(r_2))^{\frac{d-1}{2}+2\beta}\Theta_\frac12^{\frac{d-2}{2}+2\beta}(r_1,r_2)^{d-1}\,\mathrm{d} r_1\mathrm{d} r_2.
\end{align*}
Here, the set $\mathcal O_\delta$ is defined by (see also Figure \ref{fig:counter example})
\[
\mathcal{O}_\delta
=
\left\{
(r_1,r_2):\frac{1}{2\delta}<r_1<\frac{2}{\delta},\frac{\delta}{2}<r_2<2\delta
\right\}.
\]
Now, for $(r_1,r_2)\in\mathcal{O}_\delta$, by taking the limit $\delta\to0$ represented by $\phi_s(r_1)\to \infty$ and $\phi_s(r_2)\to s$, it follows that
\begin{align*}
\Theta_\frac12^{\frac{d-2}{2}+2\beta}(r_1,r_2)
\to
|\mathbb S^{d-1}|^2.
\end{align*}
Therefore, for sufficiently small $\delta>0$,
\begin{align*}
&\mathbf{KG}(\beta,d)
\int_{\mathbb{R}^{2d}}
|\widehat{f_A}(\eta_1)|^2|\widehat{g_A}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2\\
&\quad=
(2\pi)^{2d}
\mathbf{KG}(\beta,d)
\|\phi_s(\sqrt{-\Delta})^{\frac{d-1}{4}+\beta}f_A\|_{L^2(\mathbb{R}^d)}^2
\|\phi_s(\sqrt{-\Delta})^{\frac {d-1}{4}+\beta}g_A\|_{L^2(\mathbb{R}^d)}^2,
\end{align*}
and it is enough to show
\begin{align}\label{ineq:counterexample}
(2\pi)^{2d}\mathbf{KG}(\beta,d)
>
\mathbf{F}(\beta,d)
\end{align}
for $d\geq2$ and $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$. By the formula \eqref{eq:volume of d-sphere} and the definitions of constants, this can be simplified as
\[
B(\tfrac{3d-5}{4}+\beta,\tfrac{3d-3}{4}+\beta)>B(d-2+2\beta,\tfrac d2)
\]
which, if fact, follows from Lemma \ref{lem:B<B} by letting $p=\frac{3d-5}{4}+\beta$, $m=\frac{3d-3}{4}+\beta$ and $k=\frac{3-d}{4}-\beta$ for $d\geq2$ and $\beta\in (\frac{3-d}{4},\frac{5-d}{4})$. Note that for the specific triple $(p,m,k)$, the hypothesis \eqref{cond:B<B} is equivalent to considering $\beta$ from the gap $(\frac{3-d}{4},\frac{5-d}{4})$ when $d\geq2$.
\end{proof}
\subsection{Contributions of radial symmetry}
Here, we observe for general (not necessary radially symmetric) data $f$ and $g$ the inequality \eqref{ineq:biest D_-D_+ } holds only if $\beta\geq\frac{3-d}{4}$, in other words, the radial symmetry condition on $f$ and $g$ widens the range of the regularity parameter $\beta$. The proof is based on the Knapp type argument in \cite{FK00} where they proved $\beta_-\geq\frac{3-d}{4}$ is necessary for \eqref{ineq:null +-} to hold.
\begin{proposition}\label{prop:necessary condition}
Let $\beta<\frac{3-d}{4}$. For any $C_*>0$, there exists $f,g\in H^{\frac{d-1}{4}+\beta}(\mathbb{R}^d)$ such that
\begin{align}
&\||\square|^{\beta}(e^{it\phi_s(\sqrt{-\Delta})}f\overline{e^{it\phi_s(\sqrt{-\Delta})}g})\|_{L^2(\mathbb{R}^{d+1})}^2\\
&\quad>
C_*
\|\phi_s(\sqrt{-\Delta})^{\frac{d-1}{4}+\beta}f\|_{L^2(\mathbb{R}^d)}^2
\|\phi_s(\sqrt{-\Delta})^{\frac {d-1}{4}+\beta}g\|_{L^2(\mathbb{R}^d)}^2.\nonumber
\end{align}
\end{proposition}
\begin{proof
For $\eta_1\in\mathbb{R}^d$ (similarly, for $\eta_2\in\mathbb R^d$), we set indices $(1), \ldots, (d)$ to indicate components of vectors, namely, $\eta_1=(\eta_{1(1)},\ldots,\eta_{1(d)})$. Also, denote $\eta'_1=(\eta_{1(2)},\ldots,\eta_{1(d)})\in\mathbb{R}^{d-1}$ and $\eta''_1=(\eta_{1(3)},\ldots,\eta_{1(d)})\in\mathbb{R}^{d-2}$. Now, for large $L>0$, eventually sent to infinity, define sets $\mathfrak F$ and $\mathfrak G$ by
\[
\mathfrak F=\{\eta\in\mathbb{R}^d:L\le\eta_{(1)}\le 2L, 1\le\eta_{(2)}\le2,|\eta''|\le1\}
\]
and
\[
\mathfrak G=\{\eta\in\mathbb{R}^d:L\le\eta_{(1)}\le 2L, -1\le\eta_{(2)}\le-2,|\eta''|\le1\}.
\]
For such $f$ and $g$
\[
\left|
|\square|^\beta
(e^{it\phi_s(D)}f(x)\overline{e^{it\phi_s(D)}g(x)})
\right|
\sim
\left|
\int_\mathfrak F\int_\mathfrak G e^{i\Phi_s(x,t:\eta_1,\eta_2)}\mathcal K_{-\beta}^0(\eta_1,\eta_2)\,\mathrm{d}\eta_1\mathrm{d}\eta_2
\right|,
\]
where
\[
\Phi_s(x,t:\eta_1,\eta_2)=x\cdot(\eta_1-\eta_2)+t(\phi_s(|\eta_1|)-\phi_s(|\eta_2|)).
\]
Now, we follow the idea of Knapp's example to derive a lower bound. From the setting
we have $|\eta_1|\sim|\eta_2|\sim\phi_s(|\eta_1|)\sim\phi_s(|\eta_2|)\sim|\eta_1+\eta_2|\sim L$, $|\eta_{1(1)}-\eta_{2(1)}|\sim\theta\sim L^{-1}$, $|\phi_s(|\eta_1|)-\eta_{1(1)}|\sim|\eta_1'|^2|\eta_1|^{-1}\sim |\phi_s(|\eta_2|)-\eta_{2(1)}|\sim|\eta_2'|^2|\eta_2|^{-1}\sim L^{-1}$, and $|\eta_1'+\eta_2'|\lesssim1$ for $(\eta_1,\eta_2)\in \mathfrak F\times \mathfrak G$. Then, it follows that
\begin{align}\label{ineq:W beta pre-estimate}
(\phi_s(|\eta_1|)\phi_s(|\eta_2|))^2-(\eta_1\cdot\eta_2-s^2)^2
\sim
s^2|\eta_1+\eta_2|^2+|\eta_1|^2|\eta_2|^2\sin^2\theta
\sim
L^2
\end{align}
and hence
\[
\mathcal K_{-\beta}^0(\eta_1,\eta_2)
\sim
\left(
\frac{(\phi_s(|\eta_1|)\phi_s(|\eta_2|))^2-(\eta_1\cdot\eta_2-s^2)^2}{\phi_s(|\eta_1|)\phi_s(|\eta_2|)+\eta_1\cdot\eta_2+s^2}
\right)^\beta
\sim
1.
\]
Moreover, for the phase, then it follows that
\begin{align*}
&|\Phi_s(x,t:\eta_1,\eta_2)|\\
&\quad=
|t(\phi_s(|\eta_1|)-\eta_{1(1)}-\phi_s(|\eta_2|)+\eta_{2(1)})+(x_{(1)}+t)(\eta_{1(1)}-\eta_{2(1)})+x'\cdot(\eta_1'+\eta_2')|\\
&\quad\le
|t|L^{-1}+|x_{(1)}+t|L^{-1}+|x'|<\frac\pi3
\end{align*}
for $(x,t)=(x_{(1)},x',t)$ in a slab $\mathfrak R=[-L^{-1},L^{-1}]\times[-1,1]^{d-1}\times[-L,L]$ whose volume is the order of $1$. Hence,
\[
|\square|^\beta
(e^{it\phi_s(D)}f(x)\overline{e^{it\phi_s(D)}g(x)})
\gtrsim
|\mathfrak F||\mathfrak G|\chi_\mathfrak R(x,t)
\]
and so
\[
\||\square|^{\beta}
(e^{it\phi_s(D)}f(x)\overline{e^{it\phi_s(D)}g(x)})
\|_{L^2(\mathbb R^{d+1})}^2
\gtrsim
|\mathfrak F|^2|\mathfrak G|^2|\mathfrak R|\sim|\mathfrak F|^2|\mathfrak G|^2.
\]
On the other hand, we have
\[
\|\phi_s(\sqrt{-\Delta})^{\frac{d-1}{4}+\beta}f\|_{L^2(\mathbb{R}^d)}^2
\|\phi_s(\sqrt{-\Delta})^{\frac {d-1}{4}+\beta}g\|_{L^2(\mathbb{R}^d)}^2\lesssim L^{d-1+4\beta}|\mathfrak F||\mathfrak G|.
\]
Therefore, it is implied that
\[
|\mathfrak F|^2|\mathfrak G|^2\lesssim L^{d-1+4\beta}|\mathfrak F||\mathfrak G|.
\]
The fact $|\mathfrak F|\sim|\mathfrak G|\sim L$ and letting $L\to\infty$ result in the desired necessary condition
\[
\frac{3-d}{4}\le\beta.
\]
\end{proof}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[rotate around x=-60, rotate around y=0,rotate around z=-10,scale=0.45]
\fill [fill=lightgray] (10,1,0)--(10,2,0)--(18,2,0)--(18,1,0);
\fill [fill=lightgray] (10,-1,0)--(10,-2,0)--(18,-2,0)--(18,-1,0);
\fill [fill=red!30!,opacity=0.5] (10,1,1)--(10,2,1)--(18,2,1)--(18,2,-1)--(18,1,-1)--(10,1,-1);
\fill [fill=blue!30!,opacity=0.5] (10,-1,1)--(10,-2,1)--(10,-2,-1)--(18,-2,-1)--(18,-1,-1)--(18,-1,1);
\draw [->] (-1,0,0)--(22,0,0); \node [right] at (22,0,0) {$\eta_{(1)}$};
\draw [->] (0,-4,0)--(0,4,0); \node [above] at (0,4,0) {$\eta_{(2)}$};
\draw [->] (0,0,-3)--(0,0,3); \node [above] at (0,0,3) {$\eta''$};
\draw [dotted] (0,2,0)--(18,2,0);
\draw [dotted] (0,1,0)--(18,1,0);
\draw [dotted] (10,0,0)--(10,2,0);
\draw [dotted] (18,0,0)--(18,2,0)--(0,2,0);
\draw [dotted] (0,-2,0)--(18,-2,0);
\draw [dotted] (0,-1,0)--(18,-1,0);
\draw [dotted] (10,0,0)--(10,-2,0);
\draw [dotted] (18,0,0)--(18,-2,0)--(0,-2,0);
\draw (10,1,-1)--(10,2,-1)--(18,2,-1)--(18,1,-1)--(10,1,-1);
\draw (10,1,1)--(10,2,1)--(18,2,1)--(18,1,1)--(10,1,1);
\draw (10,1,-1)--(10,2,-1)--(10,2,1)--(10,1,1)--(10,1,-1);
\draw (18,1,-1)--(18,2,-1)--(18,2,1)--(18,1,1)--(18,1,-1);
\draw [thick] (18,1,-1)--(18,2,-1)--(18,2,1)--(18,1,1)--(18,1,-1);
\draw [thick] (18,1,-1)--(10,1,-1)--(10,1,1)--(18,1,1);
\draw [thick] (10,1,1)--(10,2,1)--(18,2,1);
\draw (10,-1,-1)--(10,-2,-1)--(18,-2,-1)--(18,-1,-1)--(10,-1,-1);
\draw (10,-1,1)--(10,-2,1)--(18,-2,1)--(18,-1,1)--(10,-1,1);
\draw (10,-1,-1)--(10,-2,-1)--(10,-2,1)--(10,-1,1)--(10,-1,-1);
\draw (18,-1,-1)--(18,-2,-1)--(18,-2,1)--(18,-1,1)--(18,-1,-1);
\draw [thick] (18,-1,-1)--(18,-2,-1)--(18,-2,1)--(18,-1,1)--(18,-1,-1);
\draw [thick] (18,-2,1)--(10,-2,1)--(10,-1,1)--(18,-1,1)--(18,-2,1);
\draw [thick] (10,-2,1)--(10,-2,-1)--(18,-2,-1);
\draw (0,0,0)--(16,1.5,-0.5);
\fill (16,1.5,-0.5) circle (2pt);
\draw (0,0,0)--(12,-1.3,0.5);
\fill (12,-1.3,0.5) circle (2pt);
\fill [fill=white] (5.6,0.1,0)--(6.4,0.1,0)--(6.4,-0.1,0)--(5.6,-0.1,0);
\draw (5,-0.6,0.3) to[out=20, in=-70] (5,0.35,0);
\node at (6,0,0) {$\theta$};
\node at (12,3,1) {$\textcolor{red}{\mathfrak F}$};
\node at (12,-3,-1) {$\textcolor{blue}{\mathfrak G}$};
\node[below] at (19,0,0) {$2L$};
\node [left] at (0,1,0) {$1$};
\node [left] at (0,2,0) {$2$};
\node [left] at (-0.2,0,-0.5) {$O$};
\end{tikzpicture}
\caption{The sets $\mathfrak F$ and $\mathfrak G$, which are sent away from the origin along $\eta_{(1)}$-axis.} \label{fig:setF and setG}
\end{center}
\end{figure}
\section{Sharpness of constants}\label{sec:sharpness of constants}
Let us begin with some supplemental discussions on the refined Strichartz estimates \eqref{ineq:Strichartz refined general J} and \eqref{ineq:Strichartz refined non-wave}. Then, we focus on completing our proof of Corollaries \ref{cor:biest radial}, \ref{cor:wave} and \ref{cor:non-wave} by proving that the stated constants are optimal and non-existence of extremisers. We achieve optimality of constants by considering the functions $f_a$ given by \eqref{extremisers}; this is a natural guess given that such functions are extremisers for \eqref{ineq:main}, as shown in our proof of Theorem \ref{thm:main}. Before proceeding, we introduce the following useful notation.
\[
\mathrm{L}_a(\beta)
:=
\int_{4as}^\infty
e^{-\rho}
\int_0^{(2a)^{-1}\sqrt{\rho^2-(4as)^2}}
\frac{(\rho^2-(2ar)^2-(4as)^2)^{\frac{d-2}{2}+2\beta}}{\rho^2-(2ar)^2}r^{d-1}
\,\mathrm{d} r\mathrm{d}\rho
\]
and
\[
\mathrm{R}_a(\beta,b(\beta))
:=
\left(
\int_{2as}^\infty e^{-\rho}\rho^{b(\beta)}(\rho^2-(2as)^2)^{\frac{d-2}{2}}\,\mathrm{d}\rho
\right)^2.
\]
\subsection{On the refined Strichartz estimates}
It is straightforward that the estimates \eqref{ineq:corollaries} with claimed constants in Corollaries \ref{cor:wave} and \ref{cor:non-wave} when $(\alpha,\beta)=(\frac12,\frac{3-d}{4})$ and $(\alpha,\beta)=(\frac12,\frac{2-d}{4})$ coincide with the results obtained by applying the kernel estimates \eqref{ineq:kernelest wave} and \eqref{ineq:kernelest non-wave} to \eqref{ineq:main}, respectively. To obtain the estimates \eqref{ineq:Strichartz refined general J} and \eqref{ineq:Strichartz refined non-wave}, we require the additional fact that
\[
\int_{\mathbb R^{2d}}f(x)f(y)x\cdot y\,\mathrm{d} x\mathrm{d} y\geq0.
\]
Indeed, in the wave regime, after we apply the kernel estimate \eqref{ineq:kernelest wave} to \eqref{ineq:main}, it follows that
\begin{align*}
&\int_{\mathbb R^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{f}(\eta_2)|^2
\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_0^1(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2\\
&\quad\leq
\int_{\mathbb R^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{f}(\eta_2)|^2
\phi_s(|\eta_1|)\phi_s(|\eta_2|)
(\phi_s(|\eta_1|)\phi_s(|\eta_2|)-s^2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2,
\end{align*}
which immediately yields \eqref{ineq:Strichartz refined general J}. Similarly, one can deduce \eqref{ineq:Strichartz refined non-wave} in the non-wave regime. Finally, the estimate \eqref{ineq:corollaries} with $C=\mathbf{F}(\frac{5-d}{4},d)^\frac12$ when $(\alpha,\beta)=(1,\frac{5-d}{4})$ is obtained by further estimating the kernel of \eqref{ineq:Strichartz refined general J} as
\[
\phi_s(|\eta_1|)\phi_s(|\eta_2|)-s^2
\leq
\phi_s(|\eta_1|)\phi_s(|\eta_2|).
\]
Again, we will see the sharpness of constants below. Of course, by a similar argument to the above, one can easily obtain the estimate \eqref{ineq:corollaries} with
\begin{equation}\label{const:non-wave}
C=\frac{2^{-d+1}\pi^{\frac{-d+2}{2}}}{s\Gamma(\frac{d+2}{2})}
\end{equation}
when $(\alpha,\beta)=(1,\frac{4-d}{4})$ from \eqref{ineq:Strichartz refined non-wave} in the non-wave regime, and it is natural to hope that the constant is still optimal. We do not, however, know whether or not the constant \eqref{const:non-wave} is optimal, which will become clear from the following argument on the sharpness of constants.\\
\subsection{Wave regime}
Recall $\beta_d=\max\{\frac{1-d}{4},\frac{2-d}{2}\}$. We shall consider \eqref{ineq:corollaries} with $(\alpha,\beta)=(\frac{d-1}{4}+\beta,\beta)$ for $\beta\in(\beta_d,\infty)$.
For $f_a$ given by \eqref{extremisers}, one can observe that
\[
\||\square|^\beta|e^{it\phi_s(D)}f_a|^2\|_{L^2(\mathbb R^{d+1})}^2
=
2^{\frac{-3d+7}{2}-2\beta}|\mathbb S^{d-1}|\mathbf{KG}(\beta,d)(2a)^{-2d+5-4\beta}
\mathrm{L}_a(\beta)
\]
and
\begin{equation}\label{Strichartz RHS D}
\|\phi_s(D)^{\frac{d-1}{4}+\beta}f_a\|_{L^2(\mathbb R^d)}^4
=
(2\pi)^{-2d}|\mathbb S^{d-1}|^2(2a)^{-3d+5-4\beta}\mathrm{R}_a(\beta,\tfrac{d-3}{2}+2\beta),
\end{equation}
and so it is enough to show
\begin{equation}\label{eq:LHS/RHStoF}
\lim_{a\to0}
\frac{\||\square|^\beta|e^{it\phi_s(D)}f_a|^2\|_{L^2(\mathbb R^{d+1})}^2}{\|\phi_s(D)^{\frac{d-1}{4}+\beta}f_a\|_{L^2(\mathbb R^d)}^4}
=
\lim_{a\to0}(2a)^d C(\beta,d)\frac{\mathrm{L}_a(\beta)}{\mathrm{R}_a(\beta,\tfrac{d-3}{2}+2\beta)}
=
\mathbf{F}(\beta,d),
\end{equation}
where
\[
C(\beta,d)
=
2^{-2(d-2)}\pi^{\frac{-d+1}{2}}\frac{\Gamma(\frac{d-1}{2}+2\beta)}{\Gamma(d-1+2\beta)}.
\]
Since we have, by appropriate change of variables,
\[
\mathrm{L}_a(\beta)
=
e^{-4as}
(2a)^{-d}
\int_0^\infty
e^{-\rho}\rho^{\frac32d-2+2\beta}(\rho+8as)^{\frac32d-2+2\beta}
\int_0^1
\frac{(1-\nu^2)^{d-2+2\beta}\nu^{d-1}}{(\rho+4as)^2(1-\nu^2)+(4as)^2\nu^2}
\,\mathrm{d}\nu\mathrm{d}\rho
\]
and
\[
\mathrm{R}_a(\beta,\tfrac{d-3}{2}+2\beta)
=
e^{-4as}
\left(
\int_0^\infty
e^{-\rho}
(\rho+2as)^{\frac{d-3}{2}+2\beta}
\rho^{\frac{d-2}{2}}(\rho+4as)^{\frac{d-2}{2}}
\,\mathrm{d}\rho
\right)^2,
\]
one may deduce
\[
\lim_{a\to0}
(2a)^d\frac{\mathrm{L}_a(\beta)}{\mathrm{R}_a(\beta,\tfrac{d-3}{2}+2\beta)}
=
\frac{\Gamma(3d-5+4\beta)B(d-2+2\beta,\frac d2))}{2\Gamma(\frac{3d-5}{2}+2\beta)^2},
\]
which leads to \eqref{eq:LHS/RHStoF}.
In order to show the constant $\mathbf{F}(\frac{5-d}{4},d)^\frac12$ is sharp in \eqref{ineq:Strichartz refined general J}, we apply a similar calculation. In particular, one may note that the right-hand side of \eqref{ineq:Strichartz refined general J} can be written as
\begin{equation}
(2\pi)^{-2d}|\mathbb S^{d-1}|^2(2a)^{-2d}[\mathrm{R}_a(\tfrac{5-d}{4},1)-(2as)^2\mathrm{R}_a(\tfrac{5-d}{4},0)],
\end{equation}
instead of \eqref{Strichartz RHS D}. One can also see the second term is negligible in the sense of the optimal constant since it vanishes while $a$ tends to $0$.
\subsection{Non-wave regime}\label{s:sharpness non-wave}
Let $f_a$ satisfy \eqref{extremisers}. Note that in the non-wave regime the right-hand side of \eqref{ineq:corollaries} for the pair $(\frac d4+\beta,\beta)$ is expressed as
\begin{equation}
\|\phi_s(D)^{\frac{d}{4}+\beta}f_a\|_{L^2(\mathbb R^d)}^4
=
(2\pi)^{-2d}|\mathbb S^{d-1}|^2(2a)^{-3d+4-4\beta}\mathrm{R}_a(\beta,\tfrac{d-2}{2}+2\beta).
\end{equation}
Then, as we have done above, reform $\mathrm{L}_a(\beta)$ and $\mathrm{R}_a(\beta,\tfrac{d-2}{2}+2\beta)$ as follows by some appropriate change of variables:
\[
\mathrm{L}_a(\beta)
=
e^{-4as}
(2a)^{\frac d2-4+2\beta}
\int_0^\infty
e^{-\rho}\rho^{\frac32d-2+2\beta}(\frac{\rho}{2a}+4s)^{\frac32d-2+2\beta}
\int_0^1
\frac{(1-\nu^2)^{d-2+2\beta}\nu^{d-1}}{(\frac{\rho}{2a}+2s)^2(1-\nu^2)+(2s)^2\nu^2}
\,\mathrm{d}\nu\mathrm{d}\rho
\]
and
\[
\mathrm{R}_a(\beta,\tfrac{d-2}{2}+2\beta)
=
e^{-4as}
(2a)^{2d-4+4\beta}
\left(
\int_0^\infty
e^{-\rho}
(\frac{\rho}{2a}+s)^{\frac{d-2}{2}+2\beta}
\rho^{\frac{d-2}{2}}(\frac{\rho}{2a}+2s)^{\frac{d-2}{2}}
\,\mathrm{d}\rho
\right)^2.
\]
First, we shall consider \eqref{ineq:corollaries} with $(\alpha,\beta)=(0, \frac{2-d}{4})$.
By a similar argument to the wave regime above, one can easily check that
\[
\lim_{a\to\infty}(2a)^{d+1}\frac{\mathrm{L}_a(\tfrac{2-d}{4})}{\mathrm{R}_a(\tfrac{2-d}{4},0)}=2^{d-3}s^{-1}
\]
holds, from which it follows that
\[
\lim_{a\to\infty}
\frac{\||\square|^{\frac{2-d}{4}}|e^{it\phi_s(D)}f_a|^2\|_{L^2(\mathbb R^{d+1})}^2}{\|\phi_s(D)^{\frac12}f_a\|_{L^2(\mathbb R^d)}^4}
=
\lim_{a\to\infty}(2a)^{d+1} C(\tfrac{2-d}{4},d)\frac{\mathrm{L}_a(\tfrac{2-d}{4})}{\mathrm{R}_a(\tfrac{2-d}{4},0)}
=
\frac{2^{-d+1}\pi^{\frac{-d+2}{2}}}{s\Gamma(\frac d2)}.
\]\\
We now turn to \eqref{ineq:Strichartz refined non-wave} ,where $(\alpha,\beta)=(1,\frac{4-d}{4})$, and the argument goes almost the same as above. In this case, we have
$$C(\tfrac{4-d}{4},d)
=
\frac{2^{-d+2}\pi^{\frac{-d+2}{2}}}{\Gamma(\frac{d+2}{2})},$$
\[
\mathrm{L}_a(\tfrac{4-d}{4})
=
e^{-4a}(2a)^{-2}
\int_0^\infty e^{-\rho}\rho^d\left(\frac{\rho}{2a}+4s\right)^d\left(\int_0^\infty\frac{(1-\nu^2)^\frac s2\nu^{d-1}}{(\frac{\rho}{2a}+2s)^2(1-\nu^2)+(2s)^2\nu^2}\,\mathrm{d}\nu\right)\,\mathrm{d}\rho
\]
and
\begin{align*}
&
((2a)^{d-2}e^{-4as})^{-1}
\left(
\mathrm{R}_a(\tfrac{4-d}{4},1)
-
(2as)^2
\mathrm{R}_a(\tfrac{4-d}{4},0)
\right)\\
&\quad=
\left(
\int_0^\infty e^{-\rho}(\rho+2as)\rho^{\frac{d-2}{2}}\left(\frac{\rho}{2a}+2s\right)^{\frac{d-2}{2}}\,\mathrm{d}\rho
\right)^2
-
\left(
(2as)
\int_0^\infty e^{-\rho}\rho^{\frac{d-2}{2}}\left(\frac{\rho}{2a}+2s\right)^{\frac{d-2}{2}}\,\mathrm{d}\rho
\right)^2\\
&\quad=
(2a)
\left(
\int_0^\infty e^{-\rho}\rho^{\frac{d-2}{2}}\left(\frac{\rho}{2a}+2s\right)^{\frac{d}{2}}\,\mathrm{d}\rho
\right)
\left(
\int_0^\infty e^{-\rho}\rho^{\frac{d}{2}}\left(\frac{\rho}{2a}+2s\right)^{\frac{d-2}{2}}\,\mathrm{d}\rho
\right).
\end{align*}
Hence, one can easily check that
\begin{align}\label{eq:LHS/RHStoJ}
&\lim_{a\to\infty}
\frac{\||\square|^{\frac{4-d}{4}}|e^{it\phi_s(D)}f_a|^2\|_{L^2(\mathbb R^{d+1})}^2}{\|\phi_s(D)f_a\|_{L^2(\mathbb R^d)}^4-(2as)^2\|\phi_s(D)^\frac12f_a\|_{L^2(\mathbb R^d)}^4}\nonumber\\
&\quad=
\lim_{a\to\infty}(2a)^{d+1} C(\tfrac{4-d}{4},d)\frac{\mathrm{L}_a(\tfrac{4-d}{4})}{\mathrm{R}_a(\tfrac{4-d}{4},1)-(2as)^2\mathrm{R}_a(\tfrac{4-d}{4},0)}\nonumber\\
&\quad=
\frac{2^{-d+2}\pi^{\frac{-d+2}{2}}\Gamma(d+1)\left(\int_0^\infty(1-\nu^2)^\frac d2\nu^{d-1}\,\mathrm{d}\nu\right)}{s\Gamma(\frac{d+2}{2})\Gamma(\frac d2)\Gamma(\frac d2+1)}\\
&\quad=
\frac{2^{-d+1}\pi^{\frac{-d+2}{2}}}{s\Gamma(\frac {d+2}{2})}\nonumber
\end{align}
by noticing $\int_0^\infty(1-\nu^2)^\frac d2\nu^{d-1}\,\mathrm{d}\nu=\frac12B(\frac d2+1,\frac d2)$.
In contrast to the wave regime, the squared right-hand side of \eqref{ineq:Strichartz refined general J} without the constant can be expressed as
\[
(2\pi)^{-2d}|\mathbb S^{d-1}|^2(2a)^{-2d}(\mathrm{R}_a(\tfrac{4-d}{4},1)-(2as)^2\mathrm{R}_a(\tfrac{4-d}{4},0)).
\]
Unlike the wave regime, however, $a$ is sent to $\infty$ (instead of $0$) to derive \eqref{eq:LHS/RHStoJ} and the second term of \eqref{ineq:Strichartz refined general J} does not vanish. Hence, we cannot follow the argument for the wave regime and do not know whether the constant \eqref{const:non-wave} is still optimal for \eqref{ineq:corollaries} when $(\alpha,\beta)=(1,\frac{4-d}{4})$.
\subsection{Non-existence of an extremiser}
Suppose there were non-trivial $f$ and $g$ that satisfy any of the statements in Corollary \ref{cor:non-wave} with equality.
From our proof via Theorem \ref{thm:main}, it would be required that
\begin{align*}
&\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d} \eta_1\mathrm{d} \eta_2\\
&\quad=
2^{-\frac12}s^{-1}
\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_0^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d} \eta_1\mathrm{d} \eta_2
\end{align*}
holds. Then,
\[
\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\left(
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
-
2^{-\frac12}s^{-1}
\mathcal K_0^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\right)
\,\mathrm{d} \eta_1\mathrm{d} \eta_2
=0
\]
would hold. Since $f$, $g$ are assumed to be non-trivial $\widehat{f}$, $\widehat{g}\not=0$ on some set $\mathfrak F\times \mathfrak G\subseteq\mathbb R^{2d}$ with $|\mathfrak F|$, $|\mathfrak G|>0$, it would be deduced that
\begin{equation}\label{eq:kernerls}
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
-
2^{-\frac12}s^{-1}
\mathcal K_0^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
=
0
\end{equation}
on $(\mathfrak F\times \mathfrak G)\setminus \mathfrak N$ where $\mathfrak N\subseteq\mathbb R^{2d}$ is a null set. However, \eqref{eq:kernerls} would hold only on the diagonal line $\{(\eta_1,\eta_2):\eta_1=\eta_2\}$ (the equality condition of \eqref{ineq:kernelest non-wave}), which is a null set and so is $\{(\eta_1,\eta_2):\eta_1=\eta_2\}\cap(\mathfrak F\times \mathfrak G)$. This is a contradiction. \\
For Corollary \ref{cor:biest radial}, Corollary \ref{cor:wave}, similar arguments above can be carried. In particular, for equality in the wave regime, the formula \eqref{eq:kernerls} might be replaced by
\[
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
-
\mathcal K_0^{\frac{d-3}{2}+2\beta}(\eta_1,\eta_2)
=
0
\]
on $(\mathfrak F\times \mathfrak G)\setminus \mathfrak N$, which would only occur when $s=0$ (the equality condition of \eqref{ineq:keyest wave}).
\section{Analogous results for $(++)$ case}\label{sec:analogous results ++}
Here we note the analogous versions of our main results in the $(++)$ case. Here we observe an interesting phenomenon; the null-form $|\square-(2s)^2|$ instead of $|\square|$ somehow fits into the estimate in this framework, which does not occur in similar discussions for the wave propagators in \cite{BJO16}.
\begin{theorem}\label{thm:main ++}
For $d\ge2$ and $\beta>\frac{3-2d}{4}$, we have the sharp estimate
\begin{align}\label{ineq:main ++}
&\||\square-(2s)^2|^\beta (e^{it\phi_s(D)}fe^{it\phi_s(D)}g)\|_{L^2(\mathbb{R}^{d+1})}^2\\
&\quad\le
\mathbf{KG}_{(++)}(\beta,d)
\int_{\mathbb{R}^{2d}}
|\widehat{f}(\eta_1)|^2|\widehat{g}(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)
\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)
\,\mathrm{d}\eta_1\mathrm{d}\eta_2,\nonumber
\end{align}
where
\[
\mathbf{KG}_{(++)} (\beta,d)
:=
2^{\frac{-5d+1}{2}+2\beta}\pi^{\frac{-5d+1}{2}}\frac{\Gamma(\tfrac{d-1}{2})}{\Gamma(d-1)}.
\]
Moreover, the constant $\mathbf{KG}_{(++)}(\beta,d)$ is sharp.
\end{theorem}
One may note that by invoking the Legendre duplication formula; $\Gamma(z)\Gamma(z+\tfrac12)=2^{1-2z}\pi^\frac12\Gamma(2z)$,
\[
\mathbf{KG}_{(++)}(\beta,d)=\frac{2^{\frac{-7d+5}{2}+2\beta}\pi^{\frac{-5d+2}{2}}}{\Gamma(\tfrac d2)}
\]
holds, which is the same constant introduced in \cite{BJO16}.
An extra symmetry in the $(++)$ case allows \eqref{ineq:main ++} a wider range of $\beta$ than $\beta>\frac{1-d}{4}$ for \eqref{ineq:main}. Indeed, the condition $\beta>\frac{1-d}{4}$ is no longer imposed because of the form of the sharp constant $\mathbf{KG}_{(++)}(\beta,d)$, and the alternative lower bound of $\beta$ emerges from the kernel $\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}$. To see this, let us consider an extremiser $f=g=f_1$ in \eqref{extremisers} and after applying polar coordinates to the right-hand side of \eqref{ineq:main ++} (without the constant) we obtain
\begin{align}\label{f:RHS}
&\iint
e^{-(\phi_s(r_1)+\phi_s(r_2))}(\phi_s(r_1)\phi_s(r_2))^{-1}(r_1r_2)^{d-1}\\
&\qquad\times
\int_{-1}^1
\frac{(\phi_s(r_1)\phi_s(r_2)-s^2-r_1r_2\lambda)^{\frac{d-2}{2}+2\beta}}{(\phi_s(r_1)\phi_s(r_2)+s^2-r_1r_2\lambda)^\frac12}(1-\lambda^2)^\frac{d-3}{2}\,\mathrm{d}\lambda\mathrm{d} r_1\mathrm{d} r_2.\nonumber
\end{align}
By \eqref{ineq:keyest non-wave}, one may observe that \eqref{f:RHS} is bounded (up to some constant) by
\[
s^{-3}
\iint
e^{-(\phi_s(r_1)+\phi_s(r_2))}(\phi_s(r_1)\phi_s(r_2))^{\frac{3d-4}{2}+2\beta}
\int_{-1}^1
\left(
1-\frac{r_1r_2}{\phi_s(r_1)\phi_s(r_2)-s^2}\lambda
\right)^{\frac{d-2}{2}+2\beta}
(1-\lambda^2)^\frac{d-3}{2}\,\mathrm{d}\lambda\mathrm{d} r_1\mathrm{d} r_2
\]
and finite whenever $\beta>\frac{3-2d}{4}$ by applying Lemma \ref{lem:monotonicity}.\\
In order to state the various results which follow from Theorem \ref{thm:main ++}, here we introduce the following constant for the wave regime
\[
\mathbf{F}_{(++)}(\beta,d)
=
2^{-1+4\beta}\pi^{\frac{-d+1}{2}}\frac{\Gamma(d-2+2\beta)}{\Gamma(\frac{3d-5}{2}+2\beta)}.
\]
\begin{corollary}\label{cor:biest radial ++}
Let $d\ge2$, $\beta\geq\frac{2-d}{4}$. Then, there exists a constant $C>0$ such that
\begin{align}\label{ineq:biest D_-D_+ ++}
\||\square-(2s)^2|^{\beta}(e^{it\phi_s(D)}f e^{it\phi_s(D)}g)\|_{L^2(\mathbb{R}^{d+1})}
\le
C
\|\phi_s(D)^{\frac{d-1}{4}+\beta}f\|_{L^2(\mathbb{R}^d)}
\|\phi_s(D)^{\frac {d-1}{4}+\beta}g\|_{L^2(\mathbb{R}^d)}
\end{align}
holds whenever $f$ and $g$ are radially symmetric.
Moreover, for $\beta\in[\frac{2-d}{4},\tfrac{3-d}{4}]\cup[\tfrac{5-d}{4},\infty)$, the optimal constant in \eqref{ineq:biest D_-D_+ ++} for radially symmetric $f$ and $g$ is $\mathbf{F}_{(++)}(\beta,d)^\frac12$, but there does not exist a non-trivial pair of functions $(f,g)$ that attains equality.
\end{corollary}
We remark that, following the argument in Section \ref{sec:arg. on gap}, once we apply Theorem \ref{thm:main ++} as a first step, it is not possible to obtain the constant $\mathbf{F}_{(++)}(\beta,d)^\frac12$ for $\beta\in(\frac{3-d}{4},\frac{5-d}{4})$.
\begin{corollary}\label{cor:wave ++}
Let $d\ge2$.
\begin{enumerate}[(i)]
\item
The estimate
\begin{equation}\label{ineq:corollaries++}
\||\square-(2s)^2|^\beta (e^{it\phi_s(D)}f)^2\|_{L^2(\mathbb{R}^{d+1})}
\le
C
\|\phi_s(D)^\alpha f\|_{L^2(\mathbb R^d)}^2
\end{equation}
holds with the optimal constant $C=\mathbf{F}_{(++)}(\beta,d)^\frac12$
for $(\alpha,\beta)=(\frac12,\frac{3-d}{4})$ and $(\alpha,\beta)=(1,\frac{5-d}{4})$, but there are no extremisers. Furthermore, when $(\alpha,\beta)=(1,\frac{5-d}{4})$, we have the refined Strichartz estimate
\begin{align*}
\||\square-(2s)^2|^\frac{5-d}{4} (e^{it\phi_s(D)}f)^2\|_{L^2(\mathbb{R}^{d+1})}
\le
\mathbf{F}_{(++)}(\tfrac{5-d}{4},d)^\frac12
\left(
\|\phi_s(D)f\|_{L^2(\mathbb R^d)}^4
-s^2
\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^d)}^4
\right)^\frac12,
\end{align*}
where the constant is optimal and there are no extremisers.
\item
The estimate \eqref{ineq:corollaries++} holds with the optimal constant
$$
C
=
\left(
2^{1-d}\pi^{\frac{-d+1}{2}}\frac{\Gamma(\frac{d-1}{2})}{\Gamma(\frac d2)}s^{-1}
\right)^\frac12
$$
for $(\alpha,\beta)=(\frac12,\frac{2-d}{4})$, but there are no extremisers. Furthermore, when $(\alpha,\beta)=(1,\frac{4-d}{4})$, we have the refined Strichartz estimate
\begin{align*}
&\||\square-(2s)^2|^\frac{4-d}{4} (e^{it\phi_s(D)}f)^2\|_{L^2(\mathbb{R}^{d+1})}\\
&\quad\le
\left(
2^{2-d}\pi^{\frac{-d+1}{2}}\frac{\Gamma(\frac{d-1}{2})}{\Gamma(\frac{d+2}{2})}s^{-1}
\right)^\frac12
\left(
\|\phi_s(D)f\|_{L^2(\mathbb R^d)}^4
-s^2
\|\phi_s(D)^\frac12f\|_{L^2(\mathbb R^d)}^4
\right)^\frac12,
\end{align*}
where the constant is optimal and there are no extremisers.
\end{enumerate}
\end{corollary}
Theorem \ref{thm:main ++} follows from adapting the calculation by Jeavons \cite{Jv14} without any major difficulty. Indeed, he has computed by the Cauchy--Schwarz inequality that
\[
|\widetilde{(uv)}(\tau,\xi)|^2
\leq
\frac{J^0(\tau,\xi)}{(2\pi)^{2d-2}}\int_{(\mathbb R^d)^2}|F(\eta_1,\eta_2)|^2\delta{\tau-\phi_s(|\eta_1|)-\phi_s(|\eta_2|)\choose\xi-\eta_1-\eta_2}\,\mathrm{d}\eta_1\mathrm{d}\eta_2.
\]
Here, $u(t,x)=e^{it\phi_s(\sqrt{-\Delta})}f(x)$, $v(t,x)=e^{it\phi_s(\sqrt{-\Delta})}g(x)$, $$F(\eta_1,\eta_2)=\widehat f(\eta_1)\widehat g(\eta_2)\phi_s(|\eta_1|)^\frac12\phi_s(|\eta_2|)^\frac12,$$
and $J^\beta$ is given by \eqref{eq:reformation of J}.
In terms of the Lorentz invariant measure $\mathrm{d}\sigma_s(t,x)=\frac{\delta(t-\phi_s(|x|))}{\phi_s(|x|)}\,\mathrm{d} x\mathrm{d} t$, $J^0(\tau,\xi)$ is also written as
\[
J^0(\tau,\xi)
=
\sigma_s*\sigma_s(\tau,\xi).
\]
Invoking Lemma 1 in \cite{Jv14}, we have
\[
J^0(\tau,\xi)
=
\frac{|\mathbb S^{d-1}|}{2^{d-2}}\frac{(\tau^2-|\xi|^2-(2s)^2)^{\frac{d-3}{2}}}{(\tau^2-|\xi|^2)^\frac12}
\]
so that
\begin{align*}
&\||\square-(2s)^2|^\beta (e^{it\phi_s(D)}fe^{it\phi_s(D)}g)\|_{L^2(\mathbb{R}^{d+1})}^2\\
&\quad=
(2\pi)^{-(d+1)}\int_{\mathbb R^{d+1}}|\tau^2-|\xi|^2-(2s)^2|^{2\beta}|\widetilde{(uv)}(\tau,\xi)|^2\,\mathrm{d}\xi\mathrm{d}\tau\\
&\quad\leq
\frac{2^{\frac{-7d+3}{2}+2\beta}\pi^{-2d+1}}{\Gamma(\tfrac d2)}|\mathbb S^{d-1}|
\int_{(\mathbb R^d)^2}|\widehat f(\eta_1)|^2|\widehat g(\eta_2)|^2\phi_s(|\eta_1|)\phi_s(|\eta_2|)\mathcal K_\frac12^{\frac{d-2}{2}+2\beta}(\eta_1,\eta_2)\,\mathrm{d}\eta_1\mathrm{d}\eta_2,
\end{align*}
which is what we desired.\qed
|
1,108,101,565,512 | arxiv | \section{Introduction}
\label{sec:intro}
A comprehensive description for various phenomenon in different high-energy experiments \cite{Tawfik:2014eba,Tawfik:2010aq} can be given by extensive [such as Boltzmann-Gibbs (BG)] and nonextensive (such as Tsallis) statistics. In both statistical approaches, the thermodynamic consistency should be guaranteed. In proving that the fireballs or heavy resonances lead to a bootstrap approach, i.e. further fireballs consist of smaller fireballs and so on, extensive statistics was utilized by Hagedorn \cite{Hagedorn1965}. Assuming that the characteristic distribution-function gets variations from a possible symmetrical change, etc., nonextensive concept was first introduced for the particle productions \cite{Tawfik:2010uh}. The implementation of nonextensive Tsallis statistics was first introduced in Refs. \cite{ReFF1,ReFF2,Tsallis:1987eu,Prato:1999jj}, with a clear emphasize that the phase space plays an essential role. It has been argued that substituting the Boltzmann factor by $q$-exponential function with $q>1$ leads to a good agreement with the experimental results at high energies, especially the transverse momentum spectra. Is is widely accepted that the statistical fits of the the transverse momentum spectra characterize the kinetic freezeout, which is conjectured to take place later (in the sense that the system cools down) after the chemical freezeout \cite{Tawfik:2014eba}. Recently, Tawfik explained that this procedure is not necessarily fully incorporating the nonextensivity in the particle productions, even at the kinetic freezeout \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul,Cleymans:2011in,Azmi:2014dwa}. The long-range fluctuations, the correlations, and the interactions besides the possible modifications in the phase space of the particle production are not properly incorporated through Tsallis algebra. In long-range interactions, both thermodynamic and long-time limits do not commute. Therefore, generic nonextensive statistics (GNS) was introduced in Ref. \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul}, in which the phase space becomes responsible in determining the degree of (non)extensivity. It was shown that the lattice thermodynamics is well reproduced when the proposed GNS become characterized by extensive critical exponents ($1, 1$), while the heavy-ion particle ratios are only reproduced when the proposed GNS become nonextensive critical exponents, e.g. neither $0$ nor $1$. The latter differs from Tsallis \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul}.
Nonextensive statistics becomes the relevant approach for nonequilibrium stationary states. While zeroth law of thermodynamics in equilibrium introduces {\it "the temperature"}, a so-called {\it "physical temperature"} was proposed when utilizing Tsallis-algebra, see for instance \cite{AbeRef,Refeee1,Refeee2,Refeee3}. It was concluded that if the inverse Lagrange multiplier associated with constrained internal energy is regarded as {\it "the temperature"}, both Tsallis and Clausius entropies become identical. This temperature is believed to differ from the {\it "physical"} one. Based on this assumption, the {\it "physical temperature"} was conjectured to be extrapolated to the real {\it "temperature"}, as it is assumed that this quantity is identical to the freezeout temperature. More details shall be elaborated in section \ref{sec:extraTq}. It was concluded that, the assumption on inverse Lagrange multiplier \cite{AbeRef} follows the same path that nonextensivity. It is only fulfilled through modifications in the thermodynamic intensive and extensive quantities, such as internal energy and temperature. In the present work, we introduce a GNS approach, which assumes that the phase space determines the degree of (non)extensivity depending on two critical exponents defining equivalence classes $(c,d)$, while the intensive and extensive thermodynamic quantities shouldn't necessarily to be modified. To summarize, the ($c, d$)-temperature seems not distinguished from the $T$ parameter of Tsallis statistics.
In the present paper, generalized thermostatistics (GTT) is applied to investigate the chemical freezout. The main line of the critical remarks on Tsallis type statistics is related to the interpretation of the resulting temperature. The temperature is a well elaborated concept in GTT. Characterizing the statistical nature of the particle production is the main focus of the present work. As introduced, the proposed GNS approach is able to find out either the particle production is to be described by extensive or nonextensive statistics. This basically differs from an {\it ad hoc} implementation of BG and Tsallis approaches. In Ref. \cite{Tawfik:2016pwz}, one of the authors (AT) confronted calculations from the hadron resonance gas (HRG) model with GNS to the corresponding lattice QCD thermodynamics and to various particle ratios measured at $7.7$ and $200~$GeV. He concluded that the lattice QCD simulations are best reproduced when the scaling exponents ($c, d$) get the extensive values, i.e. both are unity, while the particle production seems to have nonextensive classes, i.e. both differ from unity. In other words, the proposed GNS approach seemed to {\it "suggest"} that the lattice calculations are extensive while the particle ratios are stemming from nonextensive processes. Let us first emphasize that this conclusion is fully correct, as the lattice QCD simulations are actually based on extensivity (additivity) of subsystems (quarks and glouns proceeded and communicated by CPUs). On the other hand, the nature of the particle productions in the relativistic heavy-ion collisions might be different. The present work is devoted to a detailed analysis of the particle production in a wide range of beam energies. Accordingly, the statistical approach whether extensive or nonextensive, which well reproduces the chemical freezeout parameters, shall be defined. For the sake of completeness, we highlight that the lattice freezeout parameters are found compatible to the ones deduced from extensive fits of HRG calculations to various particle ratios measured in a wide range of beam energies. This is a solid confirmation that the proposed statistical approach, GNS, works well for the particle productions.
The present paper is organized as follows. The proposed approaches shall be shortly discussed in section \ref{sec:app}. A reminder to GNS shall be given in section \ref{sec:gns}. Sections \ref{sec:cd} and \ref{sec:extraTq} shall be devoted to the characterizations of ($c, d$) nonoextensive-entropy and the extrapolation of $T_q$ to $T_{\mathrm{ch}}$, respectively. GNS in HRG model shall be detailed in section \ref{sec:Generic}. In section \ref{sec:fittingPR}, the statistical fits of various particle-ratios at different beam-energies shall be outlined. The freezeout parameters, the temperatures and the baryon chemical potentials, deduced from the statistical fits of GNS grand-canonical partition function to various particle-ratios measured in heavy-ion experiments shall be determined and compared with the results deduced from extensive HRG calculations. The resulting equivalence classes $(c, d)$ shall be discussed in \ref{sec:resultingcd}. Section \ref{sec:cncl} outlines the conclusions and the final remarks.
\section{Approaches}
\label{sec:app}
To explore the phase diagram of hot and dense nuclear-matters which can be estimated at the beam-energies
$3.8$, $4.2$, $5$, $6.4$, $7.7$, $8$, $9$, $11.5$, $12$, $17$, $19.6$, $27$, $39$, $62.4$, $130$, $200$, and $2760~$GeV play an essential role. It was assumed that this can be accomplished from the statistical fit of thermal models, such as HRG, in which GNS approach is implemented \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul} to various particle-ratios measured in relativistic heavy-ion collisions.
A detailed descriptions for the generic nonextensive statistics can be found at \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul}. In the section that follows we give a short reminder to the generic nonextensive statistics.
\subsection{Reminder to generic nonextensive statistics}
\label{sec:gns}
Shannon entropy ($S_1$) is uniquely characterized by four axioms, Khinchin axioms:
\begin{enumerate}
\item SK1, continuity: for any $n\in \mathbb{N}$, $S_1(p)$ is continuous with respect to $p\in\Delta_n$, i.e. the entropy depending on $p$, contentiously. Accordingly, function $g(p)$ becomes continuous, as well.
\item SK2, maximality: for given $n\in \mathbb{N}$ and for $(p_1,\cdots,p_n)\in\Delta_n$, $S_1(p_1,\cdots,p_n)$ reaches maximum at $p_i=1/n$, where $i=1,\cdots,n$. Consequently, for equi-distribution ($p$), the entropy becomes maximum and the function $g(p)$ become a concave function.
\item SK3, expandability: $S_1(p_{1},\cdots,p_{n},0) = S_1(p_{1},\cdots,p_{n})$, i.e. adding a non-zero probability state does not change the entropy so that $g(0)=0$.
\item SK4, generalized additivity: if $p_{ij} \geq 0$, $p_i=\sum_{j=1}^{m_i} p_{ij}$, and $\sum_{i=1}^{n} p_{i}=1$, where $i=1,\cdots,n$ and $j=1,\cdots,m_i$, then $S_1(p_{11},\cdots,p_{nm_n}) = S_1(p_{1},\cdots,p_{n}) + \sum_{i=1}^{n} p_i S_1\left(\frac{p_{i1}}{p_i}\cdots,\frac{p_{i m_i}}{p_i}\right)$. This means that the entropy of a system, which is divided into two subsystems $A$ and $B$, is given as $S_A$ plus the expectation value of $S_B$ conditional on $A$.
\end{enumerate}
Fulfilling the four axioms characterizes ($c, d$) extensive entropy, where the equivalence classes ($1, 1$) recall BG statistics. Violating the fourth axiom, individually, characterizes ($c, d$) nonextensive entropy. The latter can be Tsallis, where ($q, 0$), or GNS, where ($c, d$) take other values. We recall again that, the scaling exponents $c$ and $d$ are corresponding to two independent asymptotic properties of the entropy functional ($s$), namely \cite{Thurner1,Thurner2}
\begin{eqnarray}
z^c &=& \lim_{x\rightarrow 0} \frac{s(z x)}{s(x)}, \\
(1+a)^d &=& \lim_{x\rightarrow 0} \frac{s(x^{1+a})}{x^{a c}s(x)},
\end{eqnarray}
with $a$ and $z$ are arbitrary variables ranging between $0$ and $1$, but not affecting the (non)extensivity.
\subsection{Characterizations of ($c$,$d$) nonextensive entropy}
\label{sec:cd}
From {\it formed} processes, on which the phase space depends, the extensive entropy can be computed, where the phase space linearly increases with the increase in the system size ($N$). On the other hand, for {\it deformed} processes, one needs a general representation of the entropy functional ($s$). Generalized families of class-representation can be achieved through two-time differentiable, monotonically increasing functions. For instance, a generalization for Shannon entropy was proposed as $H(p)=\sum_{i=1}^{W} h(p_i)$, where $W$ gives the number of states of the phase-space processes \cite{Thurner1}. From the asymptotic properties of the trace-form entropy functionals, following representations have been proposed \cite{Thurner1}
\begin{enumerate}
\item $G$-representation: $G(p)=\sum_{i=1}^{W} g(p_i)$, where
\begin{eqnarray}
g(p_i) &\equiv & - p_i \Lambda(p_i), \label{eq:genG}
\end{eqnarray}
and $\Lambda$ is a generalized logarithm, see Eq. (\ref{eq:genLog}).
\item $S$-representation: From $-\int_0^{p_i} dy \log(y)=-p_i \log(p_i) + p_i$, the second term in right-hand side can be absorbed in $-\alpha(\sum_i p_1-1)$, when the normalization constrains of the maximum-entropy-principle is implemented. $\alpha$ is a Lagrangian multiplier. Accordingly, $h(p_i)=-p_i \log(p_i)$ and
\begin{eqnarray}
s(p_i) &=& -\int_0^{p_i} dx\, \Lambda(x). \label{eq:genH}
\end{eqnarray}
\end{enumerate}
Both representations should give equivalent ($c$,$d$) entropy classes
\begin{eqnarray}
s_{(c,d,r)}(x) &=& \frac{r}{c} A^{-d} \exp(A) \Gamma(d+1,A-c \log(x)) - r x,
\end{eqnarray}
where $A=c d r /(1-r(1-c))$. Correspondingly, both exponential and logarithm should be generalized. The earlier shall be introduced in Eq. (\ref{eq:epsln}), while the latter reads
\begin{eqnarray}
\Lambda_{(c,d,r)}(x) &=& r \left[1-x^{c-1}\left(1- \frac{1-r(1-c)}{d r } \log(x)\right)^d\right]. \label{eq:genLog}
\end{eqnarray}
Now we are left with the fifth property of ($c$,$d$) nonextensive entropy. This should be characterized by vanishing entropy property,
\begin{eqnarray}
0 &=& \int_{0}^{b_{(c,d,r)}} dx \Lambda_{(c,d,r)}(x), \label{eq:ve1}\\
S_{(c,d,r)}(p) &=& \sum_{i=1}^{W} s_{(c,d,r)}(b_{(c,d,r)} p_i), \label{eq:ve2}
\end{eqnarray}
where $b_{(c,d,r)}$ optimizes the asymptotic extensivity. When scaling $s_{(c,d,r)}$, even the representation given in Eq. (\ref{eq:genG}) fulfils vanishing-entropy property, as well.
To summarize, besides SK1, SK2, and SK3, the ($c$,$d$) nonextensive entropy is characterized by:
\begin{itemize}
\item trace-form of the entropy functional, $S(p)=\sum_{i=1}^W s(p_i)$, and
\item vanishing entropy property, Eqs. (\ref{eq:ve1})-(\ref{eq:ve2}).
\end{itemize}
\subsection{Extrapolation of $T_q$ to $T_{\mathrm{ch}}$}
\label{sec:extraTq}
An {\it ab initio} assumption was made that the Tsallis {\it freezeout} temperature can be extrapolated to the BG-temperature ($T_{\mathrm{ch}}$) deduced from extensive thermal-statistical models \cite{deppmn0,deppmn1},
\begin{eqnarray}
T_q &=& T_{\mathrm{ch}} + (q-1) k, \label{eq:Tq1}
\end{eqnarray}
where $q$ is Tsallis nonextensive parameter, i.e. $q>1$. $k$ is conjectured to depend on the energy transfer between a source and the surrounding. It was found that at $T_{\mathrm{ch}} =192\pm 15~$MeV, $k=-(950\pm10)~$MeV \cite{deppmn2}. As mentioned in section \ref{sec:intro}, The temperature deduced from the statistical fits of the transverse momentum distributions was interpreted as the kinetic freezeout temperature. To remain within the scope of the present paper, we leave this point without any further discussion.
In a classical ideal gas, the Tsallis entropy, which describes an exact entropy for microcanonical distribution of such gas, was proposed to relate Tsallis temperature, the $q$-temperature, ($T_q$) to the BG freezeout temperature ($T_{\mathrm{ch}}$) \cite{tamas1}
\begin{eqnarray}
T_q &=& T_{\mathrm{ch}} \, \exp\left(\frac{S_q}{C}\right), \label{eq:Tq2}
\end{eqnarray}
where $C$ is the heat capacity of reservoir system and $S_q$ is Tsallis-entropy.
To summarize, the $q$-temperature, which as discussed in section \ref{sec:intro} is know as {\it "physical temperature"}, is about $2-3$ times smaller than $T_{\mathrm{ch}}$ know as {\it "the temperature"} \cite{Deppmann2015}. Accordingly, the resulting freezeout phase-diagram largely differs from the one drawn from the extensive statistical fits of various measured particles-ratios and recent lattice QCD simulations, especially the freezeout temperature \cite{Deppmann2015}. Despite the well-know sign-problem in the lattice QCD calculations at finite chemical potential ($\mu_b$), where the Monte Carlo techniques likely fails, both deconfinement and freezeout temperatures are conjectured to be compatible with each other, especially at small chemical potentials \cite{Ding:2015ona}. This range of $\mu_b$ is to be related to RHIC and LHC energies, where precise measurements for different particle-ratios are available \cite{Tawfik:2013bza,Tawfik:2014dha}. What is obtained so-far is that at vanishing $\mu_b$ and at the corresponding $T_q$, the resulting HRG thermodynamics, even when applying Tsallis-algebra, considerably differs from the one based on the first-principle lattice calculations. This can be understood from the fact that the latter assume extensivity and additivity besides an overall thermal equilibrium. In light of this, it is apparent that the Tsallis-type nonextensivity shouldn't be utilized in order to reproduce of the lattice calculations. In this regards, we first remind with the remarkable success of HRG with BG statistics in reproducing the first-principle lattice thermodynamics \cite{Karsch:2003vd,Karsch:2003zq,Redlich:2004gp,Tawfik:2004vv,Tawfik:2004sw,Tawfik:2005gk,Tawfik:2005qh}. This is another argumentation of why the proposed GNS approach is the proper one. It reproduces the lattice calculations only when the equivalence classes ($c, d$) get extensive values, i.e. ($1,1$). The success in characterizing the statistical nature of the lattice calculations manifests the fact that the proposed approach (GNS) is generic and its applicability is much wider than that of BG and Tsallis.
As discussed, comprehensive works have been conducted in order to bring {\it "physical temperature"} closer to the {\it "the temperature"}, for instance Eqs. (\ref{eq:Tq1})-(\ref{eq:Tq2}). These works assume that the nonextensivity in the particle production can be characterized through a radical change in intensive or extensive thermodynamic quantities, such as the internal energy and the temperature. The resulting $T_q$, which is lower than $T_{\mathrm{ch}}$, are extrapolated. One of authors (AT) has first introduced GNS approach to the statistical fits of particle ratios at $200$ and $2760~$GeV \cite{Tawfik:2016pwz}. The present work covers a wider range of beam energies. Accordingly, the two critical exponents defining the equivalence classes $(c, d)$ can be determined. While both intensive or extensive thermodynamic quantities remain almost unchanged, $(c, d)$ unambiguously characterize the nonextensive nature of the particle production. In a future work, we shall study the energy dependence of $(c, d)$.
\subsection{GNS in HRG model}
\label{sec:Generic}
As introduced in Ref. \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul}, both extensive and nonextensive statistical properties can be determined by two scaling exponents defining equivalence classes $(c, d)$ for both correlated and uncorrelated systems, for instance. In its thermodynamic limit, the statistical nature of the system of interest can be characterized, unambiguously. The remarkable success of the thermal models (extensive) \cite{Tawfik:2014eba,Tawfik:2010aq} in describing the particle production and that of the Tsallis nonextensive algebra \cite{ReFF1,ReFF2} simply mean that the cost which should be paid, namely the radical reduction in the freezeout temperature, was not necessarily. Such an {\it ad hoc} implementation of statistical approaches is not the best way to decide whether the particle ratios or transverse momentum distributions or any other phenomenological spectra are extensive or nonextensive. A direct confrontation with GNS is this \cite{Tawfik:2015}. Here, the system is not enforced to be biased towards either extensive or nonextensive statistical description. With a generalized exponential function, Eq. (\ref{eq:epsln}),
\begin{eqnarray}
\ln\, Z(T) &=& \pm V\, \sum_i^{N_{\mathtt{M|B}}}\, \frac{g_i}{(2\, \pi)^3}\, \int_0^{\infty}\, \ln\left[1\pm\varepsilon_{c,d,r}(x_i)\right]\; d^3\, {\bf p}, \label{eq1}
\end{eqnarray}
where $V$ is the fireball volume and $x_i=[\mu_{i}-E_i({\bf p})]/T$ with $E_i(p)=\sqrt{{\bf p}^2+m_i^2}$ being the dispersion relation of $i$-th state (particle) and $\pm$ represent fermions and bosons, respectively.
The exponential function $\varepsilon_{c,d,r}(x_i)$ is generalized as \cite{Thurner1,Thurner2}
\begin{eqnarray}
\varepsilon_{(c,d,r)}(x) &=& \exp\left\{-\frac{d}{c-a}\left[W_k\left(B\left(1-x/r\right)^{1/d}\right)-W_k(B)\right]\right\}, \label{eq:epsln}
\end{eqnarray}
where $W_k$ is Lambert $W$-function which has real solutions at $k=0$ with $d\geq 0$ and at $k=1$ with $d<0$, and
\begin{equation}
B=\frac{(1-c)r}{1-(1-c)r} \exp\left[\frac{(1-c)r}{1-(1-c)r}\right],
\end{equation}
where $c$ and $d$ are two critical exponents defining equivalence classes for all interacting and noninteracting systems. They give estimations for two scaling functions with two asymptotic properties. $r$ is almost a free parameter. It is assumed not affecting the ($c$, $d$)-class.
\begin{itemize}
\item for $d>0$, $r<1/(1-c)$,
\item for $d=0$, $r=1/(1-c)$, and
\item for $d<0$, $r>1/(1-c)$.
\end{itemize}
Alternatively, a particular function for $r$, namely $r=(1-c+c\,d)^{-1}$ was proposed \cite{Thurner1,Thurner2}.
From the partition function, Eq. \eqref{eq1}, the thermodynamical properties such as pressure ($p$) and number density ($n$) \cite{RAFELSKI} can be derived,
\begin{eqnarray}
\textit{p} &=& \sum_{i=1}^{N_{\mathtt{M|B}}}\, \frac{g_i\, T}{2 \pi^2} \int_0^\infty \ln\left[1\pm\varepsilon_{c,d,r}(x_i)\right]\; \textbf{p}^2 d\textbf{p}, \label{FDp}\\
\textit{n} &=& \pm \sum_{i=1}^{N_{\mathtt{M|B}}}\, \frac{g_i}{2 \pi^2} \int_0^\infty \frac{\varepsilon_{c,d,r}(x_i) \; W_0\left[B(1-\frac{x_i}{r})^{\frac{1}{d}}\right]}{(1-c)\left[1\pm\varepsilon_{c,d,r}(x_i)\right] \left(r-x_i\right) \left(1+W_0\left[B(1-\frac{x_i}{r})^{\frac{1}{d}}\right]\right)}\; \textbf{p}^2 d\textbf{p}. \label{FDn1}
\end{eqnarray}
It should be remarked that in Eq. (\ref{eq1}), the logarithmic function should also be generalized. This was given in Eq. (\ref{eq:genLog}), when assuming $G$-representation.
In the present calculations, the possible decay channels of heavy resonances are taken into consideration. The $k$-th particle final number density (identical expressions for other thermodynamic quantities, such as pressure, entropy and energy density, can be deduced) is given as
\begin{eqnarray}
n_k^{final} &=&n_k +\sum_{l\neq k}^{N_{\mathtt{M|B}}}\; b_{l \rightarrow k}\, n_l, \label{eq:nn1}
\end{eqnarray}
where $b_{l \rightarrow k}$ is the effective branching ratio of $l$-th hadron resonance into the $k$-th particle of interest. It is noteworthy mentioning that the resonance decays might be seen as correlations and interactions among the produced particles and thus the introduction of nonextensivity to the particle production in final state becomes eligible.
\section{Results}
\label{sec:reslt}
\subsection{Chemical freezeout phase-diagram}
\label{sec:fittingPR}
The particle ratios measured by the E866, NA44, NA49, and NA57 experiments at the Superproton Synchrotron (SPS) \cite{Ahle:2000wq,Klay:2001tf,Ahle:1999uy,Andronic:2005yp} at
$3.8$, $4.2$, $5$, $6.4$, $8$, $9$, $12$, and $17~$GeV and by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC) at $7.7$, $11.5$, $19.6$, $27$, $39$, $130$, and $200~$GeV shall be compared with the HRG calculations. Together with the ALICE measurements, we construct a data set covering beam energies from $3.8$ up to $2760~$GeV.
For the same set of particle ratios, we fit our calculations based on GNS, section \ref{sec:Generic}, Eq. (\ref{FDn1}), to the experimental results. The free parameters are $T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$, besides the scaling exponents ($c$, $d$). There values are deduced where chemical freezeout conditions, such as $s/T^3=7$ \cite{Tawfik:2014eba,Tawfik:2016jzk,Tawfik:2015fda,Tawfik:2013eua,Tawfik:2013dba,Tawfik:2012si,Tawfik:2005qn,Tawfik:2004ss} is fulfilled and minimum $\chi^2$ is obtained, simultaneously. The results are shown in Figs. \ref{Fig1}, \ref{RHIC}, and \ref{SPS}.
Results on the statistical fits for various particle-ratios measured by ALICE at $2760~$GeV (LHC) [left panel (a)] and by STAR experiment at $200~$GeV (RHIC) [right panel (b)] are depicted in Fig. \ref{Fig1}. The resulting $T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$ are summarized in Tab. \ref{Tab1} and shall be depicted in Fig. \ref{Phase_diagram}. It is obvious that they are very compatible with the ones deduced from BG statistics \cite{Tawfik:2013bza}. But, the resulting exponents ($c$,$d$) for first and second nonextensive property, respectively, differ from BG ($1$, $1$) and Tsallis ($q$, $0$).
So-far, we conclude that the resulting $T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$ seem very compatible with the ones assuming extensive statistics (BG). Does this alone mean that the particle productions in heavy-ion collisions is based on extensive processes? The answer is apparently {\it "no"} \cite{Tawfik:2016pwz,Tawfik:2016jol,Tawfik:2017bul}. A key measure would be the reproduction of first-principle calculations, the lattice QCD simulations on thermodynamics, for instance. While HRG with BG statistics excellently models the lattice thermodynamics below the critical temperature and describes well the particle productions in heavy-ion collisions \cite{Karsch:2003vd,Karsch:2003zq,Redlich:2004gp,Tawfik:2004vv,Tawfik:2004sw,Tawfik:2005gk,Tawfik:2005qh}, the Tsallis nonextensivity doesn't do this job. Only, it is conjectured to characterize the kinetic freezeout and fitted well with the measured transverse-momentum distribution \cite{ReFF1,ReFF2}. How to unify these together, especially, when the Tsallis temperature is remarkably smaller than the freezeout temperature? First, let us recall that the Tsallis temperature fails to reproduce the lattice thermodynamics \cite{deppmn0,deppmn1}. The application of the GNS to lattice QCD thermodynamics and particle ratios which measured at different energies of RHIC was already done by one of the authors (AT) in Ref. \cite{1712.04807}. Second, the answer has been delivered, couple years ago. In previous sections, we discussed on this that, the Tsallis resulting temperature should be extrapolated to the freezeout one, $T_{\mathrm{ch}}$. The present work proposes another answer.
Figure \ref{Fig1} compares experimental results on various particle-ratios (circles), $\mathrm{\pi}^-/\mathrm{\pi}^+$, $\mathrm{K}^-/\mathrm{K}^+$, $\bar{\mathrm{p}}/\mathrm{p}$, $\bar{\mathrm{\Lambda}}/\mathrm{\Lambda}$, $\bar{\mathrm{\Omega}}/\mathrm{\Omega}$, $\bar{\mathrm{\Xi}}/\mathrm{\Xi}$, $\mathrm{K}^-/\mathrm{\pi}^-$, $\mathrm{K}^+/\mathrm{\pi}^+$, $\bar{\mathrm{p}}/\mathrm{\pi}^-$, $\mathrm{p}/\mathrm{\pi}^+$, $\mathrm{\Lambda}/\mathrm{\pi}^-$, $\mathrm{\Omega}/\mathrm{\pi}^-$, and $\bar{\mathrm{\Xi}}/\mathrm{\pi}^+$ measured by ALICE- and STAR-experiment at energies $2760~$GeV and $200~$GeV, left and right panel, respectively, are statistically fitted by means of the HRG model in which GNS is implemented. In our HRG calculations, all possible decay channels yielding the particles of interest and the related branching ratios are taken into account, Eq. (\ref{eq:nn1}). For the decay channels with not-yet-measured probabilities, the rules given in Ref. \cite{Andronic:2005yp} have been applied. But no finite-size correction was applied \cite{r6}. We sum up contributions from all hadron resonances listed in recent particle data group compilation with masses $\leq 2~$GeV. This refers to $388$ different states of mesons and baryons besides their anti-particles. For further details, interested readers can consult Ref. \cite{Tawfik:2014eba}. The number density can be derived from the partition function and accordingly the particle ratios, Eq. (\ref{FDn1}), can be determined. Other GNS fits are illustrated in Fig. \ref{RHIC} (RHIC) and Fig. \ref{SPS} (SPS).
In Fig. \ref{Phase_diagram}, the freezeout parameters, $T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$ (closed circles) as deduced from the statistical fits of the generic-nonextensive HRG are compared with the ones from the extensive HRG (dashed line). At a finite chemical potential, the freezeout temperature in both cases is determined at constant $s/T^3$, where $s$ is the entropy density \cite{Tawfik:2014eba}. Apparently, both approaches are very compatible with each other. In other words, there is almost no difference between the resulting {\it "physical temperature"} and the {\it "temperature"} ($T_{\mathrm{ch}}$). We also compare with other freezeout parameters (symbols) determined in different phenomenologies; Andronic {\it et al.} \cite{Andronic:2005yp}, Tawfik {\it et al.} \cite{Tawfik:2013bza,Tawfik:2014dha}, and UrQMD \cite{UrQMDppr}. With the latter we mean various simulations by hybrid UrQMD version $3.4$ at varying chemical potentials. At each value of the chemical potential, the simulated particle ratios are fitted by means of extensive HRG. In other words, UrQMD simulations - in this case - are taken as experimental results. To this end, it was shown that the UrQMD simulations agree well with measured particle-ratios. Further details can be taken from Ref. \cite{UrQMDppr}.
The resulting freezeout temperature and chemical potential are almost the same when extensive and GNS fits are applied. As discussed in earlier sections, it was widely believed that the nonextensivity in the particle production should be accompanied by a radical change in intensive or extensive thermodynamic quantities, such as the internal energy and the temperature. Therefore, the resulting temperature, for instance, was taken as a so-called {\it "physical"} one, which afterwards should be extrapolated to the freezeout temperature. The short-cuts of these assumptions shall be discussed in a forthcoming work. To remain within the scope of the present word, we report on nonextensive-statistical fit of various particle-ratios, where the intensive and extensive thermodynamic properties of the strongly interacting system remain almost unchanged, while the nonextensitivity is defined by the equivalence classes $(c, d)$. Their values are found close to unity, Tab. \ref{Tab1}. This refers to a generic nonextensivity, which is apparently not of Tsallis-type. The latter is characterized by ($q, 0$) with $q>1$.
It should be remarked that the excellent fits of various particle-ratios to the GNS statistical approach shouldn't be interpreted due to adding extra parameters ($c, d$). This should be related to rightly implementing generic statistical approach. Should the statistical nature of producing various particle-ratios were really Tsallis nonextensivity, the added extra parameters should be ($q, 0$). Should this were extensive, ($1, 1$) should be resulted in. On the other hand, the success of the GNS statistical approach in describing the lattice thermodynamics at ($1, 1$) confirms the capability of the proposed GNS to determine whether the system of interest has an extensive or a nonextensive statistical nature and apparently allows its implementation in other ambiguous processes.
\subsection{Equivalence classes $(c, d)$}
\label{sec:resultingcd}
Table \ref{Tab1} summarizes the freezeout parameters ($T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$) and the scaling exponents ($c$, $d$) as determined from the statistical fits of HRG with the proposed generic nonextensivity, GNS, to various experimental results. Both freezeout parameters are graphically illustrated in Fig. \ref{Phase_diagram} and compared with other results, section \ref{sec:fittingPR}.
The resulting classes ($c$, $d$) differ from unity referring to Lambert-$W$ exponentials characterizing the entropic equivalence classes. The generalized logarithm and exponential functions were given in Eq. (\ref{eq:genLog}) and (\ref{eq:epsln}), respectively. This means that the particle productions has a non-BG statistical nature. Over the whole energy-range, the entropic equivalence classes can be expressed as Lambert-$W$ exponentials, which is generic. Both resulting classes are positive but less than unity, except at LHC energy. We observe a general trend that the exponent $c$ is closer to unity than $d$. At $\sqrt{s}\approx 20~$GeV, $d$ reaches minimum, while $c$ nearly maximum values. The differences between the two exponents becomes great at low-SPS energies.
The maximum error in the exponent is only $10\%$. Taking into account a wide energy range, during which relevant microscopic physics radically changes, this is excellently precise.
Let us take $d$ as resulted, i.e. $d>0$, and assume that the resulting $c$ can be approximated to unity. Accordingly, the resulting entropy is characterized by stretched exponential,
\begin{eqnarray}
\varepsilon_{(1,d,r)}(x) &=& \exp\left\{-d\, r \left[\left(1-\frac{x}{r}\right)^{1/d} - 1\right]\right\}.
\end{eqnarray}
Mathematically, this function can be interpreted as a fractional power law. A {\it critical} exponent is inserted into an {\it ordinary} exponential function. Accordingly, in a disordered system, for instance, the particle production, this type of entropy characterizes a delayed relaxation.
\section{Conclusions and final remarks}
\label{sec:cncl}
In the present work, we have introduced a systematic study for (non)extensive properties characterizing relativistic heavy-ion collisions, which are assumed to hadronize and then i.e. forming hadrons or particles. With the nowadays detector technologies, the latter - in turn - can very precisely be detected. In doing this, we imply GNS on the well-known statistical-thermal models, such as the HRG model. It is assumed that the degree of (non)extensivity can be determined by the phase space characterizing the (dis)ordered system of interest. In other words, the proposed approach is able to determine whether the system of interest has extensive or nonextensive properties. Furthermore, more details about the nonextensivity can be also determined.
The particle ratios at energies ranging between $3.8$ and $2760~$GeV are best reproduced by GNS, where the equivalence classes ($c$,$d$) range between $\sim0.9$ and $\sim1$. This leads to a crucial conclusion that the statistics describing the particle production at a wide range of beam energies remarkably differ from extensive BG ($1, 1$) or nonextensive Tsallis ($q, 0$). It is certainly interesting to see how the economic approach (extensive BG), as it deals with smallest free parameters, works quite well. In fact, this is the main conclusion of this present work, either generic statistical approach with $(c,d)=(\sim1, \sim1)$ and the extensive BG approach reproduces well various particle ratios in a wide range of energies. Therefore, when comparing our results with the $q$-entropies, we can propose to refute all previous relevant works in the literature. Such a simplification will strengthen the discussion. The advantage of the present work is now simply to reassure that the particle ratios have extensive statistical nature but to illustrate that the proposed generic approach indeed manifests the degree of (non)extensivity depending on the resulting equivalence classes $(c, d)$. In a previous work, we have examined this with the lattice QCD thermodynamics, which is per definition assumes additivity (extensivity). Now, we present another examination with the particle ratios produced in heavy-ion collisions at various beam energies, which were ambiguously analyzed by thermal models assuming an {\it ac hoc} extensive statistics.
Furthermore, the resulting equivalence classes imply that the system of interest can best be described by an extended exponential entropy, which basically differs from the well-know extensive BG and from the well-know nonextensive Tsallis entropy. Both are very special cases defined by specific values of the equivalence classes $(1, 1)$ and $(q, 0)$, respectively. We conclude that the particle production, in terms of the produced particle-ratios, in relativistic heavy-ion collisions seems to be originated from a dynamically disordered system.
Our results propose a plausible interpretation why the resulting Tsallis freezeout-temperature, which is called {\it "physical temperature"}, differs from the resulting BG freezeout-temperature, which is called the {\it "temperature"}. We believe that the insist to imply Tsallis-type nonextensivity to the high-energy particle productions seems to be accompanied with a high price; a radical change in intensive and/or extensive thermodynamic quantities such as temperature and internal energy. Our results propose that the resulting freezeout parameters are compatible with the ones obtained when BG statistics is implied. Accordingly, the well-known freezeout phase-diagram excellently agrees with the recent lattice predictions and with the freezeout phase-diagram based on extensive BG statistics. The latter is very obvious, especially when comparing the equivalence classes obtained in our calculations and the ones characterizing BG. We conclude that the nonextensivity characterizing the high-energy particle productions is solely through two critical exponents defining equivalence classes $(c, d)$.
A few final remark on the excellent fits reported in this paper is now in order. First, as discussed, that the GNS partition function, Eq. (\ref{eq1}), was implemented, in which an additional pair of free parameters, ($c,d$), was added, shouldn't be thought as an explanation based on statistical precision, which likely increases with increasing the number of the free parameters. The other pair of free parameters, $T_{\mathrm{ch}}$ and $\mu_{\mathrm{b}}$, agrees well with the extensive fits, while the resulting ($c,d$) can be approximated to unity, i.e. almost extensive. Thus, the GNS partition function, Eq. (\ref{eq1}), with its four free parameters is not just a statistical precision. It manifests a generic statistical approach that seems to characterize various aspects for the particle productions. Because of its generic nature, both special cases, BG and Tsallis, are also included. Its equivalence classes $(c, d)$ define the degree of (non)extensivity.
It is obvious that the statistical physics approaches to the high-energy phenomena have to be understood as an economic description of the phenomena of interest in terms of a small number of variables including the thermodynamic variables. This is true as long as the statistical nature hasn't be changed. Tsallis statistics is not only introducing an additional parameter, $q$, but also an special concept of nonextensivity. In Euclidean field theory or lattice field theory, change of exponential factor implies change of quantum theory, which does not seem to be possible as the extensivity is fundamentally imposed. The present paper presents an interpretation to the high-energy phenomena, the production of various particle ratios, through suggesting changes of the foundations of statistical physics.
|
1,108,101,565,513 | arxiv | \section{Introduction}
Loop Quantum Gravity (LQG) is a candidate for background independent and non-perturbative theory of quantum gravity \cite{book,review,review1,rovelli2014covariant}. Among successful sub-areas in LQG, applying LQG to cosmology is a fruitful direction in which LQG gives physical predictions and phenomenological impacts. Most studies of cosmology in LQG is based on Loop Quantum Cosmology (LQC): a LQG-like quantization of symmetry reduced model (quantization of homogeneous and isotropic degrees of freedom) \cite{Ashtekar:2006wn,Bojowald:2001xe,Agullo:2016tjh}. However, in this paper, we apply the full theory of LQG (quantizing all degrees of freedom) to cosmology and present a top-down derivation of cosmological perturbation theory from LQG.
A key tool in our work is the new path integral formulation of LQG proposed in \cite{Han:2019vpw}. This path integral is derived from the reduced phase space formulation of canonical LQG. The reduced phase space formulation couples gravity to matter fields such as dusts or scalar fields (clock fields), followed by a deparametrization procedure, in which gravity Dirac observables are parametrized by values of clock fields, and constraints are solved classically. The dynamics of Dirac observables is governed by the physical Hamiltonian ${\bf H}_0$ generating physical time evolution (the physical time is the value of a clock field) in the reduced phase space. Our work considers two popular scenarios of deparametrization: coupling gravity to Brown-Kucha\v{r} and Gaussian dusts \cite{Giesel:2007wi,Giesel:2007wk,Giesel:2007wn,Giesel:2012rb}. The path integral formulation is derived from discretizing the theory on a cubic lattice $\gamma$, followed by quantizing the reduced phase space and the Hamiltonian evolution generated by ${\bf H}_0$. We refer the readers to \cite{Han:2019vpw} for detailed derivation of the path integral formulation, and to \cite{Han2020} for the comparison with spin foam formulation.
The semiclassical approximation $\hbar\to0$ of LQG can be studied in this path integral formulation using the stationary phase analysis. It is shown in \cite{Han2020} that semiclassical equations of motion (EOMs) from the path integral consistently reproduces the classical reduced phase space EOMs of the gravity-dust system. These semiclassical EOMs take into account all degrees of freedom (DOFs) on $\gamma$, and govern the semiclassical dynamics of the full LQG. In addition, \cite{Han:2019vpw} shows that semiclassical EOMs contain the unique solution satisfying the homogeneous and isotropic symmetry. The solution reproduces the effective dynamics of $\mu_0$-scheme LQC, i.e. it recovers the Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmology at low energy density while replacing the big-bang singularity by a bounce at high energy density.
In this work, we study perturbations on the homogeneous and isotropic cosmology in this path integral formulation of full LQG. We focus on the cosmological perturbation theory at the semiclassical level. The dynamics of perturbations are studied by taking the above homogeneous and isotropic as the background and linearizing semiclassical EOMs of the full LQG. The resulting linearized EOMs are in terms of (perturbative) holonomies and fluxes on the cubic lattice $\gamma$. The initial condition of EOMs is imposed by the semiclassical initial state of the path integral, and uniquely determines a solution. In practice, we solve these linearized EOMs numerically and extract the physics of cosmological perturbations. The perturbation theory developed here is manifestly gauge invariant because it is derived from the reduced phase space formulation.
There are cosmological perturbation theories based on LQC instead of the full LQG, including the dressed metric approach \cite{Agullo:2016hap,Ashtekar:2016pqn,Ashtekar:2020gec}, deformed algebra approach \cite{Bojowald2008,Cailleteau2011,Mielczarek2011,Mielczarek2010} and the hybrid model \cite{Gomar:2015oea,ElizagaNavascues:2016vqw}. In all those approaches, LQC quantum dynamics serves as the background for perturbations. However the dynamics of LQC is ambiguous by different treatments of Lorentzian terms in the Hamiltonian constraint. The ambiguity can have no nontrivial effects on predictions \cite{paramtalk,Li:2019qzr}. Our approach derives the cosmological perturbation theory from the full LQG Hamiltonian (proposed by Giesel and Thiemann \cite{Giesel:2007wn}) which specifies the Lorentzian term from the start. So ambiguities mentioned in \cite{paramtalk,Li:2019qzr} do not present in our approach.
As a consistency check, we take the continuum limit of linearized EOMs by refining the lattice $\gamma$, and find results agree with perturbative EOMs in \cite{Giesel:2007wk}, where the gauge-invariant cosmological perturbation theory is developed from classical gravity-dust theory on the continuum. Our result provides an example confirming the semiclassical consistency of the reduced phase space LQG. The cosmological perturbation theory from the reduced phase space formulation closely relates to the standard gauge-invariant treatment of cosmological perturbations \cite{Giesel:2007wk}.
Our top-down approach of the cosmological perturbation theory opens a new window for extracting physical predication from the full LQG and contacting with phenomenology. As the first step, we relate holonomy and flux perturbations to the standard decomposition into scalar, vector, and tensor modes, and numerically study their power spectrums determined by the semiclassical dynamics of LQG. Resulting power spectrums are compared with predictions from the classical theory on the continuum. This comparison demonstrates physical effects implied by the lattice discreteness and cosmic bounce in LQG.
Our analysis of power spectrums mainly focuses on scalar and tensor modes, since they have more phenomenological impact. Concretely, we study the power spectrum of the Bardeen potential $\Psi$ for the scalar mode perturbation (see Section \ref{Power Spectrum}), and the power spectrum of metric perturbations of the tensor mode (see Section \ref{Tensor Mode Perturbations}). Power spectrums are obtained by numerically evolving perturbations from certain initial conditions imposed at early time.
Firstly it is clear that predictions from LQG semiclassical EOMs are very different from the continuum classical theory in case that the wavelength is as short as the lattice spacing. However when we focus on wavelengths much longer than the lattice spacing, differences in power spectrums between LQG and the classical theory are much larger in the ultra-long wavelength regime than they are in the regime where the wavelength is relatively short (but still much longer than the lattice spacing). Power spectrums from LQG coincide with the classical theory in the regime where the wavelength is relatively short. At late time, this difference of scalar mode power spectrums becomes smaller, while the difference of tensor mode power spectrums becomes larger. For the tensor mode, the long wavelength correction from LQG in the power spectrum has a similar reason as in the dressed metric approach \cite{Agullo:2016hap,Ashtekar:2020gec}, i.e. it is due to the LQG correction to the cosmological background. For the Bardeen potential $\Psi$, the difference of power spectrums is resulting from $\Psi\sim \text{wavelength}\times\text{perturbation}$ where corrections to perturbations from the lattice discreteness are amplified by ultra-long wavelengths. Differences in power spectrums between LQG and the classical theory vanish in the lattice continuum limit. Some more discussions about comparison are given in Sections \ref{Power Spectrum} and \ref{Tensor Mode Perturbations}.
At late time, tensor mode perturbations from LQG give a wave equation of spin-2 gravitons with a modified dispersion relation $\o(k)^2=k^2[1+O(k^2)]$ (see Section \ref{Tensor Mode Perturbations} for the expression). $\o(k)^2$ reduces to the usual dispersion relation of graviton in the long wavelength limit or small $k$. For larger $k$, gravitons travel in a speed less than the speed of light. Our result confirms that spin-2 gravitons are low energy excitations of LQG. It is in agreement with a recent result from the spin foam formulation \cite{Han:2018fmu}. The modified dispersion relation is in agreement with a recent result in \cite{Dapor:2020jvc} obtained from expanding the LQG Hamiltonian on the flat spacetime. The modified dispersion relation indicates an apparently spurious mode at large $k$ (at the wavelength comparable to the lattice scale). But in our opinion, the large $k$ is beyond the regime to valid our effective theory, so the dispersion relation should only be trusted in the long wavelength regime (see Section \ref{Tensor Mode Perturbations} for discussion).
As another difference between LQG and the classical theory, the cosmological perturbation theory from LQG contain couplings among scalar, tensor, and vector modes, although these couplings are suppressed by the lattice continuum limit. For instance, the initial condition containing only scalar mode can excite tensor and vector modes in the time evolution at the discrete level. These tensor and vector modes have small amplitudes vanishing in the lattice continuum limit.
As a preliminary step toward making the full LQG theory contact with phenomenology, this work has following limitations: Firstly, our model focuses on pure gravity coupled to dusts, and does not take into account the radiative matter and inflation. However various matter couplings in the reduced phase space LQG have been worked out in \cite{Giesel:2007wn}. Deriving matter couplings in the path integral formulation is straight-forward. Generalizing the cosmological perturbation theory to including radiative matter and inflation is a work currently undergoing. Secondly, this work focuses on the semiclassical analysis, and does not take into account any $O(\ell_P^2)$ quantum correction (although effects from discreteness are discussed). By taking into account quantum corrections, the continuum limit at the quantum level is expected to be better understood.
Main computations in this work are carried out with Mathematica on High-Performance-Computing (HPC) servers. Some intermediate steps and results contain long formulae that cannot be shown in the paper. Mathematica codes and formulae can be downloaded from \cite{github}.
This paper is organized as follows: Section \ref{BK} reviews the reduced phase space formulation of LQG and the path integral formulation. Section \ref{Semiclassical Equations of Motion} discusses the semiclassical approximation of the path integral and semiclassical EOMs. Section \ref{Cosmological Background and Perturbations} discuss the cosmological solution, linearization of EOMs with cosmological perturbations, and lattice continuum limit. Section \ref{Power Spectrum} focuses on scalar mode perturbations, and discusses the initial condition and the power spectrum. Section \ref{Tensor Mode Perturbations} focuses on tensor mode perturbations, including discussions of the late time dispersion relation and the power spectrum.
\section{Reduced Phase Space Formulation of LQG}\label{BK}
\subsection{Classical Framework}\label{Classical Framework}
Reduced phase space formulations of LQG need to couple gravity to various matter fields at classical level. In this paper, we focus on two scenarios of matter field couplings: Brown-Kucha\v{r} (BK) dust and Gaussian dust \cite{Brown:1994py,Kuchar:1990vy,Giesel:2007wn,Giesel:2012rb}.
The action of BK dust model reads
\begin{eqnarray}
S_{BKD}[\rho,g_{\mu\nu},T,S^j,W_j]&=& -\frac{1}{2}\int\mathrm d^4x\ \sqrt{|\det(g)|}\ \rho\ [g^{\mu\nu}U_\mu U_\nu+1],\label{dustaction}\\
U_\mu&=&-\partial_\mu T+W_j\partial_\mu S^j,
\end{eqnarray}
where $T, S^{j=1,2,3}$ are scalars (dust coordinates of time and space) to parametrize physical fields, and $\rho,\ W_j$ are Lagrangian multipliers. $\rho$ is interpreted as the dust energy density. Coupling $S_{BKD}$ to gravity (or gravity coupled to some other matter fields) and carrying out Hamiltonian analysis \cite{Giesel:2012rb}, we obtain following constraints:
\begin{eqnarray}
\mathcal C^{tot}&=&\mathcal C+\frac{1}{2}\left[\frac{P^{2} / \rho}{\sqrt{\operatorname{det}(q)}}+\sqrt{\operatorname{det}(q)} \rho\left(q^{\a \b} U_{\a} U_{\b}+1\right)\right]=0,\label{C}\\
\mathcal C^{tot}_\a&=&\mathcal C_\a+PT_{,\a}-P_jS^j_{,\a}=0,\label{Ca}\\
\rho^2&=&\frac{P^2}{\det(q)}\left(1+q^{\a\b}U_\a U_\b\right)^{-1},\label{rhoP}\\
W_j&=&P_j/P,\label{WP}
\end{eqnarray}
where $\a,\b=1,2,3$ are spatial indices, $P,P_j$ are momenta conjugate to $T,S^j$, and $\mathcal C,\mathcal C_\a$ are Hamiltonian and diffeomorphism constraints of gravity (or gravity coupled to some other matter fields). Eq.\Ref{rhoP} is solved by
\begin{eqnarray}
\rho=\varepsilon\frac{P}{\sqrt{\det(q)}}\left(1+q^{\a\b}U_\a U_\b\right)^{-1/2}, \quad \varepsilon=\pm1.
\end{eqnarray}
The dust 4-velocity $U$ being timelike and future pointing fixes $\varepsilon=1$ \cite{Giesel:2007wi}, so $\mathrm{sgn}(P)=\mathrm{sgn}(\rho)$. Inserting this solution to Eq.\Ref{C} and using Eq.\Ref{WP} lead to
\begin{eqnarray}
\mathcal C=-P\sqrt{1+q^{\a\b}\mathcal C_\a \mathcal C_\b/P^2}.
\end{eqnarray}
Thus $-\mathrm{sgn}(\mathcal C)=\mathrm{sgn}(P)=\mathrm{sgn}(\rho)$. For dust coupling to pure gravity, we must have $\mathcal C<0$ and the physical dust $\rho,P>0$ to fulfill the energy condition \cite{Brown:1994py}. However, in presence of additional matter fields (e.g. scalars, fermions, gauge fields, etc), they can make $\mathcal C>0$ and $\rho,P<0$ corresponding to the phantom dust \cite{Giesel:2007wn,Giesel:2007wi}. The case of phantom dust may not violate the usual energy condition due to presence of other matter fields. We solve $P,P_j$ from Eqs.\Ref{C} and \Ref{Ca}:
\begin{eqnarray}
&&P=\begin{cases} h &\ \text{physical dust}, \\
-h &\ \text{phantom dust},
\end{cases} \quad h=\sqrt{\mathcal C^2-q^{\a\b}\mathcal C_\a \mathcal C_\b},\label{P=-h}\\
&&P_j=-S^\a_j\left(\mathcal C_\a-hT_{,\a}\right),\label{Pj=cc}
\end{eqnarray}
are strongly Poisson commutative constraints. $S^\a_j$ is the inverse matrix of $\partial_\a S^j$ ($\a=1,2,3$). An intermediate step of the above derivation shows that $P^2 = \mathcal C^2- q^{\a\b}\mathcal C_\a \mathcal C_\b>0$. It constrains the argument of the square root to be positive. Moreover the physical dust requires $\mathcal C<0$ while the phantom dust requires $\mathcal C>0$.
We use $A^a_\a(x),E^\a_a(x)$ as canonical variables of gravity. $A^a_\a(x)$ is the Ashtekar-Barbero connection and $E^\a_a(x)=\sqrt{\det q}\, e^\a_a(x)$ is the densitized triad. $a=1,2,3$ is the Lie algebra index of su(2). Gauge invariant Dirac observables are constructed relationally by parametrizing $(A,E)$ with values of dust fields $T(x)\equiv\t,S^j(x)\equiv\sigma^j$, i.e. $A_j^a(\sigma,\t)=A_j^a(x)|_{T(x)\equiv\t,\,S^j(x)\equiv\sigma^j}$ and $E^j_a(\sigma,\t)=E^j_a(x)|_{T(x)\equiv\t,\,S^j(x)\equiv\sigma^j}$, where $\sigma,\t$ are dust space and time coordinates, and $j=1,2,3$ is the dust coordinate index (e.g. $A_j=A_\a S^\a_j$).
$A_j^a(\sigma,\t)$ and $E^j_a(\sigma,\t)$ satisfy the standard Poisson bracket in the dust frame:
\begin{eqnarray}
\{E^i_a(\sigma,\t),A_j^b(\sigma',\t)\}=\frac{1}{2}\kappa \b\ \delta^{i}_j\delta^b_a\delta^{3}(\sigma,\sigma'),\quad \kappa=16\pi G
\end{eqnarray}
where $\b$ is the Barbero-Immirzi parameter. The phase space $\mathcal P$ of $A_j^a(\sigma,\t),E^j_a(\sigma,\t)$ is free of Hamiltonian and diffeomorphism constraints. All SU(2) gauge invariant phase space functions are Dirac observables.
Physical time evolution in $\t$ is generated by the physical Hamiltonian ${\bf H}_0$ given by integrating $h$ on the constant $T=\t$ slice $\mathcal S$. The constant $\t$ slice $\mathcal S$ is coordinated by the value of dust scalars $S^j=\sigma^j$ thus is called the dust space \cite{Giesel:2007wn,Giesel:2012rb}. By Eq.\Ref{P=-h}, ${\bf H}_0$ is negative (positive) for the physical (phantom) dust. We flip the direction of the time flow $\t\to -\t$ thus ${\bf H}_0 \to -{\bf H}_0$ for the physical dust. So we obtain positive Hamiltonians in both cases:
\begin{eqnarray}
{\bf H}_0=\int_\mathcal S\mathrm d^3\sigma\, \sqrt{\mathcal C(\sigma,\t)^2-\frac{1}{4}\sum_{a=1}^3\mathcal C_a(\sigma,\t)^2}.\label{bigH0}
\end{eqnarray}
$\mathcal C$ and $\mathcal C_a=2e_a^\a \mathcal C_\a$ are parametrized in the dust frame, and expressed in terms of $A_j^a(\sigma,\t)$ and $E^j_a(\sigma,\t)$:
\begin{eqnarray}
\mathcal C&=&\frac{1}{\kappa}\left[F^a_{jk}-\left({\b^2+1}\right)\varepsilon_{ade}K^d_{j}K^e_{k}\right]\varepsilon_{abc}\frac{E^j_bE^k_c}{\sqrt{\det(q)}}+\frac{2\L}{\kappa}\sqrt{\det(q)}\\
\mathcal C_a&=&\frac{4}{\kappa\b}F^b_{jk} \frac{E^j_aE^k_b}{\sqrt{\det(q)}},
\end{eqnarray}
where $\L$ is the cosmological constant.
Coupling gravity to Gaussian dust model is similar, so we don't present the details here (while details can be found in \cite{Giesel:2012rb}). As a result the physical Hamiltonian has a simpler expression
\begin{eqnarray}
\mathbf{H}_0=\int_\mathcal S\mathrm d^3\sigma\, \mathcal C(\sigma,\t). \label{gaussd}
\end{eqnarray}
The following Hamiltonian unifies both scenarios of the BK and Gaussian dusts:
\begin{eqnarray}
\mathbf{H}_0&=&\int_\mathcal S\mathrm d^3\sigma\, h(\sigma,\t),\label{ham1}\\
h(\sigma,\t)&=&\sqrt{\mathcal C(\sigma,\t)^2-\frac{\a}{4}\sum_{a=1}^3\mathcal C_a(\sigma,\t)^2},\quad \begin{cases}
\a =1& \text{BK dust},\\
\a =0& \text{Gaussian dust}.
\end{cases} \nonumber
\end{eqnarray}
This physical Hamiltonian ${\bf H}_0$ is manifestly positive. However when $\mathcal C<0$, Eq.\Ref{ham1} is different from Eq.\Ref{gaussd} by an overall minus sign, thus reverses the time flow $\t\to-\t$ for the Gaussian dust.
The physical Hamiltonian $\mathbf{H}_0$ generates the $\t$ evolution:
\begin{eqnarray}
\frac{\mathrm d f}{\mathrm d\t}=\left\{ f, \mathbf{H}_0\right\},
\end{eqnarray}
for all phase space function $f$. In particular, the Hamilton's equations are
\begin{eqnarray}
\frac{\mathrm d A^a_j(\sigma,\t)}{\mathrm d\t}=-\frac{\kappa\b}{2}\frac{\delta\mathbf{H}_0}{\delta E^j_a(\sigma,\t)},\quad \frac{\mathrm d E^j_a(\sigma,\t)}{\mathrm d\t}=\frac{\kappa\b}{2}\frac{\delta\mathbf{H}_0}{\delta A^a_j(\sigma,\t)}.\label{hamitoncon0}
\end{eqnarray}
Functional derivatives on the right-hand sides of Eq.\Ref{hamitoncon0} give
\begin{eqnarray}
\delta{\bf H}_0=\int_\mathcal S\mathrm d^3\sigma \left(\frac{\mathcal C}{h}\delta \mathcal C -{\a}q^{ij}\frac{\mathcal C_i}{h}\delta\mathcal C_j+\frac{\a}{2h}q^{ij}q^{kl}\mathcal C_j\mathcal C_l\delta q_{ik}\right),
\end{eqnarray}
where ${C}/{h}$ is negative (positive) for physical (phantom) dust. In this work we focus on the cosmological perturbation theory $q_{ij}=q^0_{ij}+h_{ij}$ ($q^0_{ij}$ is the homogeneous and isotropic cosmological background and $h_{ij}$ is the perturbation) and linearized EOMs. The last term gives $\frac{\a}{2h}q^{ij}q^{kl}\mathcal C_j\mathcal C_l= O(h_{ij}^2)$ since $\mathcal C_j(q^0)=0$, thus does not affect linearized EOMs. Compare $\delta{\bf H}$ to the variation of Hamiltonian $H_{GR}$ of pure gravity in absence of dust motivates us to identify (dynamical) lapse function and shift vector
\begin{eqnarray}
N=\frac{\mathcal C}{h}, \quad N_j=-{\a}\frac{\mathcal C_j}{h}.\label{lapseshift0}
\end{eqnarray}
$N$ is negative (positive) for the physical (phantom) dust. Negative lapse indicates that $\t$ in Eq.\Ref{hamitoncon0} flows from future to past. Its origin is the flip $\t\to-\t$ before Eq.\Ref{bigH0}. In this paper we focus on gravity coupled to the physical dust. When we discuss the cosmological perturbation theory from the semiclassical limit of LQG, we are going to flip $\t\to-\t$ back such that $\t$ flows to the future again. In that case, the dynamical lapse function and shift vector Eq.\Ref{lapseshift0} have to change to
\begin{eqnarray}
N=-\frac{\mathcal C}{h}, \quad N_j={\a}\frac{\mathcal C_j}{h}.\label{lapseshift}
\end{eqnarray}
They can be obtained directly from the variation $\delta(-{\bf H}_0)$ ($-{\bf H}_0$ is the physical Hamiltonian of physical dust if we don't flip $\t\to-\t$ before Eq.\Ref{bigH0}.
In the gravity-dust models, we have resolved the Hamiltonian and diffeomorphism constraints classically, while the SU(2) Gauss constraint $\mathcal G_a(\sigma,\t)=D_j E^j_a(\sigma,\t)=0$ still has to be imposed to the phase space. In addition, There are non-holonomic constraints: $\mathcal C(\sigma,\t)^2-\frac{\a}{4}\sum_{a=1}^3\mathcal C_a(\sigma,\t)^2\geq 0$ and $\mathcal C<0$ for physical dust ($\mathcal C>0$ for phantom dust).
These constraints are preserved by $\t$-evolution for gravity coupling to the BK dust. Indeed, firstly $\t$-evolution cannot break Gauss constraint since $\left\{\mathcal G_a(\sigma,\t),\,{\bf H}_0\right\}=0$. Secondly both $h(\sigma,\t)$ and $\mathcal C_j(\sigma,\t)$ are conserved densities (thus $N_j$ is conserved) \cite{Giesel:2007wn}:
\begin{eqnarray}
\frac{\mathrm d h(\sigma,\t)}{\mathrm d \t}=\left\{h(\sigma,\t),\,{\bf H}_0\right\}=0,\quad \frac{\mathrm d \mathcal C_j(\sigma,\t)}{\mathrm d \t}=\left\{\mathcal C_j(\sigma,\t),\,{\bf H}_0\right\}=0\label{conserv0}
\end{eqnarray}
Therefore $\mathcal C(\sigma,\t)^2-\frac{1}{4}\sum_{a=1}^3\mathcal C_a(\sigma,\t)^2\geq 0$ is conserved. About $\mathcal C<0$ ($\mathcal C>0$), suppose $\mathcal C<0$ ($\mathcal C>0$) was violated in $\t$-evolution, there would exist a certain $\t_0$ that $\mathcal C(\sigma,\t_0)=0$, but then $\mathcal C(\sigma,\t)^2-\frac{1}{4}\sum_{a=1}^3\mathcal C_a(\sigma,\t)^2$ would becomes negative if $\mathcal C_j(\sigma,\t)\neq 0$, contradicting the conservation of $h(\sigma,\t)$ and the other nonholonomic constraint. If the conserved $\mathcal C_j(\sigma,\t)=0$, $h(\sigma,\t)^2=\mathcal C(\sigma,\t)^2$ is conserved and thus cannot evolve from nonzero to zero. For gravity coupled to the Gaussian dust, $\mathcal C_j(\sigma,\t)$ is conserved. $h(\sigma,\t)$ and $\mathcal C(\sigma,\t)$ are conserved only when $\mathcal C_j(\sigma,\t)=0$. $\mathcal C<0$ ($\mathcal C>0$) may be violated in $\t$-evolution for coupling to the Gaussian dust if $\mathcal C_j(\sigma,\t)\neq 0$.
In our following discussion, we focus on pure gravity coupled to dusts, thus we only work with physical dusts in order not to violate the energy condition.
\subsection{Quantization}
We construct a fixed finite cubic lattice $\gamma$ which partitions the dust space $\mathcal S$. In this work, $\mathcal S$ is compact and has no boundary. $E(\gamma)$ and $V(\gamma)$ denote sets of (oriented) edges and vertices in $\gamma$. By the dust coordinate on $\mathcal S$, every edge has a constant coordinate length $\mu$. $\mu\to 0$ relates to the lattice continuum limit. Every vertex $v\in V(\gamma)$ is 6-valent, having 3 outgoing edges $e_I(v)$ ($I=1,2,3$) and 3 incoming edges $e_I(v-\mu \hat{I})$ where $\hat{I}$ is the coordinate basis vector along the $I$-th direction. It is sometimes convenient to orient all 6 edges to be outgoing from $v$, and denote them by $e_{v;I,s}$ ($s=\pm$):
\begin{eqnarray}
e_{v;I,+}=e_I(v),\quad e_{v;I,-}=e_I(v-\mu \hat{I})^{-1}.
\end{eqnarray}
Canonical variables $A^a_j(\sigma,\t),E^j_a(\sigma,\t)$ are regularized by holonomy $h(e)$ and gauge covariant flux $p^a(e)$ at every $e\in E(\gamma)$:
\begin{eqnarray}
h(e)&:=&\mathcal P \exp \int_{e}A^a\t^a/2,\nonumber\\
p^a(e)&:=&-\frac{1}{2\b a^2}\mathrm{tr}\left[\t^a\int_{S_e}\varepsilon_{ijk}\mathrm d \sigma^i\wedge\mathrm d \sigma^j\ h\left(\rho_e(\sigma)\right)\, E_b^k(\sigma)\t^b\, h\left(\rho_e(\sigma)\right)^{-1}\right],\label{hpvari}
\end{eqnarray}
where $\t^a=-i(\text{Pauli matrix})^a$. $S_e$ is a 2-face intersecting $e$ in the dual lattice $\gamma^*$. $\rho_e$ is a path starting at the source of $e$ and traveling along $e$ until $e\cap S_e$, then running in $S_e$ until $\vec{\sigma}$. $a$ is a length unit for making $p^a(e)$ dimensionless. Because $p^a(e)$ is gauge covariant flux, we have
\begin{eqnarray}
p^{a}\left(e_{v ; I,-}\right)=\frac{1}{2} \operatorname{Tr}\left[\tau^{a} h\left(e_{v-\hat{I} ; I,+}\right)^{-1} p^{b}\left(e_{v-\hat{I} ; I,+}\right) \tau^{b} h\left(e_{v-\hat{I} ; I,+}\right)\right].
\end{eqnarray}
The Poisson algebra of $h(e)$ and $p^a(e)$ are called the holonomy-flux algebra:
\begin{eqnarray}
\left\{h(e), h\left(e^{\prime}\right)\right\} &=&0 ,\label{handh}\\
\left\{p^{a}(e), h\left(e^{\prime}\right)\right\} &=&\frac{\kappa}{a^{2}} \delta_{e, e^{\prime}} \frac{\tau^{a}}{2} h\left(e^{\prime}\right) ,\label{pandtheta}\\
\left\{p^{a}(e), p^{b}\left(e^{\prime}\right)\right\} &=&-\frac{\kappa}{a^{2}} \delta_{e, e^{\prime}} \varepsilon_{a b c} p^{c}\left(e^{\prime}\right),\label{pandp}
\end{eqnarray}
$h(e)$ and $p^a(e)$ are coordinates of the reduced phase space $\mathcal P_\gamma$ for the theory discretized on $\gamma$.
In quantum theory, the Hilbert space $\mathcal H_\gamma$ is spanned by gauge invariant (complex valued) functions of all $h(e)$'s, and is a proper subspace of $\mathcal H_\gamma^0=\otimes_e L^2(\mathrm{SU}(2))$. $\hat{h}(e)$ becomes multiplication operators on functions in $\mathcal H^0_\gamma$. $\hat{p}^a(e)=i t\,{R}_e^a/2$ where ${R}_e^a$ is the right invariant vector field on SU(2): $R^a f(h)=\frac{\mathrm d}{\mathrm d \varepsilon}\big|_{\varepsilon=0} f(e^{\varepsilon\t^a}h)$. $t=\ell^2_p/a^2$ is a dimensionless semiclassicality parameter ($\ell^2_p=\hbar\kappa$). $\hat{h}(e),\hat{p}^a(e)$ satisfy the commutation relations:
\begin{eqnarray}
\left[\hat{h}(e),\hat{h}(e')\right] &=&0\nonumber\\
\left[\hat{p}^a(e),\hat{h}(e')\right] &=&i t \delta_{e,e'} \frac{\t^a}{2} {h}(e')\nonumber\\
\left[\hat{p}^a(e),\hat{p}^b(e')\right]&=&-it \delta_{e,e'} \varepsilon_{abc} {p}^c(e'), \label{ph}
\end{eqnarray}
as quantization of the holonomy-flux algebra.
The physical Hamiltonian operators $\hat{\bf H}$ are given by \cite{Giesel:2007wn}:
\begin{eqnarray}
\hat{\mathbf{H}}&=&\sum_{v\in V(\gamma)}\hat{H}_v,\quad \hat{H}_v:=\left[\hat{M}_-^\dagger(v) \hat{M}_-(v)\right]^{1/4},\label{physHam}\\
\hat{M}_-(v)&=&\hat{C}_{v}^{\ \dagger}\hat{C}_{v}-\frac{\a}{4}\sum_{a=1}^3\hat{C}_{a,v}^{\ \dagger}\hat{C}_{a,v},\quad \a=\begin{cases}
1,&\text{BK dust,}\\
0,&\text{Gaussian dust.}
\end{cases}
\end{eqnarray}
In our notation, ${\bf H}_0=\int_\mathcal S\mathrm d^3\sigma\, h$, $\mathcal C$, and $\mathcal C_{a}$ are the physical Hamiltonian, scalar constraint, and vector constraint on the continuum. ${\bf H}=\sum_v H_v$, $C_v$, and $C_{a,v}$ are their discretizations on $\gamma$. $\hat{\bf H}=\sum_v \hat{H}_v$, $\hat{C}_v$, and $\hat{C}_{a,v}$ are quantizations of ${\bf H}$, $C_v$, and $C_{a,v}$:
\begin{eqnarray}
\hat{C}_{0,v}&=&-\frac{2}{i\b\kappa\ell_p^2}\sum_{s_1,s_2,s_3=\pm1}s_1s_2s_3\ \varepsilon^{I_1I_2I_3}\ \mathrm{Tr}\Bigg(\hat{h}(\a_{v;I_1s_1,I_2s_2}) \hat{h}(e_{v;I_3s_3})\Big[\hat{h}(e_{v;I_3s_3})^{-1},\hat{V}_v\Big] \Bigg)\label{Cd}\\
\hat{C}_{a,v}&=&-\frac{4}{i\b^2\kappa\ell_p^2}\sum_{s_1,s_2,s_3=\pm1}s_1s_2s_3\ \varepsilon^{I_1I_2I_3}\ \mathrm{Tr}\Bigg(\t^a \hat{h}(\a_{v;I_1s_1,I_2s_2}) \hat{h}(e_{v;I_3s_3})\Big[\hat{h}(e_{v;I_3s_3})^{-1},\hat{V}_v\Big] \Bigg)\\
\hat{C}_v&=&\hat{C}_{0,v}+{(1+\b^2)}\hat{C}_{L,v}+\frac{2\L}{\kappa}\hat{ V}_v,\quad\quad \hat{K}=\frac{i}{\hbar\b^2}\left[\sum_{v\in V(\gamma)}\hat{C}_{0,v},\sum_{v\in V(\gamma)}V_v\right]\nonumber\\
\hat{C}_{L,v}&=&\frac{16}{\kappa\left(i\b\ell_p^2\right)^3}\sum_{s_1,s_2,s_3=\pm1}s_1s_2s_3\ \varepsilon^{I_1I_2I_3}\label{HCO}\\
&&\mathrm{Tr}\Bigg( \hat{h}(e_{v;I_1s_1})\Big[\hat{h}(e_{v;I_1s_1})^{-1},\hat{K}\Big]\ \hat{h}(e_{v;I_2s_2})\Big[\hat{h}(e_{v;I_2s_2})^{-1},\hat{K}\Big]\ \hat{h}(e_{v;I_3s_3})\Big[\hat{h}(e_{v;I_3s_3})^{-1},\hat{V}_v\Big]\ \Bigg).\nonumber
\end{eqnarray}
where $\hat{V}_v$ is the volume operator at $v$:
\begin{eqnarray}
\hat{V}_v&=&\left(\hat{Q}_v^2\right)^{1/4},\\
\hat{Q}_
&=&\b^3a^6\varepsilon_{abc}\frac{\hat{p}^a({e_{v;1+}})-\hat{p}^a({e_{v;1-}})}{4}\frac{\hat{p}^b({e_{v;2+}})-\hat{p}^b({e_{v;2-}})}{4}\frac{\hat{p}^c({e_{v;3+}})-\hat{p}^c({e_{v;3-}})}{4}.\label{Qv}
\end{eqnarray}
The Hamiltonian operator $\hat{\mathbf{H}}$ is positive semi-definite and self-adjoint because $\hat{M}_-^\dagger(v) \hat{M}_-(v)$ is manifestly positive semi-definite and Hermitian, therefore admits a canonical self-adjoint extension.
Classical discrete $C_v$, and $C_{a,v}$ can be obtained from Eqs.\Ref{C} - \Ref{HCO} by mapping operators to their classical counterparts and $[\hat{f}_1,\hat{f}_2]\to i\hbar\{f_1,f_2\} $. Hence classical discrete physical Hamiltonian ${\bf H}$ is
\begin{eqnarray}
{\bf H}=\sum_{v\in V(\gamma)} H_v,\quad H_v=\sqrt{\left|C_v^2-\frac{\a}{4}\sum_{a=1}^3 C_{a,v}^2\right|}.\label{physHamcl}
\end{eqnarray}
The absolute value in the square-root results from that ${\bf H}$ is the classical limit of $\hat{\bf H}$, while $\hat{\bf H}$ is defined on the entire $\mathcal H_\gamma$ disregarding nonholonomic constraints in particular $\mathcal C^2-\frac{\a}{4}\sum_{a=1}^3\mathcal C_a^2\geq 0$ for $\a=1$.
The transition amplitude $A_{[g],[g']}$ plays the central role in the quantum dynamics of reduced phase space LQG:
\begin{eqnarray}
A_{[g],[g']}=\langle \Psi^t_{[g]}|\,\exp\left[-\frac{i}{\hbar}T \hat{\bf H}\right]\,|\Psi^t_{[g']}\rangle.
\end{eqnarray}
We focus on the semiclassical initial and final states $\Psi^t_{[g']}, \Psi^t_{[g]}$ for the purpose of semiclassical analysis. $\Psi^t_{[g']}, \Psi^t_{[g]}$ are gauge invariant coherent states \cite{Thiemann:2000bw,Thiemann:2000ca}:
\begin{eqnarray}
\Psi^t_{[g]}(h
&=&\int_{\mathrm{SU(2)}^{|V(\gamma)|}}\mathrm d h\prod_{e\in E(\gamma)}{\psi}^t_{h_{s(e)}^{-1}g(e)h_{t(e)}}\left(h(e)\right),\quad \mathrm d h=\prod_{v\in V(\gamma)}\mathrm d\mu_H(h_v).\label{gaugeinv}
\end{eqnarray}
The gauge invariant coherent state is labelled by the gauge equivalence class $[g]$ generated by $g(e)\sim g^h(e)= h_{s(e)}^{-1}g(e)h_{t(e)}$ at all $e$. $g(e)\in\mathrm{SL}(2,\mathbb{C})$. $\psi^{t}_{g(e)}\left(h(e)\right)$ is the complexifier coherent state on the edge $e$:
\begin{eqnarray}
\psi^{t}_{g(e)}\left(h(e)\right
&=&\sum_{j_e\in\mathbb{Z}_+/2\cup\{0\}}(2j_e+1)\ e^{-tj_e(j_e+1)/2}\chi_{j_e}\left(g(e)h(e)^{-1}\right),\label{coherent}
\end{eqnarray}
where $g(e)$ is complex coordinate of $\mathcal P_\gamma$ and relates to $h(e),p^a(e)$ by
\begin{eqnarray}
g(e)=e^{-ip_a(e)\t_a/2}h(e)=
e^{-ip^a(e)\t^a/2}e^{\theta^a(e)\t^a/2}, \quad p^a(e),\ \theta^a(e)\in\mathbb{R}^3.\label{gthetap}
\end{eqnarray}
Applying Eq.\Ref{gaugeinv} and discretizing time $T=N\Delta\t$ with large $N$ and infinitesimal $\Delta\t$,
\b
A_{[g],[g']}&=&\int\mathrm d h\left\langle\psi^t_{g}\right|\left[e^{ -\frac{i}{\hbar}\Delta\t \hat{\mathbf{H}}}\right]^N |{\psi}^t_{{g'}^{h}}\rangle,\\
&=&\int\mathrm d h\prod_{i=1}^{N+1}\mathrm{d}g_{i}\,\langle\psi^t_{g}|\tilde{\psi}^t_{g_{N+1}}\rangle\langle \tilde{\psi}^t_{g_{N+1}}\big|e^{ -\frac{i\Delta\t}{\hbar} \hat{\mathbf{H}}}\big|\tilde{\psi}^t_{g_{N}}\rangle
\langle \tilde{\psi}^t_{g_{N}}\big|e^{ -\frac{i\Delta\t}{\hbar}\hat{\mathbf{H}}}\big|\tilde{\psi}^t_{g_{N-1}}\rangle\cdots\nonumber\\
&&\quad \cdots\
\langle \tilde{\psi}^t_{g_2}\big|e^{ -\frac{i\Delta\t}{\hbar}\hat{\mathbf{H}}}\big|\tilde{\psi}^t_{g_1}\rangle\langle\tilde{\psi}^t_{g_1}|{\psi}^t_{g'{}^{h}}\rangle
\end{eqnarray}
where we have inserted $N+1$ resolutions of identities with normalized coherent state $\tilde{\psi}^t_{g}=\otimes_e{\psi}^t_{g(e)}/||{\psi}^t_{g(e)}||$:
\begin{eqnarray}
\int\mathrm d g_i\ |\tilde{\psi}^{t}_{g_i}\rangle\langle\tilde{\psi}^{t}_{g_i}|=1_{\mathcal H_\gamma^0},\quad \mathrm d g_i=\left(\frac{c}{t^3}\right)^{|E(\gamma)|}\prod_{e\in E(\gamma)}\mathrm d\mu_H(h_i(e))\,\mathrm d^3p_i(e),\quad i=1,\cdots,N-1.
\end{eqnarray}
The above expression of $A_{[g],[g']}$ leads to a path integral formula (see \cite{Han:2019vpw} for derivation):
\begin{eqnarray}
A_{[g],[g']}=\left\|\psi_{g}^{t}\right\|\left\|\psi_{g^{\prime}}^{t}\right\| \int \mathrm{d} h \prod_{i=1}^{N+1} \mathrm{d} g_{i}\, \nu[g]\, e^{S[g, h] / t}\label{Agg}
\end{eqnarray}
where we find the ``effective action'' $S[g,h]$ given by
\begin{eqnarray}
S[g, h]&=&\sum_{i=0}^{N+1} K\left(g_{i+1}, g_{i}\right)-\frac{i \kappa}{a^{2}} \sum_{i=1}^{N} \Delta \tau\left[\frac{\langle\psi_{g_{i+1}}^{t}|\hat{\mathbf{H}}| \psi_{g_{i}}^{t}\rangle}{\langle\psi_{g_{i+1}}^{i} | \psi_{g_{i}}^{t}\rangle}+i \tilde{\varepsilon}_{i+1, i}\left(\frac{\Delta \tau}{\hbar}\right)\right],\label{Sgh}\\
K\left(g_{i+1}, g_{i}\right)&=&\sum_{e \in E(\gamma)}\left[z_{i+1, i}(e)^{2}-\frac{1}{2} p_{i+1}(e)^{2}-\frac{1}{2} p_{i}(e)^{2}\right]
\end{eqnarray}
with $g_{0} \equiv g^{\prime h},\ g_{N+2} \equiv g$. $\tilde{\varepsilon}_{i+1, i}\left(\frac{\Delta \tau}{\hbar}\right)\to 0$ as $\Delta\t\to0$ and is negligible. In the above, $z_{i+1,i}(e)$ and $x_{i+1,i}(e)$ are given by
\begin{eqnarray}
z_{i+1,i}(e)&=& \mathrm{arccosh}\left(x_{i+1,i}(e)\right),\quad x_{i+1,i}(e)=\frac{1}{2}\mathrm{tr}\left[g_{i+1}(e)^\dagger g_{i}(e)\right].
\end{eqnarray}
\section{Semiclassical Equations of Motion}\label{Semiclassical Equations of Motion}
In the semiclassical limit $t\to0$ (or $\ell_P\ll a$), the dominant contribution to $A_{[g],[g']}$ comes from the semiclassical trajectories satisfying the semiclassical equations of motion (EOMs). Semiclassical EOMs has been derived in \cite{Han:2019vpw} by the variational principle $\delta S[g,h]=0$ (stationary phase approximation):
\begin{itemize}
\item For $i=1,\cdots,N$, at every edge $e\in E(\gamma)$,
\begin{eqnarray}
\frac{1}{\Delta\t}\left[\frac{z_{i+1,i}(e)\,\mathrm{tr}\left[\t^a g_{i+1}(e)^\dagger g_i(e)\right]}{\sqrt{x_{i+1,i}(e)-1}\sqrt{x_{i+1,i}(e)+1}}-\frac{p_i(e)\,\mathrm{tr}\left[\t^a g_{i}(e)^\dagger g_i(e)\right]}{\sinh(p_i(e))}\right]\nonumber\\
=\frac{i\kappa }{a^2}\frac{\partial}{\partial \varepsilon_{i}^{a}(e)} \frac{\langle\psi_{g_{i+1}^{\varepsilon}}^{t}|\hat{\mathbf{H}}| \psi_{g_{i}^{\varepsilon}}^{t}\rangle}{\langle\psi_{g_{i+1}^{\varepsilon}}^{t} | \psi_{g_{i}^{\varepsilon}}^{t}\rangle}\Bigg|_{\vec{\varepsilon}=0}\label{eoms1}
\end{eqnarray}
where $g^\varepsilon(e)=g(e) e^{\varepsilon^a(e)\t^a}$ ($\varepsilon^a(e)\in\mathbb{C}$) is a holomorphic deformation.
\item For $i=2,\cdots,N+1$, at every edge $e\in E(\gamma)$,
\begin{eqnarray}
\frac{1}{\Delta\t}\left[\frac{z_{i,i-1}(e)\,\mathrm{tr}\left[\t^a g_{i}(e)^\dagger g_{i-1}(e)\right]}{\sqrt{x_{i,i-1}(e)-1}\sqrt{x_{i,i-1}(e)+1}}-\frac{p_i(e)\,\mathrm{tr}\left[\t^a g_{i}(e)^\dagger g_i(e)\right]}{\sinh(p_i(e))}\right]\nonumber\\
=-\frac{i\kappa }{a^2}\frac{\partial}{\partial \bar{\varepsilon}_{i}^{a}(e)} \frac{\langle\psi_{g_{i}^{\varepsilon}}^{t}|\hat{\mathbf{H}}| \psi_{g_{i-1}^{\varepsilon}}^{t}\rangle}{\langle\psi_{g_{i}^{\varepsilon}}^{t} | \psi_{g_{i-1}^{\varepsilon}}^{t}\rangle}\Bigg|_{\vec{\varepsilon}=0}.\label{eoms2}
\end{eqnarray}
\item The closure condition at every vertex $v\in V(\gamma)$ for initial data:
\begin{eqnarray}
G_v^a\equiv-\sum_{e, s(e)=v}p_1^a(e)+\sum_{e, t(e)=v}\L^a_{\ b}\left(\vec{\theta}_1(e)\right)\,p_1^b(e)=0.\label{closure0}
\end{eqnarray}
where $\L^a_{\ b}(\vec{\theta})\in\mathrm{SO}(3)$ is given by $e^{\theta^c\t^c/2}\t^a e^{-\theta^c\t^c/2}=\L^a_{\ b}(\vec{\theta})\t^b$.
\end{itemize}
\noindent
The initial and final conditions are given by $g_{1}=g'^h$ and $g_{N+1}=g$. Eqs.\Ref{eoms1} and \Ref{eoms2} come from $\delta S/\delta g=0$ and $\delta S/\delta\bar{g}=0$, while Eq.\Ref{closure0} comes from $\delta S/\delta h=0$. These semiclassical EOMs govern the semiclassical dynamics of LQG in the reduced phase space formulation.
We can take $\Delta\t\to0$ in these semiclassial EOMs since $\Delta\t$ is arbitrarily small. Solutions of EOMs with $\Delta\t\to0$ are time-continuous approximation of solutions of Eqs.\Ref{eoms1} - \Ref{closure0}.
It is proven in \cite{Han:2019vpw} that Eqs.\Ref{eoms1} - \Ref{eoms2} implies $g_{i}\to g_{i+1}$ as $\Delta\t\to0$, i.e. $g_i=g(\t)$ is a continuous function of $\t$. Therefore, matrix elements $\langle\psi_{g_{i}^{\varepsilon}}^{t}|\hat{\mathbf{H}}| \psi_{g_{i-1}^{\varepsilon}}^{t}\rangle$ on right-hand sides of Eqs.\Ref{eoms1} - \Ref{eoms2} reduces to the expectation values $\langle\psi_{g^{\varepsilon}}^{t}|\hat{\mathbf{H}}| \psi_{g^{\varepsilon}}^{t}\rangle$ as $\Delta\t\to0$. Coherent state expectation values of $\hat{\bf H}$ have correct semiclassical limit\footnote{Firstly we apply the semiclassical perturbation theory of \cite{Giesel:2006um} to $\hat{O}\equiv\hat{{H}}_v^4$ (recall Eq.\Ref{physHam}) and all $\hat{O}^n$ ($n>1$): $\langle\tilde{\psi}_{g}^{t}|\hat{O}^n| \tilde{\psi}_{g}^{t}\rangle={O}[g]^n+O(t)$. By Theorem 3.6 of \cite{Thiemann:2000bx}, $\lim_{t\to0}\langle\tilde{\psi}_{g}^{t}|f(\hat{O})| \tilde{\psi}_{g}^{t}\rangle=f({O}[g])$ for any any Borel measurable function on $\mathbb{R}$ such that $\langle\tilde{\psi}_{g}^{t}|f(\hat{O})^\dagger f(\hat{O})| \tilde{\psi}_{g}^{t}\rangle<\infty$.}
\begin{eqnarray}
\lim_{t\to0}\langle\tilde{\psi}_{g}^{t}|\hat{\mathbf{H}}| \tilde{\psi}_{g}^{t}\rangle={\bf H}[g]
\end{eqnarray}
where ${\bf H}[g]$ is the classical discrete Hamiltonian \Ref{physHamcl} evaluated at $p^a(e),h(e)$ determined by $g(e)$ in Eq.\Ref{gthetap}. Note that the above semiclassical behavior of $\langle\tilde{\psi}_{g}^{t}|\hat{\mathbf{H}}| \tilde{\psi}_{g}^{t}\rangle$ relies on the following semiclassical expansion of volume operator $\hat{V}_v$ \cite{Giesel:2006um}:
\begin{eqnarray}
\hat{V}_v=\langle \hat{Q}_v\rangle^{2q}\left[1+\sum_{n=1}^{2k+1}(-1)^{n+1}\frac{q(1-q)\cdots(n-1+q)}{n!}\left(\frac{\hat{Q}_v^2}{\langle\hat{Q}_v\rangle^2}-1\right)^n\right],\quad q=1/4\label{expandvolume}
\end{eqnarray}
where $\langle \hat{Q}_v\rangle=\langle\psi^t_g|\hat{Q}_v|\psi^t_g\rangle$, and this expansion is valid when $\langle \hat{Q}_v\rangle\gg \ell_P^6$.
The time continuous limit of semiclassical EOMs is computed in \cite{Han2020} and expressed in terms of ${\bm p}(e)=(p^1(e),p^2(e),p^3(e))^T$ and ${\bm \theta}(e)=(\theta^1(e),\theta^2(e),\theta^3(e))^T$ and their time derivatives:
\begin{eqnarray}
\left( \begin{array}{l} {\mathrm d {\bm p}}(e)/{\mathrm d \tau} \\ {\mathrm d \bm{\theta}}(e)/{\mathrm d \tau} \end{array} \right) = \frac{i\kappa}{a^2}\, {T}\left({\bm p},{\bm \theta}\right) ^{-1}\left( \begin{array}{l} {\partial {\bf H} }/{\partial {\bm p} (e)} \\ {\partial {\bf H} }/{\partial \bm{\theta} (e)} \end{array} \right) .\label{eom0}
\end{eqnarray}
The matrix elements $T$ is lengthy, and are given explicitly in \cite{github0}. It is shown in \cite{Han2020} that Eq.\Ref{eom0} is equivalent to that for any phase space function $f$ on $\mathcal P_\gamma$, its $\t$-evolution is given by the Hamiltonian flow generated by ${\bf H} $:
\begin{eqnarray}
\frac{\mathrm d f}{\mathrm d \tau}=\left\{f, \ {\bf H}\right\}.\label{hamilton}
\end{eqnarray}
The closure condition is preserved by $\t$-evolution by $\{G^a_v,\,{\bf H}\}=0$.
The lattice continuum limit of Eq.\Ref{eom0} is studied in \cite{Han2020}. We define $\mu$ to be the coordinate length of every lattice edge, the lattice continuum limit is formally given by $\mu\to0$ and $|V(\gamma)|\to\infty$ while keeping $\mu^3|V(\gamma)|$ fixed. More precisely, recall that Eq.\Ref{eom0} are derived with $t=\ell_P^2/a^2\to0$ and the assumption $\langle \hat{Q}_v\rangle\sim \mu^6\gg \ell_P^6$ (see Eq.\Ref{expandvolume}), the lattice continuum limit are taken in the regime
\begin{eqnarray}
\ell_P\ll\mu\ll a,
\end{eqnarray}
where $a$ is a macroscopic unit, e.g. 1 mm. When keeping $a$ fixed, the lattice continuum limit sends $\mu\to0$ after the semiclassical limit $\ell_P\to0$ so $\ell_P\ll\mu$ is kept. In the lattice continuum limit, EOMs.\Ref{eom0} reduce to the EOMs \Ref{hamitoncon0} of the continuum theory, when suitable initial conditions are imposed (see \cite{Han2020} for details).
\section{Cosmological Background and Perturbations} \label{Cosmological Background and Perturbations}
\subsection{Cosmological Background}
As in \cite{Han:2019vpw}, we apply the following (homogeneous and isotropic) cosmological ansatz to the semiclassical EOMs
\begin{eqnarray}
\theta^a(e_I(v))=\mu \b K_0\delta^a_I,\quad p^a(e_I(v))=\frac{2\mu^2}{\b a^2}P_0\delta^a_I
\end{eqnarray}
Here $K_0=K_0(\t)$ and $P_0=P_0(\t)$ are constant on $\gamma$ but evolve with the dust time $\t$. Inserting the ansatz, left hand sides of EOMs \Ref{eom0} contain (1) $\mathrm d p^a(e_I(v))/\mathrm d \t$ and $\mathrm d \theta^a(e_I(v))/\mathrm d \t$ with $a= I$, which are proportional to $\dot{P}_0=\mathrm d P_0/\mathrm d\t$ and $\dot{K}_0=\mathrm d K_0/\mathrm d\t$, and (2) $\mathrm d p^a(e_I(v))/\mathrm d \t$ and $\mathrm d \theta^a(e_I(v))/\mathrm d \t$ with $a\neq I$, which are zero.
\begin{itemize}
\item EOMs of case (1) reduce to
\begin{eqnarray}
\frac{4 \beta ^2 \left[-2 \mu ^2 \sqrt{{P}_0} \dot{K_0}+\sin ^4(\beta \mu {K_0})+\Lambda \mu ^2 {P_0}\right]-\sin ^2(2 \beta \mu {K_0})}{\sqrt{{P_0}}}&=&0,\label{cosm1}\\
\sqrt{{P_0}} \left[2 \beta ^2 \sin (2 \beta \mu {K_0})-\left(\beta ^2+1\right) \sin (4 \beta \mu {K_0})\right]+2 \beta \mu \dot{P_0}&=&0.\label{cosm2}
\end{eqnarray}
where an effective Hamiltonian of cosmology can be extracted
\begin{eqnarray}
H_{eff}(P_0,K_0)=\frac{\left(\beta ^2+1\right) \sqrt{{P_0}} \sin ^2(2 \beta \mu {K_0})}{4 \beta ^2 \mu ^2}-\frac{\sqrt{{P_0}} \sin ^2(\beta \mu {K_0})}{\mu ^2}-\frac{1}{3} \Lambda P_0^{3/2}.
\end{eqnarray}
Eqs.\Ref{cosm1} and \Ref{cosm2} can be written as Hamilton's equations
\begin{eqnarray}
\dot{P_0}=\frac{\partial H_{eff}}{\partial K_0},\quad \dot{K_0}=-\frac{\partial H_{eff}}{\partial P_0}.\label{cosmhameq}
\end{eqnarray}
\item EOMs of case (2) are satisfied automatically, thus do not impose any constraints \cite{Han:2019vpw}.
\item Closure condition \Ref{closure0} is satisfied automatically.
\end{itemize}
By Hamilton's equations \Ref{cosmhameq}, $H_{eff}=H_v/6$ is conserved in $\t$-evolution:
\begin{eqnarray}
\mu^3\left[\frac{\left(\beta ^2+1\right) \sqrt{{P_0}} \sin ^2(2 \beta \mu {K_0})}{4 \beta ^2 \mu ^2}-\frac{\sqrt{{P_0}} \sin ^2(\beta \mu {K_0})}{\mu ^2}-\frac{1}{3} \Lambda P_0^{3/2}\right]=\frac{\kappa\mathscr{E}}{6}=\frac{\kappa\rho V}{6}.\label{scrE}
\end{eqnarray}
where $\mathscr{E}>0$ is the dust energy per lattice site, and $\rho=\mathscr{E}/V$ is the dust energy density (recall Eq.\Ref{P=-h}). $V=\mu^3{P_0}^{3/2}$ is the volume per lattice site. Both $\rho$ and $V$ evolve in $\t$ while $\mathscr{E}$ is conserved. Note that because we use the dust to deparametrize gravity, the physical lapse was negative and $\t$ flowed backward (recall Eq.\Ref{lapseshift}). But in Eqs.\Ref{cosm1}, \Ref{cosm2}, and all following equations, we have flipped the time orientation $\t\to -\t$ to make the dust time flow forward.
The effective cosmological equations \Ref{cosm1} and \Ref{cosm2} reduce to classical Friedmann equations when $V$ is large (low density $\rho\ll1$). It may be seen by the following lattice continuum limit of $H_{eff}$ as $\mu\to0$, because the lattice spacing $\mu$ becomes negligible at large scale.
\begin{eqnarray}
\lim_{\mu\to0}H_{eff}=\sqrt{P_0} K_0^2-\frac{1}{3} \Lambda {P_0}^{3/2}
\end{eqnarray}
reduces to $h/6=-\mathcal C/6$ for cosmology, and Eqs.\Ref{cosmhameq} reduce to Friedmann equations. {FIG \ref{back}} compares solution $P_0(\t)$ of Eqs.\Ref{cosm1} and \Ref{cosm2} to solution $P_0(\t)$ of Friedmann equations.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1\linewidth]{back}
\caption{The left panel plots of $P_0(\t)$ solving Eqs.\Ref{cosm1} and \Ref{cosm2} (orange curve) and $P_0(\t)$ solving Friedmann equations (blue curve). Two solutions approximately coincide in $\t>0$ except for regime near the big-bang singularity. The solution of Eqs.\Ref{cosm1} and \Ref{cosm2} replaces the singularity by a bounce. The right panel zooms in the regime near where the bounce happens. The solutions use initial conditions $P_0(1)=0.153262$, $K_0(1)=0.260992$. Values of other parameters are $\L=10^{-5}$, $\b=1$, $\kappa=1$, and $\mu=10^{-3}$. The final time is $T=100$. }
\label{back}
\end{center}
\end{figure}
Effective equations \Ref{cosm1} and \Ref{cosm2} with finite $\mu$ modify Friedmann equations at high density $\rho$ and lead to a unsymmetric bounce to replace the big bang singularity. {The critical volume and density is given by
\begin{eqnarray}
V_c=\frac{8}{27}\beta^6(\beta ^2+1)^3 \kappa ^3 \mathscr{E}^3+O(\L),\quad \rho_c=\mathscr{E}/V_c.
\end{eqnarray}
} $\rho_c$ depending on the conserved quantity $\mathscr{E}$ indicates that the cosmological effective dynamics given by Eqs.\Ref{cosm1} and \Ref{cosm2} is an analog of the $\mu_0$-scheme LQC. The predicted effective dynamics is problematic near the singularity/bounce, because $V_c$ has to be of $O(\ell_P^3)$ in order to have Planckian critical density (for finite $\mathscr{E}$), but $V_c\sim \ell_P^3$ is inconsistent with Eq.\Ref{expandvolume} and invalidate the semiclassical approximation of ${\bf H}$. Otherwise if $V_c$ is much larger than $\ell_P^3$, the bounce can happen at a low critical density, and is not physically sound.
Therefore Eqs.\Ref{cosm1} and \Ref{cosm2} are only valid at the semiclassical regime where the density $\rho$ is low. Given our purpose of the semiclassical analysis, it is sufficient for us to only focus on solutions $(P_0(\t),K_0(\t))$ of Eqs.\Ref{cosm1} and \Ref{cosm2} in the semiclassical regime, and take them as backgrounds to study perturbations.
Cosmological effective dynamics with better behavior at the bounce is given by the $\bar{\mu}$-scheme LQC, where $\rho_c$ is a Planckian constant. Its relation to the full LQG theory is suggested recently in \cite{Han:2019feb}. However in this work, we focus on the cosmological perturbation theory based on solutions of Eqs.\Ref{cosm1} and \Ref{cosm2}, as analog of $\mu_0$-scheme.
\subsection{Cosmological Perturbations}
Given a cosmological background $P_0(\t),K_0(\t)$ satisfying Eqs.\Ref{cosm1} and \Ref{cosm2}, we perturb $p^a(e_I(v)),\theta^a(e_I(v))$ on this background:
\begin{eqnarray}
\theta^a(e_I(v))=\mu \left[\b K_0\delta^a_I+\mathcal X^a(e_I(v)) \right],\quad p^a(e_I(v))=\frac{2\mu^2}{\b a^2}\left[P_0\delta^a_I+\mathcal Y^a(e_I(v))\right],\label{perturb}
\end{eqnarray}
where $\mathcal X,\mathcal Y$ are perturbations. We introduce a vector $V^\rho(v)$ to contain both perturbations $\mathcal X,\mathcal Y$ at $v$:
\begin{eqnarray}
V^\rho(v)=\left(\mathcal Y^a(e_I(v)),\mathcal X^a(e_I(v))\right)^T,\quad \rho=1,\cdots,18.
\end{eqnarray}
The dictionary between $V^\rho(v)$ and $\mathcal X^a(e_I(v)),\mathcal Y^a(e_I(v))$ is given below:
\begin{eqnarray}
V^1=\mathcal Y^1(e_1),\quad &V^2=\mathcal Y^2(e_2),&\quad V^3=\mathcal Y^3(e_3)\nonumber\\
V^4=\mathcal Y^2(e_1),\quad &V^5=\mathcal Y^3(e_1),&\quad V^6=\mathcal Y^3(e_2)\nonumber\\
V^7=\mathcal Y^1(e_2),\quad &V^8=\mathcal Y^1(e_3),&\quad V^9=\mathcal Y^2(e_3)\nonumber\\
V^{10}=\mathcal X^1(e_1),\quad &V^{11}=\mathcal X^2(e_2),&\quad V^{12}=\mathcal X^3(e_3)\nonumber\\
V^{13}=\mathcal X^2(e_1),\quad &V^{14}=\mathcal X^3(e_1),&\quad V^{15}=\mathcal X^3(e_2)\nonumber\\
V^{16}=\mathcal X^1(e_2),\quad &V^{17}=\mathcal X^1(e_3),&\quad V^{18}=\mathcal X^2(e_3).
\end{eqnarray}
Thanks to the spatial homogeneity of $P_0(\t),K_0(\t)$, we make the following Fourier transformation on the cubic lattice $\gamma$
\begin{eqnarray}
V^\rho(\t,\vec{\sigma})=\int_{-{\pi}/{\mu}}^{{\pi}/{\mu}}\frac{\mathrm d^3 k}{(2\pi)^3}\, e^{i \vec{k}\cdot \vec{\sigma}}V^\rho(\t,\vec{k}),\quad \vec{\sigma}\in (\mu\mathbb{Z})^3,\label{fourier00}
\end{eqnarray}
where $\vec{\sigma}$ are 3d coordinates at the vertex $v$.
Inserting perturbations Eq.\Ref{perturb} in semiclassical EOMs \Ref{eom0}, and applying Fourier transformation, we obtain the following linearized EOMs for each mode ${k}$:
\begin{eqnarray}
\frac{\mathrm d V^\rho\left(\t,{k}\right)}{\mathrm d\t}={\bf U}^\rho_{\ \nu}\left(\mu,\t,{k}\right)\,V^\nu\left(\t,{k}\right).\label{lineareom}
\end{eqnarray}
For simplicity we have assumed that
\begin{eqnarray}
\vec{k}=(k,0,0)
\end{eqnarray}
has the only nonzero component $k^x=k$. Our discussion mainly focuses on the semiclassical regime where $\mu$ is negligible, this assumption doesn't lose generality in the continuum limit $\mu\to0$, because the background is $P_0(\t),K_0(\t)$ isotropic, the coordinate can always be chosen such that $\vec{k}=(k,0,0)$.
The computation of ${\bf U}^\rho_{\ \nu}(\mu,\t,{k})$ is carried out by expanding ${\bf H}$ up to quadratic order in perturbations followed by derivatives, and ${\bf H}$ contains $C_v$ with Lorentzian term shown in \Ref{HCO}. This computation is carried out on a HPC server and uses the parallel computing environment of Mathematica with 48 parallel kernels. The entire computation lasts for about 2 days. All Mathematica codes can be downloaded in \cite{github}. The explicit expression of $18\times 18$ matrix ${\bf U}^\rho_{\ \nu}(\mu,\t,{k})$ is too long to be shown in this paper but can be found in \cite{github}. Appendix \ref{H0} expands ${\bf U}^\rho_{\ \nu}(\mu,\t,{k})={\bf U}_0{}^\rho_{\ \nu}(\t,{k})+\mu\,{\bf U}_1{}^\rho_{\ \nu}(\t,{k})+O(\mu^2)$, and shows explicitly matrices ${\bf U}_0{}^\rho_{\ \nu}(\t,{k})$ and ${\bf U}_1{}^\rho_{\ \nu}(\t,{k})$.
The linearized closure condition Eq.\Ref{closure0} reads
{\begin{eqnarray}
0&=&P_0 \big[(V^{15}-V^{18}) \sin (\beta \mu K_0)-(V^{16}+V^{17}) (\cos (\beta \mu K_0)-1)\big]\nonumber\\
&&+\ \beta K_0 \big[-i V^{1} \sin (k \mu )+V^{1} \cos (k \mu )-V^{6} \sin (\beta \mu K_0)+V^{9} \sin (\beta \mu K_0)\nonumber\\
&&+\ V^{7} \cos (\beta \mu K_0)+V^{8} \cos (\beta \mu K_0)-V^{1}-V^{7}-V^{8}\big],\nonumber\\
0&=&P_0 \big[\cos (k \mu ) (V^{14} \sin (\beta \mu K_0)+V^{13} \cos (\beta \mu K_0)-V^{13})-i \sin (k \mu ) (V^{14} \sin (\beta \mu K_0)\nonumber\\
&&+\ V^{13} \cos (\beta \mu K_0)-V^{13})-V^{17} \sin (\beta \mu K_0)+V^{18} \cos (\beta \mu K_0)-V^{18}\big]\nonumber\\
&&+\ \beta K_0 \big[i V^{5} \sin (k \mu ) \sin (\beta \mu K_0)-\cos (k \mu ) (V^{5} \sin (\beta \mu K_0)+V^{4} \cos (\beta \mu K_0))\nonumber\\
&&+\ (-V^{9}+i V^{4} \sin (k \mu )) \cos (\beta \mu K_0)+V^{8} \sin (\beta \mu K_0)+V^{4}+V^{9}\big],\nonumber\\
0&=&P_0 \big[-\cos (k \mu ) (V^{13} \sin (\beta \mu K_0)-V^{14} (\cos (\beta \mu K_0)-1))+i \sin (k \mu ) (V^{13} \sin (\beta \mu K_0)\nonumber\\
&&-\ V^{14} \cos (\beta \mu K_0)+V^{14})+V^{16} \sin (\beta \mu K_0)+V^{15} \cos (\beta \mu K_0)-V^{15}\big]\nonumber\\
&&+\ \beta K_0 \big[\cos (k \mu ) (V^{4} \sin (\beta \mu K_0)-V^{5} \cos (\beta \mu K_0))-i \sin (k \mu ) (V^{4} \sin (\beta \mu K_0)\nonumber\\
&&-\ V^{5} \cos (\beta \mu K_0))-V^{7} \sin (\beta \mu K_0)-V^{6} \cos (\beta \mu K_0)+V^{5}+V^{6}\big],\label{linearclosure}
\end{eqnarray}}
where $V^\rho=V^\rho(\t,k)$. Closure condition is preserved by $\t$-evolution, because of $\{G^a_v,\,{\bf H}\}=0$ and Eq.\Ref{hamilton}.
Eqs.\Ref{lineareom} and \Ref{linearclosure}, derived from the full LQG, govern the dynamics of cosmological perturbations. Given initial conditions of $V^{\rho=1,\cdots, 18}$ satisfying the closure condition \Ref{linearclosure}, the $\t$-evolution of $V^\rho$'s can be computed by numerically solving Eqs.\Ref{lineareom}. Some results of numerical solutions are discussed in Sections \ref{Power Spectrum} and \ref{Tensor Mode Perturbations}.
\subsection{Continuum Limit and Second Order Perturbative Equations}
Before we actually solve Eqs.\Ref{lineareom} and \Ref{linearclosure}, we would like to firstly derive their lattice continuum limits $\mu\to 0$ (keeping $k$ fixed), and compare with some existing results of the gauge invariant cosmological perturbation theory.
First of all, the continuum limit of $C_v$, $C_{a,v}$, and $H_v$ reproduce $\mathcal C$, $\mathcal C_{a}$, and $h$:
\begin{eqnarray}
C_v&=&\mu^3 \mathcal C(v)+O(\mu^4),\label{Ccc}\\
C_{a,v}&=&\mu^3 \mathcal C_{a}(v)+O(\mu^4),\label{Cjccj}\\
H_v&=&\mu^3 h(v)+O(\mu^4)=\mu^3 \sqrt{\left|\mathcal C(v)^2-\frac{\a}{4}\sum_{a=1}^3\mathcal C_{a}(v)^2\right|}+O(\mu^4)
\end{eqnarray}
The above relations not only can be checked perturbatively up to $O(V^2)$ but also can be derived even non-perturbatively as in \cite{Han2020}. Note that the absolute-value in $H_v$ can be remove here at the perturbative level.
The lattice continuum limit $\mu\to0$ of linearized EOMs \Ref{lineareom} gives
\begin{eqnarray}
\frac{\mathrm d V^\rho(\t,{k})}{\mathrm d\t}+{\bf U}_0{}^\rho_{\ \nu}(\t,{k})\,V^\nu(\t,{k})=0,\quad {\bf U}_0{}^\rho_{\ \nu}(\t,{k})=\lim_{\mu\to0}{\bf U}{}^\rho_{\ \nu}\left(\mu,\t,{k}\right)\label{leomcon1}.
\end{eqnarray}
Matrix elements of ${\bf U}_0{}^\rho_{\ \nu}(\t,{k})$ are given explicitly in Appendix \ref{H0}. It is clear from Eq.\Ref{perturb} that in the continuum limit, $V^{\rho=1,\cdots,9}$ and $V^{\rho=10,\cdots,18}$ correspond to perturbations of $E^I_a$ and $A_I^a$ respectively.
\begin{eqnarray}
&&E^I_a(\t,\sigma)=P_0(\t)\delta^I_a+\delta E^I_a(\t,\sigma), \qquad\quad A_I^a(\t,\sigma)=\b K_0(\t)\delta^I_a+\delta A_I^a(\t,\sigma)\\
&&\delta E^I_a(\t,\vec{\sigma})=\int_\infty^\infty\frac{\mathrm d^3 k}{(2\pi)^3}e^{i \vec{k}\cdot \vec{\sigma}}\delta E^I_a(\t,\vec{k}), \quad\delta A_I^a(\t,\vec{\sigma})=\int_\infty^\infty\frac{\mathrm d^3 k}{(2\pi)^3}e^{i \vec{k}\cdot \vec{\sigma}}\delta A_I^a(\t,\vec{k})\\
&&V^{\rho}(\t,k)=\left(\delta E^I_a(\t,{k}),\delta A_I^a(\t,{k})\right)\left[1+O(\mu k)\right],\quad k\in\left[-{\pi}/{\mu},{\pi}/{\mu}\right]
\end{eqnarray}
We ignore the difference between $V^{\rho}(\t,k)$ and $(\delta E^I_a(\t,{k}),\delta A_I^a(\t,{k}))$ in the context of lattice continuum limit $\mu\to0$ (fixing $k$).
Here we choose the dust coordinate adapted to the lattice $\gamma$ so that $I=1,2,3$ is the coordinate index, i.e. the tangent vector of $e_I$ is the $I$-th coordinate basis.
The linearized closure condition \Ref{linearclosure} when $\mu\to0$ gives
\begin{eqnarray}
0&=&i k {V^{1}}+\beta K_0 ( {V^{6}}- {V^{9}} )-{V^{15}}+{V^{18}},\\
0&=&i k {V^{4}}-\beta K_0 ( {V^{5}}- {V^{8}} )+{V^{14}}-{V^{17}},\\
0&=&i k {V^{5}}+\beta K_0 ( {V^{4}}- {V^{7}} )-{V^{13}}+{V^{16}},
\label{leomcon2}
\end{eqnarray}
which coincide to the linearized Gauss constraint.
We solve linear equations \Ref{leomcon1} with $\rho=1,\cdots,9$ (containing $\mathrm d \mathcal Y^a(e_I(v))/\mathrm d\t$) for $\mathcal X^a(e_I(v))$ (perturbations of $\theta^a(e_I(v))$). Inserting solutions of $\mathcal X^a(e_I(v))$ into Eqs \Ref{leomcon1} with $\rho=10,\cdots,18$ (containing $\mathrm d \mathcal X^a(e_I(v))/\mathrm d\t$) we can obtain $\mathrm d \mathcal X^a(e_I(v))/\mathrm d\t$ as functions of $\mathcal Y^a(e_I(v))$ and $\mathrm d \mathcal Y^a(e_I(v))/\mathrm d\t$. Then by taking time derivative to Eqs \Ref{leomcon1} with $\rho=1,\cdots,9$ and inserting solutions of $\mathcal X^a(e_I(v))$ and $\mathrm d \mathcal X^a(e_I(v))/\mathrm d\t$, we obtain $9$ linear second order differential equations of $\mathcal Y^a(e_I(v))=V^\rho(v),\ \rho=1,\cdots,9$ (perturbations of $p^a(e_I(v))$):
\begin{eqnarray}
\frac{\mathrm d^2 V^\rho(\t,{k})}{\mathrm d\t^2}+\Fa^\rho_{\ \nu}(\t,{k})\frac{\mathrm d V^\nu(\t,{k})}{\mathrm d\t}+\Fb^\rho_{\ \nu}(\t,{k})V^\nu(\t,{k})=0,\quad \rho,\nu=1,\cdots,9.\label{2ndeom}
\end{eqnarray}
Inserting solutions of $\mathcal X^a(e_I(v))$ into linearized closure condition \Ref{leomcon2} gives 3 first order differential equations of $\mathcal Y^a(e_I(v))=V^\rho(v),\ \rho=1,\cdots,9$
\begin{eqnarray}
G^a(\t,{k})=\Fc^a_{\ \nu}(\t,{k})\frac{\mathrm d V^\nu(\t,{k})}{\mathrm d\t}+\Fd^a_{\ \nu}(\t,{k})V^\nu(\t,{k})=0,\quad \nu=1,\cdots,9,\quad a=1,2,3.\label{2ndclosure}
\end{eqnarray}
\cite{github} contains explicit expressions of Eqs.\Ref{2ndeom} and \Ref{2ndclosure} and Mathematica codes for following derivations.
In order to relate to the standard language of cosmological perturbation theory, we construct spatial metric perturbations from the continuum limit of Eq.\Ref{perturb}
\begin{eqnarray}
q_{IJ}(\t,k)=P_0(\t)\delta_{IJ}+ \delta h_{IJ}(\t,k).
\end{eqnarray}
where $\delta h_{IJ}$ is linear to $V^{\rho=1,\cdots,9}$.
\begin{eqnarray}
\delta h_{IJ}=\left(
\begin{array}{ccc}
-V^{1}+V^{2}+V^{3} & -V^{4}-V^{7} & -V^{5}-V^{8} \\
-V^{4}-V^{7} & V^{1}-V^{2}+V^{3} & -V^{6}-V^{9} \\
-V^{5}-V^{8} & -V^{6}-V^{9} & V^{1}+V^{2}-V^{3} \\
\end{array}
\right).\label{hVVVV}
\end{eqnarray}
It is standard to decompose $\delta h_{IJ}$ into components corresponding to scalar, tensor, vector modes
\begin{eqnarray}
\delta h_{IJ}=P_0\left(h^S_{IJ}+h^T_{IJ}+h^V_{IJ}\right),
\end{eqnarray}
each of which correspond to certain set of components of $V^\rho$ (see follows):
\begin{description}
\item[Scalar modes:] We impose the following ansatz
\begin{eqnarray}
V^\rho=0 \ \text{except for}\ \rho=1,2,3,6,9, \quad V^6 = - V^9, \quad V^2=V^3\equiv V^{1}-{ k^2 P_0}\mathcal E.\label{scalaransatz}
\end{eqnarray}
$V^2-V^3$ and $V^6+V^9$ belongs to tensor modes (see below). The linearized closure condition Eq.\Ref{2ndclosure} gives only one nontrivial equation
\begin{eqnarray}
\frac{\mathrm d }{\mathrm d \t} \left( \frac{V^9(\tau,k)}{P_0(\t)} \right) = \frac{4 i \alpha \beta k \sqrt{P_0(\tau)} \dot{P}_0(\t)}{4 \Lambda P_0(\tau)^2-3 \dot{P}_0(\t)^2} \frac{\mathrm d\psi(\tau,k)}{\mathrm d \t}, \quad \psi(\tau,k) &=& \frac{V^{1}(\t,k)}{2 P_0(\t)} \label{closmode}
\end{eqnarray}
Metric perturbations in scalar modes read
\begin{eqnarray}
h^S_{IJ}(\t, k)=\left(
\begin{array}{ccc}
2\psi(k)-2k^2\mathcal E(\t, k) & 0 & 0 \\
0 & 2\psi(k) & 0 \\
0 & 0 & 2\psi( k) \\
\end{array}
\right)
\end{eqnarray}
$V^6,V^9$ doesn't appear in metric perturbations. Then Eq.\Ref{2ndeom} reduces to
\begin{eqnarray}
\frac{\mathrm d^2 \psi(\t,k)}{\mathrm d\t^2} &=& -\frac{3 \dot{P}_0(\t) }{2 P_0(\t)} \frac{\mathrm d \psi(\t,k)}{\mathrm d\t} \, , \label{Smode0}\\
\frac{\mathrm d^2 \mathcal E(\t,k)}{\mathrm d\t^2} &=& \frac{\dot{P}_0(\t)}{P_0(\t) } \left(\frac{4 \alpha P_0(\t) }{3\dot{P}_0(\t)^2-4 \Lambda P_0(\t)^2} \frac{\mathrm d \psi(\t,k)}{\mathrm d\t} - \frac{3}{2} \frac{\mathrm d \mathcal E(\t,k)}{\mathrm d\t} \right) + \frac{ \psi (\t,k) }{P_0(\t) } \label{Smode}
\end{eqnarray}
plus a few other equations indicating the conservation law of closure condition \Ref{closmode}.
This result holds for both BK and Gaussian dusts.
\item[Tensor modes:] We impose the following ansatz
\begin{eqnarray}
V^\rho=0 \ \text{except for}\ \rho=2,3,6,9,\quad V^9=V^6\quad V^{3}=-V^2\ (\text{traceless}).\label{ansatzT}
\end{eqnarray}
Note that the mode $V^6-V^9$ has been considered above in scalar modes. The linearized closure condition Eq.\Ref{2ndclosure} is satisfied by the ansatz.
Metric perturbations in tensor modes read
\begin{eqnarray}
h^T_{IJ}(\t, k)=\frac{1}{P_0(\t)}\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 2V^3(\t, k) & -2V^9(\t, k)+C(k) \\
0 & -2V^9(\t, k)+C(k)P_0(\t) & -2V^3(\t, k)P_0(\t) \\
\end{array}
\right)
\end{eqnarray}
Eq.\Ref{2ndeom} reduces to
\begin{eqnarray}
k^2 h^T_{IJ}(\t,k)+\frac{3}{2} \dot{P}_0(\t) \frac{\mathrm d h^T_{IJ}(\t,k)}{\mathrm d\t}+P_0(\t) \frac{\mathrm d^2 h^T_{IJ}(\t,k)}{\mathrm d\t^2}=0.\label{Tmode}
\end{eqnarray}
This result holds for both BK and Gaussian dusts.
\item[Vector modes:] We impose the following ansatz
\begin{eqnarray}
V^\rho=0 \ \text{except for}\ \rho=4,5,7,8.\label{vectoransatz}
\end{eqnarray}
Metric perturbations in vector modes read
\begin{eqnarray}
h^V_{IJ}(\t, k)=-\frac{1}{P_0(\t)}\left(
\begin{array}{ccc}
0 & V^4(\t, k)+V^7(\t, k) & V^5(\t, k)+V^8(\t, k) \\
V^4(\t, k)+V^7(\t, k) & 0 & 0 \\
V^5(\t, k)+V^8(\t, k) & 0 & 0 \\
\end{array}
\right),\label{vectorhV}
\end{eqnarray}
Firstly, we insert the ansatz \Ref{vectoransatz} and make the replacements $V^4\to -h^V_{12}-V^7$ and $V^5\to-h^V_{13}-V^8$ in both Eqs.\Ref{2ndeom} and \Ref{2ndclosure}. Secondly we solve the linearized closure condition \Ref{2ndclosure} for $\dot{V}^7,\dot{V}^8$. Thirdly, we insert solutions of $\dot{V}^7,\dot{V}^8$ in the resulting Eq.\Ref{2ndeom} from above replacements. As a result, we obtain in total 4 nontrivial equations, in which 2 equations can be expressed only in terms of $h_{IJ}$:
\begin{eqnarray}
\frac{\dot{P}_0(\t) \left[4\a {P}_0(\t) \left(k^2-3 \Lambda {P}_0(\t)\right)+9 \dot{P}_0(\t)^2\right]}{4\a {P}_0(\t) \left(k^2-2 \Lambda {P}_0(\t)\right)+6 \dot{P}_0(\t)^2}\frac{\mathrm d h^V_{IJ}(\t,k)}{\mathrm d \t} +{P}_0(\t) \frac{\mathrm d^2 h^V_{IJ}(\t,k)}{\mathrm d\t^2}=0,\label{Vmode}
\end{eqnarray}
where $\a=1,0$ corresponds to the BK or Gaussian dust. Other 2 equations with explicit ${V}^7,{V}^8$ are the conservation law of the closure condition.
\end{description}
We count DOFs of $V^\rho$ (before imposing closure condition): Scalar modes have 3 DOFs ($\rho=1,2,6$), tensor modes have 2 DOFs ($\rho=3,9$), and vector modes have 4 DOFs ($\rho=4,5,7,8$). In total $2+3+4=9$ exhausts all DOFs of $V^{\rho=1,\cdots,9}$.
Scalar, tensor, and vector mode EOMs \Ref{Smode}, \Ref{Tmode}, and \Ref{Vmode} coincide with the ones derived in \cite{Giesel:2007wk}, where they are derived from classical gravity deparametrized by the BK dust and cosmological perturbations. Some details of comparing Eqs.\Ref{Smode}, \Ref{Tmode}, and \Ref{Vmode} to results in \cite{Giesel:2007wk} are presented in Section \ref{tina}. These results indicates that our cosmological perturbation theory derived from LQG has the correct semiclassical limit.
\subsection{Comparison with Results in \cite{Giesel:2007wk}}\label{tina}
This subsection focuses on the lattice continuum limit $\mu\to0$ (keeping $k$ fixed) of linearized semiclassical EOMs, and compares them to the results in \cite{Giesel:2007wk}.
The metric perturbation $\delta h_{IJ}$ can be decomposed into scalar, tensor, and vector modes \cite{Giesel:2007wk}:
\begin{eqnarray}
\delta h_{IJ} = P_0 (2 \psi \delta_{IJ}+2 \partial_{I} \partial_{J} \mathcal E + 2 \partial_{(I} \mathcal F_{J)} + h^T_{IJ})
\end{eqnarray}
where $\psi,\mathcal E$ parametrize scalar modes, and $\mathcal F,\ h^T$ parametrize vector and tensor modes. The above decomposition is in position space, while their Fourier transformations e.g. $\mathcal E(\t,\vec{k})=\int_{-\infty}^{\infty}{\mathrm d^3 \sigma}\,e^{-i \vec{k}\cdot \vec{\sigma}}\mathcal E(\t,\vec{\sigma})$ are given by $\partial_I\to ik_I$ and
\begin{eqnarray}
\psi &=& \frac{1}{2 P_0} V^{1},\label{littlepsi}\\
\mathcal E&=&-\frac{-2 V^{1}+V^{2}+V^{3}}{2 k^2 P_0},\label{huaE}\\
\mathcal F&=&\left(0,\ \frac{i (V^{4}+V^{7})}{k P_0},\ \frac{i (V^{5}+V^{8})}{k P_0}\right)^T\label{huaF}\\
h^T&=&\frac{1}{P_0}\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & V^{1}-V^{2}+V^{3} & -V^{6}-V^{9} \\
0 & -V^{6}-V^{9} & V^{1}+V^{2}-V^{3} \\
\end{array}\label{hTVVV}
\right)
\end{eqnarray}
by comparing to Eq.\Ref{hVVVV}. Here we have assumed the only nonzero component of $\vec{k}$ is $k^x = k$.
Following the standard cosmological perturbation theory, we define
\begin{eqnarray}
B&=&-\frac{-\beta K_0 V^{1}+ P_0 ( V^{11}+V^{12} )}{\beta P_0 \left(\Lambda P_0-3 K_0^2\right)}\\
S&=&\left(0,\ -\frac{2i k (\beta K_0 V^{4}+ P_0 V^{16})}{\beta P_0 \left(3 K_0^2-\Lambda P_0\right)},\ -\frac{2i k (\beta K_0 V^5 + P_0 V^{17} )}{\beta P_0 \left(3 K_0^2-\Lambda P_0\right)}\right)^T
\end{eqnarray}
For gravity coupled to BK dust, the dynamical shift vector $N_I=\mathcal C_I/h$ is conserved (see Eqs.\Ref{conserv0} and \Ref{lapseshift}). The background $\mathcal C_I=0$ so $N_I=\delta N_I$. $ \delta N_{I}$ can be parametrized by $B$ and $S_I$:
\begin{eqnarray}
\delta N_{I} = \sqrt{P_0}(ik_{I} B + S_{I})
\end{eqnarray}
We are going to express our linearized EOMs in terms of the conformal time $\eta$ by
\begin{eqnarray}
\frac{\mathrm d f(\eta)}{\mathrm d \eta} \equiv f'(\eta) = \sqrt{P_0(\tau)} \frac{\mathrm d f(\tau)}{\mathrm d \tau}=\sqrt{P_0(\tau)} \dot{f}(\t).
\end{eqnarray}
\begin{description}
\item[Scalar modes:] Eq.\Ref{scalaransatz} is equivalent to
\begin{eqnarray}
\delta h_{IJ} = 2 P_0 (\psi \delta_{IJ} - k_{I} k_{J} \mathcal E ), \quad S_J = 0
\end{eqnarray}
where $\psi$ and $\mathcal E$ coincide to \Ref{scalaransatz} and \Ref{closmode} respectively. The ansatz implies
\begin{eqnarray}
B = -\frac{8 {P_0}^{3/2} \dot{\psi} }{4 \Lambda P_0^2-3 \dot{P}_0^2} \label{Bpsi}
\end{eqnarray}
Using conformal time $\eta$ and changing variables, Eqs.\Ref{Smode0} and \Ref{Smode} can be rewrite as
\begin{eqnarray}
2 \mathcal H(\eta) \frac{\mathrm d \psi(\eta,k)}{\mathrm d\eta} +\frac{\mathrm d^2 \psi(\eta,k)}{\mathrm d\eta^2} &=& 0, \label{Smode1}\\
\frac{\mathrm d^2 \mathcal E(\eta,k)}{\mathrm d\eta^2} + 2 \mathcal H(\eta) \frac{\mathrm d \mathcal E(\eta,k)}{\mathrm d\eta} -\alpha \mathcal H(\eta) B(\eta,k ) -\psi (\eta,k ) &=& 0,\label{Smode2}
\end{eqnarray}
Eqs.\Ref{Smode1} and \Ref{Smode2} at $\a=1$ recover scalar mode equations (3.38) in \cite{Giesel:2007wk} when the additional scalar field is absent.
\item[Tensor modes:] Eq.\Ref{ansatzT} is equivalent to $\delta h_{IJ} =P_0 h^T_{IJ},\ B = 0$, and $S_{J} = 0$.
Eq.\Ref{Tmode} can be rewritten in terms of conformal time
\begin{eqnarray}
k^2 h^T_{IJ}(\eta,k)+ 2 \mathcal H \frac{\mathrm d h^T_{IJ}(\eta,k)}{\mathrm d\eta}+ \frac{\mathrm d^2 h^T_{IJ}(\eta,k)}{\mathrm d\eta^2}=0.\label{k2hT}
\end{eqnarray}
where $\mathcal H$ is the Hubble parameter in conformal time $\eta$:
\begin{eqnarray}
\mathcal H = \frac{1}{\sqrt{P_0(\eta)}} \frac{\mathrm d \sqrt{P_0(\eta)}}{\mathrm d\eta}\, \label{hubblec}.
\end{eqnarray}
This equation is the Fourier transform of Eq.(3.31) in \cite{Giesel:2007wk}:
\begin{eqnarray}
-\nabla^2 h^T_{IJ}+ 2 \mathcal H \frac{\mathrm d h^T_{IJ}}{\mathrm d\eta}+ \frac{\mathrm d^2 h^T_{IJ}}{\mathrm d\eta^2}=0.\label{nablahT}
\end{eqnarray}
\item[Vector modes:] Eq.\Ref{vectoransatz} is equivalent to
\begin{eqnarray}
\delta h_{IJ} = 2 P_0 \partial_{(I} \mathcal F_{J)}, \quad B = 0
\end{eqnarray}
After inserting solution of the linearized closure condition to $S_J$, we have
{ \begin{eqnarray}
S_1&=&0,\quad S_2=-\frac{4 i k P_0 \sqrt{P_0} }{2 k^2 P_0+3 \dot{P}_0^2-4 \Lambda P_0^2} \frac{\mathrm d \partial_{(1} \mathcal F_{2)}(\tau,k)}{\mathrm d \tau},\\
S_3&=&-\frac{4 i k P_0 \sqrt{P_0} }{2 k^2 P_0+3 \dot{P}_0^2-4 \Lambda P_0^2} \frac{\mathrm d \partial_{(1} \mathcal F_{3)}(\tau,k)}{\mathrm d \tau}.
\end{eqnarray}
We check that Eq. \Ref{Vmode}, and can be rewrite as
\begin{eqnarray}
2 \mathcal H(\eta) \frac{\mathrm d \partial_{(I}\mathcal F_{J)}(\eta,k)}{\mathrm d \eta} + \frac{\mathrm d \partial_{(I}\mathcal F_{J)}(\eta,k)}{\mathrm d \eta^2} - \a\mathcal H(\eta) {\partial_{(I} S_{J)}(\eta,k)} = 0 ,\label{FS=0}
\end{eqnarray}
which is the same as the vector mode equation (3.33) in \cite{Giesel:2007wk} when $\a=1$} Here e.g. $\partial_{(I}\mathcal F_{J)}(\eta,k)=i k_{(I}\mathcal F_{J)}(\eta,k)$. Furthermore, the conservation law $\frac{\mathrm d \delta N_J(\tau)}{\mathrm d \tau} = \frac{\mathrm d (\sqrt{P_0} S_J)}{\mathrm d \tau} = 0$ reduces Eq.\Ref{FS=0} with $\a=1$ to
\begin{eqnarray}
2 \mathcal{H} \partial_{(I}\mathcal V_{J)}+\partial_{(I}\mathcal V_{J)}^{\prime}=0,
\end{eqnarray}
where $\mathcal V_I=S_I-\mathcal F_I'$.
\end{description}
\section{Scalar Mode Perturbations}\label{Power Spectrum}
\subsection{Scalar Mode Perturbation Theory}\label{Scalar Mode Perturbation Theory}
In this subsection, we make some further analysis on scalar mode EOMs on the continuum. Entire Section \ref{Power Spectrum} specifically focus on gravity coupling to BK dust with $\a=1$. We define Bardeen potentials $\Phi$ and $\Psi$ which are used in the standard gauge-invariant cosmological perturbation theory,
\begin{eqnarray}
\Phi = - (\mathcal H(B - \mathcal E') + (B -\mathcal E')')= \mathcal H \mathcal E' + \mathcal E '' , \qquad \Psi = \psi + \mathcal H (B - \mathcal E').\label{PhiPsi}
\end{eqnarray}
Eqs.\Ref{Smode1} and \Ref{Smode2} can be expressed in terms of $\Phi$ and $\Psi$:
\begin{eqnarray}
2 \Phi \mathcal H'+\mathcal H \left( \Phi'+2 \Psi'\right)+\mathcal H^2 \Phi + \Psi'' &=& 0, \label{Smode3}\\
\Phi - \Psi &=& 0,\label{Smode4}
\end{eqnarray}
where we have used {$ -\mathcal H''+\mathcal H \mathcal H'+\mathcal H^3 =0$ from background EOMs\footnote{Background EOMs $\dot{P}_0=2K_0\sqrt{P_0},\ 2\sqrt{P_0}\dot{K}_0=-K_0^2+\L P_0$ are given by continuum limits $\mu\to0$ of Eqs.\Ref{cosm1} and \Ref{cosm2}. Using conformal time, the 1st equation is written as $K_0=\mathcal H$ while the 2nd equation is $2\mathcal H'+\mathcal H^2=\L P_0$, whose derivative gives $\mathcal H''+\mathcal H\ch'=\L P_0\mathcal H$. Inserting $\L P_0=2\mathcal H'+\mathcal H^2$ in $\mathcal H''+\mathcal H\ch'=\L P_0\mathcal H$ gives $\mathcal H''-\mathcal H\ch'-\mathcal H^3=0$.} and $\mathcal H B + B' =0$ from the conservation law $(\delta N_{I})' = 0$}.
Moreover, recall that we have conserved quantities $h$ and $\mathcal C_I$:
\begin{eqnarray}
h(k)&=&{\epsilon}_0+\delta\epsilon(k),\label{Econslin}\\
\delta N_1(k)&=&\delta\epsilon_1(k)/\epsilon_0,\quad \delta N_2(k)=\delta N_3(k)=0.\label{Nconslin}
\end{eqnarray}
${\epsilon}_0,\ \delta\epsilon(k)$, and $\delta\epsilon_1(k)$ are conserved. ${\epsilon}_0=\mathscr{E}/\mu^3$ is the coordinate energy density and $\delta\epsilon,\delta\epsilon_1$ are perturbations. $\delta N_2(k)=\delta N_3(k)=0$ because of $\vec{k}=(k,0,0)$. $h(k),\delta N_I(k)$ are Fourier transformations of $h(\sigma),\delta N_I(\sigma)$. Conservation laws \Ref{Econslin} and \Ref{Nconslin} can be expressed in terms of $\Phi,\ \mathcal E$, and $\psi$:
{\begin{eqnarray}
k^2 \Phi+3 \mathcal{H} \Phi^{\prime}+P_0{ \Lambda} \Ph
&=&\frac{\kappa}{4\sqrt{P_0}}\left[\delta \epsilon-{\epsilon}_0\left(5 \Phi-k^2 \mathcal E\right)\right],\label{Gpotential}\\
ik\, \psi^{\prime}&=&\frac{\kappa}{4 P_0} \delta \epsilon_{1},\label{psiepsilon}
\end{eqnarray}}
where $k^2 \Phi$ and $ik\,\psi^{\prime}$ are Fourier transformations of $-\nabla^2\Phi$ and $\partial_I\psi^{\prime}$. In deriving above relations, we have used $\Psi=\Phi$, $\mathcal H B + B' =0$, background EOMs $\dot{P}_0=2K_0\sqrt{P_0},\ 2\sqrt{P_0}\dot{K}_0=-K_0^2+\L P_0$ (continuum limits $\mu\to0$ of Eqs.\Ref{cosm1} and \Ref{cosm2}), and the background conservation law $3\sqrt{P_0} (2K_0^2-\Lambda P_0)=\kappa \epsilon_0$ (continuum limit of Eq.\Ref{scrE}).
Background EOMs $\dot{P}_0=2K_0\sqrt{P_0},\ 2\sqrt{P_0}\dot{K}_0=-K_0^2+\L P_0$ can be solved analytically by
\begin{eqnarray}
{P_0}(\t)= \left(\frac{\kappa \epsilon_0}{2 \Lambda }\right)^{\frac{2}{3}} \sinh ^{\frac{4}{3}}\left[\frac{\sqrt{3\Lambda }}{2}
(\t-\t_0)\right],\quad {K_0}(\t)=\left(\frac{\kappa \epsilon_0}{2 \Lambda }\right)^{\frac{1}{3}}\frac{\sqrt{\Lambda } \cosh \left(\frac{\sqrt{3\Lambda }}{2} (\t-\t_0)\right)}{\sqrt{3}\, {\sinh^{\frac{1}{3}} \left(\frac{\sqrt{3\Lambda }}{2} (\t-\t_0)\right)}}
\end{eqnarray}
where the integration constant $\t_0$ is the dust time at big-bang. Prefactors of $P_0,K_0$ are determined by the background conservation law $3\sqrt{P_0} (2K_0^2-\Lambda P_0)=\kappa \epsilon_0$. Applying the background solution $P_0(\t)$ to Eq.\Ref{Smode0}, we can solve Eq.\Ref{Smode0} for $\psi(\t,k)$
\begin{eqnarray}
\psi(\t,k)=C_2(k)-{ C_1(k) \frac{\sqrt{3\Lambda }}{2} \coth \left[\frac{\sqrt{3\Lambda }}{2} (\t-\t_0)\right]},\label{psisol}
\end{eqnarray}
where $C_1(k),C_2(k)$ are arbitrary functions of $k$. Then the conservation law Eq.\Ref{psiepsilon} implies
\begin{eqnarray}
\delta \epsilon_1= \frac{3 i k C_1(k) \epsilon_0}{2}.\label{epsilonC1}
\end{eqnarray}
Furthermore, inserting the solution $\psi(\t ,k)$ into Eq.\Ref{Bpsi} and $\Phi=\Psi = \psi + \mathcal H (B - \mathcal E')$, we obtain $\mathcal E'$ in terms of $\Phi$. Moreover we obtain $\Phi'$ in terms of $\Phi$ by $\Phi' = \psi' + [\mathcal H (B - \mathcal E')]'$ and $ \Phi-\mathcal H \mathcal E' = \mathcal E ''$. Inserting resulting $\Phi'$ in Eq.\Ref{Gpotential}, we solve $\mathcal E$ in terms of $\Phi$. Resulting $\mathcal E$ and $\dot{\mathcal E}=\mathcal E'/\sqrt{P_0}$ read
\begin{eqnarray}
\mathcal E(\t,k)&=&\frac{3 C_2(k)}{k^2}-\frac{\delta \epsilon }{k^2 \epsilon_0}+\frac{2\times {2}^{\frac{2}{3}} \Phi (\t,k) \sinh ^{\frac{2}{3}}\left[\frac{\sqrt{3\Lambda }}{2} (\t-\t_0)\right]}{\kappa ^{2/3} {\Lambda }^{1/3} \epsilon_0^{2/3}}\\
\frac{\mathrm d\mathcal E(\t,k)}{\mathrm d\t}&=&\frac{2\times 2^{\frac{2}{3}} \sqrt{3\Lambda } \left[C_2(k)-\Phi (\t,k)\right] \sinh ^{\frac{2}{3}}\left[\frac{\sqrt{3\Lambda }}{2} (\t-\t_0)\right] \text{csch}\left[\sqrt{3\Lambda } (\t-\t_0)\right]}{(\kappa \epsilon_0\sqrt{\L})^{2/3}}.
\end{eqnarray}
By above relations, a complete set of initial conditions is given by values of $\delta\epsilon_1,\ \delta\epsilon$ and the initial values $\Phi(\t_i,k)$ and $\psi(\t_i,k)$ ($\t_i$ is the initial time). In practically applying these initial conditions, $\delta\epsilon_1$ specifies $C_1(k)$ by Eq.\Ref{epsilonC1}, $\psi(\t_i,k)$ specifies $C_2(k)$, then $\delta\epsilon,\ \Phi(\t_i,k),\ C_2(k) $ determines $\mathcal E(\t_i,k),\ \dot{\mathcal E}(\t_i,k)$. After that, the solution $\psi(\t,k)$ is determined by $C_{1}(k),\ C_2(k)$ via Eq.\Ref{psisol}. The time evolution of $\mathcal E(\t,k)$ is determined by Eq.\Ref{Smode} and initial values of $\mathcal E(\t_i,k),\ \dot{\mathcal E}(\t_i,k)$.
\subsection{Initial Condition}
The time evolution of perturbations $V^\rho$ is determined by initial conditions. Our strategy for initial conditions is to firstly study initial conditions of the continuum theory discussed above, then translate these initial conditions to EOMs \Ref{lineareom} with finite $\mu$. In this section, we firstly focus on scalar mode perturbations.
Here is our choice of initial conditions for scalar modes: Firstly we require following properties of matter (the dust in our case) are not changed by perturbations:
\begin{eqnarray}
\delta\epsilon=\delta\epsilon_1=0,\label{initial1}
\end{eqnarray}
where $\delta\epsilon$ relates to the dust density, and $\delta\epsilon_1$ is the perturbation of $\mathcal C_I$ and relates to the velocity of the dust (recall Eq.\Ref{Pj=cc}). Here $\delta\epsilon=0$ means that there is no additional matter energy\footnote{Here the notion of energy is fixed by our foliation with dust coordinates.} pumped into the background cosmological spacetime, and is an analog of the initial vaccum state of matter often used in cosmological perturbation theory. $ \delta\epsilon_1=0$ implies $C_1(k)=0$, then $\psi=C_2(k)$ is independent of $\t$.
Furthermore we assume that the Bardeen potential vanishes at initial time $\t_i$:
\begin{eqnarray}
\Psi(\t_i,k)=\Phi(\t_i,k)=0,
\end{eqnarray}
and the initial value of $\psi$ is a constant:
\begin{eqnarray}
\psi(\t_i,k)=C_2\label{psiC2}
\end{eqnarray}
Therefore $C_2(k)=C_2$ is a constant independent of $k$, and the solution $\psi(\t,k)=C_2$ is a constant at all time.
The above specifies a complete set of initial conditions. They determine
\begin{eqnarray}
\mathcal E(\t_i,k)&=&\frac{3 C_2}{k^2},\label{ceinitial1}\\
\frac{\mathrm d\mathcal E(\t_i,k)}{\mathrm d\t}&=&\frac{2\times 2^{\frac{2}{3}} \sqrt{3\Lambda }\, C_2 \sinh ^{\frac{2}{3}}\left[\frac{\sqrt{3\Lambda }}{2} (\t_i-\t_0)\right] \text{csch}\left[\sqrt{3\Lambda } (\t_i-\t_0)\right]}{(\kappa \epsilon_0\sqrt{\L})^{2/3}},\label{ceinitial2}
\end{eqnarray}
as the initial condition for Eq.\Ref{Smode}.
We translate the above initial condition in the continuum to the initial condition for Eq.\Ref{lineareom} with finite $\mu$: Firstly we make following setup for initial values of $V^\rho,\dot{V}^\rho$ ($\rho=1,\cdots,9$) at the discrete level by relating to above initial values of $\psi,\dot{\psi},\mathcal E,\dot{\mathcal E}$:
\begin{eqnarray}
&&V^{4,7,5,8,6,9}(\t_i,k)=0,\\
&&V^2(\t_i,k)=V^3(\t_i,k)=2P_0(\t_i)\psi(\t_i,k)-k^2P_0(\t_i)\mathcal E(\t_i,k),\\
&& V^1(\t_i,k)=2P_0(\t_i)\psi(\t_i,k)\\
&&\dot{V}^{4,7,5,8,6,9}(\t_i,k)=0,\\
&&\dot{V}^2(\t_i,k)=\dot{V}^3(\t_i,k)=\frac{\mathrm d}{\mathrm d\t}\left[2P_0(\t)\psi(\t,k)-k^2P_0(\t)\mathcal E(\t,k)\right]_{\t=\t_i},\\
&& \dot{V}^1(\t_i,k)=\frac{\mathrm d}{\mathrm d\t}\left[2P_0(\t)\psi(\t,k)\right]_{\t=\t_i}
\end{eqnarray}
Next, we use components in Eq.\Ref{lineareom} with $\dot{V}^{\rho=1,\cdots,9}$ to solve $V^{\rho=10,\cdots,18}$ as a function of ${V}^{\rho=1,\cdots,9}$ and $\dot{V}^{\rho=1,\cdots,9}$. Initial values of $V^{\rho=10,\cdots,18}$ can be determined by using initial values of ${V}^{\rho=1,\cdots,9}$ and $\dot{V}^{\rho=1,\cdots,9}$. Initial values of $V^\rho$ solves linearized closure condition \Ref{linearclosure} approximately up to $O(\mu^4)$.
\subsection{Scalar Mode Power Spectrum}
We evolve with Eq.\Ref{lineareom} from the initial condition of $V^{\rho=1,\cdots,18}$ using 4th-order implicit Runge-Kutta method. With the solution $V^\rho(\t,k)$, we obtain $\mathcal E,\psi$ using Eqs.\Ref{littlepsi} and \Ref{huaE}, and Bardeen potential $\Psi$ using Eq.\Ref{PhiPsi}. FIG.\ref{scalarpower} demonstrates the power spectrum $P_\Psi=|\Psi(\t,k)|^2$ as a function of $k$ and how $P_\Psi$ evolves in time.
Note that in obtaining $\Psi$ at the discrete level, we apply Eqs.\Ref{littlepsi} and \Ref{huaE} to the discrete theory. Moreover we define a discrete version of shift vector $\delta N_I(v)=\frac{1}{2} (C_{a,v}/H_v)\sqrt{P_0}$ linearized in perturbations $V^\rho$, followed by Fourier transform $\delta N_I(v)\to \delta N_I(k)$ as in \Ref{fourier00}. We define $B(\t,k):=\delta N_1(k)/(ik\sqrt{P_0})$ for the discrete theory. $\mathcal H$ is given by Eq.\Ref{hubblec} with background $P_0$ from Eqs.\Ref{cosm1} and \Ref{cosm2}.
FIG.\ref{scalarpower} compares $P_\Psi$ from discrete EOMs \Ref{lineareom} (from LQG) and $P_\Psi$ from the continuum theory (in Section \ref{Scalar Mode Perturbation Theory}). We find that two $P_\Psi$'s coincide for relatively large $k$ while different for small $k$. The difference comes from $\mathcal E\sim V^\rho/(k^2P_0)$ by Eq.\Ref{huaE}: although differences between the discrete and continuum $V^\rho$'s are small and of $O(\mu)$, the small $k^2P_0$ amplifies these differences in $\mathcal E$. As shown in FIG.\ref{scalarpowerc}, the correction $|\epsilon(k^2 \mathcal E)| =| k^2 \left(\mathcal E - \mathcal E |_{\mu \to 0}\right)|$ of $k^2 \mathcal E \sim h_{11}^S - h_{22}^S$ is approximately time independent but depends on $k^2$ for relatively large $k$. However $|\epsilon(k^2 \mathcal E)|$ becomes independent of $k$ for small $k$ where the $\mu$ corrections mainly come from the cosmological background, e.g. from terms of $O(\mu K_0)$ in semiclassical EOMs\footnote{If we expand EOMs \Ref{lineareom} in $\mu$, $O(\mu)$ terms are proportional either to $\mu k$ or to $\mu K_0$.}. This leads to the fact that, at late time when $K_0$ becomes smaller, $|\epsilon(k^2 \mathcal E)|$ at small $k$ becomes smaller.
\begin{figure}[h]
\begin{minipage}[t]{0.58\textwidth}
\centering\vspace{2.5em}
\subfigure[\,]{ \includegraphics[width = 1\textwidth]{newdiscretephimu3} }
\end{minipage}
\begin{minipage}[t]{0.38\textwidth}
\subfigure[\,]{ \includegraphics[width = 1\textwidth]{smodeke} }
\subfigure[\,]{ \includegraphics[width = 1\textwidth]{diffsmodeke} \label{scalarpowerc}}
\end{minipage}
\caption{(a): Comparing the scalar mode power spectrum $P_\Psi$ of Bardeen potential between the classical continuum theory and the discrete theory from LQG. Dashed lines are $P_\Psi$ from the classical continuum theory, while solid curves are from the discrete theory. Different colors illustrate $P_\Psi$ (as functions of $k$) at different time $\t$. (b): Plots of $|k^2 \mathcal E| = | (h_{11}^S - h_{22}^S)/2 | $ vers $k^2$ at $\t=10$. In the continuum limit $\mathcal E$ does not depend on $k$. (c): Plots of $ |\epsilon(k^2 \mathcal E)|$ where $\epsilon(k^2 \mathcal E) = k^2 (\mathcal E - \mathcal E |_{\mu \to 0})$ are differences between solutions of discrete EOMs and classical continuum theory. Orange dashed line separates approximately the $k$ dominant region and the background dominant region. Initial condition of those plots are imposed at $\t_i=1$. Initial values of $\psi,\dot{\psi},\mathcal E,\dot{\mathcal E}$ are given by Eqs.\Ref{psiC2}, \Ref{ceinitial1}, and \Ref{ceinitial2} with $C_2=0.001$, $\epsilon_0=0.16$, and $\t_0=0$. Values of other parameters used in numerical computations are $\L=10^{-5}$, $\a=1$, $\b=1$, $\kappa=1$, and $\mu=10^{-3}$.}
\label{scalarpower}
\end{figure}
Note that the ultra-large $k$ with $k\mu\sim1$ breaks the approximation to the continuum theory, and cause differences between the discrete and continuum $V^\rho$'s. Thus the discrete and continuum theory give different $P_\Psi$'s in the ultra-large $k$ regime, although this difference is not shown in FIG.\ref{scalarpower}.
Eq.\Ref{lineareom} with finite $\mu$ couples vector and tensor modes to scalar modes, while these couplings are turned off by the continuum limit $\mu\to0$. With finite $\mu$, the scalar model initial condition can excite tensor and vector modes in the time evolution. FIGs.\ref{phitvmode} plots power spectrums $P_T=|h^T_{23}(\t,k)|^2$ and $P_\mathcal V=|\vec{\mathcal V}(\t,k)|^2$ at different time $\t$ evolved from the scalar mode initial condition. Here $h^T_{IJ}$ are given by Eq.\Ref{hTVVV} with $V^\rho$ satisfying discrete EOMs. $\mathcal V_I=S_I-\mathcal F_I'$ where $\mathcal F_I$ are given by Eq.\Ref{huaF} and $S=(0,\delta N_2,\delta N_3)/\sqrt{P_0}$ with $V^\rho$, $\delta N_I$, and $P_0$ satisfying EOMs with finite $\mu$. FIGs.\ref{phitvmode} demonstrates that the scalar mode initial condition excites both tensor and vector mode perturbations by EOMs with finite $\mu$. These tensor and vector modes are all small and of higher order in $k\mu$ (since $\mu$ is of length dimension, $\mu$-expansion is the same as $k\mu$-expansion for relatively large $k$), while they can smoothly grow as $k$ becoming large. FIG.\ref{phiclo} plots the error of linearized closure condition in the $\t$-evolution and finds that it is much smaller than $\mu^4$.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.46\textwidth]{newdphitmodemu3}\quad
\includegraphics[width = 0.46\textwidth]{newphivmodemu3}
\end{center}
\caption{The left panel plots tensor mode power spectrums $P_T=|h^T_{23}(\t,k)|^2$ as functions of $k$ at different $\t$, evolving from the scalar mode initial condition (the same as in FIG.\ref{scalarpower}). $P_T$ at different $\t$ are illustrated by different colors. The other $h^T_{IJ}$ component $|h^T_{22}(\t,k)|^2$ is even smaller than $|h^T_{23}(\t,k)|^2$, thus is not demonstrated. The right panel plots vector mode power spectrums $P_\mathcal V=|\vec{\mathcal V}(\t,k)|^2$ at different $\t$. }
\label{phitvmode}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{newphiclomu3}
\end{center}
\caption{This figure plots the error of closure condition $|G|^2=\sum_{i=1}^3G^2_i$ where $G_i$ are given by Eqs.\Ref{linearclosure}. $|G|^2$ at different $\t$ are illustrated by different colors. $|G|^2$ is much smaller than $\mu^4=10^{-12}$ in the plotted range of $k$.}
\label{phiclo}
\end{figure}
\section{Tensor Mode Perturbations}\label{Tensor Mode Perturbations}
\subsection{Modified Graviton Dispersion Relation}\label{Modified Graviton Dispersion Relation}
We consider Eq.\Ref{lineareom} in the late-time limit $K_0={P_0'}/{(2P_0)}\to0$ and absent of cosmological constant $\L=0$, and we insert the tensor mode ansatz: $V^{1}=0$, $V^2=-V^3$, and $V^6=V^9$ which turn off scalar modes at late time.
The closure condition Eq.\Ref{linearclosure} and the compatibility of Eq.\Ref{lineareom} at late time leads to $V^{4,5,10,13,14,16,17}= 0,\ V^{12}=-V^{11},\ V^{15}= V^{18}$. Eq.\Ref{lineareom} at late time gives the following wave equation for the tensor modes metric $h^T_{IJ}$ (valid for both $\a=0,1$):
\begin{eqnarray}
\o (k)^2 h^T_{IJ}(\eta,k)+\frac{\mathrm d^2 h^T_{IJ}(\eta,k)}{\mathrm d \eta^2}=0,\qquad \omega(k)^2=\frac{\sin ^2(k \mu ) }{\mu ^2}\left[\left(\beta ^2+1\right) \cos (k \mu )-\beta ^2\right].\label{graviton}
\end{eqnarray}
The tensor mode metric perturbation relates to $V^{2,3,6,9}$ by
\begin{eqnarray}
h^T=\frac{1}{P_0}\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & V^{2}+V^{3} & -V^{6}-V^{9} \\
0 & -V^{6}-V^{9} & V^{2}-V^{3} \\
\end{array}
\right)
\end{eqnarray}
Solutions of Eq.\Ref{graviton} are spin-2 gravitons with a modified dispersion relation $\o(k)^2$. We expand the $\o(k)^2$ in terms of $\mu$
\begin{eqnarray}
\o(k)^2=k^2\left[1-\frac{1}{6} \mu ^2k^2 \left(3 \beta ^2 +5 \right)+O\left(\mu ^3k^3\right)\right]
\end{eqnarray}
Gravitons travel in the speed of light in the continuum limit $\mu\to 0$ or the long wavelength limit $k\ll \mu^{-1}$, while less than speed of light for finite $\mu$. The finite $\mu$ generates a higher derivative term $O(k^4)$ in the wave equation of $h^T_{IJ}$
\begin{eqnarray}
\frac{\mathrm d^2 h^T_{IJ}(\eta,k)}{\mathrm d \eta^2}+k^2 h^T_{IJ}(\eta,k)-\frac{1}{6} \mu ^2k^4 \left(3 \beta ^2 +5 \right) h^T_{IJ}(\eta,k)+O\left(\mu ^3k^3\right)=0.
\end{eqnarray}
The result \Ref{graviton}, derived from top to down in the full theory of LQG, proves that LQG can give spin-2 gravitons as low energy excitations. The modified dispersion relation Eq.\Ref{graviton} is the same as the one in \cite{Dapor:2020jvc} obtained by expanding the LQG Hamiltonian on the flat spacetime.
In our opinion, the dispersion relation \Ref{graviton} is only valid in the long-wavelength regime $k\ll\mu^{-1}$. If we admit $k\sim\mu^{-1}$, there exists $k$ resulting $\o^2<0$ and corresponding to non-propagating modes. In addition, the dispersion relation \Ref{graviton} indicates that there are 2 non-negative $k$'s giving the same $\o^2\geq0$ (see Figure \ref{doubledisper}). For example, at low energy $\o^2= 0$ corresponds to
\begin{eqnarray}
k=0,\quad k=\mu^{-1}\arccos\left(\frac{\b^2}{\b^2+1}\right)
\end{eqnarray}
where $k=0$ corresponds to the graviton, but the second mode is a spurious low energy excitation. $\o^2(k)$ expanded at this spurious mode gives
\begin{eqnarray}
\o^2(k)\mu^2&=&-\left(\beta ^2+1\right) \left(1-\frac{\beta ^4}{\left(\beta ^2+1\right)^2}\right)^{3/2} \mu\, p-\frac{5}{2} \beta ^2 \left(1-\frac{\beta ^4}{\left(\beta ^2+1\right)^2}\right) \mu ^2\, p^2+O\left(\mu^3p^3\right)\\
p&=&k-{\mu^{-1} }\cos ^{-1}\left(\frac{\beta ^2}{\beta ^2+1}\right)
\end{eqnarray}
which has no analog in continuum field theories. The existence of this spurious mode should be due to the regime that $\ell_P\ll\mu$ on which our discussion have focused. Beyond this regime, e.g. in $\mu\sim\ell_P$ which may be more physically sensible, the dispersion relation \Ref{graviton} should be modified in large $k$ by $O(\ell_P^2)$ corrections so that the spurious mode may be removed/changed. Taking the continuum limit $\mu\to0$ before $\ell_P\to0$ might be physically relevant since it removes the lattice-dependence at the quantum level, but it is beyond the scope of this paper. However, the perturbation theory derived here from the semiclassical approximation $\ell_P\to0$ (while keeping $\mu$ finite) should be only viewed as an effective theory which only valid in the long wavelength regime $k\ll\mu^{-1}$, while behavior at $k\sim\mu^{-1}$ should not be trusted before $O(\hbar)$-corrections are implemented.
Eq.\Ref{lineareom} with finite $\mu$ contains another 2 nontrivial equations showing couplings between $V^{\rho=2,6}$ (tensor modes) and $V^{\rho=7,8}$ (vector modes). Defining $u^\rho\equiv V^\rho/P_0$, these equations (at late time) are shown below by expanding in $\mu$
\begin{eqnarray}
0&=&{u'^7}(\eta )+\frac{1}{2}k\mu \left[-i u'^2(\eta )+\beta k u^6(\eta )\right] \nonumber \\
& &\qquad+ \frac{2 i \alpha \beta k}{3 K_0^2 \sqrt{P_0}} \left[2 u^8(\eta )+\mu\left( i k u^2(\eta )+\beta u'^6(\eta )\right)\right] +O\left(\mu ^2\right),\label{coupleVT1}\\
0&=&{u'^8}(\eta )-\frac{1}{2}k\mu \left[-i u'^2(\eta )+\beta k u^6(\eta )\right]\nonumber \\
& &\qquad+ \frac{2 i \alpha \beta k}{3 K_0^2 \sqrt{P_0}} \left[ - 2 u^7(\eta ) + \mu \left( i k u^2(\eta )+\beta u'^6(\eta )\right)\right] +O\left(\mu ^2\right), \label{coupleVT2}
\end{eqnarray}
while equations before the $\mu$-expansion is too long to be shown here and they can be downloaded in \cite{github}. Couplings between $V^{\rho=2,6}$ and $V^{\rho=7,8}$ disappear in Eqs.\Ref{coupleVT1} and \Ref{coupleVT2} when $\mu\to0$.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{doubledisper}
\end{center}
\caption{Plot of the dispersion relation \Ref{graviton}. $\o^2=0$ corresponds to 2 non-negative $k$'s.}
\label{doubledisper}
\end{figure}
\subsection{Tensor Mode Power Spectrum}
We set $\L=0$ in the discussion of tensor mode. The background EOMs \Ref{cosm1} and \Ref{cosm2} with $\L=0$ and $\mu\to0$ can be solved analytically with $\mathcal H =4/\eta$. Then the tensor mode EOM \Ref{k2hT} at $\mu\to0$ can be written as a differential equation in terms of $x=k\eta$:
\begin{eqnarray}
h^T_{IJ}+ \frac{8}{x } \frac{\mathrm d h^T_{IJ}}{\mathrm d x}+ \frac{\mathrm d^2 h^T_{IJ}}{\mathrm d x^2}=0,\quad x=k\eta
\end{eqnarray}
Therefore solutions at the continuum limit are functions of $k\eta$: $h^T_{IJ}=h^T_{IJ}(k\eta)$.
Semiclassical EOMs with finite $\mu$ can be solved numerically for both the cosmological background and tensor mode perturbations. Both initial conditions of the background $P_0,K_0$ and tensor mode perturbations are imposed at the conformal time $\eta_i$. The tensor mode initial condition is given by {$u^{1,4,5,7,8}=0, u'^{1,4,5,7,8}=0, u^3=-u^2=u^6=u^9\neq0$ and $u'^3=-u'^2=u'^6=u'^9\neq0$}. FIG.\ref{tmodeini2} plots time evolutions of tensor mode perturbations $h_{IJ}^T$ as functions of $k\eta$ (at different $k$), where we find approximately $h_{IJ}^T=h^T_{IJ}(k\eta)$ (depending on $k$ only through $k\eta$) at late time, and $h_{IJ}^T=h^T_{IJ}(k, k\eta)$ at early time (especially when we evolve from $\eta$ toward the bounce). FIG.\ref{tmodediffh2} plots the difference $\epsilon(h^T_{IJ})=h^T_{IJ}-h^T_{IJ}|_{\mu\to0}$ between solutions of discrete and continuum EOMs, and shows that $|\epsilon(h^T_{IJ})|$ is small and less than $O(\mu)$. When we evolve from $\eta$ toward the bounce (with large curvature), $|\epsilon(h^T_{IJ})|$ becomes larger, and suggests that the continuum theory approximates well to the discrete theory only when the curvature is small.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.48\textwidth]{tmodeini222}\quad
\includegraphics[width = 0.48\textwidth]{tmodeini223}
\end{center}
\caption{Plots of $|h^T_{22}|$ and $|h^T_{23}|$ as functions of $k\eta$ at different $k$. Colored dots illustrate their initial values. The initial condition is imposed at $\eta_i=0.05$. Initial values are $u^{1,4,5,7,8}=0, u^3=-u^2=u^6=u^9=0.00999754$ and $u'^3=-u'^2=u'^6=u'^9=-0.000099816$. Values of parameters are $\L=0$, $\a=1$, $\b=1$, $\kappa=1$, and $\mu=10^{-3}$. }
\label{tmodeini2}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.46\textwidth]{tmodediffh22}\quad
\includegraphics[width = 0.46\textwidth]{tmodediffh23}
\end{center}
\caption{Plots of $|\epsilon(h^T_{22})|$ and $|\epsilon(h^T_{23})|$ where $\epsilon(h^T_{IJ})=h^T_{IJ}-h^T_{IJ}|_{\mu\to0}$ are differences between solutions of discrete and continuum EOMs. Colored dots illustrate initial values. }
\label{tmodediffh2}
\end{figure}
FIG.\ref{powertt} plots power spectrums $|h^T_{22}(\eta, k\eta)|$ and $|h^T_{23}(\eta, k\eta)|$ as functions of $k\eta$ at different conformal time $\eta$. When $k$ are relatively large (but still much smaller than $\mu^{-1}$), power spectrums with finite $\mu$ approximately coincide with results from the continuum EOM \Ref{k2hT}, but depart from the continuum results for small $k$, similar to the scalar mode power spectrum FIG.\ref{scalarpower}. To understand this departure, we recall that Eq.\Ref{graviton} is an approximation of tensor mode EOMs at the late time, so at earlier time we have
\begin{eqnarray}
\o (k)^2 h^T_{IJ}(\eta,k)+\frac{\mathrm d^2 h^T_{IJ}(\eta,k)}{\mathrm d \eta^2}+O(K_0)=0.
\end{eqnarray}
$O(K_0)$ collects terms vanishing as $K_0\to0$ while non-vanishing at earlier time. The small $k$ suppresses the first term and make the term with background $K_0$ stand out, while the background $K_0$ is different between the finite $\mu$ and $\mu\to0$. $\mu\to0$ removes the difference between discrete and continuum theory.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.46\textwidth]{powertt}\quad
\includegraphics[width = 0.46\textwidth]{powertt23}
\end{center}
\caption{Colored stars illustrate power spectrums $|h^T_{22}|$ and $|h^T_{23}|$, resulting from EOMs with $\mu=10^{-3}$, as functions of $k\eta$ at different conformal times $\eta$. Different colors label different $\eta$. The blue curve is the power spectrum from the continuum theory. The initial condition is the same as in FIG.\ref{tmodeini2}.}
\label{powertt}
\end{figure}
Semiclassical EOMs couples tensor modes to scalar and vector modes when $\mu$ is finite. FIGs.\ref{tmodesini2} and \ref{tmodevini2} plot scalar mode perturbations $h^S_{11}=(-V^1+V^2+V^3)/P_0,\ u^1=V^1/P_0$ and vector mode perturbations $h^V_{12}$ (see Eq.\Ref{vectorhV}) excited by the tensor mode initial condition. Their amplitudes $|h^S_{11}|$, $|u_1|$, and $|h^V_{12}|$ are all less than $O(\mu)$, and suppressed by the lattice continuum limit $\mu\to0$. On the other hand, fixing the value of $\mu$, small effects from $\mu$ can accumulate and increase $|h^S_{11}|$, $|u_1|$, and $|h^V_{12}|$ when the evolution time is long.
We note a different between the analysis here and in subsection \ref{Modified Graviton Dispersion Relation}: Here the tensor-mode initial condition is at early time, and there are scalar mode perturbations excited at late time, while in the discussion in subsection \ref{Modified Graviton Dispersion Relation}, we turn off scalar modes at late time.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.46\textwidth]{tmodesini2}\quad
\includegraphics[width = 0.46\textwidth]{tmodeuini2}
\end{center}
\caption{Time evolution of the scalar modes $h^S_{11}=(-V^1+V^2+V^3)/P_0$ and $u^1=V^1/P_0$ excited by the tensor mode at different $k$, with the same initial condition as in FIG.\ref{tmodeini2}.}
\label{tmodesini2}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{tmodevini2}
\end{center}
\caption{Time evolution of the vector mode $h^V_{12}$ excited by the tensor mode at different $k$, with the same initial condition as in FIG.\ref{tmodeini2}.}
\label{tmodevini2}
\end{figure}
\section{Conclusion and Outlook}
In this work we derive the cosmological perturbation theory from the path integral formulation of the full LQG and the semiclassical approximation. In the lattice continuum limit, the result is consistent with the classical gravity-dust theory. Numerical studies of discrete semiclassical EOMs indicate some interesting corrections to power spectrums especially in the regime where wavelengths are very long. Our result provides a new routine of extracting physical predictions in cosmology from the full theory of LQG.
Our approach is a preliminary step toward relating LQG to observations, and at present has a few open issues which should be addressed in the future. These issues are summarized below:
\begin{enumerate}
\item This work focuses on pure gravity coupling to dusts, while neglecting radiative matter. This work also doesn't take into account the inflation. We have to generalize our work to include these perspectives in order to make contact with observations of Cosmic Microwave Background (CMB). Fortunately, it is straight-forward to generalize the reduced phase space LQG to standard-model matter couplings \cite{Giesel:2007wn}. Deriving matter couplings in the path integral is a work currently undergoing. Therefore in the near future, we should be able to include the radiative matter and inflation in our analysis. The result should be compared with the recent work \cite{Giesel:2020bht}, where the inflationary cosmological perturbation theory is studied in the classical theory of gravity and matter coupling to dust.
\item The initial state plays a crucial role in the cosmological perturbation theory. In above discussions, initial conditions of perturbations are translated from corresponding initial conditions in the classical continuum theory. We have neglect impacts on the initial condition of $O(\mu)$ from the discreteness and of $O(\ell_P^2)$ from quantum effects, while both of them are nontrivial at early time in cosmology. Therefore choices of initial states for cosmology, including their semiclassical and quantum properties, should be an important aspect to be understood in the future.
\end{enumerate}
\section*{Acknowledgements}
This work receives support from the National Science Foundation through grant PHY-1912278. Computations in this work is mainly carried out on the HPC server at Fudan University in China and the KoKo HPC server at Florida Atlantic University. The authors acknowledge Ling-Yan Hung for sharing the computational resource at Fudan University.
|
1,108,101,565,514 | arxiv | \section{Introduction}
Uncertainty\cite{uncertainty,kennard1927,robertson1929,Busch2007} and entanglement\cite{schrodinger-35,text,horodecki09} are two major cornerstones of quantum mechanics. These characteristics cause quantum mechanics to differ from classical mechanics. Quantum uncertainty provides a limit on the
precision of measurement for incompatible observables. The most typical expression of uncertainty relation is $\Delta x \Delta p \geq \hbar / 2$, where $\Delta$
is the standard deviation. Recently, researchers have analyzed different expressions of uncertainty relations, such as entropic uncertainty relations\cite{Wehner2010,Coles2017}
from the context of quantum information and generalized uncertainty principle\cite{GUP} from the context of Planck scale physics. Even though entanglement has been studied
since the discovery of quantum mechanics\cite{schrodinger-35}, it has been extensively explored for the last few decades with the development of quantum technology.
Entanglement is used as a physical resource in various quantum information processing, such as quantum teleportation\cite{teleportation,Luo2019},
superdense coding\cite{superdense}, quantum cloning\cite{clon}, quantum cryptography\cite{cryptography,cryptography2}, quantum
metrology\cite{metro17}, and quantum computer\cite{qcreview,computer}. Furthermore, with many researchers trying to realize such quantum information processing in the laboratory for the last few decades, quantum cryptography and quantum computer seems to approaching the commercial level\cite{white,ibm}.
Although these two phenomena seem to be distinct properties of quantum mechanics, there is some connection, albeit unclear, between them because of the fact that both are strongly dependent on the interaction between subsystems. For example, the uncertainty of a given system was computed in Ref.\cite{han97,han99} to discuss on the effect of the Feynman's rest of universe\cite{feyman72}. The ignoring of the effect of the rest of the universe was shown to increase
uncertainty and entropy in the target system (system in which we are interested). In other words, if the target system is one of subsystems of a whole system and it interacts with other subsystems, its
uncertainty and entanglement monotonically increase with increasing interaction strength. More specifically, let us consider two coupled harmonic
oscillator system, with the following Hamiltonian:
\begin{equation}
\label{hamil-2}
H_2 = \frac{1}{2} \left( p_1^2 + p_2^2 \right) + \frac{1}{2} \left[k_0 (x_1^2 + x_2^2) + J (x_1 - x_2)^2 \right].
\end{equation}
If we assume that the two oscillators, say $A$ and $B$, were in each ground state, the uncertainty and entanglement of
formation (EoF)\cite{benn96} are both given by\cite{park18}
\begin{equation}
\label{two-vacuum}
\left( \Delta x \Delta p \right)_{A,B}^2 = \frac{1}{4} \left( \frac{1 + \xi}{1 - \xi} \right)^2 \hspace{1.0cm}
{\cal E}_F = - \ln (1 - \xi) - \frac{\xi}{1 - \xi} \ln \xi
\end{equation}
where $\hbar = 1$ and $\xi = \left\{(\sqrt{k_0 + 2 J} - \sqrt{k_0}) / (\sqrt{k_0 + 2 J} + \sqrt{k_0}) \right\}^2$. This shows that both
$\left( \Delta x \Delta p \right)_{A,B}^2$ and ${\cal E}_F$ increase with increasing the coupling constant $J$. Thus, in this case, uncertainty and entanglement are implicitly related to each other via $\xi$. Duan et al.\cite{criterion} used quantum uncertainty to provide a sufficient criterion for entanglement in continuous variable systems.
Mandilara and Cerf\cite{cerf12} showed that the uncertainty relation for all eigenstates in the single harmonic oscillator system is saturated with respect
to Gaussianity.
So far, EoF cannot be exactly computed in the coupled harmonic oscillator system except in the ground state because of the non-Gaussian nature of exciting
states\footnote{The R\'enyi-$\alpha$ entropies of few non-Gaussian states have been derived in \cite{Ilki-18}.}.
As EoF and
uncertainty exhibit similar behavior, as shown in Eq. (\ref{two-vacuum}), the uncertainty may be used as a measure of entanglement after appropriate rescaling if EoF cannot be computed exactly.
Therefore, in order to understand the entanglement more profoundly in the continuous variable system,
it is important to examine the uncertainty of the arbitrary excited states in the coupled
harmonic oscillator system.
In there any other similarity between EoF and uncertainty? EoF is believed to have the additivity property\cite{open05}, even though the property has still not been solved
completely. For mixed states, EoF is generally defined by a convex-roof
method\cite{benn96,uhlmann99-1} as follows:
\begin{equation}
\label{roof}
{\cal E}_F (\rho) = \min \sum_i p_i {\cal E}_F (\rho_i),
\end{equation}
where the minimum is taken over all possible ensembles of pure states with $\sum_i p_i = 1$. Let $\rho^{(i)} \hspace{.2cm} (i=1,2)$ be two bipartite density
matrices, and $\rho = \rho^{(1)} \otimes \rho^{(2)}$. If we regard $\rho$ as a bipartite state, where $\rho^{(1)}$ and $\rho^{(2)}$ belong to each party,
Eq. (\ref{roof}) guarantees ${\cal E}_F (\rho) \leq {\cal E}_F (\rho^{(1)}) + {\cal E}_F (\rho^{(2)})$. The additivity conjecture of EoF is that the
equality always holds; this has been demonstrated through various examples in \cite{additivity}.
In this paper, we show that uncertainty in the coupled harmonic oscillator system also has a particular additive property, which we call the sum rule.
We present this sum rule in the coupled harmonic oscillator system when the parameters are arbitrarily time-dependent.
The paper is organized as follows. In Sec. II we derive $(\Delta x)^2$ and $(\Delta p)^2$ of arbitrary excited states by making use of explicit Wigner distribution in the
single harmonic oscillator system when the frequency is arbitrary time-dependent. It is shown that the time-dependence of frequency as well as energy level increase
the uncertainty $\Delta x \Delta p$. In Sec. III we examine the uncertainties in the two-coupled harmonic oscillator system when the parameters are arbitrarily time-dependent.
In this section we derive the uncertainties $(\Delta x)^2$ and $(\Delta p)^2$ of the first or second oscillator when two oscillators are at the arbitrary excited states. It is
shown that $(\Delta x)^2$ and $(\Delta p)^2$ are just the arithmetic average of uncertainties of two single oscillators. We call this additive property as ''sum rule of quantum uncertainties''.
As a by-product, the purity function of the reduced state
is explicitly computed in this section by making use of the reduced Wigner distribution function. In Sec. IV we examine the uncertainties in the $N$-coupled harmonic oscillator system when
the parameters are arbitrarily time-dependent. It is shown that the arithmetic average property of uncertainties arising at $N=2$ is not generally maintained when $N \geq 3$. However, this additive
property is recovered when $(N-1)$ quantum numbers are equal. In Sec. V conclusion and further discussion are briefly given.
\section{Uncertainty for arbitrary excited state of single harmonic oscillator with arbitrary time-dependent frequency}
We start with a simple single harmonic oscillator Hamiltonian with arbitrary time-dependent frequency: $H_1 = \frac{p^2}{2} + \frac{1}{2} \omega^2 (t) x^2$.
This simple model is important for studying the squeezed states, which appear in various branches of physics, such as quantum
optics\cite{walls-83,loudon-87,wu-87,schnabel-17} and cosmology\cite{grishchuk-90,grishchuk-93,einhorn-03,kiefer-07}.
The time-dependent Schr\"{o}dinger equation (TDSE) of this system was examined in detail in \cite{lewis68,rhode89,lohe09,pires-19}. The linearly independent solutions $\psi_n (x, t) \hspace{.1cm} (n=0, 1, \cdots)$ are expressed
in the following form\cite{lewis68,lohe09}:
\begin{equation}
\label{TDSE-1}
\psi_n (x, t) = e^{-i E_n \tau(t)} \frac{1}{\sqrt{2^n n!}} \left(\frac{\omega'}{\pi} \right)^{1/4}
H_n (\sqrt{\omega'} x) e^{-\frac{v}{2} x^2}
\end{equation}
where $\omega' = \frac{\omega(0)}{b^2}$ and
\begin{equation}
\label{TDSE-2}
v = \omega' - i \frac{\dot{b}}{b}
\hspace{1.0cm} E_n = \left( n + \frac{1}{2} \right) \omega(0) \hspace{1.0cm} \tau (t) = \int_0^t \frac{d s}{b^2 (s)}.
\end{equation}
In Eq. (\ref{TDSE-1}) $H_n (z)$ is the $n^{th}$-order Hermite polynomial and $b(t)$ satisfies the nonlinear Ermakov equation,
\begin{equation}
\label{ermakov-1}
\ddot{b} + \omega^2 (t) b = \frac{\omega^2 (0)}{b^3}
\end{equation}
with $b(0) = 1$ and $\dot{b} (0) = 0$. As shown in Eq. (\ref{TDSE-1}), $b(t)$ plays the role of scaling the frequency. Solutions of the Ermakov equation were discussed in \cite{lohe09,pinney50,gritsev-10,campo-16}. If $\omega(t)$ is time independent, $b(t)$ is simply one.
If $\omega (t)$ is instantly changed as follows:
\begin{eqnarray}
\label{instant-1}
\omega (t) = \left\{ \begin{array}{cc}
\omega_i & \hspace{1.0cm} t = 0 \\
\omega_f & \hspace{1.0cm} t > 0,
\end{array} \right.
\end{eqnarray}
then $b(t)$ becomes
\begin{equation}
\label{scale-1}
b(t) = \sqrt{ \frac{\omega_f^2 - \omega_i^2}{2 \omega_f^2} \cos (2 \omega_f t) + \frac{\omega_f^2 + \omega_i^2}{2 \omega_f^2}}.
\end{equation}
Of course, for a more general case of $\omega (t)$, the nonlinear Ermakov equation should be solved numerically or approximately.
The $d$-dimensional Wigner distribution function\cite{feyman72,noz91} is defined in terms of the phase space variables in the following form:
\begin{equation}
\label{dwigner}
W ({\bm x}, {\bm p}: t) = \frac{1}{\pi^d} \int d {\bm z} e^{-2 i {\bm p} \cdot {\bm z}} \Psi^* ({\bm x} + {\bm z}: t)
\Psi ({\bm x} - {\bm z}: t)
\end{equation}
where ${\bm x} = (x_1, x_2, \cdots, x_d)$, ${\bm p} = (p_1, p_2, \cdots, p_d)$, and $ \Psi ({\bm r}: t)$ is a wave function of a given system.
The Wigner distribution function is used to compute the expectation
values. For example, the expectation value of $f(x_1, p_1)$ can be computed by
\begin{equation}
\label{expectation}
\langle f(x_1, p_1) \rangle = \int d {\bm x} d {\bm p} f(x_1, p_1) W ({\bm x}, {\bm p}: t).
\end{equation}
Moreover, the Wigner distribution function has information on the substate of density matrix $\rho ({\bm x}, {\bm x'}: t) = \Psi ({\bm x}: t) \Psi^* ({\bm x'}: t)$. If $\rho_A (x_1, x_1': t) = \mbox{Tr}_{2, 3, \cdots, d} \rho ({\bm x}, {\bm x'}: t)$, the purity function of $\rho_A$ can be computed as
\begin{equation}
\label{purity}
P_A (t) \equiv \mbox{Tr} \rho_A^2 = 2 \pi \int dx_1 dp_1 W^2 (x_1, p_1: t),
\end{equation}
where $W(x_1, p_1: t) = \int dx_2 \cdots dx_d dp_2 \cdots dp_d W ({\bm x}, {\bm p}: t)$.
To explicitly compute the Wigner distribution function of $H_1$, we set $d=1$ and $\Psi = \psi_n (x, t)$ of Eq. (\ref{TDSE-1}) in Eq. (\ref{dwigner}).
The integral in Eq. (\ref{dwigner}) can be computed by using\cite{integral}
\begin{eqnarray}
\label{main}
&&\int_{-\infty}^{\infty} dx e^{-p x^2 + 2 q x} H_m (a x + b) H_n (c x + d) \\ \nonumber
&&= \sqrt{\frac{\pi}{p}} e^{\frac{q^2}{p}} \sum_{k=0}^{\min (m,n)} \left( \begin{array}{c} m \\ k \end{array} \right)
\left( \begin{array}{c} n \\ k \end{array} \right) k! \left( 1 - \frac{a^2}{p} \right)^{\frac{m-k}{2}} \left( 1 - \frac{c^2}{p} \right)^{\frac{n-k}{2}}
\left( \frac{2 a c}{p} \right)^k \\ \nonumber
&& \hspace{3.0cm} \times H_{m-k} \left(\frac{b + \frac{a q }{p}}{\sqrt{1 - \frac{a^2}{p}}} \right) H_{n-k} \left(\frac{d + \frac{c q }{p}}{\sqrt{1 - \frac{c^2}{p}}} \right).
\end{eqnarray}
Then, the Wigner distribution function for $H_1$ can be written as follows:
\begin{eqnarray}
\label{wigner-H1}
&&W_n (x, p: t) = \frac{1}{\pi} \exp \left[ - \omega' x^2 - \frac{1}{\omega'} \left( p + \frac{\dot{b}}{b} x \right)^2 \right] \\ \nonumber
&&\hspace{3.0cm} \times\sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array} \right) (-1)^k \frac{2^{n-k}}{(n-k)!}
\left[ \omega' x^2 + \frac{1}{\omega'} \left( p + \frac{\dot{b}}{b} x \right)^2 \right]^{n - k} \\ \nonumber
&& = \frac{1}{n! \pi} \exp \left[ - \omega' x^2 - \frac{1}{\omega'} \left( p + \frac{\dot{b}}{b} x \right)^2 \right]
U \left(-n, 1, 2 \left[ \omega' x^2 + \frac{1}{\omega'} \left( p + \frac{\dot{b}}{b} x \right)^2 \right] \right),
\end{eqnarray}
where $U(a, b, z)$ is a confluent hypergeometric function.
It is straightforward to show that $\int dx dp W_n (x, p: t) = 2 \pi \int dx dp W_n^2 (x, p: t) = 1$, which guarantees $\psi_n (x, t)$ is the normalized pure state.
By using the Wigner distribution function, it is straightforward to show that for non-negative integer $m$,
$\langle x^{2 m+1} \rangle = \langle p^{2m + 1} \rangle = 0$ and
\begin{eqnarray}
\label{single-expect}
&&\langle x^{2 m} \rangle = \frac{2^n (m + n)!}{m! n! \sqrt{\pi} \omega'^m} \Gamma \left( \frac{2 m + 1}{2} \right) {_2F_1} \left( -n, -n: -n - m: 1 / 2 \right) \\ \nonumber
&&\langle p^{2 m} \rangle = \frac{2^n (m + n)!}{m! n! \sqrt{\pi}} \Gamma \left( \frac{2 m + 1}{2} \right)
\left[ \omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right]^m {_2F_1} \left( -n, -n: -n - m: 1 / 2 \right),
\end{eqnarray}
where $\Gamma (z)$ and ${_2F_1} (a, b: c: z)$ are gamma and hypergeometric functions. Thus, the uncertainties for $x$ and $p$ are
\begin{equation}
\label{uncertainty-1}
\left(\Delta x\right)^2 = \frac{n + \frac{1}{2}}{\omega'} \hspace{1.0cm}
\left(\Delta p\right)^2 = \left( n + \frac{1}{2} \right) \left[\omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right],
\end{equation}
which yield an uncertainty relation
\begin{equation}
\label{uncertainty-2}
\left( \Delta x \Delta p \right)^2 = \left( n + \frac{1}{2} \right)^2 \left[1 + \frac{1}{\omega'^2} \left( \frac{\dot{b}}{b} \right)^2 \right].
\end{equation}
Thus, the time-dependence of $\omega$ as well as energy level $n$ increase the uncertainty $\Delta x \Delta p$.
\section{Uncertainty for arbitrary excited state of two-coupled harmonic oscillator system with arbitrary time-dependent parameters}
Now, let us consider the Hamiltonian (\ref{hamil-2}) again when $k_0$ and $J$ are arbitrarily time dependent. It is not difficult to show that the
Hamiltonian is diagonalized by introducing normal coordinates $y_1 = (x_1 + x_2) / \sqrt{2}$ and $y_2 = (-x_1 + x_2) / \sqrt{2}$, and their conjugate momenta $\pi_1$
and $\pi_2$ with normal mode frequencies $\omega_1 = \sqrt{k_0}$ and $\omega_2 = \sqrt{k_0 + 2 J}$, respectively. If two oscillators are in the
$n^{th}$ and $m^{th}$ states, we will show in the following that the uncertainties for $x_j$ and $p_j$ ($j = 1, 2$) are just the arithmetic mean of two single oscillators;
that is,
\begin{eqnarray}
\label{two-coupled}
&&\left(\Delta x_1\right)^2 = \left(\Delta x_2\right)^2 = \frac{1}{2} \left[ \frac{2 n + 1}{2 \omega_1'} + \frac{2 m + 1}{2 \omega_2'} \right]
\\ \nonumber
&&\left(\Delta p_1\right)^2 = \left(\Delta p_2\right)^2 = \frac{1}{2} \left[ \frac{2 n + 1}{2} \left\{ \omega_1' + \frac{1}{\omega_1'} \left( \frac{\dot{b}_1}{b_1} \right)^2 \right\} +
\frac{2 m + 1}{2} \left\{ \omega_2' + \frac{1}{\omega_2'} \left( \frac{\dot{b}_2}{b_2} \right)^2 \right\} \right],
\end{eqnarray}
where $\omega_j' = \omega_j (0) / b_j^2$ ($j=1,2$), and $b_j$ satisfy their own nonlinear Ermakov equations, $\ddot{b}_j + \omega_j^2 (t) b_j = \frac{\omega_j^2 (0)}{b_j^2}$ with $\dot{b}_j (0) = 0$ and $b_j (0) = 1$. We will call this arithmetic
average additivity as ``sum rule of quantum uncertainty''.
To prove Eq. (\ref{two-coupled}), we start with solutions of TDSE for $H_2$ in terms of $y_j$, which is
\begin{eqnarray}
\label{TDSE-3}
&&\psi_{n,m} (x_1, x_2: t) = \frac{1}{\sqrt{2^{(n+m)} n! m!}} \left( \frac{\omega_1' \omega_2'}{\pi^2} \right)^{1/4} H_n (\sqrt{\omega_1'} y_1)
H_m (\sqrt{\omega_2'} y_2) \\ \nonumber
&& \hspace{3.0cm} \times \exp \left[ -i (E_{n,1} \tau_1 + E_{m,2} \tau_2) - \frac{1}{2} \left(v_1 y_1^2 + v_2 y_2^2 \right) \right],
\end{eqnarray}
where $E_{m,j} = \left( m + \frac{1}{2} \right) \omega_j (0)$, $\tau_j = \int_0^t \frac{ds}{b_j^2(s)}$, and
$v_j = \omega_j' - i \frac{\dot{b}_j}{b_j}$. Now, let us compute the Wigner distribution functions of the $H_2$ system by setting
$\Psi ({\bm x}: t) = \psi_{n,m} (x_1, x_2: t)$ in Eq. (\ref{dwigner}). If we change Eq. (\ref{TDSE-3}) into the original phase space variables
$x_j$ and $p_j$, and insert them into Eq. (\ref{dwigner}), the computation of the Wigner distribution function is highly complicated. However, this difficulty can be avoided.
Since $y_j$ are orthogonal normal modes, they preserve the inner product and $2$-dimensional volume elements. Thus, the Wigner distribution function for
$H_2$ is simply reduced to
\begin{equation}
\label{wigner-H2}
W_{n,m} (x_1, x_2: p_1, p_2: t) = W_n (y_1, \pi_1: t) \bigg|_{\omega' \rightarrow \omega_1', b \rightarrow b_1}
\times W_m (y_2, \pi_2: t) \bigg|_{ \omega' \rightarrow \omega_2' , b \rightarrow b_2 },
\end{equation}
where $W_n$ is a Wigner distribution function of a single harmonic oscillator given in Eq. (\ref{wigner-H1}).
At this stage we want to digress little bit. Sometimes, we need to derive the lower-dimensional reduced Wigner distribution function to explore the properties of
the reduced quantum state. Although we can compute the $2$-dimensional Wigner distribution function quickly by using the normal mode,
the derivation of the reduced $1$-dimensional Wigner distribution function is very complicated problem. For example, let us consider
$W_{n,m} (x_1, p_1: t) \equiv \int dx_2 dp_2 W_{n,m} (x_1, x_2: p_1, p_2: t)$; here, the difficulty arises because $dx_2 dp_2$ is not invariant measure
in the normal modes. Thus, we should compute the reduced Wigner distribution function by using the original coordinates and conjugate momenta.
After long and tedious calculation, it is possible to show that
\begin{eqnarray}
\label{reduced-wigner-1}
&&W_{n,m} (x_1, p_1: t) = \frac{\sqrt{4 \omega_1' \omega_2'}}{\pi} \sum_{k=0}^n \sum_{\ell = 0}^m
\left( \begin{array}{c} n \\ k \end{array} \right) \left( \begin{array}{c} m \\ \ell \end{array} \right)
\frac{(-1)^{k + \ell}}{(n - k)! (m - \ell)!} 2^{(n + m) - (k + \ell)} \\ \nonumber
&& \times \left( - \frac{\partial}{\partial \mu_1} \right)^{n-k} \left( - \frac{\partial}{\partial \mu_2} \right)^{m-\ell}
\frac{1}{\sqrt{\Omega (\mu_1, \mu_2: t)}} \exp \left[ - 2 \frac{\Theta (x_1, p_1: \mu_1, \mu_2: t)}{\Omega (\mu_1, \mu_2: t)} \right]
\Bigg|_{\mu_1 = \mu_2 = 1},
\end{eqnarray}
where
\begin{eqnarray}
\label{reduced-wigner-2}
&&\Omega (\mu_1, \mu_2: t) = \omega_1' \omega_2' (\mu_1^2 + \mu_2^2) + \left[ \omega_1'^2 + \omega_2'^2 +
\left(\frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right] \mu_1 \mu_2 \\ \nonumber
&&\Theta (x_1, p_1: \mu_1, \mu_2: t) = \omega_1' \left[ \omega_2'^2 x_1^2 + \left( p_1 + \frac{\dot{b}_2}{b_2} x_1 \right)^2 \right] \mu_1^2
\mu_2 + \omega_2' \left[ \omega_1'^2 x_1^2 + \left( p_1 + \frac{\dot{b}_1}{b_1} x_1 \right)^2 \right] \mu_1 \mu_2^2.
\end{eqnarray}
Thus, the reduced Wigner distribution function for $n = m = 0$ is easily computed by
\begin{equation}
\label{reduced-wigner-3}
W_{0, 0} (x_1, p_1: t) = \frac{1}{\pi} \sqrt{\frac{4 \omega_1' \omega_2'}{\Omega(1, 1: t)}} e^{-2 \Theta (x_1, p_1: 1, 1,: t) / \Omega(1, 1: t)}.
\end{equation}
The purity function of the $A$-oscillator is defined as $P_{n, m}^A (t) = \mbox{tr} \rho_{n, m}^2 (x_1, x_1': t)$, where $\rho_{n, m} (x_1, x_1': t)$ is an effective state of the $A$-oscillator
derived by taking a partial trace to $\rho_{n,m} (x_1, x_2: x_1', x_2': t) = \psi_{n, m} (x_1, x_2: t) \psi_{n,m}^* (x_1', x_2' :t)$ over $B$-oscillator. Then,
$P_{0,0}^A (t)$ can be computed from $W_{0,0} (x_1, p_1: t)$ as follows:
\begin{equation}
\label{purity-1}
P_{0, 0}^A (t) = 2 \pi \int dx_1 dp_1 W_{0, 0}^2(x_1, p_1: t) = 2 \sqrt{z}
\end{equation}
where $z = \omega_1' \omega_2' / \Omega(1, 1: t)$. From Eq. (\ref{reduced-wigner-1}) one can show directly
$\int dx_1 dp_1 W_{n,m}(x_1, p_1: t) = 1$ by making use of simple binomial formula. Furthermore, it is possible to show that
\begin{eqnarray}
\label{purity-mn}
&& 2 \pi \int dx_1 dp_1 W_{m,n}^2 (x_1, p_1: t)
= 4 \sqrt{\omega_1' \omega_2'} \sum_{k,k' = 0}^n \sum_{\ell,\ell' = 0}^m
\left( \begin{array}{c} n \\ k \end{array} \right) \left( \begin{array}{c} n \\ k' \end{array} \right) \\ \nonumber
&&\times \left( \begin{array}{c} m \\ \ell \end{array} \right) \left( \begin{array}{c} m \\ \ell' \end{array} \right)
\frac{(-1)^{k+k'+\ell+\ell'}}{(n-k)! (n-k')! (m-\ell)! (m- \ell')!}
2^{2(n+m) - (k+k'+\ell+\ell')} \\ \nonumber
&& \left( - \frac{\partial}{\partial \mu_1} \right)^{n-k}
\left( - \frac{\partial}{\partial \nu_1} \right)^{n-k'} \left( - \frac{\partial}{\partial \mu_2} \right)^{m-\ell}
\left( - \frac{\partial}{\partial \nu_2} \right)^{m-\ell'} \frac{1}{\sqrt{\Gamma (\mu_1, \mu_2: \nu_1, \nu_2)}}
\Bigg|_{\mu_1 = \mu_2 = \nu_1 = \nu_2 = 1}
\end{eqnarray}
where
\begin{eqnarray}
\label{Big-Gamma}
&&\Gamma(\mu_1, \mu_2: \nu_1, \nu_2 ) = \omega_1' \omega_2' \left[ \mu_1^2 \nu_1^2 (\mu_2 + \nu_2)^2 + \mu_2^2 \nu_2^2 (\mu_1 + \nu_1)^2 \right] \\ \nonumber
&&\hspace{2.0cm} + \mu_1 \mu_2 \nu_1 \nu_2 (\mu_1 + \nu_1) (\mu_2 + \nu_2) \left[ \omega_1'^2 + \omega_2'^2 + \left( \frac{\dot{b}_1}{b_1} - \frac{\dot{b}_2}{b_2} \right)^2 \right].
\end{eqnarray}
If we define the ratios
\begin{equation}
\label{ratio}
\gamma_n = \frac{P_{n,0}^A (t)}{P_{0, 0}^A (t)} \hspace{1.0cm}
\delta_n = \frac{P_{n,n}^A (t)}{P_{0, 0}^A (t)} ,
\end{equation}
they are summarized at Table I. We expect that $\gamma_n$ and $\delta_n$ decrease with increasing $n$ because more excited states seem to be
more mixed.
\begin{center}
\begin{tabular}{c|c|c} \hline \hline
$n$ & $\gamma_n$ & $\delta_n$ \\ \hline \hline
$1$ & $\frac{1}{4} (3 - 4 z)$ & $\frac{1}{16} (9 - 40 z + 144 z^2)$ \\
$2$ & $\frac{1}{64} (41 - 104 z + 144 z^2)$ & $\frac{1}{4096} (1681 - 19344 z + 256608 z^2 - 1440000 z^3 + 2822400 z^4)$ \\
$3 $ & $\frac{1}{256} (147 - 540 z + 1488 z^2 - 1600z^3)$ & too long \\ \hline
\end{tabular}
\vspace{0.3cm}
Table I: The ratios $\gamma_n$ and $\delta_n$ for $n=1, 2, 3$
\end{center}
\vspace{0.5cm}
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=5.0cm]{fig1a.pdf} \hspace{0.5cm}
\includegraphics[height=5.0cm]{fig1b.pdf}
\caption[fig1]{(Color online) The time dependence of the ratios (a) $\gamma_n$ and (b) $\delta_n$
when $k_0 (0) = J (0) = 1$ and $k_0 (t) = J(t) = 2 \hspace{.2cm} (t>0)$.
As expected, the figures exhibit that the effective states for $A$-oscillator are more and more mixed with increasing $n$. }
\end{center}
\end{figure}
The time dependence of $\gamma_n$ and $\delta_n$ is plotted in Fig. 1(a) and Fig. 1(b) when $k_0 (0) = J (0) = 1$ and $k_0 (t) = J(t) = 2 \hspace{.2cm} (t>0)$.
As expected, the figures exhibit that the effective states for the $A$-oscillator is more and more mixed with increasing $n$.
Remarkably, this figure shows that the reduced state of $\rho_{n,n}$ is more mixed than that of $\rho_{n,0}$.
Now, let us return to discuss about the uncertainties. From Eq. (\ref{wigner-H2}), it is easy to show that
$\langle y_j^{2 m + 1} \rangle = \langle \pi_j^{2 m + 1} \rangle = 0$, and $\langle y_j^{2 m}\rangle$ and $\langle \pi_j^{2 m} \rangle$ are equal to
$\langle x^{2 m} \rangle$ and $\langle p^{2 m} \rangle$, respectively, in Eq. (\ref{single-expect}) with changing $\omega' \rightarrow \omega_j'$ and
$b \rightarrow b_j$. Accordingly, by using this fact and the normal modes, it is easy to prove Eq. (\ref{two-coupled}).
\section{Uncertainty for arbitrary excited state of $N$-coupled harmonic oscillator system with arbitrary time-dependent parameters}
To check whether the property of arithmetic average for uncertainties is maintained in a multi-coupled harmonic oscillator system or not, we first consider a three-coupled
harmonic oscillator system with the following Hamiltonian:
\begin{equation}
\label{hamil-3}
H_3 = \frac{1}{2} (p_1^2 + p_2^2 + p_3^2) + \frac{1}{2}\left[ k_0 (t) (x_1^2 + x_2^2 + x_3^2) + J (t) \left\{ (x_1 - x_2)^2 + (x_1 - x_3)^2 + (x_2 - x_3)^2 \right\} \right].
\end{equation}
The normal mode coordinates of $H_3$ is $y_1 = (x_1 + x_2 + x_3) / \sqrt{3}$, $y_2 = (x_1 - x_2) / \sqrt{2}$, and $y_3 = (x_1 + x_2 - 2 x_3) / \sqrt{6}$
with normal mode frequencies $\omega_1 = \sqrt{k_0}$ and $\omega_2 = \omega_3 = \sqrt{k_0 + 3 J} \equiv \omega$. If three oscillators
are in the $n^{th}$, $m^{th}$, and $\ell^{th}$ states, the $3$-dimensional Wigner distribution function can be computed as follows:
\begin{equation}
\label{wigner-H3}
W_{n,m} (x_1, x_2, x_3: p_1, p_2, p_3: t) = W_n (y_1, \pi_1: t) \bigg|_{\omega' \rightarrow \omega_1', b \rightarrow b_1}
\times W_m (y_2, \pi_2: t) \times W_{\ell} (y_3, \pi_3: t)
\end{equation}
where $\pi_j$ represent the conjugate momenta of $y_j$ and $W_n$ is the Wigner distribution function of the single harmonic oscillator given in Eq. (\ref{wigner-H1}).
Of course, $b_1(t)$ and $b(t)$ are solutions for Ermakov equations for $\omega_1$ and $\omega$, and $\omega_1' = \omega_1 (0) / b_1^2 (t)$ and
$\omega' = \omega (0) / b^2 (t)$. Thus, Eq. (\ref{uncertainty-1}) and Wigner distribution function (\ref{wigner-H3}) imply
\begin{eqnarray}
\label{revise-1}
&&\left(\Delta y_1\right)^2 = \frac{n + \frac{1}{2}}{\omega_1'} \hspace{1.0cm}
\left(\Delta \pi_1 \right)^2 = \left( n + \frac{1}{2} \right) \left[\omega_1' + \frac{1}{\omega_1'} \left( \frac{\dot{b_1}}{b_1} \right)^2 \right] \\ \nonumber
&&\left(\Delta y_2\right)^2 = \frac{m + \frac{1}{2}}{\omega'} \hspace{1.0cm}
\left(\Delta \pi_2 \right)^2 = \left( m + \frac{1}{2} \right) \left[\omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right] \\ \nonumber
&&\left(\Delta y_3\right)^2 = \frac{\ell + \frac{1}{2}}{\omega'} \hspace{1.0cm}
\left(\Delta \pi_3 \right)^2 = \left( \ell + \frac{1}{2} \right) \left[\omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right].
\end{eqnarray}
Then, it is straightforward to show
\begin{eqnarray}
\label{three-coupled}
&& \left(\Delta x_1 \right)^2 = \left( \Delta x_2 \right)^2 = \frac{1}{3} \left[ \frac{2 n + 1}{2 \omega_1'} + \frac{3 (2 m + 1) + (2 \ell + 1)}{4 \omega'} \right] \\ \nonumber
&& \left( \Delta x_3 \right)^2 = \frac{1}{3} \left[ \frac{2 n + 1}{2 \omega_1'} + 2 \frac{2 \ell + 1}{2 \omega'} \right] \\ \nonumber
&& \left( \Delta p_1 \right)^2 = \left( \Delta p_2 \right)^2 = \frac{1}{3} \left[ \frac{2 n + 1}{2} \left\{ \omega_1' + \frac{1}{\omega_1'} \left( \frac{\dot{b}_1}{b_1} \right)^2 \right\} + \frac{3 (2 m + 1) + (2 \ell + 1)}{4} \left\{ \omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right\} \right] \\ \nonumber
&& \left( \Delta p_3 \right)^2 = \frac{1}{3} \left[ \frac{2 n + 1}{2} \left\{ \omega_1' + \frac{1}{\omega_1'} \left( \frac{\dot{b}_1}{b_1} \right)^2 \right\} + 2 \frac{2 \ell + 1}{2} \left\{ \omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right\} \right].
\end{eqnarray}
Thus, the property of the arithmetic average in uncertainties is not maintained when $N=3$. However, this property is recovered when $m = \ell$.
Finally, let us consider the $N$-coupled harmonic oscillator system with the following Hamiltonian:
\begin{equation}
\label{hamil-N}
H_N = \frac{1}{2} \sum_{i=1}^N p_i^2 + \frac{1}{2} \left[ k_0 (t) \sum_{i=1}^N x_i^2 + J(t) \sum_{i<j}^N (x_i - x_j)^2 \right].
\end{equation}
This system is diagonalized by introducing the normal mode coordinates $y_1 = (x_1 + x_2 + \cdots + x_N) / \sqrt{N}$ and $y_j = (x_1 + x_2 + \cdots + x_{j-1} - (j-1) x_j) / \sqrt{j (j-1)} \hspace{.2cm} (j = 2, 3, \cdots, N)$ with normal mode frequencies $\omega_1 = \sqrt{k_0}$ and $\omega_2 = \omega_3 = \cdots = \omega_N = \sqrt{k_0 + N J} \equiv \omega$. If $N$ oscillators are in the $n_1^{th}, n_2^{th}, \cdots, n_N^{th}$ states, the $N$-dimensional
Wigner distribution function can be written as follows:
\begin{equation}
\label{wigner-H-N}
W_{n_1, n_2, \cdots, n_N} ({\bm x}, {\bm p}: t) = W_{n_1} (y_1, \pi_1: t) \bigg|_{\omega' \rightarrow \omega_1', b \rightarrow b_1}
\times \prod_{j=2}^N W_{n_j} (y_j, \pi_j: t),
\end{equation}
where $\pi_j$ represent the conjugate momenta of $y_j$ and $W_n$ is the Wigner distribution function of the single harmonic oscillator given in Eq. (\ref{wigner-H1}).
Then, it is straightforward to show that
\begin{eqnarray}
\label{N-coupled}
&&\left( \Delta x_j \right)^2 = \frac{1}{N} \left[ \frac{2 n_1+1}{2 \omega_1'} + \frac{1}{2 \omega'} \left\{ \frac{2 N (j - 1)}{j} n_j + 2 N \sum_{k = j+1}^N \frac{n_k}{k (k - 1)} + (N-1) \right\} \right] \\ \nonumber
&&\left( \Delta p_j \right)^2 = \frac{1}{N} \Bigg[ \frac{2 n_1+1}{2} \left\{ \omega_1' + \frac{1}{\omega_1'} \left( \frac{\dot{b}_1}{b_1} \right)^2 \right\} \\ \nonumber
&&\hspace{2.0cm}+ \frac{1}{2} \left\{ \frac{2 N (j - 1)}{j} n_j + 2 N \sum_{k = j+1}^N \frac{n_k}{k (k - 1)} + (N-1) \right\}
\left\{ \omega' + \frac{1}{\omega'} \left( \frac{\dot{b}}{b} \right)^2 \right\} \Bigg].
\end{eqnarray}
Eq. (\ref{N-coupled}) can be shown to reproduces Eq. (\ref{two-coupled}) and Eq. (\ref{three-coupled}) when $N=2$ and $N=3$ if the quantum numbers
$n_1$, $n_2$, and $n_3$ are replaced by $n$, $m$, and $\ell$, respectively. If $n_2 = n_3 = \cdots = n_N$, one can show that $\left( \Delta x_j \right)^2$ and
$\left( \Delta p_j \right)^2$ are independent of $j$ and they are just arithmetic average of uncertainties for each oscillator.
\section{Conclusions}
In this paper we computed the uncertainties of $(\Delta x)^2$ and $(\Delta p)^2$ analytically in an $N$-coupled harmonic oscillator system. When $N = 2$,
these uncertainties are just the arithmetic average of uncertainties of two single harmonic oscillators. We call this
property as ``sum rule of quantum uncertainty''. However, this additive property is not generally
maintained when $N \geq 3$ but is recovered in an $N$-coupled oscillator system only when $(N-1)$ quantum numbers are equal.
Our calculation can be generalized to a more general case. For example, let us consider the following Hamiltonian
\begin{eqnarray}
\label{H3-tilde}
&&\tilde{H}_3 = \frac{1}{2} \left( p_1^2 + p_2^2 + p_3^2 \right) + \frac{1}{2} \bigg[ k_0 (t) \left( x_1^2 + x_2^2 + x_3^2 \right) + J_{12} (t) (x_1 - x_2)^2
\\ \nonumber
&&\hspace{6.0cm} + J_{13} (t) (x_1 - x_3)^2 + J_{23} (t) (x_2 - x_3)^2 \bigg].
\end{eqnarray}
In this case, the normal mode coordinates become
\begin{eqnarray}
\label{normal-H3-tilde}
&& y_1 = \frac{1}{\sqrt{3}} (x_1 + x_2 + x_3) \\ \nonumber
&& y_+ = A_+ (-J_{12} + J_{23} - \zeta) x_1 + A_+ (J_{12} - J_{13} + \zeta) x_2 + A_+ (J_{13} - J_{23}) x_3 \\ \nonumber
&& y_- = A_- (-J_{12} + J_{23} + \zeta) x_1 + A_- (J_{12} - J_{13} - \zeta) x_2 + A_- (J_{13} - J_{23}) x_3
\end{eqnarray}
with $\zeta = \sqrt{J_{12}^2 + J_{13}^2 + J_{23}^2 - (J_{12} J_{13} + J_{12} J_{23} + J_{13} J_{23})}$ and
\begin{equation}
\label{Apm}
A_{\pm} = \frac{1}{J_{13} - J_{23}}\sqrt{\frac{2 \zeta \pm (J_{13} + J_{23} - 2 J_{12})}{6 \zeta}}.
\end{equation}
Moreover, the normal mode frequencies are given by $\omega_1 = \sqrt{k_0}$ and $\omega_{\pm} = \sqrt{k_0 + J_{12} + J_{13} + J_{23} \pm \zeta}$.
If the three oscillators are in the $n^{th}$, $m^{th}$, and $\ell^{th}$ exciting states, our procedure yields
\begin{eqnarray}
\label{uncertainty-H3-tilde}
&&\left( \Delta x_1 \right)^2
= \frac{1}{3} \frac{2 n + 1}{2 \omega_1'} + A_+^2 u_-^2 \frac{2 m + 1}{2 \omega_+'} +
A_-^2 u_+^2 \frac{2 \ell + 1}{2 \omega_-'} \\ \nonumber
&&\left( \Delta x_2 \right)^2
= \frac{1}{3} \frac{2 n + 1}{2 \omega_1'} + A_+^2 v_+^2 \frac{2 m + 1}{2 \omega_+'} +
A_-^2 v_-^2 \frac{2 \ell + 1}{2 \omega_-'} \\ \nonumber
&&\left( \Delta x_3 \right)^2 = \frac{1}{3} \frac{2 n + 1}{2 \omega_1'}
+ (J_{13} - J_{23})^2 \left[ A_+^2 \frac{2 m + 1}{2 \omega_+'} + A_-^2 \frac{2 \ell + 1}{2 \omega_-'} \right],
\end{eqnarray}
where $u_{\pm} = -J_{12} + J_{23} \pm \zeta$, $v_{\pm} = J_{12} - J_{13} \pm \zeta$, and
$\omega_{j}' = \omega_j / b_j^2 (t) \hspace{.2cm} (j = 1, \pm)$. Of course $b_j$ are the scaling factors of $\omega_j$. Similarly, the uncertainties
$\left( \Delta p_j \right)^2$ can be computed explicitly by following the same procedure.
We do not know whether or not the sum rule of quantum uncertainty arising at $N = 2$ is realized in other continuous variable systems such as $1/x$-potential system.
Also, we do not clearly understand whether or not the sum rule of uncertainty may have some implication on the additivity of entanglement. We hope to explore
these issues in the future.
Quantum information processing with continuous variables has attracted considerable attention from both theoretical and experimental aspects\cite{braunstein-2005,adesso-2014}.
Quantum uncertainties are closely connected to the inseparability criterion of a continuous-variable quantum system\cite{criterion,simon-2000}.
Furthermore, the distillation protocols to a maximally entangled state have already been suggested
in Duan et al.\cite{duan-99-p} and Giedke et al.\cite{giedke-2000}.
We hope that our results on the explicit expressions of uncertainties may give valuable insight into the problem of
continuous-variable quantum information processing.
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